tutorial les 2010 meneveau

8/12/2019 Tutorial LES 2010 Meneveau

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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)


3/43

!

!

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


4/43

!

!

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

" # $

% & '


5/43

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 )


6/43

!

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


7/43

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


8/43


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


9/43


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


10/43


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


11/43

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


12/43

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


13/43

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 )


14/43

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

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


15/43

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


16/43

E(k)

k

LES resolved SGS



ui u j ! u i u j = u iu j - u iu j + u iu j ! u iu j


17/43

E(k)

k

LES resolved SGS

L

T



T ij = ! ij + Lij


Lij ! (T ij ! " ij ) = 0


18/43

E(k)

k

LES resolved SGS

L

T



T ij = ! ij + Lij


! 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

( )


19/43


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


20/43



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


21/43

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


22/43

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


23/43

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

!


24/43

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


25/43


26/43

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


27/43

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.


28/43

! 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)


29/43

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)


30/43

E(k)

k

LES resolved SGS

L

T



T ij = ! ij + Lij


! 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( )


31/43



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?)


32/43

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 '


33/43



! 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 '


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 '


34/43



! 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 '


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 )


35/43

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 .


36/43

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:


37/43

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

Examples:


38/43

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

Examples:


39/43

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:


40/43

Diurnal cycle: start stably stratified, then heating.

Examples:

Resulting dynamic

coeffi cient (averaged):

Consistent wi th HATSfield measurements:


41/43

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


42/43

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


43/43

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).

tutorial les 2010 meneveau

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