customization of les turbulence model in openfoam
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Customiza*on of LES turbulence model in OpenFOAM
yotakagi77 Open CAE Local User Groups in Japan
@Kansai June 13, 2015, Osaka University
Agenda
• Basic informa*on on turbulence model • Tensor mathema*cs • Exercise 1: Compiling and execu*on of WALE model
• Exercise 2: Implementa*on of coherent structure Smagorisky model
• Addi*onal works
Basic informa*on on turbulence model
Turbulent flow simula*on
DNS LES RANS Modeling No Subgrid scale Reynolds average Accuracy ◎ ○ △
Cost × ○ ◎
Vortex (eddy) field Reynolds average 4
Turbulent flow simula*on
DNS LES RANS Modeling No Subgrid scale Reynolds average Accuracy ◎ ○ △
Cost × ○ ◎
DNS grid, u Reynolds average 5
Turbulent flow simula*on
DNS LES RANS Modeling No Subgrid scale Reynolds average Accuracy ◎ ○ △
Cost × ○ ◎
LES grid, u = u – u’ Reynolds average 6
Turbulent flow simula*on
DNS LES RANS Modeling No Subgrid scale Reynolds average Accuracy ◎ ○ △
Cost × ○ ◎
Filtering approach Reynolds average 7
Detached-‐eddy simula*on (DES) • P. R. Spalart (1997):
– We name the new approach “Detached-‐Eddy Simula8on” (DES) to emphasize its dis8nct treatments of a?ached and separated regions.
Super-‐Region Region
Euler (ER)
RANS (RR) Viscous (VR)
Outer (OR)
LES (LR) Viscous (VR)
Focus (FR)
Departure (DR) 8
Spalart (2001)
Detached-‐eddy simula*on (DES) • P. R. Spalart (1997):
– We name the new approach “Detached-‐Eddy Simula8on” (DES) to emphasize its dis8nct treatments of a?ached and separated regions.
Super-‐Region Region
Euler (ER)
RANS (RR) Viscous (VR)
Outer (OR)
LES (LR) Viscous (VR)
Focus (FR)
Departure (DR)
Spalart (2001)
9
Coupling with momentum equa*on through viscosity
• RANS
• LES
€
∂U∂t
+∇⋅ UU( ) −∇⋅ ν+ νt( ) ∇U + (∇U)T( )( ) =∇p
€
∂U∂t
+∇⋅ UU( ) −∇⋅ ν+ νSGS( ) ∇U + (∇U)T( )( ) =∇p
Turbulent viscosity
Sub-grid scale viscosity
Only change viscosity!
10
Significant problem: difference of filtering (average) approaches
LES Filtering Reynolds average
Spa*al Temporal
Inconsistency at the interface between LES and RANS regions 11
Standard SGS model in OpenFOAM
Library name Note
Smagorinksy Smagorinsky model
Smagorinksy2 Smagorinsky model with 3-‐D filter
homogeneousDynSmagorinsky
Homogeneous dynamic Smagorinsky model
dynLagragian Lagrangian two equa*on eddy-‐viscosity model
scaleSimilarity Scale similarity model
mixedSmagorinsky Mixed Smagorinsky / scale similarity model
homogeneousDynOneEqEddy
One Equa*on Eddy Viscosity Model for incompressible flows
laminar Simply returns laminar proper*es
kOmegaSSTSAS k-‐ω SST scale adap*ve simula*on (SAS) model
Standard SGS model in OpenFOAM
Library name Note
oneEqEddy k-‐equa*on eddy-‐viscosity model
dynOneEqEddy Dynamic k-‐equa*on eddy-‐viscosity model
spectEddyVisc Spectral eddy viscosity model
LRDDiffStress LRR differen*al stress model
DeardorffDiffStress Deardorff differen*al stress model
SpalartAllmaras Spalart-‐Allmaras model
SpalartAllmarasDDES Spalart-‐Allmaras delayed detached eddy simula*on (DDES) model
SpalartAllmarasIDDES Spalart-‐Allmaras improved DDES (IDDES) model
vanDriestDelta Simple cube-‐root of cell volume delta used in incompressible LES models
Tensor mathema*cs
Tensor
• Rank 0: ‘scalar’, e.g. volume V, pressure p. • Rank 1: ‘vector’, e.g. velocity vector u, surface vector S. Descrip*on: a = ai = (a1, a2, a3).
