LABORATOIRE D’ETUDES AERODYNAMIQUES (LEA) Université de Poitiers , CNRS , ENSMA

Progress in Wall Turbulence: Understanding and modelling

Lille, France, April 21-23, 2009

OutlineOutline Introduction of wall effects into Explicit Algebraic Stress Models

Explicit Algebraic Methodology

Results

Conclusion

Introduction of the wall effects Choice of the basis

Channel Flows and boundary layer Couette – Poiseuille Flows Shear – free turbulent boundary layer

Explicit Algebraic Methodology Explicit Algebraic Methodology

Anisotropy tensor :

Rodi 1976

3

Weak Equilibrium : 0ijdb

dt

ij ij

k

D

D k

Implicit algebraic equation : *( ) ( )ij ijij ij ijP P

k k

Explicit algebraic Model (EASM ) :

Galerkin Projection :

4

N

i ii

b T

( , , , )i

k Pf

R

Introduction of the Elliptic Blending into EASM

Accounting for the blocking effect of the wall

Elliptic Elliptic BlendingBlending Reynolds stress model Reynolds stress model ( Manceau &Hanjalic,2002 )

EB-RSM Based on elliptic relaxation concept of Durbin , 1991

Numerical robustness and reduction of the number of equations

5

* 3 3(1 ) w hij ij ij h SSG

ij ij

orientation of the wall

n : pseudo wall – normal vector

25 ( )

31 22 3

+M M I M Mτ τ- τ τ- w

jki k

Blending function α • obtained from elliptic relaxation equation :

2 2 1L 0 • At the wall

1 • Far from the wall

Standard EASMStandard EASM

6

Elliptic Blending EASMElliptic Blending EASM

7

8

9

Choice of the basis Choice of the basis Incomplete representation unavoidable (even in 2D)

Selected Models

EB-EASM #1 b= β1S+β2(SW-WS)+β3(S²-1/3{S²}I)

EB-EASM #2 b= β1S+β2M

Nonlinear

Linear

10

Several possibilities investigated

• Exact representation in 1D• Exact representation in 2D (singularities possible)• Approximate representation in 3D

• Approximate representation

Galerkine Projection

N

i ii

b T

2 SWMQ SMP

solution of the form :

New invariants introduced by the near-wall model

Q Boundary layer Invariant

P Impingement Invariant

Channel flow

Impinging jet

11

( , , , , , ),i

k Pf

R P Q

EB-EASM#1 : b= β1S+β2(SW-WS)+β3(S²-1/3{S²}I)

y+ 12

Results in channel flows and boundary Results in channel flows and boundary layer layer

Channel flow at Reτ= 590 (Moser et al.)

y+

ij+

Boundary layer at Re = 20800

Lille experiment

13

y+

14

Channel flows (Moser et al.;

Hoyas & Jimenez)

EB-EASM#2 : b= β1S+β2 M

y+15

Channel flow at Reτ= 590

y+

16

Couette-Poiseuille Flows (DNS: Orlandi)

Uw

-h

hy

x

y/h

PT : Poiseuille-type flowIT : Intermediate-type flowCT : Couette-type flow

17

y/h

PT : Uw = 0.75 Ub IT : Uw = 1.2 Ub

CT : Uw = 1.5 Ub

18

y/h

Intermediate- type (IT) at Reτ= 182

Poiseuille- type (PT) at Reτ= 204

Couette- type (CT) at Reτ= 207

Shear free turbulent boundary Shear free turbulent boundary layer layer

S=W= 0 everywhere in the boundary layer

19

Far from the wall : 2 0

at the wall :

25 4 5

2 2 25 4 5 4

3( 3 ) 1

18 12 2 2

a a a

a a a a

10 0

32

0 03

10 0

3

M

1 0 06

10 03

10 06

b

20

CONCLUSION

21

Introduction of wall blocage: • Through invariants involving • Implications in terms of tensorial representation• Polynomial representation is not possible with less than 6-term bases• Singularities may be faced

Applications to channels flows, boudary layers, Couette-Poiseuille flows:

• No singularities faced• Accurate representation of the anisotropy

Simplified model (2-term basis)• Linear model• Partial representation of the anisotropy• 2-component limit• Similar to the V2F model, but with more physics

More complex flows

2 SWMQ SMP