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Large Eddy Simulation, DynamicModel, and Applications
Charles Meneveau
Department of Mechanical EngineeringCenter for Environmental and Applied Fluid Mechanics
Johns Hopkins University
Mechanical Engineering
Turbulence Summer SchoolMay 2010
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Turbulence modeling:
Large-eddy-simulation (LES)
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!
!
G(x): Filter
Large-eddy-simulation (LES) and fil tering:
N-S equations:
! !u j!
t
+ !u k ! !u j! x k
= " 1
#
! ! p! x j
+ $ %2 !u j "
!
! x k
& jk
Filtered N-S equations:
! u j! t
+! u k u j
! x k = "
1
#
! p! x j
+ $ %2u j
! !u j! t
+! u k u j"
! x k = "
1
#
! ! p! x j
+ $ %2 !u j
where SGS stress tensor is:
! ij = uiu j!
" "ui "
u j
! u j! x j
= 0
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!
!
G(x): Filter
Energetics (kinetic energy):
! 12 u j u j! t
+ u k
! 12 u j u j! x k
= " !
! x j....
( )" 2 #
S jk
S jk " " $
jk
S jk ( )
!"
= # $ jk S jk
Inertial-range flux
Effects of ij upon resolved motions:
E(k)
k LES resolved SGS
!"
!
"
! =
"u 3
L
!S ij =1
2
! !u i! x j
+! !u j! xi
" # $
% & '
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SGS energy dissipation :
!"
= # $ jk S jk
If we wish to controldissipation of energy we canset ij proportional to -S ij
E.g. Smagorinsky-Lilly model:
! ijd = "# sgs
$!u i
$ x
j
+$!u j
$ x
i
%
& '
(
) * = "2 # sgs
!S ij
! sgs
= ?? = (velocity " scale ) # (length " scale )
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!
ij
d =
"#
sgs
$!u i$ x j
+$!u j$ xi
% & '
( ) *
=
"2 #
sgs
!S ij
! sgs
= cs"
( )
2| S |
Length-scale: ~ " (instead of L),Velocity-scale ~ " |S|
! sgs ~ "2 | !S |
c s : Smagorinsky constant
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Theoretical calibration of c s (D.K. Li lly, 1967):
E(k)
k LES resolved SGS
!"
!
"
! =
"u 3
L
E (k ) = c K ! 2 / 3 k " 5 /3
! " = # = $ % ij !S ij
# = cs2
"2
2 | !S |
!S ij
!S ij
# & cs2" 2 2 3/ 2 !S ij !S ij3/ 2
!S ij!S ij =
1
2
' !u i
' x j' !u i
' x j+
' !u j
' xi(
) *
+
, - =
=1
2[k j
2. ii (k ) + k i k j. ij (k )]d 3k =|k |< / / "000 12 [k
2 ( E (k )4 / k 2
(1 ii $ k 2
k 2 )) + 0] d 3k
|k |< / / "000
= cK # 2 /3 1
2k $5 / 3 + 2
3 $14 / k 2
4 / k 2dk 0
/ / "
0 = cK # 2 /3 k 1/ 3 dk 0
/ / "
0 = cK # 2 /3 34/
"
( ) *
+ , -
4 / 3
# & cs2" 2 2 3/ 2 cK # 2 /3 3
4
/
"( ) *
+ , -
4 / 3( ) *
+ , -
3/ 2
! 1 " c s2 # 2 3 c
K
2
$ % &
' ( )
3/ 2
! c s =3 c
K
2
$ % &
' ( )
*3/ 4
# *1
cK
= 1.6 ! c s " 0.16
! ij = " 2( c s # )2 | !S | !S ij
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Theoretical calibration of c s (D.K. Li lly, 1967):
E(k)
k LES resolved SGS
!"
!
