Large Eddy Simulation, Dynamic Model, and Applications...Large Eddy Simulation, Dynamic Model, and...

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

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



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



Tij = ! ij + Lij


Lij ! (Tij ! " ij ) = 0

E(k)

k

LES resolved SGS

L τT



Tij = ! ij + Lij


!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( )


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



Lij ! cs2 Mij = 0


! = 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



Tij = ! ij + Lij


!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( )



Lij ! cs2 Mij = 0


! = 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 '



! E = 0 " Cs2 =

LijMij #$

t

%1T

e#

(t - t' )T dt'

Mij Mij #$

t

%1T

e#

(t - t' )T dt'


!"

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 '



! E = 0 " Cs2 =

LijMij #$

t

%1T

e#

(t - t' )T dt'

Mij Mij #$

t

%1T

e#

(t - t' )T dt'


!"

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).

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