Elliptic relaxation for near wall turbulence models

Elliptic relaxation for near wall
turbulence models
J.C. Uribe
University of Manchester
School of Mechanical, Aerospace & Civil Engineering
Elliptic relaxation for near wall turbulence models – p. 1/22

Introduction
Outline
The elliptic relaxation approach
The v 2 − f model
The ϕ − f model
Test cases and applications
Conclusion
Elliptic relaxation for near wall turbulence models – p. 2/22

Introduction
Why modelling the near-wall region?
In the near-wall region, viscosity and nohomogeneties
are dominant.
High shear and large rates of turbulence production
are present.
Here is where the skin friction and heat transfer are
controlled, therefore, of vital importance for
engineering applications that require these quantities.
The wall normal fluctuations are reduced therefore
reducing mixing.
Elliptic relaxation for near wall turbulence models – p. 3/22

The wall effects:
Introduction
No-slip: The boundary condition on the mean
velocities creates large gradients where the turbulent
production originates.
Elliptic relaxation for near wall turbulence models – p. 4/22

The wall effects:
Introduction
No-slip: The boundary condition on the mean
velocities creates large gradients where the turbulent
production originates.
5
4
3
2
1
k
-uv
0
0 100 200 300 400
y+
0.25
0.2
0.15
0.1
0.05
P k
0
0 100 200 300 400
y+
Elliptic relaxation for near wall turbulence models – p. 4/22

The wall effects:
Introduction
No-slip: The boundary condition on the mean
velocities creates large gradients where the turbulent
production originates.
5
4
3
2
1
k
-uv
0
0 100 200 300 400
y+
0.25
0.2
0.15
0.1
0.05
P k
0
0 100 200 300 400
y+
Low Reynolds number effects: Interaction between
energetic and dissipative scales.
Elliptic relaxation for near wall turbulence models – p. 4/22

The wall effects:
Introduction
Blocking effect: The impermeability condition affects
the flow by adjusting the pressure field to ensure the
incompressibility condition.
Wall echo: Image term in Green’s function at the other
side of the wall produces an increase in the pressure.
Elliptic relaxation for near wall turbulence models – p. 5/22

The wall effects:
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Introduction
Blocking effect: The impermeability condition affects
the flow by adjusting the pressure field to ensure the
incompressibility condition.
Wall echo: Image term in Green’s function at the other
side of the wall produces an increase in the pressure.
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Elliptic relaxation for near wall turbulence models – p. 5/22
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The wall effects:
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Introduction
Blocking effect: The impermeability condition affects
the flow by adjusting the pressure field to ensure the
incompressibility condition.
Wall echo: Image term in Green’s function at the other
side of the wall produces an increase in the pressure.
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p i (x, t) = − 1
4π
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S(x ′ , t)
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Elliptic relaxation for near wall turbulence models – p. 5/22
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Elliptic relaxation approach
Starting from the Reynolds-stress equation:
with
Duiui
Dt
+ εuiuj
k = Pij + ℘ij + ∂
∂xk
℘ij = −
νTkl
Πij + εij − uiuj
∂uiuj
∂xl
ε
k
+ ν ∂2 uiuj
∂x 2 k
To solve ℘ij a non-homogeneous elliptic equation is
derived. By integrating the pressure equation and assuming
that the two point correlation Ψij can be approximated as
Ψij(x, x ′ ) = Ψij(x ′ , x ′ ) exp(−r/L) (Durbin,1993)
Elliptic relaxation for near wall turbulence models – p. 6/22

Elliptic relaxation approach
The solution is the modified Helmotz-type equation:
℘ij − L 2 ∇ 2 ℘ij = ℘ h ij
Where any quasi homogenous model can be used for ℘ h ij .
In fact the model solves for fij = ℘ij/k in order to enforce
correct behaviour at the wall.
The length scale is prescribed as:
L = CL max
k 3/2
ε
, Cη
ν 3
ε
1/4
Elliptic relaxation for near wall turbulence models – p. 7/22

Elliptic relaxation approach
The model accurately represents the asymptotic
behaviour of the stresses even in the near wall layer.
There is no need to prescribe the distance from the
wall, so no ill defined in complex geometries.
No use of empiricall damping functions.
Good representation of the stresses in channel flows
and boundary layers.
Requires 13 transport equations ( 6 for uiuj, 6 for fij
and one for ε).
Numerical difficulties with wall boundary conditions for
f22, f12, f23.
Equations required to be solved coupled.
Elliptic relaxation for near wall turbulence models – p. 8/22

