Table of Contents - Basement

Transcription

SYSTEM
MANUALS
o
fBASEMENT
System Manuals BASEMENT
This page has been intentionally left blank.
VAW ETH Zürich
System Manuals of BASEMENT
CREDITS
VERSION 1.6
October, 2009
Project Team
Prof. Dr. R. Boes
Steering Committee Member Project “Integrales Flussgebietsmanagement”, Director VAW
Dr. R. Fäh, Dipl. Ing. ETH
Committee Member of Project “Integrales Flussgebietsmanagement”, Scientific Supervisor
R. Müller, Dipl. Ing. EPFL, Software Development, Scientific Associate
P. Rousselot, Dipl. Rech. Wiss. ETH, Software Development, Scientific Associate
Dr. R. Veprek, Dipl. Rech. Wiss. ETH, Software Development, Scientific Associate
C. Volz, Dipl. Ing., Software Development, Scientific Associate
D. Vetsch, Dipl. Ing. ETH, Project Supervisor, Scientific Associate
Dr.-Ing. D. Farshi, M. Sc., External Advisor and Tester
Art Design and Layout
W. Thürig, D. Vetsch
Former Project Members
em. Prof. Dr.-Ing. H.-E. Minor,
Member of the steering committee of Rhone-Thur Project 2002-2007
Director of VAW 1998-2008
Dr.-Ing. D. Farshi, M. Sc., Software Development, Scientific Associate, 2002-2007
Commissioned and co-financed by
Swiss Federal Office for the Environment (FOEN)
Contact
basement@ethz.ch
http://www.basement.ethz.ch
© ETH Zurich, VAW,
Faeh R., Mueller R., Rousselot P., Veprek R., Vetsch D., Volz C., Farshi D., 2006 – 2009
VAW ETH Zürich
Version 10/29/2009
CREDITS
Citation Advice
For System Manuals:
Faeh, R, Mueller, R, Rousselot, P, Veprek R., Vetsch, D, Volz, C, Farshi, D 2009. System
Manuals of BASEMENT, Version 1.6. Laboratory of Hydraulics, Glaciology and Hydrology
(VAW). ETH Zurich. Available from <http://www.basement.ethz.ch>. [date of access].
For Website:
BASEMENT – Basic Simulation Environment for Computation of Environmental Flow and
Natural Hazard Simulation, 2009. http://www.basement.ethz.ch
For Software:
BASEMENT – Basic Simulation Environment for Computation of Environmental Flow and
Natural Hazard Simulation. Version 1.6. © VAW, ETH Zurich, Faeh, R., Mueller, R.,
Rousselot, P., Veprek R., Vetsch, D., Volz, C., Farshi, D., 2006-2009.
VAW ETH Zürich
Version 10/29/2009
PREFACE
Preface to Versions 1.0 – 1.3
The development of computer programs for solving demanding hydraulic or hydrological
problems has an almost thirty-year tradition at VAW. Many projects have been carried out
with the application of “home-made” numerical codes and were successfully finished. The
according software development and its applications were primarily promoted by the
individual initiative of scientific associates of VAW and financed by federal instances or the
private sector. Most often, the programs were tailored for a specific application and adapted
to fulfil costumer needs. Consequently, the software grew in functionality but with little
documentation. Due to limited temporal and personal resources to absolve an according
project, a single point of knowledge concerning the details of the software was inevitable in
most of the cases.
In 2002, the applied numerics group of VAW was invited by the Swiss federal office for
water and geology (BWG, nowadays Swiss Federal Office for the Environment FOEN) to
offer for participation in the trans-disciplinary “Rhone-Thur” project. With the idea to build up
a new software tool based on the knowledge gained by former numerical codes - while
eliminating their shortcomings and expanding their functionality - a proposal was submitted.
The bidding being successful a partnership in terms of co-financing was established. By the
end of 2002, a newly formed team took up the work to build the so-called “BASic
EnvironMENT for simulation of environmental flow and natural hazard simulation –
BASEMENT”.
From the beginning, the objectives for the new project were ambitious: developing a
software system from scratch, containing all the experience of many years as well as stateof-the-art numerics with general applicability and providing the ability to simulate sediment
transport. Additionally, professional documentation is a must. As to meet all these demands,
a part wise reengineering of existing codes (Floris, 2dmb) has been carried out, while
merging it with modern and new numerical approaches. From a software-technical point of
view, an object-oriented approach has been chosen, with the aim to provide reusability,
reliability, robustness, extensibility and maintainability of the software to be developed.
After four years of designing, implementing and testing, the software system BASEMENT
has reached a state to go public. The documentation at hand confirms the invested diligence
to create a transparent software system of high quality. The software, in terms of an
executable computer program, and its documentation are available free of charge. It can be
used by anyone who wants to run numerical simulations of rivers and sediment transport –
either for training or for commercial purposes.
VAW ETH Zürich
Version 10/1/2008
PREFACE
The further development of the software tends to new approaches for sediment transport
simulation, carried out within the scope of scientific studies on one hand side. On the other
hand, effectiveness and composite modelling are the goals. On either side, a reliable
software system BASEMENT will have to meet expectations of the practical engineer and
the scientist at the same time.
em. Prof. Dr.-Ing. H.-E. Minor
Member of the steering committee of Rhone-Thur Project 2002-2007
Director of VAW, 1998-2008
October, 2006
VAW ETH Zürich
Version 10/1/2008
PREFACE
Preface to Version 1.4
The work since the first release of the software in October 2006 was exciting and
challenging. To go public is paired with interests and demands of users – although user
support for the software never was intended. But interchange with users is definitely one of
the most crucial factors of successful software development. Feedback from academic or
professional users conveys a different point of view and enables the development team to
achieve costumer proximity as well as to consolidate experience. Accordingly, the project
team tried to meet the demands as effectively as possible. In version 1.3 of BASEMENT,
which was released in April 2007, there were some errors fixed, a few new features added
and the documentation was completed. Since then, many things have changed: on the
personnel, on the project as well as on the software technical level.
In summer 2007 one of our main software developers, Dr. Davood Farshi, left VAW and
changed to an international hydraulic consultant. Dr. Farshi supported our team from 2002 to
2007 as a profound numeric specialist and was mainly involved in the development of
BASEplane. At his own request, he is still engaged in the development of BASEMENT as
external advisor and tester. Dr. Farshi’s position in the project team was reoccupied by
Christian Volz, an environmental engineer from southern Germany. Mr. Volz has broad
experience in numerical modelling as well as object-oriented programming.
On the project level the framework slightly changed. The initial scope within BASEMENT
was developed, the “Rhone-Thur” project, has been finalized by the end of 2007. The
sequel is called “Integrales Flussgebietsmanagement”. It has the same co-financer as its
predecessor, namely the Swiss federal office for the environment (FOEN), and basically the
same participating institutions (EAWAG, WSL, LCH(EPFL) and VAW(ETHZ)). The funding
runs until the end of 2011. Due to the retirement of Prof. Dr.-Ing. H.-E. Minor in summer
2008, our laboratory is solely represented in the project committee by Dr. R. Fäh at the
moment.
The emphases of the new proposal for the further development of BASEMENT are
advanced topics of hydraulics and sediment transport, such as secondary currents and
lateral erosion. Furthermore, the efficiency of the software should be increased by the
implementation of appropriate parallelisation and coupling approaches.
Since the last minor release a long time passed, which was mainly consumed by a general
revision of the software. After five years of development a diligent consolidation was
expedient. In addition, the coincidence of a new team member offered an unbiased
reflection of the source code. All in all it was very worthwhile.
Last but not least, there are numerous bugs fixed and some new features in the current
version. Mainly the efficiency of the software has been improved. The first stage of
parallelisation is completed. The current implementation of the code includes the OpenMP
interface which allows for parallel execution of the basic computation loops. In other words,
the software is now able to exploit the power of current multi-core processors with a
VAW ETH Zürich
Version 10/1/2008
PREFACE
convincing speedup. Furthermore, the revision of some data structures and output routines
as well as the application of an optimised compiler led to a reduction in execution time.
Concerning sediment transport, the one-dimensional model BASEchain now supports the
modelling of fine material, either as suspended or bed load. Also the advanced models for
boundary conditions are worth mentioning. On the one hand, it is now possible to model
domain boundaries with momentum and on the other hand, special boundary conditions
inside the computational region, such as a weir or a gate, are implemented.
The fact, that the version 1.4 of BASEMENT is also available for the Linux operating system
the first time, rounds off the new additions and features of the software package at hand.
Summarised one may say that the release 1.4 of BASEMENT is a major release due to all
the different kinds of changes, but it’s still a minor release concerning the new features –
let’s call it a “major minor” release. We are looking forward to Version 2.0 of BASEMENT,
which is planned for next year.
D. Vetsch
Project Supervisor
October, 2008
VAW ETH Zürich
Version 10/1/2008
LICENSE AGREEMENT
SOFTWARE LICENSE
between
ETH Zurich
Rämistrasse 101
8092 Zürich
Represented by Prof. Dr. Robert Boes
VAW
(Licensor)
and
Licensee
1. Definition of the Software
The Software system BASEMENT is composed of the executable (binary) file BASEMENT and its documentation
files (System Manuals), together herein after referred to as “Software”. Not included is the source code.
Its purpose is the simulation of water flow, sediment and pollutant transport and according interaction in
consideration of movable boundaries and morphological changes.
2. License of ETH Zurich
ETH Zurich hereby grants a single, non-exclusive, world-wide, royalty-free license to use Software to the licensee
subject to all the terms and conditions of this Agreement.
3. The scope of the license
a. Use
The licensee may use the Software:
- according to the intended purpose of the Software as defined in provision 1
- by the licensee and his employees
- for commercial and non-commercial purposes
The generation of essential temporary backups is allowed.
b. Reproduction / Modification
Neither reproduction (other than plain backup copies) nor modification is permitted with the following
exceptions:
Decoding according to article 21 URG [Bundesgesetz über das Urheberrecht, SR 231.1)
If the licensee intends to access the program with other interoperative programs according to article 21 URG, he
is to contact licensor explaining his requirement.
If the licensor neither provides according support for the interoperative programs nor makes the necessary
source code available within 30 days, licensee is entitled, after reminding the licensor once, to obtain the
information for the above mentioned intentions by source code generation through decompilation.
c. Adaptation
On his own risk, the licensee has the right to parameterize the Software or to access the Software with
interoperable programs within the aforementioned scope of the licence.
d. Distribution of Software to sub licensees
Licensee may transfer this Software in its original form to sub licensees. Sub licensees have to agree to all
terms and conditions of this Agreement. It is prohibited to impose any further restrictions on the sub licensees’
exercise of the rights granted herein.
No fees may be charged for use, reproduction, modification or distribution of this Software, neither in
unmodified nor incorporated forms, with the exception of a fee for the physical act of transferring a copy or for
an additional warranty protection.
4. Obligations of licensee
a. Copyright Notice
Software as well as interactively generated output must conspicuously and appropriately quote the following
copyright notices:
Copyright by ETH Zurich, VAW, Faeh R., Mueller R., Rousselot P., Veprek R., Vetsch D., Volz C., Farshi D., 20062009
VAW ETH Zürich
Version 10/28/2009
LICENSE AGREEMENT
5. Intellectual property and other rights
The licensee obtains all rights granted in this Agreement and retains all rights to results from the use of the
Software.
Ownership, intellectual property rights and all other rights in and to the Software shall remain with ETH Zurich
(licensor).
6. Installation, maintenance, support, upgrades or new releases
a. Installation
The licensee may download the Software from the web page http://www.basement.ethz.ch or access it from
the distributed CD.
b. Maintenance, support, upgrades or new releases
ETH Zurich doesn’t have any obligation of maintenance, support, upgrades or new releases, and disclaims all
costs associated with service, repair or correction.
7. Warranty
ETH Zurich does not make any warranty concerning the:
- warranty of merchantability, satisfactory quality and fitness for a particular purpose
- warranty of accuracy of results, of the quality and performance of the Software;
- warranty of noninfringement of intellectual property rights of third parties.
8. Liability
ETH Zurich disclaims all liabilities. ETH Zurich shall not have any liability for any direct or indirect damage except for
the provisions of the applicable law (article 100 OR [Schweizerisches Obligationenrecht]).
9. Termination
This Agreement may be terminated by ETH Zurich at any time, in case of a fundamental breach of the provisions of
this Agreement by the licensee.
10. No transfer of rights and duties
Rights and duties derived from this Agreement shall not be transferred to third parties without the written
acceptance of the licensor. In particular, the Software cannot be sold, licensed or rented out to third parties by the
licensee.
11. No implied grant of rights
The parties shall not infer from this Agreement any other rights, including licenses, than those that are explicitly
stated herein.
12. Severability
If any provisions of this Agreement will become invalid or unenforceable, such invalidity or enforceability shall not
affect the other provisions of Agreement. These shall remain in full force and effect, provided that the basic intent
of the parties is preserved. The parties will in good faith negotiate substitute provisions to replace invalid or
unenforceable provisions which reflect the original intentions of the parties as closely as possible and maintain the
economic balance between the parties.
13. Applicable law
This Agreement as well as any and all matters arising out of it shall exclusively be governed by and interpreted in
accordance with the laws of Switzerland, excluding its principles of conflict of laws.
14. Jurisdiction
If any dispute, controversy or difference arises between the Parties in connection with this Agreement, the parties
shall first attempt to settle it amicably.
Should settlement not be achieved, the Courts of Zurich-City shall have exclusive jurisdiction. This provision shall
only apply to licenses between ETH Zurich and foreign licensees
By using this software you indicate your acceptance.
(license version: 10/27/2009)
VAW ETH Zürich
Version 10/28/2009
LICENSE AGREEMENT
THIRD PARTY SOFTWARE COPYRIGHT NOTICES
The Visualization Toolkit (VTK)
Copyright (c) 1993-2008 Ken Martin, Will Schroeder, Bill Lorensen. All rights reserved.
Redistribution and use in source and binary forms, with or without modification, are permitted provided that the
following conditions are met:
* Redistributions of source code must retain the above copyright notice, this list of conditions and the following
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following disclaimer in the documentation and/or other materials provided with the distribution.
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to endorse or promote products derived from this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS ``AS IS'' AND ANY
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CVM Class Library
Copyright (c) 1992-2009 Sergei Nikolaev
VAW ETH Zürich
Version 10/28/2009
VAW ETH Zürich
VAW ETH Zürich
VAW ETH Zürich
Reference Manual BASEMENT
MATHEMATICAL MODELS
Table of Contents
1
Governing Flow Equations
1.1
Saint-Venant Equations .................................................................................... 1.1-1
Introduction .................................................................................................. 1.1-1
1.1.1
Conservative Form of SVE............................................................................ 1.1-1
1.1.2
Mass Conservation................................................................................ 1.1-1
1.1.2.1
Momentum Conservation ..................................................................... 1.1-2
1.1.2.2
Source Terms ............................................................................................... 1.1-6
1.1.3
Closure Conditions ....................................................................................... 1.1-6
1.1.4
Determination of Friction Slope ............................................................ 1.1-6
1.1.4.1
Boundary Conditions .................................................................................... 1.1-9
1.1.5
Shallow Water Equations ................................................................................. 1.2-1
1.2
Introduction .................................................................................................. 1.2-1
1.2.1
Closure Conditions ....................................................................................... 1.2-6
1.2.2
Turbulence ............................................................................................. 1.2-6
1.2.2.1
Bed Shear Stress................................................................................... 1.2-6
1.2.2.2
Momentum Dispersion ......................................................................... 1.2-6
1.2.2.3
Conservative Form of SWE .......................................................................... 1.2-7
1.2.3
Source Terms ............................................................................................... 1.2-8
1.2.4
Boundary Conditions .................................................................................. 1.2-10
1.2.5
Closed Boundary ................................................................................. 1.2-10
1.2.5.1
Open Boundary ................................................................................... 1.2-11
1.2.5.2
2
Sediment and Pollutant Transport
2.1
Fundamentals of Sediment Motion ................................................................. 2.1-1
Threshold Condition for Sediment Transport ............................................... 2.1-1
2.1.1
Determination of the Critical Shear Stress ............................................ 2.1-1
2.1.1.1
Influence of Local Slope on Incipient Motion ........................................ 2.1-2
2.1.1.2
Influence of Bed Forms on Bottom Shear Stress ........................................ 2.1-3
2.1.2
Bed Armouring ............................................................................................. 2.1-4
2.1.3
Settling Velocities of Particles ...................................................................... 2.1-5
2.1.4
Bed load Propagation Velocity ...................................................................... 2.1-6
2.1.5
Suspended Sediment and Pollutant Transport .............................................. 2.2-1
2.2
One Dimensional Advection-Diffusion-Equation .......................................... 2.2-1
2.2.1
Two Dimensional Advection-Diffusion-Equation .......................................... 2.2-1
2.2.2
Source Terms ............................................................................................... 2.2-1
2.2.3
Bed Load Transport ........................................................................................... 2.3-1
2.3
One Dimensional Bed Load Transport ......................................................... 2.3-1
2.3.1
Component of the Bed Load Flux ......................................................... 2.3-1
2.3.1.1
2.3.1.2
Evaluation of Bed Load Transport due to Stream Forces ..................... 2.3-1
Bed Material Sorting.............................................................................. 2.3-1
2.3.1.3
Global Mass Conservation .................................................................... 2.3-1
2.3.1.4
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MATHEMATICAL MODELS
2.3.2
Two Dimensional Bed Load Transport ......................................................... 2.3-2
Components of the Bed Load Flux ....................................................... 2.3-2
2.3.2.1
Evaluation of Equilibrium Bed Load Transport ...................................... 2.3-2
2.3.2.2
Contribution due to Lateral Transport ................................................... 2.3-2
2.3.2.3
Bed Material Sorting.............................................................................. 2.3-2
2.3.2.4
Global Mass Conservation .................................................................... 2.3-4
2.3.2.5
Sublayer Source Term .................................................................................. 2.3-4
2.3.3
Closures for Bed Load Transport .................................................................. 2.3-4
2.3.4
Meyer-Peter and Müller (MPM) ............................................................ 2.3-4
2.3.4.1
Parker .................................................................................................... 2.3-5
2.3.4.2
Hunziker (MPM-H) ................................................................................. 2.3-6
2.3.4.3
Günter’s two grains model .................................................................... 2.3-7
2.3.4.4
Smart-Jäggi ........................................................................................... 2.3-7
2.3.4.5
Rickenmann ........................................................................................... 2.3-8
2.3.4.6
Wu Wang and Jia .................................................................................. 2.3-8
2.3.4.7
Power Law ............................................................................................ 2.3-9
2.3.4.8
List of Figures
Fig. 1:
Definition Sketch .............................................................................................................. 1.1-1
Fig. 2:
Computational Domain and Boundaries ....................................................................... 1.2-10
Fig. 3:
Determination of critical shear stress for a given grain diameter ................................... 2.1-1
Fig. 4:
Definition sketch of overall control volume of bed material sorting equation (red) ........ 2.3-3
List of Tables
Tab. 1:
Number of needed boundary conditions .......................................................................... 1.1-9
Tab. 2:
The Correct Number of Boundary Conditions in SWE .................................................. 1.2-11
R I - ii
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MATHEMATICAL MODELS
MATHEMATICAL MODELS
I
1 Governing Flow Equations
1.1
Saint-Venant Equations
1.1.1
Introduction
The BASEchain module is based on the Saint Venant Equations (SVE) for unsteady one
dimensional flow. The validity of these equations implies the following conditions and
assumptions:
-
Hydrostatic distribution of pressure: this is fulfilled if the streamline curvatures
are small and the vertical accelerations are negligible.
-
Uniform velocity over the cross section and horizontal water surface across the
section.
-
Small slope of the channel bottom, so that the cosine of the angle of the bottom
with the horizontal can be assumed to be 1.
-
Steady-state resistance laws are applicable for unsteady flow.
The flow conditions at a channel cross section can be defined by two flow variables.
Therefore, two of the three conservation laws are needed to analyze a flow situation. If the
flow variables are not continuous, these must be the mass and the momentum
conservation laws (Cunge, Holly et al. 1980).
1.1.2
1.1.2.1
Conservative Form of SVE
Mass Conservation
Fig. 1: Definition Sketch
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R I - 1.1-1
MATHEMATICAL MODELS
For the control volume illustrated in Fig. 1, assuming the mass density
be written:
ρ
is constant, it can
d x2
0 (RI-1.1)
Adx + Qout − Qin − ql ( x2 − x1 ) =
dt ∫x1
where:
[m2]
[m3/s]
[m2/s]
[m3]
[m]
[s]
A
Q
ql
V
x
t
wetted cross section area
discharge
lateral discharge per meter of length
volume
distance
time
Applying Leibnitz’s rule and the mean value theorem:
d x2
Adx
=
dt ∫x1
x2
∂A
dx
∫=
∂t
x1
and then dividing by
∂A
( x2 − x1 ) ,
∂t
( x2 − x1 ) and
Qout − Qin ∂Q
,
=
( x 2 − x1 )
∂x
the divergent form of the continuity equation is obtained:
∂A ∂Q
+
− ql =
0
∂t ∂x
1.1.2.2
Momentum Conservation
With the momentum
dM
= m=
a
dt
where:
(RI-1.2)
M
m
u
a
F
p = mu and according to Newton’s second law of motion
∑F
[kg m/s]
[kg]
[m/s]
[m/s2]
[N]
momentum
mass
velocity
acceleration
force
Making use of the Reynolds transport theorem (Chaudhry 1993) and referring to the control
volume in Fig. 1
x
dM
d 2
F
=
=
∑ dt ∫ u ρ Adx + u2 ρ A2u2 − u1ρ A1u1 − ux ρ ql ( x2 − x1 )
dt
x1
is obtained, where:
[m/s]
ux
[kg/m3]
ρ
R I - 1.1-2
(RI-1.3)
velocity in x direction (direction of flow) of lateral source
mass density
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MATHEMATICAL MODELS
Further simplification is achieved by applying Leibnitz’s rule and writing
Q / A = u resulting in:
∂Q
Q2
∑ F= ∫ ρ ∂t dx + ρ A
x1
x2
out
Q2
−ρ
A
− u x ρ ql ( x2 − x1 )
Application of the mean value theorem:
 Q2

 A
−
out
Q2
A
(RI-1.4)
in
x2
and division by
Q = Au and
∂Q
dx
∫ ρ=
∂t
x1
ρ ( x2 − x1 ) and with
∂Q
( x2 − x1 ) ρ
∂t

∂  Q2 
1
=



( x2 − x1 ) ∂x  A 
in 
leads to:
∑F
∂Q ∂  Q 2 
=+ 
−qu
ρ ( x2 − x1 ) ∂t ∂x  A  l x
For the determination of
considered:
(RI-1.5)
∑ F the following forces acting on the control volume have to be
-
pressure force upstream and downstream:
=
F1
-
weight of water in x-direction:
ρ=
gA1h1 and F2 ρ gA2 h2
x2
F3 = ρ g ∫ AS B dx
x1
x2
frictional force:
-
F4 = ρ g ∫ AS f dx , where S f =
x1
SB
Sf
K
g
kst
R
[-]
[-]
[m3/s]
[m/s2]
[m1/3/s]
[m]
bottom slope
friction slope
conveyance factor
gravity
Strickler factor
hydraulic radius
K = k st AR
QQ
K2
2
3
Introducing the sum of these forces in equation (RI-1.5) and applying the mean value
theorem:
x2
∫ A(S
B
− S f )dx = A( S B − S f )( x2 − x1 )
x1
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MATHEMATICAL MODELS
the momentum equation in the conservation form is obtained:
∂Q ∂  Q 2 
∂
+ 
−g ( Ah ) + gA( S B − S f )
 − ql u x =
∂t ∂x  A 
∂x
(RI-1.6)
As
1
2

∆ (=
Ah )  A ( h + ∆h ) − B ( ∆ )  − Ah
2


and furthermore neglecting higher order terms and letting
∂
∂
∂
∂h
∂h
.
∆h → 0,
=
Ah ) A and=
(gAh ) g =
( Ah )
gA
(
∂h
∂x
∂h
∂x
∂x
Equation (RI-1.6) becomes:
∂Q ∂  Q 2 
∂h
+ 
− gA( S B − S f ) − ql u x =
0
 + gA
∂t ∂x  A 
∂x
(RI-1.7)
As h and S B are unknown for irregular cross sections, they are eliminated from equation
(RI-1.7) by the following transformation:
h = zS − z B and
∂h ∂zS ∂z B ∂zS
=
−
=
+ SB
∂x ∂x ∂x
∂x
Insertion into equation (RI-1.7) results in:
∂zS
∂Q ∂  Q 2 
+ 
+ gAS f − ql u x =
0
 + gA
∂t ∂x  A 
∂x
(RI-1.8)
If only the cross sectional area where the water actually flows and therefore contributes to
the momentum balance shall be used, and introducing a factor β accounting for the
velocity distribution in the cross section (Cunge, Holly et al. 1980), equation (RI-1.9) is
obtained:
∂zS
∂Q ∂  Q 2 
+ β
+ gAred S f − ql u x =
0
 + gAred
∂t ∂x  Ared 
∂x
where
(RI-1.9)
Ared [m2] is the reduced area, i.e. the part of the cross section area where water
flows.
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MATHEMATICAL MODELS
If Strickler values are used to define the friction
β=
2
A∑ kstr
h 7/3bi
i i
i

5/3 
 ∑ kstri h bi 
 i

2
(RI-1.10)
If the equivalent roughness height is used
β=
A∑ ks2i hi2bi
i

3/2 
 ∑ k si hi bi 
 i

VAW ETH Zürich
Version 10/28/2009
2
(RI-1.11)
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1.1.3
MATHEMATICAL MODELS
Source Terms
With the given formulation of the flow equation there are 4 source terms:
For the continuity equation:

The lateral in- or outflow
ql
For the momentum equation:

The bed slope
W = gA

∂zS
∂x
(RI-1.12)
The bottom friction:
Fr = gAred S f

(RI-1.13)
The influence of lateral in- or outflow:
(RI-1.14)
ql u x
However, in BASEchain, the influence of the lateral inflow on the momentum equation is
neglected.
1.1.4
Closure Conditions
1.1.4.1
Determination of Friction Slope
The relation between the friction slope
S f and the bottom shear stress is:
τB
= gRS f
ρ
As the unit of
u* =
τB
ρ
(RI-1.15)
τB / ρ
is the square of a velocity, a shear stress velocity can be defined as:
.
(RI-1.16)
The velocity in the channel is proportional to the shear flow velocity and thus:
u = c f gRS f
where
If
(RI-1.17)
c f is the Chézy coefficient.
u is replaced by Q / A :
Q
= cf g R S f
A
(RI-1.18)
results, with
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Sf =
QQ
.
gA2 c 2f R
(RI-1.19)
Introducing the conveyance
=
K
Sf =
MATHEMATICAL MODELS
K:
Q
= Ac f R g
Sf
(RI-1.20)
QQ
K2
(RI-1.21)
The determination of the friction coefficient is based either on the approach by ManningStrickler k str or on the equivalent sand roughness k s of Nikuradse.
Manning-Strickler
To define the channel roughness, both notations,
k str or n , can be used. For conversion a
simple relation holds:
kstr =
1
n
The coefficient
cf =
kstr R
1
c f is calculated as follows:
6
(RI-1.22)
g
Equivalent Sand Roughness
The following approaches are implemented to determine the coefficient
cf :
Chézy:
 12 R 
c f = 5.75log 

 ks 
(RI-1.23)
Darcy-Weissbach:
cf =
8
f
VAW ETH Zürich
Version 10/28/2009
with
f =
0.24
 12 R 
log 

 ks 
(RI-1.24)
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MATHEMATICAL MODELS
In the case of a defined soil, the determination of the friction coefficient can be determined
by the grain size distribution of the river bed itself:
kstr =
factor
6 d
90
default value of
factor = 21.1
(RI-1.25)
factor = 3
(RI-1.26)
and
=
ks factor ⋅ d90
default value of
The default factors mentioned above correspond to a soil with various grain sizes and
exposed coarse components.
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1.1.5
MATHEMATICAL MODELS
Boundary Conditions
At the upper and lower end of the channel it is necessary to know the influence of the
region outside on the flow within the computational domain. The influenced area depends
on the propagation velocity of a perturbation. The propagation velocity in standing water is
c = gh
(RI-1.27)
If a one dimensional flow is considered this propagation takes place in two directions:
upstream ( −c ) and downstream ( + c ). These velocities must then be added to the flow
velocities in the channel, giving the upstream ( C− ) and downstream ( C+ ) characteristics:
C−=
dx
= u −c
dt
(RI-1.28)
C+=
dx
= u+c
dt
(RI-1.29)
With these functions it is possible to determine which region is influenced by a perturbation
and which region influences a given point after a given time.
In particular it can be said that if c < u the information will not be able to spread in
upstream direction, thus the condition in a point cannot influence any upstream point, and a
point cannot receive any information from downstream. This is the case for a supercritical
flow.
In contrast, if c > u , which is the case for sub-critical flow, the information spreads in both
directions, upstream and downstream. This fact substantiates the necessity and usefulness
of information at the boundaries. As there are two equations to solve, two variables are
needed for the solution.
If the flow conditions are sub-critical, the flow is influenced from downstream. Thus at the
inflow boundary one condition can be taken from the flow region itself and only one
additional boundary condition is needed. At the outflow boundary, the flow is influenced
from outside and so one boundary condition is needed.
If the flow is supercritical, no information arrives from downstream. Therefore, two
boundary conditions are needed at the inflow end. In contrast, as it cannot influence the
flow within the computational domain, it is not useful to have a boundary condition at the
downstream end.
Flow type
Sub critical flow ( Fr < 1 )
Supercritical flow ( Fr > 1 )
Number of boundary conditions
Inflow
Outflow
1
1
2
0
Tab. 1: Number of needed boundary conditions
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MATHEMATICAL MODELS
At the inflow boundary the given value is usually Q . If the flow is supercritical the second
variable A is determined by a flow resistance law (slope is needed!).
At the outflow boundary there are several possibilities to provide the necessary information
at the boundary:
-
determine an out flowing discharge by a weir or a gate;
-
set the water surface elevation as a function of time;
-
set the water surface elevation as a function of the discharge (rating curve).
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1.2
MATHEMATICAL MODELS
Shallow Water Equations
1.2.1
Introduction
Mathematical models of the so-called shallow water type govern a wide variety of physical
phenomena. For reasons of simplicity, the shallow water equations will from here on be
abbreviated as SWE. An important class of problems of practical interest involves water
flows with a free surface under the influence of gravity. It includes:
-
Tides in oceans
-
Flood waves in rivers
-
Dam break waves
The validity of the SWE implies the following conditions and assumptions:
-
Hydrostatic distribution of pressure: this is fulfilled if the vertical accelerations
are negligible.
-
Small slope of the channel bottom, so that the cosine of the angle between the
bottom and the horizontal can be assumed to be 1.
-
Steady-state resistance laws are applicable for unsteady flow.
A key assumption made in derivation of the approximate shallow water theory concerns the
first aspect, the hydrostatic pressure distribution. Supposing that the vertical velocity
acceleration of water particles is negligible, a hydrostatic pressure distribution can be
assumed. This eventually allows for integration over the flow depth, which results in a nonlinear initial value problem, namely the shallow water equations. They form a timedependent two-dimensional system of non-linear partial differential equations of hyperbolic
type.
There are two approaches for the derivation of shallow water equations:
-
Integrating the three-dimensional system of Navier-Stokes equations over the
depth
-
Direct approach by considering a three-dimensional control volume element of
fluid
In the following the derivation of the depth integrated mass and momentum conservation
equations from the Reynolds-averaged 3D Navier-Stokes equation is briefly presented.
Following boundary conditions are imposed:
1) At the top of water surface:
Kinetic boundary condition (this condition describes that no flow across the water
surface can take place):
∂z
∂z
∂z
dz
wS = s + uS s + vS s = s
∂t
∂x
∂y
dt
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(RI-2.1)
R I - 1.2-1
MATHEMATICAL MODELS
Dynamic boundary condition:
=
τ
,τ ) and P
(τ=
Sx
Sy
(RI-2.2)
Patm
2) At the bottom of water body:
Kinetic boundary condition (this condition describes that no flow through the bed
surface can take place):
=
wB uB
∂z B
∂z
+ vB B
∂x
∂y
(RI-2.3)
The derivation makes use of Leibniz’s integration rule, which is used here to remove the
partial derivatives from the integral. It can be written generally as
∂z
∂z
∂f ( x, y, z )
∂ S
=
d
z
f ( x, y, z )d z+ f ( x, y, z B ) B − f ( x, y, zS ) S
∫
∫
∂x
∂x zB ( x , y )
∂x
∂x
zB ( x , y )
ZS ( x, y )
Z ( x, y )
Derivation of mass conservation
The 3D Reynolds-averaged mass conservation equation is integrated over the flow depth
from the bed bottom z B to the water surface z S .
ZS
 ∂u
∂w 
∂v
0
∫  ∂x + ∂y + ∂z dz =
(RI-2.4)
zB
Applying Leibniz’s rule on the first term, the velocity gradient in x-direction, leads to
ZS
∫
zB
∂z
∂z
∂u
∂ S
dz=
udz + uB B − vS S
∫
∂x
∂x zB
∂x
∂x
Z
whereas u B , uS and vB , vS are the velocities at the bottom and at the surface in x- and ydirections respectively. The second term of eq. (RI-2.4) is treated analogous.
The third term can be evaluated exactly by applying the fundamental theorem of calculus. It
results in the difference of the vertical velocity at the surface and the bottom:
ZS
∫
zB
∂w
dz
= wS − wB .
∂z
Assembling these terms, one can identify and eliminate the kinematic boundary conditions
as stated above.
∂ S
∂
udz +
∫
∂x zB
∂y
Z
ZS
∂z
∂z
∂z B
∂z
0
+ vB B − wB − uS S − vS S + wS =
∂x
∂y
∂x
∂y
 
∫ vdz + u
zB
B
= 0
R I - 1.2-2
=
∂zS ∂h
=
∂t ∂t
VAW ETH Zürich
Version 10/28/2009
MATHEMATICAL MODELS
Evaluating the integrals over the depth as
ZS
∫ udz = uh
zB
finally leads to the depth integrated formulation of mass conservation:
∂h ∂ ( uh ) ∂ ( uh )
+
+
=
0
∂t
∂x
∂y
Derivation of momentum conservation
The Reynolds-averaged 3D momentum equation in x-direction of the Navier-Stokes equation
is integrated over the depth
ZS
 1 ∂p 1 ∂τ xx 1 ∂τ xy 1 ∂τ xz 
 ∂u ∂u 2 ∂uv ∂uw 
+
+
+
dz
=


∫ ∂t ∂x ∂y ∂z  z∫ − ρ ∂x + ρ ∂x + ρ ∂y + ρ ∂y dz
zB 
B
ZS
(RI-2.5)
whereas the momentum equation in y-direction can be treated in an analogous manner.
The first term, the time derivative of the x-velocity, is transformed with Leibniz’s rule and
integrated over the depth.
ZS
∫
zB
∂u
∂
dz =
∂t
∂t
ZS
∫ udz + u
zB
B
∂z
∂z
∂z B
∂z
∂uh
− uS S = + u B B − uS S
∂t
∂t
∂t
∂t
∂t
The Leibniz rule is also applied on the advective terms as follows:
ZS
∫
zB
∂z
∂z
∂u 2
∂ S 2
dz=
u dz + uB2 B − uS2 S ,
∫
∂x
∂x zB
∂x
∂x
Z
ZS
∫
zB
∂uv
∂
dz =
∂y
∂y
ZS
∫ uvdz + u v
B B
zB
∂z
∂z B
− uS vS S .
∂y
∂y
And the fundamental theorem of calculus allows the evaluation of the fourth term on the left
hand side.
ZS
∫
zB
∂uw
=
dz uS wS − uB wB
∂z
All terms of the left hand side of eq. (RI-2.5) now can be assembled, and again, the
kinematic boundary conditions are identified and can be eliminated. The left hand side is
reduced to three remaining terms.
∂ S 2
∂
uh +
u dz +
∫
∂x zB
∂y
Z
ZS
∫ uvdz + u
zB
B
 ∂z

∂z
∂z
∂z
∂z
 ∂z B

+ uB B + vB B − wB  − uS  S uS S + vS S + wS 

∂t
∂
∂x
∂t
∂x
∂y
x




=0
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MATHEMATICAL MODELS
With the assumption of a hydrostatic pressure distribution, the pressure term on the right
hand side can be evaluated. Furthermore, the water surface elevation is replaced by the
bottom elevation and the depth.
1
ZS
∂p
=
dz
ρ ∫ ∂x
zB
∂zS
∂h 
 ∂z
gh
=
gh  B +  .
∂x
 ∂x ∂x 
The depth integration of the first two shear stresses of the right hand side yields the depth
integrated viscous and turbulent stresses, which require additional closure conditions to be
evaluated.
1
ZS
ρ ∫
zB
∂τ xx
1
dz +
∂x
ρ
ZS
∫
zB
∂τ yx
1 ∂τ h 1 ∂τ xy h
dz = xx +
∂y
ρ ∂x
ρ ∂y
The third shear stress term can be integrated over the depth and introduces the bottom and
surface shear stresses at the domain boundaries, which again need additional closure
conditions. The surface shear stresses τ Bx , e.g. due to wind flow over the water surface,
are neglected from here on.
1
ZS
ρ ∫
zB
∂τ zx
1
(τ − τ )
=
dz
∂z
ρ Sx Bx
Putting the terms together one obtains
∂uh ∂ S 2
∂
+
u dz +
∫
∂t ∂x zB
∂y
Z
ZS
∂h
∂z B
− gh
∫ uvdz + gh ∂x =
∂x
zB
−
1
ρ
τ Bx +
1 ∂τ xx h 1 ∂τ xy h
+
ρ ∂x
ρ ∂y
The depth integrals of the advective terms still need to be solved. By dividing the velocities
in a mean velocity u and a deviation from the mean u ′ , similar to the Reynolds averaging
procedure, the advective terms can be evaluated as follows. The depth integration
introduces new dispersion terms, which describe the effects of the non-uniformity of the
velocity distribution.
∂ S 2
∂u 2 h ∂u ′u ′h ∂u 2 h 1 ∂Dxx h
=
=
+
=
+
u u + u′ ⇒
u
dz
∂x z∫B
∂x
∂x
∂x
ρ ∂x
Z
In the end, after dividing the equations by the water depth, the depth integrated xmomentum equation of the SWE is obtained in the following formulation:
∂z
1
1 ∂  h (τ xx + Dxx )  1 ∂  h (τ xy + Dyx ) 
∂u
∂u
∂u
∂h
τ Bx +
+u
+v
+g
=
−g B −
+
ρh
ρh
dx
dy
∂t
∂x
∂y
∂x
∂x ρ h
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MATHEMATICAL MODELS
Conclusive, as shown before, the complete set of SWE is derived in the form:
∂h ∂ ( uh ) ∂ ( vh )
+
+
=
0
∂t
∂x
∂y
(RI-2.6)
∂z
1
1 ∂  h (τ xx + Dxx )  1 ∂  h (τ xy + Dxy ) 
∂u
∂u
∂u
∂h
τ Bx +
+u
+v
+g
=
−g B −
+
ρh
ρh
dx
dy
∂t
∂x
∂y
∂x
∂x ρ h
(RI-2.7)
∂z
1
1 ∂  h (τ yx + Dyx )  1 ∂  h (τ yy + Dyy ) 
∂v
∂v
∂v
∂h
τ By +
+u
+v
+g
=
−g B −
+
ρh
ρh
dx
dy
∂t
∂x
∂y
∂y
∂y ρ h
(RI-2.8)
where:
h
g
P
u
uS
uB
v
vS
vB
wS
wB
wB
zB
zs
τ S , x ,τ S , y
τ B , x ,τ B , y
τ xx ,τ xy ,τ yx ,τ yy
Dxx , Dxy , Dyx , Dyy
[m]
[m/s2]
[Pa]
[m/s]
[m/s]
[m/s]
[m/s]
[m/s]
[m/s]
[m/s]
[m/s]
[m]
[m]
[m]
[N/m2]
[N/m2]
[N/m2]
[N/m2]
water depth
gravity acceleration
pressure
depth averaged velocity in x direction
velocity in x direction at water surface;
velocity in x direction at bottom (usually equal zero)
depth averaged velocity in y direction
velocity in y direction at water surface
velocity in y direction at bottom (usually equal zero)
velocity in z direction at water surface
velocity in z direction at bottom (usually equal zero)
velocity in z direction at bottom (usually equal zero)
bottom elevation
water surface elevation
surface shear stress in x- and y direction (here neglected)
bed shear stress in x- and y direction
depth averaged viscous and turbulent stresses
momentum dispersion terms
For brevity, the over bars indicating depth averaged values will be dropped from here on.
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1.2.2
MATHEMATICAL MODELS
Closure Conditions
For the shallow water equations there are mainly three different types of closure conditions
needed.
1.2.2.1
Turbulence
The turbulent and viscous shear stresses can be quantified in accordance with the
Boussinesq eddy viscosity concept, which can be expressed as
=
τ xx 2 ρν
 ∂u ∂v 
∂u
∂v
=
, τ yy 2 ρν =
, τ xy ρν  + 
∂x
∂y
 ∂y ∂x 
(RI-2.9)
If the flow is dominated by the friction forces, the total viscosity is the sum of the eddy
viscosity (quantity due to turbulence modelling) and molecular viscosity (kinematic viscosity
of the fluid): ν = ν t + ν m .
Turbulent eddy viscosity may be dynamically calculated as ν t = κ u* h 6 with the friction
velocity u* = τ B ρ .
The molecular viscosity is a physical property of the fluid and is constant due to the
assumption of an isothermal fluid.
1.2.2.2
Bed Shear Stress
The bed shear stresses are related to the depth–averaged velocities by the quadratic friction
law
uu
uv
=
τ Bx ρ=
, τ By ρ 2
2
cf
cf
(RI-2.10)
in which =
u
u + v is the magnitude of the velocity vector. The friction coefficient
can be determined by any friction law.
2
1.2.2.3
2
cf
Momentum Dispersion
The momentum dispersion terms account for the dispersion of momentum transport due to
the vertical non-uniformity of flow velocities.
At the moment the momentum dispersion terms are not explicitly modelled here. Usually, in
straight channels, these dispersive effects can be accounted for by adapting the turbulent
viscosity in the determination of the turbulent stresses (Wu 2007).
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1.2.3
MATHEMATICAL MODELS
Conservative Form of SWE
Various forms of SWE can be distinguished with their primitive variables. The proper choice
of these variables and the corresponding set of equations plays an extremely important role
in numerical modelling. It is well known that the conservative form is preferred over the
non-conservative one if strong changes or discontinuities in a solution are to be expected. In
flooding and dam break problems, this is usually the case.
(Bechteler, Nujić et al. 1993) showed that the equation sets in conservative form, with
(h, uh, vh) as independent and primitive variables produce best results. The conservative
form can be derived by multiplying the continuity equation with u and v and adding to
momentum equations in x and y direction respectively. This set of equations can be written
in the following form:
U t + ∇ ⋅ ( F, G ) + S = 0
(RI-2.11)
where U , F (U ), G (U ) and S are the vectors of conserved variables, fluxes in the x and y
directions and sources respectively, given by:
h
 
U =  uh 
 vh 
 
(RI-2.12)








uh
vh




1 2
∂u 
∂v
 2


F = u h + gh −ν h  ; G =
uvh −ν h

2
∂x
∂x




1 2
∂u
∂v 
2



uvh −ν h


 v h + 2 gh −ν h ∂y 
∂y




(RI-2.13)


0



S
gh ( S fx − S Bx ) 
=


 gh ( S fy − S By ) 


(RI-2.14)
where ν is the total viscosity.
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1.2.4
MATHEMATICAL MODELS
Source Terms
Equation (RI-2.14) has two source terms: the bed shear stress ( τ B ρ = ghS f ) and the bed
slope term ( ghS B ). The viscous stresses in the flux of equation (RI-2.15) are also treated as
flux term.
The bed shear stress is the most important physical parameter besides water depth and
velocity field of a hydro- and morphodynamic model. It causes the turbulence and is
responsible for sediment transport and has a non-linear effect of retarding the flow. When
the effect of turbulence grows, the effect of molecular viscosity becomes relatively smaller,
while viscous boundary layer near a solid boundary becomes thinner and may even appear
not to exist. It means that the bed stress (friction) is equal to the bed turbulent stress.
However, bed stress is usually estimated by using an empirical or semi-empirical formula
since the vertical distribution of velocity cannot be readily obtained.
In the one dimensional system of equations the term bed stress can be expressed as
gRS f , where S f denotes the energy slope. Assuming that the frictional force in a two
dimensional unsteady open flow can be estimated by referring to the formulas for one
dimensional flows in open channel it can be written:
Sf =
uu
g c 2f R
(RI-2.16)
where u = velocity; cf = friction coefficient; R = hydraulic radius. It can be easily seen that
the above formula can be approximately generalized to the two dimensional system. In onedimensional flows it is not distinguished between bottom and lateral (side wall) friction,
while in two dimensional flows it is often taken a unit width channel ( R = h ). For the two
dimensional system the Equation (RI-2.16) has the following forms
u u 2 + v2
v u 2 + v2
S fx =
; S fy =
g c 2f R
g c 2f R
(RI-2.17)
The coefficient c f can be calculated by different empirical approaches as in the one
dimensional system, e.g. using the Manning or Strickler coefficients. See chapter 1.1.4 for
further details on the determination of the friction coefficient c f . If the friction value is
calculated from the bed composition, the d90 diameter of the mixture is determined. In case
of single grain simulations the d90 diameter is estimated by d 90 = 3* d mean .
The bed stress terms need additional closures equation (see chapter 1.2.2.2) to be
determined. Bed slope terms represent the gravity forces
S B , x = − ∂z B ∂x ; S B , y = − ∂z B ∂y
(RI-2.18)
The viscous fluxes are treated as source terms (The superscript ‘d’ refers to the diffusion)
Sd
=
∂F d ∂G d
+
∂x
∂y
where the diffusive fluxes read
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MATHEMATICAL MODELS






 0 
0 



∂u  d  ∂u 
d

νh
,G =
F =
 νh 
 ∂x 
 ∂y 
 ∂v 
 ∂v 
 νh 
 νh 
 ∂x 
 ∂y 
The total viscosity ν is the sum of the kinematic viscosity and turbulent eddy viscosity. The
kinematic viscosity is a physical property of the fluid and is set to a constant value.
The turbulent eddy viscosity can either be set to a constant value or calculated dynamically
as Turbulent eddy viscosity may be dynamically calculated as ν t = κ u* h 6 with the friction
velocity u* = τ B ρ .
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1.2.5
MATHEMATICAL MODELS
Boundary Conditions
SWE provide a model to describe dynamic fluid processes of various natural phenomena
and find therefore widespread application in science and engineering. Solving SWE needs
the appropriate boundary conditions like any other partial differential equations. In particular,
the issue of which kind of boundary conditions are allowed is still not completely understood
(Agoshkov, Quarteroni et al. 1994). However several sets of boundary conditions of physical
interest that are admissible from the mathematical viewpoint will be discussed here.
The physical boundaries can be divided into two sets: one closed ( Γ c ), the other open ( Γ o )
(Fig. 2). The former generally expresses that no mass can flow through the boundary. The
latter is an imaginary fluid-fluid boundary and includes two different inflow and outflow
types.
1.2.5.1
Closed Boundary
The following relations are often described on the closed boundary, say
∂u
∂n
u ⋅ n 0; = 0
ρ=
where n
u
[m]
[m/s]
Γc :
(RI-2.19)
the normal (directed outward) unit vector on
velocity vector = (u , v)
Γc
Fig. 2: Computational Domain and Boundaries
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1.2.5.2
MATHEMATICAL MODELS
Open Boundary
The number of boundaries of a partial differential equations system depends on the type of
the system. From the mathematical point of view, the SWE establish a quasi-linear
hyperbolic differential equations system. If the temporal derivatives vanish, the system is
elliptical for Fr ≤ 1.0 and hyperbolical for Fr ≥ 1.0 , where Fr is Froude number.
On the open boundary (Γo ) the two types inflow and outflow can be respectively
distinguished as follows:
Γin=
Γ out=
( x ∈ Γ0 ; u ⋅ n < 0 )
(RI-2.20)
( x ∈ Γ0 ; u ⋅ n > 0 )
(RI-2.21)
Based on the behaviour of the system of equations, the theoretical number of open
boundary conditions is listed in Tab. 2 (Agoshkov, Quarteroni et al. 1994) (Beffa 1994):
Flow type
Sub critical flow ( Fr < 1 )
Supercritical flow ( Fr > 1 )
Number of boundary conditions
Inflow
Outflow
2
1
3
0
Tab. 2: The Correct Number of Boundary Conditions in SWE
However in practical application of boundary conditions, the number of the implemented
conditions is often higher or lower than the theoretical criteria (Nujić 1998).
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MATHEMATICAL MODELS
2 Sediment and Pollutant Transport
2.1
Fundamentals of Sediment Motion
2.1.1
2.1.1.1
Threshold Condition for Sediment Transport
Determination of the Critical Shear Stress
=
τ Bcr θ cr ( ρ s − ρ ) gd g is the threshold for the initiation of motion
The critical shear stress
of the grain class g and is derived from the according Shields parameter θ cr , which is a
*
function of the shear Reynolds number Re see (Shields 1936). Meyer-Peter and Müller
(1948) proposed a constant Shields parameter of 0.047 for the fully turbulent area
*
3
*
( Re > 10 ). In the numerical models of BASEMENT, the Shields parameter θ cr = f D
*
is evaluated as function of the dimensionless grain diameter D according to van Rijn
(1984) (see Fig. 3).
( )
Fig. 3: Determination of critical shear stress for a given grain diameter
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MATHEMATICAL MODELS
The critical Shields parameter θ cr can be set to a constant value or can be determined in
the numerical simulations using the following parameterization of the Shield’s curve.
=
θ cr
=
θ cr
=
θ cr
=
θ cr
=
θ cr
0.24( D* ) −1
0.14( D* ) −0.64
0.04( D* ) −0.1
0.013( D* )0.29
0.055
for 1 ≤ D* ≤ 4
for 4 < D* ≤ 10
for 10 < D* ≤ 20
for 20 < D* ≤ 150
for D* > 150
The dimensionless grain diameter
(RI-2.22)
D* thereby defines as
1
 (ρ − ρ ) g  3
D = s
d
ρ υ 2 

