Post on 10-Aug-2020
Large Eddy Simulation
Turbulence Modellingk-ε Models- Short computational time- Simple, robust- Limited range of validity
Reynolds Stress Models, RSM- More general, still not universal- More complex, seven PDE:s- Extensive modeling- Longer computational time
Large Eddy Simulation, LES- Resolving large scales- Modeling small scale effects- Long computational time
Direct Numerical Simulation, DNS- No Model, resolves all scales - Very long computational time- Limited to low Re
Approximations
The conceptual steps
• Filtering• Govering equations including Sub-Grid
Scale (residual) stress tensor• Closure by modelling SGS stress
tensor• Numerical solution
∫= ')'()',( dxxuxxGu
Computational Grid
u u
• Spatial filtering rather than ensemble average
Large-Eddy Simulation for Turbulent Flows
Turbulent kinetic energy spectrum
(Kolmogorov theory for isotropic & homogenous turbulence)
log(
E(k)
)
log(k)
Dissipationsubrange
Inertialsubrange
Largescales
Universal range
-5/3
log(∆)
Filter cutoff
LES-spectrum
−∆
∆= rHrG
211)(
Box filter
∆
∆
=κ
κκ
21
21sin
)(Ĝ
Gaussian filter
∆
−∆
= 22
2/12
6exp)6()( rrGπ
∆−=
24exp)(ˆ
22κrG
Sharp spectral filter
( )rrrGππ ∆
=/sin)(
−∆∆
= κπκ HG 1)(ˆ
Filtering in 3D
{2/02//1
´)(3
∆>∆≤∆
=−i
i
xforxfor
zzG”Box” filter
−
∆−
∆=− 22
2/32 ´
6exp)6(´)( zzzzGπ
”Gaussian” filter
Filtered governing equations
0=i
i
xu∂∂
jj
i
ij
jii
xxu
xp
xuu
tu
∂∂∂ν
∂∂
ρ∂∂
∂∂
+−=+1
Mass:
Momentum
Now consider the non-linear term
Filtered governing equations
Momentum ( )j
ij
jj
i
iji
j
i
xxxu
xpuu
xtu
∂∂τ
∂∂∂ν
∂∂
ρ∂∂
∂∂
−+−=+1
jijiij uuuu −=τSub-grid stresses:
( ) ( )jijij
jijj
ji uuuux
uuxx
uu−+=
∂∂
∂∂
∂∂
Leonards decomposition
jijiij uuuu −=τTurbulent stresses
Leonard’s decomposition ( )( ) jiijjijijjiiji uuuuuuuuuuuuuu ′′+′+′+=′+′+=
ijijijjijiij RCLuuuu ++=−=τ
jijiij uuuuL −=
ijjiij uuuuC ′−′=
jiij uuR ′′=
Leonard tensor
Cross-term tensor
Reynolds stresses
Filtered governing equations
Momentum
j
ij
jj
i
ij
ij
i
xxxu
xp
xuu
tu
∂∂τ
∂∂∂ν
∂∂
ρ∂∂
∂∂
−+−=+1
Leonard stresses:
( ) ( ) ( )jijij
jij
jij
uuuux
uux
uux
−+=∂∂
∂∂
∂∂
jijiij uuuuL −=
( )j
ji
j
ijji
j xuu
xuuuu
x ∂∂
∂∂
∂∂
+=
jijiij uuuu −=τSub-grid stresses:
Germano’s decomposition
jijioij uuuuL −=
ijjiijjioij uuuuuuuuC ′−′−′−′=
jijioij uuuuR ′′−′′=
Leonard tensor
Cross-term tensor
Reynolds stresses
Interaction among resolved scales
Interaction between resolved and non-resolved scales
Interaction among non-resolved scales
Governing Equations
LES modelling of non-linearities:
“Convective”: a. Gradient diffusion type (Smagorinsky)b. Scale similarity typec. Dynamic typesd. Others
Examples for incompressible SGS models
Governing Equations - LES
The Space Filtered Governing Equations
Transport Equation for an Inert Additive
j
j
jjj
j
xxxcD
xcu
tc
∂∂
−∂∂
∂=∂
∂+∂∂ ϕρρ 2
Subgrid Scale Mixing )( cucu jjj −= ρϕ
A note on kinetic energy
rfijijijjij
fj
f PpSuxx
Eu
tE
−−=
−−
∂∂
−∂∂
+∂∂
εδρ
τν2
ijijf SSνε 2=
ijijr SP τ−=
iiuuE 21
=
rf kEE +=
iif uuE 21
=
Filtered kinetic energy
Decomposition
Kinetic energy of the resolved scales
Small if η>>∆
Can be negative
Subgrid Scale Models
What do they do?
