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LABORATOIRE D’ETUDES AERODYNAMIQUES (LEA) Université de Poitiers , CNRS , ENSMA. Progress in Wall Turbulence: Understanding and modelling Lille, France, April 21-23, 2009. Accounting for wall effects in explicit algebraic stress models. Thomas GATSKI. - PowerPoint PPT Presentation
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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