Large Eddy Simulation, DynamicModel, and Applications

Charles Meneveau

Department of Mechanical Engineering Center for Environmental and Applied Fluid Mechanics

Johns Hopkins University

Mechanical Engineering

Turbulence Summer SchoolMay 2010

Turbulence modeling:

Large-eddy-simulation (LES)

Δ

Δ

G(x): Filter

Large-eddy-simulation (LES) and filtering:

N-S equations:

! !uj

!t+ !uk

! !u j

!xk

= "1#!!p!x j

+ $%2 !u j "!!xk

& jk

Filtered N-S equations:

!u j

!t+!uku j

!xk

= "1#!p!x j

+ $%2uj

! !uj

!t+!uku j"

!xk

= "1#!!p!x j

+ $%2 !u j

where SGS stress tensor is:

! ij = uiu j! " "ui "u j

!uj

!x j= 0

Δ

Δ

G(x): Filter

Energetics (kinetic energy):

! 12 ˜ u j ˜ u j!t

+ ˜ u k! 1

2 ˜ u j ˜ u j!xk

= "!!x j

....( ) " 2# ˜ S jk ˜ S jk " "$ jk˜ S jk( )

!" = # $ jk˜ S jk

Inertial-range flux

Effects of τij upon resolved motions:

E(k)

kLES resolved SGS

!"

!"

! = "u 3

L

!Sij =12

! !ui

!x j

+! !uj

!xi

"

#$%

&'

“SGS energy dissipation”:

!" = # $ jk˜ S jk

If we wish to “control”dissipation of energy we canset τij proportional to -Sij

E.g. Smagorinsky-Lilly model:

! ij

d = "# sgs$ !ui

$x j

+$ !uj

$xi

%

&'(

)*= "2# sgs

!Sij

! sgs = ?? = (velocity " scale) # (length " scale)

! ij

d = "# sgs$ !ui

$x j

+$ !uj

$xi

%

&'(

)*= "2# sgs

!Sij

!sgs = cs"( ) 2 | ˜ S |

Length-scale: ~ Δ (instead of L), Velocity-scale ~ Δ |S|

! sgs ~ "2 | !S |

cs: “Smagorinsky constant”

Theoretical calibration of cs (D.K. Lilly, 1967):

E(k)

kLES resolved SGS

!"

!"

! = "u 3

LE(k) = cK!

2 /3k"5 /3

!" = # = $ % ij!Sij

# = cs2"2 2 | !S | !Sij

!Sij

# & cs2"2 23/2 !Sij

!Sij

3/2

!Sij!Sij =

12

' !ui

'x j

' !ui

'x j

+' !uj

'xi

(

)*+

,-=

=12

[kj2.ii (k) + kik j.ij (k)]d 3k =

|k |</ /"000

12

[k2 ( E(k)4/k2 (1 ii $

k2

k2 )) + 0]d 3k|k |</ /"000

= cK#2 /3 1

2k$5 /3+2 3$1

4/k2 4/k2dk0

/ /"

0 = cK#2 /3 k1/3dk

0

/ /"

0 = cK#2 /3 3

4/"

()*

+,-

4 /3

# & cs2"2 23/2 cK#

2 /3 34

/"

()*

+,-

4 /3(

)*+

,-

3/2

!1 " cs2# 2 3cK

2$%&

'()

3/2

! cs =3cK

2$%&

'()*3/4

# *1

cK = 1.6 ! cs " 0.16

! ij = "2(cs#)2 | !S | !Sij

Theoretical calibration of cs (D.K. Lilly, 1967):

E(k)

kLES resolved SGS

!"

!"

! = "u 3

LE(k) = cK!

2 /3k"5 /3

!" = # = $ % ij!Sij

# = cs2"2 2 | !S | !Sij

!Sij

# & cs2"2 23/2 !Sij

!Sij

3/2

!Sij!Sij =

12

' !ui

'x j

' !ui

'x j

+' !uj

'xi

(

)*+

,-=

=12

[kj2.ii (k) + kik j.ij (k)]d 3k =

|k |</ /"000

12

[k2 ( E(k)4/k2 (1 ii $

k2

k2 )) + 0]d 3k|k |</ /"000

= cK#2 /3 1

2k$5 /3+2 3$1

4/k2 4/k2dk0

/ /"

0 = cK#2 /3 k1/3dk

0

/ /"

0 = cK#2 /3 3

4/"

()*

+,-

4 /3

# & cs2"2 23/2 cK#

2 /3 34

/"

()*

+,-

4 /3(

)*+

,-

3/2

!1 " cs2# 2 3cK

2$%&

'()

3/2

! cs =3cK

2$%&

'()*3/4

# *1

cK = 1.6 ! cs " 0.16

! ij = "2(cs#)2 | !S | !Sij

Theoretical calibration of cs (D.K. Lilly, 1967):

E(k)

kLES resolved SGS

!"

