Perspectives in m odeling w all effects in the RANS approach - - PowerPoint PPT Presentation

Svetlana V. Poroseva

Perspectives in m odeling w all effects in the RANS approach

Mechanical Engineering Departm ent University of New Mexico

National I nstitute of Aerospace Future Directions in CFD Research, A Modeling and Sim ulation Conference

August 6 -8 , 2 0 1 2

Outline

• Identification of problems with the RANS approach
• Possible directions for their solution

RANS approach: identity crisis

General belief: RANS models are one- or two-equation turbulence models that require the modeling of wall effects

2

1

i j i i i j j j i j j

u u U U U P U t x x x x x ν ρ ∂ < > ∂ ∂ ∂ ∂ + + = − + ∂ ∂ ∂ ∂ ∂ ∂

2 3

j i i j T ij j i

U U u u k x x ν δ   ∂ ∂ < > = − + −     ∂ ∂  

, , , , , / , .... k l k k k k k k multi scale ω ε ξ ϕ ε ω − − − − − − − −

RANS approach: reality

RANS model is any statistical closure obtained from the infinite set

• f the Reynolds-Averaged Navier-Stokes equations

i j i j

u u DU Dt x ∂ < > = − +⋅⋅⋅ ∂

⋅ ⋅ ⋅ + − + ∂ > < ∂ − = > <

ij ij k k j i j i

x u u u Dt u u D ε Π ⋅ ⋅ ⋅ + − + ∂ > < ∂ − = > <

ijk ijk l l k j i k j i

x u u u u Dt u u u D ε Π

2 nd order 3 rd order 1 st order The order of a closure is determined by the order of the moments for which the equations are solved. Higher the order, higher the m odel fidelity.

• Two-equation models are the first-order closures. They are the

simplest from the RANS family of models with not so much physics left.

• The purpose of any corrections in such models is not to bring

more physics, but to compensate for its lack.

W hy not to increase the closure order instead?

The infinite set of the Reynolds-Averaged Navier-Stokes equations describes com pletely the turbulent flow physics from the statistics point of view

Second-order closures ( RSTMs)

2

1

i j i i i j j j i j j

u u U U U P U t x x x x x ν ρ ∂ < > ∂ ∂ ∂ ∂ + + = − + ∂ ∂ ∂ ∂ ∂ ∂

2

1 2

i j i j j i k j k i k k k k i j k j i k i j i j j i k k k k

u u u u U U U u u u u t x x x u u u p p u u x x x u u u u x x x x ρ ν ν ∂ < > ∂ < > ∂   ∂ + = − < > + < >   ∂ ∂ ∂ ∂     ∂ < > ∂ ∂ − − < > + < >   ∂ ∂ ∂     ∂ < > ∂ ∂ + − < > ∂ ∂ ∂ ∂

ij

P

( ) T ij ij

D + Π

( ) M ij ij

D ε −

Algebraic models did not show much potential in flows of practical importance

Current state-of-the art in RSTMs

Term s to m odel Modeling directions

ij

ε

( ) i j k T ij k

u u u D x ∂ < > = ∂

1 ( )

ij i j j i

p p u u x x Π ρ ∂ ∂ = − < > + < > ∂ ∂

1 ( )

j i ij ij j i

pu pu x x Φ Π ρ ∂ < > ∂ < > = − > + ∂ ∂

i j k

u u u < >

Third-order m om ents m odeling

is a symmetric tensor, so its model should be symmetric Standard m odel: Daly & Harlow (1970): Hanjalić & Launder (1972):

i k i k l l m m

u u u u u u u x ∂ < > < > < > ∂ 

Better choice:

i k k l i l i k l l m i m k m m m m

u u u u u u u u u u u u u u u x x x ∂ < > ∂ < > ∂ < > < > < > + < > + < > ∂ ∂ ∂ 

i k l

u u u < >

A tensor can only be m odeled as a tensor of the sam e tensor rank, index order, covariance, and sym m etry

Relevance to w all effects

. . 5 1 . r / R . . 5 1 . < u v > N = N = . 6 u

2 *

. . 5 1 . r / R . . 5 1 . < v > N = N = . 6 < v >

2 2

= >

W W

Daly & Harlow (1970)

