University of Bristol PhD thesis by Kristín Martha Hákonardóttir on

Transcription

THE INTERACTION BETWEEN
SNOW AVALANCHES AND DAMS
Kristín Martha Hákonardóttir
School of Mathematics
March 2004
A DISSERTATION SUBMITTED TO THE U NIVERSITY
OF
B RISTOL
IN ACCORDANCE WITH THE REQUIREMENTS OF THE DEGREE
OF
D OCTOR
OF
P HILOSOPHY
IN THE
FACULTY OF S CIENCE
Abstract
A series of laboratory experiments has been conducted to study the deflecting and retarding effects
of avalanche defence structures. Incompressible, shallow-layer shock dynamics have been found
to give an accurate description of the interaction between high Froude number, incompressible,
granular flows and relatively tall obstacles, such as deflecting and catching dams. Stationary, oblique
shocks are formed in the two-dimensional interaction of flows with deflecting dams, while normal
shocks that travel upstream from the dams form in the interaction with catching dams. As the height
of a catching dam is lowered compared with the depth of the approaching flow, some flow may
over-top the dam and the upstream motion of the granular bore slows. Flow is projected over dams
in a supercritical flow state when the dams are too low for a change of flow state to occur at their
upstream face. Jets that are launched over catching dams are found to be accurately described as
ballistic trajectories with negligible air drag and the deflection of the jets can be approximated by
the use of ideal fluid flow theory under negligible gravitational effects.
The interaction between high Froude number, granular flows and low catching dams and braking
mounds, where shocks are not present upstream of the obstacles, have also been studied. Laboratory
experiments at a variety of physical scales using different granular materials suggest that the flow
dynamics around such obstacles are governed, to leading order, by the geometry of the obstacles and
the large-scale properties of the flowing avalanche, described by the Froude number, rather than the
micro-scale properties of the granular current. Small-scale laboratory experiments may therefore be
useful tools to study the retarding effect of protection dams and provide guidance in the design of
natural-scale avalanche protection measures.
iii
Acknowledgements
Dear supervisors, Andrew J. Hogg and Tómas Jóhannesson, thank you.
Many thanks to the Swiss Federal Institute of Snow and Avalanche Research, SLF, for the opportunity of visiting and conducting experiments at Weissfluhjoch and in Davos. Special thanks to Felix
Tiefenbacher and Martin Kern at the SLF, and everyone that helped out with the snow shovelling.
Thank you Dieter Issler; Howell Peregrine; Jim McElwaine; Brian, Matt, Steve and Sarah;
Trausti og Þóranna; Jeremy Phillips; Fred and Mike at the workshop; Hallgrímur smiður; Stevie
Haston; Brian the porter; Mokka; Undercover Rock; Veðurstofan; Audrey Fuller; Belle & Sebastian; mamma og pabbi; Hjalti Rafn.
I would like to acknowledge the financial support of the University of Bristol; the Icelandic Research Council; the Icelandic Avalanche Fund; and the European Union avalanche research projects,
Cadzie and Satsie.
v
Author’s Declaration
I declare that the work in this thesis was carried out in accordance with
the Regulations of the University of Bristol. The work is original except
where indicated by special reference in the text and no part of the dissertation has been submitted for any other degree. Any views expressed
in the dissertation are those of the author and do not necessarily represent those of the University of Bristol. The thesis has not been presented
to any other university for examination either in the United Kingdom or
overseas.
Kristín Martha Hákonardóttir
Date: June 8, 2004
vii
Contents
Abstract
iii
Acknowledgements
v
Author’s Declaration
vii
1 Introduction
1
1.1
Snow avalanches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Protection structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.3
Scaling of granular flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.3.1
Non-dimensional parameters . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.3.2
Characteristics of the dense core of dry-snow avalanches . . . . . . . . . . .
7
The thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.4
2 One-dimensional granular flow
11
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2
Formulation of mathematical models of granular flows . . . . . . . . . . . . . . . . 13
2.3
Spatially developing flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4
Experimental setup and design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5
Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.6
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 The interaction between supercritical shallow flows and dams:
a theoretical study
23
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2
Point-mass model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3
Dynamics of shocks in shallow-layer flow . . . . . . . . . . . . . . . . . . . . . . . 29
3.3.1
Hydraulic jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3.2
Normal shocks: catching dams . . . . . . . . . . . . . . . . . . . . . . . . . 31
ix
Contents
3.3.3
3.4
Oblique shocks: deflecting dams . . . . . . . . . . . . . . . . . . . . . . . . 40
Pressure impulse theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4.1
Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4.2
Pressure impulse of a semi-infinite, rectangular wave . . . . . . . . . . . . . 47
3.4.3
The velocity field and the splash-up . . . . . . . . . . . . . . . . . . . . . . 49
4 The interaction between supercritical shallow flows and dams:
an experimental study
52
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2
Short granular flows interacting with deflecting dams . . . . . . . . . . . . . . . . . 53
4.3
4.4
4.5
4.6
4.2.1
Experimental setup and design . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2.2
Flow description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2.3
Results: flow completely deflected by the dams . . . . . . . . . . . . . . . . 57
4.2.4
Results: flow over-topping dams . . . . . . . . . . . . . . . . . . . . . . . . 61
Steady water flows interacting with deflecting dams . . . . . . . . . . . . . . . . . . 64
4.3.1
4.3.2
Results: steady, oblique hydraulic jumps . . . . . . . . . . . . . . . . . . . . 65
4.3.3
Results: splash-up of the flow front . . . . . . . . . . . . . . . . . . . . . . 70
Steady granular flows interacting with deflecting dams . . . . . . . . . . . . . . . . 77
4.4.1
4.4.2
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Steady granular flows interacting with catching dams . . . . . . . . . . . . . . . . . 85
4.5.1
4.5.2
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5 The deflection of a high Froude number granular jet by a dam
91
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2
The deflection of a fluid jet: mathematical formulation . . . . . . . . . . . . . . . . 92
5.3
Experimental setup and design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.4
5.4.1
5.5
Comparison with theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6 The interaction between supercritical shallow flows and braking mounds
105
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.2
Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
x
Contents
6.2.1
Experiments with dry granular flows . . . . . . . . . . . . . . . . . . . . . . 110
6.2.2
Experiments with snow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.3
Experimental design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.4
6.5
6.4.1
Flow description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.4.2
The interaction between the flow and the obstacles . . . . . . . . . . . . . . 115
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7 Conclusions
127
A Mohr-Coulomb failure criterion
131
A.1 Stresses and failure in soil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
A.2 Failure in a moving pile of granular material . . . . . . . . . . . . . . . . . . . . . . 133
B The Schwarz-Christoffel transformation
135
xi
List of Figures
1.1
A satellite image of Iceland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
Schematic plan-view of protection structures in the run-out zone of an avalanche. . .
3
1.3
Photographs of recently constructed defence structures above Flateyri and Neskaupstaður in Iceland. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
4
Schematic diagram of the geometry of a finite mass of granular material moving
along a rigid plane inclined at an angle ξ to the horizontal. . . . . . . . . . . . . . . 13
2.2
Schematic diagram of the change in the speed of a one-dimensional current flowing
down an inclined plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3
Schematic diagram of the change in depth of a one-dimensional current flowing
down an inclined plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4
Schematic side-view of the experimental chutes. . . . . . . . . . . . . . . . . . . . . 18
2.5
Plot of the Froude number of flows down one-dimensional chutes as a function of
the mass of material released down the chutes. . . . . . . . . . . . . . . . . . . . . . 20
3.1
Schematic diagram for calculating the height of a catching dam. . . . . . . . . . . . 23
3.2
Schematic diagram of the geometry used in the point-mass model. . . . . . . . . . . 25
3.3
Schematic diagram of the path of a point-mass on a dam face. . . . . . . . . . . . . 27
3.4
The computed path of a point-mass on a deflecting dam. . . . . . . . . . . . . . . . 28
3.5
Schematic diagram of a hydraulic jump. . . . . . . . . . . . . . . . . . . . . . . . . 29
3.6
Schematic diagram of supercritical flow over a bump. . . . . . . . . . . . . . . . . . 31
3.7
Plot of the computed obstacle height necessary for supercritical, frictionless flow to
become critical at the top of the obstacle, as a function of the Froude number. . . . . 32
3.8
Schematic diagram of a bore travelling upstream from a catching dam. . . . . . . . . 33
3.9
Plot of: (a) the theoretical depth ratio between the bore and the approaching flow
as a function of the Froude number; (b) the theoretical speed ratio between the bore
and the oncoming flow as a function of the Froude number. . . . . . . . . . . . . . . 34
xiii
List of Figures
3.10 Schematic diagram of a granular bore travelling up an inclined plane, away from a
catching dam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.11 Plot of the bore speed as a function of the distance travelled upstream by a bore. . . . 36
3.12 Schematic diagrams of the interaction of supercritical flow with dams of different
heights. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.13 Plot of the minimum height of a dam for a bore to be maintained upstream of the
dam as a function of the Froude number. . . . . . . . . . . . . . . . . . . . . . . . . 39
3.14 Plot of the calculated height of a deflecting dam for supercritical, frictionless flow to
become critical at the top of the dam. . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.15 Schematic diagram of steady, uniform flow interacting with a deflecting dam and
forming a stationary oblique shock. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.16 Plot of the shock angle as a function of the deflecting angle for different values of
the Froude number. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.17 Plots of the theoretical shock angle and shock height in a stationary, oblique shock
as functions of the Froude number and the deflecting angle. . . . . . . . . . . . . . . 44
3.18 Schematic diagram of impact pressures on a dam in the dense flow of a dry-snow
avalanche. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.19 Schematic diagram of a wave impact of a rectangle of fluid on a wall. . . . . . . . . 47
3.20 Plot of the computed pressure impulse at a wall for a semi-infinite, rectangular wave.
48
3.21 Plot of the computed pressure induced velocity at the free surface of a semi-infinite
wave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.22 Schematic diagrams of a deflecting dam on a slope. . . . . . . . . . . . . . . . . . . 50
4.1
Schematic diagrams of the experimental chute in the short duration, granular, deflecting dam experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2
Measured flow speed plotted as a function of time for the short, granular flows. . . . 55
4.3
Plot of the measured leading edge run-out and the front speed as a function of the
sequential number of the experiments for the short, granular flows. . . . . . . . . . . 56
4.4
Plot of the run-up profiles of the short, granular flows on deflecting dams. . . . . . . 57
4.5
Non-dimensional run-up profiles of the short, granular flows on deflecting dams, for
dams of different lengths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.6
Measured non-dimensional run-up profiles for the short granular flows on deflecting
dams. Also plotted are the theoretical non-dimensional run-up of a point mass and
the theoretical non-dimensional shock height. . . . . . . . . . . . . . . . . . . . . . 60
4.7
Measured and theoretical shock angles plotted as a function of the deflecting angle
for the short, granular flows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
xiv
List of Figures
4.8
Schematic plan-view of flow over-topping a deflecting dam. . . . . . . . . . . . . . 62
4.9
Plot of the overflow angle as a function of the ratio between the dam height and (a)
the depth of the oncoming flow; (b) the theoretical shock depth. . . . . . . . . . . . . 63
4.10 Photograph of the experimental setup for rapid flows of water. . . . . . . . . . . . . 64
4.11 Photographs of stationary, oblique shocks of supercritical shallow-water. . . . . . . . 66
4.12 Non-dimensional run-up profiles of steady water flow along deflecting dams at three
different Froude numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.13 Schematic diagram of the adjustment region along a deflecting dam. . . . . . . . . . 69
4.14 Plot of the shock angle as a function of the deflecting angle for the three different
Froude numbers in the water experiments. . . . . . . . . . . . . . . . . . . . . . . . 70
4.15 Measured non-dimensional run-up profiles for water flow at Fr 1 = 4:5 plotted along
with the theoretical non-dimensional shock depth. . . . . . . . . . . . . . . . . . . . 71
4.18 Measured non-dimensional splash-up of the flow front of water plotted along dams
at different deflecting angles, along with the theoretical prediction from the pressure
impulse theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.19 Measured non-dimensional maximum splash-up plotted as a function of the deflecting angle of the dams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.20 Photograph of the experimental setup for steady granular flows. . . . . . . . . . . . . 77
4.21 Photographs of stationary, oblique, granular shocks. . . . . . . . . . . . . . . . . . . 78
4.22 Measured non-dimensional run-up profiles along deflecting dams at different deflecting angles for steady granular flow at three different Froude numbers, Fr 1 = 5 12 14
80
4.23 The shock angle plotted as a function of the deflecting angle for the three different
Froude numbers of the steady granular flows. . . . . . . . . . . . . . . . . . . . . . 80
4.24 Measured non-dimensional run-up profiles for steady granular flow at Fr 1 = 5 plotted
along with the theoretical non-dimensional shock height. . . . . . . . . . . . . . . . 82
4.25 Measured non-dimensional run-up profiles for steady granular flow at Fr 1 = 12 plotted along with the theoretical non-dimensional shock height. . . . . . . . . . . . . . 83
4.26 Measured non-dimensional run-up profiles for steady granular flow at Fr 1 = 14 plotted along with the theoretical non-dimensional shock height. . . . . . . . . . . . . . 84
4.27 Photographs of steady granular flow at Fr 1 = 12 interacting with catching dams of
different heights. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
xv
List of Figures
4.28 Measured bore-speed plotted as a function of the distance travelled upstream from a
catching dam by a bore, for steady granular flows at Fr 1 = 12 and 14. . . . . . . . . 87
5.1
Schematic diagram of the deflection of a jet by a dam. . . . . . . . . . . . . . . . . . 92
5.2
Schematic diagram of the geometrical mappings of the deflection of a jet from the z
plane to the f plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.3
Plot of the theoretical throw angle as a function of the non-dimensional dam height. . 97
5.4
Schematic diagram of an experimental chute along with the side-view of a dam. . . . 98
5.5
Photograph of a granular jet as it detaches from the top of an obstacle. . . . . . . . . 99
5.6
Schematic side-view of the trajectory of a jet of particles being launched from the
top of a dam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.7
Plot of the geometry of jets over dams of different heights. . . . . . . . . . . . . . . 100
5.8
Plots of measured and calculated throw angles of jets launched from the top of dams
as a function of the non-dimensional dam height. . . . . . . . . . . . . . . . . . . . 101
5.9
Plot of measured and calculated throw angles of jets launched from the top of dams
as a function of the non-dimensional dam height. . . . . . . . . . . . . . . . . . . . 102
6.1
Schematic diagram of the laboratory chutes. . . . . . . . . . . . . . . . . . . . . . . 108
6.2
Schematic diagram of the experimental chute at Weissfluhjoch. . . . . . . . . . . . . 108
6.3
Sketch of the different obstacle setups. . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.4
Schematic diagram of a supercritical granular current jumping over a dam. . . . . . . 115
6.5
Plot of the geometry of jets over dams of different heights. . . . . . . . . . . . . . . 116
6.6
Plot of the throw angle as a function of the non-dimensional dam height. . . . . . . . 117
6.7
Photograph of snow being deflected over and around two mounds at the Weissfluhjoch chute. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.8
Photograph of wedges left upstream of mounds after a wet-snow experiment on the
Weissfluhjoch chute. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.9
Plot of the ratio of the flow’s mechanical energy before and after interacting with a
dam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.10 Plot of the measured leading edge run-out as a function of the obstacle height. . . . . 121
6.11 Plot of the measured mass centre run-out as a function of the obstacle height. . . . . 122
6.12 Plot of the ratio between the flow speed at the upper and lower sensors on the Weissfluhjoch chute as a function of the obstacle height.
. . . . . . . . . . . . . . . . . . 123
6.13 Photograph of one braking mound above Neskaupstaður, Iceland. . . . . . . . . . . . 125
A.1 Schematic diagram of stresses on a soil element. . . . . . . . . . . . . . . . . . . . . 131
A.2 Schematic diagram of two-dimensional principal stresses. . . . . . . . . . . . . . . . 132
xvi
A.3 Schematic diagram of Mohr’s circle of stress for cohesive and non-cohesive materials.132
A.4 Schematic diagram of Mohr’s circle of stress for a moving pile of cohesionless granular material, with sliding on the bed surface. . . . . . . . . . . . . . . . . . . . . . 134
B.1 Schematic diagram of the Schwarz-Christoffel transformation. . . . . . . . . . . . . 136
List of Tables
2.1
Flow properties of the different granular flows. . . . . . . . . . . . . . . . . . . . . 18
4.1
The estimated interior speed of the short duration granular flows during the steady
flow phase for the different deflecting angles. . . . . . . . . . . . . . . . . . . . . . 57
4.2
The flow regime of the steady stream of water flow. . . . . . . . . . . . . . . . . . . 65
4.3
The flow regime of the steady stream of granular flow. . . . . . . . . . . . . . . . . . 79
4.4
The measured and calculated stopping position of a bore. . . . . . . . . . . . . . . . 88
5.1
The flow parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.1
The dimensions of the different experimental chutes. . . . . . . . . . . . . . . . . . 109
6.2
Material properties of the different granular flows. . . . . . . . . . . . . . . . . . . . 109
6.3
The geometry of the different obstacle setups. . . . . . . . . . . . . . . . . . . . . . 113
6.4
The flow regime in each experimental series. . . . . . . . . . . . . . . . . . . . . . . 114
xix
List of Notations
The symbols listed below are followed by a brief statement of their meaning, a reference to an
illustrative figure when applicable and the number of the page on which they are defined in the text.
u, flow speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
g, gravitational acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
h, flow depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
H, height of an obstacle measured normal to a surface, see Figure 5.1 . . . . . . . . . . . .
5
c, speed of sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
s, shear strength of granular material . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
ρ, density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
µeff , effective friction coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
ξ, slope inclination measured relative to the horizontal, see Figure 3.2 . . . . . . . . . . .
5
d, particle diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
Fr, Froude number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
Frb , obstacle Froude number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
Ma, Mach number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
e, coefficient of restitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
Re, Reynolds number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
ν, kinematic viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
H =B, aspect ratio of an obstacle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
A0 , proportion of a flow path covered by obstacles . . . . . . . . . . . . . . . . . . . . . .
7
µ, Coulomb friction coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
CD , dimensionless coefficient representing the effect of turbulent friction . . . . . . . . . . 11
t, time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
λ, dimensionless constant representing dissipation in the impact of flow with a catching dam 24
γ, deflecting angle of a dam, see Figure 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 26
m, mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
xxi
α, inclination of an upstream dam face relative to the surface, see Figures 3.2 . . . . . . . 26
D, dimensional friction coefficient representing the effect of turbulent friction . . . . . . . 26
∆l, length of a transition zone of a hydraulic jump, see Figure 3.5 . . . . . . . . . . . . . . 29
Fr1 , Froude number of the flow approaching a granular/hydraulic jump . . . . . . . . . . . 29
]], jump brackets
g? ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
the component of the gravitational acceleration normal to a surface . . . . . . . . . . . 30
U, upstream propagation speed of a bore, see Figure 3.8 . . . . . . . . . . . . . . . . . . . 33
β, shock angle of an oblique shock measured relative to the direction of the approaching
flow, see Figure 3.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
γmax , the maximum possible deflection for an oblique shock to remain attached to a deflecting dam, see Figure 3.16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
P, pressure impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
θ, the angle at which a jet detaches from the top of a dam, also termed throw angle, see
Figure 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
l, length of an experimental chute, see Figure 6.1 . . . . . . . . . . . . . . . . . . . . . . 109
b, width of an experimental chute, see Figure 6.1 . . . . . . . . . . . . . . . . . . . . . . 109
φ, internal friction angle of a material, see Figure A.3 . . . . . . . . . . . . . . . . . . . . 132
C, cohesion, see Figure A.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
δ, dynamic bed friction angle, see Figure A.4 . . . . . . . . . . . . . . . . . . . . . . . . 133
K, earth pressure coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
P, pressure tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
pi j , the ij-th component of the pressure tensor . . . . . . . . . . . . . . . . . . . . . . . . 133
Chapter 1
Introduction
Snow avalanches have caused many catastrophic accidents and severe economic losses in Iceland
since the country was settled in the nineth century, and they pose a threat to most settled areas (see
Figure 1.1). Nearly 60% of the 193 people that have been killed in snow avalanches, slush flows
and landslides since 1901 were killed in buildings, at work sites or within towns (Jóhannesson and
Arnalds, 2001). The remaining 40% were killed on roads or travelling in backcountry areas. Snow
avalanches and slush flows claimed most of the lives, while 27 people were killed by landslides.
The economic loss caused by avalanches in Iceland is enormous. Direct loss and cost of rescue
and relief operations during 1974–2000 have been estimated 41.5 million USD (August 2000 price
levels, Jóhannesson and Arnalds, 2001).
Two catastrophic avalanches in the north west of Iceland in 1995, claiming 34 lives, changed
the view regarding avalanche safety in settled areas in Iceland. After those two extreme events, the
government decided to construct avalanche defence measures for hazard areas and/or to purchase
endangered property, in order to reduce the risk of being caught in an avalanche.
However the fundamental dynamics of the interaction between snow avalanches and defence
structures remain poorly understood. There is therefore a pressing need to improve our understanding of these flows for the successful design of protection structures against catastrophic avalanches.
1.1 Snow avalanches
Snow avalanches are rapid gravity-driven currents. They are much longer than they are deep (shallow) and travel at high speeds. Snow avalanches can be wet or dry and a distinction is often made
between powder and dense dry-snow avalanches.
Powder-snow avalanches are thought to consist of a highly dispersed snow and air mass with
a density of the order 10 kg m ;3 , moving as one snow-air cloud (McClung and Schaerer, 1993;
1
Chapter 1. Introduction
Figure 1.1: A satellite image of Iceland on the 28th of January, 2004. The island is settled all along
the coast by a population of just under 300,000 people. The image was obtained at the nasa website:
http://earthobservatory.nasa.gov/.
Bozhinskiy and Losev, 1998).
Pressure measurements of dense dry-snow avalanches have, on the other hand, revealed a layered
structure of such avalanches, hidden from sight in a white cloud (Salm, 1964; Kotlyakov et al.,
1977; Schaerer and Salway, 1980). A dense core is present at the bottom, a saltation layer above
and possibly also ahead of the avalanche, and a suspension layer is travelling on top. The dense core
has been measured between 1–3 m thick with a density in the range 250–500 kg m ;3 . The saltation
layer may be 2–5 m thick with densities in the range 10 to 50 kg m ;3 . It consists of powder and
lumps of snow, tossed upward by turbulent motion at the surface of the dense flow. The suspension
layer on the top of the avalanche consists of powder, free of snow lumps, which may reach densities
up to 5 kg m;3 and a thickness of over 20 m. The saltation and suspension layers may eventually
move ahead of the dense flow. Flow speeds in large avalanches range between 30–60 m s ;1 , but peak
velocities up to 80 m s;1 have been measured (Issler, 2003). The dense core of dry-snow avalanches
is the densest part of the avalanche. It is therefore a very destructive part of the avalanche and is of
main interest to this study.
Wet-snow avalanches (and slush flows) are denser than dry-snow avalanches. They tend to flow
as one layer, without the saltation and suspension layers that are observed in dry-snow avalanches.
The friction at the sliding surface is higher, which causes them to flow more slowly than dry-snow
avalanches (McClung and Schaerer, 1993). Wet-snow avalanches follow features in the terrain more
closely than dry-snow avalanches, which makes it easier to divert and stop the motion of such
2
1.2. Protection structures
Avalanche
Avalanche
Braking mounds
γ
De
fle
cti
ng
da
m
Catching dam
Figure 1.2: A schematic plan-view of protection structures in the run-out zone of an avalanche.
avalanches. These types of avalanches are, therefore, not the most hazardous ones and will not
be discussed further in the thesis.
1.2 Protection structures
Snow avalanche protection structures in the run-out zone of an avalanche can be divided into three
categories: deflecting dams; catching dams; and braking mounds (Figure 1.2). Deflecting dams are
built to completely deflect an avalanche. The height of the structures is often between 10 and 20 m,
depending on the deflecting angle of the dam. Catching dams are constructed perpendicular to the
flow direction of an avalanche and are higher structures than deflecting dams. They are intended to
completely stop avalanches, usually when deflecting the avalanche is not an option. Those structures
may be up to 30 m high. Braking mounds are protection measures built to slow avalanches down.
They are small catching dams with aspect ratios, here defined as the ratio of their height to width,
close to one. Braking mounds are arranged in rows that are perpendicular to the flow direction. The
mounds are usually lower than dams, with a height up to 10 m. Several rows of braking mounds are
typically positioned upstream of a catching dam, in order to reduce the necessary height of the dam,
designed to stop the avalanche. Figure 1.3 shows photographs of recently constructed deflecting and
catching dams above the village Flateyri, Iceland, and two rows of braking mounds upstream of a
catching dam above the town Neskaupstaður, Iceland.
The dynamics of the interaction between granular flows such as snow avalanches and obstacles
remain poorly understood and a basis for the design of avalanche braking mounds, catching dams
and deflecting dams is therefore lacking. Important questions regarding the design of dams and
mounds address the height, size and shape of the structures and how these control the retarding
effect.
3
Figure 1.3: The photograph on the left hand side is of two deflecting dams connected by a catching
dam above the town Flateyri in western Iceland. They were constructed in 1997, two years after a
catastrophic avalanche hit the town. Each deflecting dam is approximately 600 m long and 15–20 m
high. The catching dam is 10 m high and approximately 350 m long (Sigurðsson et al., 1998). The
photograph on the right shows two rows of 10 m high and 30 m wide (measured at the base) braking
mounds upstream of a 17 m high catching dam above the town Neskaupstaður in eastern Iceland
(Tómasson et al., 1998). The construction of the protection measures was completed in 2001.
There is only one full-scale avalanche test site equipped with a catching dam. The site is located
in Ryggfonn in western Norway. A study of the retarding effect of the Ryggfonn dam for avalanches
released in the period 1983–2000 has recently been published by the Norwegian Geotechnical Institute (Lied et al., 2002). The study is based on observed avalanche deposits and measurements and
estimates of flow velocities and the thickness of the avalanches before hitting the dam, and provides
an extremely valuable data-set. The experimental site at Ryggfonn is currently being reequipped (as
a part of the European Union project Satsie) with radars for velocity and thickness measurements of
the avalanches.
Experiments that entail full-scale avalanches are, however, expensive, dangerous and highly dependent on weather conditions. For these reasons, experiments on smaller scales using dry granular materials and snow can be useful in order to study the dynamics of the much larger natural
snow avalanches. The motion of dry granular currents has been studied experimentally by a number of researchers, including e.g. Savage and Hutter, (1989); Johnson and Jackson (1990); Hutter
et al. (1995); Wieland et al. (1999); Louge and Keast (2001); and Pouliquen and Forterre (2002).
Experiments on the interaction between dry granular flows and obstacles have received less attention, but include Chu et al. (1995); Rericha et al. (2002); Faug et al. (2002) and 2003); and Gray
et al. (2003). Furthermore small-scale, laboratory experiments on the motion of fluidised snow and
ice spheres have been conducted by Nishimura (1990); Nishimura et al. (1991) and (1993); Bouchet
et al. (2003); and Tiefenbacher and Kern (2004). Salm (1964) studied impact forces on obstacles
