talk dynamicmodel course ii

7/31/2019 Talk DynamicModel Course II

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Large Eddy Simulation, Dynamic

Model, and ApplicationsCharles Meneveau

Department of Mechanical Engineering

Center for Environmental and Applied Fluid Mechanics

Johns Hopkins University

Mechanical Engineering

SCAT 1st Latin-American Summer School

January 9th, 2007

Part II:

!

!

G(x): Filter

Large-eddy-simulation (LES) and filtering:

N-S equations:

!uj

!t+ uk

!uj

!xk= "

!p

!xj+ #$

2uj "

!

!xk%jk

Filtered N-S equations:

!uj

!t+

!ukuj

!xk= "

!p

!x j+ #$

2

u j

!uj

!t+

!ukuj

!xk= "

!p

!x j+ #$

2

uj

where SGS stress tensor is:

!ij= u

iu

j" u

iu

j

!uj

!xj

= 0

!

!

G(x): Filter

Energetics (kinetic energy):

! 12ujuj

!t+ uk

! 12ujuj

!xk= "

!

!xj....( ) " 2#SjkSjk " "$jkSjk( )

!"= # $jk

Sjk

Inertial-range flux

(most attention)

Effects of ijupon resolved motions:

E(k)

k

LES resolved SGS

!"

!

"

! ="u3

L

!Sij =

1

2

! !ui!xj

+

! !uj!xi

"

#$

%

&'

SGS energy dissipation:

!"= # $jk

Sjk

If we wish to control

dissipation of energy we can

setij

proportional to-Sij

E.g. Smagorinsky-Lilly model:

!

ij

d= "#

sgs

$ !ui$xj

+

$ !uj$xi

%

&'

(

)*= "2#sgs !Sij

!sgs =

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


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!ij

d= "#sgs

$ !ui

$xj

+

$ !uj$x

i

%

&'

(

)*= "2#sgs !Sij

!sgs = cs"( )2

| S|

Length-scale: ~ " (instead ofL),

Velocity-scale ~ " |S|

!sgs ~ "2 | !S |

cs: Smagorinsky constant

Theoretical calibration ofcs (D.K. Lilly, 1967):

E(k)

kLES resolved SGS

!"

!

"

! ="u3

L

E(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'x

i

()* +,-=

=1

2[kj

2.ii (k)+ kikj.ij(k)]d3k =

|k|


3/9

Measure empirical Smagorinsky coefficient for atmospheric surface layer

as function of height and stability (thermal forcing or damping):

cs=

! "jkSjk2#2 | S| SijSij

$

%&&

'

())

1/2

Example result: effect of atmospheric

stability on coefficient from sonic

anemometer measurements in

atmospheric surface layer

(Kleissl et al., J. Atmos. Sci. 2003)

HATS - 2000(with NCAR)Kettleman City

(Central Valley, CA)

Stable stratification

Neutralstratification

cs= c

s(x, t)

How to avoid tuning and case-by-case

adjustments of model coefficient in LES?

The Dynamic Model

(Germano et al. Physics of Fluids, 1991)

E(k)

k

LES resolved SGS

L

T

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

Exact (rare in turbulence):

uiuj ! uiuj = uiuj ! uiuj

E(k)

k

LES resolved SGS

L

T



uiuj ! ui uj = uiuj - uiuj + uiuj ! uiuj


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E(k)

k

LES resolved SGS

L

T



Tij=

!ij + Lijuiuj ! uiuj = uiuj - uiuj + uiuj ! uiuj

Lij! (T

ij! "

ij)= 0

E(k)

k

LES resolved SGS

L

T



Tij=

!ij + Lij

uiuj ! ui uj = uiuj - uiuj + uiuj ! uiuj

!2 cs2"( )2

| S | Sij

!2 cs"( )2

| S| Sij

Assumes scale-invariance:

Lij! (T

ij! "

ij)= 0

where

Lij! cs2Mij = 0

Mij= 2!2| S| Sij" 4 | S | Sij

( )


Minimized when:Averaging over regions of

statistical homogeneity

or fluid trajectories

Lagrangian dynamic model (CM, Tom Lund & Bill Cabot, JFM 1996):

Average in time, following fluid particles for Galilean invariance:

A = A(t')

!"

t

#1

Te!(t-t')

T dt

Lij! cs2Mij = 0

Over-determined system:solve in some average sense

(minimize error, Lilly 1992):

!= Lij" cs2M

ij( )2

cs

2

=

LijM

ij

MijM

ij

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

u

x

uCSSCT

!

!

!

!"+"#=

~~2

~~)2(2

22

ijS

k

j

k

inl

mnl

ij SSCx

u

x

uC

~~)(2

~~22

!"#

#

#

#!=$

Two-parameter dynamic mixed model jijiijijij uuuuTL~~~~

!=!" #

!"ij

d

Sij

Similarity, tensor eddy-viscosity, and mixed models


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Reality check:

Do simulations with these closures produce

realistic statistics ofui(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 Grid

M= 6 "

1x

2x

M20 M30 M40 M48

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 = 0.01mh3 = 0.02m

!3 = 0.04m

!

Corrsin Wind-tunnel & active grid:

Makita (1991)

Mydlarski & Warhaft (1996) Grid Size M: 6 inches

Winglet Speed # [208, 417]

rpm selected with uniform

probability randomly in

both directions.

Randomly rotating winglets

X-Wire Probe Array and

Automatic Calibration System

* Four X-wire Probes

!L/d = 200, d = 2.5 m! Measurement

volume = 0.5 mm

* "= 0.01, 0.02, 0.04, 0.08 m* 36,000,000 data points / probe

* with a sampling rate of 40 kHz


6/9

1x

2x

M20 M30 M40 M48

1u

2u

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

(m3

s-2

)

(m-1)

Resu

lts

LES

Temporally Decaying Turbulence

3-D Filter

Experiment

Spatially Decaying Grid Turbulence

2-D Box Filter tux 11 =

TaylorHypothesis

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 atx1/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:

ijS

k

j

k

inl

mnnl

ij SSCx

u

x

uC

~~)(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

!ij

dyn"Smag= "2

LijMij

MijMij#

2SS

ij


7/9

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 filterat 2!

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

(! = 0.04 m)

cutoff test filterat 2!

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

2

112

~/

rmsu!

Dynamic mixed tensoreddy-viscmodelpredicts PDF oftheSGS stress accurately.

212112

~~uuuu !"#

(LES vs experiment) Lagrangian dynamic model (M, Lund & Cabot, JFM 1996):Average in time, following fluid particles for Galilean invariance:

! E = 0 " Cs2=

LijMij#$

t

%1

Te# (t-t ')

T dt'

MijMij#$

t

%1

Te#(t-t ')

T dt'

E = Lij !Cs2Mij( )

2

!"

t

#1

Te!(t-t')

T dt'

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

!MM = MijMij"#

t

$1

Te"(t-t')

T dt'

!LM = LijMij"#

t

$ 1Te"

(t-t')

T dt'

!"LM

!t+ uk

!"LM

!xk=

1

T(LijMij# "LM)

!"MM

!t+ uk

!"MM

!xk=

1

T(MijMij # "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.

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: 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 #=Cs(2")/Cs(")

Constant (non-dynamic) SGS Prandtl numberPrsgs=0.4 Imposed surface flux of sensible heat on ground Diurnal cycle: start stably stratified, then heating.

Imposed ground

heat flux during day:


9/9

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 CONTAMINATION

AND 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 using

immersed 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., 2000

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