• Rank 2: ‘tensor’, e.g. strain rate tensor Sij, rota*on tensor Ωij.
Descrip*on:
T = Tij =T11 T12 T13T21 T22 T23T31 T32 T33
!
"
###
$
%
&&&
Symmetric/an*symmetric tensor
• Velocity gradient tensor is decomposed into strain rate tensor (symmetric) and vor*city tensor (an*symmetric, skew).
• In turbulence modeling, Sij and Ωij are usually used.
Dij =∂ui∂x j, Sij =
12∂ui∂x j
+∂uj∂xi
"
#$$
%
&'',Ωij =
12∂ui∂x j
−∂uj∂xi
"
#$$
%
&''
Dij = Sij +Ωij
Opera*ons exclusive to tensors of rank 2
T = 12(T+TT )+ 1
2(T−TT ) = symmT+ skewT,
trT = T11 +T22 +T33,diagT = (T11,T22,T33),
T = T− 13(trT)I+ 1
3(trT)I = devT+ hydT,
detT =T11T12 T13T21T22 T23T31T32 T33
OpenFOAM tensor classes Opera2on Mathema2cal Class
Addi*on a + b a + b
Subtrac*on a – b a – b
Scalar mul*plica*on sa s * a
Scalar division a / s a / s
Outer product a b a * b
Inner product a ·∙ b a & b
Double inner product a : b a && b
Cross product a × b a ^ b
Square a2 sqr(a)
Magnitude squared |a|2 magSqr(a)
Magnitude |a| mag(a)
Power an pow(a, n)
OpenFOAM tensor classes Opera2on Mathema2cal Class
Transpose TT T.T()
Diagonal diag T diag(T)
Trace tr T tr(T)
Deviatoric component dev T dev(T)
Symmetric component symm T symm(T)
Skew-‐symmetric component
skew T skew(T)
Determinant det T det(T)
Cofactors cof T cof(T)
Inverse inv T inv(T)
Exercise 1: Compiling and execu*on of WALE model
Governing equa*on for incompressible LES
• Filtered con*nuity and Navier-‐Stokes equa*ons
where
∂ui∂xi
= 0,
∂ui∂t
+uj∂ui∂x j
= −1ρ∂p∂xi
+∂∂xi(−τ ij + 2νSij )
τ ij = uiuj −uiuj
SGS eddy viscosity model
• Decomposi*on of kine*c energy
• Conserva*on of GS energy kGS
• Conserva*on of SGS energy kSGS
∂kGS∂t
+uj∂kGS∂x j
= τ ijSij −εGS +∂∂xi
−uiτ ij −pujρ
+ν∂kGS∂x j
#
$%%
&
'((
k = 12ukuk =
12ukuk +
12(ukuk −ukuk )
kGS kSGS
∂kSGS∂t
+uj∂kSGS∂x j
= −τ ijSij −εSGS +∂∂xi
uiτ ij −12(uiuiuj +uj uiui )−
puj − pujρ
+ν∂kSGS∂x j
#
$%%
&
'((
Smagorinsky model • Local equilibrium between SGS produc*on rate and SGS energy dissipa*on:
• Eddy viscosity approxima*on:
• Aper dimensional analysis and scaling,
εSGS ≡ν∂ui∂x j
∂ui∂uj
−ν∂ui∂x j
∂ui∂x j
$
%&&
'
())= −τ ijSij
€
τija = −2vSGS S ij
€
νSGS = (CSΔ)2 | S |, | S |= 2S ijS ij , CS : Smagorinsky constant
WALE model
• Traceless symmetric part of the square of the velocity gradient tensor:
• Eddy viscosity of WALE model:
Sijd =12(Dij
2 +Dji2 )− 1
3δijDkk
2
= SikSkj +ΩikΩkj −13δij SmnSmn −ΩmnΩmn#$ %&
SijdSij
d =16(S2S2 +Ω2Ω2 )+ 2
3S2Ω2 + 2IVSΩ,
S2 = SijSij,Ω2 =ΩijΩij, IVSΩ = SikSkjΩ jlΩli
€
νSGS = (CwΔ)2 (SijdSij
d )3 / 2
(S ijS ij )5 / 2 + (Sij
dSijd )5 / 4
(Nicoud and Ducros, 1999)
Model parameters of WALE model
Field a Field b Field c Field d Field e Field f
Cw2/Cs2 10.81 10.52 10.84 10.55 10.70 11.27
If CS = 0.18, 0.55≤CW ≤ 0.6.If CS = 0.1, 0.32 ≤CW ≤ 0.34.