1. Account for the effects of small scales on the resolved ones
1. Dissipate the energy transferred to the smallest scales
Subgrid Scale Models• Subgrid Scale Stress Tensor
- No explicit model- Smagorinsky (“Eddy diffusivity”)- Stress Similarity Model- Dynamic Divergence Model- Exact Differential Model
• Subgrid Scale Mixing- No Explicit Model- Eddy Diffusivity type- Scale Similarity Model - Exact Differential Model
Subgrid Scale ModelsSubgrid Scale Stress Tensor
Smagorinsky
SSM
DDM
EDM
)( jijiij uuuu −= ρτ
( )jijiLkkijij uuuuC −−= ρτδτ 31
jijm
jkkij SSSC ,2
1
klkl2)(
,jij, ) 2(23
1 ∆−
= ρτδτ
)(qfq =
Liu, S., Meneveau, C. And Katz, J., J. Fluid Mech., 1994.
Held, J and Fuchs, L., AIAA paper, 97-1931, 1997.
Fuchs, L., Kluwer Academic Publishers, 1996.
)(21;)(
31 2/1
i
j
j
iijijklklLkkijij x
uxuSSSSC
∂∂
+∂∂
=−= ρτδτ
SGS Model
4
43
3
1
||41
j
ij
j
jxuxu
∂∂
∆− ∑=
ρ
No explicit model: an example
The largest term in the truncation error of the third order upwind scheme proposed by Kawamura and Kuwahara
Kawamura, T. and Kuwahara, K., “Computation of High Reynolds Number Flow around a Circular Cylinder with Surface Roughness”, AIAA-84-0340, 1984.
IMM
Advantages
+ Simple and fast, (no explicit SGS Model used) + Correct asymptotic behavior (h->0, LES->DNS)
Disadvantages
- Not physically related
SGS Model
)(21;)(
31 2/1
i
j
j
iijijklklLkkijij x
uxuSSSSC
∂∂
+∂∂
=−= ρτδτ
Smagorinsky
Momentum:
Scalar
jcj x
CA∂∂
= ρϕ
)( cucu jjj −= ρϕ
)( jijiij uuuu −= ρτ
Smagorinsky
Advantages
+ Simple + Dissipative
Disadvantages
- A “free” model parameter: Not universal- Does not account for “back-scatter”- Near wall treatment
SGS Model
( )131
=
−−=
L
jijiLkkijij
C
uuuuCρτδτ
The Stress Similarity Model (SSM)
The Liu et al. type of model
(SSMC)
1)(
=
−=
c
jjcj
CcucuCρϕ
SSM
Advantages
+ Simple + Correct asymptotic behavior in the inertial subrange+ Accounts for some “back-scatter”
Disadvantages
- A “free” model parameter - Requires an additional filter- Not absolutely dissipative
SGS ModelThe Dynamic Smagorinsky Model (Germano’s model)
) 2(2 31 2
1
klkl2
ijij ijijkkij SSSC ∆−=+
= ραατδτ
S)2(2 31
ij21
2ij klklijkkijij SSCTT ∆−=+
= ρββδ
mnmnklkl
ijijg
jijiijijij
SSSL
C
uuuuTL
αβ
τ
−=
−=−= )(
Sub-grid scale stresses:
Sub-test scale stresses:
Germano’s identity
ijklklgkkijij SSSC2/12 )(
31
∆−= ρτδτ
StressTurbulent Resolved = Lij
SGS ModelThe Dynamic Smagorinsky Model (Germano’s model)
mnmnklkl
ijijg
jijiijijij
SSSL
C
uuuuTL
αβ
τ
−=
−=−= )(Germano’s identity
StressTurbulent Resolved = Lij
Gives 6 values of the model parameter, choose by minimizing the error:
( )( )2ijijij CCLQ αβ −−=
( )( )( )mnmnmnmn
ijijijg
LC
αβαβαβ
−−−
=
Germano’s model
Advantages
+ No “free” model parameter+ Correct asymptotic behavior (h->0, LES->DNS) Disadvantages
- Requires an additional filter- May require limiting of the model parameters- Isotropic coefficient
SGS ModelThe Dynamic Divergence Model (DDM)
jijj
kkij C ,,
jij, 31 ατδτ +
=
31
,,
, jijj
kkijjij CTT βδ +
=
( )
lilkik
jiji
jjijijijjijjij
LC
uuuuTL
,,
,
,,,, )(
αβ
τ
−=
−=−=
Sub-grid scale stresses:
Sub-test scale stresses:
One model parameter for each direction
Model directly the divergence of the SGS stresses:j
ijjij x∂
∂=
ττ ,
SGS ModelThe Dynamic Divergence Model (DDM)
Bounding of the model parameter
Total dissipation must be positive 0≥++ numSGSvisc εεε
( )
∂∂
+
∆
=≥
j
iijklkl
i
xuS
h
SSCC
3
21
2min
)2(2
1 ν
( )
−
∆∆
=≤ νt
h
SSCC
klkl
i2
21
2max
)2(2
1
DDM
Advantages
+ No “free” model parameter+ Correct asymptotic behavior (h->0, LES->DNS) + Non-isotropic coefficients
Disadvantages
- Requires an additional filter- May require limiting of the model parameters
SGS ModelKinetic energy SGS model
Subgrid kinetic energy: ( )kkkksgs uuuuk −= 21
The SGS-stresses: ijsgskijsgsij SkCk
T
ν
δτ 2/1)(232
∆−=−
∂∂
∂∂
+∆
−∂∂
−=∂∂
+∂
∂
j
sgs
k
T
j
sgs
j
iij
j
sgsj
sgs
xk
xk
Cxu
xk
ut
kσντ ε
2/3
Accounts for the transport of SGS turbulent kinetic energy
SGS Model
The Exact Differential Model (EDM)
∫ −= ξξξ dqxGq )()(
We look for filters (G) such that the “error” of filtering becomes:
qDPqq )(−=−
Where P(D) is a polynomial of differentials.