by flowing snow on a large snow chute and Bozhinskiy and Losev (1998) describe similar studies
where the obstacles are moving and the snow is stationary.
4
1.3. Scaling of granular flows
1.3 Scaling of granular flows
There remain a number of unresolved issues when using laboratory-scale experiments to understand
the dynamics of natural-scale snow avalanches. In contrast to fluid flow, it remains unclear how to
maintain dynamical similarity in granular flow.
1.3.1 Non-dimensional parameters
Various non-dimensional parameters can be formulated from combinations of parameters governing
the flow of the dense core of an avalanche over an obstacle (Bozhinskiy and Losev, 1987; Lied et al.,
2002). The parameters include the flow speed, u; the gravitational acceleration, g; the flow depth,
h; the obstacle height, H; the speed of sound in the avalanche, c; the shear strength of the granular
material, s; the flow density, ρ; the effective friction coefficient, µeff ; the slope inclination, ξ; the
particle diameter, d; and the coefficient of restitution, e. Non-dimensional combinations of these
parameters are:
u2
;
g cos ξh
s
;
ρu2
u
;
c
sin ξ;
µeff ;
h=d;
H =h;
e:
The Froude number of the flow is defined by
Fr2 =
u2
g cos ξh
and is commonly used to scale free-surface fluid flow, i.e. if viscous effects are negligible, it is
assumed that flows with a similar Froude number exhibit similar dynamical features. The Froude
number represents the ratio of the flow velocity to the maximum speed of free surface gravity waves
in the flow. The flow is supercritical when Fr > 1 and disturbances cannot travel upstream in the flow,
leading to a possibility of shock formation (hydraulic/granular jumps) during the interaction with an
obstacle (Whitham, 1999). The flow is subcritical when Fr < 1 and disturbances can propagate
upstream in the flow.
A more careful study of the dimensionless numbers listed above for granular flows reveals the
importance of the Froude number scaling in free-surface granular flows. The effective friction coefficient, µeff , can be related to the Froude number if Coulomb-type friction ( hρg cosξ) and turbulent
drag ( ρu2 ) are the dominant retarding forces in avalanche flow (see discussion in Chapter 2):
µeff =
FT R
FN
+ ρCDu = µ + C Fr2
= ρgµ cosξh
D
ρg cosξh
2
where FTR is the resistive force in the flow direction, FN is the normal force on the underlying
boundary, and µ and CD are non-dimensional friction coefficients. The ratio of material strength
;
to dynamical stresses, s= ρu2 , can also be written as a function of the Froude number if the shear
strength of granular materials obeys the Coulomb law, and hydrostatic, isotropic pressure distribution
5
in the flow is assumed (see Appendix A):
s
ρu2
tan φ
φ
= C + ρhgρucosξ
= ρuC2 + tan
2
2
Fr
where C is the cohesion of the granular material and φ is the internal friction angle of the material.
Another important dimensionless number is the ratio between the height of an obstacle and the depth
of the flow approaching the obstacle, H =h. This ratio will be referred to as the ‘non-dimensional
obstacle height’. The value of the obstacle Froude number, which is defined by
Fr2b =
u2
g cos ξH
can be determined from the internal Froude number of the flow and the non-dimensional obstacle
height (Fr2b = Fr2 h=H).
The remaining non-dimensional numbers include sin ξ, which represents the slope of the underlying boundary; and h=d which is the ratio between the flow depth and the particle diameter. Furthermore terrain roughness relative to the particle diameter may affect the velocity profile through
the depth of the flow. The compressibility of the flow is represented by the Mach number
Ma =
u
:
c
The Mach number represents the ratio between the flow speed and the speed of pressure disturbances
in the flow (speed of sound in the flow, c). If Ma exceeds unity, the flow is supersonic and compression shocks may propagate in the flow. The coefficient of restitution accounts for energy loss in
inelastic collisions between grains. It is defined by the ratio of the relative initial velocities to the
relative final velocities in a head-on particle collision
e=
jv2 ; v1j
ju 2 ; u 1 j
where u1 and u2 are the initial velocities of particles 1 and 2 and v1 and v2 are the final velocities.
The value of the coefficient ranges between 0 and 1 and is nearly constant for any two bodies. The
Reynolds number in fluid flow is given by
Re =
uh
ν
where ν is kinematic viscosity. If Re is not large, then viscous forces play a dynamical role in
the flow, but when Re is sufficiently large, viscous forces are negligible, sufficiently far from solid
boundaries. It is, however, unclear how to determine viscosity in granular flows.
It is also important to account for geometrical similarity of the obstacles. Non-dimensional
combinations of the obstacles’ dimensions include:
H =B;
A0 ;
6
α
1.3. Scaling of granular flows
where H =B is the aspect ratio of an obstacle, defined as the height to width ratio of an obstacle, A 0
is the proportion of the flow path of an avalanche (or the width of an avalanche) which is covered
by obstacles, and α is the inclination of the upstream face of an obstacle relative to the underlying
boundary.
1.3.2 Characteristics of the dense core of dry-snow avalanches
A detailed paper reviewing experimental information on the dynamics of dry-snow avalanches, has
recently been published (Issler, 2003). Issler’s (2003) main results regarding the values of the nondimensional numbers, discussed above, characterising the flow of the dense core of natural dry-snow
avalanches are summarised below.
The dense core of a natural dry-snow avalanche is turbulent and supercritical with a Froude
number in the range 5–10.
The compressibility of the dense core is not well understood. The difference between the density of snow in the starting zone of an avalanche and the density of avalanche debris may indicate the compressibility to some extent. The density of dry-snow avalanche debris rarely exceeds
600 kg m;3 . The avalanches that interacted with the catching dam in Ryggfonn, Norway were found
to increase in density by 43 %, from 300 kg m ;3 in the starting zone to 430 kg m ;3 near the dam
(Lied et al., 2002). Whether snow avalanches are sub- or supersonic is open to investigation. Dense
dry-snow avalanches do not need to be highly compressible judging by an avalanche which interacted
with the eastern deflecting dam in Flateyri, Iceland in 1999 (Jóhannesson, 2001). The avalanche was
most likely not highly compressible (1 < Ma) since the channelised part of the avalanche at the dam
face was much thicker than the depth of the undisturbed avalanche further away from the dam. It
is, however, possible that fast flowing and dilute natural dry-snow avalanches may be supersonic
(Briukhanov et al., 1967).
The cohesion of snow depends strongly on temperature, but is typically on the order of 0.5–1 kPa
(Issler, 2003). Dynamical stresses in avalanche flow are much larger than the cohesion during the
flowing phase of the motion, but it may be important once the flow has slowed down sufficiently and
is close to stopping. The coefficient of restitution does also depend on the properties of the avalanche
snow, and will be larger for ice crystals than wet snow.
Snow avalanches can consist of snow lumps with diameters up to 0.1 m down to powder and
crystals with diameters on the order of 0.001 m. The ratio between the depth of the dense core and
the particle diameter in snow avalanches may therefore be in the range 100–1000.
Measurements of velocity profiles within the dense core show a shear layer at the bottom, extending over 10–30 % of the flow depth, and a region of little or no shear above. The measurements
do not have sufficient spatial resolution to determine whether there is a finite slip velocity at the
base. Issler (2003) concludes that the measurements indicate that the depth-averaged velocity of the
7
avalanche is 5–15 % smaller than the maximum velocity at the surface and the approximation of a
rectangular velocity profile through the depth of the avalanche may be justified as a first approximation.
Defence structures, such as dams and braking mounds, are usually positioned in the run-out zone
of snow avalanches, where the terrain is sloping at less than 10 to the horizontal. A typical value of
the inclination of the starting zone of dense dry-snow avalanches is between 35–45 (McClung and
Schaerer, 1993).
1.4 The thesis
The study presented here is motivated by the need for a better understanding of the physical behaviour of the interaction of the dense core between natural dry-snow avalanches and defence structures. The main objective of this study is to identify the flow behaviour in the interaction between
dense granular flows and obstacles through small-scale laboratory experiments. The laboratory flows
are scaled with the internal Froude number of the flows in order to maintain dynamical similarity
with the dense core of natural dry-snow avalanches. The study is restricted to relatively incompressible flows, and the experimental observations are interpreted by analogy with supercritical, inviscid,
shallow, free-surface fluid flows. The experiments are conducted at different physical scales in order
to investigate whether Froude number scaling accounts for dynamical similarity in the interaction of
shallow, supercritical granular flows with obstacles of different sizes and shapes.
The organisation of the thesis is as follows. The spatial development of one-dimensional, shallow, steady granular flows down fixed inclines is studied in Chapter 2 through depth averaged equations of motion of granular flows. From there on, the thesis is dedicated to the interaction between
granular flows and obstacles. Different theoretical approaches to modelling the one-dimensional
interaction of supercritical flows with high catching dams and the two-dimensional interaction with
deflecting dams are reviewed in Chapter 3. Chapter 4 describes experiments with dry granular flows
and water flows, designed to study the interaction between the flows and deflecting dams and high
catching dams, and to compare the observations with the theories described in the previous chapter.
The interaction between granular flows and braking mounds and low catching dams is studied in
Chapters 5 and 6. Chapter 5 reviews a theoretical study used to predict the deflection of an inviscid
fluid jet by an obstacle under negligible gravitational effects. The theory is compared with results
from an experimental study of the deflection of high Froude number, granular flows by low catching dams. The experimental study finishes with Chapter 6, in which the retarding effect of braking
mounds is studied on different physical scales. Each chapter starts with an introduction containing a
description of the contents of the different sections within the chapter, and most chapters finish with
a conclusions section where experimental results are summarised and interpreted in terms of natural
8
1.4. The thesis
dry-snow avalanches. The thesis ends with concluding remarks regarding the design of natural-scale
avalanche protection dams in Chapter 7.
9
Chapter 2
One-dimensional granular flow
2.1 Introduction
There has been a longstanding debate within the scientific community studying granular flows regarding the nature of frictional forces in such flows. If granular materials slide on the underlying
base, they are subject to a Coulomb-like frictional force, which is proportional to the normal force on
the base, and is similar to the resistance experienced by a sliding solid object. If there is no sliding
on the base, shearing in the flow close to the base will generate fluid-like friction that is proportional
to the square of the flow velocity, if the flow is sufficiently rapid. Alternatively, the friction could be
a combination of both types.
The Swiss engineer, Voellmy (1955), proposed an equation for the flow of a snow avalanche
down a slope, of the form
du
dt
= g sin ξ ; µg cosξ ; ChD u2
(2.1)
where u is the speed of the avalanche, ξ is the inclination of the slope, g is the gravitational acceleration, h is the depth of the avalanche, µ is a friction parameter representing Coulomb type of
friction at the base, here µ = tan δ and δ is the dynamic friction angle between the avalanche and
the underlying slope. A dimensionless constant representing the effect of turbulent friction on the
flow is denoted by CD . According to Voellmy, the flow is therefore subject to both types of frictional
forces. The drag term proportional to u 2 is often termed ‘turbulent friction’.
From to equation (2.1), if the turbulent friction term is omitted, then the flow is unable to reach a
steady state on a slope unless µ = tan ξ, which implies that δ = ξ. Measurements of snow avalanche
motion show that most avalanches accelerate until they reach a terminal velocity in the track and
travel at an approximately constant speed down-slope, even though ξ > δ (Issler, 2003). This therefore provides some evidence of the importance of velocity-dependent drag to large-scale flows.
11
Chapter 2. One-dimensional granular flow
Salm (1993) interprets the turbulent friction term as shear resistance due to inelastic impacts between snow spheres close to the ground and the ground itself. He suggests that the turbulent friction
coefficient, CD , depends upon the terrain roughness which he finds is consistent with observations
of the flow behaviour of natural dry-snow avalanches. According to Salm (1993) the flow of such
avalanches can be divided into three regimes: in a fracture zone with smooth terrain, Coulomb friction dominates and leads to high accelerations; as soon as rugged terrain is reached the resistance
increases, due to additional velocity-dependent friction and the avalanche reaches a terminal velocity; at lower velocities in the run-out zone, the avalanche decelerates and the motion becomes
dominated by Coulomb friction.
According to Voellmy’s equation, a model omitting the turbulent friction term (C D = 0), would be
independent of the flow depth (size) of an avalanche (if the avalanche does not deform, i.e. ∂h=∂x =
0). That is in contradiction with run-out measurements of avalanches of different sizes, where larger
avalanches are observed to have longer run-outs than smaller avalanches (Issler, 2003). This could,
however, be compensated in the model by the use of a non-constant basal friction angle, prescribed
to diminish with increasing avalanche size. Issler (2003) finds that this only highlights the fact that
such a model does not capture the dominant processes in avalanche flow.
Ancey and Meunier (2004) have studied the effective frictional force experienced by 15 documented avalanches of different sizes. They observed the same three frictional regimes in all of
the studied avalanches: an inertial regime, where the frictional force was low; a velocity-dependent
regime, where the frictional force revealed a complex relationship to the flow speed; and a Coulomb
regime, where the frictional force was independent of the flow speed. The complex relationship of
the frictional force in the velocity dependent regime had no universal scaling with a drag term proportional un , where n is a constant and u is the flow speed. The physical meaning of these results,
however, remains unanswered.
In this chapter, we review different theoretical approaches to modelling granular flows, and formulate a depth averaged approach (§2.2). The flow behaviour of granular flows is explored in §2.3,
by studying the spatial development of one-dimensional, steady, granular flows down fixed inclines,
through the depth averaged, shallow-layer equations of motion for granular flows that are subject to
both Coulomb friction and turbulent friction at the base. We show that when the granular flow has
reached a uniform state down the slope, the Froude number of the flow is determined by the slope
angle, the dynamic friction angle between the flow and the underlying base, and a turbulent friction
coefficient. This theoretical result is then compared with experimental studies of one-dimensional
granular flows down fixed inclines at different physical scales. The experimental setup and design
is described in §2.4 and experimental observations are presented in §2.5 along with observations of
flowing snow down a large chute studied by Kern et al. (submitted).
12
2.2. Formulation of mathematical models of granular flows
below, since we make use of this type of a model in the following section.
The conservation equations of mass and momentum for a granular continuum with a constant
bulk density are
∇u
∂u
+(u ∇)u
∂t
= 0
= ; ρ1 ∇ P + g
where u is the velocity vector, ρ is density, P is the pressure tensor, g is gravitational acceleration
and t is time.
The material is regarded as a cohesionless granular body obeying a Mohr-Coulomb yield criterion with a constant internal friction angle, φ (see Appendix A). Boundary conditions at the free surface are expressed in terms of a function FS (x t ) which is zero for a particle there, i.e. at FS (x t ) = 0
we have
∂FS
∂t
+ u ∇FS = 0
which is a kinematic statement that the free surface is materially conserved and
P n̂ = 0
which expresses stress-free conditions at the free surface where n̂ is the exterior unit normal. At the
basal interface, friction is tangential and is assumed to consist of two components
S = SC + SV where SC = jNj tan δu=juj and N is the normal pressure acting on the interface and δ is the dynamic
bed friction angle. The second term, SV , is a turbulent drag term, defined by SV
= ρ(CD h)ujuj,
=
where CD is a dimensionless friction coefficient and h is the flow depth. The equations are formulated
in a fixed cartesian coordinate system, Oxz, with the x-axis inclined along the downslope direction, at
an angle ξ to the horizontal, the z-axis being the upward pointing normal, perpendicular to the slope,
as shown in Figure 2.1, and the flow velocity u = (u w). Due to the shallowness of the flow, vertical
accelerations are negligible and so the normal pressure within the avalanche is hydrostatic while the
component of the pressure field parallel with the slope, p xx , is linked to the normal pressure, pzz ,
through an earth pressure coefficient, defined by
K=
pxx
:
pzz
Values of the earth pressure coefficient may be derived assuming a Mohr-Coulomb plastic behaviour
for the yield on the basal sliding surface,
K=
8
< K
act
for
∂u=∂x > 0
: K
pass
for
∂u=∂x < 0
14
2.3. Spatially developing flows
with
Kact=pass = 2 1 q
1 ; cos2 φ= cos2 δ sec2 φ ; 1
see §A.2 for a derivation of this expression. The equations are non-dimensionalised by
(x z) = (L] x H ] y )
(u w t ) =
( pxx
pzz pxz )
p
gL]u
Hp
gL w L
= ρg cos ξH ]( pxx
"s # !
L t
g
pzz tan ξpxz )
where L] is a typical streamwise extent of the avalanche and H ] is a typical depth. Because of
the shallowness of the flow the aspect ratio ε = H ]=L] is small. The longitudinal velocity scale
pgL] indicates that the phenomenon is governed chiefly by free fall rather than by surface waves.
The quantities with an asterisk are dimensionless. By incorporating the non-dimensionalisation,
taking the long-wave limit ε ! 0, and depth averaging the one-dimensional equations of motion, two
evolution equations for the dimensionless depth h(x t ) and the dimensionless longitudinal velocity
u(x t ) are obtained
∂h ∂ (hu)
+ ∂x
∂t
∂u
∂u
+
u
∂t
∂x
=
0
=
sin ξ ; tanδ
(2.2)
u
∂h
CD
cos ξ ;
ujuj; εK cos ξ :
juj
εh
∂x
(2.3)
The velocity profile through the depth of the flow has been assumed to be uniform, and the asterisks
have been omitted. In this equation, the dominant balance is between the acceleration of the material,
the gravitational acceleration, the Coulomb friction and the turbulent-resistance if CD
ε. As the
flow slows, its deformation (∂h=∂x) may begin to influence the motion.
These equations are similar to the shallow-water equations, but the constitutive properties complicate the model with a non-linear earth-pressure coefficient. Another approach to formulating
the constitutive behaviour of granular materials involves a direct analogy with shallow water flows,
where pressures within granular materials are assumed isotropic and hydrostatic, thus K = 1 (Eglit,
1983; Gray, 2003). This approach has been found by some experimental investigators to be a more
convenient and accurate way of describing the motion of granular materials down fixed inclines
(e.g. Pouliquen, 2002; Gray et al., 2003). Gray et al. (2003) also argue that the discontinuity in the
pressure, introduced through the earth pressure coefficient, is unphysical and leads to a discontinuity
in the flow depth.
2.3 Spatially developing flows
The spatial development of a steady granular flow down a fixed incline, such as the one in Figure 2.1,
can be analysed through equations (2.2) and (2.3). The steady, dimensional form of equations (2.2)
15
and (2.3) is
uh
∂u
u
∂x
=
=
q = constant
(2.4)
g sinξ ; tanδg cosξ ; CD
2
u
h
; g cosξ ∂h
∂x
(2.5)
on the assumption that K = 1. Equation (2.4) leads to
∂h
∂x
= ; uq2 ∂u
∂x
(2.6)
:
By combining equations (2.4), (2.5) and (2.6), the flow speed down the plane can be analysed from
the expression
∂u
∂x
=
where
u0 =
;
gq sin ξ ; CD u3 + tanδgq cosξ u2
(u3 ; gq cosξ) q
; 3
3
u ; u u2
CD 03
(u ; u3∞) q gq(sinξ ; tanδ cos ξ)
CD
1=3
and
(2.7)
u∞ = (gq cosξ)1=3 :
The flow is uniform (flowing at a constant speed) if ∂u=∂x = 0, and hence u = u 0 . The flow never
reaches a uniform state at any point during its development if ∂u=∂x ! ∞, i.e. the denominator of
equation (2.7) is zero, hence u = u ∞ . If this occurs it implies that only an unsteady evolution can
occur. The Froude number of the flow can be written as a function of u and q,
Fr =
u3=2
(qg cosξ)1=2
=
u
u∞
3=2
(2.8)
and the development of the flow down a fixed incline can be subsequently analysed for supercritical
and subcritical flows. If the flow is supercritical, Fr > 1, it follows from equation (2.8) that u > u ∞ .
If the flow is subcritical, Fr < 1, then u < u∞ .
The sign of ∂u=∂x indicates whether the flow is accelerating or decelerating along the chute.
We study the evolution of the flow speed for the two possible cases, (i) u ∞
see Figure 2.2. For the first case (i) where u∞
<
<
u0 and (ii) u∞
>
u0 ,
u0 we find that if the starting speed on the chute
is greater than the constant flow speed, u > u0 , then ∂u=∂x < 0 and the flow decelerates until the
constant speed, u0 , is reached. If the starting speed on the slope is u∞ < u < u0 , then ∂u=∂x > 0 and
the flow accelerates down the chute until u0 is reached. In both of the above cases the flow on the
chute is supercritical. For subcritical flow where u < u∞ the flow evolves to 0. For the second case
(ii) where u∞ > u0 we find that if the flow is supercritical, u > u∞ , the flow evolves towards u∞ and
only an unsteady evolution is possible. If the flow is subcritical with u0 < u < u∞ the flow evolves
towards u∞ and again only an unsteady evolution is possible. If u < u 0 the flow decelerates to 0.
We have shown that subcritical flows (u < u∞ ) do not approach a uniform flow state as x ! ∞.
Downstream conditions may, however, affect what is observed upstream in subcritical flows and the
16
2.3. Spatially developing flows
(i) u0 > u∞
u
0
u∞
u0
(ii) u0 < u∞
u
0
u0
u∞
Figure 2.2: The change in the speed of a one-dimensional current flowing down an inclined plane
for different starting values of the speed. The flow is supercritical for u > u ∞ .
h
(i) h0 < h∞
h0
0
h∞
(ii) h0 > h∞
h
h∞
0
h0
Figure 2.3: The change in depth of a one-dimensional current flowing down an inclined plane for
different starting depths. The flow is supercritical for h < h ∞ .
flow at the top of the chute can not be specified independently of the conditions at the bottom. On the
other hand, we find that supercritical flows (u > u ∞ ) evolve towards a constant flow state if u0 > u∞ .
The flow depth evolves as the inverse of the flow speed h = q=u (see Figure 2.3) and the gradient in
the flow depth down the slope is given by
∂h
∂x
=
;
h3 g sinξ ; CD q2 + tanδg cos ξh3
h3 g cos ξ ; q2
:
(2.9)
It has previously been observed in experiments with supercritical, dry granular flows down
‘rough’ chutes (basal roughness comparable to the diameter of the granular particles) that such flows
reach uniform states on the chutes for a range of chute inclinations (see Pouliquen, 1999, and references therein). Good agreement has also been found between experiments conducted on very
smooth chutes and model predictions, by omitting the turbulent friction term (C D = 0), and the flows
are not observed to reach a uniform state on the chutes (Hutter et al., 1995; Wieland et al., 1999).
This is consistent with Salm’s (1993) interpretation of the nature of the turbulent friction term, and
shows that the turbulent friction coefficient may depend on terrain roughness (relative to the size of
the particles within the flow).
If a supercritical flow has reached a constant/uniform state, ∂u=∂x = 0 and ∂h=∂x = 0, equation
17
Gate
g
l
Fl
ow
ξ
Figure 2.4: Schematic side-view of the experimental chutes.
l [m]
1.5
3.0
5.0
b [m]
0.2
0.3
0.5
ξ [ ]
43
37
45
δ [ ]
17:5 0:5
21 0:5
20 1
φ [ ]
22 1
25 1
22 1
M [kg]
0.8–2.4
3–8.5
35–70
u [m s;1 ]
2:7 0:1*
3:1–3:6 0:1
4:5–5:5 0:1
h [m]
0:006 0:001*
0:007–0:011 0:0005
0:019–0:025 0:002
Table 2.1: Chute dimensions, material properties and the flow regime of the different granular flows
are listed. The length of a chute is denoted by l, b is the width of the chute, ξ is the chute inclination,
δ is the dynamic bed friction angle, φ is the angle of repose of the ballotini beads (internal friction
angle), M is the range of mass of particles released, u is the resulting flow speed, and h is the resulting
flow depth. *The flow speed and depth corresponds to a release mass of 2 kg.
(2.5) can be rewritten as
0
=
sin ξ ; tanδ cos ξ ; CD
u2
gh
(2.10)
and solved for the Froude number
Fr2 =
tan ξ ; tanδ
CD
= constant
:
(2.11)
The Froude number for flows that have reached a uniform state should therefore be constant for a
given experimental setup, i.e. for fixed values of ξ, δ and CD . This result will be referred to as the
‘Froude number condition’.
2.4 Experimental setup and design
Three sets of experiments were conducted in order to test the Froude number condition for supercritical, dry granular flows.
18
2.5. Experimental results
Different amounts of ballotini (glass beads) were released by the rapid opening of a lock gate
down three different channelled wooden chutes with fixed inclines, and the flow speed and depth
were measured. If the flows reach a terminal speed on the chutes, the Froude number of the flows
during the constant flow state should be independent of the amount of material released down the
chute, according to the Froude number condition (2.11).
The ballotini beads were approximately spherically shaped, had an average diameter of 90 µm,
a bulk density of 1600 kg m ;3 and a particle density of 2500 kg m ;3 . The chutes were of different
lengths and widths as shown in Figure 2.4 and listed in Table 2.1. The two smaller chutes (1.5 m
and 3 m long) had a plywood base, constructed of one single plywood board. The base of the largest
chute (5 m long) was constructed of painted wooden boards, and was not as regular as the base of
the smooth plywood chutes. A mass in the range 35–70 kg was released down the 5 m long chute.
Between 3 kg and 8.5 kg of ballotini were released down the 3 m long chute, and between 0.8 kg and
2.4 kg were released down the 1.5 m long chute.
The flow depth and speed were measured close to the downstream end of the chutes. The flow
speed was measured from video recordings by tracking the flow front. The video cameras used in the
experiments recorded at 25 frames per second. The flow depth was measured by fixing a gate in the
flow path at a known distance from the base of the chute. The distance was increased systematically
until all of the current flowed under the gate without touching it. The internal friction angle, φ, was
measured by building up a cone of particles on top of a horizontal layer of the same particles, and
repeatedly measuring the height and the diameter of the cone. The dynamic bed friction angle, δ,
was measured by tilting a plane with a thin layer of moving particles and determining the angle at
which the layer came to rest.
The slope angle exceeded the value of the dynamic bed friction angle on all of the experimental
chutes, as listed in Table 2.1. Table 2.1 also shows values of the flow speed and depth for one release
mass on each of the three chutes.
2.5 Experimental results
The flows rapidly accelerated to a constant speed on the two larger experimental chutes. As the
release mass of ballotini beads was increased, the flows thickened and the flow speed increased in
such a way that the Froude number was approximately independent of the mass of particles released,
as shown in Figure 2.5. The flow on the smallest chute had not fully reached a uniform state at the
end of the chute, which was reflected in a somewhat mass-dependent Froude number. The chute
may not have been long enough for the flow to reach a uniform state. Alternatively, the base of the
chute may have been too smooth for the flow to attain a constant flow state on the chute.
A study of velocity and energy balance of flowing snow on a large snow chute on Weissfluhjoch
19
Fr
16
14
12
10
8
6
4
5 m chute
2
0
0
10
3 m chute
20
30
40
50
60
70
80
0
1
2
3
4
5
6
Mass [kg]
7
8
9
16
14
12
10
8
6
4
1.5 m chute
2
0
0
0.5
1
1.5
2
Mass [kg]
2.5
3
Figure 2.5: The Froude numbers of the ballotini particle currents (measured close to the downstream
end of the upper chute section) as a function of the mass of material released down the experimental
chutes.
20
2.6. Conclusions
in Switzerland was undertaken during the winter 2003 (Kern et al., submitted). The snow chute is
34 m long and 2.5 m wide and is sketched in Figure 6.2 and described in §6.2.2. The flows on the
chute can reach speeds up to 10 m s ;1 and a flow depth of up to half a metre. The study showed
that the flowing snow also evolved toward a uniform state on the chute. Rubber bars were fixed to
the base of the chute in order to agitate the flow. Without these rubber bars, the snow slid down
the chute as a ‘plug’ layer (a layer where the flow speed is constant through the depth of the layer).
Kern et al. (submitted) found that the 0.4 m deep flow on the chute consisted of slip at the base, an
approximately 0.05 m deep shear layer and an overlying 0.35 m deep plug layer. The flow reached
a terminal speed on the chute after 8 m and remained approximately constant on the section of
the chute, which was inclined at 32 to the horizontal, shown in Figure 6.2. This was confirmed
by calculating the energetics of the flow, i.e. energy dissipation in the shear layer and at the basal
surface were shown to be in balance with the rate of work done by gravity on the 32 chute section.
2.6 Conclusions
The depth-averaged equations of motion of steady, one-dimensional granular flows show that if a
turbulent friction term is included in the equations, supercritical granular flows can reach a constant,
uniform state down a fixed incline, even though the slope angle exceeds the dynamic bed friction
angle. This is observed in the granular flow experiments, reported here, and also in flows of natural,
dry-snow avalanches. This emphasises the need of using models that incorporate a turbulent friction
term to describe physical flows on rugged terrain, such as snow avalanches.
Once the flows have reached a constant, uniform state, the Froude number of the granular currents is determined by the terrain parameters: the slope angle, ξ; the dynamic bed friction angle, δ;
and the coefficient of turbulent friction, CD , according to equation (2.11). Experiments on smooth
chutes (Hutter et al., 1995; Wieland et al., 1999) and rough chutes (Pouliquen, 1999) and observations of snow avalanche motion (Salm and Gubler, 1985) suggest that the value of the turbulent
friction parameter depends on terrain roughness. Salm (1993) suggests that it should also be velocity dependent, but that has not been explored in the experiments presented here, at least not for a
sufficiently wide range of flow speeds on each experimental chute.
21
Chapter 3
The interaction between supercritical
shallow flows and dams: a theoretical
study
3.1 Introduction
The traditional design of dams, which are used to deflect or stop dense snow avalanches, has been
based on simple considerations of the energy of a point-mass in the flow (Salm et al., 1990; McClung
and Schaerer, 1993).
The required height of a catching dam so that no material overflows it, has been determined
based on the equation
H=
u2
+ hs + hd λ2g
where H is the height of the dam, u is the speed of the avalanche, g is gravitational acceleration, h s is