"
! =
"u 3
L
E (k ) = c K ! 2 / 3 k " 5 /3
! " = # = $ % ij !S ij
# = cs2
"2
2 | !S |
!S ij
!S ij
# & cs2" 2 2 3/ 2 !S ij !S ij3/ 2
!S ij!S ij =
1
2
' !u i
' x j' !u i
' x j+
' !u j
' xi(
) *
+
, - =
=1
2[k j
2. ii (k ) + k i k j. ij (k )]d 3k =|k |< / / "000 12 [k
2 ( E (k )4 / k 2
(1 ii $ k 2
k 2 )) + 0] d 3k
|k |< / / "000
= cK # 2 /3 1
2k $5 / 3 + 2
3 $14 / k 2
4 / k 2dk 0
/ / "
0 = cK # 2 /3 k 1/ 3 dk 0
/ / "
0 = cK # 2 /3 34/
"
( ) *
+ , -
4 / 3
# & cs2" 2 2 3/ 2 cK # 2 /3 3
4
/
"( ) *
+ , -
4 / 3( ) *
+ , -
3/ 2
! 1 " c s2 # 2 3 c
K
2
$ % &
' ( )
3/ 2
! c s =3 c
K
2
$ % &
' ( )
*3/ 4
# *1
cK
= 1.6 ! c s " 0.16
! ij = " 2( c s # )2 | !S | !S ij
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Theoretical calibration of c s (D.K. Li lly, 1967):
E(k)
k LES resolved SGS
!"
!
"
! =
"u 3
L
E (k ) = c K ! 2 / 3 k " 5 /3
! " = # = $ % ij !S ij
# = cs2
"2
2 | !S |
!S ij
!S ij
# & cs2" 2 2 3/ 2 !S ij !S ij3/ 2
!S ij!S ij =
1
2
' !u i
' x j' !u i
' x j+
' !u j
' xi(
) *
+
, - =
=1
2[k j
2. ii (k ) + k i k j. ij (k )]d 3k =|k |< / / "000 12 [k
2 ( E (k )4 / k 2
(1 ii $ k 2
k 2 )) + 0] d 3k
|k |< / / "000
= cK # 2 /3 1
2k $5 / 3 + 2
3 $14 / k 2
4 / k 2dk 0
/ / "
0 = cK # 2 /3 k 1/ 3 dk 0
/ / "
0 = cK # 2 /3 34/
"
( ) *
+ , -
4 / 3
# & cs2" 2 2 3/ 2 cK # 2 /3 3
4
/
"( ) *
+ , -
4 / 3( ) *
+ , -
3/ 2
! 1 " c s2 # 2 3 c
K
2
$ % &
' ( )
3/ 2
! c s =3 c
K
2
$ % &
' ( )
*3/ 4
# *1
cK
= 1.6 ! c s " 0.16
! ij = " 2( c s # )2 | !S | !S ij
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Theoretical calibration of c s (D.K. Li lly, 1967):
E(k)
k LES resolved SGS
!"
!
"
! =
"u 3
L
E (k ) = c K ! 2 / 3 k " 5 /3
! " = # = $ % ij !S ij
# =
cs2
"2
2 | !S |
!S ij
!S ij
# & cs2" 2 2 3/ 2 !S ij !S ij3/ 2
!S ij!S ij =
1
2
' !u i
' x j' !u i
' x j+
' !u j
' xi(
) *
+
, - =
=1
2[k j
2. ii (k ) + k i k j. ij (k )]d 3k =|k |< / / "000 12 [k
2 ( E (k )4 / k 2
(1 ii $ k 2
k 2 )) + 0] d 3k
|k |< / / "000
= cK # 2 /3 1
2k $5 / 3 + 2
3 $14 / k 2
4 / k 2dk 0
/ / "
0 = cK # 2 /3 k 1/ 3 dk 0
/ / "
0 = cK # 2 /3 34/
"( ) *
+ , -
4 / 3
# & cs2" 2 2 3/ 2 cK # 2 /3 3
4
/
"( ) *
+ , -
4 / 3( ) *
+ , -
3/ 2
! 1 " c s2 # 2 3 c
K
2
$ % &
' ( )
3/ 2
! c s =3 c
K
2
$ % &
' ( )
*3/ 4
# *1
cK
= 1.6 ! c s " 0.16
! ij = " 2( c s # )2 | !S | !S ij
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But in practice(complex flows)
cs
= cs (
x
,t
) Ad-hoc tuning?
