turbulence modeling for atmospheric simulationsturbulence modeling for atmospheric simulations...
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Turbulence Modeling for Atmospheric Simulations
Geophysical Turbulence Phenomena
NCAR, Boulder, CO, 25 July 2008
Tom Lund
NorthWest Research Associates, Colorado Research Associates DivisionBoulder Colorado, [email protected]
http://www.cora.nwra.com/~lund
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OUTLINE
• Review of the turbulence energy spectrum - enormous range of length scales.
• Reduction of scale range via spatial filtering - Large Eddy Simulation (LES).
• Closure problem - required SubGrid-Scale (SGS) models for flows with significantbuoyancy effects.
• Eddy viscosity models.
• Modeling via transport equations.
• Dynamic modeling.
• Failure of LES near the earth’s surface - what to do about it.
• Sample simulation results.
• Summary.
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TURBULENCE ENERGY SPECTRUM
Re
DissipationRange
Dissipation scale, Integral scale, L
Energy Containing
Range
η
1e−10
1e−08
1e−06
0.0001
0.01
1
100
0.01 0.1 1 10 100 1000 10000
En
erg
y,
E(k
)
Wavenumber, k
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MESH POINT REQUIREMENTS FOR DNS
• From Turbulence theory, we have the following scaling relation:
(L/η) ∼ Re3/4
• Using this information, we can form the following estimate:
Box size ∼ L
Grid size ∼ η
Number of mesh points in each direction ∼ (L/η) ∼ Re3/4
Number of mesh points in 3D ∼ (L/η)3 ∼ (Re3/4)3 ∼ Re9/4
• Based on experienceN ' 6Re9/4
• For Re = 106, this estimate indicates that order 1014 mesh points are required!
• It is simply not possible to simulate all the relevant scales of motion at thehigh Reynolds numbers found in atmospheric flows.
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LARGE EDDY SIMULATION
• Our simulations can only resolve the large eddies. We must account for theeffects of the unresolved motions via a turbulence model.
Resolved
UnresolvedSmall Scales
Large Eddies
Inertial Range
1e−10
1e−08
1e−06
0.0001
0.01
1
100
0.01 0.1 1 10 100 1000 10000
Ener
gy, E
(k)
Wavenumber, k
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WHAT CAN WE RESOLVE?
• The kinetic energy and energy dissipation resolved up to wavenumber k are
Energy =∫ k
0
E(k′) dk′ Dissipation = ν
∫ k
0
k′2E(k′) dk′
Resolved Unresolved
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.01 0.1 1 10 100 1000 10000
To
tal
En
erg
y,
To
tal
Dis
sip
atio
n
Wavenumber, k
EnergyDissipation
• We can resolve nearly all of the kinetic energy (and turbulent transport)but very little of the energy dissipation.
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FILTERING
• The separation into large and small scales is accomplished via a spatial filteringoperation. In Fourier space this amounts to multiplication by a filter transferfunction
u = G(k)u(k)Using the convolution theorem, the equivalent operation in physical space is
u =∫ L
0
f(x− x′)u(x′) dx′
Where f(x) and G(k) are Fourier transform pairs. Spatial filtering amounts to aweighted average of the function over a set region in space (filter length scale).
0
0.2
0.4
0.6
0.8
1
0.01 0.1 1 10 100 1000 10000
Fil
ter
Tra
nsf
er
Fu
ncti
on
, G
(k)
Wavenumber, k
Width
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
−5 −4 −3 −2 −1 0 1 2 3 4 5
Fil
ter
Wei
gh
t, f
(x)
x
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FILTERING EXAMPLE
In Fourier space
1e−10
1e−08
1e−06
0.0001
0.01
1
100
0.01 0.1 1 10 100 1000 10000
Ener
gy, E
(k)
Wavenumber, k
Filter→
1e−10
1e−08
1e−06
0.0001
0.01
1
100
0.01 0.1 1 10 100 1000 10000
Ener
gy, E
(k)
Wavenumber, k
In physical space
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 20 40 60 80 100 120 140 160 180 200
u
x
Filter→
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0 20 40 60 80 100 120 140 160 180 200
u
x
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DERIVATION OF THE LES EQUATIONS
The spatial average (filter) operator is applied to the mass, momentum, andpotential temperature equations to give
∂ui
∂xi= 0
∂ui
∂t+
∂
∂xj(uiuj) = − 1
ρ0
∂p
∂xi− ∂τij
∂xj+
g
θ0θδi3 +
∂
∂xj
(2νSij
)∂θ
∂t+
∂
∂xj
(θuj
)= −∂qj
∂xj+
∂
∂xj
(κ
∂θ
∂xj
)Where τij and qi are the unresolved (SubGrid-Scale, SGS) stress and heat flux
τij ≡ uiuj − uiuj qj ≡ θuj − θuj
and where Sij is the resolved rate of strain
Sij =12
(∂ui
∂xj+
∂uj
∂xi
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CLOSURE (TURBULENCE MODEL) REQUIRED
• The SGS stress (τij) and SGS heat flux (qj) appear as unknowns in the filteredmomentum and temperature equations.