!"

! = "u 3

LE(k) = cK!

2 /3k"5 /3

!" = # = $ % ij!Sij

# = cs2"2 2 | !S | !Sij

!Sij

# & cs2"2 23/2 !Sij

!Sij

3/2

!Sij!Sij =

12

' !ui

'x j

' !ui

'x j

+' !uj

'xi

(

)*+

,-=

=12

[kj2.ii (k) + kik j.ij (k)]d 3k =

|k |</ /"000

12

[k2 ( E(k)4/k2 (1 ii $

k2

k2 )) + 0]d 3k|k |</ /"000

= cK#2 /3 1

2k$5 /3+2 3$1

4/k2 4/k2dk0

/ /"

0 = cK#2 /3 k1/3dk

0

/ /"

0 = cK#2 /3 3

4/"

()*

+,-

4 /3

# & cs2"2 23/2 cK#

2 /3 34

/"

()*

+,-

4 /3(

)*+

,-

3/2

!1 " cs2# 2 3cK

2$%&

'()

3/2

! cs =3cK

2$%&

'()*3/4

# *1

cK = 1.6 ! cs " 0.16

! ij = "2(cs#)2 | !S | !Sij

Theoretical calibration of cs (D.K. Lilly, 1967):

E(k)

kLES resolved SGS

!"

!"

! = "u 3

LE(k) = cK!

2 /3k"5 /3

!" = # = $ % ij!Sij

# = cs2"2 2 | !S | !Sij

!Sij

# & cs2"2 23/2 !Sij

!Sij

3/2

!Sij!Sij =

12

' !ui

'x j

' !ui

'x j

+' !uj

'xi

(

)*+

,-=

=12

[kj2.ii (k) + kik j.ij (k)]d 3k =

|k |</ /"000

12

[k2 ( E(k)4/k2 (1 ii $

k2

k2 )) + 0]d 3k|k |</ /"000

= cK#2 /3 1

2k$5 /3+2 3$1

4/k2 4/k2dk0

/ /"

0 = cK#2 /3 k1/3dk

0

/ /"

0 = cK#2 /3 3

4/"

()*

+,-

4 /3

# & cs2"2 23/2 cK#

2 /3 34

/"

()*

+,-

4 /3(

)*+

,-

3/2

!1 " cs2# 2 3cK

2$%&

'()

3/2

! cs =3cK

2$%&

'()*3/4

# *1

cK = 1.6 ! cs " 0.16

! ij = "2(cs#)2 | !S | !Sij

But in practice (complex flows)

cs = cs (x, t)Ad-hoc tuning?

cs=0.16 works well for isotropic,high Reynolds number turbulence

Examples: Transitional pipe flow: from 0 to 0.16

Near wall damping for wall boundary layers (Piomelli et al 1989)

cs = cs (y)

ycs = 0.16

Measure:

!" = # $ jk˜ S jk

How does cs vary under realistic conditions?Interrogate data:

Measure:

Obtain“empirical” Smagorinsky coefficient = f(x,conditions…):

cs =! " jk

˜ S jk2#2 | ˜ S | ˜ S ij ˜ S ij

$

% & &

'

( ) )

1/ 2

An example result from atmospheric turbulence…:

!"Smag

cs2 = 2"2 | ˜ S | ˜ S ij ˜ S ij

Measure “empirical” Smagorinsky coefficient for atmospheric surface layeras function of height and stability (thermal forcing or damping):

cs =! " jk

˜ S jk2#2 | ˜ S | ˜ S ij ˜ S ij

$

% & &

'

( ) )

1/ 2

Example result: effect of atmosphericstability on coefficient from sonicanemometer measurements inatmospheric surface layer(Kleissl et al., J. Atmos. Sci. 2003)

HATS - 2000(with NCARresearchers:

Horst, Sullivan)Kettleman City

(Central Valley, CA)

Stable stratification

Neu

tral

str

atifi

catio

n

cs = cs (x, t)

How to avoid “tuning” and case-by-caseadjustments of model coefficient in LES?