*
2.1.1.2
(RI-2.23)
Influence of Local Slope on Incipient Motion
The investigations on incipient motion by Shields were made for almost horizontal beds. In
cases of sloped beds with slopes in flow direction or transverse to it, the stability of grains
can either be reduced or enlarged by the acting gravity forces. One approach to consider the
effects of local slopes on the threshold for incipient motion is to correct the critical shear
stresses for incipient motion. Here k β and kγ are correction factors for slopes in flow
direction and in transversal direction and τ B cr , Shields is the critical shear stress for almost
horizontal beds as derived by Shields. The corrected critical shear stress then is determined
as
τ B = kβ kγ τ B
cr
(RI-2.24)
cr , Shields
The correction factors are calculated as suggested by van Rijn (1989):
 sin(γ − β )
if
 sin γ
kβ = 
 sin(γ + β ) if
 sin γ
slope<0
slope>0
(RI-2.25)

tan 2 δ
kγ cos δ 1 −
=
tan 2 γ

where β is the angle between the horizontal and the bed in flow direction, δ is the slope
angle transversal to the flow direction and γ is the angle of repose of the sediment
material.
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2.1.2
MATHEMATICAL MODELS
Influence of Bed Forms on Bottom Shear Stress
In presence of bed forms, like ripples, sand dunes or gravel banks, additional friction losses
can occur due to complex flow conditions around these bed forms and the formation of
turbulent eddies. In such cases the dimensionless bottom shear stress θ determined from
the present flow conditions can differ from the effective dimensionless bottom shear stress
θ ′ , which is relevant for the transport of the sediment particles. It is usually assumed that
the determination of the effective shear stress should be based upon the grain friction
losses only and should exclude additional form losses, to prevent too large sediment
transport rates. Therefore a reduction factor µ is introduced for the determination of the
effective bottom shear stress θ ′ from the bottom shear stress θ as
µ=
τ B′ θ ′
=
τB θ
 µ = 1.0 (no bed forms)
with 
 µ < 1.0 (bed forms)
(RI-2.26)
This reduction factor (also called “ripple factor”) can be given a constant value if the bed
forms are distributed uniformly over the simulation domain. After Jäggi this factor should be
'
set between 0.8 and 0.85. If there are no bed forms present one can consider that θ = θ ,
i.e. µ =1.0. Generally can be said, the larger the form resistance, the smaller becomes the
reduction factor µ .
Another approach is to calculate the reduction factor by introducing a reduced energy slope
J ' , compared to the energy slope J , due to the presence of the bed forms as done by
Meyer-Peter and Müller. This approach is in particular suitable if ripples are present at the
river bed and finally leads to the following estimation of the reduction factor.
k 
µ =  str 
′ 
 k str
3/2
(RI-2.27)
′ corresponds to the definition of the Strickler coefficient for experiments with
Here, k str
Nikuradse-roughness (Jäggi 1995) and can be calculated from the grain sizes using the d90
diameter as detailed in section 1.1.4. k str is the calibrated Strickler coefficient used in the
hydraulic calculations which includes the form friction effects.
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2.1.3
MATHEMATICAL MODELS
Bed Armouring
In morphological simulations with fractional sediment transport the forming or destroying of
bed armouring layers can be simulated by modelling sorting effects without special features.
But there are also some types of protection layers which need a special treatment, like e.g.
grass layers or geotextiles. Furthermore, for single grain simulations the bed material sorting
effects cannot be captured and therefore a special treatment is needed if effects of bed
armouring shall be considered.
The effects of such a protection layer can be considered using two methods:
•
A critical shear stress τ cr , start of the protection layer can be specified, which must
be exceeded at least once before erosion of the substrate can take place. This
method is suited for simulations with one or multiple grain classes.
Start of erosion: τ B > τ cr , start
•
Another approach is to define the d90 grain diameter of the bed armouring layer.
The dimensionless critical shear stress θ cr , armour of this bed armour is then
estimated as
θ cr ,armour
d 
= θ cr  90 
d 
2/3
where d 90 is the specified d90 grain diameter of the bed armour and d and θ cr are
the diameter and critical shear stress of the substrate. But be aware that in case the
bed armour is eroded once, it cannot be built up again using this approach in single
grain computations.
If sediment has accumulated above the protection layer, the armouring condition is not
applied until this sediment is totally eroded.
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2.1.4
MATHEMATICAL MODELS
Settling Velocities of Particles
The settling velocity w of sediment particles is an important parameter to determine which
particles are transported as bed load or as suspended load. Many different empirical or
semi-empirical relations for the determination of settling velocities in dependence of the
grain diameter have been suggested in literature.
Approach of van Rijn
The sink rate can be determined against the grain diameter after van Rijn (1984).
w=
( s − 1) gd 2
18ν

10ν 
0.01( s − 1) gd 3
w=
−
1
 1+


d 
ν3

=
w 1.1 ( s − 1) gd
d is the diameter of the grain,
density.
ν
for 0.001 < d ≤0.1 mm
(RI-2.28)
for 0.1 < d ≤1 mm
(RI-2.29)
for d ≥1mm
(RI-2.30)
is the kinematic viscosity and
s  s  the specific
Approach of Wu and Wang
A newer approach for the computation of the sink velocity is the one of Wu and Wang
(2006):
Mν  1  4 N


w=
( D* )3 
+
2
Nd  4  3M


1/ n
1
− 
2

n
(RI-2.31)
where:
−0.65 S
p
M = 53.5e
−2.5 S p
N = 5.65e
=
n 0.7 + 0.9 S p
S p is the Corey shape factor, with a value for natural sediments of about 0.7 (0.3 - 0.9).
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2.1.5
MATHEMATICAL MODELS
Bed load Propagation Velocity
The propagation velocity of sediment material is an important parameter to characterize the
bed load transport in rivers. In some numerical approaches for morphological simulations
this velocity is a useful input parameter.
Several empirical investigations have been made to measure the velocity of bed load
material in experimental flumes. One recent approach for the determination of the
propagation velocity is the semi-empirical equation based on probability considerations by
Zhilin Sun and John Donahue (Sun and Donahue 2000) as
(
uB =
7.5 θ ′ − C0 θ cr
)
( s − 1) gd
(RI-2.32)
where θ ′ is the dimensionless effective bottom shear stress and θ cr is the critical Shields
parameter for incipient motion for a grain of diameter d . Furthermore, C0 is a coefficient
less than 1 and s is specific density of the bedload material. As can be seen, the
propagation velocity will increase if the difference of the actual shear stress and the criticial
shear stress enlarges as well as in the case of larger grain diameters.
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2.2
MATHEMATICAL MODELS
Suspended Sediment and Pollutant Transport
2.2.1
One Dimensional Advection-Diffusion-Equation
For a channel with irregular cross section area A (Fig.1) the following advection-diffusion
equation for to the number of pollutant species or grain size classes ng holds:
∂ ( ACg )
∂t
+
∂ (QCg )
∂x
−
∂Cg 
∂ 
− Slg 0
 AΓ
 − Sg =
∂x 
∂x 
Introducing the continuity equation
A
∂Cg
∂t
2.2.2
+Q
∂Cg
∂x
−
=
for
g 1,..., ng
(RI-2.33)
∂A δ Q
+
=
0 in equation (RI-2.33) one becomes:
δt δ x
∂Cg 
∂ 
− Slg 0
 AΓ
 − Sg =
∂x 
∂x 
=
for
g 1,..., ng
(RI-2.34)
Two Dimensional Advection-Diffusion-Equation
According to the number of pollutant species or grain size classes, ng advection-diffusion
equations for transport of the suspended material are provided as follows:
∂Cg  ∂ 
∂Cg 
∂
∂ 
0 =
Cg h +  Cg q − hΓ
for g 1,..., ng
 +  Cg r − hΓ
 − S=
g − Sl g
∂t
∂x 
∂x  ∂y 
∂y 
(RI-2.35)
where
2.2.3
Cg is the concentration of each grain size class and Γ is the eddy diffusivity.
Source Terms
For both suspended sediment and pollutant transport there can be a local sediment or
pollutant source: Slg given by a volume.
For suspended sediment transport an additional source term S g representing the sediment
exchange with the bed has to be considered. This term appears also in the bed load
equations if they are applied in combination with suspended load.
This term is calculated by the difference between the deposition rate
suspension (entry) rate. qek
S=
qeg − qd g
g
qdk and the
(RI-2.36)
The deposition rate is expressed as convection flux of the sink rate:
qd g = wg Cd g
(RI-2.37)
wg is the sink rate of grain class g and Cd g its concentration near the bed.
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MATHEMATICAL MODELS
According to a suggestion of Bennet und Nordin (1977) the suspension entry is formulated
in line with the deposition rate employing empirical relations:
qeg = wg β g Ceg
(RI-2.38)
The outcome of this is:
(
=
S g wg β g Ceg − Cd g
)
(RI-2.39)
The sink rate wg can be determined against the grain diameter after one of the relations
given in chapter 0.
The reference concentration for the suspension entry can be calculated as follows after Van
Rijn (1984):
Ceg = 0.015
d g Tg1.5
(RI-2.40)
a ( D* )0.3
g
where:
Tg
a
Dg*
is the dimensionless characteristic number
for the bottom shear stress of grain class g.
is the reference height above the mean bed bottom.
is the dimensionless diameter of grain class g.
Another approach is the one of Zyserman and Fredsøe (1994):
0.331(θ ′ − 0.045)1.75
Ceg =
1 + 0.72(θ ′ − 0.045)1.75
with the dimensionless effective bottom shear stress
=
θ ′ u*2 / ( s − 1) gd 
The reference concentration for the deposition rate is calculated after Lin (1984):

 wg  
=
Cd g  3.25 + 0.55ln 
  Cg
u
κ


*

(RI-2.41)
where:
Cg
u*
κ
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is the mean concentration over depth of suspended particles of grain class g,
is the shear velocity,
the Von Karmann constant.
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2.3
MATHEMATICAL MODELS
Bed Load Transport
2.3.1
2.3.1.1
One Dimensional Bed Load Transport
Component of the Bed Load Flux
The bed load flux for the one-dimensional case consists of one single component for each
grain size, namely the specific bed load flux in stream wise direction qB
g
2.3.1.2
Evaluation of Bed Load Transport due to Stream Forces
In the one-dimensional case, the total specific bed load flux due to stream forces is
evaluated as follows:
qBg = β g qB (ξ g )
(RI-2.42)
Details about the transport laws for evaluation of
the hiding factor ξ g can be found in chapter 2.3.4.
2.3.1.3
qB with or without the consideration of
Bed Material Sorting
For each fraction g a mass conservation equation can be written, the so called “bedmaterial sorting equation”:
∂
∂t
(1 − p ) ( β g ⋅ hm ) +
∂qB g
∂x
slBg 0
+ sg − sf g −
=
for g 1,..., ng
=
(RI-2.43)
where p = porosity of bed material (assumed to be constant), qB = total bed load flux per
g
unit width, sf g = specific flux through the bottom of the active layer due to its movement
and slB = source term per unit width to specify a local input or output of material (e.g. rock
g
fall, dredging). The term sg describes the exchange per unit width between the sediment
and the suspended material (see chapter 2.2.3).
2.3.1.4
Global Mass Conservation
Finally, the global bed material conservation equation is obtained by adding up the masses
of all sediment material layers between the bed surface and a reference level for all
fractions (Exner-equation) directly resulting in the elevation change of the actual bed level:
(1 − p )
∂z B ng  ∂qBg ∂qBg
+ ∑
+
+ sg − slBg
∂t g =1  ∂x
∂y
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
0
 =

(RI-2.44)
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2.3.2
2.3.2.1
MATHEMATICAL MODELS
Two Dimensional Bed Load Transport
Components of the Bed Load Flux
The specific bed load flux in x direction is composed of three parts (an analogous relation
exists in y direction):
qBg , x =
qBg , xx + qBg , xlateral + qBg , xgrav
(RI-2.45)
where (quantities per unit width) qB , xx = bed load transport due to flow in x direction,
g
qBg , xlateral = lateral transport in x-direction (due to bed load transport in y direction on sloped
bed) and qB , xgrav = pure gravity induced transport (e.g. due to collapse of a side slope).
g
2.3.2.2
Evaluation of Equilibrium Bed Load Transport
In the two-dimensional case, bed load transport is evaluated as follows
qBg , xx = β g qB (ξ g ) ⋅ e x
(RI-2.46)
th
The specific bed load discharge qB of the g grain class has to be evaluated by a suitable
g
transport law. Approaches for the bed load discharge qB (ξ g ) with or without the
g
consideration of a hiding factor ξ g are discussed in section 2.3.4.
2.3.2.3
Contribution due to Lateral Transport
The empirical bed load formulas were usually derived for situations where the bed gradient
equals the flow direction. But in situations where a transverse bed slope exists, the
direction of the moving sediment particle no longer exactly equals the flow direction.
Therefore a correction term in form of a lateral transport component is needed. The
gravitational induced lateral transport occurring on a transverse bed slope is determined
according to the empirical approach of Ikeda (1982):


τ Bcr ,g
qBg ,lateral = 1.5
Stransverse  qBg


τ B′


(RI-2.47)
where: Stransverse = bed slope in transverse direction to the flow, qB = bed load of fraction
g
g in flow direction and τ Bcr ,g = critical shear stress of the individual grain class. From this
the lateral transport components qB ,lateral in x- and y-directions can be computed.
g
2.3.2.4
Bed Material Sorting
The change of volume of a grain class g is balanced over the bed load control volume
and the underneath layer volume Vsub , as it is illustrated in Fig. 4.
Vg
g
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Fig. 4: Definition sketch of overall control volume of bed material sorting equation (red)
Depending on the bedload in- and outflows, the composition of the grain fractions in the
bedload control volume can change. Futhermore three source terms are distinguished:
•
•
•
External sediment sources or sinks can be specified ( Slg ).
An exchange of sediment with the water column can take place ( S g ).
The movement of the bedload control volume bottom z F can lead to changes of
the grain compositions within the bedload control volume and the underneath soil
layer ( Sf g ). (This is a special kind of source term, because it does not change the
overall grain volume within the control volume indicated in Fig. 4. It is not related
with a physical movement of particles.)
For each grain class g a mass conservation equation can be written, the so called “bedmaterial sorting equation”, which is used to determine the grain fractions β g at the new
time level




∂ ( β g ⋅ hm )
∂ ( β sub , g ⋅ ( z F − zsub ) ) 
∂qB g , x ∂qB g , y
∂V

=
+
=
−
−
− sg + slBg
(1 − p ) 


∂t
∂t 
∂t
∂x
∂y 






 change of grain volume in change of grain volume in layer 1   source terms
fluxes over boundary
= sf g
 bedload control volume

Rearranging this sorting equation leads to following formulation which is used from here on
∂
∂t
(1 − p ) ( β g ⋅ hm ) +
∂qB g , x
∂x
+
∂qB g , y
∂y
+ sg − sf=
0
g − slBg
=
for g 1,..., ng
(RI-2.48)
where hm = thickness of bedload control volume, p = porosity of bed material (assumed to
be constant), qB g , x , qB g , y = components of total bed load flux per unit width, sf g = flux
through the bottom of the bedload control volume due to its movement and slB = source
g
term to specify a local input or output of material (e.g. rock fall, dredging).
(
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MATHEMATICAL MODELS
Global Mass Conservation
The global bed material conservation equation, which is often called Exner-equation, is
obtained by adding up the masses of all sediment material layers between the bed surface
and a reference level. This is done for all grain fractions and directly results in the elevation
change of the actual bed level z B :
(1 − p )
∂z B ng  ∂qBg , x ∂qBg , y
+ ∑
+
+ sg − slBg
∂t g =1  ∂x
∂y
2.3.3
Sublayer Source Term

0
 =

(RI-2.49)
The bottom elevation of the bed load control volume z F is identical to the top level of the
underneath layer. If z F moves up, sediment flows into this underneath layer and leads to
changes in its grain compositions. The exchange of sediment particles between the bed
load control volume and the underlying layer is expressed by the source term:
sf g =−(1 − p )
2.3.4
2.3.4.1
∂
(( z F − zsub ) β subg )
∂t
(RI-2.50)
Closures for Bed Load Transport
Meyer-Peter and Müller (MPM)
The (Meyer-Peter and Müller 1948) formula can be written as follows:
qBg
3
 τ B − τ Bcr ,g
=
 0.25 ρ

 2 1 
 

  ( s − 1) g 
(RI-2.51)
Herein, τ B is the effective shear stress induced by the flow and s = ρ s ρ the sediment
density coefficient. In the dimensionless shape the formula can be written as
qB =
Φ B ( s − 1) gd m3/ 2
(RI-2.52)
with
Φ B= 8(θ ′ − θ cr )
3
2
Meyer-Peter and Müller observed in their experiments that fist grains moved already for
θ cr = 0.03 . But as their experiments took place with steady conditions they used a value for
which already 50% of the grains where moving. They proposed the value of 0.047.
However for very unsteady conditions one should use values for which the grains really start
to move (Fäh 1997) like the values given by the shields diagram.
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MATHEMATICAL MODELS
The formula of Meyer-Peter and Müller is applicable in particular for coarse sand and gravel
with grain diameters above 1 mm (Malcherek 2001).
2.3.4.2
Parker
(Parker 1990) has extended his empirical substrate-based bed load relation for gravel
mixtures (Parker, Klingeman et al. 1982), which was developed solely with reference to field
data and suitable for near equilibrium mobile bed conditions, into a surfaced-based relation.
The new relation is proper for the non-equilibrium processes.
Based on the fact that the rough equality of bed load and substrate size distribution is
attained by means of selective transport of surface material and the surface material is the
source for bed load, Parker has developed the new relation based on the surface material.
An important assumption in deriving the new relation is suspension cut-off size. Parker
supposes that during flow conditions at which significant amounts of gravel are moved, it is
commonly (but not universally) found that the sand moves essentially in suspension (1 to 6
mm). There for Parker has excluded sand from his analysis. In his free access Excel file, he
has explicitly emphasised that the formula is valid only for the size larger than 2 mm.
Regarding to the Oak Creek data, the original relation predicted 13% of the bed load as
sand. For consistency it has to be corrected for the exclusion of sand and finer material.
Wsi* = 0.00218 G ξ sωφsg 0  ; Wsi* =
Rgqbi
(RI-2.53)
3/2
(τ B ρ ) Fi
where
d
ξs =  i
d
 g



−0.0951
;
φ50 =
τ sg*
*
τ rsg
0
;
τ sg* =
τB
ρ Rgd g
;
*
τ rsg
0 = 0.0386
 ln ( di d g ) 
σ
ω0 (φsg 0 ) − 1 ; σ = ∑ Fi 
ω=
1+


σ 0 (φsg 0 ) 
 ln ( 2 ) 
2
;
d g = e∑
Fi ln ( di )
ξs
is a “reduced” hiding function and differs from the one of Einstein. The Einstein hiding
factor adjusts the mobility of each grain di in a mixture relative to the value that would be
realized if the bed were covered with uniform material of size di. The new function adjusts
the mobility of each grain di relative to the d50 or dg , where dg denotes the surface geometric
mean size.
Although the above formulation does not contain a critical shields stress, the reference
*
Schields stress τ rsg 0 makes up for it, in that transport rates are exceedingly small for
*
τ sg* < τ rsg
0 . Regarding to the fact that parker’s relation is based on field data and field data
are often in case of low flow rates, the relation calculates low bed load rates (Marti 2006).
VAW ETH Zürich
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2.3.4.3
MATHEMATICAL MODELS
Hunziker (MPM-H)
To map transport processes of mixed sediments and to calculate according bed load
discharge, corrections or special formulas for graded sediments have to be applied. Besides
the ability to model grain sorting in the active layer, the exposure of bigger grains to the flow
and the involved shielding of fine sediments, the so called hiding effect has to be
considered.
A special formula for the fractional transport of graded sediments was proposed by
(Hunziker 1995):
'
qBg = 5β g ξ g (θ dms
− θ cdms ) 
(
3
2
3
( s − 1) gd ms
(RI-2.54)
)
The additional shear stress θ dms − θ cdms , regarding the mean grain size of the bed surface
material, is corrected by a corresponding hiding factor.
'
(
)
Due to the correction of the additional shear stress θ dms − θ cdms , the transport formula
bases on the concept of “equal mobility”, i.e. all grain classes start to move at the same
flow conditions. The critical Shields value is calculated with the mean grain diameters and is
given by
θ cdms
d 
= θ ce  mo 
 d ms 
'
0.33
(RI-2.55)
where θ ce = critical shear stress for incipient motion for uniform bed material. Hunziker’s
formula distinguishes thereby between two sediment layers, the upper layer which is in
interaction with the flow and an underneath sublayer. Here d ms is the mean diameter of the
upper layer and d mo is mean diameter of subsurface bed material.
The hiding factor is determined as
d 
ξg =  g 
 d ms 
−α
(RI-2.56)
where α is an empirical parameter which depends on the dimensionless shear stress of
the mixture (see also (Hunziker and Jaeggi 2002)). The parameter α is determined as
′−1.5 − 0.3
=
α 0.011θ dms
(RI-2.57)
′ this parameter can tend to negative infinity
In case of very small bed shear stresses θ dms
which can lead to numerical instabilities of the simulation. Therefore a limitation to a
minimum value α min is implemented to prevent non-physical values.
α
limited
= min(α min , α )
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MATHEMATICAL MODELS
Günter’s two grains model
(Günter 1971) suggested a two grains model, in order to be able to consider the armoured
layer forming in case of the simulation with one grain size. Based on his formulation, the
upper limit of the discharge of the armoured layer (QD) can be calculated, for which the
armoured layer resists it. If the discharge is smaller than QD, then no material will be eroded
from the bed, but the input bed load from the upstream can be transported into the
downstream.
Using the formulation of Günter one obtains
d 
> θ cr  90 
( s − 1) d m
 dm 
RS f
0.67
(RI-2.58)
Where R denotes hydraulic radius, Sf is energy slope, dm is grain size, d90 is armoured layer
grain size and θ cr denotes the critical Shield’s parameter for the grain size (dm).
(Hunziker and Jaeggi 1988) analysed Günter’s relation and concluded that it should not be
always used. If there are aggradations on an armoured layer, then it should be possible to
transport the material. In this case, the elevation of the armoured layer has to be saved and
compared with the new bed elevation.
2.3.4.5
Smart-Jäggi
Experiments for bed load transport in gravel beds were performed at VAW ETH Zurich for
bed slopes of 0.0004-0.023 by Meyer-Peter and Müller [1948] and for bed slopes of 0.03-0.2
by Smart and Jäggi [1983] and by Rickenmann [1990]. Smart and Jäggi developed a bed load
formula for steep channels using their own results and the ones of Meyer-Peter and Müller.
4  d90 
=
qB


s − 1  d30 
0.2
J 0.6 Ru ( J − J cr )
(RI-2.59)
J is the slope, s = ρ s / ρ the sediment density coefficient, R the hydraulic radius and u the
velocity.
=
J cr θ cr ( s − 1)d m R
θ cr the dimensionless shear stress at the beginning of bed load transport and
grain size.
(RI-2.60)
d m the mean
(This formula is only implemented for 1D simulations at the moment).
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MATHEMATICAL MODELS
Rickenmann
Experiments for bed load transport in gravel beds were performed at VAW ETH Zurich for
bed slopes of 0.0004-0.023 by Meyer-Peter and Müller [1948] and for bed slopes of 0.03-0.2
by Smart and Jäggi [1983] and by Rickenmann [1990]. Rickenmann [1991] developed the
following formula for the entire slope range using 252 of these experiments.
0.2
d 
−0.5
=
Φ B 3.1 90  θ ′0.5 (θ ′ − θ cr ) Fr1.1 ( s − 1)
 d30 
qB =
Φ B ( ( s − 1) gd m3 )
(RI-2.61)
0.5
(RI-2.62)
θ ′ is the dimensionless shear stress, θ cr the dimensionless shear stress at the beginning
of bed load transport, s = ρ s / ρ the sediment density coefficient, Fr the Froude number
and d m the mean grain size.
2.3.4.7
Wu Wang and Jia
Weiming Wu, Sam S.Y. Wang and Yafei Jia (2000) developed a transport formula for graded
bed materials based on a new approach for the hiding and exposure mechanism of nonuniform transport. The hiding and exposure factor is assumed to be a function of the hidden
and exposed probabilities, which are stochastically related to the size and gradation of bed
materials. Based on this concept, formulas to calculate the critical shear stress of incipient
motion and the fractional bed-load transport have been established. Different laboratory and
field data sets were used for these derivations.
d g hidden and exposed by grains di is obtained from
ng
dg
di
=
phid g ∑
=
βi
,
pexp g ∑ βi
d g + di
d g + di
=i 1 =
i 1
The probabilities of grains
ng
(RI-2.63)
The critical dimensionless shields parameter for each grain class g can be calculated with
the hiding and exposure factor η g and the shields parameter of the mean grain size θ crm as
m
θ cr
g
 pexp g 

= θ crm 
 phid 
g 


(RI-2.64)
ηg
The transport capacity now can be determined with Wu’s formula in dimensionless form as
2.2
 θ′

=
Φ Bg 0.0053 
− 1 .
θ crg

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MATHEMATICAL MODELS
Finally the bed load transport rates calculates for each grain fraction as
qbg = β g ( s − 1) gd g3 Φ Bg .
As results of their data analysis the authors recommend to set m = −0.6 and
to obtain best results.
(This formula is only implemented for 2D simulations at the moment).
2.3.4.8
θ cr = 0.03
m
Power Law
A very simple approach for the determination of bed load transport is the power law
formula. No critical shear stress is regarded in this approach and the transport only depends
on the flow velocity. Therefore it should usually not be applied in practical situations.
qb = au b
The factors
(RI-2.66)
a and b can used as calibration parameters.
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References
Agoshkov, V. I., A. Quarteroni, et al. (1994). "Recent Developments in the NumericalSimulation of Shallow-Water Equations .1. Boundary-Conditions." Applied Numerical
Mathematics 15(2): 175-200.
Bechteler, W., M. Nujić, et al. (1993). Program Package “FLOODSIM“ and ist Application.
Beffa, C. J. (1994). Praktische Lösung der tiefengemittelten Flchwassergleichungen.
Versuchsanstalt für Wasserbau,Hydrologie und Glaziologie (VAW). Zürich, ETH
Zürich.
Bennett, J.P. and Nordin C.F. (1977). Simulation of Sediment Transport and Armoring,
Hydrological Sciences Bulletin, XXII, Vol 4; No 12, 555-569.
Chaudhry, M. H. (1993). Open-Channel Flow. Englewood Cliffs, New Jersey, Prentice Hall.
Cunge, J. A., F. M. J. Holly, et al. (1980). Practical aspects of computational river hydraulics.
London, Pitman.
Fäh, R. (1997) Numerische Simulation der Strömung in offenen Gerinnen mit beweglicher
Sohle. Versuchsanstalt für Wasserbau, Hydrologie und Glaziologie(VAW). Zürich,
ETH Zürich.
Günter, A. (1971). Die kritische mittlere Sohlenschubspannung bei Geschiebemischung
unter Berücksichtigung der Deckscichtbildung und der turbulenzbedingten
Sohlenschubspannungsschwankungen. Versuchsanstalt für Wasserbau,Hydrologie
und Glaziologie (VAW). Zürich, ETH Zürich.
Hunziker, R. P. (1995). Fraktionsweiser Geschiebetransport. Versuchsanstalt für
Wasserbau,Hydrologie und Glaziologie (VAW). Zürich, ETH Zürich.
Hunziker, R. P. and M. N. R. Jaeggi (1988). Numerische Simulation des Geschiebehaushalts
der Emme. Interprävent, Graz.
Hunziker, R. P. and M. N. R. Jaeggi (2002). "Grain sorting processes." Journal of Hydraulic
Engineering-Asce 128(12): 1060-1068.
Ikeda, S. (1982). "Lateral Bed-Load Transport on Side Slopes." Journal of the Hydraulics
Division-Asce 108(11): 1369-1373.
Jäggi, M. (1995) Vorlesung: Flussbau. ETH Abt. II, VIII, und XC. Zürich, ETH Zürich.
Marti, C. (2006). Morphologie von verzweigten Gerinnen Ansätze zur Abfluss-,
Geschiebetransport und Kolktiefenberechnung. Versuchsanstalt für
Wasserbau,Hydrologie und Glaziologie (VAW). Zürich, ETH Zürich.
Lin B. (1984) Current Study of Unsteady Transport of Sediment in China, Proceedings of
Japan-China Biateral Seminar on River Hydraulics and Engineering Experiences,
Tokyo-Kyoto –Sapporo.
Malcherek A. (2001). "Sedimenttransport und Morphodynamik", Vorlesungsskript der
Universität der Bundeswehr München, München.
Meyer-Peter, E. and R. Müller (1948). Formulas for Bed-Load Transport. 2nd Meeting IAHR,
Stockholm, Sweden, IAHR.
Nujić, M. (1998). Praktischer Einsatz eines hochgenauen Verfahrens für die Berechnung von
tiefengemittelten Strömungen. Institut für Wasserwesen. München, Universität der
Bundeswehr München.
VAW ETH Zürich
Version 10/28/2009
R I - References
MATHEMATICAL MODELS
Parker, G. (1990). "Surface-based bedload transport relation for gravel rivers." Journal of
Hydraulic Reasearch 28(4): 417-436.
Parker, G., P. C. Klingeman, et al. (1982). "Bedlaod and size distribution in paved gravel-bed
streams." Journal of Hydraulic Division, ASCE(108): 544-571.
Prandtl, L. (1942). "Comments on the theory of free turbulence." Zeitschrift Fur Angewandte
Mathematik Und Mechanik 22: 241-243.
Rickenmann, D. (1990). BedLoad capacity of slurry flows at steep slopes. Versuchsanstalt
für Wasserbau,Hydrologie und Glaziologie (VAW). Zürich, ETH Zürich.
Rickenmann, D. (1991). Hyperconcentrated flow and sediment transport at steep slopes.
Journal of Hydraulic Engineering, 117(11):1419-1439.
Shields, A. (1936). Anwendungen der Ähnlichkeitsmechanik und der Turbulenzforschung auf
die Geschiebebewegungen. Preussische Versuchsanstalt für Wasserbau und
Schiffbau. Berlin, Deutschland.
Smart, G.M., and M.N.R.Jaeggi, (1983). Sediment transport on steep slopes.
Versuchsanstalt für Wasserbau,Hydrologie und Glaziologie (VAW). Zürich, ETH
Zürich.
Sun, Z. L., and Donahue, J. (2000). "Statistically derived bedload formula for any fraction of
nonuniform sediment." Journal of Hydraulic Engineering-Asce, 126(2), 105-111.
Thuc, T. (1991). "Two-dimensional morphological computations near hydraulic structures."
Dissertation, Asian Institute of Technology, Bangkok, Thailand.
Van Rijn, L. C. (1984). "Sediment Transport, Part II: Suspended Load Transport." Journal of
Hydraulic Engineering, ASCE 110(11).
Van Rijn, L. C. (1987). "Mathematical modelling of morphological processes in the case of
suspended sediment transport." Delft Hydraulics Communication No. 382., Delft
Hydraulics Laboratory, Delft, The Netherlands.
Van Rijn, L. C. (1989). "Handbook: sediment transport by current and waves". Report H 461,
Delft Hydraulics, Netherlands.
Wu, W., Wang, S. and Y.Jia, (2000). "Nonuniform sediment transport in alluvial rivers".
Journal of Hydraulic Research, 38(6), 427-434.
Wu, W. (2007). "Computational River Dynamics". Taylor & Francis, London, UK.
R I - References
VAW ETH Zürich
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NUMERICS KERNEL
Table of contents
1
General View ....................................................................................... 1-1
2
Methods for Solving the Flow Equations
2.1
Fundamentals..................................................................................................... 2.1-1
2.1.1
Finite Volume Method .................................................................................. 2.1-1
2.1.2
The Riemann Problem .................................................................................. 2.1-3
2.1.3
Exact Riemann Solvers ................................................................................. 2.1-4
2.1.4
Approximate Riemann Solvers ..................................................................... 2.1-6
2.1.2.1
HLL – Riemann Solver ........................................................................... 2.1-6
2.1.2.2
HLLC – Riemann Solver ........................................................................ 2.1-7
2.2
Saint-Venant Equations .................................................................................... 2.2-1
2.2.1
Spatial Discretisation .................................................................................... 2.2-1
2.2.2
Discrete Form of Equations .......................................................................... 2.2-1
2.2.2.1
Continuity Equation ............................................................................... 2.2-1
2.2.2.2
Momentum Equation ............................................................................ 2.2-2
2.2.3
Discretisation of Source Terms .................................................................... 2.2-2
2.2.2.1
Bed Slope Source Term ........................................................................ 2.2-2
2.2.2.2
Friction Source Term ............................................................................. 2.2-3
2.2.2.3
Hydraulic Radius .................................................................................... 2.2-3
2.2.4
Discretisation of Boundary conditions .......................................................... 2.2-7
2.2.2.1
Inlet Boundary ....................................................................................... 2.2-7
2.2.2.2
Outlet Boundary .................................................................................... 2.2-7
2.2.2.3
Moving Boundaries ............................................................................... 2.2-8
2.2.5
Solution Procedure ....................................................................................... 2.2-9
2.3
Shallow Water Equations ................................................................................. 2.3-1
2.3.1
2.3.2.1
Flux Estimation ...................................................................................... 2.3-2
2.3.2.2
Flux Correction ...................................................................................... 2.3-3
2.3.2
2.3.2.1
Friction Source Term ............................................................................. 2.3-4
2.3.2.2
Source Term for viscous and turbulent Stresses .................................. 2.3-6
2.3.2.3
Bed Slope Source Term and Bed Slope Calculation ............................. 2.3-8
2.3.3
Conservative property (C-Property) ............................................................ 2.3-13
2.3.4
Discretisation of Boundary Conditions ....................................................... 2.3-15
2.3.2.1
Inlet Boundary ..................................................................................... 2.3-15
2.3.2.2
Outlet Boundary .................................................................................. 2.3-16
2.3.2.3
Inner Boundaries ................................................................................. 2.3-19
2.3.2.4
Moving Boundaries ............................................................................. 2.3-21
2.3.5
Solution Procedure ..................................................................................... 2.3-25
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Solution of Sediment Transport Equations
3.1
General Vertical Partition ................................................................................. 3.1-1
3.1.1
Vertical Discretisation ................................................................................... 3.1-1
3.1.2
Determination of Layer Thickness ................................................................ 3.1-2
3.2
One Dimensional Sediment Transport ............................................................ 3.2-1
3.2.1
3.2.2.1
Soil Segments ....................................................................................... 3.2-1
3.2.2
3.2.2.1
Advection-Diffusion Equation ................................................................ 3.2-2
3.2.2.2
Computation of Diffusive Flux ............................................................... 3.2-3
3.2.2.3
Computation of Advective Flux ............................................................. 3.2-3
3.2.2.4
Global Bed Material Conservation Equation ........................................ 3.2-10
3.2.2.5
Bed material sorting equation ............................................................. 3.2-11
3.2.2.6
Interpolation ........................................................................................ 3.2-12
3.2.3
Discretisation of Source Terms .................................................................. 3.2-13
3.2.2.1
External Sediment Sources and Sinks ................................................ 3.2-13
3.2.2.2
Sediment Flux through Bottom of Bed Load Control Volume ............ 3.2-13
3.2.2.3
Source Term for Sediment Exchange between Water and Bottom ... 3.2-13
3.2.2.4
Splitting of Bed Load and Suspended Load Transport ........................ 3.2-14
3.2.4
Solution Procedure ..................................................................................... 3.2-15
3.3
Two Dimensional Sediment Transport ........................................................... 3.3-1
3.3.1
3.3.2
3.3.2.1
Global Bed material Conservation Equation – Exner Equation .............. 3.3-3
3.3.2.2
Computation of Bed Load Fluxes .......................................................... 3.3-3
3.3.2.3
Flux Correction ...................................................................................... 3.3-5
3.3.2.4
Bed Material Sorting Equation .............................................................. 3.3-5
3.3.3
3.3.2.1
External Sediment Sources and Sinks .................................................. 3.3-6
3.3.2.2
Sediment Flux through Bottom of Bed Load Control Volume .............. 3.3-6
3.3.2.3
Sediment Exchange between Water and Bottom ................................ 3.3-7
3.3.4
Management of Soil Layers.......................................................................... 3.3-8
3.3.5
Solution Procedure ....................................................................................... 3.3-9
4
Time Discretisation and Stability Issues
4.1
Explicit Schemes ................................................................................................ 4.1-1
4.1.1
Euler First Order ........................................................................................... 4.1-1
4.2
Determination of Time Step Size ..................................................................... 4.2-1
4.2.1
Hydrodynamic ............................................................................................... 4.2-1
4.2.2
Sediment Transport ...................................................................................... 4.2-1
4.3
Implicit Scheme ................................................................................................. 4.3-1
4.3.1
Introduction .................................................................................................. 4.3-1
4.3.2
Time Discretisation ....................................................................................... 4.3-1
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4.3.2.1
4.3.2.2
4.3.3
Solution ......................................................................................................... 4.3-2
4.3.4
Integral Terms .............................................................................................. 4.3-4
4.3.2.1
4.3.2.2
4.3.5
General Description of the Derivatives for the Matrix A .............................. 4.3-5
4.3.2.1
Derivatives of the Continuity Equation .................................................. 4.3-5
4.3.2.2
Derivatives of the Momentum Equation ............................................... 4.3-6
4.3.6
Determination of the Derivatives with Upwind Flux Determination ............ 4.3-7
4.3.2.1
Derivatives of the Continuity Equation .................................................. 4.3-7
4.3.2.2
Derivatives of the Momentum Equation ............................................... 4.3-8
4.3.7
Determination of the Derivatives with Roe Flux Determination .................. 4.3-9
4.3.2.1
Derivatives of the Continuity Equation: ............................................... 4.3-10
4.3.2.2
Derivatives of the Momentum Equation ............................................. 4.3-11
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List of Figures
Fig. 1: Triangular Finite Elements of a Two-Dimensional Domain .............................................. 2.1-1
Fig. 2: Two-Dimensional Finite Volume Mesh: (a) Cell Centered mesh (b) Cell Vertex mesh ..... 2.1-2
Fig. 3: Initial Data for Riemann Problem...................................................................................... 2.1-3
Fig. 4: Possible Wave Patterns in the Solution of Riemann Problem ............................................ 2.1-4
Fig. 5: Piecewise constant data of Godunov upwind method ........................................................ 2.1-5
Fig. 6: Principle of the HLL Riemann Solver. The analytical solution (left) is replaced by an
approximate one with a constant state U* separated by waves with estimated wave speeds (right) 2.1-6
Fig. 7: Definition sketch................................................................................................................. 2.2-1
Fig. 8: Simple cross section ........................................................................................................... 2.2-4
Fig. 9: Conveyance computation of a channel with a flat zone ..................................................... 2.2-4
Fig. 10: Cross section with flood plains and main channel ............................................................. 2.2-5
Fig. 11: Cross section with definition of a bed ................................................................................ 2.2-5
Fig. 12: Cross section with flood plains and definition of a bed bottom ......................................... 2.2-6
Fig. 13: Geometry of a Computational Cell
Ωi
in FV ................................................................... 2.3-2
Fig. 14: Wetted perimeter of a boundary element with wall friction ............................................... 2.3-5
Fig. 15: A Triangular Cell ............................................................................................................... 2.3-9
Fig. 16: Water volume over a cell.................................................................................................. 2.3-10
Fig. 17: Partially wet cell .............................................................................................................. 2.3-10
Fig. 18: Quadrilateral element and its division into four triangles ( ∆ 12C , ∆ 23C , ∆ 34C , ∆ 41C ) ... 2.3-11
Fig. 19: Edge with linearly varying water depth ........................................................................... 2.3-13
Fig. 20: Flow over a weir .............................................................................................................. 2.3-17
Fig. 21: Outlet cross section with a weir ....................................................................................... 2.3-17
Fig. 22: Reduction factor σ uv for an incomplete flow over the weir. h u is the downstream water
depth over the weir crest and h denotes the upstream flow depth over the weir crest. ................ 2.3-20
Fig. 23: Schematic representation of a mesh with dry, partial wet and wet elements ................... 2.3-21
Fig. 24: Wetting process of a element ............................................................................................ 2.3-22
Fig. 25: Definition of effective length at partially wetted edge...................................................... 2.3-23
Fig. 26: Different cases for flux computations over an edge at wet-dry interfaces ....................... 2.3-24
Fig. 27: The logical flow of data through BASEplane ................................................................... 2.3-26
Fig. 28: Data flow through the hydrodynamic routine .................................................................. 2.3-27
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Fig. 29: Data flow through the morphodynamic routine ............................................................... 2.3-28
Fig. 30: Vertical Discretisation of a computational cell.................................................................. 3.1-1
Fig. 31: Soil Discretisation in a cross section ................................................................................. 3.2-1
Fig. 32: Effect of bed load on cross section geometry ..................................................................... 3.2-2
Fig. 33: Invariance of C along the characteristic line y .................................................................. 3.2-7
Fig. 34: Function
C in ( x) ................................................................................................................ 3.2-8
Fig. 35: Distribution of sediment area change over the cross section ........................................... 3.2-11
Fig. 36: Schematic illustration of problems related to the update of bed elevations ....................... 3.3-1
Fig. 37: Dual mesh approach with separate meshes for hydrodynamics (black) and sediment
transport (green). Sediment cells have the bed elevations defined in their cell centers. .................. 3.3-2
Fig. 38: Cross sectional view of dual mesh approach. Changes in sediment volume ∆V do not affect
the neighboring cells’ sediment volumes. ......................................................................................... 3.3-2
Fig. 39: Determination of bed load flux over the sediment cell edges i,j ......................................... 3.3-4
Fig. 40: Definition of mixed composition β gmix for the eroded volume (red) .................................. 3.3-7
Fig. 41: Uncoupled asynchronous solution procedure consisting of sequential steps................... 3.3-10
Fig. 42: Composition of the given overall calculation time step
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1 General View
There is great improvement in the development of numerical models for free surface flows
and sediment movement in the last decade. The presented number of publications about
these subjects proves this clearly. The SWE and the sediment flow equation are a nonlinear,
coupled partial differential equations system. A unique analytical solution is only possible for
idealised and simple conditions. For practical cases, it is required to implement the
numerical methods. A numerical solution arises from the discretisation of the equations.
There are different methods to discretize the equations such as:
-
Finite difference………….(FD)
-
Finite volume…………….(FV)
-
Finite element…………….(FE)
-
Characteristic Method……(CM)
It is normally distinguished between temporal and spatial discretisation of continuum
equations. The latter can be undertaken on different forms of grids such as Cartesian, nonorthogonal, structured and unstructured, while the former is usually done by a FD scheme in
time direction, which can be explicit or implicit. The explicit method is usually used for
strong unsteady flows.
In FD methods the partial derivations of equations are approximated by using Taylor series.
This method is particularly appropriate for an equidistant Cartesian mesh.
In FV methods; the partial derivations of equations are not directly approximated like in FD
methods. Instead of that, the equations are integrated over a volume, which is defined by
nodes of grids on the mesh. The volume integral terms will be replaced by surface integrals
using the Gauss formula. These surface integrals define the convective and diffusive fluxes
through the surfaces. Due to the integration over the volume, the method is fully
conservative. This is an important property of FV methods. It is known that in order to
simulate discontinuous transition phenomena such as flood propagation, one must use
conservative numerical methods. In fact, 40 years ago Lax and Wendroff proved
mathematically that conservative numerical methods, if convergent, do converge to the
correct solution of the equations. More recently, Hou and LeFloch proved a complementary
theorem, which says that if a non-conservative method is used, then the wrong solution will
be computed, if this contains a discontinuity such as a shock wave (Toro 2001).
The FE methods originated from the structural analysis field as a result of many years of
research, mainly between 1940 and 1960. In this method the problem domain is ideally
subdivided into a collection of small regions, of finite dimensions, called finite elements. The
elements have either a triangular or a quadrilateral form (Fig. 1) and can be rectilinear or
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curved. After subdivision of the domain, the solution of discrete problem is assumed to have
a prescribed form. This representation of the solution is strongly linked to the geometric
division of sub domains and characterized by the prescribed nodal values of the mesh.
These prescribed nodal values must be determined in such way that the partial differential
equations are satisfied. The errors of the assumed prescribed solution are computed over
each element and then the error must be minimised over the whole system. One way to do
this is to multiply the error with some weighting function ω ( x, y ) , integrate over the region
and require that this weighted-average error is zero. The finite volume method has been
used in this study; therefore it is explained in more detailed.
In this chapter it will be reviewed how the governing equations comprising the SWE and the
bed-updating equation, can be numerically approximated with accuracy. The first section
studies the fundamentals of the applied methods, namely the finite volume method and
subsequently the hydro- and morphodynamic models are discussed. In section on the
hydrodynamic model, the numerical approximate formulation of the SWE, the flux
estimation based on the Riemann problem and solver and related problems such as
boundary conditions will be analyzed. In the section on morphodynamics, the numerical
determination of the bed shear stress and the numerical treatment of bed slope stability will
be discussed as well as numerical approximation of the bed-updating equation.
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2 Methods for Solving the Flow Equations
2.1
2.1.1
Fundamentals
Finite Volume Method
The basic laws of fluid dynamics and sediment flow are conservation equations; they are
statements that express the conservation of mass, momentum and energy in a volume
enclosed by a surface. Conversion of these laws into partial differential equations requires
sufficient regularity of the solutions.
Fig. 1: Triangular Finite Elements of a Two-Dimensional Domain
This condition of regularity cannot always undoubtedly be guaranteed. The case of occurring
of discontinuities is a situation where an accurate representation of the conservation laws is
important in a numerical method. In other words, it is extremely important that these
conservation equations are accurately represented in their integral form. The most natural
method to achieve this is obviously to discretize the integral form of the equations and not
the differential form. This is the basis of the finite volume (FV) method. Actually the finite
volume method is in fact in the classification of the weighted residual method (FE) and
hence it is conceptually different from the finite difference method. The weighted function
is chosen equal unity in the finite volume method.
In this method, the flow field or domain is subdivided, as in the finite element method, into
a set of non-overlapping cells that cover the whole domain on which the equations are
applied. On each cell the conservation laws are applied to determine the flow variables in
some discrete points of the cells, called nodes, which are at typical locations of the cells
such as cell centres (cell centred mesh) or cell-vertices (cell vertex mesh) (Fig. 2).
Obviously, there is basically considerable freedom in the choice of the cell shapes. They can
be triangular, quadrilateral etc. and generate a structured or an unstructured mesh. Due to
this unstructured form ability, very complex geometries can be handled with ease. This is
clearly an important advantage of the method. Additionally the solution on the cell is not
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strongly linked to the geometric representation of the domain. This is another important
advantage of the finite volume method in contrast to the finite element method.
Fig. 2: Two-Dimensional Finite Volume Mesh: (a) Cell Centered mesh (b) Cell Vertex mesh
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The Riemann Problem
Formally, the Riemann problem is defined as an initial-value problem (IVP):
U t + Fx ( U ) =
0


U L ∀ x < 0
U ( x, 0 ) = 

U R ∀ x > 0 
(RII-2.1)
Here equations (RI-1.34) and (RI-1.35) are considered for the x-split of SWE. The initial states
U L , U R at the left or right side of an triangle edge
 hR 
 hL 




U L =  uL hL  ; U R =  uR hR 
v h 
v h 
 R R
 L L
are constant vectors and present conditions to the left of axes
respectively (Fig. 3).
x = 0 and to the right x = 0 ,
U
UL
UR
Fig. 3: Initial Data for Riemann Problem
There are four possible wave patterns that may occur in the solution of the Riemann
problem. These are depicted in Fig. 4. In general, the left and right waves are shocks and
rarefactions, while the middle wave is always a shear wave. A shock is a discontinuity that
travels with Rankine-Hugoniot shock speed. A rarefaction is a smoothly varying solution
∂F ( U ) / ∂U
which is a function of the variable x/t only and exists if the eigenvalues of J =
are all real and the eigenvectors are complete. The water depth and the normal velocity are
both constant across the shear wave, while the tangential velocity component changes
discontinuously. (See Toro (1997) for details about Riemann problem)
By solving the IVP of Riemann (RII-2.1), the desired outward flux
origin of local axes x = 0 and the time t = 0 can be obtained.
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Shear Wave
Rarefaction
Shock
Fig. 4: Possible Wave Patterns in the Solution of Riemann Problem
2.1.3
Exact Riemann Solvers
An algorithm, which solves the Riemann initial-value problem is called Riemann solver. The
idea of Riemann solver application in numerical methods was used for the first time by
Godunov (1959). He presented a shock-capturing method, which has the ability to resolve
strong wave interaction and flows including steep gradients such as bores and shear waves.
The so called Godunov type upwind methods originate from the work of Godunov. These
methods have become a mature technology in the aerospace industry and in scientific
disciplines, such as astrophysics. The Riemann solver application to SWE is more recent. It
was first attempted by Glaister (1988) as a pioneering attack on shallow water flow
problems in 1d cases.
In the algorithm pioneered by Godunov, the initial data in each cell on either side of an
interface is presented by piecewise constant states with a discontinuity at the cell interface
(Fig. 5). At the interface the Riemann problem is solved exactly. The exact solution in each
cell is then replaced by new piecewise constant approximation. Godunov’s method is
conservative and satisfies an entropy condition (Delis et al. 2000). The solution of Godunov
is an exact solution of Riemann problem; however, the exact solution is related to a
simplified problem since the initial data is approximated in each cell.
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Fig. 5: Piecewise constant data of Godunov upwind method
To compute numerical solutions by Godunov type methods, the exact Riemann solver or
approximate Riemann solver can be used. The choice between the exact and approximate
Riemann Solvers depends on:
-
Computational cost
-
Simplicity and applicability
-
Correctness
Correctness seems to be the overriding criterion; however the applicability can be also very
important. Toro argued that for the shallow water equations, the argument of computational
cost is not as strong as for the Euler equations. For the SWE approximate Riemann solvers
may lead to savings in computation time of the order of 20%, with respect to the exact
Riemann solvers (Toro 2001). However, due to the iterative approach of the exact Riemann
solver on SWE, its implementation of this will be more complicated if some extra equations,
such as dispersion and turbulence equations, are solved together with SWE. Since for these
new equations new Riemann solvers are needed, which include new iterative approaches,
complications seem to be inevitable. Therefore, an approximate solution which retains the
relevant features of the exact solution is desirable. This led to the implementation of
approximate Riemann solvers which are non-iterative and therefore, in general, more
efficient than the exact solvers.
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Approximate Riemann Solvers
Several researchers in aerodynamics have developed approximate Riemann solvers for the
Euler equations, such as Flux Difference Splitting (FDS) by Roe (1981), Flux Vector Splitting
(FVS) by Vanleer (1982) and approximate Riemann Solver by Harten et al. (1983); Osher and
Solomon (1982). The first two approximate Riemann solvers have been frequently used in
aerodynamics as well as in hydrodynamics.
In addition to the exact Riemann solver, the HLL and HLLC approximate Riemann solvers
have been implemented in the code.
2.1.2.1
HLL – Riemann Solver
The HLL (Harten, Lax and van Leer) approximate Riemann solver devised by Harten et al.
(1983) is used widely in shallow water models. It is a Godunov-type scheme based on the
two-wave assumption. This approach assumes estimates for the wave speeds of the left
and right waves. The solution of the Riemann problem between the two waves is thereby
approximated by a constant state as indicated in Fig. 6. This two-wave assumption is only
strictly valid for the one dimensional case. When applied to two dimensional cases, the
intermediate waves are neglected in this approach.
The HLL solver is very efficient and robust for many inviscid applications such as SWE.
Fig. 6: Principle of the HLL Riemann Solver. The analytical solution (left) is replaced by an
approximate one with a constant state U* separated by waves with estimated wave speeds (right)
By applying the integral form of the conservation laws in appropriate control volumes the
HLL-flux is derived. The numerical flux over an edge is sampled between three different
cases separated by the left and right waves. The indices L and R stand for the left and right
states of the local Riemann problem.
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 FL

 hll S R FL − S L FR + S R S L (U R − U L )
Fi +1/2
= F=
SR − SL

 FR



if S L ≤ 0 ≤ S R 


if 0 ≥ S R
if
0 ≤ SL
(RII-2.2)
Furthermore, the left and right wave speed velocities are estimated using a two-rarefaction
solution. Additionally, possible occurrence of dry-bed conditions must be regarded.
uR − 2 ghR
SL = 
min(uL − ghL ; uR − ghR )
if hL =
0 

if hL ≥ 0 
(RII-2.3)
uL + 2 ghL
SR = 
max(uR + ghR ; uL + ghL )
2.1.2.2
0 
if hR =

if hR ≥ 0 
HLLC – Riemann Solver
A modification and improvement of the HLL approximate solver was proposed by Toro et al.
(1994). This so called HLLC approximate Riemann solver also accounts for the impact of
intermediate waves, such as shear waves and contact discontinuities, in two dimensional
problems. In addition to the estimates of left and right wave speeds, the HLLC solver also
requires an estimate for the speed of the middle wave. This middle wave than divides the
region between the left and right waves into two constant states.
The numerical flux over an edge is sampled regarding four different cases as
Fi +1/2
 FL
 F = F + S (U − U )