c s =0.16 works well for isotropic,high Reynolds number turbulence
Examples: Transit ional p ipe flow: from 0 to 0.16
Near wall damping for wall boundary layers (Piomelli et al 1989)
c s =
c s ( y )
yc
s = 0.16
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Measure: ! " = # $ jk S jk
How does c s vary under realist ic condi tions?Interrogate data:
Measure:
Obtain empirical Smagorinsky coefficient = f(x,conditions):
c s =! " jk S jk
2 #2
|S |
Sij
Sij
$
% &
&
'
( )
)
1/ 2
An example result f rom atmospheric turbulence:
!"Smag
c s2
= 2"2
| S | SijSij
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Measure empirical Smagorinsky coefficient for atmospheric surface layer as function of height and stability (thermal forcing or damping):
c s =! " jk S jk
2 #2 | S | Sij Sij
$
%
& &
'
(
) )
1/ 2
Example result: effect of atmosphericstability on coefficient from sonic
anemometer measurements inatmospheric su rface layer (Kleissl et al., J. Atmos. Sci. 2003)
HATS - 2000(with NCARresearchers:
Horst, Sullivan)Kettleman City
(Central Valley, CA)
Stable stratifi cation
N e u
t r a
l s
t r a
t i f i c a
t i o n
cs
= cs (x , t )
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How to avoid tuning and case-by-caseadjustments of model coefficient in LES?
The Dynamic Model(Germano et al. Physics of Fluids, 1991)
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E(k)
k
LES resolved SGS
Germano identity and dynamic model(Germano et al. 1991):
Exact (rare in turbulence):
ui u j ! u i u j = u iu j ! u iu j
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E(k)
k
LES resolved SGS
Germano identity and dynamic model(Germano et al. 1991):
Exact (rare in turbulence):
ui u j ! u i u j = u iu j - u iu j + u iu j ! u iu j
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E(k)
k
LES resolved SGS
L
T
Germano identity and dynamic model(Germano et al. 1991):
Exact (rare in turbulence):
T ij = ! ij + Lij
ui u j ! u i u j = u iu j - u iu j + u iu j ! u iu j
Lij ! (T ij ! " ij ) = 0
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E(k)
k
LES resolved SGS
L
T
Germano identity and dynamic model(Germano et al. 1991):
Exact (rare in turbulence):
T ij = ! ij + Lij
ui u j ! u i u j = u iu j - u iu j + u iu j ! u iu j
! 2 c s 2 "( )2
| S | Sij! 2 c s"( )
2| S | Sij
Assumes scale-invariance:
Lij ! (T ij ! " ij ) = 0
where
Lij ! c s2 M ij = 0
M ij = 2 !2
| S | Sij " 4 |S | Sij
( )
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Germano identity and dynamic model(Germano et al. 1991):
Minimized when:Averaging over regions of statistical homogeneityor fluid trajectories
Lij ! c s2
M ij = 0
Over-determined system:solve in some average sense(minimize error, Lilly 1992):
! = Lij
"c s
2
M ij( )2
c s2
=
Lij M ij M ij M ij
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Germano identity and dynamic model(Germano et al. 1991):
Minimized when:Averaging over regions of statistical homogeneityor fluid trajectories
Lij ! c s2
M ij = 0
Over-determined system:solve in some average sense(minimize error, Lilly 1992):
! = Lij
"c s
2
M ij( )2
c s2
=
Lij M ij M ij M ij
Exercise:
(a) Prove the above equation, and
(b) repeat entire formulation for the SGS heat flux vector
modeled using an SGS diffusivity (find C scalar )
q i = u iT ! ! "u i "T
qi = C scalar !
2 | !S | " !T
" xi
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Mixed tensor Eddy Viscosity Model: Taylor-series expansion of similarity (Bardina 1980) model (Clark 1980, Liu, Katz & Meneveau (1994), ) Deconvolution:
(Leonard 1997, Geurts et al, Stolz & Adams, Winckelmans etc..)
Significant d irect empirical evidence, experiments: Liu et al. (JFM 1999, 2-D PIV) Tao, Katz & CM (J. Fluid Mech. 2002):
tensor alignments from 3-D HPIV data Higgins, Parlange & CM (Bound Layer Met. 2003):
tensor alignments from ABL data From DNS: Horiuti 2002, Vreman et al (LES), etc
( )k
j
k
inlijS ij x
u
xu
C S S C T !