The Dynamic Model(Germano et al. Physics of Fluids, 1991)

E(k)

k

LES resolved SGS

τ

Germano identity and dynamic model(Germano et al. 1991):

Exact (“rare” in turbulence):

uiuj ! ˜ u i ˜ u j = uiu j ! ˜ u i ˜ u j

E(k)

k

LES resolved SGS

τ

Germano identity and dynamic model(Germano et al. 1991):

Exact (“rare” in turbulence):

uiuj ! ˜ u i ˜ u j = uiu j - ˜ u i ˜ u j + ˜ u i ˜ u j ! ˜ u i ˜ u j

E(k)

k

LES resolved SGS

L τT

Germano identity and dynamic model(Germano et al. 1991):

Exact (“rare” in turbulence):

Tij = ! ij + Lij

uiuj ! ˜ u i ˜ u j = uiu j - ˜ u i ˜ u j + ˜ u i ˜ u j ! ˜ u i ˜ u j

Lij ! (Tij ! " ij ) = 0

E(k)

k

LES resolved SGS

L τT

Germano identity and dynamic model(Germano et al. 1991):

Exact (“rare” in turbulence):

Tij = ! ij + Lij

uiuj ! ˜ u i ˜ u j = uiu j - ˜ u i ˜ u j + ˜ u i ˜ u j ! ˜ u i ˜ u j

!2 cs 2"( )2 | ˜ S | ˜ S ij

!2 cs"( )2 | ˜ S | ˜ S ij

Assumes scale-invariance:

Lij ! (Tij ! " ij ) = 0

where

Lij ! cs2 Mij = 0

Mij = 2!2 | ˜ S | ˜ S ij " 4 | ˜ S | ˜ S ij( )

Germano identity and dynamic model(Germano et al. 1991):

Minimized when:Averaging over regions ofstatistical homogeneityor fluid trajectories

Lij ! cs2 Mij = 0

Over-determined system:solve in “some average sense”(minimize error, Lilly 1992):

! = Lij " cs2 Mij( )2

cs2 =

Lij Mij

Mij Mij

Germano identity and dynamic model(Germano et al. 1991):

Minimized when:Averaging over regions ofstatistical homogeneityor fluid trajectories

Lij ! cs2 Mij = 0

Over-determined system:solve in “some average sense”(minimize error, Lilly 1992):

! = Lij " cs2 Mij( )2

cs2 =

Lij Mij

Mij Mij

Exercise:

(a) Prove the above equation, and

(b) repeat entire formulation for the SGS heat flux vector

modeled using an SGS diffusivity (find Cscalar)

qi = uiT! ! "ui"T

qi = Cscalar!

2 | !S | "!T

"xi

• 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 direct 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

inlijSij x

uxuCSSCT

!!

!!"+"#=

~~2~~)2(2 22

ijSk

j

k

inl

mnlij SSC

xu

xuC ~~)(2

~~22 !"

##

##!=$

Two-parameter dynamic mixed modeljijiijijij uuuuTL ~~~~ !=!" #

!" ijd

˜ S ij

Similarity, tensor eddy-viscosity, and mixed models

Reality check:

Do simulations with these closures producerealistic statistics of ui(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

Remake of Comte-Bellot & Corrsin (1967)decaying isotropic turbulence experiment

at high Reynolds number

Contractions

Active GridM = 6 "

1x2x

M20 M30 M40 M48

Test Section

1u2u X-wire probe array

Δ1 = 0.01mΔ2 = 0.02mΔ3 = 0.04mΔ4 = 0.08m

0.91mFlow

h1 = 0.005mΔ1 = 0.01m h3 = 0.02m

Δ3 = 0.04mΔ

1x2x

M20 M30 M40 M48

1u2u

Flow

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

100 101 102 103 104

E(!)

!

x1/M=20

x1/M=48

Initial condition for LES

(m3s-2)

(m-1)

Res

ults

LESTemporally Decaying Turbulence

3-D Filter

ExperimentSpatially Decaying Grid Turbulence

2-D Box Filter tux 11 =

Taylor Hypothesis

LES of Temporally Decaying Turbulence

Pseudo-spectral code: 1283 nodes, carefully dealiased (3/2N) All parameters are equivalent to those of experiments. Initial energy distribution: 3-D energy spectrum at x1/M = 20

LES Models: standard Smagorinsky-Lilly model,dynamic Smagorinsky and

dynamic mixed tensor eddy-visc. model

Results: Dynamic Model Coefficients

Dynamic Smagorinsky Dynamic Mixed tensor eddy visc:

ijSk

j

k

inl

mnnlij SSC

xu

xuC ~~)(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 orderapproximation of τij

! ijdyn"Smag = "2 Lij M ij

M ij M ij#2 ˜ S ˜ S ij

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 (Δ = 0.04 m)• cutoff test filter at 2Δ

Exp.