L
L
*L
L
=  *L
S
=
+
(
−
F
F
U
U
R
R
*R
R)
 *R
 FR
0 ≤ SL

if S L ≤ 0 ≤ S* 

if S* ≤ 0 ≤ S R 
if 0 ≥ S R

if
U * L , U * R are obtained, as proposed by Toro(1997), from the relations
1
1
 S L − uL   
 S R − uR   
U * L = hL 
  S*  ; U * R = hR 
  S*  ;
 S L − S*   v 
 S R − S*  v 
 L
 R
(RII-2.4)
The states
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The needed solutions for the water depth and velocity in the star region are derived from
solutions of a simplified Riemann problem based on the depth-positivity-condition. Here, aL
and aR are the according wave speeds gh of the left and right initial states.
h* =
1
1 (ur − uL )(hL + hR )
(hL + hR ) −
2
4
( aL + aR )
(RII-2.6)
(h − h )(a + aR )
1
u* =
(uL + uR ) − R L L
hL + hR
2
S L , S* and S R are the estimated wave speeds for the left, middle and right waves. The left
and right wave speeds are approximated with the use of a two-rarefaction solution, whereby
the dry-bed problem must be considered separately.
uR − 2 ghR
SL = 
min(uL − ghL ; u* − gh* )
if hL =
0 

if hL ≥ 0 
(RII-2.7)
uL + 2 ghL
SR = 
max(uR + ghR ; u* + gh* )
The middle wave speed
dry-bed problems as
S* =
S* is calculated as proposed by Toro(1997) in a suitable way for
S L hR (uR − S R ) − S R hL (uL − S L )
hR (uR − S R ) − hL (u L − S L )
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0 
if hR =

if hR ≥ 0 
(RII-2.8)
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Saint-Venant Equations
2.2.1
Spatial Discretisation
The time Discretisation is based on the explicit Euler schema, where the new values are
calculated considering only values from the precedent time step.
The spatial Discretisation of the Saint Venant equations is carried out by the finite volume
method, where the differential equations are integrated over the single elements.
Fig. 7: Definition sketch
It is assumed that values, which are known at the cross section location, are constant within
the element. Throughout this section it therefore can be stated that:
xiR
∫
f ( x)dx ≈ f ( xi )( xiR − xiL ) =fi ∆xi
(RII-2.9)
xL
where xiR and xiL are the positions of the edges at the east and the west side of element i,
as illustrated in Fig. 7.
2.2.2
2.2.2.1
Discrete Form of Equations
Continuity Equation
Equation 2 is integrated over the element:
xiR
 ∂A
∂Q

0
∫  ∂t + ∂x − q dx =
l
(RII-2.10)
xL
The different parts of the equation are discretized as follows:
∂Ai
∂Ai
Ait +1 − Ait
∫ ∂t dx ≈ ∂t ∆xi ≈ ∆t ∆xi
xiL
(RII-2.11)
∂Qi
dx = Q( xiR ) − Q( xiL ) = Φ c ,iR − Φ c ,iL
∂x
xiL
(RII-2.12)
xiR
xiR
∫
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( xiR − xi ) + qiL ( xi − xiL )
(RII-2.13)
xiL
Φ c ,iR and Φ c ,iL are the continuity fluxes calculated by the Riemann solver and qiR and qiL
the lateral sources in the corresponding river segments.
For the explicit time Discretisation the new value of
1
Ait +=
Ait −
2.2.2.2
A will be:
∆t
∆t
(Φ c ,iR − Φ c ,iL ) −
(qiR ( xi − xiR ) + qiL ( xiL − xi ))
∆xi
∆xi
(RII-2.14)
Momentum Equation
Assuming that the lateral in and outflows do not contribute to the momentum balance, thus
neglecting the last term of equation (RI-1.9), and integrating over the element, the
momentum equation becomes:
 ∂Q ∂  Q 2 

0
∫x  ∂t + ∂x  β Ared  + ∑ Sources  dx =


iL
xiR
(RII-2.15)
The different parts of the equation are discretized as follows:
∂Qi
∂Qi
Qit +1 − Qit
dx
≈
∆
x
≈
∆xi
i
∫x ∂t
∂
t
∆
t
iL
(RII-2.16)
∂Qi
Q2
∫ ∂x dx = β Ared
xiL
(RII-2.17)
xiR
xiR
xiR
Q2
−β
Ared
= Φ m ,iR − Φ m ,iL
xiL
Φ m ,ir and Φ m ,il are the momentum fluxes calculated by the Riemann solver.
For the explicit time Discretisation the new value of
1
Qit +=
Qit −
2.2.3
∆t
( Φ m,iR − Φ m,iL ) + ∑ Sources
∆xi
Q will be:
(RII-2.18)
Discretisation of Source Terms
2.2.2.1
Bed Slope Source Term
The bed slope source term:
W = gA
∂zS
∂x
(RII-2.19)
is discretized as follows:
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∫ gAred
xL
∂zS
∂z
dx ≈ gAredi S
∂x
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xi
−z
z

∆xi ≈ gAredi  S ,i +1 S ,i −1  ∆xi
 xi +1 − xi −1 
(RII-2.20)
With the subtraction of the bed slope source term, equation (RII-2.18) becomes:
−z
z

∆t
Φ m ,ir − Φ m ,il ) − ∆tgAredi  S ,i +1 S ,i −1 
(
∆xi
 xi +1 − xi −1 
1
Qit +=
Qit −
2.2.2.2
(RII-2.21)
Friction Source Term
The friction source term:
Fr = gAred S f
is simply calculated with the local values:
xiR
∫ gA
red
S fi dx ≈ gAredi S fi ∆xi
(RII-2.22)
xL
and
S fi =
Qit Qit
( Kit )
(RII-2.23)
2
This computation form however leads to problems if the element was dry in the precedent
time step, because in this case A , and thus K , become very small, and Sf i very large.
This leads to numerical instabilities. For this reason a semi-implicit approach has been
applied, which considers the discharge of the present time step:
S fi =
Qit +1 Qit
( Kit )
(RII-2.24)
2
Consequently instead of simply subtracting the source term from equation (RII-2.21), the
following operation is executed on the new Q :
Qit +1 =
Qit +1
1+
∆t Qit gAit
(RII-2.25)
(K )
t 2
i
In this way for a very small conveyance the discharge tends to 0.
2.2.2.3
Hydraulic Radius
The hydraulic radius and the conveyance of a cross section are calculated by different ways
for different types of cross sections, depending on the geometry which is specified by the
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user. The cross section can be simple or composed by a main channel and flood plains.
Additionally it can have a bed bottom.
Simple cross section without definition of bed (Fig. 8)
For the simple cross section the hydraulic radius is calculated by the sum of the hydraulic
radius of all slices. The conveyance is then calculated for the whole cross section.
Fig. 8: Simple cross section
Ri =
Ai
Pi
(RII-2.26)
R = ∑ Ri
(RII-2.27)
A = ∑ Ai
(RII-2.28)
K = c f gR A
(RII-2.29)
If there are slices with nearly horizontal ground however, this computation mode can lead to
jumps in the water level-conveyance graph. This is dangerous if we store the conveyance in
a table, from which values will be read by interpolation. If the conveyance is calculated by
adding the conveyances of the single slides the jump is avoided, but this method leads to an
underestimation of the conveyance for the heights higher than the step. The problem is
illustrated by the dimensionless example in Fig. 9.
calculated w ith the total hydraulic radius
sum of the conveyances of all slices
250
conveyance
200
150
100
50
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Fig. 9: Conveyance computation of a channel with a flat zone
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Such flat zones in practice occur especially for cross sections composed by a main channel
and flood plains. For this reason this type of cross section can be specified and will be
treated differently. Such a cross section is illustrated in Fig. 10.
Fig. 10: Cross section with flood plains and main channel
In this case the conveyance and the discharge are calculated separately for each part of the
cross section. The total discharge results by summation.
Cross section with definition of bed (Fig. 11)
Additionally to the distinction of flood plains and main channel, a bed bottom can be
specified for both simple and composed cross sections. In this case for the computation of
the hydraulic radius of the main channel are considered the distinct influence areas of
bottom, walls and, if existing, interfaces to the flood plains. In Fig. 11 is illustrated the case
without flood plains or a water level below them.
Fig. 11: Cross section with definition of a bed
The conveyance in this case is calculated by the following steps:
Ab = Rb Pb
(RII-2.30)
Atot = Awl + Awr + Ab
(RII-2.31)
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Rb =
k
3/ 2
stb
 Pwl
 3/ 2
 kStWl
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Atot
P
P 
+ 3/b 2 + 3/wr2 
kStb kStWr 
(RII-2.32)
K b = cf gRb Ab
(RII-2.33)
Kb
Ab
(RII-2.34)
U mb =
S
Qb = U mb Ab
(RII-2.35)
 A

=
Qw U mb  tot − Rb Pb 
 1.05

(RII-2.36)
Q
= Qb + Qw
(RII-2.37)
K =Q/ S
(RII-2.38)
K = Kb + K w =
 A

Qb Qw K b
K  A

+
=
Rb Pb + b  tot − Rb Pb  = K b + K b  tot − 1
Ab  1.05
S
S Ab

 1.05 Ab 
(RII-2.39)
Fig. 12 shows the case of a water level higher than the flood plains.
Fig. 12: Cross section with flood plains and definition of a bed bottom
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In this case two more partial areas are distinguished for the computation of the hydraulic
radius of the bed and:
Rb =
Atot
.
Pwl
Pb
Pwr
Pir 
3/ 2  Pil
kStb  3/ 2 + 3/ 2 + 3/ 2 + 3/ 2 + 3/ 2 
 kStil kStwl kStb kStwr kStir 
2.2.4
(RII-2.40)
Discretisation of Boundary conditions
While normally the edges are placed in the middle between two cross sections, which are
related to the elements, at the boundaries the left edge of the first upstream element and
the right edge of the last downstream element are situated at the same place as the cross
sections themselves. Thus there is no distance between the edge and the cross section.
2.2.2.1
Inlet Boundary
The boundary condition is applied to the left edge of the first element, where it serves to
determine the fluxes over the element side. If there is no water coming in, these fluxes are
simply set to 0.
If there is an inlet flux it must be given as a hydrograph. The given discharge is directly used
as the continuity flux, whereas for the computation of the momentum flux, Ared and β are
determined in the first cross section (which has the same location).
Φ c ,1R =
Qbound
(RII-2.41)
( Qbound )
Φ m ,1R =
β1
(RII-2.42)
Ared 1
For the computation of the bed source term in the first cell, the values in the cell are used
instead of the lacking upstream values:
z −z 
W1 = Ared 1  S ,2 S ,1 
 x2 − x1 
2.2.2.2
•
(RII-2.43)
Outlet Boundary
weir and gate:
n −1
The wetted area AN of the last cross section at the previous time step is used to
determine the water surface elevation z S , N . With z S , N , the wetted area of the weir or
gate and finally the out flowing discharge Q are calculated, which are used for the
computation of the flux over the outflow edge:
Φ c , NL =
Qweir
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(RII-2.44)
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Φ m , NL
•
(Q )
=weir
NUMERICS KERNEL
2
(RII-2.45)
Aweir
z(t) and z(Q):
If the given boundary condition is a time evolution of the water surface elevation or a
rating curve, the discharge Q is taken from the last cell and time step. In the case of a
rating curve it is used to determine the water surface elevation zS . The given elevation
is used to calculate A and β in the last cross section:
Φ c , NL =
QNt −1
Φ m , NL =
βbound
•
(RII-2.46)
(Q )
t −1 2
N
Abound
(RII-2.47)
In/out:
With the boundary condition in/out the flux over the outflow edge is just equal to the
inflow flux of the last cell:
Φ c , Ll =
Φ c , NR
(RII-2.48)
Φ m , NL =
Φ m , NR
(RII-2.49)
For the computation of the bed source term in the last cell, the values in the cell are
used instead of the lacking downstream values:
 z − zS , N −1 
WN = AredN  S , N

 xN − xN −1 
2.2.2.3
(RII-2.50)
Moving Boundaries
Boundaries are always considered to be on an edge. A moving boundary appears if one of
the cells, limited by the edge, is dry. In this case some changes have to be considered for
the computations:
•
Flux over an internal edge:
If in one of the cells the water depth at the lowest point of the cross section is lower
than the dry depth hmin , the energy level in the other cell is computed. If this is lower
than the water surface elevation in the dry cell the flow over the edge is 0. Otherwise it
is calculated normally.
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Bed source:
For the computation of the bed source term in the case of the upstream or downstream
cell being dry, the local values are used as in the following example with an upstream
dry cell:
−z 
z
Wi = Ared ,i  S ,i +1 S ,i 
 xi +1 − xi 
2.2.5
(RII-2.51)
Solution Procedure
a) General solution procedure of BASEchain in detail
Read problem setup file
Set default friction
Read topography file
Set bottoms
Define hydraulic boundaries
Define hydraulic initial conditions
Define hydraulic parameters
Define initial conditions
Define outputs
Define hydraulic sources
Make computational grid
Set limits of main channel
If: calculation mode = “table”
Yes
No
For: each cross section
-/-
Generate tables
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If: sediment transport is active ( bed load or suspended load )
Yes
No
Define general sediment parameters
-/-
Define grain classes
-/-
Define mixtures
-/-
Define soils
If: bedload is active
Yes
No
Set bedload specific
parameters
Set default transport
type factors
Define bed load
boundaries
Set bed load fluxes
Set transport
capacities
If: suspended load is active
Yes
No
Initialize suspended
load
Define suspended load
boundaries
Set advection-diffusion
fluxes
If sediment
exchange with soil is
active:
Yes
No
Crate
exchange
source
term
Print topography output
Print initial conditions to output files
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Generate the needed sources
If: boundary time series exist
Yes
Calculate new values for boundary time series
No
-/-
Calculate time step
Time Loop
If: there are monitoring points
Yes
Print monitoring point files
No
-/-
Print restart file
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b) time loop
while: total computational time <= final simulation time:
If: suspended transport or bedload transport is active:
Yes
No
Compute morphological time step
If: suspended transport is active:
Yes
No
Solve advection-diffusion equations
If: bedload transport is active:
Yes
No
Solve morphodynamic equations
If: suspended transport or sediment transport is active:
Yes
No
Calculate sediment exchange with sub layer
(source Sfk)
Solve hydrodynamic equations
If: bedload transport or suspended load with sediment exchange with the bottom is active:
Yes
No
Update tables
If: monitoring points exist:
Yes
No
Update monitoring points
Increment current computational time
Print time dependent outputs
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c) hydrodynamic equations
For: each edge
Calculate flux
For: each element
Balance fluxes
Add hydrodynamic sources
For: each element
Substitute old values with new values
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d) morphodynamic equations
For: each element
Calculate transport width of cross section
Reset transport capacities
For: each edge
Calculate sediment fluxes
For: each element
Calculate local sources
Balance global material conservation
Add local sources
Determine bed level change for cross section
For: each slice
Update layer positions
For: each grain class
Balance material sorting equation
Add local sources
Calculate sediment exchange with sub layer ( source Sf)
Add exchange with sub layer
For: each slice
Correct all 
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e) suspended load equations
For: each element
Calculate wetted width of cross section
For: each edge
Calculate advection and diffusion fluxes
For: each element
Calculate exchange sources (Sk))
Calculate local sources (Slk)
Calculate new concentrations
Solve Exner equation
Solve grain sorting equation
Update cross section geometry
Advance all values
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2.3
2.3.1
It has been seen that for the transformation of the differential and the integral equations into
discrete algebraic equations numerical methods are used. Based on the mentioned reasons,
the FV method has been used in the present work for the Discretisation of SWE.
Equation (RI-1.34) can be rewritten in the following integral form:
∫ U dΩ + ∫ ∇ ⋅ ( F, G ) dΩ + ∫ SdΩ = 0
t
Ω
Ω
(RII-2.52)
Ω
in which Ω equals the area of the calculation cell (Fig. 1)
Using the Gauss’ relation equation (RII-2.52) becomes:
∫ U dΩ + ∫ ( F, G ) ⋅ n dl + ∫ SdΩ = 0
t
s
Ω
∂Ω
Assuming
written:
Ut +
(RII-2.53)
Ω
U t and S are constant over the domain for first order accuracy, it can be
1
∫ ( F, G ) ⋅ ns dl + S =0
Ω ∂Ω
(RII-2.54)
Equation (RII-2.53) can be discretized by a two-phase scheme namely predictor – corrector
scheme as follows:
=
U in +1 U in -
∆t 3
n
( F, G )i , j × n j l j - ∆t Si
∑
Ω j =1
(RII-2.55)
where m = number of cell or element sides
( F, G )i , j = numerical flux through the side of cell
ns ,i = unit vector of cell side
The advantage of two-phase scheme is the second order accuracy in time marching.
In the FV method, the key problem is to estimate the normal flux through each side of the
domain, namely ( ( F, G ) ⋅ n s ). There are several algorithms to estimate this flux. The set of
SWE is hyperbolic and, therefore, it has an inherent directional property of propagation. For
instance, in 1d unsteady flow, information comes from both, upstream and downstream, in
sub critical cases, while information only comes from upstream in supercritical cases.
Algorithms to estimate the flux should appropriately handle this property. The Riemann
solver, which is based on characteristics theory, is such an algorithm. It is the solution of
Riemann Problem. The Riemann solver under the FV method formulation is especially
suitable for capturing discontinuities in sub critical or supercritical flow, e.g. a dam break
wave or flood propagation in a river.
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Flux Estimation
Considering the integral term of flux in equation (RII-2.54), it can be written:
) ⋅ n dl ∫ ( F ( U ) cos θ + G ( U ) sin θ ) dl
∫ ( F ( U ) , G ( U )=
(RII-2.56)
s
∂Ω
∂Ω
in which n s ≡ ( cos θ ,sin θ ) is the outward unit vector to the boundary of domain Ω i (see
Fig. 13). Based on the rotational invariance property for F ( U ) and G ( U ) on the boundary
of the domain, it can be written according to Toro (1997):
( F ( U ) , G ( U ) ) = T (θ ) F ( T (θ ) U )
−1
where θ is the angle between the vector
the x-axis. (see Fig. 13)
(RII-2.57)
n s and x-axis, measured counter clockwise from
Fig. 13: Geometry of a Computational Cell
Ω i in FV
The rotational transformation matrix
0
1

T (θ ) =  0 cos θ
 0 − sin θ

0 

sin θ 
cos θ 
(RII-2.58)
T −1 (θ ) = inverse of T (θ )
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Using (RII-2.57), (RII-2.54) can be rewritten as:
Ut +
1
0
T−1 (θ ) F ( T (θ ) U ) dl + S =
∫
Ω ∂Ω
(RII-2.59)
The quantity T (θ ) U is transformed of U , with velocity components in the normal and
tangential direction. For each cell in the computational domain, the quantity U , thus
T (θ ) U may have different values, which results in a discontinuity across the interface
between cells. Therefore, the two-dimensional problem in (RII-2.52) or (RII-2.54) can be
handled as a series of local Riemann problems in the normal direction to the cell interface
( x ) by the equation (RII-2.59).
Applying the foregoing, the flux computations over the edges are preformed in three
successive steps:
• First, the vector of conserved variables U is transformed into the local
coordinate system at the edge with the operation T (θ ) U .
• A one-dimensional, local Riemann problem is formulated and solved in the
normal direction of the edge. From this calculation results the new flux vector
over the edge F T (θ ) U  .
• The flux vector, formulated in the local coordinate system at the edge, is
−1
transformed back to Cartesian coordinates with T F T (θ ) U  . The Sum of
the fluxes of all edges of an element gives the total fluxes in x- and y directions.
2.3.2.2
Flux Correction
When the solution is advanced and the continuity and momentum equations are updated in
each cell, there may be occurring situations in which more water is removed from an
element than is actually stored in the element (overdraft). Such overdraft is mostly
experienced in situations with strongly varying topography and low water depths, e.g. near
wet-dry interfaces on irregular beds. To guarantee mass continuity and positive depths in all
elements, a correction of the depths and volumetric fluxes is applied in such situations
following the approach of Begnudelli and Sanders (2006).
The overdraft element i having a negative water depth receives water from its surrounding
element k if two conditions are fulfilled. The element k must previously have taken water
from the overdraft element and it must have enough storage available. The corrections of
the depths and volumetric fluxes of the neighbouring elements k are then calculated as
hkcorr= hk + ωk hi
Ai
Ak
(RII-2.60)
A
Flux
= Fluxk + ωk hi i
∆t
corr
k
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where hk is the water depth and Fluxk is the volumetric Flux of the neighbouring element.
hi is the (negative) water depth of the overdraft element and ωk is a weighting factor
which is obtained by weighting the volumetric fluxes of all corrected neighbouring elements
k.
ωk =
Fluxk
∑ Fluxk
(RII-2.61)
k
After the correction of the neighbouring elements, the water depth of the overdraft element
is set to zero.
2.3.2
In Eq. (RII-2.55) there are different possibilities for the evaluation of the source term S i . It
can be evaluated either with the variables of the old time step as S i ( U i ) , which is often
n +1/2
,
referred to as unsplitted scheme, or it can be evaluated with the advanced values U i
which already include changes due to the numerical fluxes computed during this time step
n +1/2
as S i ( U i
) . The use of the advanced values for the source term calculation is chosen
here because it gives better results (Toro 2001). Therefore Eq. (RII-2.55) is split in following
way
∆t 3
n
U in +1/2 =
U in − ∑ ( F,G )i , j ⋅ n j dl j
Ω j =1
(RII-2.62)
=
U in +1 U in +1/2 +∆t Si ( U in +1/2 )
But, as explained in the following, the friction source term
treatment.
2.3.2.1
Si , fr receives a special
Friction Source Term
When treating the friction source terms, a simple explicit Discretisation may cause
numerical instabilities if the water depth is very small, because the water depth is in the
denominator. Such problematic situations may occur in particular at drying-wetting
interfaces. To circumvent the numerical instabilities, the frictions terms are treated in a
semi-implicit way. Therefore the friction source term is calculated with the unknown value
U in +1 at the new time level as
n +1
U
=
U in +1/2 − ∆tSi , fr (U in +1 )
i
Considering the generalized
one obtains:
U in +1/2
U in +1 =
1 + ∆t
R II - 2.3-4
(u ) + ( v )
n 2
i
n 2
i
(RII-2.63)
c f friction coefficient and after some algebraic manipulations,
(RII-2.64)
c 2fi Ri
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Hydraulic Radius
The calculation of the friction source term requires a definition of the hydraulic radius Ri in
the element i. The hydraulic radius is defined here as water depth in the element ( Ri = hi ).
Wall friction
In cases where an element is situated at a boundary wall of the domain, the influence of the
additional wall friction on the flow can optionally be considered, as illustrated in Fig. 14. The
friction slope is extended to include additional wall friction effects, based on the approach of
Brufau and Garcia-Navarro (2000). In this implementation the friction values of the bed c f
and the wall c fw can be chosen differently.
Aw
hw
Ab
lw
boundary
wall
element
Fig. 14: Wetted perimeter of a boundary element with wall friction
The friction slope in x-direction is calculated as
ns
1
1
u u 2 + v2
( 2
)
Si , fr , x
=
+∑ 2
g
cf R
w =1 c fw Rw




bed friction
(RII-2.65)
wall friction
with Rw being the hydraulic radius of the wall friction at the boundary edge w . The first
term in eq. (RII-2.65) defines the friction losses due to bed friction and the second term
defines the additional friction losses due to the flow along the boundary wall. The friction
slope in y-direction is derived in an analogous way.
The hydraulic radius
=
Rw
Rw at the wall boundary edge is calculated as
Vwater Ab h Ab
,
= =
Aw
lw h lw
(RII-2.66)
where l w is the length of the element’s edge located at the wall boundary, Ab is bottom
area of the element and Aw is the wetted area of the wall.
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For the determination of the bottom shear stress for sediment transport computations, this
additional wall friction component is not taken into account.
2.3.2.2
Source Term for viscous and turbulent Stresses
The kinematic and turbulent stresses are treated as source term. For the derivatives, the
divergence theorem from Gauss is used similarly to the ordinary fluxes. This approach
allows a derivative to be calculated as a sum over averaged values on an edge. A potential
division by zero cannot occur. In the following, a cantered scheme for diffusive fluxes on an
unstructured grid based on the approach of Mohamadian (2004) is described.
For the Finite Volume Method, the diffusive source terms are integrated over an element:
d dΩ
∫∫ S=
Ω
∂F d ∂G d
∫∫Ω ∂x + ∂y dΩ
(RII-2.67)
By using the divergence theorem, the diffusive flux integrals are becoming boundary
integrals
∂F d ∂G d
dΩ
∫∫Ω ∂x + ∂y=
∫ F
d
n + G d nds
(RII-2.68)
∂Ω
The boundary integral is discretized by a summation over the element edges (index e)
∫ F
d
n + G d nds=
∂Ω
∑ (F
d
e
)
n e + G de n e dse
(RII-2.69)
e
The diffusive fluxes
(
)
Fed and G de on the element edges are calculated by a centred scheme
(
1
1
Fed = FRd + FLd , G de = G dR + G dL
2
2
)
(RII-2.70)
where R and L stand for a value right and left of the edge. The diffusive fluxes on the edges
read then as




0



1  ∂u   ∂u  
d
F=
  νh  +  νh  
e
2   ∂x R  ∂x L 
  ∂v   ∂v  
  νh  +  νh  
  ∂x R  ∂x L 
(RII-2.71)
and
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



0




1   ∂u   ∂u  
d
G=
 νh  +  νh 
e
2   ∂y  R  ∂y  L 


  ∂v   ∂v  
  νh ∂y  +  νh ∂y  
R 
L 

(RII-2.72)
where ν = ν m + ν t is the sum of the molecular (kinematic) and turbulent eddy viscosity.
For this approach, the velocity derivatives at the element canters are used as right and left
approximation near the edge. The values for the water depth h right and left of an edge are
reconstructed using the water surface elevation of the adjacent elements. The turbulent
eddy viscosity ν t can be either set to a constant value or calculated dynamically for each
element. Using the dynamic case, the values for ν t are taken from the right and left
element of an edge.
All that remains is to calculate the derivatives of the velocity components at the element
centres. For Finite Volume Methods, this is an easy task using again the divergence
theorem. The derivative of a general scalar variable ϕ on an arbitrary element is given by
1 ∂ϕ
 ∂ϕ 
 =

∫ dΩ ≈
 ∂x  Elem Ω Ω ∂x
 ∂ϕ 
1 ∂ϕ
 =

∫ dΩ ≈
 ∂y  Elem Ω Ω ∂y
∑ ϕ ∆y
e
e
(RII-2.73)
e
Ω
∑ ϕ ∆x
e
e
e
Ω
(RII-2.74)
where Ω is the area of the element and e stands for an edge. ϕe is a value on the edge.
As in finite volume methods, most variables are defined on an element, ϕe has to be
calculated as average of the neighbouring elements:
ϕ=
e
1
( ϕR + ϕL )
2
(RII-2.75)
The spatial differences ∆y e and ∆x e are the differences of the edge’s node-coordinates in
x and y direction. For this method to work, it is important to have the same direction of
integration along the elements edges – either clockwise or counter-clockwise.
As a result, a viscous term from equation (RII-2.76) is computed as
 ∂u 
 νh  ≈ ν L h L
 ∂x L
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∑ u ∆y
e
e
ΩL
e
(RII-2.77)
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NUMERICS KERNEL
0.5 ( u eL + u eR ) .
The depth-averaged turbulent viscosity ν t can either be set to a constant value or it is
calculated for every element using the formula
κ
ν t = u *h
6
Where
=
u*
(RII-2.78)
κ =0.4 is the von Karman constant and u * is the shear velocity which is defined as
(
cf u 2 + v 2
)
(RII-2.79)
Where cf is the bed friction coefficient derived from the same Manning- or Strickler-value
as defined for bed friction.
2.3.2.3
Bed Slope Source Term and Bed Slope Calculation
The irregularity of the topography plays an important role in real world applications and often
can have great impacts on the final accuracy of the results. A Discretisation scheme with
the elevations defined in the nodes of a cell leads to an accurate representation of the
topography. Special attention thereby is needed with regard to the C-property which is
discussed in chapter 2.3.3.
The numerical treatment of the bed slope source term here is formulated based on
Komaei’s method (Komai and Bechteler 2004). Regarding eq. (RII-2.54), it is required to
compute the integral of the bed slope source term over a element Ω i .




0
 0 




 ∂z B ( x, y ) 
B dΩ
 dΩ g ∫∫ h  − ∂x  dΩ
∫∫Ω S=
∫∫Ω  ghS Bx =
Ω




 ghS By 
 − ∂z B ( x, y ) 


∂y


(RII-2.80)
Assuming that the bed slope values are constant over a cell (RII-2.80) can be simplified to:
 0 
 0 





g  ∫∫ hdΩ  =
S
gVolwater  S Bx 
B dΩ
∫∫ S=

  Bx 
Ω
Ω
 S 
 S By 
 By 


(RII-2.81)
In order to evaluate the above integral, it is necessary to compute the bed slope of a cell and
the volume of the water over a cell. Since the numerical model allows the use of triangular
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cells as well as quadrilateral cells in hybrid meshes, these both cases need to be
distinguished.
Triangular cells
The bed slope of a triangular cell can be computed by using the finite element formulation
as given by Hinton and Owen (1979). It is assumed that zb varies linearly over the cell (
Fig. 15):
z B ( x, y ) =α1 + α 2 x + α 3 y
in which
α2 =
(RII-2.82)
∂z B
∂z
; α3 = B
∂x
∂y
Fig. 15: A Triangular Cell
The constants α 1 , α 2 and α 3 can be determined by inserting the nodal coordinates and
equating to the corresponding nodal values of z B . Solving for α 1 , α 2 and α 3 finally gives
z B ( x,=
y)
1
(a1 + b1 x + c1 y ) zb ,1 + (a2 + b2 x + c2 y ) zb ,2 + (a3 + b3 x + c3 y ) zb ,3 
2Ω 
(RII-2.83)
where
=
a1 x2 y3 − x3 y2 

b=
y2 − y3

1

c=
x3 − x2
1

(RII-2.84)
With the other coefficients given by cyclic permutation of the subscripts in the order 1,2,3.
The area Ω of the triangular element is given by
1 x1
y1
2Ω = 1 x2
1 x3
y2
y3
(RII-2.85)
One can compute the bed slopes in x- and y-direction in each cell as
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NUMERICS KERNEL
∂z ( x, y )
1

(b1 z B ,1 + b2 z B ,2 + b3 z B ,3 ) 
S Bx =
− B
=
−
2Ω
∂x


∂z B ( x, y )
1
(c1 z B ,1 + c2 z B ,2 + c3 z B ,3 ) 
S By =
−
=
−
2Ω
∂y

(RII-2.86)
The water volume over a cell can also be computed by using the parametric coordinates of
the Finite Element Method.
h +h +h 
=
Volwater  1 2 3  Ω
3


(RII-2.87)
where h1 , h2 and h3 are the water depths at the nodes 1, 2 and 3 respectively (Fig. 16).
Fig. 16: Water volume over a cell
In the case of partially wet cells (Fig. 17) the location of the wet-dry line (a and b) has to be
determined, where the water surface plane intersects the cell surface.
Fig. 17: Partially wet cell
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Using the coordinates of a and b, the water volume over the cell can be calculated as:
Volwater = Ω a32
h2 + h3
h
+ Ω 2ba 2
3
3
(RII-2.88)
In the computation of the fluxes through the edges (1-2) and (1-3), the modified lengths are
used. The modified length is computed under the assumption that the water elevation is
constant over a cell.
Quadrilateral cells
The numerical treatment of the bed source term calculation greatly increases in complexity
if one has to deal with quadrilateral elements with four nodes. In the common case these
nodes do not lie on a plane and therefore the slope of the element is not uniquely
determined and not trivially computed. Even if the nodes initially lie on a plane, this situation
can change if morphological simulations with mobile beds are performed.
To prevent complex geometric algorithms and to avoid the problematic bed slope
calculation, the quadrilateral element is divided up into four triangles. Then the calculations
outlined before for triangular cells can be applied separately on each triangle.
Fig. 18: Quadrilateral element and its division into four triangles ( ∆12C , ∆ 23C , ∆ 34C , ∆ 41C )
The four triangles are obtained by connecting each edge with the centroid C of the
element. The required bed elevation of this centroid C is thereby estimated by a weighted
distance averaging of the nodal elevations as proposed by Valiani et al. (2002):
4
zC =
∑z
k =1
4
k
∑
k =1
( xk − xC ) 2 + ( yk − yC ) 2
(RII-2.89)
( xk − xC ) + ( yk − yC )
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2
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where zC is the interpolated bed elevation of the centroid and k is the index of the four
nodes of the quadrilateral element. Following this procedure the water volumes and bed
slopes are calculated in the same way as outlined before for each of the triangles. Finally the
bed slope term of the quadrilateral element is obtained as sum over the values of the
corresponding four triangles.

 0 
4




g  ∫∫ hdΩ  =
S
g ∑ Volwater ,∆ k
B dΩ
∫∫ S=

  Bx 
k =1 
Ω
Ω
 S 
 By 

 0 


 S Bx ,k  .
 S By ,k  


(RII-2.90)
This calculation method circumvents the problematic bed slope determination for the
quadrilateral element during the calculation of the bed source term.
But for other purposes, like for morphological simulations with bed load transport, a defined
bed slope within the quadrilateral element may be needed. For such situations the bed
slope is determined by an area-weighting of the slopes of the k triangles. Replacing φ with
the x- or y-coordinate, one obtains the bed slopes as follows:
∂z B ( x, y )
1
≈
AQuad
∂φ
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 ∂z B ,k ( x, y )

A∆ ,k 
∂φ
k =1 

4
∑
(RII-2.91)
,φ =
x, y .
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NUMERICS KERNEL
Conservative property (C-Property)
The usually applied shock capturing schemes were originally designed for hyperbolic
systems without source terms. Such schemes do not guarantee the C-property in the
presence of source terms like the bed source term in the shallow water equations. At
stagnant conditions, when simulating still water above an uneven bed, unphysical fluxes and
oscillations may result from an unbalance between the flux gradients and the bed source
terms. In order to guarantee the C-property following condition, the reduced momentum
equation for stagnant conditions, must hold true:
1
=
)n dl g ∫∫ hS
∫ ( 2 gh
2
x
dΩ
Bx
dΩ |ζ =const
Ω
1 2
=
)n y dl g ∫∫ hSBy dΩ |ζ =const
∫ ( 2 gh
Ω
dΩ
(RII-2.92)
Therefore it is necessary to guarantee conservation by an appropriate treatment and
discretisation of the flux gradients and the bed source terms. Recent studies provide several
approaches for proper source term treatment, but are often either computationally complex
or cannot be easily transferred to unstructured meshes. Following the approach of Komaei,
2
the left hand terms 0.5ghmod are calculated with a modified depth at the edges. This
modified depth is calculated as integral over the linearly varying water depth at an edge.
L
hi2 + hi h j + h 2j
1 2
=
h
=
h ( x)dx
L ∫0
3
2
mod
(RII-2.93)
hi and h j are the water depths at the edge’s left and right nodes, as shown in Fig. 19.
Fig. 19: Edge with linearly varying water depth
It can be easily proved that using hmod in the determination of the flux gradients guarantees
the C-property on unstructured grids, if the bed source terms are discretized as product of
the water volume with the bed slope as shown before. This is exemplified here for a
completely wetted triangle in x-direction:
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g ∫∫ hSBx dΩ |ζ =const
= gVolwater S=
g
Bx
Ω
NUMERICS KERNEL
1
1
( h1 + h2 + h3 ) b1 zB ,1 + b2 zB ,2 + b3 zB ,3 
3
2
g
( h1 + h2 + h3 ) [b1 (ζ − h1 ) + b2 (ζ − h2 ) + b3 (ζ − h3 )]
6


g
b1 + b2 + b3 ) − b1h1 − b2 h2 − b3 h3 
=
( h1 + h2 + h3 ) ζ (
6


=0
g
=
− ( h1 + h2 + h3 ) [b1h1 + b2 h2 + b3h3 ]
6
=
l

1
1 n 
− ∫ ( gh 2 )nx dl = − g ∑  ∫ h( x)dx 
2
2 k =1  x =0
dΩ

= −g
=
−
1 1 2
1
1

(h1 + h1h2 + h22 )b3 + (h22 + h2 h3 + h32 )b1 + (h32 + h3 h1 + h12 )b2 

2 3
3
3

g
( h1 + h2 + h3 ) [b1h1 + b2 h2 + b3h3 ]
6
(RII-2.94)
Both terms lead to the same result and therefore balance exactly for stagnant flow
conditions.
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NUMERICS KERNEL
Discretisation of Boundary Conditions
The hydrodynamic model uses the essential boundary conditions, i.e. velocity and water
surface elevation are to be specified along the computational domain. The theoretical
background of the boundary condition has already been already discussed in book one
“Physical Models”. In this part the numerical treatment of the two most important boundary
types, namely inlet and outlet will be discussed separately.
2.3.2.1
•
Inlet Boundary
Hydrograph:
The hydrograph boundary condition is applied to a user defined inlet section which is
defined by a list of boundary edges. The velocity vectors are assumed to be
perpendicular to these boundary edges and the inlet section is assumed to be a straight
line with uniform water elevation (1D treatment).
If there is an incoming discharge it must be given as hydrograph. Both steady and
unsteady discharges can be specified along the inlet section, where water surface
elevation and the cross-sectional area are allowed to change with time. For an unsteady
discharge, the hydrograph is digitized in a data set of the form:
t1 Q1
…
t n Qn
…
t end Q end
n
n
where Q is the total discharge inflow at time t . The hydrograph specified in this way,
can be arbitrary in shape. . The total discharge is interpolated, based on the
corresponding time.
The water elevation at the inlet section is determined by the values of the old time step
at the adjacent elements. In case of dry conditions or supercritical flow at the inlet
section, the water elevation is calculated according the known discharge. For these
iterative h-Q calculations normal flow is assumed and an average bed slope,
perpendicular to the inlet section, must be given.
The calculated total inflow discharge is distributed over the inflow boundary edges and
the according momentum component is calculated. Thereby only the edges below the
water surface elevation receive a discharge. If some edges lie above the water elevation
they are treated as walls. To distribute the inflow discharge over the wetted inflow
boundary edges following approach is implemented.
The discharge Qi for each edge i is calculated as fraction of the total discharge
using a weighting factor Κ i as
Qi = Κ i Qin .
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(RII-2.95)
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Κ i can be calculated based on its local conveyance as
Ki =
c fi Ri g Ai
n
∑ (c
j =1
(RII-2.96)
R j g Aj )
fj
where Κ i is the discharge at edge i and the index j ranges from the first to the last
wetted edge of the inflow boundary edges. Ri , Ai and c fi are the hydraulic radius, the
wetted area and the friction factor of the corresponding elements respectively.
Alternatively, the weighting factor Κ i can be calculated based on the local wetted areas
at the edge i, which finally results in equal inflow velocities over all edges.
Ki =
Ai
∑A
j =1
2.3.2.2
•
(RII-2.97)
n
j
Outlet Boundary
Free surface elevation boundary:
As is mentioned in Tab.2 of the first part of this reference manual (RI: Mathematical
Models), just one outlet boundary condition is necessary to be defined. This could be
the flux, the water surface elevation or the water elevation-discharge curve at the
outflow section. There is often no boundary known at the outlet. In this situation, the
boundary should be modelled as a so-called free surface elevation boundary. A zero
gradient assumption at the outlet could be a good choice. This could be expressed as
follows:
∂
=0
∂n
(RII-2.98)
Although this type of boundary condition has a reflection problem, numerical
experiences have shown that this effect is limited just to five to ten grid nodes from the
boundary. Therefore it has been suggested to slightly expand the calculation domain in
the outlet region to use this type of outlet boundary (Nujić 1998).
•
Weir:
In the other possibility of outlet boundary condition namely defining a weir (Fig. 20), the
discharge at the outlet is computed based on the weir function (Chanson 1999)
=
q
3
2
C 2 g ( hup − hweir )
3
where
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hup − hweir
hweir
;=
hup
WSE − z B and h=
zweir − z B
weir
Fig. 20: Flow over a weir
The hydraulic and geometric parameters such as WSE , zB are the calculated variables on
the adjacent elements of the outlet boundary. Here it is assumed that the water surface
elevation is constant within the element. The weir elevation zweir is a time dependent
parameter. Based on the weir elevation (Fig. 21) some edges of the outlet are
considered as a weir and the others have free surface elevation boundary condition. In
order to avoid instabilities due to the water surface fluctuations, the following condition
are adopted in the program
Fig. 21: Outlet cross section with a weir
if ( hup ( t ) ≤ hweir ( t ) + k hdry ) ⇒ edge is a wall
if ( hup ( t ) > hweir ( t ) + k hdry ) ⇒ edge acts as weir or a free surface elevation boundary
k is a numerical factor and has been set to 3 in this version.
Fig. 21 also shows the effective computational width of a weir in a natural cross section.
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Gate:
The discharge over a gate boundary condition is computed according to
q = µh gate 2gh up
(RII-2.99)
Within this formula, hup is the difference between the water surface elevation upstream
of the gate and the gate level. hgate denotes the difference between gate level and soil
elevation at the gate’s location. The factor µ has to be chosen by the user. Dependent
on the actual gate geometry, the value of µ is usually around 0.6.
The gate formula is only active if the water surface elevation in the element belonging to
the boundary edge is higher than the gate elevation. Other possible states of the gate
are wall (in case of gate elevation lower than the soil elevation) and zero-gradient (in all
other cases).
•
h-Q-relation:
A water elevation–discharge relation can be applied as outflow boundary condition.
Several outflow boundary edges are therefore defined in a list. The outflow velocity
vectors are assumed to be perpendicular to the outflow boundary edges and the outflow
section is assumed to be a straight line with a uniform water elevation (1D treatment).
A relation must be given between the outflow water surface elevation and the total
outflow discharge. This h-Q relation is digitized in a data set of the form
h1 Q1
…
hn Q n
n
n
where Q is the total outflow discharge for a given water surface elevation h . The hQ-relation specified in this way, can be arbitrary in shape. The total outflow discharge is
interpolated, based on the corresponding water surface elevation.
The water surface elevation at the outflow section is determined from the values of the
elements adjacent to the boundary edges at the last time step. With this water elevation
a total outflow discharge is interpolated using the given h-Q relation. Alternatively, if no
h-Q-relation is given, the outflow discharge is calculated under the assumption of normal
flow at the outflow section. In this case an average bed slope perpendicular to the
outflow cross section must be given.
The calculated total outflow discharge is then distributed over all wetted outflow
boundary edges according to a weighting factor based on the local conveyance or the
wetted area (see 2.3.2.1). In contrast, cells which are not fully wetted are set to wall
boundary.
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•
NUMERICS KERNEL
Z-Hydrograph:
Another outflow boundary condition is to specify the time evolution of the water surface
at the outflow location. This boundary conditions aims to control the water elevation at
the outlet, e.g. at outflows to reservoirs with known water elevations.
A time evolution of the water surface elevation must be given in the form
t1 WSE1
…
t n WSE n
If the actual outlet water elevation lies below the desired reservoir elevation, than a wall
boundary is set at the outflow. On the other hand, if the actual water elevation lies
above the reservoir water elevation, a Riemann solver is applied. The Riemann problem
is defined between the outflow edges and a ghost cell outside of the domain with the
reservoir water level. (But be aware that the outlet water level is not guaranteed to be
identical to the specified water elevation.)
2.3.2.3
•
Inner Boundaries
Inner Weir:
The inner weir uses a slightly other approach than the boundary weir. If the weir crest is
higher than the water surface elevation in the neighbouring elements, the weir acts as a
wall.
If one or both of the neighbouring water surface elevations are above the weir crest, the
weir formula for discharge
=
q
3
2
µσ uv 2 g ( hup − hweir )
3
(RII-2.100)
is used. This is the classical Poleni formula for a sharp crested weir with an additional
factor σ uv which accounts for the reduction in discharge due to incomplete weir flow.
If only one side of the weir has a water surface elevation above the weir crest, then a
complete weir flow is given with σ uv = 1 .
As soon as the water surface elevation tops the weir crest level on both sides of the
weir, the incomplete case is active and the reduction factor σ uv is calculated according
to the Diagram Fig. 22.
As for the momentum, there is no momentum due to velocity accounted for in the
downstream direction. This behaviour acts as if all kinetic energy is dissipated over a
weir.
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1
0.8
0.6
s
uv
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
h /h
u
Fig. 22:
Reduction factor σ uv for an incomplete flow over the weir. h u is the downstream
water depth over the weir crest and h denotes the upstream flow depth over the weir crest.
•
Inner Gate:
Similar to the gate boundary condition, the inner gate has three modes. Either the gate
level is equal or less than the soil elevation. The gate is then closed and acts as a wall.
If the gate level is above the local soil elevation, the gate is considered as open. As long
as the water surface elevations near the gate are below the gate level, the exact
Riemann solver is used.
The gate is active as soon as one of the neighbouring water surface elevations is above
the gate level. For the calculation of the discharge,
q = µh gate 2gh up
(RII-2.101)
is used with hgate as the difference between gate level and soil elevation at the edge
and hup as the water depth upstream of the gate. The factor µ is usually between 0.5
and 0.6.
This factor is set constant for the inner gate. Therefore, it acts always as if it were a
complete flow through the gate – that means without backwater right downstream the
gate.
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As for the momentum, the velocity through the gate is taken into account in both,
downstream and upstream direction.
2.3.2.4
Moving Boundaries
Dry, partial wet and fully wet elements
Natural rivers and streams are highly irregular in both plan form and topography. Their
boundaries change with the time varying water level. The FV-based model with moving
boundary treatment is capable of handling these complex and dynamic flow problems
conveniently. The computational domain expands and contracts as the water elevation rises
and falls. Obviously, the governing equations are solved only for wet cells in the
computational domain. An important step of this method is to determine the water edge or
the instantaneous computational boundary. A criterion, hmin , is used to classify the following
two types of nodes:
1. A node is considered dry, if z s ≤ z B + hmin
2. A node is considered wet, if z s > z B + hmin
The determination of hmin is tricky, which can vary between 10-6 m and 0.1 m. Based on the
flow depth at the centre of the element we defined three different elemet categories (Fig.
23):
1) dry cells where the flow depth is below
hmin ,
2) partial wet cells where the flow depth above
cell are under water and
hmin but not all nodes of the
3) fully wet cells where all nodes included cell centre are under water.
Fig. 23: Schematic representation of a mesh with dry, partial wet and wet elements
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By comparing the water surface elevation of two adjacent elements (Fig. 24) and
determining which cell is dry it was decided whether there were or not a flux through the
edge.
Fig. 24: Wetting process of a element
Although the determination of hmin is tricky, as it mentioned above, it has been successfully
used in the past, has in the range of 0.05 ~ 0.1 m for natural rivers. It can be adjusted to
optimize the solution for particular flow and boundary conditions. It is suggested to consider
it close to min(0.1h, 0.1) .
Another problem related to partially wetted elements is the determination of the final
velocities at the end of the time step from the vector of the conserved variables U . To
calculate the velocities the conserved variables must be divided by the flow depth as
indicated below.
vx =
vy =
(v x h )
h
(v y h )
(RII-2.102)
h
For an element situated at the wetting and drying interface, the outflow of water amounts
may lead to very small water depths. Because the water depths are in the denominator,
instabilities can arise when updating the new velocities. To prevent these instabilities it is
checked if the water depth is smaller than the residual hmin . In such a case the velocities
are set to zero, since the water will not move in such a practical situation.
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Flux computations at dry-wet interfaces
When solving the shallow water equations along dry-wet interfaces, special attention is
needed and different situations must be distinguished. Some models solve the complete
equations only for completely wetted elements, where all nodes are under water. Here, in
contrast, the flux computations are also performed for partially wetted elements. This
procedure is computationally more costly and has larger programming efforts, but it leads to
more accurate results in some situations and it can reduce problems related to the wettingdrying process.
The complete flux computation is performed over a partially wetted edge if two conditions
are fulfilled:
• At least one of the both elements adjacent to the edge must be wetted, i.e. its
water depth must be above hmin .
• At least at one side of the edge, the element’s water surface elevation must be
above the average edge elevation.
The flux computations over partially wetted edges need to take into account that the flux
takes not place over the whole length of the edge (see Fig. 25). The actually over flown
effective length Leff is calculated as follows assuming a constant water level.
 H − z B ,1 
(RII-2.103)
Leff = 
L
 z − z  edge
 B ,2 B ,1 
Here H is the water surface elevation in the partially wetted element and z B ,1 , z B ,2 are the
nodal elevations of the edge.
Fig. 25: Definition of effective length at partially wetted edge
In Fig. 26 several possible configurations at dry-wet interfaces are illustrated which need to
be treated appropriately. Attention must be paid to correctly reproduce the physics and to
preserve the C-property for stagnant flow conditions.
The first case a) shows a wetted left element adjacent to a dry right element, where the left
water surface elevation H L is above the center elevation z B , R of the right element. Here,
no special treatment is needed and a Riemann problem can be formulated. But the Riemann
solver must be capable of treating dry bed conditions in an appropriate way.
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Case b) corresponds to an adverse slope at the right, dry element, where the left water
surface elevation H L is below the right bed elevation z B , R . This case has recently received
attention in the literature, as e.g. by Brufau et. al (2002). It requires a special treatment
because applying the Riemann solver in a situation with adverse slopes can produce
incorrect results. Some authors suggest to treat the dry-wet interface as a wall or to set the
velocities to zero. But these treatments are problematic because they do not always
preserve the C-property. Here a simple method is adopted whereas no Riemann problem is
formulated at the edge. But instead, only the pressure terms are evaluated which exactly
balance the bed source terms, thus guaranteeing a correct behaviour for stagnant flow
conditions.
In case c) the water elevation H L at the left element is below the average edge elevation.
In such a situation no Riemann problem is formulated as stated above. But again the
pressure term is evaluated here to preserve the C-property.
Fig. 26: Different cases for flux computations over an edge at wet-dry interfaces
Finally, in cases d), e) and f) either both elements are dry or the edge is completely dry. In
these cases neither mass fluxes nor momentum fluxes need to be evaluated.
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Solution Procedure
The logical flow of data through BASEplane from the entry of input data to the creating of
output files and the major functions of the program is illustrated in Fig. 27, Fig. 28 and Fig. 29
shows the data flow through the hydrodynamic and morphodynamic routines respectively.
Program main control data is read first, and then the mesh file, and then the sediment data
file if there is one. Initialization of the parameters and computational values is made next. If
the sediment movement computation is requested, the hydrodynamic routine will be started
in cycle steps defined by user, otherwise the hydrodynamic routine is run. After the
hydrodynamic routine, the morphodynamic routine is carried out next, if it is requested.
Results can be printed at the end of every time steps or only at the end of selected time
steps.
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a) general solution procedure of BASEplane
Reading main control file
Reading mesh files
If: there is a sediment data file
Yes
No
Reading the sediment data file
Continue
Initialization of parameters
While: Total computational time <= Final simulation time
If: there is the sediment data file “and” total time >= sediment routing start time
Yes
No
If: Total computational cycles >=
Hydrodynamic routine
hydrodynamic cycles
Yes
No
Hydrodynamic
Continue
routine
Writing the output files for the hydrodynamic computation
Advance the computational values (hydrodynamic)
If: there is the sediment data file “and” total time >= sediment routing start time
Yes
No
Morphodynamic routine
Continue
Writing the the output files for the
morphodynamic computation
Advance the computational values
(morphodynamic)
Computing the time step
Stop
Fig. 27: The logical flow of data through BASEplane
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b) hydrodynamic routine in detail
Setting the boundary conditions
Reconstruction
For: All edges
Computing the edge values
Computing Fluxes
For: All edges
Computation of fluxes
Time Evaluating
For: All elements
Balancing the fluxes
Adding source terms
Computing the conservative values
Fig. 28: Data flow through the hydrodynamic routine
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c) morphodynamic routine in detail
Setting the boundary conditions
Computing Fluxes
For: all hydraulic elements
Computation of transport capacities
For: all sediment edges
Computation of bedload fluxes
Time Evaluating
For: all elements
Balancing the bedload fluxes
Adding source terms
Computing the new bed elevation
Computing bedload control volume thickness
For: all grains
Adding source terms
Computing new compositions
Updating the nodal elevations and the element’s centre elevation
Fig. 29: Data flow through the morphodynamic routine
R II - 2.3-28
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3 Solution of Sediment Transport Equations
3.1
3.1.1
General Vertical Partition
Vertical Discretisation
A two phase system (water and solids) in which the sediment mixture can be represented
by an arbitrary number of different grain size classes is formulated. The continuous physical
domain has to be horizontally and vertically divided into control volumes to numerically solve
the governing equations for the unknown variables. Fig. 30 shows a single cell of the
numerical model with vertical partition into the three main control volumes: the upper layer
for momentum and suspended sediment transport, the active layer for bed load sediment
transport as well as bed material sorting and sub layers for sediment supply and deposition.
Fig. 30: Vertical Discretisation of a computational cell
The primary unknown variables of the upper layer are the water depth h and the specific
discharge q and r in directions of Cartesian coordinates x and y. In the active layer, qB , x
g
and qB , y are describing the specific bed load fluxes (index refers to the g -th grain size
g
class). A change of bed elevation z B can be gained by a combination of balance equations
for water and sediment and corresponding exchange terms (source terms) between the
vertical layers.
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NUMERICS KERNEL
Determination of Layer Thickness
The bed load control volume is the region where bed load transport occurs and it is
assumed to have a uniform grain distribution over the depth. Its extension is well-defined by
the bed surface at level z B and its thickness hm , which plays an important role for grain
sorting processes during morphological simulations with multiple grain classes.
Different methods are implemented for the determination of this thickness hm . It can be
determined either dynamically during the calculations (at the moment only for 2D
simulations) or it can be given a priori as a constant value for the whole simulation.
Borah’s approach
This approach is used by default, since it is able to consider the influences of the present
erosion or deposition rates as well as effects of bed armouring. With this approach the
thickness hm for degradation is different than that for aggradation. For global deposition, hm
corresponds to the thickness of the current deposition stratum. In case of degradation it is
proportional to the erosion rate with a limitation to account for the situation of an armoured
bed (Borah et al. 1982).
If the bed level increases ( ∆z B
> 0 ):
1
hmn +=
hmn + ∆z B
If the bed level decreases ( ∆z B
hm = 20∆z B +
(RII-3.1)
< 0 ):
dl
∑ β nm (1 − p)
where d l is the smallest non mobile grain size and
sediment fractions.
(RII-3.2)
∑β
nm
is the sum of the not mobile
Calculation from mean diameter
Following this approach the new thickness hm is determined proportional to the mean
diameter in the bed load control volume. The factor of proportionality can be chosen freely.
hmn +1 = factor * d mean
(RII-3.3)
But this simple approach does not take into account influences of present bottom shear
stresses or present erosion rates.
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NUMERICS KERNEL
One Dimensional Sediment Transport
3.2.1
3.2.2.1
Soil Segments
For each cross section slice a different composition of the soil can be specified. The highest
layer is the active layer. Beneath the active layer a variable number of sub layers can be
defined. Fig. 31 illustrates by example a possible distribution of soil types in a cross section.
Each colour represents for a different grain class mixture.
Fig. 31: Soil Discretisation in a cross section
The elevation changes generated by the bed load will modify the geometry of the cross
section as it is illustrated in Fig. 32. This can lead to the elimination of layers or to the
creation of new ones. The grain class mixture of depositions will be the same over the
whole wetted width.
VAW ETH Zürich
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NUMERICS KERNEL
Fig. 32: Effect of bed load on cross section geometry
3.2.2
3.2.2.1
Advection-Diffusion Equation
The one dimensional suspended sediment or pollutant transport in river channels is
described by equation (RI-2.3). This equation has to be solved for each grain class g in the
same manner. For this reason in this section g is omitted and the equation becomes:
A
∂C 
∂C
∂C ∂ 
0
+Q
−  AΓ
−S =
∂t
∂x ∂x 
∂x 
(RII-3.4)
C is the concentration of transported particles averaged over the cross-section.
For the moment the sources, which vary for different types of transport, will be set to 0.
Equation (RII-3.4) is integrated over the element (Fig. 7)
∫
xiR
xiL
 ∂C
∂C
∂C ∂ 
 A ∂t + Q ∂x − ∂x  AΓ ∂x