the thickness of the snow cover, and h d is the thickness of the dense core of the avalanche. The depths
g
H
hd
u
hs
Figure 3.1: A schematic diagram for calculating the height of a catching dam, H. The depth of the
underlying snow cover is denoted by h s and hd is the thickness of the dense core of the avalanche.
23
Chapter 3. The interaction between supercritical shallow flows and dams:
a theoretical study
are measured vertically from the base of the dam and are shown in Figure 3.1. The dimensionless
constant, λ, accounts for dissipation in the impact of the flow with the dam and is usually chosen
in the range 1–2 (Salm et al., 1990). High values of λ are selected where the potential for large
avalanches is considered small, whereas low values are chosen for avalanche paths where extreme
avalanches with a large volume may be released.
A similar empirical expression is used to calculate the required height of a deflecting dam so that
the avalanche does not over-top the dam,
H=
(u sinγ)2 + h + h
s
2g
d
(3.1)
where γ is the deflecting angle of the dam (see Figure 1.2). The expression is equivalent to that for a
catching dam where γ = 90 with the dimensionless constant, λ, set equal to unity, because energy
is assumed to be neither lost in the impact of the avalanche with the dam, nor lost due to frictional
dissipation as the avalanche travels along the dam after the impact.
Other approaches to calculating the height of dams required to arrest or fully deflect an avalanche
consider the run-up of a point-mass that is subject to the same frictional forces as are thought to be
present in avalanche flow on the upstream facing, sloping side of the dams (Irgens et al., 1998).
This approach is formulated and discussed in §3.2. McClung and Mears (1995) extend this analysis
and consider the run-up of the leading edge of the flow onto a catching dam. Through using a
conservation expression for the momentum of the flow in one dimension, they account for hydrostatic
pressure in the flow which does not enter the point-mass equations. They conclude that the pointmass theory underestimates the run-up onto a catching dam by a factor of two, if the point-mass is
interpreted as the mass centre of the avalanche. McClung and Mears do not extend their theory to
deflecting dams. Chu et al. (1995) have performed small-scale experiments and tested the theory
of McClung and Mears against their observations, while Irgens et al. (1998) used the point-mass
model to explain run-up of a natural avalanche onto a deflecting dam. Both parties comment that
energy loss in the impact of the flow with a dam may need to be accounted for in order to use the
theories to explain experimental and field observations. McClung and Mears hypothesise that the
flow loses the normal component of its approach velocity in the interaction, while Irgens et al. find
better agreement with field observations by assuming no energy loss in the impact. Both of the
approaches may, however, be too idealised as will be shown below.
More recent chute experiments and theoretical analysis (Tai et al., 2001; Gray et al., 2003) suggest a different flow behaviour in the impact of dense, supercritical granular flow with obstacles.
The experiments indicate that a shock (or a jump) is formed upstream of an obstacle, and in the direction normal to the shock, a change from a supercritical flow state to a subcritical one occurs, and
the shock may be modelled by ‘shallow-water’ jump conditions (see, for example, Whitham, 1999).
Observations of two natural snow avalanches hitting deflecting dams above the village Flateyri in
24
3.2. Point-mass model
Deflecting dam
g
z
γ
n̂
y
p̂
upm
x
ne
li
our
ŝ
nt
Co S
ur
fac
e
pla
ne
Horizontal plane
α
ξ
Figure 3.2: Schematic diagram of the geometry used in the point-mass model, when a point of mass
m and velocity upm impacts a deflecting dam positioned on a slope, inclined at an angle ξ to the
horizontal.
western Iceland reveal the same phenomena as observed in these laboratory experiments (Jóhannesson, 2001). The avalanche debris that was left behind showed that the impact had channelised a part
of, or the whole of the width of both the avalanches into thicker streams, travelling parallel to the
dam, and thereby increased the run-out of both avalanches significantly. These observations suggest
that a shock was formed at the dam face during the interaction.
The fundamental difference between the theoretical approaches which are described above indicates that further investigations of the behaviour of granular flows interacting with dams are needed.
Such studies need to provide a better understanding of the dynamics of the flows. Small-scale laboratory experiments, larger-scale experiments with snow as well as observations of natural avalanches
that hit dams are needed as a part of such investigations.
This theoretical chapter starts with a review of the point-mass model developed by Irgens et
al. (1998) to study the run-up of an avalanche onto a deflecting dam (§3.2), followed by a review
and discussion of incompressible, shallow-layer shock dynamics, §3.3. Finally in §3.4 we review the
pressure impulse theory (Cooker and Peregrine, 1995) and consider possible applications towards
understanding measured high pressures during the first few milliseconds of the impact of the dense
core of a snow avalanche with a dam.
25
a theoretical study
3.2 Point-mass model
The run-up of an avalanche of speed u pm onto a deflecting at an angle γ to the flow direction of the
avalanche and positioned on an inclined plane, as sketched in Figure 3.2, has been studied by Irgens
et al. (1998). They use a Voellmy type of a model, which is discussed in §2.2, to describe the path of
a point-mass on the upstream face of a deflecting dam. The mass is denoted by m and is subject to
the same type of frictional forces as in Voellmy’s equation (2.1). Here we reformulate their analysis
so that we may compare their results with the different approaches developed later in this chapter
and with experimental studies which are described in Chapter 4.
The coordinate system, xyz, is defined as before with the x-axis inclined along the downslope
direction at an angle ξ to the horizontal and the z-axis being the upward pointing normal. We then
formulate the problem in a fixed cartesian coordinate system 0nsp, which lies on the face of the
deflecting dam and is defined by the unit vectors in Figure 3.2. They can be written in the (x y z)
coordinate system as
n̂
ŝ
p̂
= (; sin γ sin α ; cos γ sin α cos α)
= (cos γ ; sin γ 0)
= (sin γ cos α cos γ cos α sin α)
where α is inclination of the upstream dam face relative to the surface. Gravitational acceleration
can be expressed as g = (g sin ξ 0 ;g cosξ) in the (x y z)-coordinate system, and the components
of the gravitational acceleration in the directions of the unit vectors, ŝ, p̂ and n̂ are
gs
gp
gn
=
=
=
g ŝ = g sin ξ cos γ
g p̂ = ;g(cosξ sin α ; sinξ sin γ cos α)
g n̂ = ;g(cosξ cos α + sin ξ sin γ sin α):
Newton’s second law of motion on the dam face, in the p̂ and ŝ directions, respectively, becomes
du1
dt
du2
m
dt
m
=
=
mg p ; µm(;gn)
u1
juj ; Djuju1
u2
mgs ; µm(;gn) ; Djuju2
juj
(3.2)
(3.3)
= u1p̂ + u2ŝ is the velocity of the point-mass on the dam face and D is a dimensional
friction coefficient. With u1 = u sin ψ and u2 = u cosψ, where ψ is the angle between the path line
where u
and ŝ shown in Figure 3.3, equations (3.2) and (3.3) become
du
dψ
sin ψ + u
cos ψ
dt
dt
du
dψ
cos ψ ; u
sin ψ
dt
dt
=
=
gp ;
D 2
u sin ψ + µgn sin ψ
m
D
gs ; u2 cos ψ + µgn cosψ:
m
26
3.2. Point-mass model
Path line
Dam face
p̂
u
ψ
u0
ψ0
ŝ
Figure 3.3: Schematic diagram of the path of a point-mass on a dam face.
It follows that
du
dt
dψ
u
dt
=
g p sin ψ + gs cos ψ + µgn ;
=
g p cos ψ ; gs sin ψ
D 2
u m
(3.4)
and
du
dψ
=
;
g p sin ψ + gs cos ψ + µgn ; Du2 =m u
:
g p cos ψ ; gs sin ψ
(3.5)
Equation (3.5) can be solved numerically by a fourth order Runge-Kutta scheme, subject to the
following initial conditions
u = u0
and
ψ = ψ0
at t = 0:
The angle ψ is reduced by constant, sufficiently small increments, ∆ψ, and for each new pair of
(u ψ), the point-mass moves a distance ∆l = u∆t along the dam in a direction determined by ψ. The
time increment is given by equation (3.4): ∆t = u∆ψ (g p cosψ ; gs sin ψ).
=
We may relate the initial velocity along the dam, u0 to the oncoming velocity down the plane,
upm , by making the following assumptions. If no mechanical energy is lost in the initial impact
with the dam (the point-mass moves smoothly onto the dam) and the tangential component of the
momentum (the momentum along the dam) is conserved we find that
= (u0 p)2 +(u0 s)2
upm s = u0 s
u2pm
:
It follows that u0 s = upm cos γ and u0 p = upm sin γ, and the initial values of ψ and u on the dam
face are ψ0 = γ and u0 = upm .
27
a theoretical study
z [m]
α = 90
α = 65
α = 40
α = 25
5
4
3
2
1
0
0
10
20
30
40
50
60
s [m]
Figure 3.4: The computed path of a point-mass on the upstream face of a 60 m long deflecting dam.
The flow parameters are chosen as follows: γ = 20 ; µ = 0:25; m=D = 3000 m; ξ = 10 . No energy
loss is assumed in the initial impact with the dam and the dam face is inclined at angles α to the
slope. Note that it is the projection parallel to the incident slope onto the z-axis that is plotted.
As an example, we plot the path of a point-mass on an upstream dam face, sloping at different
angles, α, in Figure 3.4. The geometry was chosen to correspond to the eastern part of the deflecting
dam above Flateyri, Iceland, where ξ = 10 and γ = 20 (Jóhannesson, 2001). The impact speed of
the point-mass was chosen upm = 30 m s;1 and energy was assumed to be conserved in the impact
with the dam. The friction parameters were chosen to be µ = 0:25 and m=D = 3000 m, which is
within the physical range of the parameters for snow avalanches (Perla et al., 1980). The inclination
of the upstream dam face was chosen between 25 and 90 , where α 40 corresponds to the
Flateyri dam.
Jóhannesson (2001) found that the model under predicts the observed run-up onto the Flateyri
dam by a few metres when the thickness of the snow cover, h s , and the depth of the dense core of the
avalanche, hd , have been added to the run-up in Figure 3.4, according to equation (3.1). He points
out that the reason for this may be that the speed of the avalanche has been underestimated. Another
possible explanation is that the highest marks on the dam do not correspond to the run-up of the bulk
of the avalanche, but rather the flow front which might run higher up on the dam (see discussion
in §3.4), or that the point-mass model does not capture the physics of the flow. A better physical
description of the interaction may be provided by analogy with hydraulic jumps, which are reviewed
in the next section, §3.3.
28
3.3. Dynamics of shocks in shallow-layer flow
g
u2
h1
h2
u1
∆l
Figure 3.5: Schematic diagram of a hydraulic jump.
3.3 Dynamics of shocks in shallow-layer flow
One approach to modelling the interaction of a granular avalanche with a dam entails modelling
the avalanche motion as a shallow supercritical flow. Such flows may undergo changes in state
(hydraulic jump) when interacting with dams. Jumps between flow states have been observed experimentally with dense granular flows down channels (Savage, 1979) and as a result of the interaction
with catching and deflecting dams in the laboratory (Gray et al., 2003) and in nature when a snow
avalanche of volume 10 5 m3 hit a deflecting dam in Flateyri, Iceland, in 1999 (Jóhannesson, 2001).
The description of the interaction between granular flow and an obstacle in terms of the dynamics of shallow-water flow involves a dramatic simplification of the physical processes and may not
provide a complete description of the physics of natural snow avalanches. The theory is, however,
developed below in considerable detail in order to provide a consistent framework for the interpretation of the experiments that have been carried out and are described in Chapter 4. Deviations
from the predictions of this theory can then provide a starting point for a more realistic theoretical
description.
3.3.1 Hydraulic jumps
A hydraulic jump is a region where shallow-water flow changes from a supercritical to a subcritical
state and thereby dissipates mechanical energy. The transition between the two states does not occur
abruptly, but rather over a finite length, ∆l, illustrated in Figure 3.5, and hereafter referred to as the
‘transition zone’.
The characteristics of a hydraulic jump are highly dependent on the Froude number of the flow
p
approaching the jump, Fr 1 , defined by Fr1 = u1 = g? h1 , where g? is the component of the gravitational acceleration normal to the underlying surface. Hager (1992) classifies hydraulic jumps according to the magnitude of Fr1 as follows. The hydraulic jump is termed ‘pre-jump’ for 1:7 < Fr 1 < 2:5.
A series of small rollers develop on the surface for Fr1
29
1:7, and are slightly intensified for increas-
a theoretical study
ing Froude numbers. The water surface is quite smooth, and the velocity distribution in the tailwater
is fairly uniform. ‘Transition jumps’ correspond to Froude numbers between 2.5 and 4.5. This type
of jump has a pulsating action. The entering jet oscillates heavily from the bottom to the surface.
Each oscillation produces a large wave of an irregular period. Jumps for 4:5 < Fr 1 < 9 are ‘stabilised
jumps’ since they have a limited tailwater wave action, a relatively high energy dissipation, and a
compact and stable appearance. For extremely high speed flows at Froude numbers above 9, the
high velocity jet is no longer able to remain near the underlying boundary. Slugs of water rolling
down the front face of the jump intermittently fall into the high-velocity jet and generate additional
tailwater waves. The surface of the jump is usually very rough and contains a considerable amount
of spray.
According to the classical analysis of two-dimensional hydraulic jumps, mass and momentum fluxes are conserved across the jump but mechanical energy is dissipated (see, for example,
Whitham, 1999). The analysis describes the conditions on either side of the transition zone but does
not resolve the complicated three-dimensional structure of the transition zone. Jump brackets,
]],
are commonly used to indicate differences in flow states upstream and downstream of the jump.
The conservation of mass and momentum fluxes across a stationary jump, such as the one in
Figure 3.5, of fluid with isotropic, hydrostatic pressure distribution, can be written as
ρhu n̂]]
1 ? 2
ρhu (u n̂)+ g ρh n̂
2
=
=
0
(3.6)
0
(3.7)
where u is the flow velocity, h is the depth of the flow, ρ is the density, n̂ is the unit normal to the
jump and g? is the component of the gravitational acceleration normal to the underlying surface. If
the density is unchanged across the jump, ρ 1 = ρ2 = ρ, the conservation equations can be written as
=
=
h1 u1
1
h1 u21 + g? h21
2
h2 u2 (3.8)
1
h2 u22 + g? h22 :
2
Furthermore it follows that
h2
h1
= 12
p
1 + 8Fr21 ; 1
:
(3.9)
The rate at which mechanical energy per unit width is dissipated over the jump is given by
(h2 ; h1)3 ρg?q
4h2 h1
(3.10)
where q = u1 h1 = u2 h2 denotes the volume flux of fluid per unit width. The rate of energy dissipation
across the hydraulic jump increases with increasing upstream Froude number and when the upstream
Froude number is high, a large proportion of the energy flux can be dissipated across the transition.
Using (3.8), (3.9) and (3.10) we find that if Fr 1 > 8 then over 66% of the energy flux is dissipated
across the hydraulic jump.
30
u2
g
h2
H
z
h1
u1
x
Figure 3.6: Schematic diagram of supercritical flow over a bump.
In the analysis that follows, we will assume that the pressure in shallow, granular flow is hydrostatic and isotropic as in water. There is, however, a debate in the granular flow literature about
whether the assumption of an isotropic pressure field is appropriate for granular flows such as snow
avalanches. For example, Savage and Hutter (1989) link the components of the pressure field parallel
with and normal to the slope, pxx and pzz , through an earth pressure coefficient, defined by
K=
pxx
pzz
as discussed in §2.2 and derived in §A.2. If the jump-conditions for stationary jumps are implemented with anisotropic pressure, then the conservation of momentum flux across the jump, given
by equation (3.7), becomes
1
hρu (u n̂)+ Kg? ρh2 n̂
2
=0
We will show in §4.2.3 that we find good agreement using K
:
(3.11)
= 1 in equation (3.7), even though
using φ and δ, the calculated values of both the passive and active earth pressure coefficients exceed
unity, Kact=pass > 1.
3.3.2 Normal shocks: catching dams
The onset of a shock
As supercritical flow passes smoothly over an obstacle, the flow slows down and thickens as shown
in Figure 3.6. If the flow slows down and thickens enough as it climbs the obstacle so that the
supercritical flow becomes subcritical, pressure disturbances in the fluid can suddenly propagate
upstream and the flow starts to accumulate upstream of the obstacle. The supercritical oncoming
flow has no knowledge of the high obstacle further downstream and a smooth transition from a
supercritical to a subcritical flow state is impossible. The transition is accomplished in a highly
turbulent and energy dissipating hydraulic jump as described in §3.3.1.
It is possible to study how high a smooth obstacle (or a bump) needs to be so that supercritical
flow becomes critical at the top of the bump, Hcr . If the bump is lower than critical the height, H
31
<
a theoretical study
Hcr =h1
100
80
60
40
20
0
2
4
6
8
10
12
14
Fr1
Figure 3.7: The bump height necessary for frictionless, supercritical flow to become critical at the
top of the bump, non-dimensionalised by the depth of the approaching flow, H cr =h1 , as a function of
the Froude number of the flow approaching the bump.
Hcr , the flow remains supercritical at the top and passes smoothly over the bump until it returns to its
original flow state downstream of the bump. If the bump is higher than the critical height, H > H cr , a
change in flow state occurs and a turbulent hydraulic jump forms upstream of the bump. Figure 3.6
shows a supercritical, steady, uniform, stream flowing smoothly over a bump. By assuming that
there is no loss of energy as the flow passes over the bump, the conservation of energy flux from the
base of the bump to its top, can be written as
Z h1 0
1
ρgz + ρu21 + p1 u1 dz =
2
Z H +h2 1
ρgz + ρu22 + p2 u2 dz
2
H
where p1 and p2 are hydrostatic pressures in the flow. In this context p1
(3.12)
= ρg (h1 ; z) and p2 =
ρg (H + h2 ; z). By assuming that the density of the flow is constant, equation (3.12) becomes
h1 u1
1
gh1 + u21
2
= h2 u2
1
g (h2 + H )+ u22
2
:
(3.13)
Mass flux is conserved in the flow,
u1 h1 = u2 h2 :
If the flow at the top of the bump is critical, u22 = gh2 , and equation (3.13) can be rewritten as
1
3 2=3 Hcr
1 + Fr21 ; Fr1 =
:
2
2
h1
(3.14)
We plot the critical bump height, Hcr , non-dimensionalised with the depth of the approaching
flow as a function of the Froude number of the oncoming flow in Figure 3.7.
32
g
z
U
x
h1
u2 = 0
u1
h2
Figure 3.8: Schematic diagram of a two-dimensional bore travelling upstream from a catching dam
at a speed U.
The dynamics of normal shocks: flow completely blocked by a dam
Figure 3.8 shows a hydraulic (granular jump) which has formed by the interaction between a supercritical flow and a catching dam. A bore travels upstream from the dam with speed U, and the dam
is higher than the surface of the bore so that none of the flow over-tops the dam. By choosing a reference frame travelling with the bore at speed U to the the left, conservation of mass- and momentum
fluxes across the jump becomes
(u1 + U ) h1ρ1 =
1
h1 ρ1 (u1 + U )2 + g? ρ1 h21 =
2
Uh2 ρ2 (3.15)
1
h2 ρ2U 2 + g? ρ2 h22 :
2
(3.16)
For incompressible fluids, the density is unchanged across the bore, so that ρ 1
= ρ2 .
For granu-
lar materials we expect, on the other hand, the stationary material downstream of the jump to be
somewhat more closely packed than the dilated, flowing material, implying ρ 2 > ρ1 .
An expression, relating the ratio between the shock depth and the depth of the approaching
stream to the Froude number of the flow approaching the shock and the density ratio, ρ 2 =ρ1 , may be
obtained from equations (3.15) and (3.16)
Fr1 =
v u
h2 u
t1
h1 2
ρ2 =ρ1 ;
1
h2 =h1
;
1
h2 =h1
2
+
1
h2 =h1
3
!
1
ρ2 =ρ1
:
(3.17)
The ratio, h2 =h1 , is plotted in Figure 3.9 (a) as a function of Fr 1 for three different density ratios.
Note that a density increase across the jump lowers the height of the bore. When ρ 2 =ρ1 = 1, the bore
height is much lower than the required height of a dam, given in equation (3.14) and plotted in Figure 3.7, for the onset of a hydraulic jump upstream of the dam. The reason being that equation (3.14)
was derived for frictionless flow which passes smoothly over an obstacle without any loss of energy,
whereas a considerable loss of energy takes place over the hydraulic jump. In general we may expect
some loss of energy as flow passes over a dam, at least due to frictional effects. This energy loss will
lower the height, Hcr , which the flow needs climb to enter a critical flow state. A dam may still need
to be higher than the calculated height of a bore in order for a hydraulic/granular jump to occur during the initial interaction with the dam, h2 < H < Hcr . This may indeed be the case for some natural
33
a theoretical study
h2 =h1
20
U =u1
ρ2 =ρ1 = 1
ρ2 =ρ1 = 2
ρ2 =ρ1 = 3
0.7
(a)
ρ2 =ρ1 = 1
ρ2 =ρ1 = 2
ρ2 =ρ1 = 3
(b)
0.6
0.5
15
0.4
10
0.3
0.2
5
0.1
0
0
2
4
6
8
10
12
14
Fr1
16
18
20
2
4
6
8
10
12
14
16
18
20
Fr1
Figure 3.9: (a) The depth ratio between the bore and the approaching flow as a function of the Froude
number of the oncoming flow. (b) The ratio between the bore speed and the speed of the oncoming
flow as a function of the Froude number of the oncoming flow. The density ratios between 1 and 3
are chosen to correspond to possible density ratios in natural snow avalanches. Issler (2003) suggests
that the dense core of a flowing dry-snow avalanche has a density in the range ρ 1 =250–500 kg m ;3
while the density of the avalanche debris hardly exceeds ρ 2 = 600 kg m;3 .
snow avalanches. According to Jóhannesson (pers. comm., 2003), an avalanche in 1995 passed over
a steep, approximately 50 m high, natural dam in Kisárdalur, Iceland. The natural dam was much
higher than the back calculated height of the bore, based on estimates of the speed and depth of the
avalanche, but lower than the critical height for the onset of a hydraulic jump (h 2 < H
<
Hcr , with
the height of the dam denoted by H).
The Froude number of the oncoming flow can be obtained also as a function of the ratio of the
bore speed and the speed of the approaching flow and the density ratio,
s
1
Fr1 =
U =u1
1 + 2U =u1 ;(ρ2=ρ1 ; 1)(U =u1)2
:
2ρ2 =ρ1 (1 + U =u1)
(3.18)
This function is plotted in Figure 3.9 (b) for different density ratios. We note that at high upstream
Froude numbers, the bore speed is slow compared with the speed of the oncoming flow (U =u 1 1)
and that an increased density difference slows the bore down.
The same set of equations with g? = g cos ξ (the component of gravitational acceleration normal
to a slope inclined at an angle ξ to the horizontal) may be used to describe jumps on an inclined
plane, when the internal friction angle of the stationary material, downstream of the jump, is larger
than or equal to the slope angle, φ ξ. The use of these equations requires that the transition zone
(the length of the jump, denoted by ∆l in Figure 3.5) is thin enough so that the component of the
gravitational acceleration along the slope (the additional accelerative term, g sinξ) will not affect the
conservation of momentum flux over the bore, see Figure 3.10 (a). The free surface of a stationary
pile of granular material is stable at angles up to the internal friction angle of the pile, this angle
is also termed the angle of repose of the material, φ. The free surface of the stationary material
downstream of the shock is therefore stable at the slope angle, ξ, if φ ξ and arrests on the slope
34
z
g
x
h1
U
u1
h1
u2 =
(a) φ ξ
0
φ
U
u1
h2
u2 =
h2
ξ
(b) φ < ξ
L
0
hd
ξ
Figure 3.10: Schematic diagram of a granular bore travelling up an inclined plane, away from a
catching dam, for materials with different internal friction angles, φ.
downstream of the shock with the surface inclined at an angle ξ to the horizontal. That implies that
the depth of the stationary material upstream of the dam is h 2 everywhere.
In the case of material with φ < ξ (for example φ = 0 for fluids), the stationary material is initially stopped on the slope downstream of the jump with the free surface parallel with the slope.
The stationary material is, however, not stable at that angle. The surface of the bore is unstable and
is intermittently readjusted to the angle of repose of the material. (For granular materials this happens through thin avalanches on the free surface of the bore.) Therefore the depth of the stationary
material just upstream of the dam, hd , increases the further upstream the bore travels, as shown in
Figure 3.10 (b). The redistribution of the stationary material upstream of the dam slows the bore
down as it travels upstream, since more and more material needs to avalanche down the free surface
as it lengthens. The mass flux, previously described by equation (3.15), is therefore adjusted in order
to account for the the increased volume of stationary material upstream of the dam
u 1 h 1 ρ1 +
∂
∂
(
ρ1 h1 L) =
∂t
∂t
h2 +
hd ; h2
ρ2 L
2
which may be rewritten as
(u1 + U ) h1ρ1 = Uρ2 (h2 + L tan (ξ ; φ))+ Lρ2 ∂h∂t2
where L is the distance from the front of the bore to the dam, h d
(3.19)
= h2 + L tan (ξ ; φ), ∂L
∂t
=
= U,
and U is the average bore speed at L. Since the intermittent avalanching on the bore surface only
redistributes material that has already been transported to the bore, we assume that the depth that
the flow jumps to, h2 , does not change significantly with time, ∂h2 =∂t U and can be neglected in
equation (3.19), and the flux of momentum across the jump, given by equation (3.16), is unaffected
by the redistribution process. From equations (3.16) and(3.19), the Froude number of the approaching flow can be obtained as a function of the speed ratio of the bore to the approaching flow, U =u 1,
35
a theoretical study
U =u1
0.07
0.06
0.05
0.04
0.03
ξ;φ = 0
ξ ; φ = 10
ξ ; φ = 20
ξ ; φ = 30
0.02
0.01
0
0
20
40
60
80
100
L=h1
Figure 3.11: The ratio of the bore speed to the speed of the approaching flow as a function of
the distance travelled upstream from the dam by a bore, non-dimensionalised by the depth of the
approaching flow. The flow approaches the dam with Froude number equal to 10, and four different
values of the slope angle and the angle of repose of the material, ξ ; φ, are plotted.
and the ratio of the length of the bore to the depth of the approaching stream, L=h 1 ,
Fr1 =
1
U =u1
v
u
u1
u
t
i
h
+ 2 1 ; hL tan (ξ ; φ) U
h
2 1 +U
1
h
i
u1 ; 2 ; hL1 tan (ξ ; φ)
=
L
h1
u1 + hL1 tan (ξ ; φ)(U =u1 )2
tan (ξ ; φ)(U =u1)2
i
:
=
In this expression, the density ratio, ρ1 =ρ2 , has been set equal to unity for simplicity. The ratio of
the bore speed to the speed of the approaching flow is plotted in Figure 3.11 as a function of the
dimensionless distance travelled upstream from the dam by the bore for flow approaching the dam
at Froude number 10. We note that the bore slows down, the further upstream it travels and that the
bore speed is only a small fraction of the speed of the approaching flow.
The dynamics of normal shocks: flow over-tops a dam
As the height of the catching dam decreases relative to the depth of the approaching stream, some
flow will eventually over-top the dam, as shown in Figure 3.12 (a). A larger proportion of the flow
escapes over the dam as the height of the dam is decreased and finally a bore does no longer form
upstream of the dam and the flow is launched supercritically over the dam, as shown in Figure 3.12
(b).
A description of the flow and the bore upstream of a dam, when some part of the bore overtops the dam, may be formulated. The dam is positioned on a slope inclined at an angle ξ to the
36
g
U
z
u2
h1
u1
x
h2
ud
hd
h3
u3
H
L
ξ
(a)
u2
h1
u1
H
ξ
(b)
Figure 3.12: Schematic side-view of (a) a granular jump formed upstream of a dam, when supercritical flow interacts with the dam and some of the flow over-tops the dam; (b) supercritical flow
launched over a dam.
37
a theoretical study
horizontal. The conservation of mass and momentum fluxes across the jump in a reference frame
travelling upstream with the bore becomes
( u 1 + U ) h1
1 ? 2
g h1 +(u1 + U )2 h1
2
= ( u 2 + U ) h2
= 12 g?h22 +(u2 + U )2 h2
(3.20)
(3.21)
:
= ρ2 = ρ3 for simplicity, since the flow does not
In these expressions, we have assumed that ρ1
stop completely in front of the dam and will therefore not be as closely packed as in the case of no
over-topping where the flow is stationary in front of the dam. Mass flux is conserved and
h2 u2 = h3 u3 :
(3.22)
By assuming that the flow downstream of the dam is supercritical, we find that the flow over the dam
is critical (see, for example, Gerhart et al., 1993) and
Fr3 = 1
u3 =
which implies that
p
g? h3 :
(3.23)
The bore propagates upstream until the massflux over the jump equals the massflux over the dam
(u1 h1 = u2 h2 = u3 h3cr ) and the critical flow depth over the dam is given by
h3cr =
u1 h1
pg ?
2=3
:
The lower the dam is, the more mass is transported over the dam, and when
H
h2 ; h3cr
(3.24)
all of the approaching mass flux is transported directly over the dam and a bore does not need to
propagate upstream from the dam. In this context (U = 0 and the bore is stationary), the depth of the
jump relative to the depth of the approaching stream, h 2 =h1 , is given in equation (3.9) as a function
of the Froude number of the approaching flow, Fr 1 , and equation (3.24) can be written as
H
h1
12
q
1 + 8Fr21 ; 1
;(Fr1 )2=3
:
(3.25)
This height ratio, H =h1 , is plotted in Figure 3.13 along with the non-dimensional height of the jump
determined by equation (3.9) as a function of the Froude number of the flow approaching the dam.
From Figure 3.13 we note that the minimum height of a dam for a bore to be present upstream of
the dam is H
= 9h1 for flow approaching the dam at Froude number 10.
If the dam is lower, the
flow is launched directly over the dam. Jets launched over such low dams that have a height that is
comparable to or a few times the depth of the approaching flow are described in Chapter 5 and by
Hákonardóttir et al. (2003a) and (2003b).
The flow downstream of the jump is subcritical. Pressure disturbances can therefore propagate
upstream within the bore and the flow depth is affected by the presence of the dam. The flow is
38
z=h1
H =h1
h2 =h1
25
20
15
10
5
0
2
4
6
8
10
12
14
16
18
20
Fr1
Figure 3.13: The minimum height of a dam for a bore to be present upstream of the dam, nondimensionalised with the depth of the approaching flow, H =h 1, given in equation (3.25), plotted as
a function of the Froude number of the approaching flow, Fr 1 . The ratio of the depth of the jump
of a stationary bore to the depth of the approaching flow, h 2 =h1, is also shown as a function of the
upstream Froude number. The difference between the two lines is the critical flow depth on the top
of the dam, h3cr .
also far from being uni-directional in the vicinity of the dam, which may also affect the flow depth
upstream of the dam. By assuming that the flow is unaffected by the presence of the dam, we may
estimate the distance, L, that the bore propagates upstream from the dam.
The depth of the flow over-topping the dam can be written as h 3 = hd ; H, where hd is the depth
of the flow directly upstream of the dam, as shown in Figure 3.12 (a). The flow depth, h d , may be
calculated from
ZL
hd
= h2 + (∂h ∂x)dx
=
(3.26)
:
0
where the gradient in the flow depth, ∂h=∂x, can be determined from equation (2.9) for steady flow
that is subject to Coulomb and turbulent friction at the base. This equation may be written as
∂h
∂x
=
;
tanξ ; Fr2CD + tanδ
;
1 ; Fr2
;
(3.27)
where tan ξ represents gravitational acceleration and Fr2CD + tanδ represents friction. In this
p
context, the Froude number is defined by Fr = (u)= g? h and is a function of position within the
flow. Since the flow downstream of the jump is subcritical, Fr < 1, we find that the flow thickens
down the slope, towards the dam if friction is sufficiently small, or the slope is sufficiently steep
(tanξ > Fr2CD + tanδ). As a result, the mass flux over-topping the dam increases as the bore propa-
gates upstream, until the flow depth over the dam reaches the critical flow depth (h 3 = h3cr ) and the
39
a theoretical study
Hcr =h1
γ = 40
γ = 30
γ = 20
γ = 10
