Accounting for wall effects in explicit algebraic stress models
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Transcript of Accounting for wall effects in explicit algebraic stress models
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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
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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
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Explicit Algebraic Methodology Explicit Algebraic Methodology
Anisotropy tensor :
Rodi 1976
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Weak Equilibrium : 0ijdb
dt
ij ij
k
D
D k
Implicit algebraic equation : *( ) ( )ij ijij ij ijP P
k k
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Explicit algebraic Model (EASM ) :
Galerkin Projection :
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N
i ii
b T
( , , , )i
k Pf
R
Introduction of the Elliptic Blending into EASM
Accounting for the blocking effect of the wall
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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
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* 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
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Standard EASMStandard EASM
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Elliptic Blending EASMElliptic Blending EASM
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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
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Several possibilities investigated
• Exact representation in 1D• Exact representation in 2D (singularities possible)• Approximate representation in 3D
• Approximate representation
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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
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( , , , , , ),i
k Pf
R P Q
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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.)
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y+
ij+
Boundary layer at Re = 20800
Lille experiment
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y+
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Channel flows (Moser et al.;
Hoyas & Jimenez)
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EB-EASM#2 : b= β1S+β2 M
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Channel flow at Reτ= 590
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y+
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Couette-Poiseuille Flows (DNS: Orlandi)
Uw
-h
hy
x
y/h
PT : Poiseuille-type flowIT : Intermediate-type flowCT : Couette-type flow
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y/h
PT : Uw = 0.75 Ub IT : Uw = 1.2 Ub
CT : Uw = 1.5 Ub
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y/h
Intermediate- type (IT) at Reτ= 182
Poiseuille- type (PT) at Reτ= 204
Couette- type (CT) at Reτ= 207
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Shear free turbulent boundary Shear free turbulent boundary layer layer
S=W= 0 everywhere in the boundary layer
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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
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CONCLUSION
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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