0
 =

(RII-3.5)
and the different parts are calculated as follows:
xiR
∫
xiL
iR
∂Ci
∂C
C n +1 − Cin
∂C
dx Ai ∫
dx ≈ Ai i ∆xi ≈ Ai i
∆xi
A=
∂t
∂t
∂t
∆t
xiL
x
iR
∂C
∂C
Q
x
=
Q
d
dx = Qi ( C ( xiR ) − C ( xiL ) ) = (Φ a ,iR − Φ a ,iL )
i ∫
∫x ∂x
∂
x
x
iL
iL
xiR
R II - 3.2-2
(RII-3.6)
x
(RII-3.7)
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xiR
∂ 
∂C 

NUMERICS KERNEL
∂C
∫ ∂x  AΓ ∂x dx =  AΓ ∂x

xiL
− AΓ
X iR
∂C
∂x

 = Φ d ,iR − Φ d ,iL
X iL 
(RII-3.8)
The concentration at the new time is:
1
Cin +=
Cin −
3.2.2.2
∆t
(Φ a ,iR − Φ a ,iL − Φ d ,iR + Φ d ,iL )
∆xi Ai
(RII-3.9)
Computation of Diffusive Flux
The diffusive flux is calculated by finite differences.
Φ d ,iR = AiR Γ
Ci +1 − Ci
xi +1 − xi
(RII-3.10)
For the interpolation on the edge of the wetted area, known only in the cross sections, the
geometric mean is used:
AiR = Ai +1 Ai
If
Γ is not given by the user it is calculated as follows:
Γ=
σ
(RII-3.11)
ν Lν R
σ
(RII-3.12)
is generally assumed to be 0.5 (Celik and Rodi 1984).
The eddy viscosity averaged over the depth can be calculated by (Faeh 1997):
ν = uhκ / 6
3.2.2.3
(RII-3.13)
Computation of Advective Flux
A general problem of the computation of the advective flux is that it leads to strong
numerical diffusion. Several schemes can be used, four of them are implemented.
The first possibility to compute he advective flux over the edge (element boundary) is to
interpolate the concentration values from the neighbouring elements, considering the flow
direction, and multiply it with the water discharge over the edge.
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a) QUICK-Scheme
For a positive flow from left to right the quick scheme determines the concentration
due to advection at the upstream edge of element i by:
( x − x )( x − x )
( x − x )( x − x )
( x − x )( x − x )
Ca ,iL = iL i −1 iL i − 2 Ci + iL i − 2 iL i Ci −1 + iR i −1 iR i Ci − 2
( xi − xi −1 )( xi − xi − 2 )
( xi −1 − xi − 2 )( xi −1 − xi )
( xi − 2 − xi −1 )( xi − 2 − xi )
(RII-3.14)
More in general the concentration is:
 C + g1 (Ci +1 − Ci ) + g 2 (Ci − Ci −1 ) → u > 0
Ca ,iL =  i
Ci +1 + g3 (Ci − Ci +1 ) + g 4 (Ci +1 − Ci + 2 ) → u < 0
Between the velocities
determinant.
(RII-3.15)
ui and ui +1 the one with the larger absolute value is
g1 =
( xiR − xi )( xiR − xi −1 )
( xi +1 − xi )( xi +1 − xi −1 )
(RII-3.16)
g2 =
( xiR − xi )( xi +1 − xiR )
( xi − xi −1 )( xi +1 − xi −1 )
(RII-3.17)
g3 =
( xiR − xi +1 )( xiR − xi + 2 )
( xi − xi +1 )( xi − xi + 2 )
(RII-3.18)
g4 =
( xiR − xi +1 )( xi − xiR )
( xi +1 − xi + 2 )( xi − xi + 2 )
(RII-3.19)
The QUICK scheme tends to get instable, especially for the pure advection-equation
with explicit solution (Chen and Falconer 1992). For this reason the more stable
QUICKEST scheme (Leonard 1979) is often used:
b) QUICKEST-Scheme
1
1
CiR ,QUICKEST
= CiR ,QUICK − CriR (Ci +1 − Ci ) + CriR (Ci +1 − 2Ci + Ci −1 )
2
8
(RII-3.20)
with
CriR =
R II - 3.2-4
ui +1 + ui ∆t
∆x
2
(RII-3.21)
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c) Holly-Preissmann
The QUICKEST-scheme still leads to an important diffusion. For this reason the
Holly–Preissmann scheme (Holly and Preissmann 1977), which gives better results,
is also implemented. This scheme is based on the properties of characteristics and
can not be applied directly for the present Discretisation.
To find C ( xiR ) of equation (RII-3.7) the properties of characteristics or finite
differences are used, placing the edges on a new grid so that C ( xiR ) becomes C j
and C ( xiL ) = C j −1 .
Considering only the advection part of the equation (RII-3.4) and dividing by the
cross section area A:
∂C u∂C
+
=
0.
∂t
∂x
(RII-3.22)
and
C j n +1 − C nj
∆t
= −u
C nj − C nj −1
(RII-3.23)
x j − x j −1
Thus the new concentration is

∆t  n
∆t
C nj +1 =
C nj −1
1 − u
 C j + u
x
x
x
x
−
−
j
j −1 
j
j −1

(RII-3.24)
n +1
for a courant number
=
CFL u=
(dt / dx) 1: C j = C j −1 . This means that the
solute travels from one side of the cell to the other during the time step.
n
The Holly-Preissmann scheme calculates the values in function of
upstream value.
Y (Cr ) = ACr 3 + BCr 2 + DCr + E
CFL and the
(RII-3.25)
Y (0) = C nj ; Y (1) = C nj−1
∂C j −1
∂C j 
; Y (1) =
.
Y (0) =
∂x
∂x
n
C
n +1
j
= a1C
n
j −1
n
+ a2C + a3
=
a1 Cr 2 (3 − 2Cr )
a2 = 1 − a1
VAW ETH Zürich
Version 10/27/2009
n
j
∂C nj −1
∂x
+ a4
∂C nj
∂x
(RII-3.26)
(RII-3.27)
(RII-3.28)
R II - 3.2-5
NUMERICS KERNEL
a3= Cr 2 (1 − Cr )∆x
(RII-3.29)
a4 =
−Cr (1 − Cr ) 2
(RII-3.30)
and
∂C nj +1
∂x
= b1C
n
j −1
+ b2C + b3
n
j
∂C nj −1
∂x
+ b4
∂C nj
∂x
=
b1 6Cr (Cr − 1) / ∆x
(RII-3.31)
(RII-3.32)
b2 = −b1
(RII-3.33)
=
b3 Cr (3Cr − 2)
(RII-3.34)
b4 =
(Cr − 1)(3Cr − 1) .
(RII-3.35)
However this form is only valid for a constant velocity u . If u is not constant the
velocities in the different cells and at different times have to be considered.
n
n
n +1
The velocity u* is determined by interpolation of u j −1 , u j , u j .
uj
=
un
=
1 n
u j + u nj +1 )
(
2
( u θ ) + (1 − θ ) u
n
j −1
(RII-3.36)
n
j +1
(RII-3.37)
with
∆t
x j −1 − x j
(RII-3.38)
 1
=
u
uj +un )
(
2
(RII-3.39)
θ = uin
 ∆t
=
Cr u=
∆x
u nj +1 + u nj
x − xj
2 j −1
− u nj −1 + u nj
∆t
(RII-3.40)
The Holly-Preissmann scheme gives good results for the pure advection-diffusion
equation. But if a sediment exchange with the bed takes place, because of the
shifted grid, it does not react to the influence of the source term.
For the last 3 schemes the advective flux is computed multiplying the concentration on the
edge with the discharge over the edge:
Φ a ,iR =
QiR CiR
R II - 3.2-6
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NUMERICS KERNEL
d) Modified Discontinuous Profile Method (MDPM)
The MDPM method presented by Bradot-Nico et al. (2007) is like a transposing of
the Holly-Preissmann scheme from a finite difference to a finite volume context and
thus much more adapted for the use within BASEMENT.
In this method the advective flux is calculated directly as a sediment discharge:
t n+1
1
Φ iR = Ai ∫ n u (t )C ( xiR , t )dt
t
∆t
(RII-3.41)
Fig. 33: Invariance of C along the characteristic line y
Using the invariance property along a characterstic line (Fig. 33) this equation can be
transformed to
xiR
1
Φ iR = Ai ∫ C ( x, t n )dx
xA
∆t
n
(RII-3.42)
n
The function Ci ( x ) is reconstructed from the mean concentration in the cell Ci
n
n
and the concentration values on the edges CiL and CiR (Fig. 34) by satisfying mass
conservation in the cell.
VAW ETH Zürich
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R II - 3.2-7
NUMERICS KERNEL
n
Fig. 34: Function C i ( x)
U iLn if x ≤ xi − 1 + α i ∆xi

2
C ( x) =  n
U iR if x > xi − 1 2 + α i ∆xi
n
i
(RII-3.43)
with
αi =
and
If
Cin − CiRn
CiLn − CiRn
(RII-3.44)
α i ∈ [ 0,1] .
CiLn = CiRn or ( CiLn − Cin )( Cin − CiRn ) < 0 the following values are set:
CiLn = Cin 

CiRn = Cin 
α i = ε 
ε
is an arbitrary value between 0 and 1.
If the velocity is the flux of suspended load per unity of depth and width can now be
determined as follows:
=
hi CiLn max( xi + 1 − x − (1 − α i )∆xi , 0) + CiRn min( xi + 1 − x, (1 − α i )∆xi )
2
R II - 3.2-8
(RII-3.45)
2
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NUMERICS KERNEL
If the velocity is negative respectively:
gi =
−CiLn min( x − xi + 1 , α i +1∆xi +1 ) − CiRn max( x − xi + 1 − α i +1∆xi +1 , 0)
2
(RII-3.46)
2
The abscissa of the foot of the characteristic is given by
xA =
xi + 1 − uin+ 1 (t )∆t .
2
2
Finally the advective flux is:
Φi+ 1
2
 1
n
 ∆t hi ( x A ) if u xi+ 12 > 0
=

 1 g ( x ) if u n < 0
i +1
A
xi + 1
2
 ∆t
(RII-3.47)
Furthermore the new concentrations on the edges have to be prepared for the
computations of the next time step:
Cin++11
2
CiLn if u xn 1 (t ) > 0 and Crx ≥ 1 − α i
i+
2

n
n
CiR if u x (t ) > 0 and Crx < 1 − α i
i+ 1

2
= n
n
Ci +1L if u xi+ 12 (t ) < 0 and Crx ≥ −α i
 n
n
Ci +1R if u xi+ 1 (t ) < 0 and Crx < −α i

2
(RII-3.48)
The required Courant number is
∆t

 u x (t ) ∆x

i
Crx = 
u (t ) ∆t
 x ∆xi +1
VAW ETH Zürich
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u x (t ) > 0
(RII-3.49)
u x (t ) < 0
R II - 3.2-9
3.2.2.4
NUMERICS KERNEL
Global Bed Material Conservation Equation
As the y-direction is not considered, in the one dimensional case the mass conservation
equation (Exner-equation) becomes:
(1 − p )
∂z B  ng ∂qB
+∑
+ sg − slBg
∂t  k =1 ∂x

0
=

(RII-3.50)
qB is the sediment flux per unit channel width. Integrating equation (RII-3.50) over the
channel width, hence multiplying everything by the channel width, the following equation is
obtained:
(1 − p )
∂ASed  ng ∂QB
+∑
+ S g − SlBg
∂t
 k =1 ∂x

0
=

(RII-3.51)
The discretisation is effected exactly in the same way as for the hydraulic mass
conservation.
Equation (RII-3.51) is integrated over the element (Fig. 7):
∫
xiR
xiL


∂ASed  ng ∂QB
(1
)
0
p
−
+∑
+ S g − SlBg  dx =

∂t
 k =1 ∂x


(RII-3.52)
The parts of the equation (RII-3.52) are discretized as follows:
xiR
∂ASed ,i
xiL
∂t
(1 − p ) ∫
n +1
n
ASed
,i − ASed ,i
dx =
(1 − p )
∆x
∆t
(RII-3.53)
ng
∫
xiR
xiL
∑Q
k =1
B ,i
∂x
ng
dx = ∑ Q ( x ) − ∑ QB ( xiL ) = Φ B ,iR − Φ B ,iL
B
iR
=
k 1=
k 1
ng
∫ ∑ (S
xiR
ng
xiL
=
k 1
ng
g
(RII-3.54)
ng
− SlBg )dx =∑ S g − ∑ SlBg
(RII-3.55)
=
k 1=
k 1
Φ B ,iL and Φ B ,iR are the bed load fluxes through the west and east side of the cell. Their
determination will be discussed later (3.2.2.1).
The change of the sediment area is thus calculated by:
n +1
n
∆A=
− A=
ASedi
Sed
Sedi
ng

∆t
∆t  ng
Φ
−
Φ
−
−
S
(
B ,iR
B ,iL )
 ∑ g ∑ SlBg 
∆xi
∆=
xi  k 1 =k 1

(RII-3.56)
As the result of the sediment balance is an area, the deposition or erosion height ∆zb has
yet to be determined. The erosion or deposition is distributed over the wetted part of the
cross section. If a bed bottom is defined the deposition height is equal and constant for all
wetted slices. Only in the exterior slices the bed level difference is 0 where the cross
section becomes dry. The repartition of the soil level change is illustrated in Fig. 35.
R II - 3.2-10
VAW ETH Zürich
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NUMERICS KERNEL
Fig. 35: Distribution of sediment area change over the cross section
The change of the bed level is calculated as follows:
∆ASed
∆z B =
br bl
+ + ∑ bi
2 2
(RII-3.57)
As the sediments transported by bed load can be deposed only on the bed bottom, whilst
the sediments transported by suspended load can be deposed on the whole wetted section,
two separate values of ∆ASed and ∆z B are calculated for the two processes by splitting
equation (RII-3.56) in two parts.
The change of bed level due to bed load is:
∆ASed=
,bl

∆t
∆t  ng
Φ B ,iR − Φ B ,iL ) −
(
 ∑ SlBg 
∆xi
∆xi  k =1

(RII-3.58)
The change of bed level due to suspended load accordingly:
ng

∆t  ng
∆ASed , susp =
−
−
S
 ∑ g ∑ Slg 
∆=
xi  k 1 =k 1

3.2.2.5
(RII-3.59)
Bed material sorting equation
For the 1D computation the bed material sorting equation computation is:
(1 − p )
∂qBg
∂
( β g hB ) +
+ sg − sf g − slBg =
0
∂t
∂x
(RII-3.60)
Considering the whole width of the cross section and introducing an active layer area
equation (RII-3.60) becomes:
(1 − p )
∂QBg
∂
( β g AB ) +
0
+ S g − Sf g − SlBg =
∂t
∂x
AB ,
(RII-3.61)
Equation (RII-3.61) is integrated over the length of the control volume:
∫
xiR
xiL
∂QBg

∂
+ S g − Sf g − SlBg
 (1 − p ) ( β g AB ) +
∂t
∂x

VAW ETH Zürich
Version 10/27/2009

0
 dx =

(RII-3.62)
R II - 3.2-11
NUMERICS KERNEL
The different parts are discretized as follows:
(1 − p ) ∫
xiR
xiL
xiR
∂QBg
xiL
∂x
∫
β gn +1 ABn +1 − β gn ABn
∂
( β g AB )dx =
(1 − p )
∆x
∂t
∆t
dx = QB ( xiR ) − QB ( xiL ) = Φ g ,iR − Φ g ,iL
Then the new
βg


(RII-3.64)
at time n + 1 can be calculated for every element
1
n n
β gn +=
 (1 − p ) β g AB −
i
(RII-3.63)
i
i
i by:
∆t
∆t 
n +1
(Φ iRk − Φ iLk ) − S g − Sf g − SlBg
 / (1 − p ) ABi )
∆xi
∆xi 
(
)
(RII-3.65)
As for the global bed material conservation equation the bed material sorting equation is
also solved twice: once for the bed load and once for the suspended load.

n n
1
β gn +,=
 (1 − p ) β g AB −
bl
i

i
i
∆t
∆t 
n +1
(Φ iRk − Φ iLk ) − − Sf g − SlBg
 / (1 − p ) ABi )
∆xi
∆xi 
(RII-3.66)

n +1
 / (1 − p ) ABi )

(RII-3.67)
(
)
and

1
=  (1 − p ) β gn An − ( − Sf g − Slg )
β gn +, susp
i
3.2.2.6

i
Bi
∆t
∆xi
Interpolation
To solve the equations (RII-3.54) and (RII-3.64) the total bed load fluxes over the edges
( Φ B ,iL , Φ B ,iR ) and the fluxes for the single grain classes ( QB ,iL , QB ,iR ) are needed, but the
g
g
data for the computation of bed load are available only in the cross sections.
For this reason the values on the edges are interpolated from the values calculated for the
cross sections, depending on a weight choice of the user ( θ ):
Φ
=
(θ )QB ,i −1 + (1 − θ )QB ,i
B ,iL
(RII-3.68)
If all values for the computation of QB by a bed load formula are taken from the cross
section, the results of sediment transport tend to generate jags, as some effects of
discretisation accumulate instead of being counterbalanced. For this reason it has been
preferred not to take all values from the same location. The local discharge Q is substituted
by a mean discharge for the edge, computed with the discharges in the upstream and
downstream elements of the edge. This means that the bed load in a cross section will be
calculated twice with different values of Q .
R II - 3.2-12
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3.2.3
NUMERICS KERNEL
3.2.2.1
External Sediment Sources and Sinks
The discretisation of external sediment sources and sinks is analogous to BASEplane.
Please see 3.3.2.1
3.2.2.2
Sediment Flux through Bottom of Bed Load Control Volume
The discretisation of the sediment flux through the bottom of the bed load control volume is
analogous to BASEplane. Please see 3.3.2.2.
3.2.2.3
Source Term for Sediment Exchange between Water and Bottom
The source term S g from equation (RI-2.7) is computed in different ways depending on the
scheme used for the solution of the advection equation.
a) With MDPM scheme
The source term S g is calculated for the concentration value on the left and the
concentration value on the right according to equation (RI-2.7):
S g , L = f (CiLn )
S g , R = f (CiRn )
The volume exchanged with the bottom during ∆t is given by the sum of the
exchange on the left and the exchange on the right:
=
EL1 α S g , L B∆x∆t
(RII-3.69)
=
EL 2 α 2 S g , L B∆x∆t / 2
(RII-3.70)
E=
EL1 + EL 2
L
(RII-3.71)
=
ER1 α 2 S g , R B∆x∆t / 2
(RII-3.72)
ER 2 α 3 S g , R B∆x∆t
=
(RII-3.73)
E=
ER1 + ER 2
R
Where
α
is defined like in equation (RII-3.44),
=
α2
α=
3 max(1 − α − Crx , 0)
(RII-3.74)
min(Crx ,1 − α ) and
The final mean source term is then:
Sg =
( EL + ER )
B∆t ∆x
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The exchange values on the right and left side are used to adjust the concentration
values on the edges. The new concentrations on the edges after deposition or
erosion in the left and right part of the cell are calculated by:
CiLn +1 =
n +1
iR
C
=
CiLn Aiα∆x + EL1
Aiα∆x
(RII-3.76)
C iRn Aiα 3 ∆x + ER 2
(RII-3.77)
Aiα 3 ∆x
b) With QUICK and QUICKEST scheme
S g is calculated for each cross section according to equation (RI-2.7).
c) With Holly-Preissmann scheme
The Holly-Preissmann scheme should not be used with material erosion and
deposition.
3.2.2.4
Splitting of Bed Load and Suspended Load Transport
The same size of particles can be transported by bed load as well as by suspended load. Van
Rijn (1984) found a parameter which describes the relation between the two transport
modes depending on the shear velocity u* and the sink velocity wk determined in equations
(RI-2.8 – RI-2.10).
ϕk
0
if
u 
=
ϕk 0.25 + 0.325ln  * 
 wk 
=
ϕk 1
if 0.4 ≤
if
 u*

 wk
 u*

 wk

 < 0.4


 ≤ 10

 u* 
  > 10
 wk 
(RII-3.78)
The computation of bed load flux and the computation of the exchange flux between
suspended load and bed (RI-2.7) have to be modified as follows:
QBg= (1 − ϕ g )QBg
(
=
S g wg ϕ g β g Ceg − Cd g
R II - 3.2-14
(RII-3.79)
)
(RII-3.80)
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Solution Procedure
The solution procedure for the one dimensional sediment transport is described in chapter
2.2.5 on page 2.2-9 of this manual.
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3.3
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Two Dimensional Sediment Transport
3.3.1
The finite volume method is applied to discretise the morphodynamic equations, slightly
different from that of the hydrodynamic section. In the hydrodynamic Discretisation a cellcentered approach is applied (see Fig. 2). Thereby, the bed elevations are defined in the cell
vertexes (nodes) of each cell. This arrangement, with bed elevations defined in the nodes,
enables a more accurate representation of the topography compared to an approach with
bed elevations defined in the cell centres. A further advantage of the chosen approach is
that the slope within each cell is clearly defined by its nodal bed elevations.
But applying the same Discretisation approach for the sediment transport leads to several
problematic aspects, which can be summarized briefly as follows.
•
The change of a cell’s sediment volume ∆V would have to be distributed on all
nodes of the cell, where the bed elevations are defined. But it is not obvious by
which criteria the sediment volume must be divided upon these nodes (see left part
of Fig. 36).
•
If a nodal elevation would be changed due to a sediment inflow into a cell, this
change in bed elevation would not only affect the sediment volume of this cell, but
also the sediment volumes of all neighbouring cells (see right part of Fig. 36). This
situation is problematic regarding the conservation properties of the numerical
scheme and it induces numerical fluxes into the neighbouring cells which cause
undesired numerical diffusion.
Fig. 36: Schematic illustration of problems related to the update of bed elevations
To circumvent these problematic aspects and to ensure a fully conservative numerical
scheme, a separate mesh is used for the spatial Discretisation for the sediment transport.
Because the hydraulic and sediment simulations are performed on different meshes, this
approach is called “dual mesh morphodynamics” (DMMD) from here on. Both meshes, with
its cells and edges, are illustrated in Fig. 37
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Fig. 37: Dual mesh approach with separate meshes for hydrodynamics (black) and sediment
transport (green). Sediment cells have the bed elevations defined in their cell centers.
The cells of the sediment mesh are constructed around the nodes of the hydraulic mesh by
connecting the midpoints of the edges and the centres of the hydraulic cells. This procedure
results in the generation of median dual cells. Following this dual mesh approach all
conservative variables of the sediment transport ( zb , β g , C g ) are defined within the centres
of the sediment cells, thus forming a standard finite volume approach regarding the
sediment transport. Changes in bed elevation of a node do not influence the neighbouring
sediment elements, as it is illustrated in Fig. 38. Therefore this Discretisation approach is
conservative and no diffusive fluxes into the neighbouring cells do occur.
Fig. 38: Cross sectional view of dual mesh approach. Changes in sediment volume ∆V do not affect
the neighboring cells’ sediment volumes.
The sediment mesh is generated automatically during the program start from the hydraulic
mesh without the need of any additional information.
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3.3.2.1
Global Bed material Conservation Equation – Exner Equation
Considering the Exner equation (RI-2.4) and applying the FVM, it can be written in the
following integral form
ng  ∂q
ng
∂qBg , y 
∂z B
Bg , x
dΩ+ ∫ ∑ 
d
+ =
Ω
( slg − sBg )dΩ

∑
∫
 ∂x
∂
t
∂
y
1
1
=
=
g
g
Ω
Ω
Ω


(1 − p ) ∫
(RII-3.81)
In which Ω is the same computational area as defined in hydrodynamic model (Fig. 13).
Using the Gauss’ theory and assuming ∂zb ∂t is constant over the element, one obtains
z Bn +1 − z Bn 1 ng
1 ng
+ ∑ ∫ qBg , x nx + qBg , y n y dl = ∑ ( Slg − S Bg )
∆t
Ω g 1=
Ωg1
=
∂Ω
(
(1 − p )
Where nx and
respectively.
3.3.2.2
)
(RII-3.82)
n y are components of the unit normal vector of the edge in x and y direction
Computation of Bed Load Fluxes
Direction of bed load flux
The direction of the bed load flux equals the direction of the velocity in near bed region and
is a 2D vector in a 2D simulation. It is assumed here that this direction equals the direction
of the depth-averaged velocity, although this assumption may become invalid in particular in
curved channels with significant secondary flow motions.
A correction of this flux direction is performed on sloped bed surfaces due to the
gravitational induced lateral bed load flux component. The lateral transport component is
perpendicular to the direction of flow velocity and therefore the resulting bed load flux
vector is determined as
 cos θ 

 
qBres =
qB + qBlateral =
qB 
 + qBlateral
 sin θ 
where
θ
 sin θ 


 − cos θ 
(RII-3.83)
is the angle between the velocity vector and the x-axis.
Computation of bed load flux


The bed load transport capacity qB and the lateral transport qBlateral are calculated using the
transport formulas outlined in manual R I. Different empirical transport formulas can be used
and also fractional transport for multiple grain classes can be considered. These formulas
require the flow variables and the soil compositions as data input.
As a consequence of the Discretisation of the sediment elements as median dual cells, each
sediment edge lies completely within a hydraulic element (see Fig. 39, where sediment
edges are indicated in green color). Therefore an obvious approach is to determine the bed
load fluxes over a sediment edge with the flow variables and the bottom shear stress
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defined in this hydraulic element. This eases the computations since no interpolations of
hydraulic variables onto the sediment edges are necessary. Furthermore, it can be made
use of the clearly defined bed slope within this hydraulic element, derived from its nodal
elevations.
Following this approach the transport capacity is calculated with the flow variables defined
in the centre of the hydraulic element. But since the transport capacity calculations also
depend on the bed materials and grain compositions, this computation is repeated for every
sediment element which partially overlaps the hydraulic element. Thus, one obtains multiple
transport rates within the hydraulic element, as illustrated in Fig. 39. (In case of single grain
simulations, the bed material is the same over the hydraulic element and therefore the
transport calculation must only be done once.)
Fig. 39: Determination of bed load flux over the sediment cell edges i,j
Finally, the flux over the sediment cell edge is determined from the calculated transport
capacities to its left and right sediment elements as



=
qB ,edge (θup ) q B ,res , L + (1 − θup ) q B ,res , R  nedge

where θ up is the upwind factor and n edge is the normal vector of the edge.
(RII-3.84)
Treatment of partially wetted elements
Per default no sediment transport is calculated within partially wetted hydraulic elements.
This behaviour seems favourable in most situations. For example, in some cases it prevents
upper parts of a river bank, which are not over flown, from automatically being eroded by
sediment erosion which takes place only at the toe of the bank. But this default behaviour
can be changed for special situations.
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NUMERICS KERNEL
Flux Correction
When bed load fluxes are summed up over an element k, there may occur situations in
which more sediment mass leaves the element than is actually available. Such situations are
observed for example when the bed level reaches a fixed bed elevation or bed armour
where no further erosion can take place.
To guarantee sediment mass conservation over the whole domain a correction of the bed
load fluxes which leave the element is applied in such situations. All the outgoing computed
bed load fluxes of such an overdraft element k are reduced proportionally by a factor ωk , g .
This factor is determined in a way that limits the overall outflow to the available sediment
mass Vsed , k , g .
 ng
 ∑ Fluxout , j , g − Vsed ,k , g
 j =1
ng

Fluxout , j , g
ωk , g = 
∑
j =1


0

corr
Fluxout ,k , g= (1 − ωk , g ) Fluxout ,k , g
3.3.2.4
ng
if
Vsed ,k , g < ∑ Fluxout , j , g
j =1
(RII-3.85)
ng
if
Vsed ,k , g >= ∑ Fluxout , j , g
j =1
Bed Material Sorting Equation
The global bed material conservation equation (RII-3.82) has to be solved first, because its
results are needed to solve the bed material sorting equation. The bed material sorting
equation (RI-2.3) is also discretized using FVM. The discretized form is
(1 − p )
(h β )
m
g
n +1
− ( hm β g )
∆t
n
+
(
1
∫ qBg , x nx + qBg , y ny
Ω ∂Ω
)
n
dl +
1
1
0 (RII-3.86)
S g − SlBng − sf g* =
Ω
Ω
The bed load control volume thickness hm has to be determined before the new values of
fractions are calculated through the Eq. (RII-3.86). The determination of hm is detailed in
chapter 3.1.2. The solution of the bed material sorting equation finally yields the grain
n +1
factions β g at the new time level.
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3.3.2.1
External Sediment Sources and Sinks
The source term SlB can be used, for example, to describe a local input of sediment
g
masses into a river, dredging of sediments or mass inflow due to slope failures. This source
term can be added directly to the equations. It is specified as mass inflow with a defined
grain mixture.
3.3.2.2
Sediment Flux through Bottom of Bed Load Control Volume
The source term sf g describes the change in volume of material of grain class g which
enters or leaves the bed load control volume. The term sf g is a time dependent source
term and a function of grain fractions and the bed load control volume thickness. Therefore
it has been handled in a special form in order to consider the time variation of the
parameters. A two step method is applied, where in the first step the fractions are updated
without the sf g source term. Then, in the second step, this source term is computed with
the advanced grain fractions values. The first step can be written as
β gn +1/2=
n
∆t
1 
hm β g ) −
q n +q n
(
n +1 

hm 
(1 − p ) Ω ∂Ω∫ Bg , x x Bg , y y
(
)
n
dl +
∆t
SlBng
(1 − p ) Ω

 (RII-3.87)

is used for the calculation of the
After this predictor step, the advanced value β g
n +1/2
n +1
fractions of the new time level β g by adding the sf g ( β
) source term as
n +1/ 2
=
β gn +1 β gn +1/2 +
n +1
m
h
∆t
sf g
(1 − p )
(RII-3.88)
And the sf g source term, which describes the material which enters or leaves the bedload
control volume, is finally discretized as given below.
sf g =−(1 − p)
β gn +1/2 ( z Fn +1 − zsub ) − β gn ( z Fn − zsub )
∆t
In the calculation of this expression for the
be considered separately.
(RII-3.89)
sf g term, cases of erosion and deposition must
Erosion
In case of erosion the bottom of the bed load control volume z F moves down and the
fractions of the underneath layer enter the control volume. The fractions of this underneath
n
β g β=
β gnsub+1/2 , and the source term therefore
layer are constant over time, i.e.=
g sub
calculates as:
sf g =−(1 − p) β g
R II - 3.3-6
( z Fn +1 − z Fn )
(erosion)
∆t
(RII-3.90)
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Implementing this source term it must be paid attention to situations where the eroded bed
volume comprises more than the first underneath sub layer. In such a situation the
n
exchanged sediment does not exactly have the composition β g sub of this first underneath
layer, but a mixture β gmix of the different compositions of all affected underneath sublayers
(see Fig. 40). In this implementation the number of the affected layers nsub can be
arbitrary.
Fig. 40: Definition of mixed composition β gmix for the eroded volume (red)
The mixed grain composition
layer thicknesses as
=
β gmix
βg
mix
is determined by weighting the grain fractions with the
nsub
1
∑  β g ( z j −1 − max( z j , zFn+1 ) 
z Fn − z Fn +1 j =1  j
(RII-3.91)
Deposition
In case of deposition, the bottom of the bed load control volume z F moves up and material
n +1/ 2
leaves the bed load control volume and enters the
with the updated composition β g
underneath layer. The source term therefore calculates as:
n+
sf g =−(1 − p ) β g
1
2
( z Fn +1 − z Fn )
(deposition)
∆t
(RII-3.92)
And a likewise term is used to update the grain compositions of the first underneath layer.
3.3.2.3
Sediment Exchange between Water and Bottom
The source term S g describes the exchange between the suspended load in the water
column and the sediment surface.
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Management of Soil Layers
In case of sediment erosion, the bottom of the bed load control volume can sink below the
bottom of the underneath soil layer. In such a situation the soil layer is completely eroded
and consequently the data structure is removed. If the eroded layer was the last soil layer,
than fixed bed conditions are set.
In case of sediment deposition, the uppermost soil layer grows in its thickness. And such an
increase in layer thickness can continue up to large values during prolonged aggradation
conditions. But very thick soil layers can be problematic, because the newly deposited
sediments are completely mixed with the sediment materials over the whole layer
thickness. Therefore a dynamic creation of new soil layers in multiple grain simulations
should be enabled, which allows the formation of new soil layers.
Finding suitable general criteria for the creation of new soil layers is a difficult task. Here,
two main conditions are identified and implemented:
•
The thickness hsl of the existing soil layer must exceed a given maximum layer
thickness hmax before a new layer is created.
Generation of a new soil layer: hsl > hmax
•
The grain composition of the depositing material must be different from the present
grain composition of the soil layer. It is assumed here in a simple approach that this
condition is fulfilled if the mean diameters of the grain compositions differ more
than a given percentage P .
d m ,deposition − dlayer
dlayer
R II - 3.3-8
> P /100
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Solution Procedure
The manner - uncoupled, semi coupled or fully coupled - to solve equations (RI-1.34), (RI-2.3)
and (RI-2.4) or appropriate derivatives with minor corrections has been often discussed over
the last decade. A good overview is given by Kassem and Chaudhry (1998) or Cao et al.
(2002). Uncoupled models are often blamed for their lacking of physical and numerical
considerations. Vice versa coupled models are said to be very inefficient in computational
effort and accordingly inapplicable for practical use. Kassem and Chaudhry (1998) showed
that the difference of results calculated by coupled and semi coupled models is negligible. In
addition Belleudy (2000) found that uncoupled solutions have nearly identical performance to
coupled solutions even near critical flow conditions. Furthermore, the increasing difficulties
and stability problems of coupled models that have to be expected when applying
complicated sediment transport formulas or simulating multiple grain size classes are to be
mentioned.
According to the preliminary state of this project the model with uncoupled solution of the
water and sediment conservation equations has been chosen. This requires the assumption
that changes in bed elevation and grain size distributions at the bed surface during one
computational time step have to be slow compared to changes of the fluid variables, which
dictates an upper limit on the computational time step. Consequently, the asynchronous
solution procedure as depicted in Fig. 41 and described in Fig. 27 is justified: flow and
sediment transport equations are solved uncoupled throughout the entire simulation period,
i.e. with calculation of the flow field at the beginning of a given time step based on current
bed topography and ensuing multiple sediment transport calculations based on the same
flow field until the end of the respective time step. Thus the given duration of a calculation
cycle ∆t seq (overall time step) consists of one hydraulic time step ∆th and a resulting
number of time steps ∆t s for sediment transport (see Fig. 42).
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Fig. 41: Uncoupled asynchronous solution procedure consisting of sequential steps
Fig. 42: Composition of the given overall calculation time step ∆t seq
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4 Time Discretisation and Stability Issues
4.1
Explicit Schemes
4.1.1
Euler First Order
The explicit time discretisation method implemented in BASEMENT is based on Euler first
order method. According the method the full discretized equations are
+1
U in=
U in + RES ( U )
1
z Bn +=
z Bn + RES ( U, d )
+1
β gn=
β gn + RES ( U, d g , hm )
Where the
4.2
RES (..) is the summation of fluxes and source terms.
Determination of Time Step Size
4.2.1
Hydrodynamic
For explicit schemes, the hydraulic time step is determined according the restriction based
on the Courant number. In case of the 2D model the Courant number is defined as follows:
=
CFL
(
)
u 2 + v 2 + c ∆t
L
≤1
(RII-4.1)
L is the length of an edge with corresponding velocities of the element u , v and
c = gh . In general, the CFL number has to be smaller than unity.
Where
4.2.2
Sediment Transport
For the sediment transport another condition c >> c3 holds true, which states that the wave
speed of water c is much larger than the expansion velocity c3 of a bottom discontinuity (eg.
DeVries (1966)). Since the value of c3 depends on multiple processes like bed load, lateral
and gravity induced transport, its definitive determination is not obvious. Therefore the
global time steps have been adopted based on the hydrodynamic condition.
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R II - 4.2-2
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Implicit Scheme
Introduction
In addition to the explicit scheme, BASEchain supports implicit calculations. To evaluate the
evolution of the geometry of a channel as an effect of sediment transport, often long term
computations are necessary. Additionally the calibration of a model with sediment transport
is particularly laborious and needs many simulations. With the explicit solution of the
hydraulics the needed simulation time becomes very large. This is because the explicit
method uses a small time step, limited by the CFL-Number. The implicit method is needed
to avoid this problem, allowing much larger time steps.
This chapter describes the implicit solution of the hydrodynamics in detail. The system of
equations to solve is formed by equations RI-1.3 and RI-1.9 applied to each cross section.
4.3.2
Time Discretisation
For the time discretisation of the differential equation
∂u ∂t =f (u ) the θ -method is used:
u n +1 − u n
= θ f (u n +1 ) + (1 − θ ) f (u n )
∆t
with
0 ≤ θ ≤1.
If θ = 0 the explicit Euler method results: the values at time n+1 are computed solely from
the old values at time n with limitation of the time step by the CFL-number. For θ = 0.5 the
scheme is second-order accuracy in time.
On the contrary, if θ = 1 the solution is fully implicit. The equations are solved with the
values at the new time n+1. As they are not known, initial values are assumed and the
solution is approached to the exact solution by iteration. BASEchain uses the NewtonRaphson method, which has the quality to converge rapidly. However this is only the case if
the initial values are sufficiently close to the exact solution. Here the values of the last time
step are used as initial values for the iterations. This means that the more distant the new
time from the old one and the bigger the change of the hydraulic state, the higher is the
possibility that the solution cannot be found.
For the present implementation, the recommended value of
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θ
is between 0.5 and 1.
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NUMERICS KERNEL
Continuity Equation
The integration of the continuity equation
∫
xie
xiw
 δ Asi δ Q

+
− q dx = Fi n + Fi Φ + Fi q = 0

δx
 δt

gives the following integral terms.
xie
Fi n = ∫
xiw
Fi Φ = ∫
xie
xiw
δ Asi
dx
δt
δQ
dx
δx
xie
Fi q = − ∫ qdx
xiw
θ -method:
= Fi + θ ( Fi Φ + Fi q ) + (1 − θ )( Fi Φ 0 + Fi q 0 )
Applying the
Fi
n +1
n
4.3.2.2
(RII-4.2)
Momentum Equation
The integration of the momentum equation:
 δ Q δ  Q2 
δz
∫x  δ t + δ x  β Ared  + gAred δ x + gAred S f
iw 
xie

n
sz
sf
Φ
dx = Gi + Gi + Gi + Gi = 0

leads to the following integral terms:
δQ
dx
δt
xx
δ  Q2 
GiΦ = ∫
β
dx
δ
x
A
red


xx
δz
sz
Gi = ∫ gAred dx
δx
xx
xie
∫
G =
n
i
iwie
iw
ie
Gisf =
iw
ie
∫ gA
red
S f dx
xiw
θ -method:
= G + θ (GiΦ + Gisz + Gisf ) + (1 − θ )(GiΦ 0 + Gisz 0 + Gisf 0 )
Applying the
n +1
i
G
4.3.3
n
i
(RII-4.3)
Solution
The equation system is
F ( x) = 0
with
F = ( F1 , G1 , F2 , G2 ,..., Fn −1 , Gn −1 , Fn , Gn )
and
x = ( A1 , Q1 , A2 , Q2 ,..., An −1 , Qn −1 , An , Qn )
R II - 4.3-2
VAW ETH Zürich
Version 10/27/2009
NUMERICS KERNEL
The system is solved by the Newton-Raphson method. Starting from an approximated
k
k +1
is determined by the linear equation
solution x , the corresponding improved solution x
system.
0
Ax k +1 − c =
(RII-4.4)
with
=
c Ax k − F
and
 ∂F1
 ∂A
 red ,1
 ∂G1
 ∂A
 red ,1
 ∂F2

 ∂Ared ,1
 ∂G2

 ∂Ared ,1








A=





















∂F1
∂F1
∂F1
∂Q1
∂Ared ,2
∂Q2
∂G1
∂G1
∂G1
∂Q1
∂Ared ,2
∂Q2
∂F2
∂F2
∂F2
∂F2
∂F2
∂Q1
∂Ared ,2
∂Q2
∂Ared ,3
∂Q3
∂G2
∂G2
∂G2
∂G2
∂G2
∂Q1
∂Ared ,2
∂Q2
∂Ared ,3
∂Q3
∂F3
∂F3
∂F3
∂F3
∂F3
∂F3
∂Ared ,2
∂Q2
∂Ared ,3
∂Q3
∂Ared ,4
∂Q4
∂G3
∂G3
∂G3
∂G3
∂G3
∂G3
∂Ared ,2
∂Q2
∂Ared ,3
∂Q3
∂Ared ,4
∂Q4
VAW ETH Zürich
Version 10/27/2009
.
.
.
.
.
.
∂Fn − 2
∂Fn − 2
∂Fn − 2
∂Fn − 2
∂Fn − 2
∂Fn − 2
∂Ared , n − 3
∂Qn − 3
∂Ared , n − 2
∂Qn − 2
∂Ared , n −1
∂Qn −1
∂Gn − 2
∂Gn − 2
∂Gn − 2
∂Gn − 2
∂Gn − 2
∂Gn − 2
∂Ared , n − 3
∂Qn − 3
∂Ared , n − 2
∂Qn − 2
∂Ared , n −1
∂Qn −1
∂Fn −1
∂Fn −1
∂Fn −1
∂Fn −1
∂Fn −1
∂Ared , n − 2
∂Qn − 2
∂Ared , n −1
∂Qn −1
∂Ared , n
∂Gn −1
∂Gn −1
∂Gn −1
∂Gn −1
∂Gn −1
∂Ared , n − 2
∂Qn − 2
∂Ared , n −1
∂Qn −1
∂Ared , n
∂Fn
∂Fn
∂Fn
∂Ared , n −1
∂Qn −1
∂Ared , n
∂Gn
∂Gn
∂Gn
∂Ared , n −1
∂Qn −1
∂Ared , n
R II - 4.3-3






























∂Fn −1 
∂Qn 

∂Gn −1 
∂Qn 

∂Fn 

∂Qn

∂Gn 

∂Qn 
4.3.4
NUMERICS KERNEL
Integral Terms
4.3.2.1
Continuity Equation
The integral terms of the continuity equation are approximated as follows.
For a general cross section i:
Fi n +1 = Fi n + Fi Φ + Fi q
Asi − Asi0
∆xi
∆t
c
Fi Φ = Φ iec − Φ iw
=
Fi n
− qi− ∆xi− − qi+ ∆xi+
Fi q =
For the first cross section i = 1:
A1 − A10
∆x1
∆t
F1Φ =Φ1ce − Qin
=
F1n
F1q =
− q1+ ∆x1
For the last cross section i = N
An − An0
=
F
∆xn
∆t
Φ
c
F=
Qout − Φ iw
n
n
n
− qn− ∆xn
Fnq =
4.3.2.2
Momentum Equation
The integral terms of the continuity equation are approximated as follows.
Qi − Qi0
∆xi
∆t
m
GiΦ = Φ iem − Φ iw
z −z
Gisz = gAred ,i i +1 i −1
2
=
Gin
=
Gisf gAred ,i S fi ∆xi
S fi =
Qi Qi
K i2
=
G1n
Qin − Qin0
∆x1
∆t
R II - 4.3-4
VAW ETH Zürich
Version 10/27/2009
G1Φ =Φ1me −
NUMERICS KERNEL
β1Qin2
Ared ,in
z −z
G1sz = gAred ,1 2 1
2
=
G1sf gAred ,1S f 1∆x1
Sf1 =
Q1 Q1
K12
S fn =
Qn Qn
K n2
For the last cross section i = n
0
Qout − Qout
∆xn
∆t
2
β nQout
=
− Φ mnw
GnΦ
Ared ,out
z −z
Gnsz = gAn n n −1
2
=
Gnn
=
Gnsf gAn S fn ∆xn
4.3.5
4.3.2.1
General Description of the Derivatives for the Matrix A
Derivatives of the Continuity Equation
c
∂Fi
∂Φ iec
∂Φ iw
=
−
∂Ared ,i −1 ∂Ared ,i −1 ∂Ared ,i −1
c
∂Fi
∆xi dAi
∂Φ iec ∂Φ iw
=
+
−
∂Aredi ∆t dAredi ∂Aredi ∂Aredi
c
∂Fi
∂Φ iec
∂Φ iw
=
−
∂Ared ,i +1 ∂Ared ,i +1 ∂Ared ,i +1
c
∂Fi
∂Φ iec ∂Φ iw
=
−
∂Qi −1 ∂Qi −1 ∂Qi −1
c
∂Fi ∂Φ iec ∂Φ iw
=
−
∂Qi ∂Qi
∂Qi
c
∂Fi
∂Φ iec ∂Φ iw
=
−
∂Qi +1 ∂Qi +1 ∂Qi +1
∂Φ1ce
∂Qin
∂F1
∆x1 dA1
=
+
−
∂Ared ,1 ∆t dAred ,1 ∂Ared ,1 δ Ared ,1
VAW ETH Zürich
Version 10/27/2009
R II - 4.3-5
NUMERICS KERNEL
∂Φ1ce
∂Φ inc
∂F1
=
−
∂Ared ,2 ∂Ared ,2 ∂Ared ,2
∂Qin
∂F1 ∂Φ1ce
=
−
∂Q1=
∂Q1 in=
∂Q1 in
∂F1 ∂Φ1ce ∂Qin
=
−
∂Q2 ∂Q2 ∂Q2
For the last cross section i = n:
∂Fn
∂Qout
∂Φ cnw
=
−
∂Ared ,n −1 ∂Ared ,n −1 ∂Ared ,n −1
∂Fn
∆xn dAn
∂Qout
∂Φ cnw
=
+
−
∂Ared ,n
∆t dAred ,n ∂Ared ,n ∂Ared ,n
∂Fn
∂Qout ∂Φ cnw
=
−
∂Qn −1 ∂Qn −1 ∂Qn −1
∂Fn
∂Qout
∂Φ cnw
=
−
∂Q=
∂Qn out=
∂Qn out
n
4.3.2.2
Derivatives of the Momentum Equation
m
gA
dzi −1
∂Gi
∂Φ iem
∂Φ iw
=
−
+ red ,i
2 dAred ,i −1
∂Ared ,i −1 ∂Ared ,i −1 ∂Ared ,i −1
m
A dK i 

∂Gi
∂Φ iem
∂Φ iw
g
=
−
+ ( zi +1 − zi −1 ) + g ∆xi S fi 1 − 2 red ,i

K i dAi 
∂Ared ,i ∂Ared ,i ∂Ared ,i 2

m
A
dzi +1
∂Gi
∂Φ iem
∂Φ iw
=
−
+ g red ,i
2 dAred ,i +1
∂Ared ,i +1 ∂Ared ,i +1 ∂Ared ,i +1
m
∂Gi
∂Φ iem ∂Φ iw
=
−
∂Qi −1 ∂Qi −1 ∂Qi −1
m
∂Gi ∆xi ∂Φ iem ∂Φ iw
Sf
= +
−
+ 2 gAred ,i ∆xi i
∂Qi ∆t ∂Qi
∂Qi
Qi
m
∂Gi
∂Φ iem ∂Φ iw
=
−
∂Qi +1 ∂Qi +1 ∂Qi +1