120
100
80
60
40
20
0
2
4
6
8
10
12
14
16
18
20
Fr1
Figure 3.14: The height of a smooth deflecting dam, Hcr , for supercritical, frictionless flow to become critical at the top of the dam, non-dimensionalised with the depth of the oncoming flow, as a
function of the Froude number of the flow approaching the dam for dams positioned at four different
deflecting angles, γ.
bore stops. The lower the dam is, the shorter the distance that the bore travels becomes. Experimental observations of bores over-topping dams are described in §4.5 and compared to the theoretical
framework formulated above.
3.3.3 Oblique shocks: deflecting dams
The onset of an oblique shock
The critical height, Hcr , of a smooth deflecting dam, so that the approaching supercritical flow becomes critical at the top of the dam, may be calculated as for normal shocks (see §3.3.2). The dam
is positioned at an angle γ to the direction of the approaching flow and the speed and depth of the
oncoming flow is u1 and h1 , respectively (see Figure 3.2). The flow is assumed to pass over the
dam without any loss of energy and the speed of the flow that climbs the dam is taken to equal the
component of the velocity of the approaching flow normal to the dam, u 1 sin γ. Equations (3.12) to
(3.14), with u1 replaced by u1 sin γ, lead to
Hcr
h1
= 1 + 12 (Fr1 sin γ)2 ; 32 (Fr1 sin γ)2=3
:
(3.28)
The height ratio, Hcr =h1 , in equation (3.28) is plotted in Figure 3.14 as a function of the Froude
number of the approaching flow, for different deflecting angles, γ.
40
n̂
u1 = (u1 0)
ŝ
u1
u2 = (u2 cosγ u2 sin γ)
u2
h1
h2
n̂ = (; sin β cos β)
y
β
γ
x
ŝ = (cos β sin β)
Plan-view
Figure 3.15: Schematic diagram of uniform flow interacting with a deflecting dam positioned at an
angle γ to the flow direction, forming a stationary, oblique shock at an angle β to the direction of the
approaching flow. Unit vectors normal and tangential to the shock are denoted by n̂ and ŝ. The flow
downstream of the shock is parallel to the dam.
The dynamics of oblique shocks
Shock conditions can be formulated for the two-dimensional case of fluid flow hitting a deflecting
dam and forming an oblique, stationary shock that is flowing with a constant speed in a direction
parallel to the dam as shown in Figure 3.15. These conditions can be obtained in most text books
on gas dynamics (see, for example, Whitham, 1999; Chapman, 2000), and adapted to shallow-water
flow. We assume that there is no density difference across the shock, ρ1 = ρ2 , since the flow on both
sides of the shock is in a dilated flowing state. The conservation of mass flux across the jump is
given be equation (3.6) and takes the form
h1 u1 sin β = h2 u2 sin (β ; γ)
(3.29)
where β is the shock angle, measured relative to the direction of the approaching flow. Conservation
of momentum flux across the jump is given by equation (3.7) and in the directions normal and
tangential to the shock becomes
1 ? 2
g h1 + h1 u21 sin2 β
2
h1 u21 sin β cosβ
=
=
1 ? 2
g h2 + h2 u22 sin2 (β ; γ)
2
h2 u22 sin (β ; γ) cos (β ; γ):
(3.30)
(3.31)
Equation (3.31) together with (3.29) implies that the tangential velocity along the shock must be
continuous
u1 cosβ = u2 cos (β ; γ):
By solving (3.29), (3.30) and (3.31) for a given oncoming speed, flow depth and deflecting angle,
the flow speed and depth downstream of the shock can be calculated along with the shock angle. An
expression, relating the shock angle to the known flow parameters (the Froude number of the flow
and the deflecting angle of the dam) is given by
Fr1 =
s
1
sin β
tanβ
2 tan (β ; γ)
41
tanβ
+1
tan (β ; γ)
:
(3.32)
a theoretical study
For a given Froude number and deflecting angle, two shock angles are possible. The shocks corresponding to the smaller and larger angles are termed ‘weak shocks’ and ‘strong shocks’, respectively.
It is, therefore, convenient to rewrite equation (3.32) as
tan γ =
;
4 sin β cos β 1 ; Fr21 sin2 β
;3 + 4 cos2 β ; 4 cos2 βFr21 + 4 cos4 βFr21 ;
q
1 + 8Fr21 sin2 β
(3.33)
since for a fixed Froude number and shock angle, only one deflecting angle is possible. As γ ! 0
the weak shock tends to β = sin;1 (1=Fr1 ) and the strong shock tends to a normal shock, for which
β = 90.
The Froude number of the flow downstream of the shock can be obtained as a function of the
Froude number of the approaching flow, the deflecting angle and the shock angle,
Fr2 =
cos β
cos (β ; γ)
s
Fr21
1
tan2 β
+ 2 sin
1; 2
β
tan (β ; γ)
:
The flow downstream of a weak shock is nearly always supercritical, while the flow becomes subcritical downstream of a strong shock. The shock angle is plotted as a function of the deflecting
angle for different values of the upstream Froude number in Figure 3.16. The points in the figure
separate supercritical flow from subcritical flow. We note that the flow is subcritical downstream of
weak shocks for deflecting angles that are approximately 2 smaller than the maximum deflecting
angle, γmax , for upstream Froude numbers in the range 5–15. The deflecting angle, γ max , defines the
maximum possible deflection for an oblique shock to remain attached to the deflecting dam and is
shown in Figure 3.16 as a dashed curve. For larger deflecting angles (γ > γ max ) the shock becomes
detached from the corner and has curved streamlines (Chapman, 2000). As the upstream Froude
number varies, the solution curves, (γ β)
β [ ]
80
Fr1
strong
70
Fr1
60
Fr1
= 15
Fr1 ! ∞
70
80
= 10
=5
weak
50
40
30
20
10
0
0
10
20
30
40
50
γ [ ]
60
90
Figure 3.16: The shock angle, β, plotted as a function of the deflecting angle, γ, for different values
of the upstream Froude number. The maximum possible deflecting angle for each Froude number,
γmax , lies on the dashed curve. The dashed curve, furthermore, separates the weak shocks from
the strong shocks. The points (+) separate subcritical flow from supercritical flow, with subcritical
conditions occurring for shock angles in excess of the marked values on each curve.
and as a function of the deflecting angle for constant Froude numbers in Figure 3.17 (d). We note
that the shock depth increases relative to the depth of the approaching flow with increasing Froude
number flows and larger deflecting angles. Furthermore, in Figure 3.17 (c), the lines terminate where
there is no solution for an attached shock for smaller Froude numbers, as shown in Figure 3.16.
43
a theoretical study
β ; γ [ ]
β ; γ [ ]
(a)
25
γ = 10
γ = 20
γ = 30
γ = 40
20
Fr1 = 5
Fr1 = 10
Fr1 = 15
25
(b)
20
15
15
10
10
5
5
0
0
5
0
10
h2 =h1
γ = 10
γ = 20
γ = 30
γ = 40
15
Fr1
15
20
0
5
10
15
20
h2 =h1
Fr1 = 5
Fr1 = 10
Fr1 = 15
(c)
15
10
10
5
5
0
25
γ [ ]
30
35
40
30
35
40
(d)
0
0
5
10
15
20
0
Fr1
5
10
15
20
25
γ [ ]
Figure 3.17: The shock angle relative to the deflecting dam plotted: (a) as a function of the Froude
number for constant deflecting angles; (b) as a function of the deflecting angle for constant Froude
numbers. The shock height non-dimensionalised by the depth of the approaching flow plotted: (c)
as a function of the Froude number for constant deflecting angles; (d) as a function of the deflecting
angle for constant Froude numbers.
44
3.4. Pressure impulse theory
a theoretical study
dictably between apparently identical wave impacts. Entrained air increases the compressibility of
water significantly and seems to cushion the most violent impacts.
3.4.1 Mathematical formulation
A mathematical model has been derived for the high pressures and sudden velocity changes which
may occur during the impact between a region of incompressible liquid and either a solid surface
or a second liquid region (Cooker and Peregrine, 1995). The theory rests upon the idea of pressure
impulse, for the sudden initiation of fluid motion in incompressible fluids. The pressure impulse is
defined by
P(x) =
Z ta
tb
p(x t )dt (3.34)
where tb and ta are the times immediately before and after impact, respectively. The dense core of
an avalanche can be regarded as a steep, long, shallow wave. We therefore investigate the pressure
impulse approach in order to study the peak pressures, measured during the first milliseconds of an
impact between the dense core of an avalanche and a dam and the splash-up on the dam due to the
rapid acceleration of the flow front during impact.
According to the analysis of Cooker and Peregrine (1995), the change in velocity during the
impulsive event takes place over such a short time that the non-linear resistive and advective terms
in the equation of motion are negligible compared with the time derivative. Thus the equation of
motion reduces to
∂u
∂t
= ; ρ1 ∇p
:
(3.35)
Equation (3.35) is integrated with respect to time over the impact interval
1
ua ; ub = ; ∇P
ρ
(3.36)
where P is the pressure impulse defined by equation (3.34), u b and ua are the velocities immediately
before and after impact, respectively, and ∇ u a and ∇ ub both vanish since the flow is incompressible. The divergence of equation (3.36) leads to Laplace’s equation
∇2 P = 0:
(3.37)
The following boundary conditions are applied to Laplace’s equation: at a free surface, the pressure
is constant and taken to be a zero reference pressure, P = 0; at a stationary rigid boundary, in contact
with the liquid before and after the impulse, the normal velocity is unchanged so that
∂P
∂n
= 0;
46
g
z
P=0
0
∂P
∂x
= ;ρunb
;h1
b
x
unb
∂P
∂z
∇2 P = 0
P=0
=0
Figure 3.19: A schematic diagram of a wave impact of a rectangle of fluid on a vertical wall at x = 0.
where liquid meets a solid boundary during impact the change in normal velocity gives the normal
derivative of the pressure impulse, for a stationary rigid boundary
unb =
1 ∂P
ρ ∂n
where unb is the normal component of the approach velocity of the liquid.
3.4.2 Pressure impulse of a semi-infinite, rectangular wave
Cooker and Peregrine solved (3.37) subject to the boundary conditions shown in Figure 3.19. The
boundary conditions correspond to the impact of a rectangular wave of length b and depth h 1 with a
solid, vertical wall. Using separation of variables and Fourier analysis, they found that
∞
P(x z) = ρh1 ∑ an sin (λn z=h1 )
n=1
;
sinh λn (b ; x) =h1 ]
cosh (λn b=h1 )
for ;h1 z 0, and 0 x b where λn = n ; 12 π, and the constants an are
an = ;
2unb
:
λ2n
They further showed that it is only the region closest to the wall (x=h 1 1, where x is distance from
the wall) that is affected by the wall. The length of the wave does not affect the pressure impulse
significantly if b h1 and the solution for b=h 1 = 1 is a fair approximation for a semi-infinite wave,
b ! ∞.
Cooker and Peregrine (1995) found that the maximum value of the pressure impulse at the wall
was P = 0:742ρunbh1 for the rectangular wave. The pressure-impulse profile on the wall is shown
in Figure 3.20. An estimate of the pressure field can be made if the duration of the violent impact,
∆t = ta ; tb , is known. By approximating the peak pressure as a function of time as triangular (see
47
a theoretical study
0.1
0.2
P= (ρunbh1 )
0.3
0.4
0.5
0.6
0.7
0
–0.2
–0.4
z=h1
–0.6
–0.8
–1
Figure 3.20: The pressure-impulse profile on the impact wall, x = 0, for a semi-infinite, rectangular
wave, b ! ∞. The distance down the wall is denoted by z.
Figure 3.18) we find that
P=
and the peak pressure becomes p peak
Z ta
tb
2P
=
1
p dt ∆t ppeak 2
∆t. Peregrine (2003) notes that the impact duration
increases with the physical size of the waves, but that scaling from the laboratory to the natural-scale
is uncertain for violent impacts.
Using this framework, we may estimate the peak pressure during the impact of the dense core of a
snow avalanches with a vertical catching dam, on the assumption that the dense core of an avalanches
is incompressible. Typical values of density, flow speed and depth of a natural dry-snow avalanche
are given by: 250 kg m ;3 < ρ < 500 kg m;3 ; 30 m s;1 < unb < 60 m s;1 ; and 1 m< h1 < 3 m, and
the rise and fall in the peak pressure lasts for 10–30 ms (Issler, 2003), leading to peak pressures
in the range 400 kPa < p peak < 6700 kPa. These predicted peak pressures are of the same order of
magnitude as the measured pressures. It should be noted that in most cases the dam face is not
vertical (normal to the ground), but is inclined at some angle, usually α 40 to the horizontal (see
Figure 3.2). This reduces the steepness of the flow front compared with the dam face and the impact
becomes less violent. The measured value of the pressure will furthermore depend on the position
and the size of the pressure sensor (load plate) on the dam, since the computed peak pressures occur
close to the base of the wall. Compressibility of the avalanche will furthermore reduce the values of
the peak pressures, as discussed above.
48
wa =ua
3
2.5
2
1.5
1
0.5
0
0
0.2
0.4
0.6
0.8
1
x=h1
Figure 3.21: The ratio between the normal and streamwise components of the pressure induced
velocity at the free surface of a semi-infinite wave, as a function of the non-dimensional distance
from the wall.
3.4.3 The velocity field and the splash-up
The velocity immediately after impact can be obtained from equation (3.36), and at the free surface
of a semi-infinite wave it becomes
ua (x 0)
=
wa (x 0) =
unb ∞
1 ;λn x=h1
e
:
λ
n=1 n
2unb ∑
Cooker and Peregrine (1995) showed that at the free surface and near the origin
wa (x 0) ! ;(2unb=π) log πx=(4h1 )] as x=h1 ! 0:
(3.38)
The ratio between the normal and streamwise components of the pressure induced velocity at the
free surface is plotted in Figure 3.21 and indicates the shape of the free surface near the wall, soon
after impact. The high velocity change may indicate the strength of the splash onto the wall. The
theory does, however, not describe the ‘inner’ region, closer to the wall, where the splash form, and
there is a singularity in the vertical component of the free surface velocity after impact at x = 0.
We continue by estimating the height of the splash, h a , from the pressure induced velocity at the
free surface after impact, ua , on a deflecting dam positioned normal to a sloping surface inclined at
an angle ξ to the horizontal, as shown in Figure 3.22. The normal component of the flow velocity
at the dam is unb
= u1 (;n̂) = u1 sin γ, where n̂ is the unit normal to the dam.
If there is no
energy dissipation in the impact and changes in the free surface elevation during impact are small,
49
a theoretical study
Side-view
Plan-view
y
g
z
x
x
h1
Defl
n̂
u1
ecti
ŝ
u1
ng
dam
ŝ
ξ
γ
ha
De
fle
ctin
gd
am
ẑ
Figure 3.22: Schematic diagrams of a deflecting dam on a slope inclined at an angle ξ to the horizontal. The avalanche flows down a channel with a speed u 1 before interacting with a deflecting
dam. The dam is positioned at an angle γ to the flow direction. Unit vectors normal and parallel with
the dam are given by n̂ = (; sin γ cos γ 0), ŝ = (cos γ sin γ 0) and ẑ = (0 0 1).
Bernoulli’s theorem along the surface streamline yields
1; 2
ua + w2a = g cosξha 2
which from equation (3.38) leads to
ha =
u2a
+ =
w2a
πx
u21 sin2 γ 1 + π42 log2 ( 4h
)
1
2g cosξ
2g cosξ
:
(3.39)
The normal component of the pressure induced, free-surface velocity, w a , is given as a function of
the distance from the dam, x (see Figure 3.21). How this distance may be determined for the surface
streamline will not be discussed further here, but rather in context with water experiments in §4.3.
The splash-up on the deflecting dam may be expected to move ballistically along the dam face
since the pressure impulse acts over an extremely short period. The components of the gravitational
acceleration in the x, y and z directions (defined in Figure 3.22) are g = (g sin ξ 0 ;g cosξ). We
take n̂ and ŝ to be unit vectors normal and parallel to the dam given by n̂ = (; sin γ cos γ 0), ŝ =
(cos γ sin γ 0), and ẑ = (0 0 1).
If dissipation is ignored, we may write Newton’s second law of
motion on the dam face as
d2 s
dt 2
d2 z
dt 2
=
g sin ξ cos γ
= ;g cosξ
:
50
(3.40)
(3.41)
We solve equations (3.40) and (3.41) subject to the initial conditions:
d
s (0)
dt
d
z (0)
dt
=
=
u1 cos γ;
q
w2a + u2a ;
s(0) = 0;
z(0) = h1 πx
where w2a + u2a = u21 sin2 γ 1 + π42 log2 ( 4h
) . It follows that
1
s
=
z
=
1
g cos γ sin ξt 2 + u1 cos γt 2
q
; 12 g cosξt 2 + w2a + u2at + h1:
(3.42)
(3.43)
Equations (3.42) and (3.43) describe the path taken by the surface of a fluid, impacting the dam
at (n s z)
= (0 0 h1).
The impact between the flow front and the dam, however, takes place all
along the dam at different times, leading to the possibility that flow on its way down the dam face
interacts with flow on the way up from an impact further down along the dam. The simplest way to
include this effect is to treat the splash-up on the dam as the locus of points of maximum height, h a ,
from impacts all along the dam face. See further discussion in connection with water experiments in
§4.3.3.
51
Chapter 4
shallow flows and dams: an
experimental study
4.1 Introduction
Three series of experiments to study the interaction between shallow, supercritical, granular flows
and dams in small-scale laboratory chutes are described in this chapter. The experiments involve
supercritical flows of small glass beads down steep slopes of both relatively long and short duration
and supercritical flows of water on shallow slopes.
The key dimensionless parameter that characterises these experimental flows and the natural
p
avalanche flow is the internal Froude number, Fr = u= g? h, where u is the flow speed, g? = g cos ξ
is the component of the gravitational acceleration normal to a slope inclined at an angle ξ to the
horizontal and h is the depth of the flow. The Froude numbers of the experimental flows were in
range of 5–15. The experiments at the lower Froude numbers are comparable with experimental
results of flows interacting with dams reported by Gray et al. (2003). These can be compared with
the higher Froude number flows in order to draw conclusions about the expected flow behaviour
when the dense core of natural dry-snow avalanches interacts with dams (5 < Fr < 10).
The first experimental series is described in §4.2. It involves experiments with granular flows at
Froude numbers of order 10, conducted to identify the flow behaviour during the interaction with
a deflecting dam. The avalanches in this series had a relatively short duration, that is the length to
depth ratio of the currents was approximately 200. The experimental observations gave rise to questions that were addressed in the following two experimental series where avalanches with a longer
52
4.2. Short granular flows interacting with deflecting dams
flow duration were used (a length to depth ratio of over 10000). Experiments with steady flows of
water on shallow slopes with Froude numbers close to 5 are described in §4.3. They were designed
to investigate how well shallow-water jump conditions agreed with shallow-water experiments. The
splash-up of the first front of the flows onto dams is also studied in §4.3 and compared with predictions of pressure impulse theory. The third and final experimental series is described in §4.4. These
experiments involved steady granular flows at three different Froude numbers, 5, 12 and 14, linking
the previous two experimental series. The three experimental series are finally compared in §4.6
and some general conclusions drawn regarding the physics of the interaction of high Froude number
flows with dams.
4.2 Short duration granular flows interacting with
deflecting dams
The experiments were designed to study the deflection of high Froude number granular flows by
dams and to compare the flow behaviour during this interaction with the point mass theory and the
shallow-water jump-conditions, described in §3.2 and §3.3, respectively. The height of a deflecting
dam, required to fully deflect the oncoming flow, was quantified empirically for dams positioned at
different deflecting angles to the flow direction. The flow behaviour in situations where some of the
flow escaped over the dam was also studied.
4.2.1 Experimental setup and design
The experiments were performed on a 6 m long plywood chute consisting of two straight sections,
one inclined at 38 to the horizontal and the other horizontal, as shown in Figure 4.1. The sections
were connected by a thin plywood sheet to obtain a smooth transition between the sections. The
chute had 0.20 m high side walls made of perspex to allow for observations from the sides.
The deflecting dams were constructed of plywood and positioned at the upper section of the
chute, 1.70 m downslope from the release gate. The dams had a planar upstream face, normal to the
base of the chute. They were positioned at different angles, γ, to the direction of the approaching
flow. The angle, γ, is shown in Figure 4.1 (b) and will be referred to as the ‘deflecting angle’. The
dams in the experiments were positioned at five different deflecting angles, γ = 8 , 15 , 24 , 32 and
44 . The dams had a height to flow depth ratio from 1 to 20, or up to the dam height required to fully
deflect the flow at each deflecting angle.
The experiments were designed so that the granular current had a Froude number of order 10.
Glass beads (ballotini) of mean size 90 µm, density 2500 kg m ;3 (bulk density of 1600 kg m ;3 ) and
an approximately spherical shape were used. In each experiment, 6 kg of particles were released
53
g
Gate
(a) side-view
Camera 1
38
1.7
0m
z
s
3:1
0m
Deflecting dam
0.6
0m
3:10 m
(b) plan-view
L = 0:325 m Perspex
0:975 m
Flow
Perspex
Deflecting dam
1.70 m
n
Camera 2
s
γ
Curved sheet
l
2.30 m
Run-out zone
1.52 m
Figure 4.1: Schematic diagrams of the experimental chute that was used in the short duration, granular, deflecting dam experiments: (a) a side-view of the chute; (b) a plan-view of the chute section
next to the deflecting dam. The run-out zone is defined to start where the slope angle changes, at the
curved sheet. The channelled section of the chute was approximately 1.70 m long, measured from
the lock gate, and the curved plywood sheet was positioned 2.30 m down from the gate.
54
u [m s;1 ]
83
84
85
86
3.5
3
2.5
2
1.5
1
Main body
Head
Steady flow phase
Tail
0.5
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
t [s]
Figure 4.2: Flow speed plotted as a function of time for experiments number 83 to 86. The speed
was measured at a fixed position, close to the downstream end of the deflecting dams.
from the top of the chute by the rapid opening of a lock gate. The interaction between the flow and
the deflecting dams was recorded from above and from the side using two video cameras, recording
at 25 frames per second.
The depth profile along the deflecting dams was measured by analysing the side-view video
footage of the experiments. The angles of the deflected and the over-flowing parts of the stream
relative to the deflecting dam were analysed from the video footage provided by the camera that was
positioned above the dams. The run-out length and distribution of the deposited particles were also
measured.
The speed of the flow approaching the dams was measured over a length of 0.40 m, where the
dams would later be positioned. The speed was measured by tracking the first front of the flow
and tracer particles for interior free-surface speeds. The depth of the flow approaching the dams was
measured by fixing a gate in the flow path at the downstream end of the channelled chute section. The
gate was positioned at a known height above the base of the chute. The distance of the gate from
the chute was then increased systematically until all of the current flowed under the gate without
touching it. The internal friction angle of the ballotini beads was measured φ 21 by building up
a cone of particles on a horizontal plane and repeatedly measuring the height and diameter of the
cone. The dynamic bed friction angle was measured δ 20 by tilting a plane with a thin layer of
particles and the angle at which the particles came to a rest was determined.
55
u [m s;1 ]
Lmax [m]
4
2.5
3.5
3
2
2.5
1.5
2
1
1.5
1
0.5
0.5
0
0
20
40
60
No.
80
100
0
0
20
40
60
No.
80
100
Figure 4.3: The measured leading edge run-out, L max , and the front speed, u, as a function of the
sequential number of the experiment performed (+) with a linear fit through the observations. The
leading edge run-out was measured from the start of the run-out zone (the curved sheet, shown in
Figure 4.1).
4.2.2 Flow description
The flow consisted of a short, turbulent and thick ‘head’, followed by a thinner, denser and longer
‘main body’ and a thinner and slower ‘tail’. These are common flow features of Boussinesq gravity
currents (Simpson, 1987) and also snow avalanches (Issler, 2003) and ping-pong ball avalanches
(McElwaine and Nishimura, 2001). The flow quickly reached a terminal speed on the upper section
of the chute and flowed down the chute with a constant speed and a constant depth. The interior
speed of the flow was measured at the end of the experimental period and is plotted in Figure 4.2.
The head of the flow was about 0.50 m long, 0.04 m thick and had a speed of (3:7 0:3) m s ;1 . It was
dilute (it could be seen through when looked at from above) and turbulent, with eddies suspending
the ballotini beads. The flow immediately following was much denser with a constant depth of
(0 009 0 001) m and speed of (3 5 0 1) m s
:
:
:
:
1,
;
resulting in a Froude number of 13. The dense
flow phase was maintained for 0.6 s, so the flow was roughly 2.1 m long. This flow phase will be
referred to as the ‘steady flow phase’. The flow rapidly thinned and slowed down after that.
The ballotini beads changed slightly in character during the experimental period, leading to
increased mobility of the flow. The run-out (measured from the start of the run-out zone, shown
in Figure 4.1) lengthened by approximately 5% during the period. The front speed of the flow was
also found to increase through the experimental period as shown in Figure 4.3. The interior speed of
the steady flow phase was only measured at the end of the experimental period. The measurements
(plotted in Figure 4.2) showed that the speed of the steady flow phase was approximately 0.2 m s ;1
lower than the front speed. It was thus assumed that the speed of the steady flow phase increased
throughout the experimental period in the same fashion as the front speed, but was 0.2 m s ;1 lower
than the front speed. The values of the interior speeds during the steady flow phase that will be used
in the following calculations, for the different deflecting angles, is given in Table 4.1.
56
γ ]
8:0 0:5
15:0 0:5
24:0 0:5
32:0 0:5
44:0 0:5
90:0 0:1
No.
65
60
47
39
55
11
u [m s;1 ]
3.3
3.3
3.1
3.0
3.2
2.7
Table 4.1: The estimated speed during the steady flow phase, u, for the different deflecting angles, γ.
The number of each experiment is denoted by No.
z=h1
γ = 44
γ = 32
γ = 24
γ = 15
γ=8
20
15
10
5
0
0
0.1
0.2
0.3
0.4
0.5
s [m]
Figure 4.4: The measured semi-steady run-up profiles on the deflecting dams non-dimensionalised
with the depth of the oncoming stream, as a function of the length along the dams. Each curve
corresponds to a fixed deflecting angle, γ. The uncertainty in the non-dimensional run-up is estimated
1 due to the short duration of the steady flow phase and an inaccuracy in drawing/visualising the
curves from the video recordings.
4.2.3 Results: flow completely deflected by the dams
The dilute flow front splashed onto the deflecting dams up to a similar height as the denser part of
the flow and formed a semi-steady profile on the dams. The width of the stream flowing along the
dam increased downstream and finally formed a jet at the end of the dams in a direction parallel with
the dam. The semi-steady run-up profiles on the dams are plotted in Figure 4.4. It shows that the
flow depth at the upstream face of the dams grew with increasing deflecting angles.
The way in which subjecting larger and smaller proportions of the avalanches to the deflecting
dams affected the run-up on the dams (long and short dams) was examined for three different deflecting angles. Figure 4.5 shows a plot of the semi-steady run-up profiles on the deflecting dams
for deflecting dams covering different proportions of the width of the avalanches, 0:2 l =L 1:1,
57
z=h1
l =L = 0:2
γ = 15
l =L = 0:6
15
l =L = 0:4
γ = 24
l =L = 0:4
l =L = 0:6
l =L = 0:9
10
5
0
0
20
0.1
0.2
0.3
0.4
s [m]
0.5
0.6
0.7
l =L = 0:5
γ = 32
l =L = 0:8
l =L = 1:1
15
10
5
0
0
0.1
0.2
0.3
0.4
s [m]
0.5
0.6
0.7
Figure 4.5: The measured non-dimensional run-up profiles of the avalanches on the deflecting dams
plotted as a function of the length along the dams. The deflecting dams covered different proportions of the width of the avalanches, l =L (see Figure 4.1 for definition), between 0.2 and 1.1. The
maximum run-up was reached during the steady flow phase.
where L is the width of an avalanche and l is the deflected width, defined in Figure 4.1. The maximum run-up height along the dams was always reached after a distance along the dams that was
shorter than the length of the dams. Changing the length of the dams did therefore not alter the
maximum run-up on the dams. It was furthermore concluded that the downslope end of the dams,
where the particles are no longer supported and the pressure suddenly drops, did not affect the run-up
profiles on the dams significantly. This is consistent with the flow being in a supercritical flow state
along the dams, since conditions upstream remain unaffected by changes to flow state downstream.
Comparison with existing models
The point mass model (PM) was used to calculate the path of a point-mass, positioned on the surface
of the flow, along a deflecting dam. The model was developed by Irgens et al. (1998) and is reviewed
in §3.2. Energy was assumed to be conserved in the impact with the dam and the friction coefficients
in the model (µ and m=D) were determined as follows. The Coulomb friction coefficient was taken to
be µ = tanδ, where δ is the dynamic friction angle between the flow and the base of the experimental
58
chute, as is common practice for granular flows (see §A.2). The dynamic bed friction angle was
measured approximately 20 , implying µ 0:4. The dimensional coefficient representing turbulent
drag, m=D, was determined from the steady state of the Voellmy equation (2.1), where m=D in the
point mass model equals h=CD in Voellmy’s equation,
u2 = (m=D)g(sinξ ; µ cosξ):
(4.1)
By substituting the measured, steady flow speed on the chute into equation (4.1) we obtain m=D 4 m. The physical range of these parameters for snow is discussed by Perla et al. (1980).
Figure 4.6 shows that for small deflecting angles, the point mass model predicts the run-up on
the deflecting dams well, but over-shoots when the deflecting angle increases. No parameter values, within a reasonable physical limit of the two parameters, µ and m=D, could reproduce all the
observed curves at once. The reason might be that a part of the mechanical energy of the flow is
dissipated in the impact with the dams. This dissipation can be assumed to increase with larger deflecting angles as the velocity component normal to the dams increases. McClung and Mears (1995)
suggest that the flow should lose its component of momentum normal to a dam in the impact with a
catching dam. That implies that there should not be run-up onto dams that have an upstream face that
is normal to the slope on which they are positioned. That is in contradiction with the experimental
results presented here. Chu et al. (1995), however, note that the leading edge model of McClung
and Mears is not sophisticated enough to adequately model the flow behaviour resulting from an
abrupt slope change. Irgens et al. (1998) suggest that only a fraction of the normal component of the
flow’s momentum is lost in the impact. That would, however, lead to an underestimate of the run-up
onto the deflecting dams at the smaller deflecting angles. The two models do, therefore, not seem to
capture the physics of the interaction correctly.
The granular flow consists of a thin, dense layer of numerous interacting point masses and can
therefore be viewed to some approximation as a fluid or gas-like current. The flows are shallow,
supercritical and not highly compressible, which gives rise to the possibility of shocks in the flow
depth forming upstream of the dams (granular jumps) as described in §3.3. The depth of a steady,
oblique shock, h2 , was calculated from the shallow-water jump conditions, equations (3.29), (3.30),
and (3.31), and is plotted in Figure 4.6 as a dashed line. We see that the observed run-up profiles (+)
are higher than the predicted shock heights close to the upstream end of the deflecting dams. The
maximum run-up then decreases along the dams and the theoretically derived shock height predicts
the run-up close to the downstream end of the deflecting dams. By introducing an earth pressure
coefficient, K, into the jump conditions (equation (3.11)) the theoretical shock height is lowered,
since Kact=pass > 1 (using φ = 21 and δ = 20 , Kact = 1:05 and Kpass = 1:55), and worse agreement