Model parameter Cw is dependent on Smagorinsky constant CS.
(Nicoud and Ducros, 1999)
Source code of WALE model
• V&V working group, Open CAE Society of Japan hsps://github.com/opencae/VandV/tree/master/
OpenFOAM/2.2.x/src/libraries/incompressibleWALE
• OpenFOAM-‐dev hsps://github.com/OpenFOAM/OpenFOAM-‐dev/tree/
master/src/TurbulenceModels/turbulenceModels/LES/WALE
Download and compile
$ mkdir –p $FOAM_RUN $ cd $ git clone https://github.com/opencae/VandV $ cd VandV/OpenFOAM/OpenFOAM-‐BenchmarkTest/
channelReTau110 $ cp –r src $FOAM_RUN/.. $ run $ cd ../src/libraries/incompressibleWALE $ wmake libso $ ls $FOAM_USER_LIBBIN
1. Download the source code of WALE model from the V&V repository, and compile the WALE model library.
Simula*on of channel flow
$ run $ cp –r $FOAM_TUTORIALS/incompressible/pimpleFoam/
channel395/ ./ReTau395WALE $ cd ReTau395WALE
2. The standard tutorial case of channel flow at Reτ = 395: copy the tutorial case file into your run directory.
3. Edit constant/LESProper*es and system/controlDict $ gedit constant/LESProperties
LESModel WALE; printCoeffs on; delta cubeRootVol; ...
Simula*on of channel flow $ gedit system/controlDict
... libs ("libincompressibleWALE.so");
This line is necessary to call the new WALE library in solver.
$ ./Allrun
4. Aper checking other numerical condi*ons and parameter, run the solver.
5. If the solver calcula*on is normally finished, you check the logs and visualize the flow field with ParaView, and plot the fields profile generated by postChannel.
Simula*on of channel flow at Reτ = 110
$ run $ cp –r ~/VandV/OpenFOAM/OpenFOAM-‐BenchmarkTest/
channelReTau110/template $FOAM_RUN/ReTau110WALE $ cd ReTau110WALE $ gedit caseSettings
6. If you use the test case of channel flow supplied in the V&V repository, copy the template case and edit the seung.
controlDict { deltaT 0.002; endTime 0.022; libs "libincompressibleWALE.so"; }
Simula*on of channel flow at Reτ = 110 turbulenceProperties { simulationType LESModel; } LESProperties { LESModel WALE; delta cubeRootVol; }
The original caseSeungs is for DNS simula*on on large parallel machine. You had beser to change other parameters in blockMeshDict and decomposeParDict.
Simula*on of channel flow at Reτ = 110
$ ./Allrun
7. Aper checking other numerical condi*ons and parameter, run the solver.
8. If the solver calcula*on is normally finished, you check the logs and visualize the flow field with ParaView. If the integra*on *me is not sufficient for the flow field to become fully developed state, run longer simula*ons.