∂Φ1me
∂Φ inm g
∂G1
A dK1 
=
−
+ ( z2 − z1 ) + g ∆x1S fi 1 − 2 1


∂Ared ,1 ∂Ared ,1 ∂Ared ,1 2
K1 dAred ,1 

R II - 4.3-6
VAW ETH Zürich
Version 10/27/2009
NUMERICS KERNEL
∂Φ1me
∂Φ inm
∂G1
A dz
=
−
+g 1 2
∂Ared ,2 ∂Ared ,2 ∂Ared ,2
2 dA2
∂
β1Qin2
Sf1
Ared ,in
∂G1 ∆x1 ∂Φ1me
= +
−
+ 2 gAred 1∆x1
∂Q1 ∆t ∂Qin
∂Qin
Qin
∂G1 ∂Φ1me
=
−
∂Q2 ∂Q2
∂
β1Qin 2
Ared ,in
∂Q2
For the last cross section i = n:
m
A
∂Gn
∂Φ out
∂Φ1mw
dzn −1
=
−
+ g red ,n
2 dAred ,n −1
∂Ared ,n −1 ∂Ared ,n −1 ∂Ared ,n −1
m

∂Gn
∂Φ out
∂Φ mnw g
A dK n 
=
−
+ ( zn − zn −1 ) + g ∆xn S fn 1 − 2 n

K n dAredn 
∂Aredn ∂Aredn ∂Aredn 2

m
∂Gn
∂Φ out
∂Φ mnw
=
−
∂Qn −1 ∂Qn −1 ∂Qn −1
m
∂Gn ∆xn ∂Φ out
∂Φ mnw
Sf
= +
−
+ 2 gAredn ∆xn n
∂Qn
∆t
∂Qn
∂Qn
Qn
4.3.6
Determination of the Derivatives with Upwind Flux Determination
With the upwind method the flux is defined as follows:
f ( xie ) = Γiei fi + Γiei +1 fi +1
(RII-4.5)
with
1 and Γiei+1 =
Γiei =
0 if Qi + Qi +1 ≥ 0
ie
ie
0 and Γi+1 =
1 if Qi + Qi +1 < 0
Γi =
4.3.2.1
Derivatives of the Continuity Equation
For a general cross section:
∂Fi
=0
∂Ared ,i −1
∂Fi
∆x dAi
= i
∂Aredi ∆t dAredi
∂Fi
=0
∂Ared ,i +1
VAW ETH Zürich
Version 10/27/2009
R II - 4.3-7
NUMERICS KERNEL
∂Fi
=−θΓii −−10.5
∂Qi −1
∂Fi
= θ (Γiw − Γie )
∂Qi
∂Fi
= θΓie+1
∂Qi +1
∂Fi
= θΓie+1
∂Qi +1
For cross section i =1:
∆x dAi
∂F1
= i
∂Ared 1 ∆t dAredi
∂F1
= θ (Γ1e − 1)
∂Q1
∂F1
= θΓ e2
∂Q2
For cross section i =n:
∂Fn ∆xn dAn
=
∂An
∆t dAred ,n
∂Fn
=−θΓ nw−1
∂Qn −1
∂Fn
= θ (1 − Γ nw )
∂Qn
4.3.2.2
 w  Q2   1
Aredi dzi −1
∂Gi
1 d βi −1  
θ  Γi −1  β
=
−
  − g

2 dAredi −1
∂Aredi −1
 Ared   Aredi −1 βi −1 dAredi −1  


 Q2   1

∂Gi
Aredi dK i  
1 d βi  g
z
z
g
x
S
(
)
1
2
= θ  ( Γiw − Γie )  β
−
+
−
+
∆
−





i +1
i −1
i fi

K i dAredi  
∂Aredi
 Ared  i  Aredi βi dAredi  2



 Q2   1
A
dzi +1 
∂Gi
1 d βi +1 
= θ  −Γie+1  β
−
+ g redi





2 dAredi +1 
∂Aredi +1
 Ared  i +1  Aredi +1 βi +1 dAredi 

∂Gi
1  Q2 
=−2θΓiw−1
β

Qi −1  Ared  i −1
∂Qi −1
R II - 4.3-8
VAW ETH Zürich
Version 10/27/2009
NUMERICS KERNEL

∂Gi ∆xi
1
=
+ 2θ  ( Γie − Γiw )

Qi
∂Qi ∆t

 Q2 
Sfi 

β
 + 2 gAredi ∆xi
Qi 
 Ared  i
∂Gi
1  Q2 
= 2θΓie+1
β

Qi +1  Ared  i +1
∂Qi +1
For cross section i =1:

dz1 
Q2   1
1 d β1  g
e 
1
)
−
Γ
−
β
(
)

 + ( z2 − z1 − Ared 1
 
i 
dAred 1 
∂G1
 Ared 1  Ared 1 β1 dAred 1  2

=θ 

∂Ared ,1

Ared 1 dK1 


+ g ∆x1S f 1 1 − 2



K
dA
red 1 
1




 Q2   1
A
∂G1
dz2 
1 d β2 
= θ  −Γ e2  β
−
+ g red 2




A



∂Ared 2
2
β
A
dA
dA
red
red
red
red
2
2
,2
2



2


2

∂G1 ∆x1
Sf 
1  Q 
=
+ 2θ  ( Γ1e − 1)  β
+ 2 gAred 1∆x1 1 


Q1  Ared 1
Q1 
∂Q1 ∆t

∂G1
1  Q2 
= 2θΓ e2
β

Q2  Ared  2
∂Q2
For cross section i =n:

 Q2   1
An dzn −1
∂Gn
1 d βi −1  
θ  Γ nw−1  β
=
−

  − g

2 dAred ,n −1
∂Ared ,n −1
 A  n −1  An −1 β n −1 dAn −1  



 Q2   1
dz
A dK n  
∂Gn
1 d βn  g
= θ  ( Γ nw − 1)  β
−
+ ( zn − zn −1 + An n ) + g ∆xn S fn 1 − 2 n





dAn
K n dAn  
∂Ared ,n
 A  n  An β n dAn  2


∂Gn
1  Q2 
β
=−2θΓ nw−1
Qn −1  A  n −1
∂Qn −1

∂Gn ∆xn
1
=
+ 2θ  (1 − Γiw )

Qn
∂Qn
∆t

4.3.7
 Q2 
Sf n 
gA
x
β
+
∆

n
n
 A
Qn 

n
Determination of the Derivatives with Roe Flux Determination
Fluxes and derivatives of the fluxes of the continuity equation:
c
)
Q(ie
Φ=
=
ie
VAW ETH Zürich
Version 10/27/2009
1
(Qi + Qi +1 ) − RcA ( Ai +1 − Ai ) − RcQ (Qi +1 − Qi )
2
(RII-4.6)
R II - 4.3-9
∂Φ iec
= 0
∂Ai −1
NUMERICS KERNEL
∂Φ iec
ie
= RcA
∂Ai
∂Φ iec
ie
= − RcA
∂Ai +1
∂Φ iec
∂Φ iec
ie
=
=
0
0.5 + RcQ
∂Qi −1
∂Qi
∂Φ iec
ie
=
0.5 − RcQ
∂Qi +1
c
Φ iw
= Qiw = 0.5(Qi −1 + Qi ) − RcA ( Ai − Ai −1 ) − RcQ (Qi − Qi −1 )
(RII-4.7)
c
c
c
∂Φ iw
∂Φ iw
∂Φ iw
iw
iw
=
RcA
=
− RcA
=
0
∂Ai −1
∂Ai
∂Ai +1
c
∂Φ iw
iw
=
0.5 + RcQ
∂Qi −1
c
∂Φ iw
iw
=
0.5 − RcQ
∂Qi
c
∂Φ iw
=
0
∂Qi +1
Fluxes and derivatives of the fluxes of the momentum equation:
 Qi2
Qi2+1 
Q2
=
Φ
+ βi +1
β = 0.5  βi
 − RmA ( Ai +1 − Ai ) − RmQ (Qi +1 − Qi )
A ie
Ai +1 
 Ai
∂Φ iem
∂Φ iem
Qi2
∂Φ iem
Qi2+1
ie
ie
=
0
=
−0.5βi 2 + RmA
=
−0.5β i +1 2 − RmA
∂Ai −1
∂Ai
Ai
∂Ai +1
Ai +1
m
ie
Q
∂Φ iem
∂Φ iem
ie
=
=
0
βi i + RmQ
∂Qi −1
∂Qi
Ai
(RII-4.8)
Q
∂Φ iem
ie
=
βi +1 i +1 − RmQ
∂Qi +1
Ai +1
 Qi2−1
Qi2 
Q2
=
Φ
+ βi
β = 0.5  βi
 − RmA ( Ai − Ai −1 ) − RmQ (Qi − Qi −1 )
A iw
Ai 
 Ai − 1
m
m
∂Φ iw
Qi2−1
∂Φ iw
Qi2
∂Φ iem
iw
iw
=
−0.5βi −1 2 + RmA
=
−0.5βi 2 − RmA
=
0
∂Ai −1
∂Ai
∂Ai +1
Ai −1
Ai
m
iw
m
∂Φ iw
Q
iw
=
βi −1 i −1 + RmQ
∂Qi −1
Ai −1
4.3.2.1
m
∂Φ iw
Q
iw
=
βi i − RmQ
∂Qi
Ai
(RII-4.9)
∂Φ iem
=
0
∂Qi +1
Derivatives of the Continuity Equation:
∂Fi
∂Ared ,i −1
∂Fi
=
∂Aredi
∂Fi
∂Ared ,i +1
iw
= − RcA
∆xi dAi
ie
iw
+ RcA
+ RcA
∆t dAredi
ie
= − RcA
R II - 4.3-10
VAW ETH Zürich
Version 10/27/2009
NUMERICS KERNEL
∂Fi
1
iw
=− − RcQ
∂Qi −1
2
∂Fi
ie
iw
= RcQ
− RcQ
∂Qi
∂Fi
1
ie
=
− RcQ
∂Qi +1 2
For the first cross section i=1:
∂Qin
∂F1
∆x1 dA1
ie
=
+ RcA
−
∂Ared 1 ∆t dAred 1
∂Ared 1
∂F1
ie
= − RcA
∂Ared ,2
∂F1
ie
= RcQ
− 0.5
∂Q1=in
∂F1 1
ie
=
− RcQ
∂Q2 2
For the last cross section i=n:
∂Fn
iw
= − RcA
∂An −1
∂Fn ∆xn dAn
∂Q
iw
=
+ RcA
+ out
∂An
∆t dAredn
∂Aredn
∂Fn
1
iw
=− − RcA
2
∂Qn −1
∂Fn
iw
= 0.5 + RcQ
∂Qn
4.3.2.2
A
∂Gi
dzi −1
1
iw
βi −1U i2−1 − RmA
=
+ g red ,i
2 dAred ,i −1
∂Ared ,i −1 2

A
dK i 
∂Gi
g
1
1
ie
iw
=
− βiU i2 + RmA
+ βiU i2 + RmA
+ ( zi +1 − zi −1 ) + g ∆xi S fi 1 − 2 red ,i


K i dAred ,i 
2
2
2
∂Ared ,i

A
∂Gi
dzi +1
1
ie
=
− βi +1U i2+1 − RmA
+ g red ,i
2
2 dAred ,i +1
∂Ared ,i +1
∂Gi
iw
βi −1U i −1 − RmQ
=
∂Qi −1
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NUMERICS KERNEL
∂Gi ∆xi
Sf
ie
ie
=
+ RmQ
+ RmQ
+ 2 gAred ,i ∆xi i
∂Qi ∆t
Qi
∂Gi
ie
βi +1U i +1 − RmQ
=
∂Qi +1
For the first cross section i=1:

A
∂Φ inm g
dK1 
∂G1
1
1e
=− β1U12 + RmA
−
+ ( z1+1 − z1 ) + g ∆x1S fi 1 − 2 red ,1


2
K1 dAred ,1 
∂Ared ,1
∂Ared ,1 2

A
∂G1
dz
1
1e
=
− β 2U 22 − RmA
+ g red ,1 2
2
2 dA2
∂Ared ,2
∂Φ inm
Sf
∆x1
∂G1
1e
=
=+
β1U1 + RmQ
−
+ 2 gA1∆x1 1
Q1
∂Q1
∆t
∂Q1=in
∂G1
1e
β 2U 2 − RmQ
=
∂Q2
For the last cross section i=n
A
∂Gn
dzn −1
1
nw
β n −1U n2−1 − RmA
=
+ g red ,n
2 dAred ,n −1
∂Ared ,n −1 2

A
dK n 
∂Gn
g
1
1
ne
nw
=
− β nU n2 + RmA
+ β nU n2 + RmA
+ ( zn − zn −1 ) + g ∆xn S fn 1 − 2 red ,n