with the experimental observations is obtained.
The theoretical shock angle relative to the dams is plotted in Figure 4.7 along with the observed
59
z=h1
γ = 8
PM
Data
Shock
20
γ = 15
PM
Data
Shock
15
10
5
0
25
γ = 24
PM
Data
Shock
20
γ = 32
PM
Data
Shock
15
10
5
0
0
25
0.2
0.3
0.4
s [m]
0.5
0.6
0.7
γ = 44
PM
Data
Shock
20
0.1
15
10
5
0
0
0.1
0.2
0.3
0.4
s [m]
0.5
0.6
0.7
Figure 4.6: Non-dimensional run-up profiles for granular flows at Fr 1 = 13. Each graph shows
the observed depth profile along the deflecting dams, non-dimensionalised with the depth of the
approaching stream, for a fixed deflecting angle, γ (+). The theoretical non-dimensional run-up of
a point mass is shown with a solid curve and the theoretical non-dimensional shock height with a
dashed line.
60
β ; γ]
Data
Theory
7
6
5
4
3
2
1
0
0
10
20
30
γ ]
40
50
Figure 4.7: The shock angle relative to the dam, β ; γ, plotted as a function of the deflecting angle,
γ, calculated from the shallow-water jump-conditions (dashed curve) and experimentally observed
(+).
widening of the stream along the deflecting dams for the different deflecting angles. A surprisingly
good agreement is found between the two.
The agreement of the shock theory with the experimental observations indicates that a shock
forms upstream of the deflecting dams as is expected for supercritical flow of incompressible fluids
interacting with deflectors. What still remains unclear from these experiments is why the flow ran
higher up on the dams than the jump conditions predicted and had a tendency to turn over in a
backward rotating motion along the first 0.4–0.5 m of the dams. It is also unclear whether gravity
disturbs the shock formation or influences the shock height along the dams due to a subsequent along
slope acceleration, since the experiments were conducted with steady flows on a sloping surface,
whereas the theory was derived for fluid flow on a horizontal plane where the flow states on both
sides of the shock are constant. (It should be noted that the flows were flowing at a constant speed
down the chute as they interacted with the dams.) There is, furthermore, a possibility that friction at
the upstream face of the dams affects the relatively narrow shock. The steady state flow phase was
also quite short ( 0:6 s) and the run-up onto the dams may not have been fully developed. Some of
these issues are addressed in the other experimental studies described later in this chapter.
4.2.4 Results: flow over-topping dams
Flow over-topping deflecting dams is perhaps not of direct practical interest since deflecting dams
are always designed to completely deflect the oncoming flow. The over-topping is therefore only
described here experimentally but not studied in detail.
61
Flow
β
k
oc
γ
ψ
De
fle
cti
ng
da
m
Sh
Figure 4.8: A schematic plan-view of flow over-topping a deflecting dam.
Over-topping of flow interacting with deflecting dams was studied by measuring the ‘overflow
angle’, ψ, shown in Figure 4.8, during the steady flow phase when the flow was not fully deflected
by the dams. The deflecting dams were of various heights, H, and were positioned at different
deflecting angles, γ, to the direction of the oncoming flow. The ratio of the overflow angle to the
deflecting angle is plotted in Figure 4.9 (a) as a function of the ratio of the dam height to the depth
of the oncoming flow.
The experiments showed that for low dams the current shot over the dams in a direction close to
that of the approaching stream, ψ 0. The jets were then turned more in the direction of the deflecting dams as the dams became higher and the overflow angle of the jets approached the deflecting
angle of the dams as the height of the dams approached the height needed to fully deflect the current.
If a shock is formed upstream of the dams, the height needed to fully deflect the current is the
shock height, h2 , implying that ψ ! γ as H ! h2 . This effect is illustrated in Figure 4.9 where ψ=γ
is plotted as a function of H =h2 for the five deflecting angles. The data does not collapse completely
onto a single curve. From the figure, we note that the dams need to be higher relative to the height of
the shock to fully deflect the flow for larger deflecting angles, γ. The large error bars in Figures 4.9 (a)
and (b) are due to the short duration of the steady flow phase.
62
ψ=γ
(a)
0.8
0.6
γ = 44
γ = 32
γ = 24
γ = 15
γ=8
0.4
0.2
0
0
2
4
6
8
ψ=γ
H =h1
10
12
14
(b)
0.8
0.6
γ = 44
γ = 32
γ = 24
γ = 15
γ=8
0.4
0.2
0
0
0.2
0.4
0.6
0.8
H =h2
1
1.2
1.4
Figure 4.9: The ratio of the measured overflow angle to the deflecting angle, ψ=γ, plotted as a
function of the ratio between the dam height and (a) the depth of the approaching stream, H =h 1 ; (b)
the calculated shock height, H =h2.
63
Figure 4.10: A photograph of the experimental setup for rapid flows of water. A steady stream of
water was supplied through 3 hoses and stored in a reservoir. The water flowed over a weir and under
a gate to minimise turbulence and wave action, and continued down a 0.2 m wide perspex channel
before finally hitting a deflecting dam, also made of perspex. The deflecting dams were 0.3 m long
and did not ever completely span the width of the channel.
4.3 Steady water flows interacting with deflecting dams
Two series of water experiments were conducted to study how well the shallow-water jump-conditions,
derived in §3.3.3, agreed with observations of rapid, shallow-water flow. The upstream face of the
dams was normal to the slope of the channel in all of the water experiments. The run-up of the first
front of the water flow on the dams was compared with run-up obtained by pressure impulse theory
for incompressible flows, described in §3.4, in order to explain the observed high run-up. In this
context the run-up is also termed splash-up.
The experiments involved a steady stream of supercritical water flow down a 0.2 m wide channel at
a speed u1 and a depth h1 . This stream hit a deflecting dam and formed a steady, oblique shock at
an angle β ; γ to the deflecting dam. The shock had a depth h 2 and speed u2 along the dam, see
Figure 4.10. The flow ended up in a large tank that gradually filled during the experiments, since the
draining system did not fully cope with the flux of water into the tank. It was therefore necessary to
split the experiments into three series: series (I) to measure the height of the hydraulic jump; series
64
4.3. Steady water flows interacting with deflecting dams
Series
I
II
III
ξ [ ]
3
6
9
3
6
9
9
u1 [m s;1 ]
1:05 0:05
1:25 0:08
1:4 0:1
0:95 0:05
1:18 0:07
1:25 0:08
1:54 0:13
h1 [m]
0:00575 0:00025
0:00475 0:00025
0:00425 0:00025
0:0055 0:0005
0:0045 0:0005
0:004 0:0005
0:011 0:001
Fr1
4:4 0:3
5:8 0:5
7:0 0:8
4:1 0:4
5:6 0:7
6:4 0:8
4:7 0:7
Re
6100
6000
6100
5200
5300
5000
17000
Table 4.2: The flow regime of the steady stream of water flow for the three experimental series.
Series I was used to measure the depth profiles along the dams, series II to measure the shock angles
and series III to measure the splash-up of the flow front on the dams.
(II) to measure the shock angle; and series (III) to measure the splash-up of the flow front.
The steady depth profiles on the dams and the shock angles were photographed with a digital
camera through the side of the dams and from above (see Figure 4.11). The depth profiles and shock
angles were measured for water flow on three different slopes, inclined at ξ = 3 , 6 and 9 to the
horizontal. The flow speed, u1 , and depth, h1 , upstream of the dams were different for the different
channel slopes, and consequently the oncoming flow had different Froude numbers on the slopes.
The splash-up on the dams was recorded using a digital video camera, recording at 25 frames per
second, since the splash was unsteady. The water was dyed and the maximum run-up (or the splashup) traced along the dams.
4.3.2 Results: steady, oblique hydraulic jumps
The speed, depth and Froude number of the flows approaching the dams in the three experimental
series are listed in Table 4.2. All of the hydraulic jumps were ‘stabilised’, 4:5
<
Fr 1
<
9, and
relatively free of wave action, apart from the experiments on the 3 slope where the hydraulic jumps
were on the boundary of being ‘stabilised’ jumps and might be categorised as a ‘transitional’ jumps
with pulsating action (Hager, 1992). The Reynolds number, defined by Re = ρuh=µ, where ρ is
density and µ is viscosity, is sufficiently large so that viscous effects may be neglected (in these
experiments).
The run-up profiles along the side faces of the dams are plotted in Figure 4.12. The flow depth
along the dams was found to be nearly constant, for dams at γ 20 . For dams at the largest
deflecting angles, γ > 20 , there was a zone at the upstream end of the dams with higher run-up
and overturning of the flow. The flow ran up, reached a maximum height and fell back upon the
oncoming stream in a rotating motion. The flow depth was approximately constant further along the
dams (see bottom photograph in Figure 4.11).
This zone of higher run-up and overturning of the flow may be regarded as an adjustment region
and that when the flow falls back on the oncoming stream the hydraulic jump is initiated. Another
65
Figure 4.11: Photographs of stationary, oblique shocks of supercritical shallow-water at Fr 1 = 4:5,
γ = 20 (top left) and Fr1 = 7, γ = 40 (top right), looking up into the streams along the deflecting
dams. The bottom photograph shows the same shock as the top right photograph, but looking onto
the side of the deflecting dam. It can be observed that the flow runs higher up on the deflecting
dam at the upstream end of the dam and then falls back on itself and the shock widens and becomes
shallower.
66
z=h1
Fr1 = 4:5
Fr1 = 6:0
8
6
4
2
0
0
10
0.05
0.1
0.15
0.2
s [m]
0.25
0.3
Fr1 = 7:0
8
6
4
2
0
0
0.05
0.1
0.15
0.2
s [m]
0.25
0.3
Figure 4.12: Non-dimensional run-up profiles of steady water flows at different Froude numbers.
Each curve shows the measured depth of the steady, oblique shock at different positions along the
dam face, s, for a fixed deflecting angle. The deflecting angles tested were γ = 5 , 10 , 15 , 20 , 30
and 40 . The shallowest profile on each graph corresponds to the 5 deflection and the deepest to
the 40 deflection.
67
possible explanation for the higher run-up and overturning of the flow at the upstream end of the
dams lies in the finite width of the highly turbulent transition zone. The width of the transition zone
is denoted by ∆l in Figure 4.13. The higher run-up on the dam face occurs where the transition
zone intersects with the dam face. Namely at the upstream end of the dams. The significant vertical
accelerations in the transition zone may push the water higher up on the dams than the depth of
the hydraulic jump further along the dams. This higher run-up is visible from Figure 4.12 for the
largest deflecting angles, where the velocity component of the approaching flow normal to the dams,
u1 sin γ, is largest. The flow may follow ballistic trajectories along the side of the dam in this region,
due to high accelerations and turbulence in the flow. The region along the upstream end of a dam
where the dam and the transition zone intersect will be referred to as the ‘adjustment region’, also
illustrated in Figure 4.13.
The length of the adjustment region may be expected to increase with firstly smaller shock angles relative to the dam and secondly a wider transition zone. Both of these quantities (β ; γ, ∆l)
are directly related to the Froude number of the oncoming flow, Fr 1 . Figure 3.17 (a) and (b) shows
that the shock angle relative to the dam decreases with increasing Froude number, but is relatively
independent of the deflecting angle for Froude numbers above 5. We may therefore expect decreasing shock angles, β ; γ, with higher Froude number flows for any deflecting angle. It follows that
the adjustment region should lengthen with higher Froude number flows. Furthermore, studies of
hydraulic jumps show that the transition zone widens as the depth of the jump, h 2 , increases for
Froude numbers up to approximately 7. (For 1 < Fr 1 < 7, 4 < ∆l =h2 < 6, see, for example, Gerhart
et al., 1993.) Figure 3.17 (c) and (d) shows that the shock depth increases with increasing deflecting
angles and increasing Froude numbers flows. Hence, the adjustment region should also lengthen
with larger deflecting angles and higher Froude numbers. We therefore conclude that the length of
the adjustment region increases with higher Froude numbers of the oncoming flow and with larger
deflecting angles of the dams.
These conclusions are supported by the experimental observations. An increase in the length
of the adjustment region with higher Froude number flows is not obvious from the run-up profiles
in Figure 4.12, perhaps due to the narrow range of the Froude numbers. The transition zone in the
experiments was, on the other hand, observed to widen as a function of the deflecting angle (and
consequently as a function of the depth of the hydraulic jump) from being just under 0:01 m for
γ = 5 to approximately 0:03 m for γ = 40 . This widening leads to a longer adjustment region
for the experiments at the largest deflecting angles, and is observable from the run-up profiles in
Figure 4.12.
We now compare the experimental results with predictions of the jump-conditions for stationary,
weak, oblique shocks in some detail. The system of equations (3.29), (3.30) and (3.31) was solved
numerically for the shock depth, h 2 , speed, u2 , and angle, β, for a given oncoming flow depth,
68
u2 h2
B
∆l
Transition zone
B
u1 h1
A
γ
β
A
Plan-view
j.
Ad ion
e
rg
Transition zone
∆l
Deflecting dam
Deflecting dam
u2
h2
h1
h1
Section AA
Section BB
Figure 4.13: Schematic diagram of the adjustment region along a deflecting dam, caused by the
intersection of the turbulent transitions zone with the deflecting dam.
69
β ; γ ]
Fr1
Fr1
Fr1
25
= 4:0
= 5:5
= 6:5
20
15
10
5
0
0
5
10
15
γ ]
20
25
30
35
40
45
Figure 4.14: The shock angle relative to the dam plotted as a function of the deflecting angle for the
three different Froude numbers of the approaching stream. The lines show the theoretical predictions
and the points are the experimental observations.
h1 , speed, u1 , and deflecting angle, γ. The shock angle relative to the dams, β ; γ, is plotted in
Figure 4.14 as a function of the deflecting angle. There is good agreement between the experimental
results and the theory for the three experimental sets. The predicted shock depths are plotted as
dashed lines in Figures 4.15, 4.16 and 4.17 and the observed run-up profiles are plotted as points.
The agreement between experiments and theory is fairly good apart from the observed higher run-up
at the upstream end of the dams.
We conclude that the hydraulic jumps on shallow slopes are well described by the shallow-water
jump-conditions where constant flow states are assumed on both sides of the hydraulic jump. There
is, however, a region close to the upstream end of the dams where the flow runs higher up on the
dams than the theory predicts.
4.3.3 Results: splash-up of the flow front
The run-up of the flow front on the dams was studied in experimental series (III) (see Table 4.2). The
observed run-up was considerably higher than the predicted height of an oblique shock. The run-up
was also found to be higher than the predicted run-up onto the dam, determined from Bernoulli’s
theorem along the surface streamline
hb =
(u1 sin γ)2
2g cosξ
(4.2)
where hb is measured normal to the base of the experimental chute, from the surface of the approaching flow, the chute is inclined at an angle ξ to the horizontal and u 1 sin γ is the component of the flow
70
z=h1
8
γ = 5
γ = 10
γ = 15
γ = 20
γ = 30
γ = 40
6
4
2
0
10
8
6
4
2
0
10
8
6
4
2
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0
0.05
0.1
0.15
0.2
s [m]
0.25
0.3
Figure 4.15: Non-dimensional run-up profiles for water flow at Fr 1 = 4:5. Each graph is a plot of the
observed depth profile along a deflecting dam, non-dimensionalised with the depth of the oncoming
flow at a fixed deflecting angle, γ (+). The theoretical non-dimensional shock depth, h 2 =h1 , is shown
with a dashed line.
71
z=h1
8
γ = 5
γ = 10
γ = 15
γ = 20
γ = 30
γ = 40
6
4
2
0
10
8
6
4
2
0
10
8
6
4
2
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0
0.05
0.1
0.15
0.2
s [m]
0.25
0.3
flow, at a fixed deflecting angle, γ (+). The theoretical non-dimensional shock depth, h 2 =h1 , is shown
as a dashed line.
72
z=h1
8
γ = 5
γ = 10
γ = 15
γ = 20
6
4
2
0
10
8
6
4
2
0
10
γ = 40
γ = 30
8
6
4
2
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0
0.05
0.1
0.15
0.2
s [m]
0.25
0.3
flow, at a fixed deflecting angle, γ (+). The theoretical non-dimensional shock depth, h 2 =h1 , is shown
as a dashed line.
73
velocity normal to the deflecting dam. We use the theoretical run-up, derived from the pressure
impulse theory in §3.4, to explain this high run-up of the flow front on the dams.
The theory requires the distance of the free surface from the dam immediately after impact (x=h 1
in equation (3.39)) to be determined in order to obtain the normal component of the pressure induced
velocity after impact, wa (see Figure 3.21). This distance was chosen such that the theoretical splashup on a catching dam (γ = 90 ), calculated from equation (3.39), would match the observed splash-
up on the dam. The required distance turned out to be x=h 1 0:3 which corresponds to wa =ua 1,
and equals approximately 3 mm from the dam in dimensional units. The thickness is similar to the
observed thickness of the jet travelling up the dam.
The locus of points of maximum run-up at each location along the dam face is referred to as the
splash-up on the dam. The profiles of the observed splash-up on the dams for the different deflecting
angles are plotted in Figure 4.18. The theoretical profiles are also plotted in Figure 4.18. They were
obtained by allowing the fluid to evolve ballistically along the dam face after the impact according
to equations (3.42) and (3.43), and plotting the locus of points of maximum height of all possible
trajectories along each dam. We find that the theory slightly under-predicts the splash-up on the
deflecting dams, but follows the observed experimental trend. The under-prediction of the theory
is not all together surprising, since we only plot the locus of maximum points of all the trajectories
along each dam, but do not account for the interaction between water trajectories on the way down
the dam face and water trajectories flowing up from an impact further downstream along the dam.
The values of the calculated, pressure induced, maximum splash-up on dams at different deflecting angles to the flow direction, γ, are plotted in Figure 4.19 non-dimensionalised with the depth
of the oncoming flow, (h a + h1 ) =h1 . The pressure induced splash-up, h a , is determined from equation (3.39) with x=h1 = 0:3. Also plotted is the maximum splash-up without any pressure induced
velocity changes in the impact, (hb + h1) =h1 , where hb is derived from Bernoulli’s theorem, given in
equation (4.2), along with the experimentally observed maximum splash-up. A much better agreement is obtained between theory and experimental observations when allowing for pressure induced
velocity change in the impact. Considerable pressure induced velocity changes may therefore take
place in the impact between the first front of the flow and the dams. The drawback of determining
the splash-up from the pressure-impulse theory is that the pressure induced velocity is obtained as
a function of a distance from the dam face, which is unknown. One constraint on this length-scale
might be that it should equal the thickness of the jet moving up the wall. The thickness of the jet will,
however, in most cases not be determined beforehand, so that it becomes necessary to determine the
length-scale empirically.
74
20
15
10
5
0
zmax =h1
Data
Impulse
Bernoulli
20
15
10
5
0
0
20
40
60
γ [ ]
80
100
Figure 4.19: Non-dimensional, maximum splash-up, z max =h1 , plotted as a function of the deflecting
angle of the dams, γ. The experimental observations (+) are compared with: the pressure impulse
theory (upper dashed line), where x=h 1 was chosen such that the theoretical maximum splash-up
would fit the observed splash-up on the catching dam (γ = 90 ); and the theoretical maximum splashup without any velocity change during impact (lower dashed line).
76
4.4. Steady granular flows interacting with deflecting dams
Figure 4.20: Photograph of the experimental setup for steady granular flows.
4.4 Steady granular flows interacting with deflecting dams
A series of experiments at three different Froude numbers was designed to study the formation of a
steady, oblique granular jump along a deflecting dam, positioned on a sloping plane. The objective
of the experiments was to maintain a steady flow for a few seconds in order to link the experimental
observations of granular jumps on steep slopes, described in §4.2, to the observations of hydraulic
jumps on shallow slopes, described in §4.3. In addition, the effect that tilting of the upstream dam
faces had on the oblique jumps was also investigated.
The same setup and the same particles were used as in the short granular experiments described in
§4.2.1, but with the channel narrowed by 0.1 m, to 0.225 m. The release mechanism was adjusted so
that it was possible to control the depth of the flow out of the hopper, in order to obtain thinner flows
with a longer flow duration. As before, the flow was released by the abrupt opening of a lock gate,
see Figure 4.20.
The experiments within each set were conducted during a 12 hour period to minimise the effect
of changes of the humidity in the laboratory on the ballotini beads. The flow speed and depth were
77
Figure 4.21: Photographs of oblique, granular shocks, looking downstream onto deflecting dams
which are positioned at γ = 24 to the direction of the approaching flow. The slope on the left hand
side photograph is inclined at an angle ξ = 26 to the horizontal and the flow has a Froude number of
Fr1 = 5. The slope on the right hand side photograph is inclined at an angle ξ = 30 to the horizontal
and the flow has a Froude number of Fr 1 = 12. The shock is narrower on the photograph to the right
(Fr1 = 12) and the flow flips over and falls back on the oncoming stream at the upstream end of the
dam.
measured at the start and end of each experimental set. The flow speed was measured by tracking
tracer particles in the flow and the flow depth was measured using an optical distance sensor (ODS
96, from Leuze electronic) that uses infrared light to measure the distance to a surface.
The experiments were conducted at three different Froude numbers, 5 12 and 14, overlapping the
values in the previous experiments with water and granular flows, and for five different deflecting
angles; namely γ = 8 , 15 , 24 , 32 and 44 . The dams had a planar upstream face which was
normal to the experimental chutfftF53st-204(td)-28(and)-216(ths)-288(e)12(xperim0sor)6(angle)]TJ/F37.15Tf6.4850 TD()Tj/T3 1 Tf0.1
ξ [ ]
26:5 0:5
30:3 0:5
30:3 0:5
u1 [m s;1 ]
0:7 0:01
1:85 0:07
1:52 0:05
h1 [m]
0:0020 0:0002
0:0027 0:0002
0:0013 0:0002
Fr1
5
12
14
Table 4.3: The flow regime of the steady stream of granular flow for the three experimental sets.
4.4.2 Results
The characteristics of the steady flow are listed in Table 4.3. The flow reached a terminal speed
quickly (within the first tens of centimetres from the release) on the upper section of the experimental chute. The steady stream hit the deflecting dam approximately 2 m further down the chute
and a stationary oblique shock was formed within a fraction of a second of the initial impact, see
Figure 4.21. The dilute flow front splashed up to a similar height on the dams as the shock did.
The run-up profiles along the dams are plotted in Figure 4.22 for the three different Froude
number flows. The oblique shocks were stationary for all but the largest deflecting angles. The
dams for the 32 and 44 deflections extended out of the flow path of the avalanches leading to a
significant slowing of the current along the downstream end of the dams. Subsequently, material
started to pile up in front of the dams, starting at the downstream end. This was observed for flows
at Fr1 = 5 with γ = 32 and also for flows at Fr1 = 12 and 14 with γ = 44 . In the case of the flow at
Fr1 = 5 with γ = 44 , the shock was observed to detach from the upstream corner of the deflecting
dam. The shock then propagated upstream until the mass flux down the chute waned. The plotted
profiles of the unsteady flows are those that were observed before the material started to pile up in
front of the dams.
If these unsteady profiles are not considered, the run-up profiles along the dams show an approximately constant flow depth along the dams. The profiles follow the same trend as previously
observed for the water and granular flows: larger Froude numbers and larger deflecting angles lead
to higher run-up on the dams.
Experimental observations of the narrow shocks (Fr 1 = 12, 14) showed that there was a region
at the upstream end of the dams where the flow turned over and fell back on the oncoming stream,
see Figure 4.21. This was also observed in the previous granular experiments and in the water
experiments, while the run-up onto the dams was unaffected. This may be due to a thinner transition
zone in these granular jumps (always less than 0.01 m wide) than in the hydraulic jumps of water.
The experimental results will now be compared with the shallow-water jump-conditions for
weak, oblique shocks in some detail. Equations (3.29), (3.30) and (3.31) were solved numerically
for the shock depth, speed and angle, h 2 , u2 and β, for a given oncoming flow depth, speed and
deflecting angle, namely h1 , u1 and γ. The shock angle relative to the dam is plotted as a function of
the deflecting angle in Figure 4.23. The shock conditions accurately predict the experimental results
79
z=h1
Fr1 = 5
20
Fr1 = 12
15
10
5
0
0
0
0.1
0.2
0.3
s [m]
0.4
0.5
Fr1 = 14
20
15
10
5
0
0
0.1
0.2
0.3
s [m]
0.4
0.5
Figure 4.22: Non-dimensional run-up profiles of steady granular flow along deflecting dams at three
different Froude numbers, Fr 1 = 5, 12 and 14. Each curve shows the steady depth of the shock,
measured at the dam face, for a fixed deflecting angle. The different curves correspond to different
deflecting angles, γ = 8 , 15 , 24 , 32 and 44 . The shallowest profile on each graph corresponds
to the 8 deflection and the deepest to the 44 deflection. The profiles for Fr1 = 5 with γ = 32 and
44 and for Fr1 = 12 and 14 with γ = 44 were unsteady.
β ; γ ]
16
=5
= 14
Fr1
Fr1
Fr1
= 12
12
8
4
0
0
10
20
30
γ ]
40
50
0
10
20
30
γ ]
40
50
Figure 4.23: The shock angle relative to the dam, β ; γ, plotted as a function of the deflecting angle
for flows approaching the dams at three different Froude numbers. The lines show the theoretical
predictions and the points are the experimental results.
80
for all three Froude numbers. The observed depth profiles along the dams are plotted in Figures 4.24,
4.25 and 4.26, and show good agreement with the predicted depth of the granular jump. The depth
profiles for the flow with Fr1 = 5 show a slight increase in the flow depth along the dams while there
is no increase in the flow depth for the higher Froude number flows.
For flow with Fr1
= 5, the maximum possible deflecting angle for a shock to be attached to a
deflecting dam is approximately 46 , as shown in Figure 3.16. We found this detachment to occur
experimentally at γ = (44 1) , which is close to the theoretical value.
The effect of tilting the front face of the dams between α = 45 and α = 90 for flows at Fr1 = 12
and γ = 24 did not affect the oblique shocks. The shock angle and the depth, measured normal to
the base of the chute, remained the same as for a deflecting dam with an upstream face normal to
the base of the chute, α = 90. The depth profiles and the shock angles for dams with α = 90 are
plotted in Figures 4.22 and 4.23 and coincide with the observed shock angles and depth profiles for
the other inclinations of the dam face (45 α 90 ).
We conclude that shallow-water jump-conditions can be used to describe granular jumps along
deflecting dams, and that tilting the upstream dam faces does not affect the jumps, at least not for
the high Froude number flow, Fr 1 = 12, that was studied.
81
z=h1
20
γ = 8
γ = 15
γ = 24
γ = 32
15
10
5
0
25
20
15
10
5
0
0
0.1
0.2
0.3
s [m]
0.4
0.5
γ = 44
20
15
10
5
0
0
0.1
0.2
0.3
s [m]
0.4
0.5
Figure 4.24: Non-dimensional run-up profiles for granular flow at Fr 1 = 5. Each graph is a plot
of the observed depth profile along a deflecting dam, non-dimensionalised with the depth of the
approaching stream, at a fixed deflecting angle, γ (+). The theoretical non-dimensional shock height,
h2 =h1 , is shown as a dashed line. The shocks were unsteady for γ = 32 and 44. The theoretical
shock height corresponding to these unsteady profiles is shown as a solid line.
82
04
z=h1
20
γ = 8
γ = 15
15
10
5
0
25
20
γ = 24
15
10
5
0
0
0.1
0.2
0.3
z=h1
20
γ = 8
γ = 15
γ = 24
γ = 32
15
10
5
0
25
20
15
10
5
0
0
0.1
0.2
0.3
s [m]
0.4
0.5
γ = 44
20
15
10
5
0
0
0.1
0.2
0.3
s [m]
0.4
0.5
Figure 4.26: Non-dimensional run-up profiles for granular flow at Fr 1 = 14. Each graph is a plot
of the observed depth profile along a deflecting dam, non-dimensionalised with the depth of the
approaching stream, at a fixed deflecting angle, γ (+). The theoretical non-dimensional shock height,
h2 =h1 , is shown as a dashed line. The shock was unsteady for γ = 44 . The corresponding theoretical
shock height is shown as a solid line.
84
4.5. Steady granular flows interacting with catching dams
4.5 Steady granular flows interacting with catching dams
A series of experiments with high Froude number flows was designed to study the formation of
a granular jump upstream of a catching dam. The upstream propagation of the granular bore was
studied for various dam heights, and the dam height for which a bore was no longer observed was
determined and compared with theoretical predictions described in §3.3.2.
The experimental setup was the same as in the steady granular experiments described in the previous
section (§4.4) and the same ballotini beads were used. The experiments were conducted on slopes
inclined at an angle ξ = 30 to the horizontal and the flows had an internal Froude number of Fr 1 = 12
and 14. The flow regime of the steady, uniform granular flows are listed in Table 4.3. The angle
of repose of the ballotini was measured φ = (21 0:5) , by building up a cone of particles on a
horizontal plane and repeatedly measuring the height and diameter of the cone. The dynamic bed
friction angle was measured δ = (24 2) by tilting a plane with a thin layer of moving particles
until the angle at which the particles came to rest was determined. We note that δ > φ and the
assumption that failure in a moving pile of granular material occurs at the bed is based on the bed
friction angle being smaller than the angle of internal friction of the material, δ < φ, see §A.2.
The catching dams had a planar upstream face that was normal to the base of the experimental
chute. The height of the dams was systematically lowered and the speed and depth of the bore,
travelling upstream, was measured. The depth of the flow over-topping the dam was measured when
over-flow occurred.
4.5.2 Results
Flow completely blocked by a dam
The experiments revealed that granular bores propagated upstream from catching dams when the
dams were high enough, as shown in Figure 4.27 (a). The dilute first front of the flow splashed
higher up on the dams than the subsequent granular jump. The angle of repose of the granular
material (ballotini beads) was smaller than the slope of the chute, φ < ξ. The free surface of the
stationary material upstream of the dams (downstream of the jumps) was intermittently levelled out
to the angle of repose of the ballotini beads by thin avalanches on the free surface, propagating from
the jump and down to the catching dam. The bore slowed down during the upstream propagation,
as more and more material was required to level out the lengthening free surface of the material
downstream of the jump.
This scenario is described by equations (3.16) and (3.19) for φ
85
<
ξ. The depth of the bore
(a)
(b)
(c)
(d)
Figure 4.27: Photographs of granular flow at Fr 1 = 12, travelling from right to left down a 30 incline
and interacting with (a) a high catching dam (on the left) and forming a bore that travels upstream;
(b) a catching dam, H =h1 = 18, with some of the flow over-topping the dam and a stationary bore
upstream of the dam; (c) a catching dam, H =h 1 = 15, where the stationary bore is closer to the dam;
(d) a catching dam, H =h1 = 6:7, where all of the flow over-tops the dam. Note that the incoming flow