(1)
(2)
SGS Model
Fourier transform ( ) relations (1) and (2):
qikPqqandqGq ˆ)(ˆˆˆˆˆ −=−=
We look for filters (G) such that the “error” of filtering becomes:
*̂
)(11ˆ
ikPG
−=
For realizability require that P is an even polynomial.
To lowest order (P(D)= ∆) one gets:
SGS Model
~
~
~~
~
~ tu~
=q~ where ~
2
22i2
j,ij
2
22222
−+∆+++∆−=
=∇∆−=∇∆−=
j
i
ij
ji
j
ji
j
ji
j
xu
xp
xuu
xuu
xuu
xqqqqqqq
∂∂
µ∂∂
∂ρ∂
∂ρ∂
∂ρ∂
∂ρ∂
τ
∂∂
The Exact Differential Model (EDM)
An additional term enters into the continuity equation
0~
2 =∂∂
∆−∂∂
k
k
i
i
xu
xu
EDMAdvantages
+ Exact expression for the SGS term + No model parameter+ Enhanced understanding of SGS roles + Can estimate the contribution of each term + Correct asymptotic behavior (h/D ->0 for all flows!
h = cell size and D = filter width)
Disadvantages
- Additional (model dependent) boundary conditions- Boundary conditions have to be computed implicitly - Direct use may be numerically unstable
Issues: Inflow/outflow boundary conditions
Handling walls and near wall effects
Accuracy
Modelling of Turbulent Flows
Model comparisonsImpinging jet
Impinging Jet
Impinging Jet
Impinging Jet
EXAMPLES
LES of turbulent flows
Results
Concentration
Velocity
Area shown in the animation
Results
Instantaneous velocity field in the wall jet.
Instantaneous concentration field.
Impinging jets
Impinging Jets: Scalar Mode 0,1,2
Largest variance (instead of energy)
Asymmetric
Impinging Jets
Mode 0,1,2 (the most energetic)
Coherent vortices
T
vu
vu
R
=
Vector valued u:
49
Unsteady Impinging Jet
Basic case H/D=2, S=0, Re=20000, LES
x
y
z
Free jet region
Wall jet region
Stagnation region
50
Centreline (x/D=0) frequency spectra u-vel.
Unsteady Impinging Jet
(-): y/D=0.004
(--): y/D=1
St2
St1
• Vortex shedding at x/D=0.5
• St1: fundamental freq.
• St2: sub-harmonic freq
• Spectra normalized with local RMS and energy levels
• Initially inviscid instabilities (inflection point, laminar flow + perturbations)
• Rolling up of vortices, growing downstream
• Vortex shedding, vortex pairing
LES Case1 (H/D=2, Re=20000, S=0)
Space/frequency relation
St2 St1 St2
St1
Energy levels of St1 and St2Dominant frequencies at x/D=0.5
y / D
St E
Thomas Hällqvist
LES (H/D=2, Re=20000, S=0,1)
Space/frequency relation
St2 St1
Dominant frequencies at x/D=0.5
y / D
Thomas Hällqvist
Bildnummer 1Turbulence Modelling The conceptual stepsBildnummer 4Bildnummer 5Bildnummer 6Bildnummer 7Bildnummer 8Bildnummer 9Bildnummer 10Bildnummer 11Bildnummer 12Bildnummer 13Bildnummer 14Bildnummer 15Bildnummer 16Bildnummer 17Bildnummer 18Bildnummer 19Bildnummer 20Bildnummer 21Bildnummer 22Bildnummer 23Bildnummer 24Bildnummer 25Bildnummer 26Bildnummer 27Bildnummer 28Bildnummer 29Bildnummer 30Bildnummer 31Bildnummer 32Bildnummer 33Bildnummer 34Bildnummer 35Bildnummer 36Bildnummer 37Modelling of Turbulent FlowsModel comparisonsImpinging JetImpinging JetImpinging JetBildnummer 43ResultsResultsBildnummer 46Bildnummer 47Bildnummer 48Bildnummer 49Bildnummer 50Bildnummer 51Bildnummer 52