K n dAred ,n 
2
2
2
∂Ared ,n

∂Gn
nw
β n −1U n −1 − RmQ
=
∂Qn −1
∂Gn ∆xn
Sf
∆x
Sf
ne
nw
ne
ne
=
+ β nU n + RmQ
− β nU n + RmQ
+ 2 gAn ∆xn n = n + RmQ
+ RmQ
+ 2 gAn ∆xn n
∂Qn
∆t
Qn
∆t
Qn
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NUMERICS KERNEL
References
Begnudelli, L., and Sanders, B.F (2006). "Unstructured Grid Finite-Volume Algorithm for
Shallow-Water Flow and Scalar Transport with Wetting and Drying" Journal of
Hydraulic Engineering - ASCE, 132(4), 371-384.
Belleudy, P. (2000). "Numerical simulation of sediment mixture deposition part 1: analysis of
a flume experiment." Journal of Hydraulic Research, 38(6), 417-425.
Borah, D. K., Alonso, C. V., and Prasad, S. N. (1982). "Routing Graded Sediments in Streams
- Formulations." Journal of the Hydraulics Division-Asce, 108(12), 1486-1503.
Bradot-Nico F.,Brissaud F., Guinot V. (2007). A finite volume upwind scheme for the solution
of the linear advection-diffusion equation with sharp gradients in multiple
dimensions. Advances in Water Resources 30 (2007), 2002-2025.
Brufau P., García-Navarro P. (2000). "Two-dimensional dam break flow simulation."
International Journal for Numerical Methods in Fluids, 33 (2000), 35-57.
Brufau P, Vázquez-Cendón, M.E., and P. García-Navarro (2002). "A numerical model for the
flooding and drying of irregular domains". International Journal for Numerical
Methods in Fluids, 39 (2002), 247-275.
Cao, Z., Day, R., and Egashira, S. (2002). "Coupled and Decoupled Numerical Modeling of
Flow and Morphological Evolution in Alluvial Rivers." Journal of Hydraulic
Engineering, 128(3), 306-321.
Celik I., and Rodi, W. (1984). "A Deposition-Entrainement Model for Suspended Sediment
Transport", Sonderforschungsbereich 210, Universität Karlsruhe.
Chanson, H. (1999). The Hydraulics of Open Channel Flow. Butterworth-Heinemann.
Chen, Y. P., and Falconer, R. A. (1992). "Advection Diffusion Modeling Using the Modified
Quick Scheme." International Journal for Numerical Methods in Fluids, 15(10), 11711196.
Delis, A. I., Skeels, C. P., and Ryrie, S. C. (2000). "Evaluation of some approximate Riemann
solvers for transient open channel flows." Journal of Hydraulic Research, 38(3), 217231.
DeVries, M. (1966). "Application of Luminophores in Sandtransport – Studies." 39, Delft
Hydraulics Laboratory, Delft, Netherlands.
Faeh, R. (1997). "Numerische Simulation der Strömung in offenen Gerinnen mit beweglicher
Sohle" Mitteilung Nr. 153 der Versuchsanstalt für Wasserbau, Hydrologie und
Glaziologie an der ETH, Zürich
Glaister, P. (1988). "Approximate Riemann Solutions of the Shallow-Water Equations."
Journal of Hydraulic Research, 26(3), 293-306.
Godunov, S. K. (1959). "A difference method for numerical calculation of discontinuous
solutions of the equations of hydrodynamics." Mat. Sb. (N.S.), 47 (89), 271--306.
VAW ETH Zürich
Version 10/27/2009
R II - References
NUMERICS KERNEL
Harten, A., Lax, P. D., and Vanleer, B. (1983). "On Upstream Differencing and Godunov-Type
Schemes for Hyperbolic Conservation-Laws." Siam Review, 25(1), 35-61.
Hinton, E., and Owen, D. R. J. (1979). An Introduction to Finite Element Computations,
Pineridge Press, Swansea.
Holly, F. M., and Preissmann, A. (1977). "Accurate Calculation of Transport in 2 Dimensions."
Journal of the Hydraulics Division-Asce, 103(11), 1259-1277.
Kassem, A. A., and Chaudhry, M. H. (1998). "Comparison of coupled and semicoupled
numerical models for alluvial channels." Journal of Hydraulic Engineering-Asce,
124(8), 794-802.
Komai, S., and Bechteler, W. "An Improved, Robust Implicit Solution for the Two
Dimensional Shallow Water Equations on Unstructured Grids." River Flow 2004,
Napoli, Italy.
Leonard, B. P. (1979). "Stable and Accurate Convective Modeling Procedure Based on
Quadratic Upstream Interpolation." Computer Methods in Applied Mechanics and
Engineering, 19(1), 59-98.
Mohamadian, A., Le Roux, D.Y., Tajrishi, M. and Mazaheri, K. (2004) "A mass conservative
scheme for simulating shallow flows over variable topographies using unstructured
grid" Advances in Water Resources 28 (2005) 523-539
Nujić, M. (1998). "Praktischer Einsatz eines hochgenauen Verfahrens für die Berechnung von
tiefengemittelten Strömungen," Universität der Bundeswehr München, München.
Osher, S., and Solomon, F. (1982). "Upwind Difference-Schemes for Hyperbolic Systems of
Conservation-Laws." Mathematics of Computation, 38(158), 339-374.
Roe, P. L. (1981). "Approximate Riemann Solvers, Parameter Vectors, and DifferenceSchemes." Journal of Computational Physics, 43(2), 357-372.
Sun, Z. L., and Donahue, J. (2000). "Statistically derived bedload formula for any fraction of
nonuniform sediment." Journal of Hydraulic Engineering-Asce, 126(2), 105-111.
Toro, E. F., M. Spruce, and W. Speares (1994). Restoration of the Contact Surface in the
HLL-Riemann Solver, Shock Waves, 4:25-34
Toro, E. F. (1997). Riemann Solvers and Numerical Methods for Fluid Dynamics, SpringerVerlag, Berlin.
Toro, E. F. (2001). Shock-Capturing Methods for Free-Surface Shallow Flows, John Wiley &
Sons, Ltd., U.K.
Valiani, A., Caleffi, V. and Zanni A. (2002). "Case Study: Malpasset Dam-Break Simulation
using a Two-Dimensional Finite Volume Method", Journal of Hydraulic Engineering ASCEE, 128(5),460-472.
Vanleer, B. "Flux Vector Splitting for the Euler Equations." 8th Int. Conf. On Numer. Methods
in Fluid Dynamics, 507-512.
R II - References
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TEST CASES
Table of Contents
1
Hydraulic
1.1
Common Test Cases .......................................................................................... 1.1-1
H_1 : Dam break in a closed channel ........................................................... 1.1-1
1.1.1
H_2 : One dimensional dam break on planar bed......................................... 1.1-3
1.1.2
H_3 : One dimensional dam break on sloped bed ....................................... 1.1-4
1.1.3
H_4 : Parallel execution ................................................................................ 1.1-5
1.1.4
BASEchain Specific Test Cases ........................................................................ 1.2-1
1.2
H_BC_1 : Fluid at rest in a closed channel with strongly varying geometry 1.2-1
1.2.1
H_BC_2: bed load simulation with implicit hydraulic solution ...................... 1.2-2
1.2.2
BASEplane Specific Test Cases ........................................................................ 1.3-1
1.3
H_BP_1 : Rest Water in a closed area with strongly varying bottom........... 1.3-1
1.3.1
H_BP_2 : Rest Water in a closed area with partially wet elements ............. 1.3-3
1.3.2
H_BP_3 : Dam break within strongly bended geometry .............................. 1.3-5
1.3.3
H_BP_4 : Malpasset dam break ................................................................... 1.3-7
1.3.4
H_BP_5: Circular dam break wave ............................................................... 1.3-8
1.3.5
2
Sediment Transport
2.1
ST_1 : Soni et al: Aggradation due to overloading ........................................ 2.1-1
2.1.1
ST_2 : Saiedi ................................................................................................. 2.1-3
2.1.2
ST_3 : Guenter .............................................................................................. 2.1-4
2.1.3
2.2
ST_BC_1: Advection of suspended load ...................................................... 2.2-1
2.2.1
ST_BC_2: Advection-Diffusion ..................................................................... 2.2-3
2.2.2
ST_BC_3: Conservativity of sediment exchange with soil ........................... 2.2-5
2.2.3
3
Results of Hydraulic Test Cases
3.1
H_1: Dam break in a closed channel ............................................................ 3.1-1
3.1.1
H_2: One dimensional dambreak on planar bed........................................... 3.1-2
3.1.2
H_3: One dimensional dam break on sloped bed ........................................ 3.1-4
3.1.3
H_4 : Parallel execution ................................................................................ 3.1-7
3.1.4
3.2
H_BC_1: Fluid at rest in a closed channel with strongly varying geometry . 3.2-1
3.2.1
H_BC_2: bed load simulation with implicit hydraulic solution ...................... 3.2-1
3.2.2
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TEST CASES
3.3
BASEplane Specific Test Cases ........................................................................ 3.3-1
3.3.1
H_BP_1: Rest water in a closed area with strongly varying bottom ............ 3.3-1
H_BP_2: Rest Water in a closed area with partially wet elements .............. 3.3-1
3.3.2
H_BP_3: Dam break within strongly bended geometry ............................... 3.3-1
3.3.3
H_BP_4: Malpasset dam break .................................................................... 3.3-4
3.3.4
H_BP_5: Circular dam break ......................................................................... 3.3-6
3.3.5
4
Results of Sediment Transport Test Cases
4.1
ST_1: Soni ..................................................................................................... 4.1-1
4.1.1
ST_2: Saiedi .................................................................................................. 4.1-4
4.1.2
ST_3: Guenter ............................................................................................... 4.1-6
4.1.3
BASEchain specific Test Cases ......................................................................... 4.2-1
4.2
ST_BC_1: Advection of suspended load ...................................................... 4.2-1
4.2.1
ST_BC_2: Advection-Diffusion ..................................................................... 4.2-2
4.2.2
ST_BC_3: Conservation of sediment exchange of soil ................................ 4.2-3
4.2.3
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TEST CASES
List of Figures
Fig. 1: H_1: Dam break in a closed channel ................................................................................. 1.1-2
Fig. 2: H_2: One dimensional dam break on planar bed .............................................................. 1.1-3
Fig. 3: H_3: One dimensional dam break on sloped bed .............................................................. 1.1-4
Fig. 4: H_BC_1: Top view: cross-sections .................................................................................... 1.2-1
Fig. 6: H_BP_1: Rest Water in a closed area with strong varying bottom level ........................... 1.3-2
Fig. 7: H_BP_2: Rest Water in a closed area with partially wet elements .................................... 1.3-3
Fig. 8: H_BP_2: Computational grid ............................................................................................ 1.3-4
Fig. 9: H_BP_3: Strongly bended channel (horizontal projection and cross-section) .................. 1.3-5
Fig. 10: H_BP_4: Computational area of the malpasset dam break ............................................... 1.3-7
Fig. 11: H_BP_5: Computational area (rectangular grid) of the circular dam break .................... 1.3-8
Fig. 12: ST_BC_1: Initial condition for case A ............................................................................... 2.2-1
Fig. 13: ST_BC_1: Initial condition for case B ............................................................................... 2.2-2
Fig. 14: ST_BC_2: Initial condition for case A ............................................................................... 2.2-3
Fig. 15: ST_BC_2: Initial condition for case B ............................................................................... 2.2-4
Fig. 16: ST_BC_3: Geometry of case A ........................................................................................... 2.2-5
Fig. 17: ST_BC_3: Geometry of case B ........................................................................................... 2.2-6
Fig. 18: H_1: Initial and final water surface elevation (BC & BP) ................................................ 3.1-1
Fig. 19: H_2: Water surface elevation after 20 s (BC) .................................................................... 3.1-2
Fig. 20: H_2: Distribution of discharge after 20 s (BC).................................................................. 3.1-2
Fig. 21: H_2: Flow depth 20 sec after dam break (BP) ................................................................... 3.1-3
Fig. 22: H_2: Discharge 20 sec after dam break (BP) .................................................................... 3.1-3
Fig. 23: H_3: Water surface elevation after 10 s (BC) .................................................................... 3.1-4
Fig. 24: H_3: Temporal behaviour of water depth at x = 70.1 m (BC) ........................................... 3.1-4
Fig. 25: H_3: Temporal behavior of water depth at x = 85.4 m (BC) ............................................. 3.1-5
Fig. 26: H_3: Cross sectional water level after 10 s simulation time (BP) ..................................... 3.1-5
Fig. 27: H_3: Chronological sequence of water depth at point x = 70.1 m (BP) ............................ 3.1-6
Fig. 28: H_3: Chronological sequence of water depth at point x = 85.4 m (BP) ............................ 3.1-6
Fig. 29: H_4 (BASEchain) – Parallel speedup for 1D test case ...................................................... 3.1-7
Fig. 30: H_4 (BASEplane) – Parallel speedup for 2D test case ...................................................... 3.1-8
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TEST CASES
Fig. 31: H_BP_3: Water level and velocity vectors 5 seconds after the dam break ........................ 3.3-2
Fig. 32: H_BP_3: Water surface elevation at G1 ............................................................................ 3.3-2
Fig. 35: H_BP_5: Comparison of the water level at point G5 for a coarse and a dense grid......... 3.3-4
Fig. 36: H_BP_4: Computed water depth and velocities 300 seconds after the dam break. ........... 3.3-5
Fig. 37: H_BP_4: Computed and observed water surface elevation at different control points ..... 3.3-5
Fig. 38: H_BP_5: Perspective view of the circular dam break wave patterns ................................ 3.3-7
Fig. 39: H_BP_5: Water surface and velocity profiles along y=20m ............................................. 3.3-8
Fig. 40: H_BP_5: Water surface and velocity profiles along y = 20m ........................................... 3.3-9
Fig. 41: ST_1: Soni Test with MPM-factor = 6.44 (BC) ................................................................ 4.1-1
Fig. 42: ST_1: Soni Test – Bed aggradation and water level after 15 min (BP) ............................. 4.1-2
Fig. 45: ST_2: Saiedi Test, with MPM-factor = 13 (BC)................................................................. 4.1-4
Fig. 46: ST_2: Saiedi Test, propagation of sediment front (BP)...................................................... 4.1-5
Fig. 47: ST_3: Guenter Test: equilibrium bed level (BC) ................................................................ 4.1-6
Fig. 48: ST_3: Guenter Test: grain size distribution (BC) .............................................................. 4.1-7
Fig. 49: ST_3: Guenter Test: equilibrium bed level (BP) ................................................................ 4.1-8
Fig. 50: ST_3: Guenter Test: grain size distribution (BP)............................................................... 4.1-9
Fig. 51: ST_BC_1: case A, concentration after 9060 s................................................................... 4.2-1
Fig. 52: ST_BC_1: case B, concentration after 9772.5 s................................................................. 4.2-1
Fig. 53: ST_BC_2: case A, concentration after 605 s...................................................................... 4.2-2
Fig. 54: ST_BC_2: case B, concentration after 605 s...................................................................... 4.2-2
List of Tables
Tab. 1: H_BC_2: Performance of implicit calculations ................................................................... 3.2-1
Tab. 2: ST_BC_3: Sediment balances .............................................................................................. 4.2-3
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TEST CASES
III TEST CASES
The test catalogue is intended for the validation of the program. The catalogue consists of
different test cases, their geometric and hydraulic fundamentals and the reference data for
the comparison with the computed results. The test cases are built on each other going
from easy to more and more complex problems. First, the hydraulic solver is validated
against analytical and experimental solutions using the test cases in chapter 1. In a second
part, the sediment transport module is tested against some simple experimental test cases
described in chapter 2. Common test cases are suitable for both, 1D and 2D simulations.
Specific test cases are intended for either 1D or 2D simulations. In chapter 3, the simulation
results for all test cases are presented.
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TEST CASES
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TEST CASES
1 Hydraulic
1.1
Common Test Cases
1.1.1
1.1.1.1
H_1 : Dam break in a closed channel
Intention
The simplest test case checks verifies the conservation property of the fluid phase and
whether a stable condition can be reached for all control volumes after a finite time.
Additionally, wet and dry problematic is tested.
1.1.1.2
Description
Consider a rectangular flume without sediment. The basin is initially filled in the half length
with a certain water depth leaving the other side dry. After starting the simulation, the water
flows in the other half of the system until it reaches a quiescent state (velocity = 0, half of
initial water surface elevation (WSE) everywhere).
1.1.1.3
Geometry and Initial Conditions
Initial Condition: water at rest (all velocities zero), the water surface elevation is the same on
the half of channel. This test uses a WSE of z S = 5.0 m.
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TEST CASES
Fig. 1: H_1: Dam break in a closed channel
1.1.1.4
Boundary Conditions
No inflow and outflow. All of the boundaries are considered as a wall.
Friction: Manning’s factor n = 0.010
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1.1.2
TEST CASES
H_2 : One dimensional dam break on planar bed
1.1.2.1
Intention
This test case is similar to the previous one but this one compares the WSE a short time
after release of the dam with an analytical solution for this frictionless problem. It is based
on the verification of a 1-D transcritical flow model in channels by Tseng (1999).
1.1.2.2
Description
A planar, frictionless rectangular flume has initially a dam in the middle. On the one side,
water level is h2 , on the other side, water level is at h1 . At the beginning, the dam is
removed and a wave is propagating towards the shallow water. The shape of the wave
depends on the fraction of water depths ( h2 / h1 ) and can be compared to an analytical,
exact solution.
1.1.2.3
Two cases are considered:
-
h2 h1 = 0.001 ; h1 = 10.0 m (this avoids problems with wet/dry areas)
-
h2 h1 = 0.000 ; h1 = 10.0 m (downstream is initially dry)
The dam is exactly in the middle of channel.
Since it is a one dimensional problem, it should not depend on the width of the model,
which can be chosen arbitrary.
Fig. 2: H_2: One dimensional dam break on planar bed
1.1.2.4
Boundary Conditions
All cases are frictionless, Manning factor n=0.
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1.1.3
TEST CASES
H_3 : One dimensional dam break on sloped bed
1.1.3.1
Intention
Different from the previous cases, this problem uses a sloped bed with friction. The aim is
to reproduce an accurate wave front in time. It is based on the verification of a 1-D
transcritical flow model in channels by Tseng (1999).
1.1.3.2
Description
In the middle of a slightly sloped flume, an initial dam is removed at time zero (see Fig. 3).
The wave propagates downwards. Comparison data comes from an experimental work.
Friction is accounted for with a Manning roughness factor estimated from the experiment.
1.1.3.3
Length of flume:
Width of flume:
Slope:
Initial water level:
122 m
1.22 m
0.61 m / 122 m
0.035 m (at the dam location)
Fig. 3: H_3: One dimensional dam break on sloped bed
1.1.3.4
Boundary Conditions
Manning friction factor: 0.009
No inflow/outflow. Boundaries are considered as walls.
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1.1.4
1.1.4.1
TEST CASES
H_4 : Parallel execution
Intention
To test the parallel performance of hydraulic simulations on a multi-core shared memory
computer a test case is set up. The aim is to demonstrate the increase in performance (the
speedup) when running BASEMENT in parallel with an increasing number of cores.
1.1.4.2
Description
A hydraulic simulation is repeated with varying number of threads on a multi-core shared
memory system. The simulation times are measured and compared to the sequential
execution time in order to evaluate the speedups.
1.1.4.3
Geometry, initial conditions and boundary conditions
A simple rectangular channel with constant slope is considered. The channel is discretized
with a large number of cross sections (2000) in BASEchain and a large number of elements
(16000) in BASEplane. There is an inflow located at the upper boundary of the channel and
an outflow at the lower boundary. The inflowing discharge is constant over the time and
steady state conditions within the channel are reached.
Large sized scenarios and evenly distributed work loads (steady state conditions) are set up
in order to check for the full potential of the parallel implementation.
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TEST CASES
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1.2
TEST CASES
BASEchain Specific Test Cases
1.2.1
H_BC_1 : Fluid at rest in a closed channel with strongly varying geometry
1.2.1.1
Intention
This test case checks for the conservation of momentum. Numerical artefacts (mostly due
to geometrical reasons) could generate impulse waves although there is no acting force.
1.2.1.2
Description
The computational area consists of a rectangular channel with varying bed topology and
different cross-sections. The fluid is initially at rest with a constant water surface elevation
and there is no acting force. There should be no changes of the WSE in time.
1.2.1.3
The cross sections and bed topology were chosen as in Fig. 4 and Fig. 5. Notice the strong
variations within the geometry. The water surface elevation is set to z s = 12 m.
30
20
y [m]
10
0
-10
-20
-30
0
100
200
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
x [m]
Fig. 4: H_BC_1: Top view: cross-sections
14
height [m]
12
bed level
10
8
6
4
2
0
0
100
200
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
x [m]
Fig. 5: H_BC_1: Bed topography and water surface elevation
1.2.1.4
Boundary Conditions
All boundaries are considered as walls. There is no friction acting.
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1.2.2
TEST CASES
H_BC_2: bed load simulation with implicit hydraulic solution
1.2.2.1
Intention
The aim of this test case is to verify if the use of the implicit computation mode leads to the
expected gain of computational time for a long sediment transport simulation.
1.2.2.2
Description
The test case simulates a section of the river Thur near Altikon. At this place, there is
located a widening of which the morphological evolution should be evaluated.
1.2.2.3
Geometry and general data
The model is composed by 55 irregular cross sections, the mean grain size is 2.5 cm and
the length of the transport generating hydrograph is 338 hours. The given time steps for the
implicit simulation are 60, 120 and 180 seconds. The Program is allowed to reduce the time
step if necessary, but this leads to time loss. The precisions of the results for which the
iteration is interrupted are 0.1 m2 for the wetted area and 0.1 m3/s for the discharge.
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1.3
TEST CASES
BASEplane Specific Test Cases
1.3.1
1.3.1.1
H_BP_1 : Rest Water in a closed area with strongly varying bottom
Intention
This test in two dimensions checks for the conservation of momentum. Numerical artefacts
(mostly due to geometrical reasons) could generate impulse waves although there is no
acting force.
1.3.1.2
Description
The computational area consists of a rectangular channel with varying bottom topography.
The fluid is initially at rest with a constant water surface elevation and there is no acting
force. There should be no changes of the WSE in time.
1.3.1.3
Initial Condition: totally rest water, the WSE is the same and constant over the domain. The
WSE can be varied e.g. from zS (min) = 0.8 to zS (max) 3.0 m .
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Fig. 6: H_BP_1: Rest Water in a closed area with strong varying bottom level
1.3.1.4
Boundary Conditions
Friction: Manning’s factor n = 0.015
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1.3.2
1.3.2.1
TEST CASES
H_BP_2 : Rest Water in a closed area with partially wet elements
Intention
This test case checks the behaviour of partially wet control volumes. The wave front
respectively the border wet/dry is always the weak point in a hydraulic computation. There
should be no momentum due the partially wet elements.
1.3.2.2
Description
In a sloped, rectangular channel, the WSE is chosen to be small enough to allow for partially
wet elements. The water is at rest and no acting force is present.
1.3.2.3
Initial Condition: totally rest water, the WSE is the same and constant over the whole
domain. The WSE is zS = 1.9 m .
Fig. 7: H_BP_2: Rest Water in a closed area with partially wet elements
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1.3.2.4
TEST CASES
Boundary Conditions
Friction: Manning’s factor n = 0.015.
The mesh (grid) can be considered as following pictures. In this form, the partially wet cells
(elements) can be handled.
Fig. 8: H_BP_2: Computational grid
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1.3.3
1.3.3.1
TEST CASES
H_BP_3 : Dam break within strongly bended geometry
Intention
This test case challenges the 2D code. The geometry permits a strongly two dimensional
embossed flow. The results are compared against experimental measurements at certain
control points.
1.3.3.2
Description
In a two-dimensional geometry with a jump in bed topology, a gate between reservoir and
some outflow channel is removed at the beginning. The bended channel is initially dry and
has a free outflow discharge as boundary condition at the downstream end.
1.3.3.3
The geometry is described in Fig. 9. G1 to G6 are the measurement control points for the
water elevation.
Initial condition is a fluid at rest with a water surface elevation of 0.2 m above the channels
ground. The channel itself is dry from water. At t =0, the gate between reservoir and gate is
removed.
Fig. 9: H_BP_3: Strongly bended channel (horizontal projection and cross-section)
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TEST CASES
Boundary Conditions
No inflow. At the downstream end of the channel, a free outflow condition is employed. All
other boundaries are considered as walls.
The friction factor is set to 0.0095 (measured under stationary conditions) for the whole
computational domain.
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1.3.4
1.3.4.1
TEST CASES
H_BP_4 : Malpasset dam break
Intention
This final hydraulic test case is the well known real world data set from the Malpasset dam
break in France. The complex geometry, high velocities, often and sudden wet-dry changes
and the good documentation allow for a fundamental evaluation of the hydraulic code.
1.3.4.2
Description
The Malpasset dam was a doubly-curved equal angle arch type with variable radius. It
breached on December 2nd, 1959 all of a sudden. The entire wall collapsed nearly
completely what makes this event unique. The breach created a water flood wall 40 meters
high and moving at 70 km/h. After 20 minutes, the flood reached the village Frejus and still
had 3 m depth. The time of the breach and the flood wave can be exactly reconstructed, as
the time is known, when the power of different stations switched off.
1.3.4.3
Fig. 11 shows the computational grid used for the simulation. The points represent
“measurement” stations. The initial water surface elevation in the storage lake is set to
+100.0 m.a.s.l. and in the downstream lake to 0.0 m.a.s.l. The area downstream of the wall
is initially dry. At t =0, the dam is removed.
Fig. 10: H_BP_4: Computational area of the malpasset dam break
1.3.4.4
Boundary Conditions
The computational grid was constructed to be large enough for the water to stay within the
bounds. Friction is accounted for by a Manning factor of 0.033.
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1.3.5
TEST CASES
H_BP_5: Circular dam break wave
1.3.5.1
Intention
The circular dam break problem is a demanding two-dimensional test case with some
distinct features. It is used to evaluate the capability to correctly model complex interactions
of shock and rarefaction waves. The results are qualitatively compared to results obtained
by Toro (2001), and other numerical studies, for this idealized dam break scenario. Also the
results of the exact and approximate Riemann solvers are compared to each other.
1.3.5.2
Description
A virtual circular dam is located in the center of a computational domain. At the beginning, at
time t = 0.0s, the dam is removed. The evolution of subsequent wave patterns is examined
until about t = 5s after the dam break.
1.3.5.3
The computational domain has a width and height of 40m and is modelled with a uniform
rectangular grid, which consists of 160000 quadratic elements of the size 0.1m x 0.1m. The
grid is selected large enough to fully capture the outward propagating primary shock wave
during the simulation time of about 5s. The small element sizes shall enable the modelling
of the circular flow patterns with sufficient accuracy.
The initial height of the water column is 2.5m, whereas the surrounding initial water surface
level is set to 0.5m. The water column behind the circular dam has a radius of 2.5m. The
time step is chosen according to a CFL number of 0.65.
Fig. 11: H_BP_5: Computational area (rectangular grid) of circular dam break
1.3.5.4
Boundary Conditions
Friction is accounted for by a small Manning factor of 0.001.
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2 Sediment Transport
2.1
Common Test Cases
2.1.1
2.1.1.1
ST_1 : Soni et al: Aggradation due to overloading
Intention
This validation case is intended to reproduce equilibrium conditions for steady flow followed
by simple aggradation due to sediment overloading at the upstream end. The test is suitable
for 1D and 2D simulations and uses one single grain size.
2.1.1.2
Description
Soni et al. (1980) and Soni (1981) performed a series of experiments dealing with
aggradations. With this particular test, they determined the coefficients a and b of the
empirical power law for sediment transport. This test has been used by many researchers
as validation for numerical techniques containing sediment transport phenomenas (see e.g.
Kassem and Chaudhry (1998), Soulis (2002) and Vasquez et al.( 2005)).
The experiments were realized within a rectangular laboratory flume. First, a uniform
equilibrium flow is established as starting condition. Then the overloading with sediment
starts, resulting in aggradations.
2.1.1.3
Length of flume
Width of flume
Median grain size
Relative density
Porosity
2.1.1.4
lf
wf
dm
s
p
20
0.2
0.32
2.65
0.4
[m]
[m]
[mm]
[]
[]
Equilibrium experiments
Totally, a number of 24 experiments with different initial slopes S have been performed by
Soni. Measured values are equilibrium water depth heq and the sediments equilibrium
discharge qB ,eq . The flow velocity ueq has been computed manually by Soni.
Concerning initial conditions for the equilibrium runs, “the flume was filled with sediment up
to a depth of 0.15 m and then was given the desired slope”. The equilibrium is reached,
when a uniform flow has established respectively “when the measured bed- and water
surface profiles were parallel to each other.” In the average, the equilibrium needed about 4
to 6 hours to develop.
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Boundary conditions:
water flow:
sediment:
constant upstream flow discharge qin and a weir on level 0 at the
downstream end. Alternatively, a ghost cell may be used downstream, where
the total flux into the last cell leaves the computational domain.
periodic boundary conditions are applied. The inflow qB ,in upstream equals
the outflow qB ,out downstream.
From now on, we refer to the Soni test case #1, as the entire data for all experiments would
go beyond the scope of this section. Measured data for the equilibrium are
qin
S
heq
qB ,eq
ueq
[m3/(s*m)] x 10-3
-3
[-] x 10
3.56
[m] x 10-2
3
4.0
5.0
-6
[m /(s*m)] x 10
12.1
[m/s]
0.4
The establishment of the same equilibrium conditions is achieved by calibrating e.g. the
hydraulic friction. As soon as the simulation data looks similar to the measured results, the
aggradation can be started.
2.1.1.5
Aggradation experiments
After the equilibrium has been reached, the sediment feed was increased upstream by an
overloading factor of qB qB ,eq . This factor is different for each experiment. For test case #1,
it was set to 4.0. The flow discharge remains the same as in the equilibrium case. Due to
the massive sediment overloading, aggradation starts quickly at the upstream end.
Measured data is available for the bottom elevation at certain times.
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2.1.2.1
TEST CASES
ST_2 : Saiedi
Intention
Similar to the Soni test case, this validation deals with aggradation due to sediment
overloading. However, compared to Soni, the sediment input is considerably greater than
the carriage capacity of the flow and an aggradation shock is forming. The test verifies
whether the computational model can handle shocks within the sediment phase. Again, a
single grain size is used.
2.1.2.2
Description
The experiments were conducted in a laboratory flume located at the Water Research
Laboratory, School of Civil Engineering, University of New South Wales, Australia.
Geometric details and results are reported in Saiedi (1993 and1994b). Saiedi did two
separate tests, one in a steady flow and one in a rapidly unsteady flow with varying
sediment supply. Only the steady case is of our interest here.
2.1.2.3
The sediment used was of fairly uniform size with the median grain size d 50 = 2 mm,
relative density s = 2.6 and porosity at bed p = 0.37.
The test section was about 18 m long in a 0.61 m wide glass-sided flume with a slope of
0.1 %. The flow was fed with a sediment-supply of constant rate using a vibratory feeder.
The sediment input rate was chosen to be greater than the transport capacity of the flow,
therefore resulting in the formation of an aggradation shock sand-wave travelling
downstream.
The downstream water level was maintained constant by adjusting the downstream gate.
2.1.2.4
Calibration
Before starting the sediment feed, at first, the aim is a uniform flow which fulfils a
measured flow discharge q and the corresponding water depth h . This can be achieved by
adjusting e.g. the hydraulic friction coefficient from Manning n. From the experiments,
Saiedi proposes a calibration estimate=
of n 0.0136 + 0.0170 q . Note that there is no
sediment layer at the calibration phase.
As initial conditions, start with a fluid at rest and a uniform flow depth of h0 = 0.223 m on
the sloped flume. The flow feed rate upstream is constant at qin = 0.098 m3/s. At the
downstream boundary, the water level is kept constant at h = 0.223 m “by adjusting the
downstream gate”.
2.1.2.5
Aggradation
As soon, as a uniform flow with constant water depth is established, the sediment feed
upstream can be started. It remains constantly on qB = 3.81 kg/min.
The results for the steady state computation are available at t = 30 min and t = 120 min. To
avoid cluttering, the plots of the measured bed levels have been averaged.
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TEST CASES
ST_3 : Guenter
2.1.3.1
Intention
The Günter test series deals with degradation in a laboratory flume. The experiments were
conducted using a multiple grain size distribution. The reported results allow for a validation
of the bed topography and the behaviour of heterogeneous sediment transport models.
2.1.3.2
Description
Günter performed some tests at the Laboratory of Hydraulics, Hydrology and Glaciology at
the ETH Zurich using sediment input with a multiple grain size distribution. His aim was to
determine a critical median shear stress of such a grain composition.
Günter was interested in the behaviour of different grain classes but also in the steady
ultimate state of the bed layer, when a top layer has developed and no more degradation
occurs with the given discharge. As initial state, a sediment bed with a higher slope than the
slope in the steady case is used. There is no sediment feed but a constant water discharge.
The bed then starts rotating around the downstream end resulting in a steady, uniform state
with an armouring layer.
The validation is based on the resulting bed topography and the measured grain size
distribution of the armouring layer at the end of the simulation.
2.1.3.3
Guenter actually investigated several test cases with different grain compositions and
boundary conditions. This test case corresponds to his experiment No.3 with the grain
composition No. 1.
The experiment was conducted in a straight, rectangular flume, 40 m long, 1 m width with
vertical walls. At the downstream end, the flume opens out into a slurry tank with 8 m
length, 1.1 m width and 0.8 m depth (relative to the main flume), where the transported
sediment material is held back.
The sediment bed has an initial slope of 0.25 % with an initial grain size distribution given by
d g [cm]
0.0 – 0.102
0.102 – 0.2
0.2 – 0.31
0.31 – 0.41
0.41 - 0.52
0.52 – 0.6
2.1.3.4
Mass fraction [%]
35.9
20.8
11.9
17.5
6.7
7.2
Simulation procedure and boundary/initial conditions
First of all, the domain gets filled slowly with fluid until the highest point is wet. The weir on
the downstream end is set such that no outflow occurs. After the bed is completely under
water, the inflow discharge upstream is increased successively during 10 minutes up to the
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constant discharge of 56.0 l/s. At the same time, the weir downstream is lowered down to a
constant level until the water surface elevation remains constant in the outflow area. There
is no sediment inflow during the whole test.
Using this configuration, the bed layer should rotate around the downstream end of the soil.
The experiment needed about 4 to 6 weeks until a stationary state was reached. The data to
be compared with the experiment are final bed level and final grain size distribution.
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2.2
TEST CASES
2.2.1
ST_BC_1: Advection of suspended load
2.2.1.1
Intention
This test is intended to verify the quality of the advection schemes. The aim is to minimize
the numerical diffusion even over long distances.
2.2.1.2
Description
In a rectangular channel with steady flow conditions the advection of the suspended
material is observed, once for a vertical front of concentration (case A) and once for a
concentration with a Gaussian distribution (case B). The Quickest, Holly-Preissmann and
MDPM-Scheme are tested. The diffusion is set to 0 and there is no sediment exchange with
the soil.
2.2.1.3
•
•
•
Geometry and initial conditions
The computational area is a rectangular channel with 10 km length, 30 m width and
a slope of 0.5 ‰.The cell width ∆x is 40 m.
The initial condition for the hydraulics is a steady discharge of 10 m3/s
The initial conditions for the suspended load are the following:
Case A: The concentration is 1 for the first 200 meters of the flume and 0 for the rest.
1.2
Concentration[-]
1
0.8
0.6
0.4
0.2
0
0
100
200
300
400
500
600
700
Distance [m]
Fig. 12: ST_BC_1: Initial condition for case A
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Case B: The concentration has a Gauss-distribution on the upstream part of the flume.
Concentration [-]
1.2
1
0.8
0.6
0.4
0.2
0
0
100
200
300
400
500
600
700
800
Distance [m]
Fig. 13: ST_BC_1: Initial condition for case B
2.2.1.4
•
•
•
•
•
Boundary conditions
The friction expressed as kStr is 80.
The hydraulic upper Boundary is a hydrograph with steady discharge of 10 m3/s.
The hydraulic outflow boundary is h-q-relation directly computed with the slope.
The upper boundary condition for suspended load is a constant inflow concentration
of 1 for case A and 0 for case B.
The lower boundary condition is an outflow concentration corresponding to the
concentration in the last cell.
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2.2.2
TEST CASES
ST_BC_2: Advection-Diffusion
2.2.2.1
Intention
This test is intended to verify the combination of advection and diffusion for suspended load
for a given diffusion factor Γ (RII-3.9).
2.2.2.2
Description
In a rectangular channel with steady flow conditions the behaviour of the suspended
material is observed, once for a vertical front of concentration (case A) and once for a
concentration with a Gaussian distribution (case B). The Quickest, Holly-Preissmann and
MDPM-Scheme are tested. There is no sediment exchange with the soil.
2.2.2.3
•
•
•
The computational area is a rectangular channel with 1000 length, 20 m width and a
slope of 1 ‰. The friction expressed as kStr is 30. The cell width ∆x is 1 m.
The initial condition for the hydraulics is a steady discharge of 50 m3/s.
The initial conditions for the suspended load are:
Case A: The concentration is 1 for the first 50 meters of the flume and 0 for the rest.
Concentration [-]
1.2
1
0.8
0.6
0.4
0.2
0
0
20
40
60
80
100
120
140
Distance [m]
Fig. 14: ST_BC_2: Initial condition for case A
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Case B: The concentration has a Gauss-distribution on the upstream part of the flume.
Concentration [-]
0.03
0.025
0.02
0.015
0.01
0.005
0
0
20
40
60
80
100
120
140
Distance [m]
Fig. 15: ST_BC_2: Initial condition for case B
2.2.2.4
•
•
•
•
•
Boundary conditions
The hydraulic upper Boundary is hydrograph with steady discharge of 50 m3/s.
The hydraulic outflow boundary is h-q-relation directly computed with the slope.
of 1 for case A and 0 for case B.
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2.2.3
TEST CASES
ST_BC_3: Conservativity of sediment exchange with soil
2.2.3.1
Intention
This test is intended to verify that no sediment is created or lost by the computation in the
simulation of suspended transport with sediment exchange with the soil.
2.2.3.2
Description
The most critical situation for errors in the conservation of sediment volume is the passage
between cross sections with varying geometries and wetted areas. For this reason two
examples have been chosen which have a strong variation of geometry.
2.2.3.3
The computational area is formed by trapezoidal cross sections each 50 m. The total length
of the channel is 4.95 km.
•
•
The initial condition for the hydraulilcs is a steady discharge of 60 m3/s.
The initial concentration of suspended material is 0 and the bed is fixed.
The variation of the cross section width, at the top of the section, is shown in Fig. 16 for
case A and Fig. 17 for case B.
Cross section width
Bed elevation [m]
200
202
200
160
140
198
120
100
196
80
194
60
40
Elevation [m]
Cross section width [m]
180
192
20
0
190
1
5
9
13
17
21
25
29
33
37
41
45
49
53
57
61
65
69
73
77
81
85
89
93
97
Cross section number
Fig. 16: ST_BC_3: Geometry of case A
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Bed elevation
70
201
60
200
199
50
198
40
197
30
196
20
Elevation [m]]
Cross section width [m]
cross section width
195
10
194
0
193
1
5
9
13
17
21
25
29
33
37
41
45
49
53
57
61
65
69
73
77
81
85
89
93
97
Cross section number
Fig. 17: ST_BC_3: Geometry of case B
2.2.3.4
•
•
•
•
•
Boundary conditions
The hydraulic upper Boundary is hydrograph with steady discharge of 60 m3/s.
The hydraulic outflow boundary is h-q-relation, given in a file.
of 0.001.
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3 Results of Hydraulic Test Cases
3.1
Common Test Cases
3.1.1
H_1: Dam break in a closed channel
The conservation of mass is validated. After t = 1107 s, the discharges still have a
magnitude of 10-6 m3/s with decreasing tendency. Both, BASEchain and BASEplane deliver
similar results.
6
t = 1107 s
5
Watersurface elevation [m]
t=0s
4
3
2
1
0
0
5
10
15
20
25
30
35
40
Distance [m]
Fig. 18: H_1: Initial and final water surface elevation (BC & BP)
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H_2: One dimensional dambreak on planar bed
3.1.2.1
Results obtained by BASEchain
A comparison of the 1D test case using h2 h1 = 0.000 with the analytical solution shows an
accurate behaviour of the fluid phase in time.
12
analytical
simulated
Watersurface elevation [m]
10
initial condition
8
6
4
2
0
0
100
200
300
400
500
600
700
800
900
1000
Distance [m]
Fig. 19: H_2: Water surface elevation after 20 s (BC)
35
analytcal
simulated
30
Discharge[m3/s]
25
20
15
10
5
0
0
100
200
300
400
500
600
700
800
900
1000
Distance [m]
Fig. 20: H_2: Distribution of discharge after 20 s (BC)
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TEST CASES
Results obtained by BASEplane
In two dimension, the dam break test case was computed using h2 h1 = 0.0 and h1 = 10
m. The length of the channel was discretized using 1000 control volumes. The CFL number
was set to 0.85. The results show a good agreement between simulation and analytical
solution.
10
Depth (m)
8
Exact
HLLC
HLL
6
Initial
Analytical
4
2
0
0
100
200
300
400
500
600
700
800
900
1000
Distance (m)
Fig. 21: H_2: Flow depth 20 sec after dam break (BP)
30
25
q (m^2/sec)
20
Exact
HLLC
HLL
15
Analytical
10
5
0
0
100
200
300
400
500
600
700
800
900
1000
Distance (m)
Fig. 22: H_2: Discharge 20 sec after dam break (BP)
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H_3: One dimensional dam break on sloped bed
3.1.3.1
The dam break in a sloped bed was compared against experimental results. The agreement
is satisfying.
0.7
measured
watersurface elevation [m]
0.6
bed
0.5
simulated
0.4
0.3
0.2
0.1
0
0
20
40
60
80
100
120
Distance [m]
Fig. 23: H_3: Water surface elevation after 10 s (BC)
0.12
simulated
0.1
measured
waterdepth [m]
0.08
0.06
0.04
0.02
0
0
20
40
60
80
100
120
time [s]
Fig. 24: H_3: Temporal behaviour of water depth at x = 70.1 m (BC)
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0.1
simulated
0.09
0.08
measured
waterdepth [s]
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0
20
40
60
80
100
120
time [s]
Fig. 25: H_3: Temporal behavior of water depth at x = 85.4 m (BC)
3.1.3.2
For two dimensions, the numerical results show a good agreement with the experimental
measurements. Although in certain areas, the water depth is underestimated, the error is
just at 7 % and only for a limited time. The comparison is still successful.
10.1
10
9.9
WSE (m)
9.8
Bed
Exact
HLLC
HLL
9.7
EXP
9.6
9.5
9.4
9.3
0
20
40
60
80
100
120
Distance (m)
Fig. 26: H_3: Cross sectional water level after 10 s simulation time (BP)
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0.1
0.09
0.08
0.07
Depth (m)
0.06
Exact
HLLC
0.05
HLL
EXP
0.04
0.03
0.02
0.01
0
0
20
40
60
80
100
120
Tim e (sec)
Fig. 27: H_3: Chronological sequence of water depth at point x = 70.1 m (BP)
0.1
0.09
0.08
0.07
Depth (m)
0.06
Exact
HLLC
HLL
0.05
EXP
0.04
0.03
0.02
0.01
0
0
20
40
60
80
100
120
Time (sec)
Fig. 28: H_3: Chronological sequence of water depth at point x = 85.4 m (BP)
R III - 3.1-6
VAW ETH Zürich
Version 10/28/2009
3.1.4
3.1.4.1
TEST CASES
H_4 : Parallel execution
General
Hydraulic simulations in a rectangular channel with steady state conditions are repeated
multiple times and the execution times are measured. Thereby only the number of used
threads is varied from one simulation to another. The selected model scenarios are suited
well for parallel execution regarding size and load balancing in order to check the full
potential of the parallel execution (in many practical model setups the observed speedups
may be significantly lower).
3.1.4.2
System configuration
The simulations were performed an Intel multi-core shared memory system with 8 cores.
The used operating system was WinXP 64.
The simulation procedure is repeated with two different versions of BASEMENT. One
version (blue) is compiled with the Intel C++ compiler (V 10.1), whereas the other version
(green) is compiled with the Microsoft VC++ compiler (V14).
3.1.4.3
8
linear speedup
BASEchain (INTEL V10.1)
BASEchain (VC++ V14)
7
Speedup ( - )
6
5
4
3
2
1
1
2
3
4
5
6
7
8
Number Of Cores ( - )
Fig. 29: H_4 (BASEchain) – Parallel speedup for 1D test case
The results indicate an excellent speedup and scalability. The speedup increases nearly
linear with the number of used cores. The parallel speedups obtained by the Intel compiled
binary are superior to those obtained by the VC++ compiled binary.
VAW ETH Zürich
Version 10/28/2009
R III - 3.1-7
3.1.4.4
TEST CASES
8
linear speedup
BASEplane (Intel C++ V10.1)
BASEplane (Microsoft VC++ V14)
7
Speedup (-)
6
5
4
3
2
1
1
2
3
4
5
6
7
8
Number of cores (-)
Fig. 30: H_4 (BASEplane) – Parallel speedup for 2D test case
The results indicate a satisfactory speedup and scalability. The effect of increasing slope of
the blue curve, when running this scenario with 8 cores, is probably due to caching effects.
The parallel speedups obtained with the Intel compiled binary are superior to those obtained
with the VC++ binary, if the number of cores exceeds 2.
R III - 3.1-8
VAW ETH Zürich
Version 10/28/2009
3.2
3.2.1
TEST CASES
H_BC_1: Fluid at rest in a closed channel with strongly varying geometry
The test was successfully carried out. There was no movement at all – the water surface
elevation remained constant over the whole area. No picture will be shown as there is
nothing interesting to see.
3.2.2
H_BC_2: bed load simulation with implicit hydraulic solution
The time steps and simulation times for the explicit and implicit computations are listed in
the table below. The reduction of simulation time is satisfying.
Time step
(seconds)
Explicit
Implicit, base time step = 60
2-5
3-90
3-120
3-180
Simulation
time (hours)
13.94
0.9
0.6
0.47
Speedup
(times faster)
15
23
30
Tab. 1: H_BC_2: Performance of implicit calculations
VAW ETH Zürich
Version 10/28/2009
R III - 3.2-1
TEST CASES
R III - 3.2-2
VAW ETH Zürich
Version 10/28/2009
3.3
3.3.1
TEST CASES
BASEplane Specific Test Cases
H_BP_1: Rest water in a closed area with strongly varying bottom
3.3.2
H_BP_2: Rest Water in a closed area with partially wet elements
3.3.3
H_BP_3: Dam break within strongly bended geometry
The computational area was discretized using a mesh with 1805 elements and 1039
vertices. The vertical jump in bed topography between the reservoir and the channel was
modelled with a strongly inclined cell (a node can only have one elevation information). Fig.
12 shows the water level contours ant the velocity vectors. The following figures compare
the time evolution of the measured water level with the simulated results at the control
points G1, G3 and G5.
The results are showing an accurate behaviour expect at the control point G5, where some
bigger differences can be observed. They are caused by the use of a coarse mesh around
that area. Using a higher mesh density, the water level in the critical area after the strong
bend is better resolved and delivers more accurate results. Fig. 35 shows a comparison at
point G5 for a coarse and a dense mesh.
VAW ETH Zürich
Version 10/28/2009
R III - 3.3-1
TEST CASES
Fig. 31: H_BP_3: Water level and velocity vectors 5 seconds after the dam break
0.2
0.175
0.15
WSE (m)
0.125
EXP
Exact
HLLC
0.1
HLL
0.075
0.05
0.025
0
0
5
10
15
20
25
30
35
40
Time (sec)
Fig. 32: H_BP_3: Water surface elevation at G1
R III - 3.3-2
VAW ETH Zürich
Version 10/28/2009
TEST CASES
0.2
WSE (m)
0.15
EXP
Exact
HLLC
0.1
HLL
0.05
0
0
5
10
15
20
25
30
35
40
Time (sec)
0.2
WSE (m)
0.15
EXP
Exact
HLLC
0.1
HLL
0.05
0
0
5
10
15
20
25
30
35
40
Time (sec)
VAW ETH Zürich
Version 10/28/2009
R III - 3.3-3
TEST CASES
0.2
0.175
0.15
WSE (m)
0.125
Exp
fines Gitter
grobes Gitter
0.1
0.075
0.05
0.025
0
0
5
10
15
20
25
30
35
40
Time (sec)
Fig. 35: H_BP_5: Comparison of the water level at point G5 for a coarse and a dense grid.
3.3.4
H_BP_4: Malpasset dam break
The computational grid consists of 26000 triangulated control volumes and 13541 vertices.
The time step was chosen according to a CFL number of 0.85. Fig. 36 shows the computed
water depth and velocities 300 seconds after the dam break. Fig. 37 compares several
computed with the observed data on water level elevation at different stations.
BASEplane results show a good agreement with the observed values as also other
simulation results. The differences between computed and measured data are maximally
around 10%.
R III - 3.3-4
VAW ETH Zürich
Version 10/28/2009
TEST CASES
Fig. 36: H_BP_4: Computed water depth and velocities 300 seconds after the dam break.
100.00
90.00
80.00
Wasserspiegellage [m.ü.M]
70.00
60.00
field data
BASEplane
Yoon & Kang (2004)
Valiani et al. (2002)
50.00
40.00
30.00
20.00
10.00
0.00
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
P11
P12
P15
P17
Messstelle
Fig. 37: H_BP_4: Computed and observed water surface elevation at different control points
VAW ETH Zürich
Version 10/28/2009
R III - 3.3-5
3.3.5
TEST CASES
H_BP_5: Circular dam break
The obtained results for the circular dam break test case are plotted in two different ways. A
3D perspective view of the depth is shown in Fig. 38 to illustrate the overall flow and wave
patterns. Additional water surface and velocity profiles are plotted in Fig. 39 and Fig. 40 for
distinct times.
The following flow patterns of the 2D circular dam break are reported by Toro and other
numerical studies and are also observed here.
After the collapse of dam at t = 0.0s an outward propagating, primary shock wave is
created. A sharp depth gradient develops behind this shock wave. Also, a rarefaction wave
is generated which propagates inwards in the direction of the center of the dam break. The
rarefaction wave finally reaches the center and generates a very distinct dip of the surface
elevation at the center (t = 0.4 s) This dip travels outwards resulting in a rapid drop of the
surface elevation at the center, which falls even below the initial outer water level (t = 1.4s)
and finally nearly reaches the ground (t = 3.5s). A second shock wave develops which
propagates inwards while the primary shock wave continues to propagate outwards with
decreasing strength. This secondary shock wave converges to the center and finally
implodes. This generates a sharp jump in the surface water elevation at the center (t =
4.7s).
As a result it can be stated that BASEplane is able to reproduce these distinct features of
the flow and wave patterns of the 2D circular dam break. Comparisons with the numerical
results of Toro show qualitative agreement of the calculated water surface and velocities
profiles. Generally, a more diffusive behaviour is observed compared to Toro’s results. This
may be attributed to the use of first order Godunov methods and to the use of a lower CFL
number in the simulations. Despite the use of a rectangular grid, the cylindrical symmetry of
the wave propagations is maintained well. Only at the first moments of the dam break,
some water surface modulations can be seen at the crest of the primary shock wave which
diminish with proceeding time.
The resulting water surface and velocity profiles of the approximate Riemann solvers match
well the results of the exact Riemann solver. Both approximate Riemann solvers, HLL and
HLLC, reproduce the flow and wave patterns.
R III - 3.3-6
VAW ETH Zürich
Version 10/28/2009
TEST CASES
t = 0.0s
t = 0.4s
t = 0.7s
t = 1.4s
t = 3.5s
t = 4.7s
Fig. 38: H_BP_5: Perspective view of the circular dam break wave patterns
VAW ETH Zürich
Version 10/28/2009
R III - 3.3-7
TEST CASES
Waterlevel (t = 0.0 s)
Velocity (t = 0.0 s)
2.5
3
2
2
h - HLL
h - HLLC
h - EXACT
1
velocity [m/s]
waterlevel [m]
1
1.5
v - HLL
0
v - HLLC
0
5
10
15
20
25
30
35
40
30
35
40
v - EXACT
-1
0.5
-2
0
0
5
10
15
20
25
30
35
40
-3
x [m]
x [m]
t = 0s
t = 0s
2.5
4
3
2
1
1.5
h - HLL
h - HLLC
h - EXACT
1
velocity [m/s]
waterlevel [m]
2
v - HLL
v - HLLC
0
0
5
10
15
20
25
v - EXACT
-1
-2
0.5
-3
0
0
5
10
15
20
25
30
35
40
-4
x [m ]
x [m ]
t = 0.4s
t = 0.4s
4
2.5
3
2
1
1.5
h - HLL
h - HLLC
h - EXACT
1
velocity [m/s]
waterlevel [m]
2
v - HLL
0
v - HLLC
0
5
10
15
20
25
30
35
-1
-2
0.5
-3
0
0
5
10
15
20
25
x [m ]
t = 0.7s
30
35
40
-4
x [m ]
t = 0.7s
Fig. 39: H_BP_5: Water surface and velocity profiles along y= 20m
R III - 3.3-8
VAW ETH Zürich
Version 10/28/2009
40
v - EXACT
TEST CASES
4
2.5
3
2
1
1.5
h - HLL
h - HLLC
h - EXACT
1
velocity [m/s]
waterlevel [m]
2
v - HLL
0
v - HLLC
0
5
10
15
20
25
30
35
40
v - EXACT
-1
-2
0.5
-3
0
0
5
10
15
20
25
30
35
-4
40
x [m ]
x [m ]
t = 1.4s
t = 1.4s
4
2.5
3
2
2
h - HLL
h - HLLC
h - EXACT
1
velocity [m/s]
waterlevel [m]
1
1.5
0
v - HLLC
0
5
10
15
20
25
30
35
40
v - EXACT
-1
-2
0.5
-3
0
0
5
10
15
20
25
30
35
-4
40
x [m ]
x [m ]
t = 3.5s
t = 3.5s
4
2.5
3
2
1
1.5
h - HLL
h - HLLC
h - EXACT
1
velocity [m/s]
waterlevel [m]
2
v - HLL
0
v - HLLC
0
5
10
15
20
25
30
35
40
-1
-2
0.5
-3
0
0
5
10
15
20
25
x [m ]
t = 4.7s
30
35
40
-4
x [m ]
t = 4.7s
Fig. 40: H_BP_5: Water surface and velocity profiles along y = 20m
VAW ETH Zürich
Version 10/28/2009
R III - 3.3-9
v - EXACT
TEST CASES
R III - 3.3-10
VAW ETH Zürich
Version 10/28/2009
TEST CASES
4 Results of Sediment Transport Test Cases
4.1
Common Test Cases
4.1.1
ST_1: Soni
4.1.1.1
The Soni test case has been simulated with the MPM approach like the one used for the
next test case of Saiedi (see next chapter 4.1.2). As the equilibrium bed load of 2.42*10-7
m3/s is known, the MPM-factor has been calibrated to obtain a good agreement for the
equilibrium state. This leads to a value of 6.44 for the prefactor in the MPM-formula.
The results show a quite good agreement between experiment and simulation.
Difference of bed level from initial state [m]
0.05
mesured 15 min.
0.04
mesured 30 min.
mesured 45 min.
0.03
computed 15 min.
computed 30 min.
0.02
computed 45 min.
0.01
0
-0.01
0
2
4
6
8
10
12
14
16
18
Distance from upstream [m]
Fig. 41: ST_1: Soni Test with MPM-factor = 6.44 (BC)
VAW ETH Zürich
Version 10/28/2009
R III - 4.1-1
4.1.1.2
TEST CASES
The Soni test case was modelled as single grain computation on a mobile bed with the
transport formula of Meyer-Peter & Müller (MPM). The grain diameter is chosen as the
mean diameter of the grain mixture used by Soni. The simulations were performed on an
unstructured mesh with 1202 triangular elements.
In Soni’s experiments at first an equilibrium transport was established within the laboratory
flume. The corresponding transport rate observed by Soni is known to be 2.42*10-7 m3/s.
Then the sediment inflow was increased to 4 times the equilibrium transport.
To be able to reproduce the experiments, the transport formula must be calibrated to
achieve the same equilibrium transport rate. The calibration resulted in a reduction of the
pre-factor of the MPM formula from 8 to about 3.3. The critical dimensionless shear stress
for incipient motion in the MPM formula is not calibrated here and therefore, per default,
determined from the Shields diagram.
The numerical results for the bed aggradations and the water levels are compared with the
measurements by Soni. The situations after 15min, 30min and 40min are plotted in the
following figures. Generally the numerical results show an acceptable agreement with the
measured values. Compared to the measurements, the toe of the aggradation front shows a
less diffusive smearing behaviour. This may be attributed to the finer grain fractions within
Soni’s grain mixtures, whose behaviour is not adequately modelled in single grain
computation. Also the rather fine diameter of 0.32 mm can be seen as problematic
concerning the applicability of the MPM formula, which is best suited for coarse sands and
gravel.
Fig. 42: ST_1: Soni Test – Bed aggradation and water level after 15 min (BP)
R III - 4.1-2
VAW ETH Zürich
Version 10/28/2009
TEST CASES
VAW ETH Zürich
Version 10/28/2009
R III - 4.1-3
4.1.2
TEST CASES
ST_2: Saiedi
4.1.2.1
The Saiedi test was simulated using the Meyer-Peter Müller approach for the sediment flux,
which can also be formulated as:
qB = factor (θ − θ cr )3/ 2 ( s − 1) gd m3/ 2
The factor is usually set around 8.0 and should be between 5 and 15 (Wiberg and Smith
1989). In this case it was calibrated to 13.
Bed profile normalized by 0.223 m
0.5
measured 30 min.
0.4
measured 120 min
calculated 30 min.
0.3
calculated 120 min.
0.2
0.1
0
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Distance from upstream normalized by 18 m
Fig. 45: ST_2: Saiedi Test, with MPM-factor = 13 (BC)
The results present a good fit. This example shows that, to reproduce this type of transport,
the approach of MPM approach needs calibration.
R III - 4.1-4
VAW ETH Zürich
Version 10/28/2009
4.1.2.2
TEST CASES
The Saiedi test case was simulated using a single grain approach on an unstructured mesh
made of 768 triangles. The Meyer-Peter & Müller formula was used to determine the
bedload transport. The simulation was performed on a fixed bed, where no erosion can take
place.
To obtain a reasonable fit between the measured values and the simulation results the prefactor of the transport formula again had to be reduced from 8 to about 5. Furthermore, the
simulations show that the sediment is transported too rapidly out of the flume compared to
the experiments. To compensate for this effect the critical Shields factor for incipient motion
is increased to 0.055, instead of using the value from the Shields diagram.
With these adjustments, the shape and the propagation speed of the sediment front are
captured well and Saiedi’s results can be reproduced with good accuracy. The front of the
sediment bore does not smear over the time but remains steep during the propagation.
Fig. 46: ST_2: Saiedi Test, propagation of sediment front (BP)
VAW ETH Zürich
Version 10/28/2009
R III - 4.1-5
4.1.3
TEST CASES
ST_3: Guenter
4.1.3.1
For mixed materials, the Guenter test case has been performed. The initial slope was
chosen to be 0.25 % - this corresponds to the full experiment of Guenter.
The eroded material is eliminated from the end basin by a sediment sink. The active layer
height is 7 [cm] and the critical dimensionless shear stress (for beginning of sediment
transport) is set to the default value of 0,047. After 200 hours, there are no more important
changes recognizable. After the results of Guenter the slope at the final equilibrium state
should be the same as the initial one.
The results show a good agreement for the final slope between experiment and simulation.
However, concerning the grain size distribution, the result shows a sorting process taking
place: The finer material is obviously washed out too strongly.
There are many parameters which influence this result, especially the choice of the critical
shear stress and the height of the active layer. Again, this example shows, one has to use
different approaches for sediment transport with different values for the free parameters.
Quite often, a sensitivity analysis may give a better insight in the behaviour of certain
formulas and their parameters.
10.1
10
9.9
Bed level [m]
9.8
9.7
9.6
9.5
initial state
9.4
computed 200 hours
9.3
9.2
9.1
9
0
5
10
15
20
25
30
35
40
45
50
55
Distance from upstream [m]
Fig. 47: ST_3: Guenter Test: equilibrium bed level (BC)
R III - 4.1-6
VAW ETH Zürich
Version 10/28/2009
TEST CASES
1
initial state
0.9
equilibrium state measured
computed 200 hours
0.8
Sum of fractions
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Grain size [cm]
Fig. 48: ST_3: Guenter Test: grain size distribution (BC)
VAW ETH Zürich
Version 10/28/2009
R III - 4.1-7
4.1.3.2
TEST CASES
The Guenter test has been numerically modelled with 6 different grain classes. The
sediment transport capacity has been determined with Hunziker’s formula for graded
transport. The critical dimensionless shear stress for beginning of sediment transport is set
to the default value of 0.047 in Hunziker’s formula. This bedload formula was not further
adjusted for the simulations and used with its default values for the pre-factor (=1.0) and the
exponent of the hiding exponent (=-1.5).
The experimental flume has a mobile bed and is modelled with 312 rectangular elements.
The channel parts in front and behind the experimental flume, are set to fixed bed elevations
in this simulation. The sediment which leaves the mobile bed and enters the fixed bed
section is removed using a sediment sink. The thickness of the bedload control volume is
set to a constant value of 5 mm. The choice of the thickness of the bedload control volume
shows large impacts on the simulation results. An increased thickness here results in larger
erosion and a finer composition of the final bed armour.
The mobile bed is eroded during the simulation until finally an armouring layer of coarser
materials is formed which prevents further erosions. The final slope of the bed is nearly
constant with a value of 0.23%, which is slightly smaller than the original slope of 0.25%.
This result is in agreement with the observations made by Guenter. Also the rotation of the
bed surface around the downstream end of the flume can be observed during the
simulation.
Fig. 49: ST_3: Guenter Test: equilibrium bed level (BP)
R III - 4.1-8
VAW ETH Zürich
Version 10/28/2009
TEST CASES
Fig. 50 shows the measured and computed grain compositions, whereas the latter was
taken from the upstream end of the flume. The simulated grain size distribution (red curve)
shows a good qualitative agreement with the measured distribution by Guenter (blue dots).
But a trend can be seen that the computed composition is slightly too fine. Finally, it can be
said, that the numerical model seems capable of reproducing the sorting effects and seems
able to simulate the formation of an armouring bed layer.
Fig. 50: ST_3: Guenter Test: grain size distribution (BP)
Beside the results obtained with the Hunziker’s transport formula, in Fig. 50 presents also
the resulting grain distributions obtained with Wu’s transport formula (green curve). Here it
can be seen that the finer fractions are eroded too strongly, but a qualitative agreement is
obtained. For the simulations with the Wu formula the same critical shear stress of 0.047
was used, and the default settings of the bed load factor (=1.0) and the default exponent of
the hiding-and exposure coefficient (=-0.6) were set. But to enable similar erosion volumes
as obtained with the Hunziker’s formula, the thickness of the bed load control volume was
increased to 2.5 cm.
To check for the mass conservation properties a comparison was made between the initial
sediment volumes in the flume and the final sediment volumes, under regard of all removed
sediment volumes. Despite the long run time, the simulation proves to be conservative
regarding the sediment transport.
VAW ETH Zürich
Version 10/28/2009
R III - 4.1-9
TEST CASES
R III - 4.1-10
VAW ETH Zürich
Version 10/28/2009
4.2
TEST CASES
BASEchain specific Test Cases
4.2.1
ST_BC_1: Advection of suspended load
The Advection Test has been executed for the QUICK, the QUICKEST, the Holly-Preissmann
and the MDPM-scheme. The results of the QUICK-scheme are not illustrated as they are
very instable.
Case A:
Fig. 51 shows the concentration front after 9060 s or after 8200 m of way. The
results show that the big issue of the advection simulation, the numerical
diffusion, is almost completely avoided by the MDPM scheme.
1.4
Analytical
1.2
Concentration [-]
MDPM
1
QUICKEST
0.8
Holly-Preissmann
0.6
0.4
0.2
0
-0.2
6500
7000
7500
8000
8500
9000
9500
10000
Distance [m]
Fig. 51: ST_BC_1: case A, concentration after 9060 s
Case B:
Fig. 52 shows the concentration front after 9772.5 s or after 8843 m of way. For
the MDPM scheme there are some local deviations from the analytical solution.
These are due to the discretisation and do not increase with time and distance.
The initial maximal concentration is conserved.
1.4
Concentration [-]
1.2
1
MDPM
Analytical
0.8
Holly-Preissmann
0.6
QUICKEST
0.4
0.2
0
-0.2
8000
8200
8400
8600
8800
9000
9200
9400
9600
9800
10000
Distance [m]
Fig. 52: ST_BC_1: case B, concentration after 9772.5 s
VAW ETH Zürich
Version 10/28/2009
R III - 4.2-1
4.2.2
TEST CASES
ST_BC_2: Advection-Diffusion
The Advection Test has been executed for the QUICKEST, and the MDPM-scheme.
Fig. 53 and Fig. 54 show the concentration front of case A and the Gauss distribution for
case B after 605 s or after 788 m of way. The diffusion factor is 0.04 for case A and 0.1 for
case B. The result of the simulation with the internally computed diffusion is also illustrated,
but there is no analytical solution to compare.
1.2
Analytical
1
MDPM simulated without diffusion factor
MDPM simulated with diffusion factor 0.04
Concentration [-]
0.8
QUICKEST simulated without diffusion factor
0.6
QUICKEST simulated with diffusion factor 0.04
0.4
0.2
0
-0.2
750
770
790
810
830
850
870
890
910
930
950
Distance [m]
Fig. 53: ST_BC_2: case A, concentration after 605 s
0.025
Analytical diffusion factor 0.1
Concentration [-]
0.02
0.015
MDPM simulated with diffusion factor 0.1
MDPM simulated without diffusion factor
QUICKEST simulated with diffusion factor 0.1
0.01
QUICKEST simulated without diffusion factor
0.005
0
-0.005
600
650
700
750
800
850
900
950
1000
Distance [m]
Fig. 54: ST_BC_2: case B, concentration after 605 s
The results are satisfying. The QUICKEST scheme gives here a result which is nearly as
good as the one of the MDPM scheme, but this is due to the short simulation time. Its
deviation from the exact solution due to numerical diffusion increases with the time like
shown in test ST_BC_1.
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4.2.3
TEST CASES
ST_BC_3: Conservation of sediment exchange of soil
The computations have been executed for the QUICKEST, and the MDPM-scheme. The
sediment balances of the simulations are listed in Tab. 1.
A
B
A
B
MDPM
MDPM
QUICKEST
QUICKESTB
Sediment Inflow
[m3]
20736
20736
20736
20736
0
31
0
38
77
53
59
30
20662
20666
20686
20774
-3.45
-14.47
-8.71
-106.68
-0.01664189
-0.06981477
-0.04202272
-0.51449791
Sediment outflow
[m3]
Suspended
sediments [m3]
Deposited
sediments [m3]
Gain/Loss [m ]
3
Gain/Loss [%]
Tab. 2: ST_BC_3: Sediment balances
The conservation is satisfactory in all cases. With the MDPM-scheme the obtained results
are better.
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TEST CASES
R III - 4.2-4
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TEST CASES
References
Kassem, A. A., and Chaudhry, M. H. (1998). "Comparison of coupled and semicoupled
numerical models for alluvial channels." Journal of Hydraulic Engineering-Asce,
124(8), 794-802.
Soni, J. P. (1981). "Laboratory Study of Aggradation in Alluvial Channels." Journal of
Hydrology, 49(1-2), 87-106.
Soni, J. P., Garde, R. J., and Raju, K. G. R. (1980). "Aggradation in Streams Due to
Overloading." Journal of the Hydraulics Division-Asce, 106(1), 117-132.
Soulis, J. V. (2002). "A fully coupled numerical technique for 2D bed morphology
calculations." International Journal for Numerical Methods in Fluids, 38(1), 71-98.
Torro, F.T. (2001). "Shock-Capturing Methods for Free-Surface Shallow Flows". John Wiley &
Sons
Tseng, M. H. "Verification of 1-D Transcritical Flow Model in Channels." Natl. Sci. Counc.
ROC(A), 654-664.
Vasquez, J. A., Steffler, P. M., and Millar, R. G. "River2D Morphology, Part I: Straight alluvial
channels." 17th Canadian Hydrotechnical Conference, Edmonton, Alberta.
Wiberg, P. L., and Smith, J. D. (1989). "Model for Calculating Bed-Load Transport of
Sediment." Journal of Hydraulic Engineering-Asce, 115(1), 101-123.
VAW ETH Zürich
Version 10/28/2009
R III - References
TEST CASES
R III - References
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COMMAND
ANDI
NPUTFI
LES
o
fBASEMENT
I
V
R
VAW ETH Zürich
Reference M anual BASEM ENT
COM M AND AND INPUT FILES
Table of Contents
Part 1: The Command File of BASEMENT
1
Main structure of the command file
1.1
1.2
1.3
1.4
2
Syntax ................................................................................................................. 1.1-1
Super blocks ....................................................................................................... 1.2-1
Structure of Command File .............................................................................. 1.3-1
Block Description ............................................................................................... 1.4-1
Block List
2.1
PROJECT ............................................................................................................. 2.1-1
DOMAIN .............................................................................................................. 2.2-1
2.2
PARALLEL ........................................................................................................... 2.3-1
2.3
PHYSICAL_PROPERTIES ................................................................................... 2.4-1
2.4
BASECHAIN_1D: core block for one dimensional simulations ..................... 2.5-1
2.5
GEOMETRY .................................................................................................. 2.5-3
2.5.1
HYDRAULICS ............................................................................................... 2.5-4
2.5.2
PARAMETER ......................................................................................... 2.5-5
2.5.2.1
SECTION_COMPUTATION ............................................................ 2.5-8
2.5.2.1.1
FRICTION .............................................................................................. 2.5-9
2.5.2.2
BOUNDARY ........................................................................................ 2.5-10
2.5.2.3
INITIAL ................................................................................................ 2.5-14
2.5.2.4
SOURCE .............................................................................................. 2.5-15
2.5.2.5
2.5.2.5.1
EXTERNAL_SOURCE ................................................................... 2.5-16
MORPHOLOGY .......................................................................................... 2.5-18
2.5.3
PARAMETER ....................................................................................... 2.5-19
2.5.3.1
BEDMATERIAL ................................................................................... 2.5-21
2.5.3.2
GRAIN_CLASS ............................................................................. 2.5-22
2.5.3.2.1
MIXTURE ...................................................................................... 2.5-23
2.5.3.2.2
SOIL_DEF ..................................................................................... 2.5-24
2.5.3.2.3
2.5.3.2.3.1 SUBLAYER ................................................................................ 2.5-26
SOIL_ASSIGNMENT .................................................................... 2.5-27
2.5.3.2.4
BEDLOAD ........................................................................................... 2.5-28
2.5.3.3
PARAMETER ................................................................................ 2.5-29
2.5.3.3.1
BOUNDARY ................................................................................. 2.5-31
2.5.3.3.2
SOURCE ....................................................................................... 2.5-33
2.5.3.3.3
2.5.3.3.3.1 EXTERNAL_SOURCE ................................................................ 2.5-34
SUSPENDED_LOAD ........................................................................... 2.5-35
2.5.3.4
PARAMETER ................................................................................ 2.5-36
2.5.3.4.1
BOUNDARY ................................................................................. 2.5-37
2.5.3.4.2
INITIAL.......................................................................................... 2.5-39
2.5.3.4.3
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SOURCE ....................................................................................... 2.5-40
2.5.3.4.4.1 EXTERNAL_SOURCE ................................................................ 2.5-41
OUTPUT...................................................................................................... 2.5-42
2.5.4
SPECIAL_OUTPUT .............................................................................. 2.5-43
2.5.4.1
BASEPLANE_2D: core block for two dimensional simulations .................... 2.6-1
2.6
GEOMETRY .................................................................................................. 2.6-3
2.6.1
STRINGDEF ........................................................................................... 2.6-4
2.6.1.1
HYDRAULICS ............................................................................................... 2.6-5
2.6.2
PARAMETER ......................................................................................... 2.6-6
2.6.2.1
FRICTION .............................................................................................. 2.6-9
2.6.2.2
BOUNDARY ........................................................................................ 2.6-11
2.6.2.3
INNER_BOUNDARY ............................................................................ 2.6-14
2.6.2.4
TURBULENCE_MODEL ...................................................................... 2.6-16
2.6.2.5
INITIAL ................................................................................................ 2.6-18
2.6.2.6
SOURCE .............................................................................................. 2.6-20
2.6.2.7
EXTERNAL_SOURCE ................................................................... 2.6-21
2.6.2.7.1
MORPHOLOGY .......................................................................................... 2.6-22
2.6.3
PARAMETER ....................................................................................... 2.6-23
2.6.3.1
BEDMATERIAL ................................................................................... 2.6-25
2.6.3.2
GRAIN_CLASS ............................................................................. 2.6-26
2.6.3.2.1
MIXTURE ...................................................................................... 2.6-27
2.6.3.2.2
SOIL_DEF ..................................................................................... 2.6-28
2.6.3.2.3
2.6.3.2.3.1 LAYER ....................................................................................... 2.6-30
FIXED_BED .................................................................................. 2.6-31
2.6.3.2.4
SOIL_ASSIGNMENT .................................................................... 2.6-33
2.6.3.2.5
BEDLOAD ........................................................................................... 2.6-34
2.6.3.3
PARAMETER ................................................................................ 2.6-35
2.6.3.3.1
BOUNDARY ................................................................................. 2.6-39
2.6.3.3.2
SOURCE ....................................................................................... 2.6-41
2.6.3.3.3
2.6.3.3.3.1 EXTERNAL_SOURCE ................................................................ 2.6-42
OUTPUT...................................................................................................... 2.6-43
2.6.4
SPECIAL_OUTPUT .............................................................................. 2.6-45
2.6.4.1
3
Auxiliary input files
3.1
Introduction ........................................................................................................ 3.1-1
Elements of an auxiliary input files ............................................................... 3.1-1
3.1.1
Syntax description ........................................................................................ 3.1-1
3.1.2
Type “BC” ........................................................................................................... 3.2-1
3.2
3.3
Type “Ini1D” ....................................................................................................... 3.3-1
Type “InitialConcentration1D” ......................................................................... 3.4-1
3.4
Type “Topo1D” .................................................................................................. 3.5-1
3.5
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IV COMMAND AND INPUT FILES
The main purpose of the command file is to ensure a structured and logical definition of a
simulation independent of the chosen solvers or model dimensions (1d/2d). The command
file shall be of self explaining character and allow for a consistent integration of future
developments.
To not overload the command file, auxiliary files are necessary. One of the most often used
auxiliary file is the one describing a time series of a quantity, e.g. a hydrograph used as
boundary condition. If the command file is demanding for an auxiliary file, a reference to its
type is given in the parameter description. Following auxiliary file types are documented in
this part of the manual:
-
Type “BC” : file containing a boundary condition
Type “Ini1D” : file containing a initial condition (only used by BASECHAIN_1D)
Type “InitalConcentration1D” : file containing a initial condition for suspended
sediment transport (only used by BASECHAIN_1D)
Type “Topo1D” : topography file for BASECHAIN_1D
The detailed description of these files can be found in part two of this section of the manual.
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Part 1:
The Command File of BASEMENT
1 Main structure of the command file
1.1
Syntax
Generally, the input file has an object-oriented approach. The main keywords are treated as
classes and have keywords as members which can be kind of super classes again. Every
such block is enclosed by two curly braces, e.g.
BLOCKNAME
{
variable_name = …
variable_name = …
…
BLOCKNAME
{
variable_name = …
…
variable_list = ( var1 var2 var3 )
// this is a comment
}
…
}
All information within the two brackets {} belongs to the superclass.
A listing of variables is always embraced by two normal brackets, even if there is just one
element in the list. Note that there must be a space between opening bracket and the first
element as at the end of the list between last element and closing bracket. Between
elements in the list, there is also at least one space required.
Comments can be placed anywhere within the input file. They are indicated with two
slashes followed by a space:
//
Every information after the two slashes and the space until the end of line will be ignored.
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1.2
Super blocks
On the first level, there are two super blocks: PROJECT and DOMAIN.
The PROJECT-block contains all kind of meta-information as title, author, date, version, etc.
The DOMAIN-block then includes all sub-blocks for a simulation, namely different regions
for one or two dimensional simulations: BASECHAIN_1D and BASEPLANE_2D.
These two superblocks contain the concrete input information for BASEMENT. Although the
1D and 2D solver differ in certain parts concerning the needed information, the rough topics
are the same and will be treated as superblocks themselves:
•
•
•
•
•
•
•
Geometry
Initial Conditions
Boundary Conditions
Friction Values
General parameters
Element specific parameters
Output specifications
For a graphical illustration of the structure of the command file, see chapter 1.3.
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1.3
Structure of Command File
PROJECT
DOMAIN
PARALLEL
PHYSICAL_PROPERTIES
BASECHAIN_1D
GEOMETRY
HYDRAULICS
PARAMETER
SECTION_COMPUTATION
FRICTION
BOUNDARY
INITIAL
SOURCE
EXTERNAL_SOURCE
MORPHOLOGY
PARAMETER
BEDMATERIAL
GRAIN_CLASS
MIXTURE
SOIL_DEF
SUBLAYER
SOIL_ASSIGNMENT
BEDLOAD
PARAMETER
BOUNDARY
SOURCE
EXTERNAL_SOURCE
SUSPENDED_LOAD
PARAMETER
BOUNDARY
INITIAL
SOURCE
EXTERNAL_SOURCE
OUTPUT
SPECIAL_OUTPUT
BASEPLANE_2D
GEOMETRY
STRINGDEF
HYDRAULICS
PARAMETER
FRICTION
BOUNDARY
INNER_BOUNDARY
TURBULENCE_MODEL
INITIAL
SOURCE
EXTERNAL_SOURCE
MORPHOLOGY
PARAMETER
BEDMATERIAL
GRAIN_CLASS
MIXTURE
SOIL_DEF
LAYER
FIXED_BED
SOIL_ASSIGNMENT
BEDLOAD
PARAMETER
BOUNDARY
SOURCE
EXTERNAL_SOURCE
OUTPUT
SPECIAL_OUTPUT
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1.4
Block Description
The next chapter contains a full description of all possible blocks and variables within the
command file. The structure looks as following:
BLOCKNAME
compulsory
repeatable
Super Block
[…]
Condition:
-
[…]
Possible super blocks are listed here
Description:
Gives a short description about the use of the block and its features
Inner Blocks
[…]
Possible sub blocks are listed here
Variable
type
comp
rep
values
condition
example
string
Yes
No
-
BLOCK 3
something
integer
No
Yes
0, 1, 2
-
desc
Block names are always written in capital letters for a clear identification. Every Block (and
also variables) may be compulsory and/or repeatable or not. Condition indicates the
prerequisite of a super block.
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2 Block List
2.1
PROJECT
PROJECT
compulsory
repeatable
Condition:
-
Super Block
-
Description:
Contains some meta information about the setup. This is the only block without further inner
blocks.
Variable
type
comp
rep
values
condition
title
string
No
No
One word
-
author
string
No
No
One word
-
date
date
No
No
-
-
desc
Example:
PROJECT
{
title = Alpenrhein
// note: Alpenrhein Delta would not work
// write Alpenrhein_Delta instead
author = MeMyselfAndI
date = 10.8.2006
}
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2.2
DOMAIN
DOMAIN
compulsory
repeatable
Condition:
-
Super Block
-
Description:
Defines the whole computational area and connects different specific areas. Multriregion is
just the name for the overall area, e.g. Rhine river. Specific areas within the Rhine could be
Rhinedelta (2D) and Alp-Rhine (1D). The Domain is primarily used to connect different
computations and regions with each other.
Inner Blocks
PHYSICAL_PROPERTIES
PARALLEL
BASECHAIN_1D
BASEPLANE_2D
Variable
type
comp
Rep
values
condition
multiregion
string
Yes
No
One word
-
desc
Example:
PROJECT
{
[…]
}
DOMAIN
{
multiregion = Rhein
PHYSICAL_PROPERTIES
{
[…]
}
PARALLEL
{
[…]
}
BASECHAIN_1D
{
[…]
}
BASEPLANE_2D
{
[…]
}
BASECHAIN_1D
{
[…]
}
}
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2.3
PARALLEL
PARALLEL
compulsory
repeatable
Condition:
-
Super Block
DOMAIN
Description:
This block defines parameters concerning the parallel execution of BASEMENT. It has to be
defined only for simulations on multi-core shared memory systems when multiple cores
shall be used.
Inner Blocks
-
Variable
type
comp
Rep
values
number_threads
int
No
No
Default : 1
•
condition
desc
number_threads
The number of threads usually equals the number of used cores, although the final
mapping of the threads on the cores is managed by the operating system.
- If the number of threads is set to 1 (default), than the simulation is
performed sequentially.
- If the number of threads n > 1 than the simulation runs in parallel on n
threads.
- If the number of threads is set to zero or a negative value, it is automatically
set to the number of available cores on the system.
Example:
PARALLEL
{
number_threads = 2
}
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2.4
PHYSICAL_PROPERTIES
PHYSICAL_PROPERTIES
compulsory
repeatable
Condition:
-
Super Block
DOMAIN
Description:
Some universally valid properties for all computational areas to follow are defined here.
Inner Blocks
-
Variable
type
comp
Rep
values
gravity
real
No
No
Per default: 9.81
condition
desc
Acceleration
due to gravity
[m/s2]
viscosity
real
No
No
Per default: 0.0001
Kinematic
viscosity [m2/s]
rho_fluid
string
No
No
Per default: 1000
-
Fluid density
[kg/m3]
rho_sediment
string
No
No
Per default: 2650
-
Sediment
density [kg/m3]
Example:
PHYSICAL_PROPERTIES
{
gravity = 9.81
viscosity = 0.0002
rho_fluid = 1000
rho_sediment = 2650
}
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2.5
BASECHAIN_1D: core block for one dimensional simulations
BASECHAIN_1D
compulsory
repeatable
Condition:
Super Block
DOMAIN
Description:
This block includes all information for a 1D simulation
Inner Blocks
GEOMETRY
-
HYDRAULICS
MORPHOLOGY
OUTPUT
Variable
type
comp
rep
values
condition
region_name
string
Yes
No
One word
-
desc
The main block declaring a 1D simulation is called BASECHAIN_1D within the DOMAINblock. It contains all information about the precise simulation. To ensure a homogeneous
input structure, the sub blocks for BASECHAIN_1D and BASEPLANE_2D are roughly the
same.
A first distinction is made for geometry, hydraulic and morphologic data as the user is not
always interested in both results of fluvial and sedimentologic simulation. The geometry
however is independent of the choice of calculation. Inside the two blocks (hydraulics and
morphology), the structure covers the whole field of numerical and mathematical
information that is required for every computation.
These blocks deal with boundary conditions, initial conditions, sources and sinks, numerical
parameters and settings and output-specifications. Be aware of a slightly different behaviour
of the sub blocks dependent on the choice of 1D or 2D calculation.
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Example:
DOMAIN
{
multiregion = Rhein
PHYSICAL_PROPERTIES
{
[…]
}
BASECHAIN_1D
{
region_name = Alpenrhein
GEOMETRY
{
[…]
}
HYDRAULICS
{
[…]
}
MORPHOLOGY
{
[…]
}
OUTPUT
{
[…]
}
}
}
RIV - 2.5-2
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2.5.1
