This slide is the distribution material on the seminar, "Customization of LES turbulence model in OpenFOAM". (June 13 2015 "OpenCAE Local User Group @ Kansai") http://ofbkansai.sakura.ne.jp/

1. Customiza*on
of
LES
turbulence
model
in
OpenFOAM
yotakagi77
Open
CAE
Local
User
Groups
in
Japan
@Kansai
June
13,
2015,
Osaka
University

2. 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

3. Basic
informa*on
on
turbulence
model

4. Turbulent
flow
simula*on
DNS
LES
RANS
Modeling
No
Subgrid
scale
Reynolds
average
Accuracy
◎
○
△
Cost
×
○
◎
Vortex
(eddy)
field
Reynolds
average
4

5. Turbulent
flow
simula*on
DNS
LES
RANS
Modeling
No
Subgrid
scale
Reynolds
average
Accuracy
◎
○
△
Cost
×
○
◎
DNS
grid,
u
Reynolds
average
5

6. Turbulent
flow
simula*on
DNS
LES
RANS
Modeling
No
Subgrid
scale
Reynolds
average
Accuracy
◎
○
△
Cost
×
○
◎
LES
grid,
u
=
u
–
u’
Reynolds
average
6

7. Turbulent
flow
simula*on
DNS
LES
RANS
Modeling
No
Subgrid
scale
Reynolds
average
Accuracy
◎
○
△
Cost
×
○
◎
Filtering
approach
Reynolds
average
7

8. 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)

9. 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

10. 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

11. Significant
problem:
difference
of
filtering
(average)
approaches
LES
Filtering
Reynolds
average
Spa*al
Temporal
Inconsistency
at
the
interface
between
LES
and
RANS
regions
11

12. Standard
SGS
model
in
OpenFOAM
Library
name
Note
Smagorinksy
Smagorinsky
model
Smagorinksy2
Smagorinsky
model
with
3-­‐D
filter
homogeneousDynSmagor
insky
Homogeneous
dynamic
Smagorinsky
model
dynLagragian
Lagrangian
two
equa*on
eddy-­‐viscosity
model
scaleSimilarity
Scale
similarity
model
mixedSmagorinsky
Mixed
Smagorinsky
/
scale
similarity
model
homogeneousDynOneEqE
ddy
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

13. 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

14. Tensor
mathema*cs

15. 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 T13
T21 T22 T23
T31 T32 T33
!
"
#
#
#
$
%
&
&
&

16. 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
∂xj
, Sij =
1
2
∂ui
∂xj
+
∂uj
∂xi
"
#
$$
%
&
'', Ωij =
1
2
∂ui
∂xj
−
∂uj
∂xi
"
#
$$
%
&
''
Dij = Sij +Ωij

17. Opera*ons
exclusive
to
tensors
of
rank
2
T =
1
2
(T+ TT
)+
1
2
(T− TT
) = symmT+skewT,
trT = T11 +T22 +T33,
diagT = (T11,T22,T33),
T = T−
1
3
(trT)I+
1
3
(trT)I = devT+ hydT,
detT =
T11 T12 T13
T21 T22 T23
T31 T32 T33

18. 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)

19. 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)

20. Exercise
1:
Compiling
and
execu*on
of
WALE
model

21. Governing
equa*on
for
incompressible
LES
• Filtered
con*nuity
and
Navier-­‐Stokes
equa*ons
where
∂ui
∂xi
= 0,
∂ui
∂t
+uj
∂ui
∂xj
= −
1
ρ
∂p
∂xi
+
∂
∂xi
(−τij + 2νSij )
τij = uiuj −uiuj

22. 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
∂xj
= τijSij −εGS +
∂
∂xi
−uiτij −
puj
ρ
+ν
∂kGS
∂xj
#
$
%%
&
'
((
k =
1
2
ukuk =
1
2
ukuk +
1
2
(ukuk −ukuk )
kGS
kSGS
∂kSGS
∂t
+uj
∂kSGS
∂xj
= −τijSij −εSGS +
∂
∂xi
uiτij −
1
2
(uiuiuj +uj uiui )−
puj − puj
ρ
+ν
∂kSGS
∂xj
#
$
%
%
&
'
(
(

23. 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
∂xj
∂ui
∂uj
−ν
∂ui
∂xj
∂ui
∂xj
$
%
&&
'
(
)) = −τijSij
τij
a
= −2vSGS Sij
νSGS = (CS Δ)2
| S |, | S |= 2SijSij , CS : Smagorinsky
constant

24. WALE
model
• Traceless
symmetric
part
of
the
square
of
the
velocity
gradient
tensor:
• Eddy
viscosity
of
WALE
model:
Sij
d
=
1
2
(Dij
2
+ Dji
2
)−
1
3
δijDkk
2
= SikSkj +ΩikΩkj −
1
3
δij SmnSmn −ΩmnΩmn
#$ %&
Sij
d
Sij
d
=
1
6
(S2
S2
+Ω2
Ω2
)+
2
3
S2
Ω2
+ 2IVSΩ,
S2
= SijSij, Ω2
= ΩijΩij, IVSΩ = SikSkjΩjlΩli
νSGS = (CwΔ)2
(Sij
d
Sij
d
)3/ 2
(SijSij )5/ 2
+ (Sij
d
Sij
d
)5/ 4
(Nicoud
and
Ducros,
1999)

25. Model
parameters
of
WALE
model
Field
a
Field
b
Field
c
Field
d
Field
e
Field
f
Cw
2/Cs
2
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)

26. 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

27. 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.

28. 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;
...

29. 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.

30. 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";
}

31. 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.

32. 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.

33. Exercise
2:
Implementa*on
of
coherent
structure
Smagorisky
model

34. 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).

35. 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

36. 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 =
1
2
ΩijΩij − SijSij( )= −
1
2
∂uj
∂xi
∂ui
∂xj
Sij =
1
2
∂ui
∂xj
+
∂uj
∂xi
#
$
%%
&
'
((, Ωij =
1
2
∂ui
∂xj
−
∂uj
∂xi
#
$
%%
&
'
((

37. 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 =
Q
E
€
τij
a
= −2CΔ2
| S | Sij
(τij
a
= −2νt Sij, νt = CΔ2
| S |)
E =
1
2
ΩijΩij + SijSij( )=
1
2
∂ui
∂xj
$
%
&&
'
(
))
2

38. 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 =
1
22
, FCS =
Q
E
, FΩ =1− FCS
€
τij
a
= −2CΔ2
| S | Sij
(τij
a
= −2νt Sij, νt = CΔ2
| S |)
E =
1
2
ΩijΩij + SijSij( )=
1
2
∂ui
∂xj
$
%
&&
'
(
))
2

39. 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

40. 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
&

41. 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

42. 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

43. 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

44. 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

45. 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).

46. 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;
...

47. 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.

48. 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.

49. 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).

50. 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).