Hanjalić & Launder (1972) - - - Rotating pipe flow :

N W / U =

Kurbatskii & Poroseva, Int. J. Heat Fluid Flow, 1999

Nagano & Tagava, TSFP 7,u1991

Pipe Flat plate Back-step

Relevance to w all effects

Back-step flow : w all + separation effects

Kasagi & Matsunaga, Int. J. Heat Fluid Flow, 1995

Solution for turbulent diffusion

Higher-order closures can be a required choice:

⋅ ⋅ ⋅ + − + ∂ > < ∂ − = > <

ijk ijk l l k j i k j i

x u u u u Dt u u u D ε Π

Pressure-containing correlations m odeling

1 ( )

j i ij ij j i

pu pu x x Φ Π ρ ∂ < > ∂ < > = − > + ∂ ∂

Choices for m odeling the pressure diffusion

• neglect
• absorb in a model for the turbulent diffusion
• model separately from the pressure-strain correlations, somehow

Relevance to w all effects

cannot be neglected and particularly near a w all

Hanjalić & Launder, 2011

Relevance to w all effects

cannot be absorbed into a turbulent diffusion m odel

Spalart, JFM,1988

Flat plate, ZPG

× − ρ 1

( ) ( ) ( ) r s w i i i i i i i i

pu pu pu pu x x x x ∂ < > ∂ < > ∂ < > ∂ < > = + + ∂ ∂ ∂ ∂

i i m m

u u u

,

5 1 > <

(Lumley, Adv. Appl. Mech.,1978; Poroseva, THMT, 2000)

Pressure-strain correlations m odeling

1 ( )=

j ( r ) ( s ) ( w ) i ij ij ij ij ij j i

pu pu x x Φ Π Φ Φ Φ ρ ∂ < > ∂ < > = − > + + + ∂ ∂

Choices for m odeling

• linear
• non-linear
• other approaches (Q-model)

ij

Φ

Relevance to w all effects

Only the sim plest linear m odel required w all corrections

Rotating pipe flow : IP (Naot et al., 1970), LRR (Launder et al., JFM, 1975),

SSG (Speziale et al., JFM, 1991), LSSG (Gatski & Speziale, JFM, 1993), Q-model (Kassinos et al., Int. J. Heat Fluid Flow, 2000) (Poroseva, CTR Ann. Res. Briefs, 2001)

TCL m odel (Craft & Launder, 1996):

( ) r ij

Φ

Solution for pressure term s

• Non-linear and Q models are too complex, and still need

wall corrections

• Derived under assumption of turbulence homogeneity.
• It is unphysical to model the pressure diffusion separately

from the pressure-strain correlations.

• Finite boundary conditions for both: pressure-strain

correlations and the pressure diffusion.

Solution: m odeling Πij instead

• Πij are originally present in the RANS equations
• easy boundary conditions
• No need to model the pressure diffusion separately

Hanjalić & Launder, 2011

Modeling ideas

1

( r ) ( s ) ( w ) ij j i ij ij ij i j

p p u u x x Π Π Π Π ρ   ∂ ∂ = − < > + < > = + +     ∂ ∂  

, , , , ,

1 1 1 1 1 2 4 1 1 1 4

i m n n i m m n i mn j j j j i j i

p ' u U' u u dV u u u dV x r r p' u p' u dS' r n' n' r ρ π π π ∂ ′ ′ ′ ′ ′ ′ ′   − < >= − < > − < >   ∂ ∂ < >   ∂   − − < >     ∂ ∂    

∫∫∫ ∫∫∫ ∫∫

m - j; if m = n, then anmji = 0; if m = j, then anmji = 2 < un ui >

( ) ,

( )

M r ij nmij nmji m n

a a U Π = +

1 1 2

n i nmji m j

u' u a dV ' x' x' r π ∂ < > = − ∂ ∂

∫∫∫

this is NOT an assumption of turbulence homogeneity Poroseva, THMT, 2000

Modeling ideas ( Cont.)

