Smagorinsky

Algebraic subgrid-scale turbulence modeling in large-eddy simulation of the neutral boundary layer
3
1
2
Rica Mae Enriquez , Robert L. Street , and Francis L. Ludwig
Large-eddy Simulation
The Advanced Regional Prediction System
[ARPS] is 3D, compressible, non-hydrostatic,
parallelized, and appropriate for LES. Some
parameters are listed and further details of
the ARPS neutral boundary layer simulation
parameters can be found in Chow et al (2005)
and Ludwig et al (2009).
Horizontal Resolution:
Vertical Resolution:
Domain Size:
Geostrophic Wind:
Lateral Boundaries:
Bottom Boundary:
Roughness Length:
32 m
37.5 m average,
10 m minimum
1.28 km x 1.28 km x 1.5 km
[Ug, Vg] = [10, 0] m s-1
Periodic
Rigid wall, semi-slip
0.1 m
Eddy-Viscosity Models
The Smagorinsky model prescribes a static
coefficient, while the Dynamic Wong-Lilly
[DWL] model dynamically computes νT.
Smagorinsky:
DWL:
Cs = 0.18
νT clipped at -1.5 x 10−5 m2 s-1
Apply canopy model of Chow et al (2005)
uj
0 =
xk
∂ui
− A jk
∂xk
− Aik
∂u j u j
uj
∂xk xk
xk
Production
Πij =
− c1 ε
e
(
Aij − 2 c1 e δ ij
3
)
(
− 2 εδ ij
3
uj
uj
xk
xk
Dissipation
)
+ Πij
uj
xk
Pressure Redistribution
(
2 Pδ
2 Pδ
P
−
D
−
− c2 ij
ij − c3 eSijj − c4 ij
ij
3
Slow Pressure-Strain, φ1
3
) +{ (
c5 ε
e
}
)
Aij − 2 e δ ij + c6 Pij − c7 Dij + c8 eSij f ( z)
3
Rapid Pressure-Strain, φ2
Wall Effects, φw
We solve for Āij, the subgrid-scale [SGS] stress. Production need not be modeled, dissipation appears in its isotropic form for
high Reynolds number flows, and pressure redistribution is replaced with the Launder et al (1975) model. c1 = 1.8, c2 = 0.78, c3 =
0.27, c4 = 0.22, c5 = 0.8, c6 = 0.06, c7 = 0.06, and c8 = 0.0.
Logarithmic Velocity Law
Smagorinsky
Dynamic Wong−Lilly [DWL]
LASS
Exact
20
18
z (m )
16
14
50
12
10
10
−2
10
z/H
−1
0
1
1.2
ΦM
1.4
1.6
Resolved Vertical Velocity: “Standard Definition to HD”
Smagorinsky
Extremes
-47 39
Dynamic
Wong-Lilly
Extremes
-63 50
LASS
Extremes
-92 84
1.28 km
-50
0
50 cm s-1
Near-Wall Anisotropy
150
100
Computational Cost Factors
50
0
−1
Figure 2. Resolved vertical velocity domain snapshots at 15 m
with black contour lines of w = 0
cm s-1 for Smagorinsky, DWL,
and LASS models. Snapshots are
every 2,500 s from 260,000 s to
280,000 s, and are placed left to
right and then top to bottom. Extreme values [cm s-1] are printed
at each model square corner. The
LASS model incorporates more
physics and specifically allows
backscatter without constraint ,
so it allows representation of the
granular behavior.
Figure 3. Normalized SGS anisotropies of LASS and experimental Horizontal Array
Turbulence Study [HATS] data (Chen et al, 2009). The LASS model provides anisotropy in the near-wall region and reminds us that SGS normal stresses are anisotropic
near walls, contrary to the typical assumption of SGS stress isotropy. At this location, the LASS model anisotropies agree well with the HATS values.