Exercise 2: Implementa*on of coherent structure Smagorisky model
Original source codes for SGS model
$ src $ cd turbulenceModels/incompressible/LES/ $ ls
1. Check the original source code for SGS model.
2. Glance the codes of Smagorinsky model. $ gedit Smagorinsky/Smagorinsky.*
3. In this exercise, we look the codes of dynamic models. $ ls *[Dd]yn*
4. Compare the structures and statements of the related codes (*.C and *.H).
Private member func*ons: updateSubGridScaleFields
void Smagorinsky::updateSubGridScaleFields (const volTensorField& gradU) { nuSgs_ = ck_*delta()*sqrt(k(gradU)); nuSgs_.correctBoundaryConditions(); }
void dynLagrangian::updateSubGridScaleFields (const tmp<volTensorField>& gradU) { nuSgs_ = (flm_/fmm_)*sqr(delta())*mag(dev(symm(gradU))); nuSgs_.correctBoundaryConditions(); }
void dynOneEqEddy::updateSubGridScaleFields ( const volSymmTensorField& D, const volScalarField& KK ) { nuSgs_ = ck(D, KK)*sqrt(k_)*delta(); nuSgs_.correctBoundaryConditions(); }
In Smagorinsky.C
In dynLagrangian.C
In dynOneEqEddy.C
Understanding formula*on with codes
• What calcula*on, mathema*cal opera*on, and variable are necessary for coherent structure Smagosinsky model (CSM)? Compare the formula*on of models with the related source codes.
• In CSM, the second invariant of velocity gradient is used:
where
€
Q =12Ω ijΩ ij − S ijS ij( ) = −
12∂u j∂xi
∂u i∂x j
€
Sij =12∂u i∂x j
+∂u j∂xi
#
$ % %
&
' ( ( ,Ωij =
12∂u i∂x j
−∂u j∂xi
#
$ % %
&
' ( (
Coherent structure Smagorinsky model for non-‐rota*ng flow (NRCSM)
• Smagorinsky model (SM) based on an eddy-‐viscosity,
• The model parameter C is determined as follows:
with
where NRCSM model is invalid for rota*ng flow.
€
C = C1 | FCS |3 / 2
€
C1 =1
20, FCS =
QE
€
τija = −2CΔ2 | S | S ij
(τija = −2νtS ij , νt = CΔ2 | S |)
€
E =12Ω ijΩ ij + S ijS ij( ) =
12∂u i∂x j
$
% & &
'
( ) )
2
Coherent structure Smagorinsky model (CSM)
• Smagorinsky model (SM) based on an eddy-‐viscosity,
• The model parameter C is determined as follows:
with
where Improved CSM model is valid for rota*ng flow.
€
C = C2 | FCS |3 / 2 FΩ
€
C2 =122
, FCS =QE
, FΩ =1− FCS
€
τija = −2CΔ2 | S | S ij
(τija = −2νtS ij , νt = CΔ2 | S |)
€
E =12Ω ijΩ ij + S ijS ij( ) =
12∂u i∂x j
$
% & &
'
( ) )
2
Seung for making new library
$ run $ cd ../src/libraries $ cp -‐r incompressibleWALE/WALE/ ./NRCSM $ cp –r incompressibleWALE/Make ./NRCSM $ cd NRCSM $ rename WALE NRCSM * $ rm –r NRCSM.dep $ rm –rf Make/linux64Gcc47DPOpt $ gedit Make/files
1. Copy the source code of WALE model. Compile them.
NRCSM.C LIB = $(FOAM_USER_LIBBIN)/libNRCSM
$ sed –i ‘s/WALE/NRCSM/g’ NRCSM.C $ sed –i ‘s/WALE/NRCSM/g’ NRCSM.H
Seung for making new library $ wmake libso $ ls $FOAM_USER_LIBBIN
If you find the renamed and recompiled library (libNRCSM.so), you are ready to make a new library for the NRCSM. 2. You can easily learn the codes for calcula*ng the Q and E terms from the postProcessing u*li*es. There are two ways of calcula*ng Q, that is, with velocity gradient tensor and with SS and ΩΩ terms.
$ util $ cd postProcessing/velocityField/Q $ gedit Q.C &
Introducing model coefficient C1
$ run $ cd ../src/libraries/NRCSM/ $ gedit NRCSM.C NRCSM.H
3. Replace all ‘cw’ with ‘c1’ (gedit or sed), and change the value to 0.05.
c1_ ( dimensioned<scalar>::lookupOrAddToDict ( "c1", coeffDict_, 0.05 ) )