!
!
!"+"#=
~~2
~~)2(2
22
ijS k
j
k
inl
mnlij S S C x
u xuC ~~)(2
~~22 !"
#
#
##!=$
Two-parameter dynamic mixed model ji jiijijij uuuuT L
~~~~
!=!" #
! " ijd
Sij
Similarity, tensor eddy-viscosity, and mixed models
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Reality check:
Do simulations with these closures producerealistic statistics of u i(x,t)?
Need good data Need good simulations
Next: Summary of results from Kang et al. (JFM 2003)
Smagorinsky model, Dynamic Smagorinsky model, Dynamic 2-parameter mixed model
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Remake of Comte-Bellot & Corrsin (1967) decaying isotropic turbulence experiment
at high Reynolds number
Contractions
Active Grid M = 6 "
1 x
2 x
M 20 M 30 M 40 M 48
Test Section
1u
2u X-wire probe array!
1 = 0.01m!
2 = 0.02m!3 = 0.04m
!4 = 0.08m
0.91mFlow
h1 = 0.005m! 1 = 0.01m
h3 = 0.02m! 3 = 0.04m
!
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1 x 2 x
M 20 M 30 M 40 M 48
1u
2u
Flow
10 -7
10 -6
10 -5
10 -4
10 -3
10 -2
10 -1
10 0
10 0 10 1 10 2 10 3 10 4
E( ! )
!
x1/M=20
x1/M=48
Initial condition for LES
(m3s-2)
(m -1)
R e s u
l t s
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Results: Dynamic Model Coefficients
! Dynamic Smagorinsky ! Dynamic Mixed tensor eddy visc:
ijS k
j
k
inl
mnnlij S S C x
u
xu
C ~~
)(2~~
22 !"#
#
#
#!=$
0
0.05
0.1
0.15
0.2
0 10 20 30 40 50
Cs
x1
/M
Cs
0
0.05
0.1
0.15
0.2
0 10 20 30 40 50
Cs
Cnl
x1
/M
Cnl
Cs
Cnl
Cs
1/12 from the first order approximation of " ij
! ijdyn " Smag
= " 2 Lij M ij
M ij M ij#
2 S Sij
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3-D Energy Spectra (LES vs experiment)
! Dynamic Smagorinsky
10-5
10-4
10-3
10-2
10-1
100
100
101
102
103
E( ! )
!
x1/M=20
x1/M=48
cutoff grid filter (! = 0.04 m)
cutoff test filter
at 2 !
Exp.
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! Dynamic Mixed tensor eddy-visc. model
10-5
10 -4
10 -3
10 -2
10 -1
100
100
101
102
103
E( ! )
!
x1/M=20
x1/M=48
cutoff grid filter (! = 0.04 m)
cutoff test filter at 2 !
3-D Energy Spectra (LES vs experiment)
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PDF of SGS Stress
! SGS Stress
0
5
10
15
20
25
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
PDF
x1/M=48
Exp.
Dynamic Mixed Nonlinear
Smagorisnky
DynamicSmagorisnky
2
112
~/rms
u!
Dynamic mixed tensor ed d y-visc mod el pr ed icts PDF of theSGS str ess accurately.
212112~~ uuuu !"#
(LES vs experiment)
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E(k)
k
LES resolved SGS
L
T
Germano identity and dynamic model(Germano et al. 1991):
Exact (rare in turbulence):
T ij = ! ij + Lij
ui u j ! u i u j = u iu j - u iu j + u iu j ! u iu j
! 2 c s 2 "( )2
| S | Sij! 2 c s"( )
2| S | Sij
Assumes scale-invariance:
Lij ! (T ij ! " ij ) = 0
where
Lij ! c s2 M ij = 0
M ij = 2 !2
| S | Sij " 4 |S | Sij( )
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Germano identity and dynamic model(Germano et al. 1991):
Minimized when:Averaging over regions of statistical homogeneityor fluid trajectories
Lij ! c s2
M ij = 0Over-determined system:solve in some average sense(minimize error, Lilly 1992):
! =
Lij "
c s2
M ij( )
2
c s2
=
Lij M ij M ij M ij
Problem: what to do for non-homogeneous flowswithout directions over which to average( learn , or assimilate larger-scale statistics?)
