liu turbulence sas modeling

7/25/2019 LiU Turbulence SAS Modeling

1/41

2011 AN, I. M 9, 20141

Scale-AdaptiveSimulation (SAS)Turbulence Modeling

F.R. MenterANSYS Germany GmbH

Torbjrn VirdungANSYS Sweden AB


2/41

2011 AN, I. M 9, 20142

Unsteady RANS Based Models

URANS (Unsteady Reynolds averaged

Navier Stokes) Methods URANS gives unphysical single mode unsteadybehavior

Some improvement relative to steady state (RANS)but often not sufficient to capture main effects

Reduction of time step and refinement of mesh donot benefit the simulation

SAS (Scale-Adaptive Simulation) Method

Extends URANS to many technical flows Provides LES-content in unsteady regions Produces information on turbulent spectrum Can be used as basis for acoustics simulations

URANS

SAS-URANS


3/41

2011 AN, I. M 9, 20143

Assumptions Two-Equation

Models

Largest eddies are most effective in mixing

Two scales are minimum for statistical description oflarge turbulence scales

Two model equations of independent variables define

the two scales Equation for turbulent kinetic energy is representing the large scale

turbulent energy

Second equation ( , , kL) to close the system Each equation defines one independent scale

Both - and -equations describe the smallest (dissipate) eddies Rotta developed an exact transport equation for the large turbulent

length scales. This is a much better basis for a term-by-term modelling

approach


4/41

2011 AN, I. M 9, 20144

Classical Derivation 2 Equation Models

( ) ( )

+=

+

jk

t

j

k

j

j

x

k

xkcP

x

kU

t

k

( ) ( ) ( )

+

=

+

j

t

j

k

j

j

xxk

kP

kxU

t

k

t=

The k-equation: Can be derived exactly from the

Navier-Stokes equations

Term-by-term modelling

The - (-) equation: Exact equation for smallest

(dissipation) scales Model for large scales not based

on exact equation

Modelled in analogy to k-

equation and dimensionalanalysis

Danger that not all effects areincluded


5/41

2011 AN, I. M 9, 20145

Source Terms Equilibrium k- Model

Only one Scale in Sources (S~1/T)

Turbulence ModelInput S

Output

Output k

O ?

( )

( )

2 2

2 2

1 2

( )( )

( )( )

j tt

j j k j

j t

j j j

U kk kS c

t x x x

Uc S c

t x x x

+ = +

+ = +


6/41

2011 AN, I. M 9, 20146

Determination of L in k-Model

k-equation:

+=

+

y

kk

ycS

k

x

kU

t

k

k

k

)(

)()( 22

+=

kkccS

k 1)(0 22

2

2~0

kcSc +=

D

F

C

N ,

22~ Sk

S~ from -equation

2 2

~ ~ ~tk S

LS


7/41

2011 AN, I. M 9, 20147

Rottas Length Scale Equation

To avoid the problem that the () equation is an equation

for the smallest scales, an equation for the large (integral)

scales is needed.

This requires first a mathematical definition of an integral

length scale, L.

In Rottas (1968) approach this definition is based on two-pointcorrelations

Based on that definition of L, an exact transport equation can

be derived from the Navier-Stokes equations (the actual

equation is based on kL) This exact equation is then modelled term-by-term

R, J.C.: M B ,

A G, R. 69 A14, (1968).


8/41

2011 AN, I. M 9, 20148

Two-Point Velocity Correlations

Measurement of velocity fluctuations with two probesat two different locations

For small r, all eddies contribute

For large r, only large scales contribute

For r > L, correlation goes to zero

Integral vs. rproportional to size of large eddies L

rr

Fixed Probe

Shifted Probe

ijR~

1

r

L

Eddies

),('),('

),('),('~

txutxu

trxutxuR

ji

ji

ij rr

rrr+

=

rr


9/41

2011 AN, I. M 9, 20149

Rottas k-kL Model

Integral Length Scale:( , ) ( , , )

iiL x t c R x t r dr

= %

, , .

N.

E ( , ..) .

ry

1

L

Rii

rr


10/41

2011 AN, I. M 9, 201410

Exact Transport Equation Integral

Length-Scale (Rotta)Exact transport equations for =kL (boundary layer form):

( )( )

( ) ( ) ( )

++

+

+

+

=

+

yvpvpR

y

drrr

RdrRR

r

drRy

rxUdrRy

xU

x

U

t

ii

y

kk

iiyikiiik

k

y

y

y

k

k

''1

16

3

8

3

16

3

)(

16

3)(

16

3)()(

2

2

)(

1221

I :

)(xkL=with

y

yidrR

y

rxU12

)(

16

3

+


11/41

2011 AN, I. M 9, 201411

Expansion of Gradient Function

Important term:

R:

22 3

y yy2 3

U ( x r ) r U ( x ) U ( x ) U ( x )r ...y y y y 2

+ = + + +

+

+

+

yyyyyy

ydrRr

y

xUdrRr

y

xUdrR

y

xUdrR

y

rxU

12

2

3

3

122

2

1212

)(

2

1)()()(

0)(

122

2

=

yy drRr

y

xU

D R

ry

1Rij


12/41

2011 AN, I. M 9, 201412

I 3=0 N N

Transport Equation Integral

Length-Scale (Rotta)Transport equations for kL:

( )

+

+

=

+

yyqcc

yxUL

yxULuv

xU

t

tL

ii

k

k

2/32

3

33

32

)()()()(

E :

33

3

2

/

/

yU

yUccL l

=

P 3 :

N

N 03=


13/41

2011 AN, I. M 9, 201413

Virtual Experiment 1D Flow

?

Logarithmic layer Lt=y

2

122 0y y

Ur R dr

y

=

)(~

)(~

1212 y

II

y

IrRrR

rr

liu turbulence sas modeling

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