GEOMETRY
GEOMETRY
compulsory
repeatable
Super Blocks
BASECHAIN_1D < DOMAIN
Condition:
-
Description:
Defines the problems topography using different input formats.
Variable
type
comp
rep
values
condition
geometry_type
enum
Yes
No
{floris,hecras}
-
geofile
string
Yes
No
-
desc
Name of the
topography
file
cross_section_order
string list
Yes
No
( cs1 cs2 …
BASECHAIN_1D
… csx )
where csX
are defined
in the input
file.
bottom_file
string
No
No
BASECHAIN_1D
Name of the
file which
contains the
bottom
information
The cross section order is only used for 1D geometry. Within the input file, all crossstrings
are defined. However, you might want to use only some of these cross strings or change
their order. The active cross stings are then defined in the string cross section order.
Attention: The first string in the list will be treated internally as upstream and the last string
will be the downstream. This will be of further use with the boundary conditions.
Example:
BASECHAIN_1D
{
region_name = alpine_rhine
GEOMETRY
{
geometry_type = floris
geofile = RheinTopo.txt
cross_section_order = ( QP84.000 QP84.200 QP84.400
QP84.600 QP85.800 )
}
}
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2.5.2
HYDRAULICS
HYDRAULICS
compulsory
repeatable
Super
Condition:
-
Blocks
Description:
Defines the hydraulic specifications for the simulation. This block contains only further sub
blocks but no own variables.
Inner Blocks
BOUNDARY
INITIAL
FRICTION
SOURCE
PARAMETER
Variable
type
comp
rep
-
-
-
-
values
condition
desc
Example:
HYDRAULICS
{
BOUNDARY
{
[…]
}
INITIAL
{
[…]
}
FRICTION
{
[…]
}
SOURCE
{
[…]
}
PARAMETER
{
[…]
}
}
RIV - 2.5-4
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2.5.2.1
PARAMETER
PARAMETER
compulsory
repeatable
Super Blocks
HYDRAULICS < BASECHAIN_1D < DOMAIN
Condition:
-
Description:
This block contains some few numerical variables for hydraulic simulations.
Inner Blocks
SECTION_COMPUTATION
Variable
type
comp
rep
values
condition
desc
initial_time_step
real
Yes
no
[s]
HYDRAULICS
Time step used for the
determination
of
the
computational time step.
Must be clearly
than
the
larger
biggest
computed time step
minimum_water_depth
real
Yes
no
[m]
HYDRAULICS
CFL
real
Yes
no
0.5
<
CFL
<
Below this value, a cell is
treated as dry.
HYDRAULICS
CFL
(Courant-Friedrichs-
Lewy) number. Influences
1.0
the
stability
of
the
computation (value 0.5-1).
It should be as close as
possible
to
1,
and
diminished only in case of
stability
problems.
This
leads to an increase of
numerical diffusion..
total_run_time
real
Yes
No
[s]
HYDRAULICS
Defines the duration of
simulation_scheme
string
No
No
explicit(
HYDRAULICS
Defines if an implicit or
the simulation.
default)
explicit
implicit
executed.
computation
computations
is
Implicit
only
for
completely wet channels
and not strongly unsteady
conditions.
theta
real
No
No
0.5-1
implicit
1 is fully implicit, 0.5
accounts for one half the
solution of the previous
time step. Lower values
of theta tend to generate
instability, but if they do
not, they can give faster
convergence.
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riemann_solver
string
roe
The
use
of
the
roe
(default),
scheme is recommended.
upwind
truncation_interval_area
real
No
No
Default =
implicit
Indicates the precision of
the values of the areas for
0.001
which the iterations are
interrupted.
truncation_interval_discharge
real
No
No
Default =
implicit
0.001
Indicates the precision of
the
values
of
the
discharges for which the
iterations are interrupted.
iteration_number
int
No
No
Default =
implicit
10
Number
of
allowed
iterations. If at this point
the truncation interval is
exceeded, a message is
written in the implicit_log
file and the time step can
be reduced.
try-limit
int
No
No
Default =
implicit
0
This
value
determines
how many time the time
step can be reduced if the
asked truncation intervals
are not reached within the
allowed
number
of
iterations.
time_step_reduction
real
No
No
default
0.5
=
Inplicit,
If try_limit is > 0, the time
try_limi
step is multiplied with this
t>0
value.
Examples:
HYDRAULICS
{
[…]
PARAMETER
{
total_run_time = 10000.0
initial_time_step = 10.0
CFL = 0.7
minimum_water_depth = 0.0001
}
[…]
}
RIV - 2.5-6
VAW ETH Zürich
Version 10/28/2009
HYDRAULICS
{
[…]
PARAMETER
{
total_run_time = 600
CFL = 0.98
simulation_scheme = implicit
riemann_solver = roe
theta = 1
truncation_interval_area = 0.01
truncation_interval_discharge = 0.01
initial_time_step = 600
iteration_number = 8
try_limit = 5
time_step_reduction = 0.2
}
}
VAW ETH Zürich
Version 10/28/2009
RIV - 2.5-7
2.5.2.1.1
SECTION_COMPUTATION
SECTION_COMPUTATION
compulsory
repeatable
Super Blocks
PARAMETER < HYDRAULICS < BASECHAIN_1D < DOMAIN
Description:
Specifies the mode of section computation and the necessary parameters.
Variable
type
comp
rep
values
type
string
Yes
No
Iteration,
Type
table
computation
number_of_
int
Condition:
Yes
No
BASECHAIN_1D
condition
iteration
desc
of
After
this
cross
section
maximum
parameter
number
of
iterations the determination of WSE
iterations
from A is interrupted if the precision has
not been reached
precision
real
Yes
No
2
[m ]
iteration
The
maximum
difference
between
given and calculated A in iterative
determination of WSE.
table_name
string
Yes
No
table
Name of the file in which the table is
stored.
minInterval
real
Yes
No
[m]
table
Minimum interval between WSE for
table calculation.
maxInterval
real
Yes
No
[m]
table
Maximum interval between WSE for
table calculation.
The type of cross section computation can also be defined in a “floris”-type topography file.
Example:
PARAMETER
{
…
SECTION_COMPUTATION
{
type = table¨
table_name = RheinTable
minInterval = 0.05
maxInterval = 0.2
}
}
RIV - 2.5-8
VAW ETH Zürich
Version 10/28/2009
2.5.2.2
FRICTION
FRICTION
compulsory
repeatable
Condition:
-
Super Blocks
Description:
The FRICTION block deals with all influences on the Source term in the equations
concerning the friction
Inner Blocks
Variable
type
comp
rep
values
condition
desc
type
string
No
No
strickler,
BASECHAIN_1D
Friction law:
chezy,
strickler:Strickler
darcy,
with Strickler value.
manning
chezy: Chézy.
darcy:
Darcy-
Weissbach
manning: Manning
default_friction
Real
Yes
No
grain_size_friction
enum
No
No
{ on, off }
BEDMATERIAL
If this is “on” the
friction value of the
bottom
will
computed
be
as
a
function of the grain
size. Default is “off”.
strickler_factor
real
No
No
BEDMATERIAL
Factor
for
the
computation
friction
value
of
from
grain diameter:
k Str = factor /
6
d 90
Default value 21.1.
Example:
HYDRAULICS
{
[…]
FRICTION
{
default_friction = strickler
default_friction = 30
grain_size_friction = off
}
[…]
}
VAW ETH Zürich
Version 10/28/2009
RIV - 2.5-9
2.5.2.3
BOUNDARY
BOUNDARY
compulsory
repeatable
Super Blocks
Condition:
-
Description:
Defines boundary conditions for hydraulics. If no boundary condition is specified for
part of or the whole boundary, an infinite wall is assumed. For every different
condition, a new block BOUNDARY has to be used.
Inner Blocks
Variable
type
comp
rep
values
condition
boundary_type
string
Yes
No
{weir,
HYDRAULICS,
hqrelation, wall,
BASECHAIN_1D
desc
gate,
zero_gradient,
hydrograph,
zhydrograph}
boundary_string
string
Yes
No
{upstream,
Upstream and
downstream}
downstream are
defined by the
first resp. last
entry in the cross
section order list
for 1D geometry
boundary_file
string
No
No
File type “BC”
hydrograph, weir,
*.txt file
hqrelation, gate,
zhydrograph
precision
real
No
No
[m3/2]
hydrograph
For iterative
determination of
the wetted area
corresponding to
a discharge:
accepted
difference
between given
and calculated
discharge.
number_of_iterations
integer
No
No
hydrograph
Maximum
number of
iterations allowed
for the iterative
determination of
the wetted area,
before the
computation is
RIV - 2.5-10
VAW ETH Zürich
Version 10/28/2009
interrupted.
level
real
No
No
[m rel.]
gate, weir
For constant level
of weir or the
bottom part of
gate
width
real
No
No
[m abs.]
gate, weir
Width of the weir
crest.
embankement_slope
real
no
No
weir
Slope of the weir
embankments
(the weir must be
symmetric).
poleni_factor
real
No
No
gate, weir
Weir factor
according to
Poleni
Default: 0.65
contraction_factor
real
No
No
gate
Contraction factor
reduction_factor
real
No
No
gate
Reduction factor
of the gate.
for weir (if the
water level is
lower than the
bottom line of the
gate a weir
computation is
executed)
gate_elevation
real
No
No
[m rel.]
gate
gate_file
string
No
No
File type “BC”
gate
For constant level
of gate
[m rel.]
For timedependent
behaviour of the
gate_elevation
level_file
string
No
No
File type “BC”
gate,weir
[m rel.]
For timedependent
behaviour of weir
or lower part of
the gate
slope
real
No
No
[‰]
hydrograph,
Slope of the cross
hqrelation
section to which
the boundary
condition belongs.
VAW ETH Zürich
Version 10/28/2009
RIV - 2.5-11
Boundary types for 1D simulations:
•
hydrograph: This boundary conditions for the upstream is a hydrograph describing
the total discharge in time over a cross-string. It requires a boundary file with two
columns: time and corresponding discharge. You don’t have to specify the
discharge for each timestep – values will be interpolated between the defined times
in the input file. However, you need at least 2 discharges in time – at the beginning
and in the end. If the duration of the simulation is longer than the given hydrograph,
the last given discharge will be kept for the rest of the simulation. If the discharge
has to become 0, this must be indicated explicitly! The interpolation of the values is
linear. If the slope of the cross section is not defined in the topography file, it must
be defined here.
•
zero_gradient: This boundary condition is only used for downstream conditions.
Basically, the main variables within the last computational cell remain are constant
over the whole element. There are no further variables needed.
•
weir: This is another outflow boundary condition. It requires the definition of its
width, embankement_slope, poleni_factor and its level. For a constant level use
level, for changing level in time, use a level_file of type “BC” with two columns:
time and corresponding level. Values are interpolated in time automatically.
Data for the use of a weir as boundary condition:
2
Q = bµ 2 g H 2 / 3
3
Q : discharge;
b : width of the weir;
µ : contraction or weir factor;
H : level difference between weir crest and upstream energy level.
•
wall: This is the default boundary condition for all nodes where no other boundary
was specified. The wall is of infinite height – there is no flux across the element
borders. There are no further variables needed.
•
gate1D: A gate is like an inverted weir – hanging from the top down into the water.
It is used as a special downstream boundary condition in 1D simulations.
Compulsory variables are width, contraction_factor, reduction_factor, poleni_factor,
gate_elevation and level. Level indicates the level of the bottom floor,
gate_elevation the level of the gate. For changing level and gate elevation in time,
use the level_file and gate_file of type “BC” with two columns: time and
corresponding level. Values are interpolated in time automatically.
RIV - 2.5-12
VAW ETH Zürich
Version 10/28/2009
Following parameters are for the use of a hydraulic gate as lower boundary
condition. The flux is computed with:
=
Q bµ a 2 g ( H − µ a )
Q : discharge;
b : gate width;
a : height of gate aperture;
µ : contraction factor;
H : upstream energy level.
•
hqrelation: Knowing the relation between water depth and discharge, this
downstream boundary condition takes the discharge corresponding to the
computed water depth. The relation between h and q can be manually defined in a
boundary file of type “BC”. Values are interpolated. If no boundary file is specified,
the hq-relation is computed internally due to the given topography. In this case, if
the slope is not defined in the topography file it has to be defined here.
•
zhydrograph: Another downstream boundary condition is the behaviour of the
Water Surface Elevation in time. Time and corresponding WSE have to be defined in
the boundary_file of type “BC”.
Example:
HYDRAULICS
{
BOUNDARY
{
boundary_type = hydrograph
boundary_string = upstream
boundary_file = discharge_in_time.txt
precision = 0.001
number_of_iterations = 50
}
BOUNDARY
{
boundary_type = weir
boundary_string = downstream
level_file = weirlevel_in_time.txt
// for constant level in time use e.g. level = 4.2
width = 13.5
embankement_slope = 0.04
poleni_factor = 1.0
}
[…]
}
VAW ETH Zürich
Version 10/28/2009
RIV - 2.5-13
2.5.2.4
INITIAL
INITIAL
compulsory
repeatable
Super
Condition:
-
Blocks
Description:
Defines initial conditions for hydraulics. If no initial condition is specified for parts or the whole
computational area, a dry condition is assumed.
Variable
type
comp
rep
values
initial_type
enum
Yes
No
{backwater, dry, fileinput}
initial_file
string
No
No
Dependent on initial type
3
condition
desc
fileinput
q_out
real
Yes
No
[m /s]
backwater
WSE_out
real
Yes
no
[m rel.]
backwater
-
Initial conditions for 1D simulations:
•
•
•
dry: This is the most simple initial condition. The whole area is initially dry – no
discharge and no WSE are needed. This is the default for every computational cell
where no initial condition is specified. No further variables are needed for this
choice of initial condition.
fileinput: In 1D, for every cross-section, the WSE and discharge can be specified as
initial condition. This choice needs an input file of type “Ini1D” with three columns:
name of the cross-string (as specified in geometry), area A (corresponds to a certain
WSE), discharge Q. This is also the restart option for 1Dsimulations as the default
output file from previous simulations can directly be used for the import of initial
conditions.
backwater: This initial condition is only available for 1D simulations. By specifying
the WSE and discharge at the downstream boundary, BASEMENT then computes
the backwater (“Staukurve”) for all cross-strings. Choosing this initial condition
requires the specification of WSE_out and q_out.
Example:
HYDRAULICS
{
[…]
INITIAL
{
initial_type = backwater
q_out = 15.6
WSE_out = 1.78
}
[…]
}
RIV - 2.5-14
VAW ETH Zürich
Version 10/28/2009
2.5.2.5
SOURCE
SOURCE
compulsory
repeatable
Super Blocks
Condition:
-
Description:
The SOURCE block deals with all influences on the Source term in the equations like
external sources. The EXTERNAL_SOURCE block is used for additional sources. In 1D,
these terms represent inflow and outflow over the lateral sides of the channel.
Inner Blocks
EXTERNAL_SOURCE
Variable
type
comp
rep
values
condition
desc
Example:
HYDRAULICS
{
[…]
SOURCE
{
EXTERNAL_SOURCE
{
[…]
}
}
[…]
}
VAW ETH Zürich
Version 10/28/2009
RIV - 2.5-15
2.5.2.5.1
EXTERNAL_SOURCE
EXTERNAL_SOURCE
compulsory
repeatable
Super Blocks
SOURCE < HYDRAULICS < BASECHAIN_1D < DOMAIN
Condition:
-
Description:
Specifies sources and sinks static or in time for parts or the whole area. The external source
block is used for additional sources, mostly used for lateral inflow and outflow. Note, that
external sources for hydraulics act between two cross-sections which have to be defined
(contrary to morphologic external sources for 1D). If no source block is specified at all, the
additional source is set to zero.
Variable
type
comp
rep
values
source_type
enum
Yes
No
[off_channel,
condition
desc
qlateral]
source_file
string
No
No
File type “BC”
qlateral
[m3/s]
cross_sections
string
No
No
BASECHAIN_1D
Cross
section
name
must
have
been
specified
in
the
geometry input file
enum
side
No
No
[left, right]
BASECHAIN_1D
Specifies
side
of
from
which
the
cross
section (seen in flow
direction) the source is
acting.
Source types for 1D simulations:
•
off_channel
This is kind of a boundary condition which allows overflow over the lateral edges of
the cross sections. This option is followed with the bounding cross_sections names.
Additionally, the side token specifies whether this occurs at the left or right side.
The behaviour of an off channel is like a lateral weir with the height of the extremal
cross string points. No additional variable is needed. The off_channel flow acts
between the two defined cross-sections.
•
qlateral
This additional source simulates in- or outflows at the right or left side of the cross
section. This option needs to be followed with the key token cross_sections with a
list of the two successive cross-sections. Additionally, the side token specifies the
location of the acting source (seen in flow direction). Finally, the source file includes
the information about the temporal behaviour of the source’s discharge. The
discharge will act between the defined two cross-sections. These cross-sections
have to be defined in the geometry block successively.
RIV - 2.5-16
VAW ETH Zürich
Version 10/28/2009
Example:
SOURCE
{
EXTERNAL_SOURCE
{
source_type = qlateral¨
cross_sections = ( QP84.200 QP84.400 )
side = right
source_file = lateral_discharge_in_time.txt
}
EXTERNAL_SOURCE
{
source_type = off_channel
side = left
}
}
VAW ETH Zürich
Version 10/28/2009
RIV - 2.5-17
2.5.3
MORPHOLOGY
MORPHOLOGY
compulsory
repeatable
Super Blocks
Condition:
-
Description:
Defines the morphologic specifications for the simulation. Contrary to hydraulics, this is not
compulsory. If Morphology is not specified, sediment transport will be inactive. Contrary to the
HYDRAULICS block, there is no INITIAL block within MORPHOLOGY. Therefore, all bed load
is initially at rest by default.
Inner Blocks
BEDMATERIAL
BEDLOAD
SUSPENDED_LOAD
PARAMETER
Variable
type
comp
rep
values
condition
desc
-
-
-
-
-
-
-
Example:
MORPHOLOGY
{
BEDMATERIAL
{
[…]
}
BEDLOAD
{
[…]
}
SUSPENDED_LOAD
{
[…]
}
PARAMETER
{
[…]
}
}
RIV - 2.5-18
VAW ETH Zürich
Version 10/28/2009
2.5.3.1
PARAMETER
PARAMETER
compulsory
repeat
Condition:
-
able
Super Blocks
MORPHOLOGY < BASECHAIN_1D < DOMAIN
Description:
The PARAMETER block for morphology in 1D contains general settings dealing with
sediment transport. It contains parameters concerning both bed load and suspended
load.
Variable
density
type
real
comp
Yes
rep
values
No
3
[kg/m ]
condition
desc
MORPHOLOGY
Density of
sediment
(recommended
value 2650).
max_dz_table
real
No
No
[m]
MORPHOLOGY
porosity
real
Yes
No
[0…100] %
MORPHOLOGY
Fraction of soil
volume which is
not occupied by
the sediment.
sediment_relocation
string
No
No
wetted, bottom,
Defines where
betweenwetted
erosion or
deposition takes
place
height_active_layer
real
No
No
MORPHOLOGY
Height of the
active layer. If this
value is set to 0,
the active layer
height will be
calculated at each
time step and is
thus variable.
The values of the parameter sediment_relocation have the following meanings:
• Wetted: erosion and deposition is limited to the wetted zones of the bottom. These
can be more than one.
• Bottom: erosion and deposition take place on the whole zone defined as bottom.
• Betweenwetted: erosion or deposition take place in one zone belonging to the
defined bottom and limited by the first and last wetted slice.
VAW ETH Zürich
Version 10/28/2009
RIV - 2.5-19
Example:
MORPHOLOGY
{
[…]
PARAMETER
{
porosity = 0.37
height_active_layer = 0.05
density = 2650
[…]
}
[…]
}
RIV - 2.5-20
VAW ETH Zürich
Version 10/28/2009
2.5.3.2
BEDMATERIAL
BEDMATERIAL
compulsory
repeatable
Super Blocks
Condition:
-
Description:
This block defines the characteristics of the bed material. The GRAIN_CLASS specifies
the relevant diameters of the grain. The MIXTURE block allows producing different grain
mixtures with the defined diameters. The SOIL_DEF block defines sub layers and the
active layer. Finally, the SOIL_ASSIGNMENT block is used to assign the different soil
definitions to their corresponding areas in the topography.
Inner Blocks
GRAIN_CLASS
MIXTURE
SOIL_DEF
SOIL_ASSIGNMENT
Variable
type
comp
rep
values
condition
desc
-
-
-
-
-
-
-
Example:
MORPHOLOGY
{
BEDMATERIAL
{
GRAIN_CLASS
{
[…]
}
MIXTURE
{
[…]
}
MIXTURE
{
[…]
}
SOIL_DEF
{
[…]
}
SOIL_DEF
{
[…]
}
SOIL_ASSIGNMENT
{
[…]
}
}
}
VAW ETH Zürich
Version 10/28/2009
RIV - 2.5-21
2.5.3.2.1
GRAIN_CLASS
GRAIN_CLASS
compulsory
repeatable
Super
BEDMATERIAL < MORPHOLOGY < BASECHAIN_1D < DOMAIN
Condition:
-
Blocks
Description:
This block defines the mean diameters of ALL possibly occurring grains of the bed material in
millimetres. The number of diameters in the list automatically sets the number of grains. For
“single grain simulations”, only one diameter must be defined.
The grain sizes must be entered in ascending order from the smallest grains to the largest
grains!
Variable
type
comp
rep
values
diameters
list of reals
yes
no
[mm]
condition
desc
Example:
BEDMATERIAL
{
GRAIN_CLASS
{
// simulation with 5 grain sizes
diameters = ( 5 21 46 70 150 )
}
[…]
}
BEDMATERIAL
{
GRAIN_CLASS
{
// simulation with 1 grain size
diameters = ( 46 )
}
[…]
}
RIV - 2.5-22
VAW ETH Zürich
Version 10/28/2009
2.5.3.2.2
MIXTURE
MIXTURE
compulsory
repeatable
Super Blocks
Condition:
-
Description:
This block allows the definition of different grain mixtures. According to the number of grain
diameters from the GRAIN_CLASS, for each class, the percental fraction of volume has to
be specified using a value between 0 and 100 %. The fractions must be written in the order
corresponding to the grain class list. If a grain class is not present in a mixture, its fraction
value has to be set to 0.
The mixture_name is used later for easy-to-use assignments.
As the MIXTURE block is repeatable, several mixtures can be defined. However, this makes
only sense for multi grain size simulations.
In case of a single grain simulation (only one mean diameter in GRAIN_CLASS), the
MIXTURE block must be defined with ‘volume_fraction = ( 100 )’.
Variable
type
comp
rep
mixture_name
string
Yes
No
volume_fraction
list of reals
Yes
No
values
condition
desc
[0 to 100] %
Example:
BEDMATERIAL
{
GRAIN_CLASS
{
diameters = ( 5 21 46 )
}
MIXTURE
{
mixture_name = mixture1
volume_fraction = ( 30 50 20 )
}
MIXTURE
{
}
[…]
}
VAW ETH Zürich
Version 10/28/2009
RIV - 2.5-23
2.5.3.2.3
SOIL_DEF
SOIL_DEF
compulsory
repeatable
Super Blocks
Condition:
-
Description:
This block defines the sub-structure of the soil. At the top is the active layer. Every
further layer is defined within a SUBLAYER block. Every layer has an own mixture
assigned.
The definition of d90_armored_layer is only valid for single grain size simulations. If
d90_armored_layer is set to 0 or undefined, no armoured layer is being used.
Unlike in 2D simulations, for single grain size simulations, one SUBLAYER block is
needed to define the fixed bed by its bottom elevation.
If no SUBLAYER block is defined within a SOIL_DEF, a fixed bed without sub layer on
top is assumed at this place. The SUBLAYER blocks have to be defined ordered by
their relative distance to the surface!
Inner Blocks
SUBLAYER
Variable
type
comp
rep
soil_name
string
yes
no
active_layer_height
string
yes
no
active_layer_mixture
string
yes
no
values
condition
desc
[m]
Mixture
defined
d90_armored_layer
real
no
no
[mm]
no
2
Single grain
simulation
tau_erosion_start
RIV - 2.5-24
real
no
[N/m ]
VAW ETH Zürich
Version 10/28/2009
Example:
BEDMATERIAL
{
GRAIN_CLASS
{
[…]
}
MIXTURE
{
[…]
}
MIXTURE
{
[…]
}
SOIL_DEF
{
soil_name = soil1
active_layer_height = 0.3
active_layer_mixture = mixture1
tau_erosion_start = 0.89
SUBLAYER
{
[…]
}
SUBLAYER
{
[…]
}
}
SOIL_DEF
{
soil_name = soil2
d90_armored_layer = 6.8
SUBLAYER
{
[…]
}
SUBLAYER
{
[…]
}
}
[…]
}
VAW ETH Zürich
Version 10/28/2009
RIV - 2.5-25
2.5.3.2.3.1
SUBLAYER
SUBLAYER
compulsory
repeatable
Super Blocks
SOIL_DEF < BEDMATERIAL < MORPHOLOGY < BASECHAIN_1D < DOMAIN
Condition:
-
Description:
This block defines a sub layer. For every sub layer, a new block has to be used. In 1D, the
bottom elevation is relative to the surface topography defined in the geometry file.
Therefore, a minus-sign has to be used. In 2D however, the bottom elevation is meant to
be an absolute height!
The SUBLAYER blocks have to be defined ordered by their relative height to the surface.
In 1D the bottom of the lowest sub layer automatically defines the fixed bed. Unlike in 2D,
there is no FIXED_BED block needed.
In case of a single grain simulation (only one mean diameter in GRAIN_CLASS), only one
SUBLAYER block is needed, which automatically defines the fixed bed.
If no SUBLAYER block is defined within a SOIL_DEF block, a fixed bed without sub layer
on top is assumed at this place.
Variable
type
comp
rep
values
bottom_elevation
real
Yes
No
[m]
sublayer_mixture
string
No
No
condition
desc
Relative to surface
Mixture
defined
Example:
SOIL_DEF
{
soil_name = soil1
tau_erosion_start = 0.89
SUBLAYER
{
bottom_elevation = -0.5
sublayer_mixture = mixture1
}
SUBLAYER
{
bottom_elevation = -1.3
sublayer_mixture = mixture2
}
SUBLAYER
{
[…]
}
}
RIV - 2.5-26
// first sub layer
// second sub layer
VAW ETH Zürich
Version 10/28/2009
2.5.3.2.4
SOIL_ASSIGNMENT
SOIL_ASSIGNMENT
compulsory
repeatable
Super Blocks
Condition:
-
Description:
This block is finally used to assign the soil definitions to their corresponding areas in the
topography. For this purpose, the index-technique is used. In the geometry input file,
every cross-section (and also parts of a cross-section) can be labelled with an integer
index number which is used here to attach a soil definition.
For single grain size simulations, a soil assignment is still needed, although only one
mixture exists.
Inner Blocks
Variable
type
comp
rep
values
condition
input_type
string
Yes
No
[index_table]
index
list of integers
Yes
No
index_table
soil
list of reals
Yes
No
index_table
desc
Example:
BEDMATERIAL
{
[…]
SOIL_DEF
{
[…]
}
SOIL_DEF
{
[…]
}
SOIL_ASSIGNMENT
{
input_type = index_table
index = ( 1 2 4 6 )
soil = ( soil1 soil2 soil3
}
}
VAW ETH Zürich
Version 10/28/2009
soil1
)
RIV - 2.5-27
2.5.3.3
BEDLOAD
BEDLOAD
compulsory
repeatable
Super
Condition:
-
Blocks
Description:
Defines all needed data for bed load transport. If this block does not exist, then no bed load
transport will be simulated.
Inner Blocks
BOUNDARY
SOURCE
PARAMETER
Variable
type
comp
rep
values
condition
desc
-
-
-
-
-
-
-
Example:
BEDLOAD
{
[…]
BOUNDARY
{
[…]
}
BOUNDARY
{
[…]
}
SOURCE
{
[…]
}
PARAMETER
{
[…]
}
}
RIV - 2.5-28
VAW ETH Zürich
Version 10/28/2009
2.5.3.3.1
PARAMETER
PARAMETER
compulsory
repeatable
Super Blocks
BEDLOAD < MORPHOLOGY < BASECHAIN_1D < DOMAIN
Condition:
-
Description:
The PARAMETER block for bed load in 1D contains general settings dealing with bed load
transport.
Variable
type
comp
rep
values
condition
bedload_transport
String
Yes
No
{MPM,
BEDLOAD
desc
MPMH,
parker,
powerlaw,
rickenmann,
smartjaeggi}
factor1
real
No
No
factor2
real
No
No
d90
real
if 1k
MPM,
powerlaw
powerlaw
[mm]
Rickenmann,
smartjaeggi
d30
real
if 1k
upwind
real
Yes
[mm]
Rickenmann,
smartjaeggi
No
[0.5 … 1]
BEDLOAD
Defines which part of the
sediment flux on the edge
is taken from the
upstream cross-section.
Recommended 0.5 - 1,
depending on the stability
of the computation!
angle_of_repose
real
Yes
No
theta_critic
real
No
No
[degree]
BEDLOAD
Default value 30
MPM,
This value is taken instead
MPMH,
of the critical shear stress
rickenmann
from
the
diagram
of
Shields.
bed_forms
real
No
No
0.8-0.85
MPM,
Factor applied in case of
MPMH,
bed forms (RI 2.1).
rickenmann
Further descriptions:
• factor1: used for MPM: factor of MPM formula (eq. UIII-3.5), usually set to 8
used for power law: α, a calibration value (eq. UIII-3.6)
• factor2: used for power law: β, a calibration value (eq. UIII-3.6)
VAW ETH Zürich
Version 10/28/2009
RIV - 2.5-29
Example:
BEDLOAD
{
[…]
PARAMETER
{
bedload_transport = parker
upwind = 0.8
critical_angle = 35
}
}
BEDLOAD
{
[…]
PARAMETER
{
bedload_transport = MPM
factor1 = 8
upwind = 1
critical_angle = 30
}
}
RIV - 2.5-30
VAW ETH Zürich
Version 10/28/2009
2.5.3.3.2
BOUNDARY
BOUNDARY
compulsory
repeatable
Super Blocks
Condition:
-
Description:
Defines boundary conditions for bed load. If no boundary condition is specified for part of
or the whole boundary, an infinite wall is assumed. For every different condition, a new
block BOUNDARY has to be used. It is distinguished between upstream and downstream
boundary conditions. At the upstream, inflow conditions are valuable. They need a
mixture assigned in any case of multi grain size or single grain size simulations.
Downstream boundary conditions are for outflow – they don’t need any mixture assigned.
Variable
type
comp
rep
values
condition
boundary_type
enum
Yes
No
{sediment_discharge,
BEDLOAD
boundary_string
string
Yes
No
desc
IOUp, IODown, wall}
{upstream,
string name defined
downstream}
Upstream
and
downstream
are defined
by the first
resp. last
entry in the
string order
list for 1D
geometry
boundary_file
string
No
No
boundary_mixture
string
No
No
File type “BC”
sediment_discharge
upstream,
mixture
defined
•
sediment_discharge
This boundary condition is based on a sediment-“hydrograph” describing the
bedload inflow at the upstream bound in time. The behaviour in time is specified in
the boundary file with two columns: time and corresponding sediment discharge.
The values are interpolated in time. Additionally, the boundary_mixture must be set
to a predefined mixture name.
•
IOUp
This upstream boundary condition grants a constant bed load inflow. Basically, the
same amount of sediment leaving the first computational cell in flow direction
enters the cell from the upstream bound. This is useful to determine the equilibrium
flux. There is no boundary_mixture needed. Sediment inflow is determined by the
flows transport capability.
VAW ETH Zürich
Version 10/28/2009
RIV - 2.5-31
•
IODown
The only downstream boundary condition available for sediment transport
corresponds to the definition of IOup. All sediment entering the last computational
cell will leave the cell over the downstream boundary. There is no further
information needed for this case. If you don’t specify any downstream boundary
condition, no transport over the downstream bound will occur – a wall is assumed.
•
wall
This boundary condition can be used for upstream and downstream boundaries.
There is no sediment flow across these boundaries. This is the default setting if no
BOUNDARY block is defined.
Example:
BEDLOAD
{
[…]
BOUNDARY
{
boundary_type = sediment_discharge
boundary_file = upstream_discharge.txt
boundary_mixture = mixture2
}
BOUNDARY
{
boundary_type = IODown
}
[…]
}
RIV - 2.5-32
VAW ETH Zürich
Version 10/28/2009
2.5.3.3.3
SOURCE
SOURCE
compulsory
repeatable
Super Blocks
Condition:
-
Description:
The SOURCE block for bed load in 1D contains only external sources for lateral bed
load inflow. For each such external source, a new inner EXTERNAL_SOURCE block
has to be defined.
Inner Blocks
EXTERNAL_SOURCE
Variable
type
comp
rep
values
-
-
-
-
-
condition
desc
Example:
BEDLOAD
{
[…]
SOURCE
{
EXTERNAL_SOURCE
{
[…]
}
EXTERNAL_SOURCE
{
[…]
}
}
[…]
}
VAW ETH Zürich
Version 10/28/2009
RIV - 2.5-33
2.5.3.3.3.1
EXTERNAL_SOURCE
EXTERNAL_SOURCE
compulsory
repeatable
Super Blocks
SOURCE < BEDLOAD < MORPHOLOGY < BASECHAIN_1D < DOMAIN
Condition:
-
Description:
The external source block is used for additional lateral sources beneath the boundary
conditions, e.g. unloading of bed load or lateral sediment inflow. Note, that a external
sources for bed load acts exactly on one cross-section (contrary to hydraulic external sources
for 1D). An input file for sediment discharge can be used for the definition of inflow and
outflow (sink / negative discharge).
The source_mixture has to be specified in any case – also if the source_file defines a pure
sink.
If no source block is specified at all, the additional source is set to zero.
Inner Blocks
Variable
type
comp
rep
values
source_type
enum
Yes
No
sediment_discharge
source_file
String
Yes
No
File type “BC”
cross_section
string
Yes
No
source_mixture
string
Yes
No
condition
desc
Mixture
defined
Example:
BEDLOAD
{
[…]
SOURCE
{
EXTERNAL_SOURCE
{
source_type = sediment_discharge
source_file = source_in_time.txt
cross_section = QP84.200
source_mixture = mixture2
}
EXTERNAL_SOURCE
{
source_type = sediment_discharge
source_file = another_source_in_time.txt
}
}
[…]
}
RIV - 2.5-34
VAW ETH Zürich
Version 10/28/2009
2.5.3.4
SUSPENDED_LOAD
SUSPENDED_LOAD
compulsory
repeatable
Super
Condition:
MORPHOLOGY
Blocks
Description:
Defines all needed data for suspended transport. If this block does not exist, then no suspended
transport will be simulated.
Inner Blocks
BOUNDARY
PARAMETER
INITIAL
SOURCE
Variable
type
comp
rep
values
condition
desc
-
-
-
-
-
-
-
Example:
SUSPENDED_LOAD
{
[…]
INITIAL
{
[…]
}
BOUNDARY
{
[…]
}
BOUNDARY
{
[…]
}
PARAMETER
{
[…]
}
SOURCE
{
[…]
}
}
VAW ETH Zürich
Version 10/28/2009
RIV - 2.5-35
2.5.3.4.1
PARAMETER
PARAMETER
compulsory
repeatable
Super Blocks
SUSPENDED_LOAD < MORPHOLOGY < BASECHAIN_1D < DOMAIN
Condition:
-
Description:
The PARAMETER block for suspended load in 1D contains general settings dealing with
suspended transport.
Variable
type
comp
scheme
string
Yes
rep
values
condition
desc
{quick,
PARAMETER
The numerical scheme used
quickest,
for advection computation.
mdpm,
“mdpm” is recommended.
HollyPreiss}
diffusion_factor
real
No
No
PARAMETER
The factor representing the
eddy diffusivity Γ can be
given here. If it’s not, it is
computed by the program.
sediment_exchange
real
Yes
No
{on, off}
PARAMETE
Defines if there is exchange
of sediment with the river
bed
(deposition/entrainment)
Example:
SUSPENDED_LOAD
{
[…]
PARAMETER
{
scheme = mdpm
diffusion_factor = 0.1
sediment_exchange = on
}
[…]
}
RIV - 2.5-36
VAW ETH Zürich
Version 10/28/2009
2.5.3.4.2
BOUNDARY
BOUNDARY
compulsory
repeatable
Super Blocks
Condition:
-
Description:
Defines boundary conditions for suspended load. If no boundary condition is specified for
part of or the whole boundary, an infinite wall is assumed. For every different condition, a
new block BOUNDARY has to be used. It is distinguished between upstream and
downstream boundary conditions. At the upstream, inflow conditions are valuable. They
need a mixture assigned in any case of multi grain size or single grain size simulations.
Downstream boundary conditions are for outflow – they don’t need any mixture assigned.
Variable
type
comp
rep
values
condition
desc
boundary_type
enum
Yes
No
{suspension_discharge,
SUSPENDED_LOAD
boundary_string
string
Yes
No
out_down, wall}
{upstream,
string name defined
downstream}
Upstream
and
downstream
are defined
by the first
resp. last
entry in the
string order
list for 1D
geometry
boundary_file
string
No
No
boundary_mixture
string
No
No
File type “BC”
sediment_discharge
upstream,
mixture
defined
•
suspension_discharge
This boundary condition is based on a sediment-“hydrograph” describing the
suspended inflow at the upstream boundary in time. The behaviour in time is
specified in the boundary file with two columns: time and corresponding
concentration. The values are interpolated in time. Additionally, the
boundary_mixture must be set to a predefined mixture name.
•
out_down
The concentration of sediment in the last cell is multiplied with the hydraulic outflow
to determine the suspended sediment outflow.
VAW ETH Zürich
Version 10/28/2009
RIV - 2.5-37
•
wall
This boundary condition can be used for upstream and downstream boundaries.
There is no sediment flow across these boundaries. This is the default setting if no
BOUNDARY block is defined.
Example:
SUSPENDED_LOAD
{
[…]
BOUNDARY
{
boundary_type = suspension_discharge
boundary_file = cVar7.txt
boundary_mixture = unique
}
BOUNDARY
{
boundary_type = out_down
}
[…]
}
RIV - 2.5-38
VAW ETH Zürich
Version 10/28/2009
2.5.3.4.3
INITIAL
INITIAL
compulsory
repeatable
Super Blocks
Condition:
-
Description:
Unlike for bed load, for suspended load it is necessary to specify initial conditions.
The initial concentrations have to be defined for each cross section. This is done in a
separate file, similar to the initial condition file for hydraulics.
Variable
type
comp
rep
initial_concentration_file
string
Yes
No
values
condition
desc
fileinput
Example:
SUSPENDED_LOAD
{
[…]
INITIAL
{
initial_concentration_file = initialC.txt
}
[…]
}
VAW ETH Zürich
Version 10/28/2009
RIV - 2.5-39
2.5.3.4.4
SOURCE
SOURCE
compulsory
repeatable
Super Blocks
Condition:
-
Description:
The SOURCE block for suspended load in 1D contains only external sources for lateral
suspended load inflow. For each such external source, a new inner
EXTERNAL_SOURCE block has to be defined.
Inner Blocks
EXTERNAL_SOURCE
Variable
type
comp
rep
values
-
-
-
-
-
condition
desc
Example:
SUSPENDED_LOAD
{
[…]
SOURCE
{
EXTERNAL_SOURCE
{
[…]
}
EXTERNAL_SOURCE
{
[…]
}
}
[…]
}
RIV - 2.5-40
VAW ETH Zürich
Version 10/28/2009
2.5.3.4.4.1
EXTERNAL_SOURCE
EXTERNAL_SOURCE
compulsory
repeatable
Super Blocks
SOURCE < SUSPENDED_LOAD < MORPHOLOGY < BASECHAIN_1D < DOMAIN
Condition:
-
Description:
The external source block is used for additional lateral sources beneath the boundary
conditions. Note, that an external sources for suspended load acts exactly on one crosssection (contrary to hydraulic external sources for 1D). An input file for sediment discharge
can be used for the definition of inflow and outflow (sink / negative discharge).
The source_mixture has to be specified in any case – also if the source_file defines a pure
sink.
If no source block is specified at all, the additional source is set to zero.
Inner Blocks
Variable
type
comp
rep
values
source_type
enum
Yes
No
suspended_discharge
source_file
String
Yes
No
File type “BC”
cross_section
string
Yes
No
source_mixture
string
Yes
No
condition
desc
Mixture
defined
Example:
SUSPENDED_LOAD
{
[…]
SOURCE
{
EXTERNAL_SOURCE
{
source_type = suspended_discharge
source_file = source_in_time.txt
}
}
[…]
}
VAW ETH Zürich
Version 10/28/2009
RIV - 2.5-41
2.5.4
OUTPUT
OUTPUT
compulsory
repeatable
Super Blocks
Condition:
-
Description:
This block specifies the desired output data. In 1D, a default output file contains all
information about geometry and other variables along the flow direction. It is written
according to the output time step setting and can be used for continuation simulations.
For temporal outputs, one has to specify a SPECIAL_OUTPUT where several variables
are listed versus the time for a chosen cross-section. Additionally, it is possible to extract
minimal and maximal values or the sum of variables over the time.
Inner Blocks
SPECIAL_OUTPUT
Variable
type
comp
rep
values
output_time_step
real
Yes
No
[s]
console_time_step
real
No
No
[s]
condition
desc
If not specified, output time
step is used.
Example:
BASECHAIN_1D
{
[…]
OUTPUT
{
output_time_step = 10.0
console_time_step = 1.0
SPECIAL_OUTPUT
{
[…]
}
SPECIAL_OUTPUT
{
[…]
}
}
[…]
}
RIV - 2.5-42
VAW ETH Zürich
Version 10/28/2009
2.5.4.1
SPECIAL_OUTPUT
SPECIAL_OUTPUT
compulsory
repeatable
Super Blocks
OUTPUT < BASECHAIN_1D < DOMAIN
Condition:
-
Description:
Besides the default output, some extra output can be defined. Either one chooses the
tecplot_all output type for an exhaustive output. However, this option could produce too
much data. In this case, one may choose specifically what data for which cross-section
shall be written to an output file. This may include minimum, maximum, summation and
whole time series for the principal variables as the wetted area, discharge, sediment
discharge, water surface elevation, velocities and concentration. Each such information for
each cross-section will be written into an own file.
For graphical output of the simulation results during run-time with the BASEviz
visualization choose the BASEviz output type.
Variable
Type
comp
rep
values
type
string
Yes
No
{monitor, tecplot_all,
condition
desc
tecplot_all_bin mixtures,
delta_v_sed, delta_v_water,
transport_diagram,
delta_v_suspension, BASEviz }
real
Yes
No
cross_sections
string list
Yes
No
A
string list
No
No
{min max time}
monitor
Q
string list
No
No
{min max time sum}
monitor
Qb
string list
No
No
{min max sum}
monitor
WSE
string list
No
No
{min max time}
monitor
V
string list
No
No
{min max time}
monitor
C
string list
No
No
{min max time sum}
monitor
topology
String list
No
No
{min max time}
monitor
output_time_step
monitor
Writes the
topology to
a ascii
Tecplot file.
topology_bin
String list
No
No
{time}
monitor
Writes the
topology to
a binary
Tecplot file.
file_name
string
Yes
No
tecplot_all
Output
file
name
BASEviz:
window_interact_time
real
No
No
Should be selected as
BASEviz
Time interval for
large as possible to
interaction/refresh of the
avoid significant slow
visualization window
down of the simulation
VAW ETH Zürich
Version 10/28/2009
RIV - 2.5-43
value
string
No
No
{depth, wse, velocity,
BASEviz
Output (plot) variable
BASEviz
Enable or disable gridline
tau, bedload}
gridlines
string
No
No
{on, off}
plotting
Example:
BASECHAIN_1D
{
[…]
OUTPUT
{
SPECIAL_OUTPUT
{
type = monitor
A = ( min max )
Q = ( sum time )
Qb = ( sum )
V = ( time )
topology_bin = ( time )
}
SPECIAL_OUTPUT
{
type = monitor
cross_sections = ( QP82.100 )
A = ( time )
Q = ( sum time )
}
SPECIAL_OUTPUT
{
type = tecplot_all
file_name = tecplot_output.dat
}
SPECIAL_OUTPUT
{
type = BASEviz
window_interact_time = 200.0
variable = depth
gridlines = on
}
}
[…]
}
RIV - 2.5-44
VAW ETH Zürich
Version 10/28/2009
2.6
BASEPLANE_2D: core block for two dimensional simulations
BASEPLANE_2D
compulsory
repeatable
Condition:
Super Block
DOMAIN
Description:
This block includes all information for a 2D simulation
Inner Blocks
GEOMETRY
-
HYDRAULICS
MORPHOLOGY
OUTPUT
Variable
type
comp
rep
values
condition
region_name
string
No
No
One word
-
desc
The main block declaring a 2D simulation is called BASEPLANE_2D within the DOMAINblock. It contains all information about the precise simulation. To ensure a homogeneous
input structure, the sub blocks for BASEPLANE_2D and BASECHAIN_1D are roughly the
same.
A first distinction is made for geometry, hydraulic and morphologic data as the user is not
always interested in both results of fluvial and sedimentologic simulation. The geometry
however is independent of the choice of calculation. Inside the two blocks (hydraulics and
morphology), the structure covers the whole field of numerical and mathematical
information that is required for every computation.
These blocks deal with boundary conditions, initial conditions, friction, sources and sinks,
numerical parameters and settings and output-specifications. Be aware of a slightly different
behaviour of the sub blocks dependent on the choice of 2D or 1D calculation.
VAW ETH Zürich
Version 10/28/2009
RIV - 2.6-1
Example:
DOMAIN
{
multiregion = Rhein
PHYSICAL_PROPERTIES
{
[…]
}
BASEPLANE_2D
{
region_name = RheinDelta
GEOMETRY
{
[…]
}
HYDRAULICS
{
[…]
}
MORPHOLOGY
{
[…]
}
OUTPUT
{
[…]
}
}
}
RIV - 2.6-2
VAW ETH Zürich
Version 10/28/2009
2.6.1
GEOMETRY
GEOMETRY
compulsory
repeatable
Super Block
BASEPLAN_2D < DOMAIN
Description:
Condition:
-
Defines the problems topography using different input formats. The type of input format and the
name of the geometry input file have to be specified.
Inner
STRINGDEF
Blocks
Variable
type
comp
rep
values
condition
type
enum
Yes
No
{sms}
-
file
string
Yes
No
-
-
desc
The inner block STRINGDEF is used only for 2D simulations to define connected node
strings, e.g. for cross-string or boundary definitions. These definitions are of various uses,
mostly for the definition of boundary conditions.
A way of defining element attributes in 2D is the use of a material index. When importing an
SMS-geometry type mesh, every cell can be assigned such an index (an integer number).
This index can be used to assign the following attributes:
-
friction factor n
constant viscosity v
initial condition WSE
initial condition u
initial condition v
soil index
rock elevation (fixed bed)
The possible use of a material index is explained in the corresponding sections.
Example:
BASEPLANE_2D
{
region_name = RhineDelta
GEOMETRY
{
type = sms
file = mesh.2dm
}
[…]
}
VAW ETH Zürich
Version 10/28/2009
RIV - 2.6-3
2.6.1.1
STRINGDEF
STRINGDEF
compulsory
repeatable
Super Block
GEOMETRY < BASEPLANE_2D < DOMAIN
Condition:
BASEPLANE_2D
Description:
Defines a successive string represented by an array of node indices from the 2D-input file. Can
be used to assign node-specific stuff like boundary conditions or the definition of cross-strings for
output observation. Be aware to use continuous strings!
Variable
type
comp
rep
name
string
Yes
No
values
condition
Name for the String
desc
node_ids
integer
Yes
No
List of nodes within brackets. Must
contain at least two nodes.
file
string
No
Yes
Not implemented yet
Examples:
BASEPLANE_2D
{
region_name = RhineDelta
GEOMETRY
{
type = sms
file = mesh.2dm
STRINGDEF
{
name = cross-section1
node_ids = ( 12 22 25 72 92 )
}
STRINGDEF
{
name = boundary1
node_ids = ( 112 212 215 712 192
}
STRINGDEF
{
name = boundary2
node_ids = ( 13 23 35 73 93 )
}
}
[…]
}
RIV - 2.6-4
)
VAW ETH Zürich
Version 10/28/2009
2.6.2
HYDRAULICS
HYDRAULICS
compulsory
repeatable
Super
BASEPLANE_2D < DOMAIN
Condition:
-
Blocks
Description:
Defines the hydraulic specifications for the simulation. This block contains only further sub
blocks but no own variables.
Inner Blocks
BOUNDARY
INNER_BOUNDARY
INITIAL
SOURCE
FRICTION
PARAMETER
Variable
type
comp
rep
-
-
-
-
values
condition
desc
Example:
HYDRAULICS
{
BOUNDARY
{
[…]
}
INNER_BOUNDARY
{
[…]
}
INITIAL
{
[…]
}
SOURCE
{
[…]
}
FRICTION
{
[…]
}
PARAMETER
{
[…]
}
}
VAW ETH Zürich
Version 10/28/2009
RIV - 2.6-5
2.6.2.1
PARAMETER
PARAMETER
compulsory
repeatable
Super Blocks
HYDRAULICS < BASEPLANE_2D < DOMAIN
Condition:
-
Description:
The PARAMETER block for hydraulics is used to specify general numerical
parameters for the simulation, as time step, total run time, CFL number,
solver type etc.
Inner Blocks
Variable
type
comp
rep
values
condition
simulation_scheme
enum
Yes
No
{ exp }
HYDRAULICS
riemann_solver
enum
Yes
No
{
CFL
real
Yes
No
des
c
EXACT,
HLL,
HYDRAULICS
HLLC}
0.5 < CFL < 1.0
HYDRAULICS
initial_time_step
real
Yes
No
[s]
HYDRAULICS
miminum_time_step
real
Yes
No
[s]
HYDRAULICS
minimum_water_depth
real
Yes
No
[m]
HYDRAULICS
total_run_time
real
Yes
No
[s]
HYDRAULICS
riemann_tolerance
real
No
No
HYDRAULICS
geo_min_area_ratio
real
No
No
HYDRAULICS
geo_max_angle_quadrilateral
real
No
No
HYDRAULICS
geo_min_aspect_ration
real
No
No
HYDRAULICS
Explanation for parameters:
•
simulation_scheme
The keyword exp indicates an explicit time integration scheme (Euler method). An
implicit time integrator has not yet been implemented.
•
riemann_solver
Different solvers for the Riemann problem are applicable. By default, EXACT is
chosen if nothing else is specified.
•
CFL
Courant-Friedrichs-Lewy number. This parameter is used to control the time step.
The CFL-number is defined as the ratio between spatial resolution and time step.
Values lie between 0.5 and 1. By default, it is set to 1.
•
initial_time_step
This option only influences the time step at the beginning of a simulation. After a
few iterations, it has no more effect. By default, it is set to 0.1 s.
RIV - 2.6-6
VAW ETH Zürich
Version 10/28/2009
•
minimum_time_step
This option throws a warning on the console during the simulation, if the actual time
step is lower than the minimum time step defined here. It’s just some additional
info. There is no possibility to influence the time step using this option. Use the CFL
number if you want to in-/decrease the time step. By default, it is set to 0.001 s.
•
minimum_water_depth
This parameter is a key element in shallow waters simulation. The definition of a
minimal water depth avoids division by zero which would lead to numerical
instabilities. Every cell with water depth below this value will be treated as dry. By
default, it is set to 0.05 m.
•
total_run_time
When this time value is reached, the simulation stops and writes all output. By
default, this is set to 0 s.
•
riemann_tolerance
The exact Riemann solver performs an iterative solution of an analytical equation for
the depth. The tolerance determines the minimal deviation between two successive
iterations. By default, it is set to 1.0e-6 and usually not has to be changed.
Additional parameters for geometry checks:
•
geo_min_area_ratio
Set the minimum area ratio (i.e. element’s area divided by the average element
area) for which is checked during the geometry checks. By default, it is set to 0.05
(This should usually not be changed!).
•
geo_min_area_ratio
Set the aspect ratio for which is checked during the geometry checks. By default, it
is set to 0.06 (This should usually not be changed!).
•
geo_max_angle_quadrilateral
Set the maximum angle for ambiguous quadrilateral elements (i.e. the angle
between the normal vectors of the two triangles which result by splitting the
quadrilateral along a break line) for which is checked during the geometry checks.
By default, it is set to 45° (This should usually not be changed!).
VAW ETH Zürich
Version 10/28/2009
RIV - 2.6-7
Example:
HYDRAULICS
{
[…]
PARAMETER
{
simulation_scheme = exp
riemann_solver = EXACT
CFL = 1.0
initial_time_step = 0.01
minimum_time_step = 0.0001
total_run_time
= 3600.0
riemann_tolerance = 1E-6
}
[…]
}
RIV - 2.6-8
VAW ETH Zürich
Version 10/28/2009
2.6.2.2
FRICTION
FRICTION
compulsory
repeatable
Super Blocks
Condition:
-
Description:
Different friction values can be assigned to the computational domain using the material
index technique.
Variable
type
comp
rep
values
condition
desc
type
enum
Yes
No
{manning,
Friction law:
strickler,
manning: manning with
chezy,
manning’s coefficient,
darcy}
strickler: Strickler with
Strickler’s coefficient,
chezy: Chézy
darcy: Darcy-Weissbach
default_value
string
Yes
No
Default friction
coefficients for all
elements
input_type
string
Yes
No
index
int_list
Yes
No
friction
real
Yes
No
wall_friction
enum
No
No
index_table
Friction coefficient
{on, off}
Enables or disables wall
friction at boundary walls
wall_friction_factor
Real
No
No
Friction coefficient of side
wall friction (= bed friction
if not specified)
grain_size_friction
enum
No
No
{ on, off }
BEDMATERIAL
If this is “on” the friction
value of the bottom will
be
computed
as
a
function of the grain size.
Default is “off”.
strickler_factor
real
No
No
BEDMATERIAL
Factor
computation
for
the
of
friction
value from grain diameter:
k Str = factor /
6
d 90
Default value 21.1.
VAW ETH Zürich
Version 10/28/2009
RIV - 2.6-9
Example:
[…]
FRICTION
{
type = manning
default_friction = 0.033
input_type = index_table
index = ( 1 2 )
friction = ( 0.033 0.038 )
wall_friction = off
}
[…]
RIV - 2.6-10
// disables wall friction
VAW ETH Zürich
Version 10/28/2009
2.6.2.3
BOUNDARY
BOUNDARY
compulsory
repeatable
Super Block
Description:
Condition:
-
Defines boundary conditions for hydraulics. Every boundary is assigned to the edges
defined within the chosen STRINGDEF. If no boundary condition is specified for part
of or the whole boundary, an infinite wall is assumed. For every different condition, a
new block BOUNDARY has to be used.
Variable
type
comp
rep
values
condition
type
Enum
Yes
No
{ hydrograph,
HYDRAULICS,
weir, gate, wall,
BASEPLANE_2D
desc
zero_gradient,
hqrelation,
zhydrograph }
string_name
String
Yes
No
string name
defined in
STRINGDEF
String
file
No
No
File type “BC”
hydrograph, weir
level,
hqelation,
zhydrograph
my
Real
Yes
No
gate
slope
Real
Yes
No
hydrograph,
average slope at
hqrelation
the boundary
cross section
max_interval
Real
No
No
hydrograph
max. interval for
h-Q iterations
weighting_type
Enum
No
No
{ conveyance,
hydrograph,
Type of
area }
hqrelation
weighting
method for
determination of
Qi at edge i
number_of_iterations
Real
No
No
hydrograph
max. number of
h-Q iterations
precision
Real
No
No
hydrograph
precision of h-Q
iterations
•
wall: This is the default boundary condition for all nodes where no other boundary
was specified. The wall is of infinite height – there is no flux across the element
borders.
VAW ETH Zürich
Version 10/28/2009
RIV - 2.6-11
Inflow boundaries
•
hydrograph: This boundary conditions is a hydrograph describing the total
discharge in time over an inflow boundary-string. It requires a boundary file with two
columns: time and corresponding discharge. You don’t have to specify the
discharge for each time step – values will be interpolated between the defined
times in the input file. However, you need at least 2 discharges in time – at the
beginning and in the end. If you use a constant discharge in time, at the beginning
of the simulation, the discharge will be smoothly increased from zero to the
constant value within a certain time.
An average slope at the inflow cross section must be defined which is used for h-Q
iterations at the inflow cross section. (If no momentum flow must be considered,
alternatively, instead of this boundary type, an external source may be used).
The inflow discharge can be weighted along the edges of the cross section
according to the local conveyance (default behaviour) or according to the area. (Area
weighting is recommended if bed load enters the inflow section, because this
weighting type leads to constant velocities along the cross section.)
Outflow boundaries
•
•
•
•
•
zero_gradient: This boundary condition is used for outflow out of the computational
area. Basically, the flux entering the boundary element leaves the computational
area with the flux over the boundary. Mathematically speaking, all gradients of the
main variables waterdepth and velocities are zero in the boundary cell.
weir: This is another outflow boundary condition. It requires a boundary file with
two columns: time and corresponding weir height. Values are interpolated in time
automatically as described at the hydrograph.
gate: This is an outflow boundary condition as a counterpart to the weir. It requires
a boundary file with two columns: time and corresponding gate elevation in meters
above sea level. Values are interpolated in time automatically as described at the
hydrograph. Additionally, the factor my has to be specified. It usually takes values
around 0.6.
h-Q relation: This boundary condition can be used for defining outflow out of the
computational area. An h-Q relation can be specified, which is used to determine an
outflow discharge Q for the present water elevations at the outflow section. The hQ relation must be given in a separate file with a left column for the water elevation
and a right column for the corresponding discharge. This boundary can also be used
without specifying an h-Q relation by defining an average slope at the outflow cross
section, which is used to calculate the outflow discharge assuming normal flow.
zhydrograph: This is an outflow boundary. A time evolution of the water elevation
at the outlet must be specified in a file of type “BC”. The left column contains the
time in seconds, the second column the corresponding water elevation in absolute
values (meters).
RIV - 2.6-12
VAW ETH Zürich
Version 10/28/2009
Example:
HYDRAULICS
{
BOUNDARY
{
type = hydrograph
string_name = boundary1
// as defined in STRINGDEF
file = discharge_in_time.txt
slope = 1 // per mill
weighting_type = conveyance // default
}
BOUNDARY
{
type = weir
string_name = other_bndry // as defined in STRINGDEF
file = weirlevel_in_time.txt
}
BOUNDARY
{
type = zero_gradient
string_name = third_bndry // as defined in STRINGDEF
}
BOUNDARY
{
type = hqrelation
string_name = outflow_section
file = hq_relation.txt
// OR
slope = 1.25 // per mill
}
BOUNDARY
{
type = gate
string_name = yetanotherstring
file = gate_elevation_in_time.txt
my = 0.6
}
[…]
}
VAW ETH Zürich
Version 10/28/2009
RIV - 2.6-13
2.6.2.4
INNER_BOUNDARY
INNER_BOUNDARY
compulsory
repeatable
Super Block
Condition:
-
Description:
Defines inner boundary conditions as gate or weir for hydraulics only An inner Boundary needs
two STRINGDEF assigned. This is because after a weir, the edge elevation does not
necessarily have to be on the same Level as above the weir.
Important: The correct order of the nodes within the STRINDEF is crucial.!
Both STRINGDEF’s must contain the same number of nodes in the same order (direction)!
For a gate, it is recommended to ensure that the two corresponding edges are on the same
soil level.
The inner boundaries act in both direction.
For every different condition, a new block INNER_BOUNDARY has to be used.
Variable
type
comp
rep
values
condition
type
Enum
Yes
No
{gate, weir}
HYDRAULICS,
string_name1
String
Yes
No
desc
BASEPLANE_2D
string name
defined in
STRINGDEF
string_name2
String
Yes
No
String name
defined in
STRINGDEF
String
file
Yes
No
File
“txt”
Real
my
Yes
No
type
Weir
level,
gate
level
Gate, weir
Linear factors for a weir
(poleni) and a gate
Inner Boundary types for 2D simulations:
•
weir: This requires a boundary file with two columns: time and corresponding weir
height. Values are interpolated in time automatically as described at the hydrograph.
•
gate: This requires a boundary file with two columns: time and corresponding gate
level. Values are interpolated in time automatically as described at the hydrograph.
For a gate, my is usually between 0.5 and 0.6.
Note: At the current version v1.4, the inner boundaries have not been tested exhaustively.
They have to be considered experimental as the computation of velocity momentum flux is
not yet at its final stage. However, the discharge over this boundary condition behaves
correct.
RIV - 2.6-14
VAW ETH Zürich
Version 10/28/2009
Example:
HYDRAULICS
{
INNER_BOUNDARY
{
type = weir
my = 0.85
string_name1 = weir_in
// as defined in STRINGDEF
sting_name2 = weir_out
file = weirLevel.txt
}
INNER_BOUNDARY
{
type = gate
my = 0.7
string_name1 = gate_in // as defined in STRINGDEF
string_name2 = gate_out
file = gateLevel_in_time.txt
}
[…]
}
VAW ETH Zürich
Version 10/28/2009
RIV - 2.6-15
2.6.2.5
TURBULENCE_MODEL
TURBULENCE_MODEL
compulsory
repeatable
Super Block
Condition:
-
Description:
Variable
type
comp
rep
values
condition
turbulence_type
Enum
Yes
No
{algebraic}
HYDRAULICS,
kinematic_viscosity
Real
No
No
> 0 [m2/s]
desc
BASEPLANE_2D
Default:
1.307*10-6 (water
at 10 degrees
Celsius)
const_eddy_viscosity
Real
No
No
2
> 0 [m /s]
Usually between
0.1-1
turbulence_factor
Real
No
No
> 0 []
Default: 1
The algebraic turbulence model is the only type available at the moment. For this model,
either a constant eddy viscosity or a dynamic eddy viscosity can be chosen. If
const_eddy_viscosity > 0, then this value for turbulent viscosity is used throughout the
whole domain. If const_eddy_viscosity < 0 or not defined within the command file, the
dynamic eddy viscosity is set to active. Dynamic eddy viscosity depends on the local water
depth and velocity and is therefore different for each element. To increase the effect of
dynamic eddy viscosity, the dimensionless turbulence_factor can be set to any positive
number. Usual values are around 1 to 5.
RIV - 2.6-16
VAW ETH Zürich
Version 10/28/2009
Example:
HYDRAULICS
{
TURBULENCE_MODEL
{
// for a constant eddy viscosity
type = algebraic
kinematic_viscosity = 0.000001307
const_eddy_viscosity = 0.1
}
[…]
}
HYDRAULICS
{
TURBULENCE_MODEL
{
// for a dynamic eddy viscosity
type = algebraic
kinematic_viscosity = 0.000001307
turbulence_factor = 5.0
}
[…]
}
VAW ETH Zürich
Version 10/28/2009
// m2/s
// m2/s
// m2/s
RIV - 2.6-17
2.6.2.6
INITIAL
INITIAL
compulsory
repeatable
Super Block
Condition:
-
Description:
Defines initial conditions for hydraulics in 2D. If no initial condition is specified for parts or the
whole domain, the initial conditions is set to dry.
Initial conditions may be derived from earlier simulations where one wants to continue from an
outputfile. This filename is always called “region_name_old.sim”, where “region_name” has
been defined in the BASEPLANE_2D block.
Otherwise, the initial conditions can be set using the material-index from the geometry input file.
Variable
type
comp
rep
values
type
Enum
Yes
No
{dry, continue, index_table}
file
String
No
No
Dependent on initial type
index
Integer
No
No
wse
Real
No
No
condition
desc
continue
Index_table
[m rel.]
Index_table,
h not used
u
real
No
No
[m/s]
Index_table
v
real
No
No
[m/s]
Index_table
Initial types for 2D simulations:
•
dry
This is the simplest initial condition. The whole area is initially dry – no discharge and no
WSE is needed. This is the default for every cell in 2D where no initial condition is
specified.
•
continue
From earlier simulations, one can continue using the former output file as new initial
conditions input file. This filename is always called “region_name_old.sim”, where
“region_name” has been defined in the BASEPLANE_2D block. This option is only valid
for pure hydraulic simulations without sediment transport. If you renamed the output
file, a file name may be provided here.
•
index_table
The initial conditions for WSE, u and v can be set manually using the material index
technique. In the geometry file, every cell can be assigned an integer value which is
called the “material index”. To every one of these indices, specific values for WSE, u
and v can be assigned. All lists must have the same number of elements, which
correspond to the element in the index-list. See the example for explanation.
RIV - 2.6-18
VAW ETH Zürich
Version 10/28/2009
Example:
HYDRAULICS
{
[…]
INITIAL
{
type = index_table
index = ( 1
2
wse =
( 12.1 15.0
u
=
( 0.0 -3.2
v
=
( 0.0 0.0
}
[…]
}
3
0.0
1.5
2.1
4
0.54
0.0
3.0
5
2.3
5.6
0.1
)
)
)
)
HYDRAULICS
{
[…]
INITIAL
{
type = continue
file = beispiel_old.sim
}
[…]
}
VAW ETH Zürich
Version 10/28/2009
RIV - 2.6-19
2.6.2.7
SOURCE
SOURCE
compulsory
repeatable
Super Blocks
Condition:
-
Description:
The SOURCE block deals with all influences on the source term in the equations to
be solved. The EXTERNAL_SOURCE block is used for additional sources or sinks,
sometimes used instead of flux boundary conditions.
Inner Blocks
EXTERNAL_SOURCE
Variable
type
comp
rep
values
condition
desc
Example:
HYDRAULICS
{
[…]
SOURCE
{
EXTERNAL_SOURCE
{
[…]
}
EXTERNAL_SOURCE
{
[…]
}
}
[…]
}
RIV - 2.6-20
VAW ETH Zürich
Version 10/28/2009
2.6.2.7.1
EXTERNAL_SOURCE
EXTERNAL_SOURCE
compulsory
repeatable
Super
SOURCE < HYDRAULICS < BASEPLANE_2D < DOMAIN
Condition:
-
Blocks
Description:
The external source block is used for additional sources, mostly used for sources and sinks. The
only available source type is source_discharge which acts as a sink when delivered with
negative values in the file. All external sources act on one or more computational cells which
have to be listed in an element_ids list. The specified source will then be partitioned among the
corresponding cells weighted with their volumes. For each different external source, a new
block has to be created. The input format for the file is of file type “BC”: two columns with time
and corresponding source discharge.
Variable
type
comp
rep
values
type
enum
Yes
No
[source_discharge]
file
string
Yes
No
File type “.txt”
element_ids
integer list
Yes
No
condition
desc
source_discharge
Example:
SOURCE
{
[…]
EXTERNAL_SOURCE
{
type = source_discharge
element_ids = ( 45 ) // only one element also needs brackets
file = discharge1_in_time.txt
}
EXTERNAL_SOURCE
{
type = source_discharge
element_ids = ( 12 14 18 )
file = another_discharge.txt
}
}
VAW ETH Zürich
Version 10/28/2009
RIV - 2.6-21
2.6.3
MORPHOLOGY
MORPHOLOGY
compulsory
repeatable
Super Blocks
Condition:
-
Description:
Defines the morphologic specifications for the simulation. Contrary to hydraulics, this is not
compulsory. If MORPHOLOGY is not specified, all kind of sediment transport will be inactive.
To activate bedload transport, the BEDLOAD block has to be specified.
All information about the bed soil is defined within the BEDMATERIAL block.
Inner Blocks
BEDMATERIAL
BEDLOAD
PARAMETER
Variable
type
comp
rep
values
condition
desc
-
-
-
-
-
-
-
Example:
MORPHOLOGY
{
BEDMATERIAL
{
[…]
}
BEDLOAD
{
[…]
}
PARAMETER
{
[…]
}
}
RIV - 2.6-22
VAW ETH Zürich
Version 10/28/2009
2.6.3.1
PARAMETER
PARAMETER
compulsory
repeatable
Super Blocks
MORPHOLOGY < BASEPLANE_2D < DOMAIN
Condition:
-
Description:
The PARAMETER block for morphology in 2D contains general settings dealing
with sediment properties.
Variable
type
comp
rep
values
condition
desc
porosity
Real
No
No
[%]
density
Real
No
No
[kg/m3]
bedload_cv_type
String
No
No
{constant,
Default =
borah,
constant
dmean}
constant
dmean
bedload_cv_thickness
real
No
No
bedload_cv_factor
real
No
No
create_new_layers
Enum
No
No
{ on, off }
max_layer_thickness
Real
No
No
[m]
Default = 0.1
Default = 3.0
Default = off
create_ne
w_layers
Description of parameters:
•
porosity
This is the general bed material porosity in [%]. By default, it is set to 37 %.
•
density
The general density of the bed load has to be specified in [kg/m3]. By default, it is set to
2650.
•
bedload_cv_type (also called “active_layer”)
This parameter defines the type of the bed load control volume, which is sometimes
also called “active layer”. Available types are ‘constant’ for bed load control volumes
with constant thickness, and ‘borah’ or ‘dmean’ for bed load control volumes with
variable thicknesses (as explained in the reference manual). By default, the type
‘constant’ is set.
•
bedload_cv_thickness
The thickness of the bed load control volume can be given in [m]. This value is only
needed if the type of the bed load control volume is set to ‘constant’. The specified
thickness is valid for all elements of the simulation. In multiple-grain simulations this can
be an important calibration parameter! By default, it is set to 0.1 [m].
VAW ETH Zürich
Version 10/28/2009
RIV - 2.6-23
•
bedload_cv_factor
This parameter is only needed if the bedload_cv_type is ‘dmean’. In this case the
bedload control volume thickness calculates as: factor * dmean. Per default, it is set to
3.0.
•
create_new_layers
If this parameter is set to “on”, new soil layers can be generated during bed
aggradation. If it is set to “off”, than no soil layers are generated dynamically. Per
default this option is set to “off”.
•
max_layer_thickness
This is the maximum thickness of newly created soil layers [m]. In case of deposition, if
a layer increases in height, the maximum thickness defines when a new soil layer shall
automatically be created. By default it is set to 0.3 m. This parameter is only used if
create_new_layers is active.
Example:
MORPHOLOGY
{
[…]
PARAMETER
{
porosity = 0.37
density = 2650
bedload_cv_type = constant
bedload_cv_thickness = 0.05 // [m]
create_new_layers = off
[…]
}
[…]
}
RIV - 2.6-24
VAW ETH Zürich
Version 10/28/2009
2.6.3.2
BEDMATERIAL
BEDMATERIAL
compulsory
repeatable
Super Blocks
Condition:
-
Description:
This block defines the characteristics of the bed material. The GRAIN_CLASS specifies
the relevant diameters of the grain. The MIXTURE block allows producing different grain
mixtures using the mean diameters. The SOIL_DEF block defines all soil layers of a soil.
Finally, the SOIL_ASSIGNMENT block is used to assign the different soil definitions to
their corresponding areas within the topography.
Inner Blocks
GRAIN_CLASS
MIXTURE
SOIL_DEF
FIXED_BED
SOIL_ASSIGNMENT
Variable
type
comp
rep
values
condition
desc
Example:
MORPHOLOGY
{
BEDMATERIAL
{
GRAIN_CLASS
{
[…]
}
MIXTURE
{
[…]
}
MIXTURE
{
[…]
}
SOIL_DEF
{
[…]
}
SOIL_DEF
{
[…]
}
SOIL_ASSIGNMENT
{
[…]
}
}
}
VAW ETH Zürich
Version 10/28/2009
RIV - 2.6-25
2.6.3.2.1
GRAIN_CLASS
GRAIN_CLASS
compulsory
repeatable
Super
BEDMATERIAL < MORPHOLOGY < BASEPLANE_2D < DOMAIN
Condition:
-
Blocks
Description:
This block defines the mean diameters of the bed material in [mm]. The number of diameters in
the list automatically sets the number of grains. For “single grain simulations”, only one diameter
must be defined (see example).
The grain sizes must be entered in ascending order from the smallest grains to the largest
grains!
Variable
type
comp
rep
values
diameters
real
Yes
No
[mm]
condition
desc
Example:
BEDMATERIAL
{
GRAIN_CLASS
{
// simulation with 5 grain sizes [mm]
diameters = ( 0.5 2.1 4.6 7.0 15.0 )
}
[…]
}
BEDMATERIAL
{
GRAIN_CLASS
{
// simulation with 1 grain size [mm]
diameters = ( 4.6 )
// don’t forget to put the brackets
}
[…]
}
RIV - 2.6-26
VAW ETH Zürich
Version 10/28/2009
2.6.3.2.2
MIXTURE
MIXTURE
compulsory
repeatable
Super Blocks
Condition:
-
Description:
This block allows the definition of different grain mixtures. According to the number of grain
diameters from the GRAIN_CLASS, for each class, the volume percentage has to be
specified. The fractions must be written in the order corresponding to the grain class list. If
a grain class is not present in a mixture, its fraction value has to be set to 0.
The mixture_name is used later for easy to use assignments.
As the MIXTURE block is repeatable, several mixtures can be defined.
In case of a single grain simulation (only one mean diameter in GRAIN_CLASS), one
MIXTURE block has to be defined using ‘volume_fraction = ( 100 )’;
Variable
type
comp
rep
mixture_name
string
Yes
No
volume_fraction
list of reals
Yes
No
values
condition
desc
[%]
Example:
BEDMATERIAL
{
GRAIN_CLASS
{
diameters = ( 0.5 2.1 4.6 )
}
MIXTURE
{
}
MIXTURE
{
volume_fraction = ( 90.1 5.7 4.2
}
[…]
}
VAW ETH Zürich
Version 10/28/2009
)
RIV - 2.6-27
2.6.3.2.3
SOIL_DEF
SOIL_DEF
compulsory
repeatable
Super Blocks
Condition:
-
Description:
This block defines the structure of the soil. The soil can be divided into vertical soil
layers. Every layer is defined within a LAYER block. Every layer has an own mixture
assigned. (In contrast to the former implementation, the mixture of the bed load
control volume is now automatically set to the mixture of the uppermost layer!)
The definition of the parameter d90_armored_layer or the parameter tau_erosion_start
specifies the consideration of bed armour. It allows simulating the effects of a surface
protection layer which must be eroded once to allow the erosion of the underlying
substrate.
Inner Blocks
LAYER
Variable
type
comp
rep
soil_name
string
yes
no
d90_armored_layer
real
no
no
values
condition
desc
[mm]
Single grain
An armoured layer is set
size
at the initial elevation.
simulation
tau_erosion_start
real
no
no
[N/m2]
An armoured layer is set
at the initial elevation.
•
soil_name
This is the name of the soil. Each soil must have a unique name. The specification of
a name is obligatory.
•
d90_armored_layer
d90 of the armoured bed layer in [mm]. This armoured bed layer is set at the
beginning. If it is eroded once in an element, it is destroyed in this element for the
rest of the simulation. This parameter should only be used in single grain
simulations to model effects of an armoured bed layer.
•
tau_erosion_start
Critical bed shear stress needed to destroy the armoured bed layer in [N/m2]. This
armoured bed layer is set at the beginning. If it is eroded once in an element, it is
destroyed in this element for the rest of the simulation. This parameter can be
helpful for example to define a grass protection layer.
RIV - 2.6-28
VAW ETH Zürich
Version 10/28/2009
Example:
BEDMATERIAL
{
GRAIN_CLASS
{
[…]
}
MIXTURE
{
[…]
}
MIXTURE
{
[…]
}
SOIL_DEF
{
soil_name = soil1
d90_armored_layer = 0.1
LAYER
{
[…]
}
LAYER
{
[…]
}
}
SOIL_DEF
{
soil_name = soil2
LAYER
{
[…]
}
}
[…]
}
VAW ETH Zürich
Version 10/28/2009
RIV - 2.6-29
2.6.3.2.3.1
LAYER
LAYER
compulsory
repeatable
Super Blocks
SOIL_DEF < BEDMATERIAL < MORPHOLOGY < BASEPLANE_2D < DOMAIN
Condition:
-
Description:
This block defines a soil layer. For every soil layer, a new block has to be used. The
bottom elevation of a layer is given relative to the surface topography defined in the
geometry file. Therefore, a minus sign has to be used. The LAYER blocks have to be
defined ordered by their distance to the surface, the uppermost layers first, the lowest
layers last.
The bottom of the lowest layer automatically defines the fixed bed, where no further
erosion is possible. Therefore, in contrast to former implementations, usually no
FIXED_BED block is needed.
In case of a single grain simulation, only one LAYER block is needed, which automatically
defines the fixed bed. If no LAYER block is defined within a SOIL_DEF block, a fixed bed
is assumed at the bed surface.
Variable
type
comp
rep
values
bottom_elevation
real
Yes
No
[m]
condition
desc
Distance relative to surface
(must be negative!)
mixture
string
No
No
Mixture
Grain mixture of the soil layer
defined
Example:
SOIL_DEF
{
soil_name = soil1
LAYER
{
bottom_elevation = -5
mixture = mixture1
}
LAYER
{
bottom_elevation = -7
mixture = mixture2
}
LAYER
{
[…]
}
}
RIV - 2.6-30
// [m], relative to surface
// [m], relative to surface
VAW ETH Zürich
Version 10/28/2009
2.6.3.2.4
FIXED_BED
FIXED_BED
compulsory
repeatable
Super
Condition:
-
Blocks
Description:
In order to control the erosion process, the user can define rock elevations (fixed bed) for the
computational domain. No erosion can occur if there is no more sediment on top of the fixed
bed.
The FIXED_BED block is not obligatory, because the bottom of the lowest layer in the soil
definition automatically defines the rock elevation.
But a FIXED_BED block may be inserted optionally. There are two ways how to define fixed bed
elevations here.
An additional geometry file can be used to define rock elevations in more detail. In such a case, a
fixed bed meshfile must be given, which must have the same structure as the simulation
meshfile. Then the z-elevations of the nodes of this additional mesh file, define the fixed bed
elevations.
Or, alternatively, the FIXED_BED block can be used to define a list of node numbers with
according fixed bed elevations. If the elevations are set to values <= -100 than the surface bed
elevation is used as fixed bed.
Variable
type
comp
rep
values
condition
type
string
Yes
No
{ meshfile,
file
string
Yes
No
meshfile
node_ids
integer
Yes
No
nodes
desc
node_ids }
List of nodes within brackets.
(Nodes must not be adjacent)
zb_fix
Real
Yes
No
[m.a.s.l,
absolute]
nodes
List
of
zb_fix
values
within
brackets. If zb_fix < -100, than bed
elevation is used as zb_fix.
VAW ETH Zürich
Version 10/28/2009
RIV - 2.6-31
Example:
BEDMATERIAL
{
[…]
FIXED_BED
{
type = meshfile
file = fixed_bed_mesh.2dm
}
[…]
}
BEDMATERIAL
{
[…]
FIXED_BED
{
type = nodes
node_ids = ( 12 22 25 72 92 )
zb_fix = ( 312.2 313 313.56 -100 -100 )
// the “-100” values set the fixed bed elevation
// to the surface elevation
}
[…]
}
RIV - 2.6-32
VAW ETH Zürich
Version 10/28/2009
2.6.3.2.5
SOIL_ASSIGNMENT
SOIL_ASSIGNMENT
compulsory
repeatable
Super Blocks
Condition:
Description:
This block is finally used to assign the soil definitions to their corresponding areas in the
topography. For this purpose, the index-technique is used. In the geometry input file,
every cross-section (and also parts of a cross-section) can be labelled with an integer
index number which is used to attach a soil definition.
For single grain size simulations, a soil assignment is still needed, although there is only
one MIXTURE to assign.
Inner Blocks
Variable
type
comp
rep
values
condition
type
string
Yes
No
index_table
index
list of ints
Yes
No
index_table
soil
list of reals
Yes
No
index_table
desc
Example:
BEDMATERIAL
{
[…]
SOIL_DEF
{
[…]
}
SOIL_DEF
{
[…]
}
[…]
SOIL_ASSIGNMENT
{
type = index_table
index = ( 1
2
soil = ( soil1 soil2
}
}
VAW ETH Zürich
Version 10/28/2009
4
soil3
6
soil1
)
)
RIV - 2.6-33
2.6.3.3
BEDLOAD
BEDLOAD
compulsory
repeatable
Super
Condition:
-
Blocks
Description:
Defines all needed data for bed load transport. If this block does not exist, no bed load transport
will be simulated.
Inner Blocks
BOUNDARY
SOURCE
PARAMETER
Variable
type
comp
rep
values
condition
desc
-
Example:
MORPHOLOGY
{
BEDMATERIAL
{
[…]
}
BEDLOAD
{
SOURCE
{
[…]
}
PARAMETER
{
[…]
}
}
}
RIV - 2.6-34
VAW ETH Zürich
Version 10/28/2009
2.6.3.3.1
PARAMETER
PARAMETER
compulsory
repeatable
Super Blocks
BEDLOAD < MORPHOLOGY < BASEPLANE_2D < DOMAIN
Condition:
-
Description:
The PARAMETER block for morphology in 2D contains general settings dealing
with bedload transport.
Variable
type
comp
rep
values
bedload_transport
enum
Yes
No
{MPM, MPMH,
c
d
PARKER,
RICKENMANN,
WU, POWER_LAW}
bedload_routing_start
real
No
No
[s]
hydro_step
integer
No
No
upwind
real
No
No
[-]
bedload_factor
real
No
No
[-]
theta_critic
real
No
No
MPM, MPMH,
RICKENMANN,
WU
bed_forms
real
No
No
[-]
<=1.0
.
hiding_exponent
real
No
No
[-]
MPMH, WU
default: -1.5 (MPMH)
default: -0.6(WU)
angle_of_repose
real
No
d90
real
if
d30
real
power_law_a,
real
No
single
[degree]
[mm]
RICKENMANN
[mm]
RICKENMANN
[-]
POWER_LAW
grain
if
single
grain
No
No
power_law_b
local_slope
Enum
No
No
{ on, off }
lateral_transport
Enum
No
No
{ on, off }
wetted_factor
real
No
No
[-]
VAW ETH Zürich
Version 10/28/2009
RIV - 2.6-35
•
bedload_transport
These bedload relations are available at the moment:
- MPM
(Meyer-Peter & Müller)
- MPMH
(Meyer-Peter & Müller & Hunziker)
- PARKER
(Parker)
- RICKENMANN (Rickenmann)
- WU
(Wu,Wang & Jia)
- POWER_LAW
•
bedload_routing_start
Indicates, at which time sediment routing is started during a simulation. The default
value is set to 0.0, which means bed load transport from the beginning.
•
hydro_step
Specifies the cycle steps for hydrodynamic routing. For example, a value of 5 forces the
hydrodynamic routine to be computed after every 5 cycles of the sediment routine. The
default value is set to 0.
•
upwind
The upwind factor is implemented with a default value of 0.5. The upwind factor
remains constant during the whole simulation. (It is not used for single grain transport).
See (RII 3.3.2.2) for more details.
•
bedload_factor (important calibration factor!)
This factor is a key factor for the calibration of sediment transport. It is multiplied with
the computed transport capacity of the empirical bedload formula. Per default is set to
1.0.
•
theta_critic (important calibration factor!)
If this parameter is not specified, the critical shear stress is determined from the Shields
diagram which is the default behaviour. If the parameter is specified, this value is taken
as given by the user and set constant over the whole simulation.
•
hiding_exponent (important calibration factor!)
Factor of the exponent of the hiding-and-exposure function in Hunziker’s or Wu’s
fractional bed load formula. This factor should be set to a negative number. Per default
is set to -1.5 in Hunziker’s formula or to -0.6 in Wu’s formula. Changing this exponent
primarily has influences on the grain sorting processes. Large changes will probably
result in instabilities.
RIV - 2.6-36
VAW ETH Zürich
Version 10/28/2009
•
angle_of_repose
Angle of repose of the sediment material. Per default set to 30 degrees.
•
bed_forms
This factor is applied in case of bed forms to determine the effective bottom shear
stress θ ' from the bottom shear stress θ . This factor must be set to a value <=1.0. If
the factor is set to a positive number, then it is assumed constant over the whole
domain and simulation time. If the factor is set to a negative number, then it is
determined dynamically by the program. See chapter RI-2.2 for more information. Per
default it is set to a value < 0.0, i.e. the factor is calculated by the program.
•
d90 / d30
Diameters d90 or d30 of the mixture. These factors are only of interest if the
Rickenmann formula is used. In single grain simulations these factors are obligatory. In
multiple grain simulations they can be optionally be given. If they are not given, then
they are calculated from the mixtures in multi-grain simulations.
•
power_law_a, power_law_b
Coefficients a and b for the power law formula. Per default a=1.0, b=1.0.
•
local_slope
This flag determines if the gravitational influence of the local slope within the elements
should be considered for bed load transport. If it is considered, the bedload is increased
if there is a slope in flow direction, or the bedload is reduced if there is a counter slope.
Per default the local_slope is active and set to ‘on’.
•
lateral_transport
This flag determines if the lateral transport, caused by transversal slope in relation to the
flow velocity, shall be taken into account. Per default, the lateral transport is active and
set to ‘on’.
•
wetted_factor
Usually, this parameter should not be changed! This parameter defines if on partially
wetted cells bedload transport shall be calculated or not. If the parameter is set to 0.0
than no bedload transport is calculated on partially wetted cells. If it is, for example, set
to 1.0 than bedload is also calculated in partially wetted cells if the water elevation in
these cells is larger than 1.0 times the minimum water depth. Per default this factor is
set to 0.0.
VAW ETH Zürich
Version 10/28/2009
RIV - 2.6-37
Example:
BEDLOAD
{
[…]
PARAMETER
{
bedload_transport = MPM // Meyer-Peter&Mueller formula
bedload_routing_start = 0.0
upwind = 1
bedload_factor = 0.5 // reduces the bedload transport rate
theta_critic = 0.055
}
}
RIV - 2.6-38
VAW ETH Zürich
Version 10/28/2009
2.6.3.3.2
BOUNDARY
BOUNDARY
compulsory
repeatable
Super Blocks
Condition:
-
Description:
Defines boundary conditions for the bed load transport. For every boundary condition, a new
block BOUNDARY has to be defined. At the upstream boundaries a sediment inflow can be
specified. This type of boundary needs a mixture assigned, in any case, even if single grain
computations are performed. IOUp and IODown are special boundaries designed to facilitate
the determination of the equilibrium bed load transport.
Please note: Instead of using an inflow boundary, alternatively also an EXTERNAL_SOURCE can
be used to specify an inflow of sediment material in the simulation domain!
Variable
type
comp
rep
values
condition
desc
type
enum
Yes
No
{sediment_discharge,
Type
IOUp, IODown}
boundary
of
the
condition
string_name
String
Yes
No
string name defined
in STRINGDEF
string
file
No
No
File type “BC”
sediment_discharge
File with the
sediment
inflow in
[m3/s]. The file
needs 2
columns, time
and inflow.
string
mixture
No
No
sediment_discharge
Mixture of the
sediment
inflow
Upstream
•
sediment_discharge
This boundary condition is based on a sediment “hydrograph” describing the bed
load inflow at an upstream boundary in time. The behaviour in time is specified in
the boundary file with two columns: time and corresponding sediment discharge.
The values are interpolated in time. Additionally, the mixture must be set to a
predefined mixture name. The sediment input in [m3/s] is distributed over all given
nodes.
VAW ETH Zürich
Version 10/28/2009
RIV - 2.6-39
•
IOUp
This upstream boundary condition grants a constant bed load inflow. Basically, the
same amount of sediment leaving the first computational cell in flow direction
enters the cell from the upstream bound. This is useful to determine the equilibrium
flux. There is no boundary_mixture needed.
Downstream
•
IODown
It is the only downstream boundary condition available for sediment transport. It
corresponds to the definition of IOup. All sediment entering the last computational
cell will leave the cell over the downstream boundary. There is no further
information needed for this case.
If one does not specify any boundary condition, no transport over the boundary will occur – a
wall is assumed.
Example:
BEDLOAD
{
[…]
BOUNDARY
{
type = sediment_discharge
file = discharge.txt
mixture = mixture2
}
BOUNDARY
{
type = IODown
string_name = downstream
}
[…]
}
RIV - 2.6-40
VAW ETH Zürich
Version 10/28/2009
2.6.3.3.3
SOURCE
SOURCE
compulsory
repeatable
Super Blocks
Condition:
-
Description:
The SOURCE block for morphology in 2D contains external sources or sinks which
can be attached to any computational cell with its corresponding node_id. For each
different external source, a new inner block EXTERNAL_SOURCE has to be defined.
Inner Blocks
EXTERNAL_SOURCE
Variable
type
comp
rep
values
-
-
-
-
-
condition
desc
Example:
BEDLOAD
{
[…]
SOURCE
{
EXTERNAL_SOURCE
{
[…]
}
EXTERNAL_SOURCE
{
[…]
}
}
[…]
}
VAW ETH Zürich
Version 10/28/2009
RIV - 2.6-41
2.6.3.3.3.1
EXTERNAL_SOURCE
EXTERNAL_SOURCE
compulsory
repeatable
Super Blocks
SOURCE < BEDLOAD < MORPHOLOGY < BASEPLANE_2D < DOMAIN
Condition:
-
Description:
The EXTERNAL_SOURCE block is used for additional sources or sinks. If no source block is
specified, the additional source is set to zero and no sediment enters the computational
domain as source.
The specified source or sink in [m3/s] is distributed over all given nodes!
Please note that also for single grain size simulations, the definition of mixture is still needed.
Iin case of a sediment sink, the definition of the mixture must also be given.
Inner Blocks
AREA
Variable
type
comp
rep
values
type
enum
Yes
No
sediment_discharge
node_ids
list
Yes
No
of
condition
desc
Type of the external source
Node ids of the nodes where
the source or sink shall take
integers
place
file
string
Yes
No
File type “BC”
File with the source values
listed over time. The file
needs two columns, time
and sediment inflow [m3/s].
mixture
string
Yes
No
Mixture
Grain
defined
sediment source
mixture
of
the
Example:
MORPHOLOGY
{
[…]
SOURCE
{
EXTERNAL_SOURCE
{
file = source_in_time.txt
node_ids = ( 34 35 36 37 38 39 40 )
mixture = mixture2
}
EXTERNAL_SOURCE
{
file = another_source_in_time.txt
node_ids = ( 256 )
// don’t forget the brackets
mixture = mixture1
}
}
[…]
}
RIV - 2.6-42
VAW ETH Zürich
Version 10/28/2009
2.6.4
OUTPUT
OUTPUT
compulsory
repeatable
Super Blocks
Condition:
-
Description:
This block specifies the desired output data. In 2D, a default output file contains all
information about geometry and other variables along the flow direction. It is written
according to the output time step setting and can be used for continuation simulations.
For temporal outputs, one has to specify a SPECIAL_OUTPUT block where time histories
or grain information for selected mesh-elements can be chosen. For each different output
desired, a new SPECIAL_OUTPUT block has to be used.
Inner Blocks
SPECIAL_OUTPUT
Variable
type
comp
rep
output_time_step
real
Yes
No
console_time_step
real
no
No
result_location
enum
No
No
{node, center}
AVS-UCD
Enum
No
No
{on, off}
values
condition
desc
[s]
[s]
•
output_time_step
Specifies the time step for the default output files.
•
console_time_step
Specifies the time step for console output. If no value is set, output_time_step will
be used.
•
result_location
This parameter sets the type of output location. To plot the results, all plotting
programs need an underlying mesh. The mesh for 2D is defined by its nodes and
can be used easily for plotting. However, the results of the computation are
represented in the centre of an element.
Choosing node forces the program to extrapolate the centre-results on the nodes.
The original mesh can be used for plotting then.
Choosing center prints the results as they have been computed for the cell centres.
For a correct plotting, a mesh file will be needed, which is defined by the location of
the cell centres. Default setting is node.
•
AVS-UCD
This enables a time dependent Output File in the UCD Format which can be used by
AVS-Express for post processing. The generated data file is quite large; therefore,
this option is set to off by default.
VAW ETH Zürich
Version 10/28/2009
RIV - 2.6-43
Example:
BASECHAIN_2D
{
[…]
OUTPUT
{
result_location = node
AVS-UCD = off
SPECIAL_OUTPUT
{
[…]
}
SPECIAL_OUTPUT
{
[…]
}
}
[…]
}
RIV - 2.6-44
VAW ETH Zürich
Version 10/28/2009
2.6.4.1
SPECIAL_OUTPUT
SPECIAL_OUTPUT
compulsory
repeatable
Condition:
-
Super Blocks
OUTPUT < BASEPLANE_2D < DOMAIN
Description:
Besides the default output, some extra output for selected elements can be defined. The
choices for output type are time_history, BASEviz, tecplot_hydro and tecplot_hydro_bin,
grain, sed_balance.
Time history plots all important cell information versus the time. Grain delivers all
sediment information for the selected elements.
Each such information for each element-group will be written to an own file
filename_tHNdGlb-[i]_thnd.out, where i is the element id.
For graphical output of the simulation results during run-time with the BASEviz
visualization choose the BASEviz output type.
ASCII and binary files which can be interpreted by Tecplot can be generated using
tecplot_hydro and tecplot_hydro_bin respectively.
Special output for sediment transport is also possible by defining selected nodes. The
output type here is grain. Then two type of files are generated which contain the grain
compositions and the bed load fluxes at these nodes.
Another special output for sediment transport is the sed_balance output. If this output
type is activated, the program compares the initial sediment volume of the model with the
sediment volume at the current runtime to check for errors in the sediment mass balance.
Note: For each type of special output, only one block may be defined.
Variable
type
comp
rep
values
condition
type
enum
Yes
No
{ time_history, BASEviz,
desc
tecplot_hydro, tecplot_hydro_bin,
grain, sed_balance }
output_time_step
real
Yes
[s]
element_ids
integer
Yes
no
Maximum 30 ids
time_history
Yes
No
Maximum 30 ids
grain
list
node_ids
Integer
list
BASEviz:
window_interact_time
real
No
No
Should be selected as
BASEviz
Time interval for
large as possible to
interactions/refresh of the
avoid slow down of
visualization window
the simulation
value
string
No
No
{depth, wse, velocity,
BASEviz
Output (plot) variable
tau, bedload}
VAW ETH Zürich
Version 10/28/2009
RIV - 2.6-45
gridlines
enum
No
No
{on, off}
BASEviz
Enable or disable gridlines plotting
vectors
enum
No
No
{on, off}
BASEviz
Enable or disable velocity vector plotting
wse3D
enum
No
No
{on, off}
BASEviz
Enable or disable 3D WSE plotting (no
vector plotting is supported here)
vectors_scaling_factor
real
No
No
default =
BASEviz
Scale the length of the velocity vectors
1.0
Example:
BASECHAIN_2D
{
[. . .]
OUTPUT
{
result_location = node
SPECIAL_OUTPUT
{
type = time_history
element_ids = ( 1 3 25 94 )
}
SPECIAL_OUTPUT
{
type = BASEviz
window_interact_time = 5.0
variable = depth
gridlines = off
vectors = on
}
SPECIAL_OUTPUT
{
type = tecplot_hydro_bin
}
SPECIAL_OUTPUT
{
type = grain
// only for sediment transport
node_ids = ( 23 45 98 486 )
}
SPECIAL_OUTPUT
{
type = sed_balance
// only for sediment transport
}
}
}
RIV - 2.6-46
VAW ETH Zürich
Version 10/28/2009
Part 2:
Auxiliary Input Files
3 Auxiliary input files
3.1
3.1.1
3.1.2
Introduction
Elements of an auxiliary input files
Keyword
The keywords must be at the beginning of the line without any space
between margin and word.
The first two letters have to be either both capitals or both lower case
letters.
DefType
Format for connected parameters.
//
Comment, e.g. whole line or after a parameter.
Syntax description
General