• Although we can derive exact evolution equations for these quantities, theseequations contain additional unknown terms. This is the closure problem.
• We proceed by attempting to relate τij and qj to the resolved flow variables.This process is known as turbulence modeling.
• Unlike turbulence models for the Reynolds averaged equations (classical approachusing a long time average), the LES system requires models only for theunresolved transport.
• SGS models are thus fundamentally different from classical turbulence models.In general, simpler models will suffice for the LES system.
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THE TURBULENT KINETIC ENERGY BUDGET
• An Evolution equation for the resolved turbulent kinetic energy can be formedby taking the dot product of the resolved velocity with the resolved momentumequation
ui∂ui
∂t=
∂
∂t
(12uiui
)=
∂
∂t
(12u2
)=
− ∂
∂xj
[(12u2 + p
)uj
]+
∂
∂xj
[(2νSij − τij
)ui
]︸ ︷︷ ︸
Transport
+ uiuj∂ui
∂xj︸ ︷︷ ︸Shear Production
+g
θ0θu3︸ ︷︷ ︸
Buoyancy Production
+ τijSij︸ ︷︷ ︸SGS Dissipation
− 2νSijSij︸ ︷︷ ︸Viscous Dissipation
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BALANCING PRODUCTION AND DISSIPATION
∂
∂t
(12u2
)= ... uiuj
∂ui
∂xj︸ ︷︷ ︸Shear Production
+g
θ0θu3︸ ︷︷ ︸
Buoyancy Production
+ τijSij︸ ︷︷ ︸SGS Dissipation
− 2νSijSij︸ ︷︷ ︸Viscous Dissipation
• The resolved production is normally positive and is well represented by theresolved-scale motions.
• The resolved viscous dissipation is negative definite.
• Due to the removal of the small scales, however, the resolved dissipation is asmall fraction of the total dissipation present in the unfiltered system.
• There is thus a large imbalance between production and dissipation in the filteredsystem.
• The primary objective of the SGS model is to provide sufficient dissipationin order to balance the energy budget.
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SPECTRAL VIEW
Dissipation
uiujSij
τ ijS ij − 2νS ijS ij
S ijS ij−2ν
Flux
Production
1e−10
0.01
1
100
0.01 0.1 1 10 100 1000 10000
Ene
rgy,
E(k
)
Wavenumber, k
1e−06
1e−08
0.0001
The energy flux at the filter scale must equal the total dissipation. Thus
τijSij − 2νSijSij = −2νSijSij
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EDDY VISCOSITY MODEL
Eddy viscosity is the simplest way to ensure a proper energy flux
τij = −νtSij
where νt is the eddy viscosity. Using this model, the energy balance becomes
τijSij − 2νSijSij = −2νSijSij
−2νtSijSij − 2νSijSij = −2νSijSij
(ν + νt)S2 = νS2
Thus we can obtain the correct energy flux in order to balance production anddissipation through the correct specification of the eddy viscosity.
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SMAGORINSKY MODEL
• Smagorinsky (1961) postulated the following prescription for the SGS eddyviscosity:
νt = Cs ∆︸︷︷︸∼l
∆|S|︸︷︷︸∼u
where Cs is a non-dimensional scaling factor, ∆ is the mesh spacing, and |S| =√2S2 is the strain rate magnitude.