1 1 2

n i nmji m j

u' u a dV ' x' x r π ∂ ∂ = − < > ∂ ∂

∫∫∫

Rotta (1951), Launder, Reece, Rodi (1975)

Different integrals have different properties and their analysis results in different m odels I n m odeling Π( r)

ij :

1 1 2

n i nmji m j

u' u a dV ' x' x' r π ∂ < > = − ∂ ∂

∫∫∫

I n m odeling Φ( r)

ij : To apply the Green’s theorem to this expression,

• ne has to assume the turbulence homogeneity

( ) ) , 1 4 ( ) ( ) , , , , 1 5 5 M r a a U ij nmij nmji m n u u U u u U u u U u u U R i m m j j m m i i m j m j m i m Π = + = − < > + < > + < > + < > +

Linear m odel

1 ( ) ( ) ( ) ( ) 1 1 2 , , 1 2 , , 2 1 ( ) ( ) ( 4 ) , 1 2 , , 1 2 , 2 R k C C U U C C u u U u u U i j j i i m m j j m m i C C u u U u u U C C u u U i m j m j m i m m n m n ij δ = + + + − − < > + < > + − − < > + < > + − − < >

Poroseva, THMT, 2000

Application to lim iting states

• Transforms to LRR model in homogeneous turbulence:

1 1 2

n i nmji m j

u' u a dV ' x' x' r π ∂ < > = − ∂ ∂

∫∫∫

n i −

1

6 4 ' 55 11 C C = +

1 2

1 5 5 2 C C − − =

• Satisfies the exact solution for isotropic turbulence

subjected to sudden distortion with any value of C1 and C2.

8 2 ( ( )) 15 15

nmji ni mj nm ji nj mi

a k δ δ δ δ δ δ = − +

• Two-component turbulence ( , β = 2 or 3)

2 2 1,1 , 1 2 2 2 1,1 ,

(4 ) 2 ( ) (10 4 ) 8 ( ) U k u U k u C C U k u U k u

β β β β β β β β

− < > + − < > = − − < > + − < >        

2 1

u < >= 

• Two-component axisymmetric turbulence ( )

1 2

0.5 C C = − ⋅

2 2 2 3

u u k < >=< >=  

• Two-component axisymmetric homogeneous turbulence

2

0.4 C =

C1, C2 can be kept as const in a given flow, but vary depending

• n the flow geometry and some other parameters.

More research and data are required to suggest their functional form.

26

SSG model coefficients:

DNS data: Hoyas & Jimenez, 2006 Plot: Hanjalić & Launder, 2011

Prelim inary tests

(0.4 )

i t k i j j i k i

U Dk k C u u wall Dt x x x ν ε ν σ     ∂ ∂ ∂ = − + < > − + + +     ∂ ∂ ∂      

1 2 i t i j j i i

U D (C u u C ) Dt k x x x

ε ε ε

ν ε ε ε ε ν σ     ∂ ∂ ∂ = − < > + + +     ∂ ∂ ∂      

2

/ 1.5, 1.92, 0.09

k

C C

ε ε µ

σ σ = = =

Iaccarino & Poroseva, CTR Ann. Res. Briefs, 2001 Poroseva, THMT, 2000;

( ) ,

1 ( 0.6 )

r i j k

u p C P ρ − < > = − +

1 2

15 3 2

k

C C C = +

Diff BS stand 0.2 0.9 0.6 1 1 0.8 0.6 0.8 0.6 0.6 1.9 1.5 2.12 2.2 1.85 1.5 1.85 1.44

1

Cε

flow Wake ML RJ PJ BL1

k

C

BL2

BL1 BL2

β = 19 6 . β =

Back-step flow

new model standard k-ε model model

2

v f < > −

Skin-friction coefficient

Diffuser

new model standard k-ε model model

2

v f < > −

1

0.6, 1.5

k

C Cε = =

Skin-friction coefficient

30

Com bustion cham ber

new standard k-ε model model

2

v f < > −

1

1.7 Cε =

0 7 r / R . = 1 68 r / R . =

3 6 r / R . =

axial velocity swirl velocity

Re = 75,000

Future Direction in Turbulence Modeling

31

from

the state–of-the-art

(as based on imagination) to

the state-of–the-science

(as based on logical reasoning) high-order statistical closures

Questions?

poroseva@unm .edu