is only 0.002 m thick and hardly visible on the photographs. The dimensions of the larger grid-cells
are (0:02 0:02)m.
remained constant during the upward propagation. That is consistent with the theoretical assumption
that ∂h2 =∂t 0 in equation (3.19). The theoretical bore speed is plotted in Figure 4.28 as a function
of the distance from the dam face to the edge of the bore, L, for the same constant density on both
sides of the jump, ρ2 =ρ1 = 1. The experimental observations follow the theoretical curves, but with
a slightly higher bore speed than predicted. A larger density ratio, ρ 2 =ρ1 > 1, accounts for a closer
packing in the stationary material upstream of the dam. A larger density ratio will, however, lead to
a slower bore propagation, as shown in Figure 3.9 (b), and a worse agreement with the experimental
observations.
Gray et al. (2003) have conducted similar catching dam experiments using material with a larger
internal friction angle than the angle of the slope on which their flows take place, φ > ξ. They
find that equations (3.15) and (3.16) predict the speed and the depth of the upstream propagating
bore. This scenario is somewhat more relevant for the analysis of snow avalanches, since avalanche
protection dams will, in most cases, be positioned in the run-out zone of an avalanche path where
the terrain slope is less than 10 . The slope angle is therefore lower than the internal friction angle
of snow and snow stopped in front of a dam will be stable at the terrain angle. Cohesion and sudden
freezing of the stopped snow will also help to stabilise it. On the other hand we may expect the snow
that is stopped upstream of the dam to be more densely packed than the flowing avalanche (expected
to be 1 < ρ2 =ρ1 < 3), leading to a shallower bore and a slower bore propagation than for ρ 2 =ρ1 = 1
(see Figure 3.9).
86
4.5. Steady granular flows interacting with catching dams
U =u 1
U =u1
Fr1 = 12
0.08
Fr1 = 14
0.08
0.06
0.06
0.04
0.04
0.02
0.02
0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0
0.1
0.2
0.3
0.4
L [m]
0.5
0.6
0.7
L [m]
Figure 4.28: The bore-speed non-dimensionalised with the speed of the approaching flow, U =u 1,
plotted as a function of the distance travelled upstream by the bore, L, for flows at Fr 1 = 12 and 14.
The theoretical prediction for ξ ; φ = 9 and ρ2 =ρ1 = 1 is plotted as a curve. The experimentally
observed bore speed is averaged over the distance indicated by the L-error bars (+).
Flow over-topping a dam
When the catching dams were lower than the depth of the granular jumps, some of the flow overtopped the dams (see Figure 4.27 (b) and (c)). Experiments with flow approaching a catching dam
at Fr1
= 12 showed that the bore propagated upstream until the mass flux over the bore balanced
the mass flux over the dam at which point the bore stopped. The distance travelled upstream by the
bore shortened when the height of the dam was lowered, until a granular jump upstream of the dam
disappeared and all of the flow was launched over the dam (see Figure 4.27 (d)).
The minimum height of a dam, H, for a bore to be present upstream of the dam may be calculated
from equation (3.25),
H =h 1 =
1
2
q
1 + 8Fr21 ; 1
;(Fr1 )2=3 = 11 2
: :
This prediction agrees well with the experimental observations. A stationary bore was observed only
(0:025 0:02) m upstream from a dam with H =h 1 = 11:5, as listed in Table 4.4, and had disappeared
when H =h1 = 6:7, as can be seen in Figure 4.27 (d).
The distance, L, that the bore propagates upstream (the position of the bore when U
= 0) can be
estimated from equation (3.26), which may be written as
h2 +
where h2 = h1
p
1 + 8Fr12 ; 1
Z L ∂h
0
∂x
dx = H + h3cr
2, h3cr = h1 (Fr1 )2=3 and ∂h=∂x is the gradient in the flow depth of
=
the bore, downstream from the jump (relative to the base of the chute). Here it has been assumed that
the subcritical flow downstream of the jump is unaffected by the presence of the dam. The gradient
in the flow depth may be calculated from equation (3.27),
∂h
∂x
=
;
tan ξ ; Fr2CD + tanδ
;
1 ; Fr2
87
:
H =h1
18
15
11.5
Lmeas [m]
0:15 0:02
0:08 0:02
0:025 0:02
Lcalc [m]
0:14 0:05
0:08 0:03
0:005 0:002
Table 4.4: The measured and calculated distance travelled upstream by a bore, L meas and Lcalc ,
upstream of a dam of height H, for flow approaching the dam at Froude number Fr 1 = 12 with a
flow depth h1 = (0:0027 0:0002) m.
The flow downstream of the jump (within the bore) is subcritical, and Fr 2
friction coefficient associated with turbulent friction is small, CD
<
1.
Furthermore, the
1. We therefore find that the
flow thickens towards the dam with a constant gradient in the flow depth to leading order, ∂h=∂x =
tanξ ; tanδ = 0:13 0:04 where δ is the dynamic friction angle between the base of the chute and
the flow. The dynamic friction angle was determined experimentally δ = (24 2) and the chute was
inclined at an angle ξ = 30 to the horizontal. We note that the gradient in the flow depth is sensitive
to the exact value of the dynamic bed friction angle, which is not well defined in these experiments.
The stopping position is now determined by
L=h1 =
H =h1 +
2=3
Fr1
;
1
2
q
4.6. Conclusions
4.6 Conclusions
These experimental studies have considered steady, supercritical shallow flows of water and dry,
granular material interacting with dams. The flows were at different Froude numbers, were of different physical scales (different flow depth and speed) and took place on steep and shallow slopes,
on which the flows were uniform (not accelerating) when they interacted with the dams. Features
such as shocks were observed upstream of dams in both granular flows and water flows. A better
agreement between theory and experiments was obtained for the granular flows if the pressure was
taken to be isotropic and hydrostatic, instead of linking the longitudinal and normal pressures in the
flow through an earth pressure coefficient.
Weak, stationary, oblique shocks were formed in the interaction of the flows with deflecting
dams. They were well described by incompressible, shallow-layer jump-conditions. The jumpconditions were derived on a horizontal plane, assuming constant flow states on both sides of the
shock. Less intense wave action and turbulence were observed in the granular flows than in the
water flows, presumably owing to dissipation in inelastic collisions between grains. This may also
account for a sharper transition zone of the granular jumps. The structure of the granular jumps
was not observed to depend strongly on the Froude number of the approaching flow for the Froude
number range that was studied, 5 Fr1 14, as is the case in hydraulic jumps. Gray et al. (2003)
studied granular jumps for flows with lower upstream Froude numbers. They observed a difference
in the structure of the granular jumps between low Froude numbers and high Froude numbers of
the approaching flow. At low Froude numbers (presumably Fr 1
<
5) they describe the shock as
being diffuse, while at high Froude numbers the shock is sudden and some of the incoming particles
rebound creating a small recirculating zone on the front face of the jump that propagates upslope with
the shock. The latter description fits the granular jumps, observed in our experiments. Higher run-up
than the jump conditions predicted and turn-over of the flow was observed close to the upstream end
of the deflecting dams for the higher Froude number granular flows, and for water flows at the largest
deflecting angles and for all Froude numbers. This initial effect was more pronounced for the water
flows than the granular flows, possibly due to a wider transition zone in the hydraulic jumps of water.
Bores propagating upstream were formed upstream of catching dams when none of the flow
over-topped the dams. For lower dams with some over-flow, the bores propagated a certain distance
upstream where they then stopped. As the height of the dams was gradually lowered, the bores
stopped closer to the dams and finally, when enough material could be transported over the dams,
the flow was launched entirely over the dams and bores did not form. The granular jumps were well
described by shallow-water jump-conditions. The internal friction angle of dry snow is suggested
by Salm (1993) to be close to 25 . In most cases avalanche protection dams will be positioned in
the run-out zone of an avalanche path and not at angles steeper than 25 to the horizontal. The
89
theoretical consideration for φ ξ would therefore in most cases be appropriate for the analysis of
snow avalanches. The snow stopped by a catching dam may furthermore be more densely packed
than the flowing avalanche leading to a density difference over the shock (most likely in the range
1 < ρ2 =ρ1 < 3) and a lower shock depth at the dam than for an incompressible jump.
Highly compressible flows with Mach numbers exceeding unity were not considered in the experiments (excluding the front of the flows). There is a possibility that the dense core of some natural
dry-snow avalanches is slightly supersonic (Briukhanov et al., 1967) which may give rise to the formation of compression shocks in the interaction with dams. Such compression shocks have been
observed experimentally in dilute granular flows (Rericha et al., 2002).
The high splash-up of the flow front of the granular flows onto catching dams is consistent with
high pressures (impulse pressures) measured during the first few milliseconds of the impact of an
avalanche with a catching dam (Salm, 1964; Kotlyakov, et al., 1977; Schaerer and Salway, 1980;
Bozhinskiy and Losev, 1998) and also pressure records and observed splash-up of steep water waves
on walls (Peregrine, 2003). The study of the splash-up of the flow front of water onto dams showed
that a considerable change in velocity took place during the impact and the velocity change may be
explained by the pressure impulse theory. The splash-up of the granular flows was not compared
with this theory, since the flow fronts of the granular flows were highly compressible and the theory
was derived for incompressible flows. It was, however, noted that the front of the granular flows did
not splash any higher up on the deflecting dams (44 γ 8 ) than the granular jumps.
An important consideration in the application of these experimental results to practical design of
protection dams is the question of the formation of the shock in the initial impact with the dam. In
order for the shock to form, the flow must undergo a change in flow state from a supercritical flow
to a subcritical flow as it climbs the dam. This height may be larger than the theoretically calculated
height of the shock, depending on the dissipation of energy during the impact of the flow with the
dam. The energy loss in the impact of a natural dry-snow avalanche with a dam may depend on the
height of the dam compared with the depth of the flow, the steepness of the upstream face of the dam,
some physical properties of the flow and the dam face, such as the wetness of snow and whether the
dam face is icy, etc. This will, however, not be further discussed here, but rather in Chapter 6, in
connection with experiments using dry granular materials and snow to analyse the interaction of
dense granular flows with obstacles that have a height comparable to or a few times the depth of the
approaching flow.
90
Chapter 5
The deflection of a high Froude
number granular jet by a dam
5.1 Introduction
Experiments with catching dams of various heights are described in §4.5. They showed that in
the interaction between high Froude number flows and low dams, we may expect to see a jet of
particles launched from the top of a dam if the dam is low enough so that a bore is not present at its
upstream face. For example, we expect to see this phenomenon if H
9h 1 for flows with internal
Froude numbers of 10. Here, we have denoted the height of a dam by H and the depth of the flow
approaching the dam by h 1 . It was observed in these experiments that the jet travels a considerable
distance through the air before landing back on the slope. The formation of such an airborne jet
has important practical consequences for the use of multiple rows of defence structures to retard
avalanches, such as the two rows of mounds which are shown in Figure 1.3. The spacing between
the rows must be chosen sufficiently large so that the snow launched from the first row of mounds
does not jump over subsequent rows further down the slope. In order to obtain the range of such an
airborne jet we need to know how the mass and momentum fluxes of the flow are deflected by the
barrier.
In this study we present some laboratory experiments and theoretical analysis that investigate
how high Froude number, shallow, two-dimensional granular flows over-top an obstacle that has a
height that is comparable to, or a few times the depth of the approaching flow.
The chapter starts with theoretical considerations about the two-dimensional deflection of an
inviscid, irrotational, fluid jet by an obstacle where gravitational effects are negligible (§5.2). We
then move on to the experimental study. The experimental setup and design are described in §5.3
and the experimental results are reported in §5.4. The airborne jets which arise from the interaction
91
Chapter 5. The deflection of a high Froude number granular jet by a dam
;y
x
g
u2
u1
θ
h1
α
H
ξ
A dam
Figure 5.1: Schematic side-view of the deflection of a jet by a dam.
with dams are documented, the angle at which jets leave the dams is investigated and compared with
the theoretical calculations of the motion of an ideal fluid. This angle will be referred to as ‘throw
angle’ and denoted by θ, see Figure 5.1. The applicability of the study to the design of avalanche
braking mounds is finally discussed in §5.5.
5.2 The deflection of a fluid jet: mathematical formulation
Figure 5.1 shows flow detaching from the top of a dam at an angle, θ, to the direction of the approaching flow and becoming airborne. It is this two-dimensional deflection that we aim to model.
We formulate the problem in a coordinate system with the x-axis pointing downslope and the y-axis
being the downward pointing normal. The flow velocity in the x and y directions respectively is
u = (u v). The flow approaches the dam with a velocity of u 1 = (u1 0) and a depth h1 , and detaches
from the top of the dam with the velocity u 2 = (u2 v2 ).
We consider steady, irrotational flow of an incompressible, inviscid fluid and assume that the
flow passes smoothly over the dam. The mass flux approaching the dam is q = u 1 h1 , and the Froude
number of the oncoming flow is
pgu?1h
Fr1 =
1
where g? = g cos ξ is the component of the gravitational acceleration normal to the slope, and ξ is the
angle of the slope relative to the horizontal. The pressure is assumed constant on the free surfaces,
so Bernoulli’s equation along the surface streamline yields
1; 2
u + v2 ; g? y = const: on the free surface:
2
(5.1)
The problem is non-dimensionalised by taking h 1 as a unit length and u1 as a unit speed. In terms of
dimensionless variables, condition (5.1) becomes
1; 2
y
u + v2 ; 2
2
Fr1
= const
92
:
on the free surface:
(5.2)
5.2. The deflection of a fluid jet: mathematical formulation
For high Froude number flow, Fr 1 1, we find that gravitational effects can be neglected locally in
the vicinity of the dam, since the dimensionless height of the dam is comparable to the depth of the
approaching flow, O(H =h1) = 1. Equation (5.2) can thus be written as
1; 2
u + v2 = const: on the free surface
2
(5.3)
to leading order.
Yih (1979) has analysed the deflection of a two-dimensional fluid jet on the assumption that it
is inviscid and irrotational and gravity may be neglected. The flow is modelled by introducing a
complex potential, f
= φ + iψ, in terms of the variable z = x + iy.
The complex velocity, d f =dz,
may be derived analytically and, using conformal mappings, transformed onto a half plane of some
appropriate complex variable, as shown below. The complex potential is then mapped onto the
same half plane, and the relationship between z and f , or between d f =dz and f , is parametrically
determined, whereby an implicit relationship is obtained between the height of the dam relative to
the depth of the approaching flow, H =h 1 , the inclination of the dam face, α, and the throw angle, θ.
The key to the analysis lies in the fact that the flow region in a hodograph plane, which is the
plane of
Ω = ; ln
df
dz
is a polygon. The flow region in the complex potential plane, f , is also a special polygon with two
angles, and there is a transformation due to Schwarz and Christoffel, that transforms the interior
of a polygon into the entire upper (or lower) half plane of another variable (see Appendix B). If
gravity effects are taken into account, the flow domain in the logarithmic hodograph plane, Ω, is not
known beforehand and the Schwarz Christoffel transformation is not useful. For this situation other
techniques are needed (see Vanden-Broeck and Keller, 1986 and 1987; Dias and Vanden-Broeck,
1989; Dias and Christodoulides, 1991). In what follows we explain the gravity-free analysis in some
detail.
Figure 5.2 (a) illustrates a jet along a horizontal plane being deflected through an angle, θ, by an
object inclined at an angle, α, to the direction of the jet. The dimensionless Bernoulli equation (5.3)
along the free streamline AD yields
ju1 j = ju2 j = 1
:
The complex velocity is defined by
η = u ; iv = qe;iζ =
df
dz
where q = juj is the flow speed and ζ is the angle to the x-axis, as shown in Figures 5.2 (a) and (b).
We now define a new variable (Figure 5.2 (c))
η0 = ηπ=α :
93
This maps the point C to η0
= ;1 and flow domain to the semi circle jη j 1.
0
The logarithmic
hodograph plane is defined by
Ω = ; lnη0 = (; ln q + iζ)
π
:
α
In the Ω plane the semi circle jη0 j 1 maps to the semi-infinite strip with Re(Ω) 0 shown in Figure 5.2 (d). The Schwarz-Cristoffel transformation (see Appendix B) can now be used to transform
the interior of the polygon BADCB in the Ω plane into the entire upper half plane of the variable t
(see Figure 5.2 (e)). We choose tC = 1 and tA = ;1, and map B into a point at infinity in the t plane.
The factor associated with the vertex B in the Ω plane is thus ignored in the Schwarz-Christoffel
transformation, and
Zt
=
Ω
ds
(s ; 1)1=2 + N (s + 1)
+N
= M 0 p ds
s2 ; 1
= M cosh 1 t + N
M
1=2
0
Zt
;
where M and N are complex constants. They may be determined as follows
ΩA
ΩC
=
=
0 = M cosh;1 (;1)+ N = iMπ + N iπ = M cosh;1 (1)+ N = N and M = ;1 and N = iπ. The relationship between Ω and t is therefore determined by
Ω = ; cosh;1 t + iπ
which can be rewritten as
t
= cosh(ln η + iπ)
= ; 12 η + η1
0
0
0
:
In the complex potential plane ( f plane), the streamlines are a distance unity apart and AD is chosen
to be at ψ = 0 and ABCD at ψ = ;1. We note that
tA
tD
= ;1 as
= ; cos πθ
α
φ ! ;∞
as
φ ! ∞:
The polygon in the f plane can thus be mapped into the upper half of the t plane by
f
= ln (t + 1); ln (t + cos πθ
); i
α
:
94
5.2. The deflection of a fluid jet: mathematical formulation
;v
;y
C
D
u2
0<θ<α<π
D
θ
α
C
u1
A
α
1
H
θ
x
B
A
u
1
B
(a)
(b)
;v
Im(Ω)
0
D
C
D
πθ
α
C
;1
B
A
π
πθ
α
Re(Ω)
A
u0
B
1
(d)
(c)
B
C
ψ = ;1
Im(t)
A
A
B
;1
D
C
; cos ( πθα )
1
D
ψ=0
B
φ
Re(t)
(f)
(e)
Figure 5.2: The deflection of a jet. (a) z plane; (b) η plane: η = qe ;iζ; (c) η0 plane: η0 = ηπ=α ;
(d) Ω plane: Ω = (; lnq + iζ) π=α; (e) t plane: t = ;1=2 (η0 + 1=η0 ); (f) f plane: f = ln (t + 1);
ln (t + cos (πθ=α)); i.
95
If
α=
n
π
m
we can introduce
τ = η1=n
and
dz =
dt
= η1 ddtf dη
df
η
dη0 dη
dτ:
0
dη dτ
This can be written as
dz = ;
m
π
1
1
+
; 2 τm;n;1dτ:
τm ; eiθm=n τm ; e;iθm=n τm ; 1
By resolving the expression in parentheses into partial fractions, we have
1 m;1
dz = ; ∑
π r=0
e;i(2rα+θ)
τ ; ei(2rα+θ)=n
;i(2rα;θ)
+ τ ;e ei(2rα+θ)=n
;i2rα
; τ ;2eei2rα=n
!
dτ:
The expression is finally integrated between B and C, where
zB = 0
zC = cos α ; iH τB = 0
τC = ei n :
α
Hence
zC
=
m;1
; θ)
; π1 ∑ exp (;i(2rα + θ)) ln 1 ; exp i (α ; 2rα
n
r=0
+ θ)
+exp (;i (2rα ; θ)) ln 1 ; exp i (α ; 2rα
n
;2exp (;i2rα) ln 1 ; exp i (α ;n2rα)
!
(5.4)
and
Im(zC ) = ;H (5.5)
which yields Im(zC ) = ;H =h1 in dimensional units. The expression provides an implicit relationship
between the height of the dam relative to the flow depth, H =h 1 , the inclination of the dam face, α,
and the throw angle, θ. The throw angle is plotted in Figure 5.3 as a function of the non-dimensional
dam height for inclinations, α, of the upstream dam face between 30 and 90 . We note from the
figure that as the height of the dams increases compared with the depth of the approaching flow, the
jets get deflected at the full inclination of the dam face, whereas low dams deflect the current at a
somewhat smaller angle. Furthermore the steeper the upstream face of the dams is, the higher the
dams need to be to fully deflect the flow.
96
5.3. Experimental setup and design
θ ]
80
70
60
50
40
30
α = 90
α = 75
α = 60
α = 45
α = 30
20
10
0
0
1
2
3
4
5
6
H =h1
Figure 5.3: The throw angle, θ, as a function of the ratio of the dam height to the depth of the
approaching flow, H =h1, for different inclinations of the upstream dam face, α.
5.3 Experimental setup and design
Three series of small-scale laboratory experiments (i, ii, iii) were conducted to study the twodimensional deflection of a granular jet by a continuous dam. Small glass particles were instantaneously released down an incline to form a high Froude number granular flow.
The experiments were performed on wooden chutes consisting of two straight sections inclined
at different angles to the horizontal (see Figure 5.4 and Table 5.1). A dam with a planar upstream
face inclined at an angle α to the chute was located at the end of the upper chute section and the
interaction of the flow with dams of various heights was studied. (Experiments with mounds that
do not fully span the width of the chute are reported in Chapter 6.) The heights of the dams in the
experiments ranged between 0:5 H =h 1 5 and the upstream faces of the dams were inclined at
angles, α, between 30 and 90 to the base of the chute.
The flow comprised glass ballotini beads which were approximately spherical with mean diameter 90 µm and density 2500 kg m ;3 . The granular material had a bulk density of 1600 kg m ;3 .
In each experiment a measured quantity of particles was released from the top of the chute and the
progression down the chute and interaction with the obstacle were recorded using a video camera.
The camera recorded at 25 frames per second. The velocity field of the free surface was measured
by tracking tracer particles in the flow. The experiments were designed so that the particulate current
had an internal Froude number of the order 10, close to that of the dense core of a natural dry-snow
avalanche.
97
b1
Perspex
ξ
Position of dam
H
Side-view of a dam
α
l
10
l
b2
Figure 5.4: Schematic diagram of an experimental chute along with the side-view of a dam of height
H with an upstream face inclined at an angle α to the experimental chute.
Series
i
ii
iii
ξ [ ]
41
37*
43
l [m]
3
3
1.5
b1 [m]
0.3
0.3
0.2
b2 [m]
0.5
0.5
0.3
M [kg]
6
6
2
u1 ms;1
3:1 0:2
2:8 0:1
2:7 0:2
h1 [m]
0:008 0:002
0:01 0:001
0:006 0:001
Fr1
13
10
13
Table 5.1: The dimensions of the experimental chutes according to Figure 5.4 for each of the series.
The speed, u1 , and the flow depth, h1 , were measured upstream of the dams, and M denotes the mass
of material released. *Dams were positioned on a metal sheet connecting the two chute segments
with a slope of 30 .
98
Figure 5.5: A photograph of a granular jet as it detaches from the top of a dam. In the photograph,
the chute is inclined at 40 to the horizontal and the upstream face of the dam is at 45 to the base of
the chute.
z
g
u2
θ
h1
u1
ξ
α
hj
x
H
Figure 5.6: Schematic side-view of the trajectory of a jet of particles of width h 1 deflected through
an angle θ by a dam of height H with an upstream face inclined at an angle α to the underlying
boundary.
In all three experimental series it was observed that as the flow front hit the dam, particles were
launched from the top of the dam at an angle close to its upstream angle, α (see Figure 5.5). Thereafter the jet quickly reached a semi-steady state as the bulk of the current passed over the barrier
with a well defined jet geometry. After the main flow had passed over the dam, the jet died out. In
these experiments the entire motion was over in approximately 2 s.
The angle at which the coherent semi-steady jet leaves the dam is the main focus of this study.
The bulk of the flow passes over the dam at that angle and thus understanding how this angle is
determined is of considerable use in the design of avalanche defence structures.
By capturing images from video recordings of the experiments we found that the trajectory taken
by the semi-steady jet was similar to that of a two-dimensional ballistic projectile. We denote the
99
Height [m]
0.05
0
-0.05
-0.1
H =h1 = 0:6
H =h1 = 1:2
H =h1 = 2:4
H =h1 = 3:8
H =h1 = 5:3
-0.15
-0.2
-0.25
-0.3
0
0.1
0.2
0.3
0.4
0.5
Length [m]
0.6
0.7
0.8
Figure 5.7: Jet trajectories for experimental series i with α = 90 . The points denote the centre of the
observed trajectories and the curves are the corresponding fitted parabolas. The length and height is
measured from the top of the dam horizontally and vertically.
horizontal distance from the top of the dam by x, the vertical distance by z, the speed of the avalanche
by ju2 j and the throw angle by θ, as shown in Figure 5.6. The ballistic trajectory may be calculated
from Newton’s second law
CD dx dx F = mg ; m
h j dt dt (5.6)
where F = md2 x=dt 2 is the force exerted on the mass m, g is the gravitational acceleration, h j is
the thickness of the jet and CD is a dimensionless constant representing air drag. The photographic
images of the experiments show more spreading of the leading edge of the jet than of the flow
immediately following (see Figure 5.5). This effect may indicate entrainment of air into the leading
edge of the jet or pressure differences as the flow front displaces the air. The trajectories of the
jets were best approximated by assuming that the air resistance was negligible. Furthermore video
images from the experiments showed that the horizontal speed of the airborne jets was approximately
= 0 in equation (5.6) and
solving the equation subject to the initial conditions dx(0) dt = u 2 and x(0) = 0, we find that the
constant, which is consistent with negligible air drag. By choosing CD
=
trajectory of the jet is given by
z = x tan (θ ; ξ);
1 gx2
sec2 (θ ; ξ):
2 ju2 j2
(5.7)
The parameters that define the geometry of the semi-steady jet are the throw angle, θ, and the speed,
ju2 j. These were calculated indirectly for each experiment by fitting a parabola through the observed
jet trajectory using least squares (see, for example, Figure 5.7).
100
θ ]
80
α = 30
α = 45
α = 60
α = 75
70
60
50
40
30
20
10
0
90
80
70
60
50
40
30
20
10
0
0
1
2
3
4
H =h 1
5
6
0
1
2
3
4
H =h 1
5
6
Figure 5.8: The throw angle, θ, of a jet plotted as a function of the non-dimensional dam height,
H =h1, for different inclinations of the upstream faces of the dams: α = 30 (series i); α = 45
(series i); α = 60 (series iii); α = 75 (series iii). The points (+) denote experimental results and
the curves (–) are theoretical predictions.
We observed that the throw angle was smaller than the angle of the upstream face of the dams
for small ratios of the non-dimensional dam height, but approached the upstream angle as the height
of the dams increased relative to the depth of the oncoming flow (θ=α ! 1 as H =h 1 ! ∞).
5.4.1 Comparison with theory
The granular flows in the laboratory experiments were fast and shallow, Fr 1
10, and thus the effects
of gravity during the interaction with the obstacle are expected to be negligible. Dissipation is more
important, although the spatial extent of the region over which the flow interacts with the obstacle
before becoming airborne, is small, at least for the lowest obstacles. Furthermore dams with a small
inclination of the upstream face provide a smoother deflection of the flow than steeper dams. We may
expect more dissipation, however, as the obstacles become higher and steeper (see Hákonardóttir et
al., 2003b). This dissipation is studied in Chapter 6 for steep dams of various heights.
Figures 5.8 and 5.9 show the experimentally measured throw angles along with the theoretical
curves for the deflection of a jet of an ideal fluid, determined by equations (5.4) and (5.5). From
Figure 5.8 we note that the experiments follow the theoretical prediction closely. For the less steep
101
θ ]
80
70
60
50
40
30
Theory
Series i
Series ii
Series iii
Fluid experiment
20
10
0
0
1
2
3
4
5
6
H =h1
Figure 5.9: The deflection θ of a jet plotted against the non-dimensional dam height H =h 1 for a dam
with a upstream face inclined at α = 90. The fluid data is from Yih (1979).
dams (α = 30 and 45 ), the throw angle reaches the angle of the upstream dam face for relatively
small values of H =h1, while the steeper dams (α = 60 and α = 75 ) need to be higher for the jet to
be fully turned by the interaction.
Figure 5.9 illustrates experimental results for very steep dams with α = 90 , for all three series
of experiments (Table 5.1). Also plotted on the graph are experimental results for the free-surface
flow of a fluid jet (data from Yih, 1979) and the theoretical predictions of the ideal fluid theory. For
small values of the ratio of the height of the dam to the flow depth, the experimental results follow
the theoretical curve. As H =h1 increases, the throw angle approaches an angle of about 75 instead
of reaching 90 . A possible explanation for this effect in the context of these granular flows is that
a wedge of deposited particles is built up on the upstream faces of the obstacles, thus lowering the
effective inclination of the face. Some investigators refer to this effect as self-ramping (Chu et al.,
1995). After each experimental run we found particles deposited in this location and we hypothesise
that this occurs during the initial interaction with the obstacle so that the semi-steady jet is not
deflected at the full angle of the upstream face of the obstacle. This residue of particles is evident
for all barrier inclinations, but it is most pronounced when the barrier is at 90 to the chute. Some
of these particles were deposited at the end of the flow as the velocity wanes; they should have little
effect on the deflection of the jet. Instead it is those trapped during the initial phase that may alter the
deflection. We will revisit this problem in Chapter 6 in a slightly different setup, with experiments
on much larger physical scales adding snow to the experimental materials (see Figure 6.6).
To summarise, the gravity-free theory for the two-dimensional deflection of a jet of an ideal fluid
agrees well with the experimentally measured deflection of shallow, high Froude number granular
102
5.5. Conclusions
flows for inclinations of the upstream face of the dams of angles up to α = 75 . For these dams, the
current is fully deflected to the angle of the upstream face provided the height of the dam relative to
the depth of the flow is sufficiently large. For dams with upstream faces of 90 , the flow is not fully
turned by the dams as the theory predicts and we suggest that a wedge is formed upstream of the
dams, reducing the effective angle of the upstream face of the dams. This agreement between theory
and experiments is found in spite of the fact that speed in the experiments is reduced in the impact
with the dam (ju2 j < ju1 j), but is assumed unchanged in the theory.
5.5 Conclusions
The experiments show that an airborne jet is formed in the interaction of high Froude number granular flow with low obstacles, such as dams. The jet travels a considerable distance through the air
before landing back on the slope. We find that at this laboratory scale the jet can be described as
a two-dimensional ballistic projectile, with negligible air drag. Even though different physical interactions control the dynamics of fluid and granular flows, the experiments show that the vertical
deflection of the momentum flux by the barrier is similar to that predicted for an inviscid, twodimensional flow of a fluid jet.
This provides important guidelines for the design of avalanche defence structures, since given a
flow speed and depth of an avalanche relative to that of the obstacle, it is now possible to estimate
the range of the jet, if speed reduction due to dissipation in the impact can be estimated. At the
physical scale of snow avalanches, air drag may be important, retaining the motion and leading to
a shortening of the range of the jet. The role of air drag on the jet on large physical scales will be