• For homogeneous unstratified flows, Cs ' 0.01. For inhomogeneous and stratifiedflows, Cs may be required to vary in space and with the relative stability.
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HEAT FLUX MODEL
• The simplest heat flux model is the eddy diffusivity model
qj = −κt∂θ
∂xj
where κt is the eddy diffusivity.
• It is customary to relate κt to νt via a turbulent Prandtl number
Prt =νt
κtκt =
νt
Prt
• For homogeneous, unstratified flows, Prt ' 1. For inhomogeneous, stratifiedflows Prt may need to vary in space and vary with the relative stability.
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MORE COMPLEX EDDY VISCOSITY MODELS
• TKE model.
– Turbulent kinetic energy based eddy viscosity ala Deardorff (1980), Moeng(1984).
– Stability-corrected mixing length scale.
– Stability-corrected turbulent Prandtl number for heat flux.
• Dynamic Smagorinsky, dynamic heat flux model.
– Parameter-free computation of eddy viscosity and eddy diffusivity.
– No Prandtl number assumption is necessary.
– Effect of stability is accounted for automatically.
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TKE MODEL
τij = −2νtSij
qi = −kt∂θ
∂xi
where the eddy viscosity and eddy diffusivity are computed according to
νt = Ckle1/2
kt =(
1 +2l
∆
)︸ ︷︷ ︸
1/Prt
νt
and where l is the mixing length, e is the subgrid-scale kinetic energy, and ∆ is theLES filter width. The mixing length is computed via
l =
∆ convectively unstable
0.76e1/2“gθ0
∂θ∂z
”1/2 convectively stable
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KINETIC ENERGY TRANSPORT EQUATION(∂
∂t+ uj
∂
∂xj
)e = P + B − ε + D
where
P = −τijSij
B =g
θ0q3
ε = Cεe3/2
l
D =∂
∂xi
(2νt
∂e
∂xi
)The constants are set as follows:
Ck = 0.1 Cε = 0.93
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DYNAMIC MODEL
The dynamic model makes use of scale-similarity ideas in order to estimate SGSstresses from the stresses produced by the smallest resolved length scales.
kk
in order toestimate unresolved stresses
computestresseshere
.
E(k)
k
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SOLUTION FOR Cs
Grid level: ui; τij = uiuj − uiuj ' −2(Cs∆)2|S|Sij
Test level: ui; Tij = uiuj − uiuj ' −2(Cs∆)2|S|Sij
Germano’s identity:
Tij − τij ≡ Lij = uiuj − uiuj︸ ︷︷ ︸computable
Postulate Smagorinsky models at both the test and grid scales, substitute intoGermano’s identity
−2(Cs∆)2(
∆2
∆2|S|Sij − |S|Sij
)︸ ︷︷ ︸
Mij
= Lij −13Lkk
Least squares solution for Cs
(Cs∆)2 = −12〈LijMij〉〈MijMij〉
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SOLUTION FOR Ch
Grid level: θ; qi = uiθ − uiθ ' −(Ch∆)2|S| ∂θ∂xi
Test level: θ; Qi = uiθ − uiθ ' −(Ch∆)2|S| ∂bθ∂xi
Germano’s identity for scalar flux:
Qi − qi ≡ Hi = uiθ − uiθ︸ ︷︷ ︸computable
Postulate gradient diffusion models at both the test and grid scales, substitute intoGermano’s identity
−(Ch∆)2
∆2
∆2|S|
∂θ
∂xi−
|S| ∂θ
∂xi
︸ ︷︷ ︸
Ni
= Hi
Least squares solution for Ch
(Ch∆)2 = −〈HiNi〉〈NiNi〉
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ADVANTAGES OF THE DYNAMIC PROCEDURE
• The model constants Cs and Ch are computed as a function of space and timeas the simulation evolves.
• Stability and the necessary damping near the surface are accounted forautomatically.
• No external inputs are required.
But ...
• The filter scale must be in the inertial range.
• This requirement is almost never met near the surface for the atmosphericboundary layer or at coarse resolution away from the surface.