NRCSM.C
$ wmake libso
Q and E calcula*ons
4. In NRCSM.C, insert the calcula*on of Q and E. Copy&Paste the corresponding sec*on from Q.C. Save and Compile them.
volScalarField Q ( 0.5*(sqr(tr(gradU)) -‐ tr(((gradU)&(gradU))))
); volScalarField E ( 0.5*(gradU && gradU)
);
NRCSM.C $ gedit NRCSM.C
$ wmake libso
FCS and C calcula*ons
5. In NRCSM.C, insert the calcula*on of FCS and C (coefficient of eddy viscosity model). Save and Compile them.
volScalarField Fcs ( Q/max(E,dimensionedScalar("SMALL",E.dimensions(),SMALL))
); volScalarField ccsm_ ( c1_*pow(mag(Fcs),1.5) );
NRCSM.C
$ gedit NRCSM.C
$ wmake libso
νSGS calcula*on
6. In NRCSM.C, modify the nuSGS_ calcula*on. Look the other updateSubGridScaleFields func*ons in the dynamic models.
nuSgs_ = ccsm_*sqr(delta())*mag(dev(symm(gradU))); NRCSM.C
Save and compile them. $ wmake libso
7. Finally, comment out or delete unnecessary statements (the calcula*ons for WALE model). Save and compile them.
$ wmake libso
$ gedit NRCSM.C
kSGS calcula*on
//-‐ Return SGS kinetic energy // calculated from the given velocity gradient tmp<volScalarField> k(const tmp<volTensorField>& gradU) const { return (2.0*c1_/ce_)*sqr(delta())*magSqr(dev(symm(gradU)));
}
NRCSM.H
8. The calcula*on of kSGS is invalid, but the value of kSGS is not actually used in LES with NRCSM model. If you requires a proper kSGS, consult the paper of Kobayashi (PoF, 2005).
Valida*on with channel flow
$ run $ cp –r $FOAM_TUTORIALS/incompressible/pimpleFoam/
channel395/ ./ReTau395NRCSM $ cd ReTau395NRCSM
9. The standard tutorial case of channel flow at Reτ = 395: copy the tutorial case file into your run directory.
10. Edit constant/LESProper*es and system/controlDict $ gedit constant/LESProperties
LESModel NRCSM; printCoeffs on; delta cubeRootVol; ...
Valida*on with channel flow
$ gedit system/controlDict
... libs ("libNRCSM.so");
This line is necessary to call the new NRCSM library in solver.
$ ./Allrun
11. Aper checking other numerical condi*ons and parameter, run the solver.
12. If the solver calcula*on is normally finished, you check the logs and visualize the flow field with ParaView, and plot the fields profile generated by postChannel.
Addi*onal works
1. Compile and test the WALE model supplied from openfoam-‐dev. Prepare a Make directory by yourself.
2. Implementa*on of CSM model. Add FΩ term and C2 coefficient.
3. Calcula*on of Q and E terms with SS and ΩΩ terms. Compare the results with the solu*on of Exercise 2.
4. Valida*on of customized model with other flow fields such pipe, backstep, cylinder, and rota*ng flow.
References • OpenFOAM User Guide • OpenFOAM Programmer’s Guide • 梶島, 乱流の数値シミュレーション 改訂版, 養賢堂 (2014). • P. R. Spalart et al., “Comments on the Feasibility of LES for
Wings, and on a Hybrid RANS/LES Approach”, 1st ASOSR CONFERENCE on DNS/LES (1997).
• P. R. Spalart, “Young-‐Person’s Guide to Detached-‐Eddy Simula*on Grids”, NASA CR-‐2001-‐211032 (2001).
• F. Nicoud and F. Ducros, “Subgrid-‐scale modelling based on the square of velocity gradient tensor”, Flow, Turbulence and Combus*on, 62, pp.183-‐200 (1999).
References • 小林, “乱流構造に基づくサブグリッドスケールモデルの開
発”, ながれ, 29, pp.157-‐160 (2010). • H. Kobayashi, “The subgrid-‐scale models based on coherent
structures for rota*ng homogeneous turbulence and turbulent channel flow”, Phys. Fluids, 17, 045104 (2005).
• H. Kobayashi, F. Ham and X. Wu, “Applica*on of a local SGS model based on coherent structures to complex geometries”, Int. J. Heat Fluid Flow, 29, pp.640-‐653 (2008).