[par]
Symbol for space
optional parameter
Units (U)
e rel.
e
unit e relative to a chosen reference system (e.g. elevation).
absolute value of unit e (e.g. distance).
Parameter types (T)
r
real
i
integer
c
character
s
string
decimal number (e.g. 52.37, 8.)
integer (e.g. 103)
single character
character string (word)
Default values (D)
For some parameters a default value is assumed if it is not specified.
Repeatability (R)
Specifies if the command can be used more than once (yes or no).
VAW ETH Zürich
Version 10/28/2009
RIV - 3.1-1
RIV - 3.1-2
VAW ETH Zürich
Version 10/28/2009
3.2
Type “BC”
This file describes a typical boundary condition for the use within BASECHAIN_1D or
BASEPLANE_2D.
Definition Syntax:
// this is a comment
value1value2
Example:
// T
0.000
50000
100000
Q
10
12
11
Summary:
The first line contains information about the listed parameters. This line is not read by
the program. The value table is terminated by the keyword “end”.
Parameter Description:
Parameter
U
T
R
Description
value1
S
r
D
y
Value of the first parameter: e.g. time, water surface elevation,…
r
y
Value of the second parameter:
m rel.
value2
m3
Discharge,…
VAW ETH Zürich
Version 10/28/2009
RIV - 3.2-1
RIV - 3.2-2
VAW ETH Zürich
Version 10/28/2009
3.3
Type “Ini1D”
For use within BASECHAIN_1D super block only. This file can be a restart file generated by
a previous simulation, which has been saved as text file with the extension “.txt”. It
describes the initial hydraulic state in the channel with the discharge and the wetted area for
each cross section.
time
[CompactSerial]
Keyword
DefType
Definition Syntax:
startTime
Example:
time0
Summary:
Defines the time of the described situation.
Parameter
U
T
startTime
s
r
VAW ETH Zürich
Version 10/28/2009
D
R
Description
n
Time at which the simulation begins (usually 0).
RIV - 3.3-1
none
[Table]
Keyword
DefType
Definition Syntax:
crossSectionNameQA
Example:
cs1200150
Summary:
Defines discharge and wetted area for each cross section.
Parameter
U
crossSectionName
-
T
D
R
Description
s
y
Name of the cross section.
Q
3
m /s
r
y
Initial discharge.
A
m2
r
y
Initial wetted area.
RIV - 3.3-2
VAW ETH Zürich
Version 10/28/2009
3.4
Type “InitialConcentration1D”
For use within BASECHAIN_1D super block only. This file describes the initial situation of
suspended material in the channel with the concentration and the mixture type for each
cross section.
time
[CompactSerial]
Keyword
DefType
Definition Syntax:
startTime
Example:
time0
Summary:
Defines the time of the described situation.
Parameter
U
T
startTime
s
r
VAW ETH Zürich
Version 10/28/2009
D
R
Description
n
Time at which the simulation begins (usually 0).
RIV - 3.4-1
none
[Table]
Keyword
DefType
Definition Syntax:
crossSectionNameCmixtureName
Example:
cs10.01mixture1
Summary:
Defines concentration and mixture area for each cross section.
Parameter
U
T
crossSectionName
-
C
mixtureName
RIV - 3.4-2
D
R
Description
s
y
-
r
y
Initial value of the total concentration.
-
s
y
Name of the mixture of the suspended load.
VAW ETH Zürich
Version 10/28/2009
3.5
Type “Topo1D”
Topography File for BASECHAIN_1D.
ZCalculation
[Serial]
Keyword
DefType
Definition Syntax:
calculationType  [filename]
Example:
zcalculation
tabletableRhein
Summary:
Defines whether the hydraulic computation will be based on the iterative determination
of z(A) and direct computation of all other values, or on the linear interpolation of the
values from a previously computed table (faster for complex geometries). The option
table can only be used if there is no sediment transport.
Parameter
U
T
D
R
Description
calculationType
-
s
n
“table” (only for pure hydraulic computations) or “iteration”.
fileName
-
s
n
Name of the file in which the tables are stored. Read only for calculation
type “table“.
VAW ETH Zürich
Version 10/28/2009
RIV - 3.5-1
REsistance
[Serial]
Keyword
DefType
Definition Syntax:
resistanceType
Example:
resistance
strickler
Summary:
Defines the friction law.
Parameter
U
T
resistanceType
-
s
RIV - 3.5-2
D
R
Description
n
Friction law:
• strickler: Stricker with Strickler value;
• heightstr: Strickler with equivalent friction height;
• chezy: Chézy;
• darcy: Darcy-Weissbach;
• manning: Manning.
VAW ETH Zürich
Version 10/28/2009
VAlues repeatable
[Serial]
Keyword
DefType
Definition Syntax:
Par1Par2kMainChannelkForelandsbedSlopekBed
Example:
values
0.050.0166.66666.666,,66.666
Summary:
Defines several values, which are valid for all the following cross-sections. It can be
repeated to adopt new values for further cross-sections.
Parameter
U
T
Par1
m
r
D
R
Description
n
If calcuation type = table: maximum difference of water surface
abs.
elevation between the calculated values in the table.
/m2
If calculation type = iteration: maximum accepted difference
between given A and calculated A for iterative z(A) calculation.
Par2
m/-
r/i
n
If calculation type = table: minimum interval of water surface
elevation between the calculated values in the table.
If calculation type = iteration: maximum number of iterations allowed
for iterative z(A) calculation before the program is stopped.
kMainChannel
m1/3/s
i
n
Friction value of the main channel, if not specified in the single cross
sections or cross section slices.
or
mm
kForelands
m1/3/s
i
n
Friction value of forelands, if not specified in the single cross
sections or cross section slices.
or
mm
bedSlope
-
i
n
Slope of the channel if not replaced by values in the single cross
sections.
kBed
1/3
m /s
or
i
n
Friction value of the bed, if not specified in the single cross sections
or cross section slices.
mm
VAW ETH Zürich
Version 10/28/2009
RIV - 3.5-3
20 repeatable
[CompactSerial]
Keyword
DefType
Definition Syntax:
crossSectionNamedistance[referenceHight][kMainChannel][kForelands][slop
e][ soilCode ] …
Example:
20'Qp2'0.05,,,,0.0361
Summary:
General parameters for each cross section.
Parameter
U
T
D
R
Description
crossSectionName
-
s
n
distance
km
r
n
Distance from the upstream boundary in [km]!
referenceHight
m
r
n
Height to which the other heights of the cross section are relative.
kMainChannel
-
r
n
Friction value of the main channel, if not defined differently in the
cross section slices; depending on the specified friction law this is a
Strickler value or an equivalent friction height.
kForelands
-
r
n
Friction value of the forelands, if not defined differently in the cross
section slices; depending on the specified friction law this is a
Strickler value or an equivalent friction height.
slope
-
r
n
Slope of the cross section.
soilCode
-
i
n
Code of a soil defined in the Command file. Valuable for the whole
cross section.
RIV - 3.5-4
VAW ETH Zürich
Version 10/28/2009
21 repeatable
[CompactSerial]
Keyword
DefType
Definition Syntax:
yz[K][typeChange][flownThrough][significantPoint][soilCode]
Examples:
210.00325.8/
217.53316.760 , , , 15 /
2118.04317.230 , , , 16 /
2124.8326.160 , , , ,2 /
Summary:
Describes a point at the interface between cross section slices. For the start point only y
and z can be specified. The other parameters are always referred to the slice on the left
of the point.
Parameter
U
T
y
m
D
R
Description
r
n
Distance from left border.
r
n
Elevation.
r
n
Friction value of the previous cross section slice: depending on the
rel.
z
m
rel.
K
typeChange
m1/3/s
or
specified friction law this is a Strickler value or an equivalent friction
mm
height.
-
i
n
Change of channel type:
12: has foreland on the left
13: has main channel on the left
2: beginning of the zone considered for the computation if it is not the
first point (default).
32: end of the computational area with foreland on the left.
33: end of computational area with main channel on the left.
flownThrough
-
i
n
Indicates if the water flows in the slice or not.
0: flown through (default)
1: not flown through.
VAW ETH Zürich
Version 10/28/2009
RIV - 3.5-5
significantPoint
-
i
n
1: point will be used in the table.
2: defines the lowest point of the cross section.
3: defines the highest point of the cross section.
6: defines the point representing the left dam.
7: defines the point representing the right dam.
15: begin of the bed bottom.
16: end of the bed bottom.
soilCode
-
i
n
Code of a soil defined in the Command file. If sediment transport is “on”, at
least a soil code for the cross section must be specified. Here a soil code
can be specified for a single slice.
RIV - 3.5-6
VAW ETH Zürich
Version 10/28/2009
99
[Serial]
Keyword
DefType
Summary:
Closes the description of a cross section.
end
[Serial]
Keyword
DefType
Summary:
Terminates the list of cross sections.
VAW ETH Zürich
Version 10/28/2009
RIV - 3.5-7
RIV - 3.5-8
VAW ETH Zürich
Version 10/28/2009
VAW ETH Zürich
APPENDIX AND INDEX
Table of Contents
A. Notation
A.1.
A.2.
A.3.
A.4.
Super- and Subscripts ...................................................................................A.1-1
Differential Operators ....................................................................................A.2-1
English Symbols.............................................................................................A.3-1
Greek Symbols ...............................................................................................A.4-1
VAW ETH Zürich
Version 4/23/2007
i
APPENDIX AND INDEX
ii
VAW ETH Zürich
Version 4/23/2007
APPENDIX AND INDEX
A. Notation
A.1. Super- and Subscripts
(.) B
(.)cr
(.)i , (.) j , (.)k
(.) g
(.) L
(.)l
(.)n
(.) R
(.) S
(.) Sub
(.) x , (.) y , (.) z
(.) (.),(.)
(.)(.)(.) ,(.)
VAW ETH Zürich
Version 4/23/2007
Property of top most soil layer for bed load (active layer)
Critical value
Index corresponding to three dimensional Cartesian
3
coordinate system \ with coordinates ( x, y , z )
th
Property corresponding to g grain size class
Property on the left hand side
Lateral property
nth step of time integration
Property on the right hand side
Property at water surface
Property of bed material storage layer (sub layer)
Property corresponding to three dimensional Cartesian
3
coordinate system \ with coordinates ( x, y , z )
Dual property, i.e. QB , x = bed load discharge in x direction
Triple property, i.e. QBg , x = bed load discharge of grain size
class g in x direction
A.1-1
INDEX AND GLOSSAR
A.1-2
VAW ETH Zürich
Version 4/23/2007
APPENDIX AND INDEX
A.2. Differential Operators
d
dx
dn
dx n
∂
∂x
∂n
∂x n
∇
Differential operator for derivation with respect to variable
Differential operator for derivation of order
Version 4/23/2007
n w. r. to var. x
Partial differential operator for derivation w. r. to variable
Partial differential operator for derivation of order
x
n w. r. to var. x
Nabla operator. In three-dimensional Cartesian
coordinate system
T
⎛
⎞
\3 with coordinates ( x, y , z ): ∇ = ⎜⎜ ∂∂x , ∂∂y , ∂∂z ⎟⎟
⎝
VAW ETH Zürich
x
⎠
A.2-1
INDEX AND GLOSSAR
A.2-2
VAW ETH Zürich
Version 4/23/2007
APPENDIX AND INDEX
A.3. English Symbols
Symbol
Unit
Definition
A
a
Ared
c
cf
C
cμ
dg
dm
[m2]
[m/s2]
[m2]
[m/s]
[-]
[-]
[-]
[m]
[m]
F (U) , G (U)
F
g
h
hB
K
k st
ks
m
n
[-]
[N]
[m/s2]
[m]
[m]
[m3/s]
[m1/3/s]
[mm]
[kg]
[m/s]
ng
M
P
P
p
pB
pSub
Q
QB
qBg
qBg , x , qBg , y
qBg , xx , qBg , yy
qBg , xy , qBg , yx
ql
R
S (U)
Sf
SB
Sg
Sf g
[-]
[Ns]
[N/m2]
[m]
[-]
[-]
[-]
[m3/s]
[m3/s]
[m3/s/m]
[m3/s/m]
[m3/s/m]
[m3/s/m]
[m2/s]
[m]
[-]
[-]
[-]
[m/s]
[m/s]
Wetted cross section area
Acceleration
Reduced area
Wave speed
Friction coefficient
Concentration
Dimensionless coefficient (used for turb. kin. viscosity)
Mean grain size of the size class g
Arithmetic mean grain size according to Meyer-Peter &
Müller
Flux vectors
Force
Gravity
Water depth, flow depth
Thickness of active layer
2
Conveyance factor K = k st AR 3
Strickler factor
equivalent roughness height
Mass
Normal (directed outward) unit flow vector of a
computational cell
Total number of grain size classes
Momentum
Pressure
Hydraulic Perimeter
Porosity
Porosity of bed material in active layer
Porosity of bed material in sub layer
Stream- or surface discharge
total bed load flux for cross section
total bed load flux of grain size class g per unit width
Cartesian components of total bed load flux qBg
Cartesian comp. of bed load flux due to stream forces
Cartesian comp. of lateral bed load flux
Specific lateral discharge (discharge per meter of length)
Hydraulic radius
Vector of source terms
Friction slope
Bed slope
suspended load source per cell and grain size class
active layer floor source per cell and grain size class
VAW ETH Zürich
Version 4/23/2007
A.3-1
Sl g
t
U
u
u*
u , v, w
u,v
u B , vB , wB
uS , vS , wS
V
x,y,z
x, y , z
zB
zS
zF
A.3-2
[m/s]
[s]
[-]
[m/s]
[m/s]
[m/s]
[m/s]
[m/s]
[m/s]
[m3]
[-]
[m]
[m]
[m]
[m]
INDEX AND GLOSSAR
local sediment source per cell and grain size class
Time
Vector of conserved variables
Flow velocity vector with Cartesian components ( u , v, w )
shear stress veleocity
Cartesian components of flow velocity vector u
Cartesian components of depth averaged flow velocity
Cartesian components of flow velocity at bottom
Cartesian components of flow velocity at water surface
Volume
Cartesian coordinate axes
Distance in corresponding Cartesian direction
Bottom elevation
Water surface elevation
Elevation of active layer floor
VAW ETH Zürich
Version 4/23/2007
APPENDIX AND INDEX
A.4. Greek Symbols
Symbol
Unit
Definition
αB
[-]
Empirical parameter depending on the dimensionless
Shear stress of the mixture
volumetric fraction of grain size class g in active layer
volumetric fraction of grain size class g in active layer
Eddy diffusivity
Kármán constant
Computational time step
Time step for hydraulic sequence
Time step for sediment transport sequence
Overall, sequential time step
Grid spacing according to three dimensional Cartesian
3
Coordinate system \ with coordinates ( x, y , z )
Molecular viscosity
Kinematic viscosity, μ ρ
Isotropic eddy viscosity
*
Turbulent kinematic viscosity ν t = ν 0 + cμ u h
Base kinematics eddy viscosity
Mass density (fluid)
Bed material density
Cartesian components of bottom shear stress vector
Vector of shear stress at bottom due to water flow
Critical shear stress related to grain size d g
Cartesian components of surface shear stress vector
Vector of shear stress at water surface (e.g. due to wind)
Area of an element
Hiding factor
βg
β g , Sub
Γ
κ
Δt
Δth
Δts
Δt seq
Δx , Δy , Δz
η
ν
νε
νt
ν0
ρ
ρs
τ B , x ,τ B , y
τB
τ Bg ,crit
τ S , x ,τ S , y
τS
Ω
ξg
VAW ETH Zürich
Version 4/23/2007
[-]
[-]
[-]
[-]
[s]
[s]
[s]
[s]
[m]
[kg/ms]
[m2/s]
[m2/s]
[m2/s]
[m2/s]
[kg/m3]
[kg/m3]
[N/m2]
[N/m2]
[N/m2]
[N/m2]
[N/m2]
[m2]
[-]
A.4-1
INDEX AND GLOSSAR
A.4-2
VAW ETH Zürich
Version 4/23/2007

Table of Contents - Basement

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