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NEAR-SURFACE ISSUES
Following Sullivan et al. (1994) we reason as follows:
• In the absence of strong buoyant forcing, the size of the dominant eddies in thenear-surface region scales with the distance from the surface.
unresolvededdies
resolvededdy
w
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NEAR-SURFACE ISSUES
Following Sullivan et al. (1994) we reason as follows:
• In the absence of strong buoyant forcing, the size of the dominant eddies in thenear-surface region scales with the distance from the surface.
• We can not possibly resolve these eddies with conventional LES methods.
• The representation of the flow in the near-surface region is thus more like aReynolds Averaged Navier Stokes (RANS) computation.
• It is therefore sensible to make use of RANS modeling ideas for the near-surfacelayer.
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TWO-PART EDDY VISCOSITY MODEL
Sullivan’s ideas can be realized via a two-part eddy viscosity model
τij = − 2γ(z)νtSij︸ ︷︷ ︸LES part
− 2VT 〈Sij〉︸ ︷︷ ︸RANS part
VT = 2CR∆2|〈S〉|
• The RANS part is active mainly in the near-surface region; 〈S〉 is maximum atthe surface and falls off like 1/z with height.
• In the case of the tke model, the LES part is multiplied by γ(z), an isotropy(damping) function that reduces its influence near the surface. The dynamicmodel does this automatically and thus no isotropy function is used in this case.
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DETERMINATION OF CR
Determine CR such that the mean speed derivative matches with similarity theoryat the first grid point.
Postulate a constant stress layer near the surface. Then the computed total stressat the first grid point should satisfy
[〈τ13〉21 + 〈τ23〉21
]1/2+[〈u′w′〉21 + 〈v′w′〉21
]1/2= u∗
Mean SGS stress, neglecting the fluctuating strain at the first grid point;
〈τ13〉1 ' −2 (〈γνt〉1 + VT 1) 〈∂u
∂z〉1
〈τ23〉1 ' −2 (〈γνt〉1 + VT 1) 〈∂v
∂z〉1
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The surface stress condition then becomes
2 (〈γνt〉1 + VT 1)
[(∂〈u〉1∂z
)2
+(
∂〈v〉1∂z
)2]1/2
+[〈u′w′〉21 + 〈v′w′〉21
]1/2= u∗
Neglect the turning of the mean wind speed at the first grid point:
[(∂〈u〉1∂z
)2
+(
∂〈v〉1∂z
)2]1/2
' ∂Us1
∂z=
u∗φm(z1)kz1
solve for VT 1
VT 1 =u∗kz1
φm(z1)− 〈γνt〉1 −
kz1
u∗φm(z1)[〈u′w′〉21 + 〈v′w′〉21
]1/2
At any other height
VT = VT 1
|〈S〉||〈S〉|1
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BOUNDARY CONDITIONS
dzdw
dxdv= −[ + ]dudy
dudz =0dv
dz,
q3 specifiedu, v, w=0
P specified alaKlemp & Durran1983
θ=θ( t)
13, τ23τ ,
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SURFACE BOUNDARY CONDITIONS
τ13(x, y, z1) = −u2∗
[〈Us〉u′(x, y) + Us(x, y)〈u〉
〈Us〉√〈u〉2 + 〈v〉2
]1
τ23(x, y, z1) = −u2∗
[〈Us〉v′(x, y) + Us(x, y)〈v〉
〈Us〉√〈u〉2 + 〈v〉2
]1
q3(x, y, z1) = −Q∗
[〈Us〉θ′(x, y) + Us(x, y)〈θ − θ0〉
〈Us〉(〈θ〉 − θ0)
]1
where
u′(x, y, z) = u(x, y, z)− 〈u〉, etc. Us(x, y, z) =√
u(x, y, z)2 + v(x, y, z)2
These forms obey the constraints√〈τ13(x, y, z1)〉2 + 〈τ23(x, y, z1)〉2 = u2
∗
〈q3(x, y, z1)〉 = Q∗
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ESTIMATION OF SURFACE FLUXES
Use surface similarity theory:
〈Us〉1 =u∗
k
[log(
z1
z0
)+ βm
(z1
L
)]
〈θ〉1 − θ0 =Q∗
u∗k
[log(
z1
z0
)+ βh
(z1
L
)]Solve for u∗ and Q∗:
u∗ =〈Us〉1k
log(
z1z0
)+ βm
(z1L
)Q∗ =
(〈θ〉1 − θ0)u∗k
log(
z1z0
)+ βh
(z1L
)The constants are set as follows:
βm = 5.0 βh = 5.0 k = 0.4
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KELVIN-HELMHOLTZ TEST PROBLEM
• Simulate a wind shear event in a stable atmosphere Re = 2000, Ri = 0.05.