σ 11 /u 2*
σ 22 /u 2*
σ 33 /u 2*
HA TS Data
−0.5
02
σ/u *
0.5
1
The LASS model is a more physically complete SGS turbulence model that provides
near-wall anisotropies that eddy-viscosity
models do not. It yields proper shear stress
values in the logarithmic layer. It shows promise, as well, as a model for simulations within
the “terra incognita” where reconstruction of
the subfilter-scale stress is not feasible because of the proximity of the peak energy
containing-scale to the grid-scale.
The Generalized LASS
[GLASS] model will have
coupled SGS flux equations GLASS
for momentum, heat, and
water vapor. We will also add reconstruction
of the subfilter-scale variables (Chow et al,
2005) with GLASS to create a mixed model.
100
U M/u *
Figure 1. Comparison of normalized
mean wind speed, UM/u*, and nondimensional mean shear, ΦM, profiles
for the Smagorinsky, Dynamic WongLilly [DWL], and LASS models with the
exact logarithmic velocity law [valid
within ~10% of H, the boundary layer
depth]. The DWL and LASS models
adhere to the theoretical logarithmic
law closely in this zone; the Smagorinsky model overestimates mean velocities and shear.
Conclusions
Future Work
150
1.28 km
We are developing a non-eddy-viscosity
subgrid-scale [SGS] stress model that allows
for anisotropy and backscatter to: (1) improve
the mixed models, and (2) be a stand-alone
model for the “terra incognita.”
We use the Carati et al (2001) framework, in
which we apply a spatial filter [overbar] and a
discretization filter [wavy overbar]. This produces a Reynolds stress that can be separated
into subfilter-scale and SGS stresses and can
be parameterized with a mixed model. A
weakness of mixed models is that the SGS
model is typically an eddy-viscosity model.
The “terra incognita” (Wyngaard, 2004) lies
between (1) traditional mesoscale modeling
where most or all turbulence is parameterized, and (2) LES where most of the turbulence is resolved. In the “terra incognita,” it is
desirable to use only a SGS model, so the SGS
model must incorporate as much physics as
possible in order to represent the effects of
small scales properly.
The Linear Algebraic Subgrid-Scale Stress [LASS] Model
z (m )
Rationale
SGS Model
Turbulence
Total
Smagorinsky
DWL
LASS
1.0
3.5
4.7
1.0
1.3
1.4
References
Carati, D., Winckelmans, G.S., Jeanmart, H., 2001. On the modelling of the subgrid-scale and filtered-scale stress tensors in
large-eddy simulation. J Fluid Mech 441, 119-138.
Chen, Q., Otte, M.J., Sullivan, P.P., Tong, C., 2009. A posteriori
subgrid-scale model tests based on the conditional means of
subgrid-scale stress and its production rate. J Fluid Mech
626, 149-181.
Chow, F.K., Street, R.L., Xue, M., Ferziger, J.H., 2005. Explicit filtering and reconstruction turbulence modeling for largeeddy simulation of neutral boundary layer flow. J Atmos Sci
62, 2058-2077.
Launder, B.E., Reece, G.J., Rodi, W., 1975. Progress in the development of a Reynolds-stress turbulence closure. J Fluid
Mech 68, 537-566.
Ludwig, F.L., Chow, F.K., Street, R.L., 2009. Effect of turbulence
models and spatial resolution on resolved velocity structure
and momentum fluxes in large-eddy simulations of neutral
boundary layer flow. J Appl Meteorol Clim 48, 1161-1180.
Wyngaard, J.C., 2004. Toward numerical modeling in the `terra
incognita'. J Atmos Sci 61, 1816-1826.
Acknowledgements
We appreciate the spirited engagement of Professor Tina Chow
of UC Berkeley and Dr. Peter Sullivan of NCAR. We are grateful
for support from an NSF Graduate Research Fellowship [RME],
NSF Grant ATM-0453595, and NCAR for the computing time
used in this research and for support [RME & RLS] through the
Advanced Study Program.
1
ricae@stanford.edu
2street@stanford.edu
3
fludwig@stanford.edu