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Lagrangian dynamic model (M, Lund & Cabot, JFM 1996):
Average in time, following fluid particles for Galilean invariance:
E = Lij ! C s2 M ij( )
2
!"
t
# 1T e! (t -t')
T dt '
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Lagrangian dynamic model (M, Lund & Cabot, JFM 1996):
Average in time, following fluid particles for Galilean invariance:
! E = 0 " C s2
=
Lij M ij #$
t
% 1T e# (t-t')
T dt '
M ij M ij #$
t
% 1T e# (t-t')
T dt '
E = Lij ! C s2 M ij( )
2
!"
t
# 1T e! (t -t')
T dt '
! MM
= M ij M ij "#
t
$ 1T e" (t-t ')
T dt '
! LM
= Lij M ij "#
t
$ 1T e" (t-t')
T dt '
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Lagrangian dynamic model (M, Lund & Cabot, JFM 1996):
Average in time, following fluid particles for Galilean invariance:
! E = 0 " C s2
=
Lij M ij #$
t
% 1T e# (t-t')
T dt '
M ij M ij #$
t
% 1T e# (t-t')
T dt '
E = Lij ! C s2 M ij( )
2
!"
t
# 1T e! (t -t')
T dt '
With exponential weight-function, equivalent to relaxation forward equations:
! MM
= M ij M ij "#
t
$ 1T e" (t-t ')
T dt '
! LM
= Lij M ij "#
t
$ 1T e" (t-t')
T dt '
!" LM ! t
+ uk !" LM
! xk = 1
T ( L ij M ij # " LM )
!" MM
! t + uk
!" MM
! xk =
1
T ( M ij M ij # " MM )
cs
2=
! LM (x , t )
! MM (x , t )
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Lagrangian dynamic model has allowed applying the Germano-identity to a number of complex-geometry engineering problems
LES of flows in internal combustion engines :Haworth & Jansen (2000)Computers & Fluids 29 .
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LES of structure of impinging jets :Tsubokura et al. (2003)Int Heat Fluid Flow 24 .
LES of flow over wavy walls Armenio & Piomelli (2000)Flow, Turb. & Combustion.
Examples:
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LES of flow in thrust -reversers Blin, Hadjadi & Vervisch (2002)J. of Turbulence.
Examples:
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LES of flow in tur bomachinery Zou, Wang, Moin, Mittal. (2007)Journal of Fluid Mechanics.
Examples:
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Examples:
LES of convective atmospheric boundary l ayer: Kumar, M. & Parlange (Water Resources Research, 2006)
Transport equation for temperature Boussinesq approximation Coriolis forcing Lagrangian dynamic model with assumed # =C s (2 " )/C s(" ) Constant (non-dynamic) SGS Prandtl number Pr sgs =0.4 Imposed surface flux of sensible heat on ground Diurnal cycle: start stably stratified, then heating.
Imposed groundheat flux during day:
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Diurnal cycle: start stably stratified, then heating.
Examples:
Resulting dynamic
coeffi cient (averaged):
Consistent wi th HATSfield measurements:
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Large-eddy-Simulation of atmospheric flow over fractal trees:
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Downtown Baltimore:
URBAN CONTAMINATIONAND TRANSPORT
Yu-Heng Tseng, C. Meneveau & M. Parlange, 2006 (Env. Sci & Tech. 40 , 2653-2662)
Momentum and scalar transport equations solved using LES andLagrangian dynamic subgrid model. Buildings are simulated usingimmersed boundary method.
wind
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Useful references on LES and SGS modeling:
P. Sagaut: Large Eddy Simulation of Incompressible Flow (Springer, 3rd ed., 2006)
U. Piomelli, Progr. Aerospace Sci., 1999
C. Meneveau & J. Katz, Annu Rev. Fluid Mech. 32 , 1-32 (2000)
C. Meneveau, Scholarpedia 5, 9489 (2010).