• Compare the LES results with DNS computed on a much finer (factor of 12 ineach direction) mesh.
• Compare the TKE with the dynamic Smagorinsky model.
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RESULTS - KE DECAY
0
10
20
30
40
50
60
0 50 100 150 200 250 300
Kin
etic
Ene
rgy
Time
Kinetic energy decay, Effect of SGS Model, 60X20X120
TKE Model ResolvedDynamic Model Resolved
Unfiltered DNS
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RESULTS - MEAN STREAMWISE VELOCITY
-4
-3
-2
-1
0
1
2
3
4
-1 0 1 2 3 4 5 6 7
z
<u>
Mean Streamwise Velocity, Effect of SGS Model, 60X20X120
t=103 117 134 142 164 220 290
TKE ModelDynamic Model
DNS
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RESULTS - MEAN POTENTIAL TEMPERATURE
-4
-3
-2
-1
0
1
2
3
4
-1 0 1 2 3 4 5 6 7 8 9 10
z
<T>
Mean Temperature, Effect of SGS Model, 60X20X120
t=103 117 134 142 164 220 290
TKE ModelDynamic Model
DNS
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RESULTS - STREAMWISE VELOCITY FLUCTUATION
-10
-5
0
5
10
0 0.3 0.6 0.9
z
u_rms
U rms, Effect of SGS Model, 60X20X120
t=103 117 134 142 164 220 290
TKE ModelDynamic Model
DNS
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RESULTS - POTENTIAL TEMPERATURE FLUCTUATION
-6
-4
-2
0
2
4
6
0 1.2 2.4 3.6 4.8 6 7.2
z
T_rms
T rms, Effect of SGS Model, 60X20X120
t=103 117 134 142 164 220 290
TKE ModelDynamic Model
DNS
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RESULTS - EDDY VISCOSITY AND DIFFUSIVITY
0
1
2
3
4
5
6
7
8
9
10
100 150 200 250 300
Tur
bule
nt tr
ansp
ort c
oeffi
cien
t
Time
Transport coefficients, Effect of SGS Model, 60X20X120
TKE Model <νt>/νTKE Model <κt>/κ
Dynamic Model <νt>/νDynamic Model <κt>/κ
DNS <νt>/νDNS <κt>/κ
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RESULTS - TURBULENT PRANDTL NUMBER
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
100 150 200 250 300
Tur
bule
nt P
rand
tl nu
mbe
r
Time
Turbulent Prandtl number, Effect of SGS Model, 60X20X120
TKE ModelDynamic Model
DNS
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HOW WELL RESOLVED ARE ATMOSPHERICSIMULATIONS?
mesoscale
scalesynoptic
scaleglobal
process studies
1e−20 0.01 0.1 1 10 100 1000
Ene
rgy
E(k
)
Wavenumber, k
100000
1
1e−05
1e−10
1e−15
0.001
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IS IT REALLY LES?
• Detailed process studies are LES (by construction).
• Mesoscale simulation may barely satisfy the requirements of LES.
• Synoptic scale and larger simulations typically do not resolve the integral scaleand thus violate the underlying ideas of LES (i.e. bulk of the turbulent transportis not resolved).
• Turbulence modeling for large-scale simulations proceed along the same linesbut with lots of ad hoc modifications.
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SUMMARY
• It is impossible to resolve all relevant length scales in turbulent atmosphericflows.
• We apply a spatial filter to eliminate small scales and then account for theireffects through a turbulence model.
• The turbulence model must supply the correct energy flux at the filter cutoff inorder to balance production and dissipation.
• Simple eddy viscosity models can achieve such a balance.
• Many different approaches exist for scaling the eddy viscosity.
• Dynamic modeling is self-calibrating but requires the filter cutoff to be in theinertial range.
• The LES approach is successful if the resolution is adequate and the model iswell calibrated.
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