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 (Δ = 0.04 m)• cutoff test filter at 2Δ

3-D Energy Spectra (LES vs experiment)

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

Dynamic Smagorisnky

2112

~/ rmsu!

Dynamic mixed tensor eddy-visc model predicts PDF of theSGS stress accurately.

212112~~uuuu !"#

(LES vs experiment)

E(k)

k

LES resolved SGS

L τT

Germano identity and dynamic model(Germano et al. 1991):

Exact (“rare” in turbulence):

Tij = ! ij + Lij

uiuj ! ˜ u i ˜ u j = uiu j - ˜ u i ˜ u j + ˜ u i ˜ u j ! ˜ u i ˜ u j

!2 cs 2"( )2 | ˜ S | ˜ S ij

!2 cs"( )2 | ˜ S | ˜ S ij

Assumes scale-invariance:

Lij ! (Tij ! " ij ) = 0

where

Lij ! cs2 Mij = 0

Mij = 2!2 | ˜ S | ˜ S ij " 4 | ˜ S | ˜ S ij( )

Germano identity and dynamic model(Germano et al. 1991):

Minimized when:Averaging over regions ofstatistical homogeneityor fluid trajectories

Lij ! cs2 Mij = 0

Over-determined system:solve in “some average sense”(minimize error, Lilly 1992):

! = Lij " cs2 Mij( )2

cs2 =

Lij Mij

Mij Mij

Problem: what to do for non-homogeneous flowswithout directions over which to average(“learn”, or “assimilate” larger-scale statistics?)

Lagrangian dynamic model (M, Lund & Cabot, JFM 1996):

Average in time, following fluid particles for Galilean invariance:

E = Lij ! Cs2Mij( )2

!"

t

# 1T

e!

(t -t')T dt '

Lagrangian dynamic model (M, Lund & Cabot, JFM 1996):

Average in time, following fluid particles for Galilean invariance:

! E = 0 " Cs2 =

LijMij #$

t

%1T

e#

(t - t' )T dt'

Mij Mij #$

t

%1T

e#

(t - t' )T dt'

E = Lij ! Cs2Mij( )2

!"

t

# 1T

e!

(t -t')T dt '

!MM = Mij Mij "#

t

$ 1T

e"

(t - t' )T dt'

!LM = LijMij "#

t

$ 1T

e"

(t -t')T dt '

Lagrangian dynamic model (M, Lund & Cabot, JFM 1996):

Average in time, following fluid particles for Galilean invariance:

! E = 0 " Cs2 =

LijMij #$

t

%1T

e#

(t - t' )T dt'

Mij Mij #$

t

%1T

e#

(t - t' )T dt'

E = Lij ! Cs2Mij( )2

!"

t

# 1T

e!

(t -t')T dt '

With exponential weight-function, equivalent to relaxation forward equations:

!MM = Mij Mij "#

t

$ 1T

e"

(t - t' )T dt'

!LM = LijMij "#

t

$ 1T

e"

(t -t')T dt '

!"LM

!t+ ˜ u k

!"LM

!xk

=1T

(LijMij # "LM )

!"MM

!t+ ˜ u k

!"MM

!xk

=1T

(Mij Mij # "MM )

cs2 =

!LM (x, t)!MM (x, t)

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.

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:

LES of flow in thrust-reversers Blin, Hadjadi & Vervisch (2002)J. of Turbulence.

Examples:

LES of flow in turbomachinery Zou, Wang, Moin, Mittal. (2007)Journal of Fluid Mechanics.

Examples:

Examples:

LES of convective atmospheric boundary layer: Kumar, M. & Parlange (Water Resources Research, 2006)

• Transport equation for temperature• Boussinesq approximation• Coriolis forcing• Lagrangian dynamic model with assumed β=Cs(2 Δ)/Cs(Δ)• Constant (non-dynamic) SGS Prandtl number Prsgs=0.4• Imposed surface flux of sensible heat on ground• Diurnal cycle: start stably stratified, then heating….

Imposed ground heat flux during day:

• Diurnal cycle: start stably stratified, then heating….

Examples:

Resulting dynamic coefficient (averaged):

Consistent with HATS field measurements:

Large-eddy-Simulation of atmospheric flow over fractal trees:

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

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).