discussed in connection with braking mound experiments in Chapter 6.
The current study has considered the interaction of the flow with two-dimensional dams of a
height that is comparable to or a few times the depth of approaching flow. In the introduction to
this chapter, it was mentioned that multiple rows of braking mounds are often used to retard the
flow of snow avalanches. Such mounds are usually designed with a height that is comparable to or
a few times the depth of the dense core of an avalanche. The interaction with mounds is similar to
the interaction with dams in that coherent airborne jets form and launch the granular flow from the
top of the mounds (Hákonardóttir et al., 2003b). However mounds also introduce lateral deflection
of the flow which passes over them. This leads to the collision of streams from adjacent mounds,
resulting in additional dissipation and slowing of the flow.
Laboratory experiments indicate that the dynamics of the interaction of granular flows with
mounds is very similar to dams when the aspect ratio of the mound, defined as the ratio of its height
to width, is small. However when this aspect ratio approaches unity, different mechanisms start to
dominate the motion and coherent jet trajectories can no longer be observed (Hákonardóttir et al.,
103
2003b). The discussion of the interaction between high Froude number granular flows and braking
mounds will be continued in Chapter 6.
104
Chapter 6
shallow flows and braking mounds
6.1 Introduction
Braking mounds are defence structures designed to retard snow avalanches. At present, there exist no
accepted guidelines for the design of such structures. Mounds are nevertheless widely used for protection against dense, wet-snow avalanches and have also been built to retard dry-snow avalanches.
The mounds have often been designed with a conical shape, since that is the most convenient and
economic shape of such structures constructed from loose materials. Salm (1987) has formulated an
estimate for the reduction in the speed of an avalanche that hits several obstacles, such as buildings,
that are spread over the run-out area of the avalanche and assumed to cover a certain fraction, c, of
the cross-sectional area of the flow path. According to his expression, the speed of the avalanche is
reduced by the factor c=2, assuming that the obstacles are sufficiently strong that they are not swept
away by the avalanche. The expression indicates a substantial effect of the obstructions on the speed
of the avalanche. Voellmy (1955) proposed a similar expression for the reduction in the speed of an
avalanche that hits several rows of trees. These expressions are not derived from a theoretical model
of the three-dimensional flow around obstacles and it is not clear whether they may be expected to
apply to a rapidly moving, dry-snow avalanche.
The height of braking mounds is typically comparable to a few times the depth of the dense core
of an avalanche. The height of the mounds is therefore only a small fraction of the height-scale
corresponding to the kinetic energy of the avalanche, u 2 =(2g), where u is the speed of the avalanche
and g is gravitational acceleration. Shocks do not form upstream of such low structures, as discussed
in Chapter 3, since the Froude number of an avalanche is on the order of 10. Energy dissipation and
subsequent speed reduction of the flow caused by the mounds must therefore be brought about by
105
Chapter 6. The interaction between supercritical shallow flows and braking mounds
irregularities and mixing in the flow, introduced by the abrupt deviation of the avalanche flow over
and around the mounds. Such effects may be expected to depend to a large degree on various details
in the layout and geometry of the mounds. One may also note that the volume of the avalanche will
typically be so large that only a small fraction of the snow near the front of the avalanche is needed
to fill the space upstream of the mounds so that they become buried and the bulk of the avalanche
easily overflows the mounds. For braking mounds to be effective while the avalanche passes over
them, they must not become buried by the avalanche.
Braking mounds have been studied previously in the context of fluid flows (Peterka, 1984;
U.S.B.R., 1987; Hager, 1995; Roberson et al., 1997). The corresponding structures are called ‘baffle
blocks’ or ‘baffle piers’ and are commonly used in hydraulic waterways. The waterways, or ‘stilling
basins’ as they are also termed, are designed to dissipate mechanical energy in supercritical, freesurface flows, by introducing a stationary hydraulic jump before the flow enters the downstream
channel. Baffle blocks are sometimes placed in the stilling basins to shorten the length of the hydraulic jump by inducing disturbances to the fluid motion which enhance the energy dissipation over
the hydraulic jump. Stilling basins without the baffle blocks have been found to be approximately
25% less effective in shortening the length of the hydraulic jump (Peterka, 1984; U.S.B.R., 1987).
The physical scale of the flow in these structures is often more than an order of magnitude larger than
in the laboratory experiments described here, with flow speeds up to 30–40 m s ;1 which is similar to
the speed of natural dry-snow avalanches (Issler, 2003). Experiments with pyramidal baffle blocks
show jets launched from the structures (Gerodetti, 1985). The jets are similar to the jets observed
and studied in Chapter 5.
Experiments to study the retarding effect of avalanche braking mounds were undertaken during
1999–2001, as a part of the design of braking mounds above the town Neskaupstaður, Iceland, in the
absence of accepted design guidelines (Tómasson et al., 1998, see also Figure 1.3). The experiments
are described by Hákonardóttir et al. (2003b). They revealed a number of interesting aspects of the
interaction of high Froude number, granular flow with mounds. These include: the formation of a
coherent airborne jet, which is discussed in the previous chapter; a complicated three-dimensional
structure of the flow as the particulate current is deflected vertically and laterally from an obstacle;
and considerable mixing of streams, that are deflected from adjacent obstacles. Furthermore, the
experiments showed that one row of obstacles with a height approximately twice the flow depth, can
shorten the run-out of the flow by 30–40%.
The experiments reported here extend the previous work of Hákonardóttir et al. (2003b). The
experiments were conducted at four different length-scales in order to investigate whether Froude
number scaling of granular flows would be sufficient to maintain dynamical similarity in the interaction with obstacles of this type, and thus confirm the findings of the previous chute experiments
regarding the optimal design and retarding effect of avalanche braking mounds. The design of
106
6.2. Experimental setup
the experiments was based on Froude number similarity with the dense core of natural, dry-snow
avalanches in which Fr
10. The retarding effect of one row of braking mounds with a height of
a few times the flow depth and an upstream face normal to the flow direction was investigated and
compared to that of dams of similar heights. The aspect ratio of the mounds, here defined as the
ratio of the height to the width, and the obstructed proportion of the flow path were kept fixed while
the ratio between the obstacle height and the flow depth was changed. Small, spherical glass beads,
snow and coarse sand were used in the experiments. The granular flows were dense and not highly
compressible. These experiments, together with the earlier experiments, mentioned above, have
been used to formulate the first systematic ideas regarding the design of avalanche braking mounds
(Jóhannesson and Hákonardóttir, 2003).
The study starts with a brief description of the experimental setup (§6.2) and design (§6.3).
The experimental results are reported in §6.4 where the throw angle, the energy dissipation at the
upstream mound face, and the run-out reduction of the flow are studied. Finally in §6.5, we draw
some general conclusions regarding the interaction of natural-scale snow avalanches with braking
mounds.
6.2 Experimental setup
In the laboratory experiments, particles were instantaneously released down an inclined chute to
form a rapid granular flow, which interacted with obstacles close to the end of the upper section
of the chute, see Figures 6.1 and 6.2. The chutes ranged in length and width from 3 m to 34 m
and 0.2 m to 2.5 m, respectively. Table 6.1 summarises the dimensions of the different experimental
chutes. The original Ballotini (i) experiments were conducted on a chute in Bristol, UK, while the
Ballotini (ii) experiments were carried out on a smaller chute situated in Reykjavík, Iceland; the
Ballotini (iii) and the Sand experiments on a larger chute owned by the Swiss Federal Institute of
Snow and Avalanche Research, SLF, in Davos, Switzerland; and the snow experiments on a 34 m
long chute on Weissfluhjoh, above, Davos, also owned by the SLF. The Ballotini (ii), (iii) and Sand
experiments are described in detail by Hákonardóttir et al. (2001) and experiments with snow during
the winter 2002 are described in detail by Hákonardóttir et al. (2003d).
The particles within the flows varied from small, almost spherical and approximately uniformly
sized glass beads (ballotini) to course, angular sand and both wet and granular, and dry and powdery
snow, see Table 6.2. The barriers had a planar upstream face, normal to the experimental chute,
spanned different proportions of the chute and were of varying heights and breadths as will be
described in §6.3.
107
6.2. Experimental setup
Setup
Ballotini (i)
Ballotini (ii)
Ballotini (iii) & Sand
Snow
l [m]
3.0
1.5
5
20
D [m]
0.025
0.15
0.6
b0 [m]
0.3
0.2
0.5
2.5
b1 [m]
0.5
0.3
0.8
2.5
ξ 0 [ ]
37, 30*
43
45
45
ξ1 [ ]
10
14
10
32
Table 6.1: The dimensions of the different experimental chutes according to Figure 6.1. *The obstacles in the original experiments (Ballotini (i) reported by Hákonardóttir et al., 2003b), were positioned close to the downstream end of the upper chute section on a curved metal sheet connecting
the two sections smoothly.
Material
Ballotini (i)
Ballotini (ii)
Ballotini (iii)
Sand
Wet snow
Dry snow
ρ [kg m;3]
1600
1600
1600
1750
350–400
250–350
d [m]
0.00009
0.00009
0.00009
0.002
0.002–0.1
0.001–0.1
φ [ ]
25 1
22 1
22 1
32:5 1
> 25*
25*
δ [ ]
21:0 0:5
17:5 0:5
20 1
26 1
nm
nm
Table 6.2: Material properties of the different granular flows. The bulk density is denoted by ρ, d
is the mean diameter of the particles, φ the angle of repose (internal friction angle), δ the dynamic
bed friction angle and nm stands for not measured. Values of the dynamic bed friction coefficient
for snow avalanches, µ, were estimated by Salm (1993) to be in the range 0:155 < µ = tan δ < 0:30,
which leads to 9 < δ < 17 , based on observations of snow avalanches in the run-out zone. *The
estimated angle of repose for snow (Salm, 1993).
109
6.2.1 Experiments with dry granular flows
The progression of the dry granular flows down the experimental chutes and the interaction with the
obstacles were recorded using video cameras. The speed of the flow front was measured by video
analysis, while the surface speed of the interior of the flow was measured by tracking tracer particles
in the flow. High speed cameras were used for measurements of the internal speed of the flows in
the Ballotini (i), Ballotini (iii) and Sand experiments. In the Ballotini (i) experiments, the camera
recorded 500 frames per second, while the camera recorded at a rate of 250 frames per second in the
Ballotini (iii) and the Sand experiments. The experimental chute in Davos, where the Ballotini (iii)
and Sand experiments were conducted, was also equipped with optical velocity sensors at the basal
surface of the upper section of the chute. Basal velocity measurements of the flows were obtained
through cross correlation of the signals from the sensors and could be compared to the surface and
interior velocity measurements (see McElwaine and Tiefenbacher, 2003).
The maximum flow depth on the chutes was measured by fixing a gate in the flow path at a known
distance from the base of the chutes. The height of the gate above the base was then increased systematically until all of the current flowed under the gate without touching it. The internal friction
angle, φ, was measured by building up a cone of particles on a horizontal plane and repeatedly measuring the height and the diameter of the cone. The dynamic bed friction angle, δ, was determined
by tilting a plane with a thin layer of moving material until the angle at which the layer stopped was
found.
The flows came to an abrupt rest on the lower section of the chutes, where the run-out and deposit
thickness were measured to compare the retarding effect of the different mound and dam setups.
6.2.2 Experiments with snow
Experiments with snow were conducted on a large chute owned by the Swiss Federal Institute of
Snow and Avalanche Research (SLF). The chute is located at Weissfluhjoch near Davos, Switzerland
and was built in the 1950’s to study forces on obstacles positioned on the chute (Salm, 1964). The
chute was recently reequipped and the experiments reported here were undertaken during two weeks
in winters 2002 and 2003. The air temperature in the mountain during the experimental week in 2002
was around 0 C and the snow was granular and wet. During 2003, the temperature at Weissfluhjoch
was on average ;15C and the snow was fine, dry and powdery. A volume of approximately 5 m 3
of snow was shovelled manually onto the chute behind a lock gate. After the release of the gate, the
snow flowed down the chute and carried on down the mountainside until arresting on the skislope
underneath. Rubber bars were placed on the base of the chute between the lock gate and the barriers
to agitate the flow. A row of obstacles was placed in the flow path of the avalanche. The obstacles
had a planar upstream face, normal to the base of the experimental chute. The progression of the
110
6.3. Experimental design
avalanche down the chute was recorded using two video cameras, recording at 25 frames per second.
The flow depth was measured from the video footage of the experiments. The flow speed at
different locations upstream and downstream of the obstacles was obtained through cross correlation
of signals from optical sensors, see McElwaine and Tiefenbacher (2003) for a description of the
sensors and the measurement method. A row of basal velocity sensors and an array of sensors at the
side of the chute were positioned just upstream of the obstacles and another row of basal sensors was
positioned downstream from the obstacles, see Figure 6.2. Due to lack of space on the mountainside
at Weissfluhjoch, it was not possible to build a run-out zone which was long enough for the flow to
come to rest on the chute and thus compare the effect of different mound and dam geometries by
measurements of the run-out as for experiments on the smaller chutes. The flow speed upstream of
the obstacles and downstream of the point were the jets landed were measured instead, using the
optical velocity sensors. The difference in the measured speed downstream of the landing point of
the jet allowed for comparison between the different mound and dam setups.
A detailed description of the chute and its instrumentation is given by Tiefenbacher and Kern
(2004) and the experiments with wet snow during the winter 2002 are described by Hákonardóttir et
al. (2003c) and (2003d).
6.3 Experimental design
The experiments were scaled with the Froude number of the flow, such that the flow depth and the
flow speed squared were reduced by factors of 10 to 1000 relative to those of the dense core of a
large, natural dry-snow avalanche, while maintaining the same Froude number (Fr
10).
A number of different configurations of the obstacles were tested during the initial experiments,
Ballotini (i), which are described by Hákonardóttir et al. (2003b). These experiments revealed that
many narrow mounds were more effective in retarding the flow than either fewer and wider mounds
or even a dam spanning the entire width of the flow path, if the height of the obstacles was up to
twice the depth of the flow. The experiments further showed the importance of a steep upstream face
of the obstacles on retarding the flow. These results along with visual observations of the interaction
were used to explain the way in which mechanical energy is dissipated in the interaction between
the flow and mounds. It was concluded that energy dissipation took place in the following ways: at
the upstream face of the obstacles; as the flow was deflected sideways around the mounds (in the
plane of a wedge formed upstream of the mounds during the initial interaction) and interacted with
streams deflected from adjacent mounds; as jets launched straight over the mounds interacted; and
as the jets landed on the experimental chute below the obstacles.
The fact that mounds spanning a smaller proportion of the flow path than a continuous dam
could be more effective in shortening the run-out of the flow revealed the importance of the sideways
111
deflection and the subsequent mixing of different coherent streams within the flow. The deflection
of streams from adjacent mounds into each other leads to the loss of the some of the across slope
momentum within the flow due to inelastic collisions between grains. Furthermore, the drag forces
may then be more effective in slowing down the subsequent motion on the chute. The shortening of
the run-out may therefore to some degree depend on the material properties of the grains, such as the
coefficient of restitution and the shape of the particles. The experiments showed that a wedge was
built up at the upstream face of the mounds during the interaction, and that by changing the aspect
ratio of the mounds, different proportions of the current, initially headed towards the mounds, were
either launched straight over the mounds or deflected sideways around them. The lateral deflection
was less significant for wider mounds where most of the current was launched straight over the
mounds. Most of the oncoming current was deflected sideways when the mounds had an aspect ratio
close to one.
Experiments with a few carefully chosen obstacle configurations were repeated at different scales
in order to test the results summarised above on different physical scales. A configuration of ‘datum
mounds’ (DM) consisted of mounds that had planar upstream faces, normal to the experimental
chute, an aspect ratio of 0.5 and covered 60% of the flow path. The height of the mounds, H, relative
to the flow depth, h1 , was changed between 1 < H =h1 < 5 and the number of mounds in the flow
path was also changed. The resulting run-out and the depth profile of the deposit in the datum mound
setup were measured and compared to: a setup without mounds (control flow); one setup of more
and narrower mounds covering the same proportion of the flow path, with H =B = 1, H =h 1 2 and
A0 = 0:6 (NM), where B is the width a mound and A 0 is the proportion of the flow path covered by
mounds; dams of different heights covering the entire width of the flow path, with 1 < H =h 1 < 5
and A0 = 1:0 (Dams); and finally a setup of conical mounds with H =B 0:5, H =h 1 2 and A0 0:5
(CM), which is similar to the traditional design of braking mounds. Furthermore the jet trajectories
were analysed for continuous dams and the throw angles and dissipation at the upstream dam face
were studied. The different obstacle setups are sketched in Figure 6.3 and listed in Table 6.3.
In some of the experimental series (Ballotini (ii), (iii) and Sand), the barriers were positioned
close to the upstream edge of the lower chute section, which was inclined at an angle of approximately 10 to the horizontal in order to maintain similarity in the slope inclination between the
experiments and mounds in run-out zones of natural avalanches. In the original series, Ballotini (i),
the mounds were positioned on a curved metal sheet, inclined at 30 to the horizontal, connecting
the two chute sections smoothly. The mounds were positioned on a slope of 32 in the snow experiments. The inclination of the upper chute section was chosen such that the granular current had
an Froude number of the order 10, apart from on the Weissfluhjoch snow chute, where it was not
possible to obtain such a high Froude number as will be discussed in the following section, §6.4.1.
112
Material (series)
Ballotini (i)
Ballotini (ii)
Ballotini (iii)
Sand
Wet snow
Dry snow
M [kg]
6
2
50
50
1500–2000
1500–2000
u [m s;1 ]
2:8 0:1*
2:7 0:1
5:0 0:1
5:3 0:1
7:5 1:0*
6:5 1:0*
h [m]
0:01 0:0005*
0:006 0:001
0:0225 0:002
0:03 0:002
0:3 0:1*
0:2 0:05*
h=d
100
60
225
15
3–150
2–200
Fr
9*
11
11
10
3–6*
5*
Table 6.4: The mass of particles released in each experimental series and the resulting flow speed
and depth of the semi-steady flow phase, along with the ratio between the flow depth and the particle diameter, h=d, and the Froude numbers of the flows, Fr. The stars indicate measurements of
flow speed and depth directly upstream of the obstacles (h 1 , u1 ), while the non-starred entries are
measurements just before the break in the slope (h 0 , u0 ).
6.4.1 Flow description
Flow properties in the different experimental series are listed in Table 6.4. The snow experiments
differed from the dry granular experiments in that the snow mass contained both fine grains and
larger snow lumps and the wetness of the snow changed between individual experiments. The wetter
flows were in general thicker and slower than the dry ones. The Froude numbers in these experiments
were in the range 3–6, varying with each experimental run, depending on the condition of the snow.
These were the highest Froude numbers that could be realised with the experimental setup and they
were somewhat lower than those of large, natural, dry-snow avalanches. When the snow was released
it extended 5 to 6 m behind the lock gate. After the release the avalanche spread longitudinally on
the chute, and had a length of approximately 8 m before reaching the mounds.
The dry granular currents (sand and ballotini) had a dilute and turbulent head, followed by a
denser and thinner, quasi-steady body which flowed with a close to constant speed and depth. Finally
there was a significantly thinner and slower tail. The duration of each avalanche was approximately
2 s, of which the steady flow phase lasted for just under 1 s in all the experiments (including the snow
experiments). The ratio between the flow depth and the length of the flow was thus much smaller
in the thicker snow experiments than in the other experiments, leading to a more ‘impulsive’ event
rather than the quasi-steady flow which was observed in the experiments at the smaller experimental
scales.
It was difficult to measure the flow depth and speed upstream of the obstacles that were positioned just after the break in the slope. Instead the flow speed and depth were measured at the
downstream end of the upper chute section. The distance from the break in the slope to the obstacles was about five times the flow depth in all of the setups, chosen such that the current would have
fully changed its direction without having thickened and slowed down significantly before hitting the
114
z
g
u2
θ
x
h1
u1
H
ξ
u4
Figure 6.4: Schematic diagram of a supercritical granular current jumping over a dam with an upstream face normal to the experimental chute.
obstacles. The flow depth on the lower chute section may be theoretically determined from equation (2.9), with q = uh and CD determined from the steady flow state on the upper chute section by
equation (2.11), if the connection between the two chute sections is assumed to be perfectly smooth.
We use Maple to integrate the equation and find that the flows should only thicken by a fraction of a
millimetre over the 0.02–0.15 m before they reach the obstacles. Since the connection between the
two chute sections was not perfectly smooth, it is likely that the flow will have slowed down and
thickened more than this theoretical prediction shows before it reached the obstacles. From video
recordings of the flows, it was possible to obtain an upper bound on the flow depth upstream of the
obstacles. The maximum estimated flow depth was 0.007 m for Ballotini (ii), 0.03 m for Ballotini
(iii) and 0.04 m for the Sand. The minimum flow speed directly upstream of the dams was estimated
using an expression of mass conservation in which it is assumed that the bulk density of the flow
does not change significantly. Thus it is possible to write
u0 h0 = u1 h1 (6.1)
where u0 and h0 are the flow speed and depth just before the break in the slope and u 1 and h1 are the
flow speed and depth directly upstream of the obstacles.
6.4.2 The interaction between the flow and the obstacles
The trajectory of the airborne jet
It was observed in all of the experiments that on reaching the obstacles, a stream of particles was
projected from their top to form a coherent jet. During a fraction of a second of the interaction, the
115
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
0
0.2
0.4
0.6
0.8
1
θ ]
80
70
60
50
40
30
Ballotini (i)
Ballotini (ii)
Ballotini (iii)
Wet snow
Dry snow
Theory
20
10
0
0
1
2
3
4
5
6
H =h1
Figure 6.6: The throw angle for experiments with dams plotted as a function of the non-dimensional
height of the dams. The theoretical deflection of an inviscid, fluid jet is shown as a curve. The
dams were positioned after the break in the slope in the Ballotini (ii) and (iii) setups, leading to high
uncertainties in the flow depth directly upstream of the dams, h 1 , see discussion in §6.4.1. The point
corresponds to the measured flow depth, just before the break in the slope, while the horizontal error
bars indicate the flow depth directly upstream of the dams, estimated from the video recordings.
avalanche and are not as steep as the initial wedges. Judging by the photograph in Figure 6.8 of
snow left upstream of two mounds at the Weissfluhjoch chute, cohesion might affect the shape of
the wedges. The wedges are inclined at an angle of approximately 30 to the experimental chute
(or approximately 0 to the horizontal). An experimental study of the initial wedges or ‘dead zones’
upstream of catching dams of various heights for flows with Froude numbers in the range 2 < Fr 1 < 6
was carried out by Faug et al. (2002). They showed that the wedges become steeper as the Froude
number of the flow increases.
The other parameter determining the size and shape of the jet is the launch speed, ju 2 j. We
study its dependence on the dam height relative to the flow depth. The ratio between the mechanical
energy of the flow just before interacting with a dam and the mechanical energy at the top of a dam,
as the jet takes off, is approximately given by
E2 =
u21
u22
; 2gH cos ξ :
(6.2)
(Here a point mass approach is employed.) The square root of this ratio is plotted in Figure 6.9 for
the ballotini and the snow experiments. The measurements roughly overlap when the error bars are
taken into account. A difference in the dissipation between the ballotini and the wet and dry snow
is not noticeable (bearing in mind the large error bars associated with the snow experiments) even
though the materials have very different material properties, such as coefficients of restitution. The
117
Figure 6.7: A photograph of a 0.3 m deep, dry-snow avalanche being deflected over and around two
0.6 m high mounds at the Weissfluhjoch snow chute.
118
Figure 6.8: A photograph of wedges left upstream of the 0.4 m high and 0.6 m wide mounds after a
wet-snow experiment on the Weissfluhjoch chute, during winter 2002.
Ballotini (i)
Ballotini (ii)
Ballotini (iii)
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
6
energy dissipation, 1 ; E 2, appears to be a function of the non-dimensional dam height and can be
estimated to be approximately 50% for dams with a height about 2 to 3 times the flow depth. Further
experiments with snow and other materials at larger scales are needed to confirm these findings.
The retarding effect of the different obstacle setups
The leading edge run-out and the run-out of the centre of mass of the flow relative to the undisturbed
run-out of the control flow (no obstacles in the flow path) are plotted in Figures 6.10 and 6.11 as a
function of the non-dimensional height of the dams and the mounds. The leading edge and centre
of mass run-outs were reproducible in all the experiments. As expected, the run-out decreases with
increasing obstacle height. In the ballotini experiments, mounds lead to a larger run-out reduction,
in terms of the leading edge of the flow, than dams for obstacles with a height up to twice the flow
depth. Many narrow mounds are more efficient than fewer and wider mounds (for constant A 0 ,
where A0 is the proportion of the flow path covered by mounds) and the setup of conical mounds
is the least effective setup in the ballotini experiments. The ballotini experiments approximately
collapse, while a larger run-out reduction is measured for the sand. The larger run-out reduction
in the sand flows might be caused by locking of grains in the sand during the stopping flow phase,
due to their angular shape, or by the sand flows being more dissipative than the ballotini flows. It is
furthermore noteworthy that the narrow mounds in the sand experiments are less effective than the
conical mounds, which is unlike the ballotini experiments. The narrow mounds are, nevertheless,
more effective than the datum mounds for both ballotini and sand, as is expected. Faug et al. (2003)
have conducted similar experiments using catching dams. They used sand and ballotini particles
with the same mean diameter and found that the run-out reduction for both the sand and the ballotini
collapsed onto a single curve.
There are at least three flow parameters that are different in the sand experiments, from those in
the ballotini and snow experiments. These include: the angle of repose, which is approximately 5–
10 larger for the sand than for ballotini and snow; the ratio between the flow depth and the particle
diameter, h=d, which is much smaller in the sand experiments (see Table 6.4); and the ratio between
the particle diameter and the roughness of the base of the chute.
The angular geometry of the sand grains and the coefficient of restitution will affect the energy
dissipation in the relatively thin flow through inelastic particle collisions induced by the obstacles.
Furthermore, the particle diameter compared with the roughness of the base of the chute may affect
the velocity profile of the flow. Indeed we find this to be the case when studying the velocity profiles
of sand and ballotini on the same experimental chute. Comparison of basal and surface velocities
between sand and ballotini flows on the Davos chute (Ballotini (iii) and Sand) showed a difference
in the velocity profiles of the two flows. There was a considerably smaller difference between the
surface and basal velocities for the sand (0:5 0:4 m s ;1 ) than for the ballotini (1:2 0:5 m s ;1 )
120
x=xcont
Dams
Ballotini (i)
Ballotini (ii)
Ballotini (iii)
Sand
0.8
0.6
0.4
0.2
0
1
Mounds
0.8
NM
0.6
CM
Ballotini (i): DM
NM
Ballotini (ii): DM
NM/CM
Ballotini (iii): DM
NM/CM
Sand: DM
NM/CM
0.4
0.2
0
0
1
2
3
4
5
6
7
8
H =h1
Figure 6.10: The leading edge run-out relative to the leading edge run-out of the undisturbed control flow measured from the upstream face of the obstacles, as a function of the non-dimensional
obstacle height for experiments with dams and mounds. A line is drawn through the datum mound
experiments (DM), while the narrow and conical mounds are presented with points. The higher point
corresponds to the conical mounds (CM) while the lower point corresponds to the narrow mounds
(NM) in all but the Sand experiments where the order of points is reversed, as indicated.
121
xcm =xcmcont
Dams
Ballotini (ii)
Ballotini (iii)
Sand
0.8
0.6
0.4
0.2
0
1
Mounds
0.8
NM
0.6
CM
0.4
Ballotini (ii): DM
NM/CM
Ballotini (iii): DM
NM/CM
Sand: DM
NM/CM
0.2
0
0
1
2
3
4
5
6
7
8
H =h1
Figure 6.11: The run-out of the mass centre of the deposit relative to the centre of mass run-out of
the undisturbed control flow, as a function of the non-dimensional obstacle height for experiments
with dams and mounds. A line is drawn through the datum mound experiments (DM), while the
narrow and conical mounds are presented with points. The higher point corresponds to the conical
mounds (CM) while the lower point corresponds to the narrow mounds (NM) in all but the Sand
experiments where the order of points is reversed, as indicated.
122
ju4 j ju1j
=
Dams
Mounds
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
6
H =h1
Figure 6.12: The ratio between the flow speed in the dry-snow experiments at the upper and lower
sensors on the Weissfluhjoch chute (see Figure 6.2) plotted for the dams and mounds as a function of
the non-dimensional obstacle height. Each point is an average of two experiments. A line is drawn
through the points of the dam experiments to distinguish them from the mound experiments.
suggesting less shearing in the sand flow. This may be explained by considering the roughness of
the chute. The chute was rough compared with the diameter of the ballotini particles but smooth
compared with the larger sand particles. The sand particles were therefore sliding on the chute base,
while the ballotini particles were significantly agitated by the base of the chute, leading to shearing
near the base.
The reason why the conical mounds were more effective in retarding the sand flow than the
ballotini flow can not be analysed quantitatively based on the experiments that have been carried
out and will therefore not be discussed further here. The relatively thin sand flows (in terms of the
number of particles) may, however, have less relevance to natural-scale snow avalanches than the
ballotini flows because the flow depth of snow avalanches may be expected to consist of many more
particles than in the sand experiments.
The effectiveness of the different obstacle setups for retarding the flow in the snow experiments
was analysed by plotting the ratio of flow speed upstream of the obstacles, ju 1 j, to the flow speed
downstream of the landing point of the jet, ju 4 j, see Figure 6.12. The analysis was only carried out
for the dry-snow experiments because of difficulties with the velocity sensors during the wet-snow
experiments that were performed a year earlier. The speed ju 4 j was measured close to the end of the
25 chute section, a few metres downstream from the landing point of the jets. The fact that the jets
landed on a considerable slope, may have lead to a subsequent acceleration of the flow towards the
same, approximately constant, flow speed in all the experimental setups. (The spatial development
123
of steady granular flows down a fixed incline is formulated in §2.2.) The retarding effect of the
mound and dam setups may therefore be larger than the plot indicates, had the jets landed on low
inclines on which the flows would decelerate and stop, as in the other laboratory experiments. One
may, nevertheless, conclude from Figure 6.12 that the mounds are at least as effective in retarding
the avalanche flow as a continuous dam of the same height (when H =h 1 2).
6.5 Conclusions
We have investigated the interaction of supercritical granular flows with mounds and have found
similar dynamical behaviour across a range of physical scales. This similarity and the nature of
the interaction have been shown to depend on the following factors: the Froude number; the nondimensional obstacle height; the aspect ratio of the obstacles; the relative area covered by the obstacles; and a large enough flow depth to particle diameter ratio. The similarity is most clearly evident
in the same type of mound setups being the most effective at reducing the run-out distances of the
flows for all the materials on all experimental scales. (An exception to this was the setup of conical
mounds in the sand experiments which is discussed in the previous section.) The trajectories of the
jets and the dynamics of the interaction are relatively insensitive to the exact value of the Froude
number, since similar observations are made for the snow flows with Froude numbers in the range
3–6 and the dry granular currents with Froude numbers of the order 10.
The experiments confirm the findings of the earlier experiments (Hákonardóttir et al., 2003b)
regarding the optimal shape and layout of steep braking mounds for maximum retardation of granular
flows. Therein the following conclusions were deduced: the height of the mounds should be 2–3
times the thickness of the flow; the aspect ratio of the mounds should be chosen close to 1; and
the mounds should be placed close together for maximum interaction between jets launched from
adjacent mounds. It is interesting to note that this is almost exactly the same optimal geometry as
was found to be most effective for baffle blocks in hydraulic stilling basins (Peterka, 1984). The
most effective baffle blocks for flows at oncoming Froude numbers in the range 8–12 had a height
that was 2–2.5 times the upstream flow depth; the transverse width of the blocks and the spacing
between them was 0.75 times their height; and steep upstream faces, normal to the bottom of the
stilling basin, with relatively sharp corners, were important for effective energy dissipation, whereas
the geometry of the downstream part of the blocks was of little importance.
Air entrainment into the airborne jets was noticeable in the experiments but did not appear to
significantly influence the ballistic trajectories. The trajectories were best modelled by neglecting
air drag. Larger-scale experiments with water show, on the other hand, that between 0 and 30% of
the initial mechanical energy of the jet may be lost during the jump (USBR, 1987; Hager, 1992;
Novak, 1998). The drag term m(CD =h j )dx=dt jdx=dt j in equation (5.6) may therefore be expected to
124
6.5. Conclusions
Figure 6.13: A photograph of one braking mound above Neskaupstaður, Iceland. A person is standing in front of the mound.
become significant in avalanche flow at natural scales and energy loss due to air drag will then affect
the length of the jump (Jóhannesson and Hákonardóttir, 2003).
It is interesting to note that a similar dissipation of the mechanical energy of both snow and
ballotini flows was found at the upstream faces of the continuous dams for dams with the same
height to flow depth ratio but at a range of physical scales. The experimental set is, however, limited
and further experiments using snow, and dams with different frictional properties are needed to
confirm further these findings. For example, if the obstacles are sufficiently high, experiments with
supercritical flow of dry granular materials (Gray et al., 2003) have shown that a jump in flow depth
and speed can occur upstream of the obstacles (granular jump) as described in Chapter 3. The
dissipation of mechanical energy over such jumps only depends on the Froude number of the flow
and not on the material properties of the current to leading-order.
Mounds with a layout based on these experimental results have recently been constructed above
Neskaupstaður in eastern Iceland. Figure 6.13 shows one of the mounds. The geometry of these
mounds is different from the more common conical shape of avalanche braking mounds, which
allows for a smoother passage of the flow around the mounds.
125
Chapter 7
Conclusions
The interaction of granular flows with different types of protective structures (catching dams, deflecting dams and braking mounds) has been addressed in the thesis. The granular experiments, reported
in the previous chapters, reveal an interesting analogy between the interaction of incompressible,
high Froude number, granular and fluid flows with dams. Here we will summaries the main experimental observations and discuss their application towards protection against natural dry-snow
avalanches.
Incompressible, shallow-layer shock dynamics were found to give an accurate description of the
interaction of granular flows with relatively tall obstacles, such as deflecting and catching dams. A
change in flow state, from supercritical to subcritical flow, occurred over a shock whereby mechanical energy was dissipated. In incompressible flows, a jump in flow depth and speed occurs over the
shock. The sudden change in the flow depth is striking and such shocks are therefore also termed
jumps. The flow depth, speed and energy downstream of the jumps depend sensitively on the value
of the Froude number of the approaching flow.
Stationary, weak, oblique shocks were formed in the two-dimensional interaction of flows with
deflecting dams, while normal shocks (or bores), travelling upstream from the dams, formed in the
interaction with catching dams. As the height of the catching dams decreased relative to the depth
of the approaching flow, some flow over-topped the dams and the granular bores slowed down. The
flow was launched over the catching dams in a supercritical flow state when the dams had become
too low, and all of the mass flux approaching the dams was transported directly over the dams.
There are indications of the formation of oblique shocks in the impact of natural-scale snow
avalanches with the deflecting dams at Flateyri in Iceland (Jóhannesson, 2001). Similar observations
of the interaction with catching dams are not available to our knowledge, but a re-analysis of the dataset from Ryggfonn in Norway (Lied et al., 2002) by analogy with shock dynamics may reveal some
interesting results. An important consideration in the application of these experimental results to
127
Chapter 7. Conclusions
practical design of protection dams is the question of the formation of the shock in the first place. A
change in flow state from supercritical to subcritical needs to take place at the upstream dam face as
the flow climbs the dam if a jump is to occur. The required height of a dam so that the approaching
supercritical flow becomes subcritical as it climbs the upstream face of the dam depends on the
energy dissipation in the flow as it interacts with the dam.
The small-scale granular experiments with flows which completely over-top dams show that the
dissipation at the upstream dam face is a function of the height of the dams relative to the depth of
the approaching flow and that the dissipation increases as the dam height increases relative to the
flow depth. For steep dams with a height that is 2–3 times the flow depth, we find experimentally
that approximately 50% of the mechanical energy of the flow is dissipated in the interaction and
approximately 60% for dams that are 5 times higher than the flow depth. Previous experiments by
Hákonardóttir et al. (2003b) furthermore show that the dissipation at the upstream dam face is also
a function of the steepness of the upstream face of the dams, such that the dissipation increases with
increasing steepness. In §4.5 we calculated that bores propagate upstream of catching dams that
are approximately nine times higher than the depth of the oncoming flow, if the approaching flow
has a Froude number of 10. This is not necessarily the case for natural dry-snow avalanches. It is
conceivable that less energy is dissipated in the initial interaction of a dry-snow avalanche with a dam
than in these small-scale, granular avalanches, since, for example, friction between the avalanche and
the dam face may be less. Furthermore additional physical processes that cannot be considered at
small scales may be important at the natural-scale, such as the compressibility of the avalanche.
The whole avalanche might get launched over a higher dam without forming a shock upstream of
the dam. In fact, there is an example of a snow avalanche that completely over-flowed a natural
catching dam where the calculated bore height was lower than the height of the dam (Jóhannesson,
pers. comm., 2003). There is therefore a pressing need to better understand how much energy can
be dissipated in the initial interaction of natural dry-snow avalanches with catching dams.
The interaction of high Froude number flows with obstacles that have a height that is only a
few times the depth of the approaching flow have revealed the formation of airborne jets. Such jets
are important for the use of multiple rows of defence structures to retard avalanches. The spacing
between the rows must be chosen sufficiently large so that the avalanche does not get launched over
subsequent rows further down. Two-dimensional jets that are launched over dams are found to be
accurately described as ballistic trajectories with negligible air drag and the throw angle of the jets
can be approximated by the use of inviscid, irrotational fluid flow theory. Experiments with fluid jets
on larger scales have revealed that some dissipation due to air drag takes place during the motion. It
is likely that the larger-scale snow avalanches are also affected by air drag, shortening the distance
that such snow jets travel through the air.
Laboratory experiments on different physical scales using different granular materials suggest
128
that the three-dimensional flow dynamics around low obstacles such as braking mounds are governed to leading order by the geometry of the obstacles and the large-scale properties of the flowing
avalanche, described by the Froude number, rather than micro-scale properties of the granular current. The results do however not seem to be sensitive to the exact value of the Froude number, given
that the flows are well within the supercritical range. The retardation of the flow by the mounds is
highly sensitive to the geometry and layout of the mounds, and a row of mounds may retard the flow
more effectively than a continuous dam of the same height does.
These small-scale, laboratory experiments have provided useful information on the dynamics
of the interaction of shallow, high Froude number, granular flows with different types of retarding
structures, and thereby a starting point for the formulation of new design criteria for protection
dams. They are also useful for implementing deflection dynamics in numerical models that could be
used to evaluate the effectiveness of dams as protection measures for settlements, communication
lines and other properties and infrastructure. Larger-scale experiments and observations of natural
snow avalanches are, however, needed to confirm these findings, and reveal whether other physical
processes of the flowing motion of natural dry-snow avalanches play a significant role.
129
Appendix A
Mohr-Coulomb failure criterion
A.1 Stresses and failure in soil
Figure A.1 illustrates a general case of stresses at a point within any homogeneous, isotropic, elastic body, and a soil mass, in particular (Bowles, 1979). It is possible to orientate the soil element
in Figure A.1 such that no shear stresses exist on the sides of the element. This orientation produces principal axes and the normal stresses on the element faces are principal stresses, σ1 and σ3 ,
illustrated in Figure A.2. The normal stress, σn , and shear stress, τ, on any plane oriented at an
angle, θ, to the direction of the axis of the maximum principal stress, σ 3 , can be obtained from the
two-dimensional principal stresses. Balance of forces in the principal stress directions leads to
σ3 sin θ + τ cosθ ; σn sin θ
σ1 cos θ ; τ sinθ ; σn cos θ
Z
0
0
σz
τzx
τzy
σy
=
=
τxz
τyx
τyz
τxy
σx
X
Y
Figure A.1: Schematic diagram of stresses on a soil element.
131
Appendix A. Mohr-Coulomb failure criterion
σ1
A
A
σ3
σn
σ3
τ
θ
B
B
σ1
σ1
Figure A.2: Schematic diagram of two-dimensional principal stresses to obtain the normal stresses
on any plane AB as shown. The arrows correspond to stresses acting on each individual plane.
τ
τ
φ
R
φ
R
2θ
σ3
σn
σ1
C
2θ
σ3
σ1
σn
Figure A.3: Schematic diagram of Mohr’s circle of stress for cohesive and non-cohesive materials.
which may be solved for σn and τ by making use of the trigonometric relationships: cos 2 θ = 1 ;
sin2 θ; sin2 θ =
1
2
(1 ; cos2θ); and sin θ cos θ = 12 sin 2θ. We find that
σ1 + σ3 σ1 ; σ3
+ 2 cos 2θ
σn =
2
σ1 ; σ3
τ =
sin 2θ
:
2
These equations are parametric equations of a circle of stress in the στ plane 1 with radius
R=
and origin at
σn =
σ1 ; σ3
2
σ1 + σ3
2
τ = 0:
Slow flows of granular material with strong grains, i.e. non-deforming grains, are well described
by the Coulomb equation (Hungr and Morgenstern, 1984a and 1984b)
τ = σn tanφ
where φ is the angle of internal friction of the material. It defines the failure criterion of a granular
material. The shear strength of a cohesionless material is determined by the contact forces and
1 This
circle of stress is called Mohr’s circle since it is believed that Otto Mohr first proposed its use in 1871.
132
friction between the grains. Soil failure is governed by the normal stress that is applied to the soil
mass and the amount of shear force within the soil. As the normal stress increases the shear stress to
cause failure also increases, according to Figure A.3. For cohesive materials
τ = C + σn tan φ
where C is cohesion.
A.2 Failure in a moving pile of granular material
Another failure plane is the bed plane. Along the bed, failure occurs by sliding of a mass of particles.
This is defined by the friction angle between the bed and the material, δ.
The ratio of lateral and vertical in situ soil stresses, σx and σz , respectively may be defined by a
factor K as
K=
σx
:
σz
Two states of limiting equilibrium exist for a mass of earth: upper and lower equilibrium, i.e. active
or passive earth pressure. They may be thought of as follows. Consider a frictionless vertical wall
inserted into a horizontal mass of sand. The wall is subjected to a stress called the earth pressure at
rest. The mass on the left hand side of the wall is excavated without changing any stresses acting
on the mass at the right hand side. Moving the wall horizontally to the right produces a plastic
deformation in the sand, known as passive Rankine state. Moving the wall to the left corresponds
to active Rankine state and the stress on the wall is the active thrust. Active and passive factors,
Kactpass , are defined in the above way depending on the state of the earth (Bowles, 1979).
By adopting this way of thinking, Savage and Hutter (1989) assume that an active or passive
state of stresses is developed in a moving mass of granular material. They denote the stress tensor
by P, and pxx and pzz correspond to σx and σz , respectively. At the bed, the normal stress and the
shear stress must be such that they lie on the bed yield line. Two possible Mohr circles can be drawn
through the point corresponding to the ( p zz pxz ) stress state, see Figure A.4. The one corresponding
to a larger value of the normal stress, pxx , is associated with the passive state of stress and the other
is associated with the active state of stress. Those states are developed depending upon whether
an element of material is being elongated or compressed in the direction parallel to the bed, i.e. an
active state is observed when ∂u=∂x > 0 and a passive state when ∂u=∂x < 0. From Figure A.4 the
133
Shear stress
( pzz
pxz )
Bed yield line
δ
( pxx
φ
Normal stress
( pxx
; pxz )
; pxz )
passive case
active case
Figure A.4: Mohr diagram showing Coulomb yield criterion, bed friction angle and active and passive stress states (Savage and Hutter, 1989).
active and passive pressure coefficients can be derived
Kactpass
Kact
Kpass
=
=
=
pxx
pzz
2 1;
2 1+
p
1 ; cos2 φ= cos2 δ
p
cos2 φ
1 ; cos2 φ= cos2 δ
cos2 φ
134
;1
;1
:
Appendix B
The Schwarz-Christoffel
transformation
The Schwarz-Christoffel transformation transforms the interior of a polygon into the entire upper
(or lower) half plane of another variable. In this appendix we illustrate its use for a pentagon.
The pentagon A’B’C’D’E’A’ (Figure B.1 (a)) in the Ω plane can be transformed into the upper
half t plane (Figure B.1 (b)) by the Schwarz-Christoffel transformation
Ω=M
Zt
0
(s ; a) (s ; b)
α=π
β=π
ds
(s ; c)γ=π(s ; d )δ=π(s ; e)ε=π + N (B.1)
where a, b, c, d and e are arbitrary real numbers; α, β, γ, δ and ε are real and positive, measured
counter clockwise with
α + β + γ + ε + δ = 2π;
and M, N are complex numbers. From equation (B.1) it can be seen that every time s crosses any of
the values a, b, c, d and e the argument of dΩ=ds, and hence of dΩ (since s is real except around the
semicircles surrounding the points A, B, C, D and E), will change by α, β, γ, δ and ε. Of the seven
constants (a, b, c, d, e, M and N), three can be chosen arbitrarily and the remaining four determined
from the positions of A’, B’, C’ and D’ (or any four of the five vertices):
A0
=
B0
=
C0
=
D0
=
Za
ds
(s ; b) (s ; c)γ=π(s ; d )δ=π(s ; e)ε=π + N;
0 (s ; a )
Zb
ds
+ N;
0 (s ; a)α=π (s ; b)β=π (s ; c)γ=π (s ; d )δ=π (s ; e)ε=π
Zc
ds
α
=
π
β
=
π
(s ; b) (s ; c)γ=π(s ; d )δ=π(s ; e)ε=π + N;
0 (s ; a)
Zd
ds
+ N:
0 (s ; a)α=π (s ; b)β=π (s ; c)γ=π (s ; d )δ=π (s ; e)ε=π
α=π
β=π
As long as the polygon is closed (α + β + γ + ε + δ = 2π), there is no need to impose a condition on
the location of E’, if the locations of A’, B’, C’ and D’ are given and the lines E’A’ and D’E’ have
135
Appendix B. The Schwarz-Christoffel
transformation
(a)
δ
Origin
D’
E’
ε
(b)
γ
A’
(c)
α
C’
infinity.B.1: Schwarz-Christoffel transformation: (a) Ω plane; (b) t plane; (c) t plane with E at
Figure
B’ β
the correct inclinations. The same argument applies to a polygon of n sides. Three (usually not M or
N) of the n
E
E
A
B
A
B
C
C
D
D
+ 2 constants can always be chosen arbitrarily.
If any of the constants associated with the vertices in the t plane is infinite, the corresponding
factor should be dropped. For instance, if e is infinite, the factor (s ; e) ε=π should be dropped from
equation (B.1). This can be seen from Figure B.1 (c). Let the angle of inclination of E’A’ be zero.
The transformation then becomes
Ω=M
Zt
0
(s ; a)α=π(
ds
Bibliography
Ancey, C. and Meunier, M. (2004).
Estimating bulk rheological properties of flowing snow
avalanches from field data. Journal of Geophysical Research, 109(F01004).
Bouchet, A., Naaim, M., Ousset, F., Bellot, H., and Cauvard, D. (2003). Experimental determination of constitutive equations for dense and dry avalanches. Presentation of the set-up and first
results. Surveys in Geophysics, 24(5–6):525–541.
Bowles, J. E. (1979). Physical and Geotechnical Properties of Soils. McGraw-Hill Book Company,
USA.
Bozhinskiy, A. N. and Losev, K. S. (1998). The Fundamentals of Avalanche Science. Mitt.
Eidgenöss. Inst. Schnee- Lawinenforsch., 55.
Briukhanov, A. V., Grigorian, S. S., Miagkov, S. M., Plam, M. Y., Shurova, I. Y., Eglit, M. E., and
Yakimov, Y. L. (1967). On some new approaches to the dynamics of snow avalanches. In
Physics of Snow and Ice, volume 1, part 2, pages 1223–1241. Proc. of the Intl. Conf. on Low
Temperature Science, Sapporo 1966, The Institute of Low Temperature Science, Hokkaido
University, Sapporo Japan.
Campbell, C. S. (1990). Rapid Granular Flows. Annual Review of Fluid Mechanics, 22:57–92.
Chapman, C. J. (2000). High Speed Flow. Cambridge texts in applied mathematics. Cambridge
University Press, UK.
Chu, T., Hill, G., McClung, D. M., Ngu, R., and Sherkat, R. (1995). Experiments on granular flow
to predict avalanche runup. Canadian Geotechnical Journal, 32:285–295.
Cooker, J. M. and Peregrine, D. H. (1995). Pressure-impulse theory for liquid impact problems.
Journal of Fluid Mechanics, 297:193–214.
Dias, F. and Christodoulides (1991). Ideal jets falling under gravity. Physics of Fluids A, 7:1711–
1717.
137
Bibliography
Dias, F. and Vanden-Broeck, J.-M. (1989). Open channel flows with submerged obstructions. Journal of Fluid Mechanics, 206:155–170.
Faug, T., Lachamp, P., and Naaim, M. (2002). Experimental investigation on steady granular flows
interacting with an obstacle down an inclined channel: Study of the dead zone upstream from
the obstacle. Application to interaction between dense snow avalanches and defence structures.
Natural Hazards and Earth System Sciences, 2:187–191.
Faug, T., Naaim, M., Bertrand, D., Lachamp, P., and Naaim-Bouvet, F. (2003). Varying dam height
to shorten the run-out of dense avalanche flows: developing a scaling law from laboratory
experiments. Surveys in Geophysics, 24(5–6):555–568.
Gerhart, P. M., Gross, R. J., and Hochstein, J. I. (1993). Fundamentals of Fluid Mechanics. AddisonWesley, 2nd edition.
Gerodetti, M. (1985). Drag coefficient on pyramidal baffle blocks. Water Power & Dam Construction, 37:26–28.
Goldhirsch, I. (2003). Rapid granular flows. Annual Review of Fluid Mechanics, 35:267–293.
Gray, J. M. N. T., Tai, Y.-C., and Noelle, S. (2003). Shock waves, dead-zones and particle-free
regions in rapid granular free surface flows. Journal of Fluid Mechanics, 491:161–181.
Hager, W. H. (1992). Energy Dissipators and Hydraulic Jump. Kluwer Academic Publishers.
Hager, W. H. and Vischer, D. L. (1995). Energy dissipators. Balkema Publishers.
Hákonardóttir, K. M., Hogg, A. J., Batey, J., and Woods, A. W. (2003a). Flying avalanches. Geophys.
Res. Lett., 30(23):2191, doi:10.1029/2003GL018172.
Hákonardóttir, K. M., Hogg, A. J., Jóhannesson, T., and Tómasson, G. G. (2003b). A laboratory
study of the retarding effect of braking mounds on snow avalanches. Journal of Glaciology,
49(165):191–200.
Hákonardóttir, K. M., Hogg, A. J., Jóhannesson, T. J., Kern, M., and Tiefenbacher, F. (2003c).
Large-scale avalanches braking mound and catching dam experiments with snow: A study of
the airborne jet. Surveys in Geophysics, 24(5–6):543–554.
Hákonardóttir, K. M., Jóhannesson, T. J., Tiefenbacher, F., and Kern, M. (2001). A laboratory study
of the retarding effect of breaking mounds in 3, 6 and 9 m long chutes. Technical Report 01007,
The Icelandic Meteorological Office.
138
Bibliography
Hákonardóttir, K. M., Jóhannesson, T. J., Tiefenbacher, F., and Kern, M. (2003d). Avalanche braking mound experiments with snow Switzerland-March 2002. Technical Report 03023, The
Icelandic Meteorological Office.
Hungr, O. and Morgenstern, N. R. (1984a). Experiments on the flow behaviour of granular materials
at high velocity in an open channel flow. Géotechnique, 34:405–413.
Hungr, O. and Morgenstern, N. R. (1984b). High velocity ring shear tests on sand. Géotechnique,
34:415–421.
Hutter, K., Koch, C., Pluss, C., and Savage, S. (1995). The dynamics of granular materials from
initiation to runout. Part II: Experiments. Acta Mechanica, 109:127–165.
Irgens, F., Schieldrop, B., Harbitz, C. B., Domaas, U., and Opdahl, R. (1998). Simulations of densesnow avalanches on deflecting dams. Annals of Glaciology, 26:265–271.
Issler, D. (2003). Experimental information on the dynamics of dry-snow avalanches. In Hutter, K.
and Kirchner, N., editors, Response of granular and porous materials under large and catastrophic deformations, volume 11 of Lecture notes in applied and computational mechanics,
pages 109–160. Springer (Berlin).
Jóhannesson, T. (2001). Run-up of two avalanches on the deflecting dams at Flateyri, northwestern
Iceland. Annals of Glaciology, 32:350–354.
Jóhannesson, T. and Arnalds, T. (2001). Accidents and economic damage due to snow avalanches
and landslides in Iceland. Jökull, 50:81–94.
Jóhannesson, T. J. and Hákonardóttir, K. M. (2003). Remarks on the design of avalanche braking
mounds based on experiments in 3, 6, 9 and 34 m long chutes. Technical Report 03024, The
Icelandic Meteorological Office.
Johnson, P. C. and Jackson, R. (1990). Frictional-collisional equations of motion for particulate
flows and their application to chutes. Journal of Fluid Mechanics, 210:501–535.
Kern, M., Tiefenbacher, F., and McElwaine, J. N. (2004). Energy balance in chute flows of snow.
Cold Regions Science and Technology, submitted.
Kotlyakov, V. M., Rzhevskiy, B. N., and Samoylov, V. A. (1977). The dynamics of avalanching in
the Khibins. Journal of Glaciology, 19:431–439.
Lied, K., Moe, A., Kristensen, K., and Issler, D. (2002). Snow Avalanche Research, Programme
SIP–6. Ryggfonn. Full scale avalanche test site and the effect of the catching dam. Technical
Report 581200-35, Norwegian Geotechnical Institute.
139
Bibliography
Louge, M. Y. and Keast, S. C. (2001). On dense granular flows down flat frictional inclines. Physics
of Fluids, 13(15):1213–1233.
McClung, D. M. and Mears, A. I. (1995). Dry-flowing avalanche run-up and run-out. Journal of
Glaciology, 41:359–372.
McClung, D. M. and Schaerer, P. (1993). The Avalanche Handbook. The Mountaineers, Seattle.
McElwaine, J. and Nishimura, K. (2001). Ping-pong ball avalanche experiments. In McCaffrey,
W. D., Kneller, B. C., and Peakall, C., editors, Particulate Gravity Currents, pages 135–148.
Blackwell Science.
McElwaine, J. and Tiefenbacher, F. (2003). Calculating internal avalanche velocities from correlation with error analysis. Surveys in Geophysics, 24(5–6):499–524.
Nishimura, K. (1990). Studies on the Fluidized Snow Dynamics. Technical Report 37, The Institute
of Low Temperature Science, Hokkaido University, Sapporo, Japan.
Nishimura, K., Kosugi, K., and Nakagawa, M. (1993). Experiments on ice-sphere flows along an
inclined chute. Mechanics of Materials, 16:205–209.
Nishimura, K., Maeno, M., and Nakagawa, M. (1991). Chute flow experiments of ice spheres.
In Japan-U.S. Workshop on Snow Avalanche, Landslide, Debris Flow Prediction and Control,
pages 191–196.
Novak, P., Moffat, A. I. B., Nalluri, C., and Narayanan, R. (1989). Hydraulic Structures. Unwinn
Hyman.
Peregrine, D. H. (2003). Water-wave impact on walls. Annual Review of Fluid Mechanics, 35:23–43.
Perla, R., Cheng, T. T., and McClung, D. M. (1980). A two-parameter model of snow-avalanche
motion. Journal of Glaciology, 26:197–207.
Peterka, A. J. (1984). Hydraulic design of stilling basins and energy dissipators. Engineering
Monograph, No 25. Denver, US Department of the Interior, US Bureau of Reclamation.
Pouliquen, O. (1999). Scaling laws in granular flows down rough inclined planes. Physics of Fluids,
11(3):542–548.
Pouliquen, O. and Forterre, Y. (2002). Friction law for dense granular flows: application to the
motion of a mass down a rough inclined plane. Journal of Fluid Mechanics, 453:133–151.
Rericha, E. C., Bizon, C., Shattuck, M. D., and Swinney, H. L. (2002). Shocks in supersonic sand.
Phys. Rev. Lett., 88(014302).
140
Bibliography
Roberson, J. A., Cassidy, J. J., and Chaudhry, M. H. (1997). Hydraulic Engineering. Boston, etc.
Houghton Mifflin Company.
Salm, B. (1964). Anlage zur Untersuchung dynamischer Wirkungen von bewegtem Schnee. ZAMP,
15:357–375.
Salm, B. (1987). Schnee, Lawinen und Lawinenschutz. Technical report, ETH, Zürich.
Salm, B. (1993). Flow, flow transition and runout distances of flowing avalanches. Annals of Glaciology, 18:221–226.
Salm, B., Burkard, A., and Gubler, H. U. (1990). Berechnung von Fliesslawinen. Eine Anleitung fuer
Praktiker mit Bleispielen. Technical Report Mitteilungen Nr. 47, Mitt. Eidgenöss. Inst. SchneeLawinenforsch., Davos.
Salm, B. and Gubler, H. (1985). Measurement and analysis of the motion of dense flow avalanches.
Annals of Glaciology, 6:26–34.
Savage, S. B. (1979). Gravity flow of cohesionless granular materials in chutes and channels. Journal
of Fluid Mechanics, 92 part 1:53–96.
Savage, S. B. and Hutter, K. (1989). The motion of a finite mass of granular material down a rough
incline. Journal of Fluid Mechanics, 199:177–215.
Schaerer, P. A. and Salway, A. A. (1980). Seismic and impact-pressure monitoring of flowing
avalanches. Journal of Glaciology, 26:179–187.
Simpson, J. E. (1987). Gravity Currents In the Environment and the Laboratory. Ellis Horwood
Limited.
Tai, Y.-C., Noelle, S., Gray, J. M. N. T., and Hutter, K. (2001). An accurate shock-capturing finitedifference method to solve the Savage-Hutter equations in avalanche dynamics. Annals of
Glaciology, 32:263–267.
Tiefenbacher, F. and Kern, M. (2004). Experimental devices to determine snow avalanche basal
friction and velocity profiles. Cold Regions Science and Technology, 38:17–30.
Tómasson, G. G., Sigurðsson, F., and Rapin, F. (1998). The avalanche situation in Neskaupstaður,
Iceland: A preliminary defensive plan. In Hestness, E., editor, 25 Years of snow Avalanche Research, Voss 12-16 May 1998, number 203 in NGI Publ., pages 283–288. Norwegian Geotechnical Institute, Oslo.
US Bureau of Reclamation (1984). Design of small dams. Washington, DC, US Department of the
Interior, 3rd edition.
141
Bibliography
Vanden-Broeck, J.-M. and Keller, J. B. (1986). Pouring flows. Physics of Fluids, 29:3958–3961.
Vanden-Broeck, J.-M. and Keller, J. B. (1987). Weir flows. Journal of Fluid Mechanics, 176:283–
293.
Voellmy, A. (1955). Uber die Zerstörungskraft von Lawinen. Bauzeitung, 73:12, 15, 17, 19, 37.
Whitham, G. B. (1999). Linear and Nonlinear Waves. John Wiley & Sons, Inc.
Wieland, M., Gray, J. M. N. T., and Hutter, K. (1999). Channelised free-surface flow of cohesionless
granular avalanches in a chute with shallow lateral curvature. Journal of Fluid Mechanics,
392:73–100.
Yih, C.-S. (1979). Fluid Mechanics. West River Press, USA.
142

University of Bristol PhD thesis by Kristín Martha Hákonardóttir on

Transcription

Similar documents

0859 SKY 11 BB H - Sky Lake Sions Loft

Dam water Intake

John M Steed

Chesaning

From: Lee Sexsmith [mailto:hecubuspg@telus

Num. 23, December 2010 (pdf 3.6 MB)

New Progress for Better Development of Dams and Reservoirs

Milltown Dam Removal - Association of Conservation Engineers

Módulo 5 4: Deslizamientos, flujos de escombro (huaycos), aluviones

Wild Capeb.indd - World Wide Creative