The v 2 − f model
In order the simplify the RSM, the elliptic relaxation is
introduced to the eddy viscosity approximation
(Durbin,1995).
Use of correct velocity scale near the wall, νt = Cµv 2 T
ν t
40
30
20
10
0
0 100 200 300
y+
ν t = -uv/dU/dy
νt = C μ k 2 /ε
νt = C μ k υ 2 /ε
Elliptic relaxation for near wall turbulence models – p. 9/22

L 2 ∂2 f
∂x 2 j
The v 2 − f model
Solve transport equations for k, ε and v2 , and elliptic
equation for f22
Dv2 ε ∂
= kf − v2 + ν +
Dt k ∂xj
νt
∂v2 σk ∂xj
− f = 1
T (C1
v
− 1)
2
2 Pk
− − C2
k 3 k
v 2 has no tensorial meaning, is now a scalar. So is f.
Wall boundary condition f = − 20νv2
εy 4
Elliptic relaxation for near wall turbulence models – p. 10/22

Advantages:
The v 2 − f model
Reproduces the correct behaviour of the turbulent
viscosity near the wall
No need to include the distance from the wall. Can be
used in any geometry
Improves predictions on separating flows, as well as
heat transfer and skin friction.
Drawbacks:
One transport and one elliptic equations more than the
standard k − ε
Stiffness of the boundary condition makes it necessary
to solve v 2 − f coupled.
Elliptic relaxation for near wall turbulence models – p. 11/22

The ϕ − f model
Instead of solving v 2 , a transport equation is solved for the
ratio ϕ = v 2 /k (Laurence et al., 2004):
D(υ 2 /k)
Dt
= f − (υ2 /k)
k
P + ∂
∂xk
ν + νt
σ (υ 2 /k)
ν + νt
σ (υ 2 /k)
Where X is the "cross diffusion" term from the
transformation:
X = 2
∂(υ
k
2 /k) ∂k
∂xk ∂xk
∂(υ2
/k)
+ X
∂xk
Elliptic relaxation for near wall turbulence models – p. 12/22

The ϕ − f model
Advantages of the substitution are:
The term εv 2 /k is not present, leaving stable viscous
diffusion as a dominant mechanism near the wall.
The wall boundary condition for f is reduced to:
fw = lim
y→0
−2ν(υ 2 /k)
y 2
which is more convenient and easier to reproduce
since it has y 2 instead of y 4 of the original model.
The LDM modification (Lien and Durbin,1996)
proposed as a method to uncouple the equations,
does not ensure the correct behaviour in the region far
from the wall.
(1)
Elliptic relaxation for near wall turbulence models – p. 13/22

The ϕ − f model
A changes of variable is applied to the original model:
Leading to:
Dϕ
Dt
= f − P ϕ
k
L 2 ∇ 2 f − f = 1
T (C1 − 1)
f = f + 2ν(∇ϕ∇k)
k
2 νt ∂ϕ ∂k
+
k σk ∂xj ∂xj
ϕ − 2
3
− C2
+ ν∇ 2 ϕ (2)
+ ∂
∂xj
P
k
νt
σk
− 2ν
k
∂ϕ
∂xj
∂ϕ
∂xj
∂k
∂xj
(3)
− ν∇ 2 ϕ
Far from the wall the last two terms are negligible, ensuring
the correct behaviour.
(4)
Elliptic relaxation for near wall turbulence models – p. 14/22

The ϕ − f model
The boundary condition of f is zero at the wall, improving robustness.
The equations can be solved totally uncouple.
Useful for unstructured codes (Implemented in the industrial
Code_Saturne of EDF)
Modification ensures correct behaviour far from the wall (see
Laurence et al., 2004)
0.02
0
-0.02
-0.04
ƒ
ƒ hom
L 2 Δƒ
term neglected in LDM
term neglected in ϕ model
0.004
0.002
0
-0.002
20 40 60 80
150 200 250 300 350 400
y+
y+
a) b) Elliptic relaxation for near wall turbulence models – p. 15/22

Cf
0.005
0.0045
0.004
0.0035
0.003
0.0025
Test cases
Flat plate: Channel Flow:
(Exp: Weighardt and Tillman) (Kim et al., 1987)
Exp
LDM
ϕ - f
2e+06 4e+06 6e+06 8e+06
Re x
Elliptic relaxation for near wall turbulence models – p. 16/22

Test cases
Natural convection: Tall cavity (Exp: Betts and Bokhari,
1995).
V(m/s)
V (m/s)
V(m/s)
0.2
0.1
0
-0.1
-0.2 0.2
0.1
0
-0.1
-0.2 0.1
0.05
0
-0.05
-0.1
-0.15
Exp y/H = 0.50
ϕ - f
LDM
k-ε L-S
Exp y/H = 0.70
Exp y/H = 0.95
a)
0 0.02 0.04 0.06 0.08
x (m)
V (m/s)
V (m/s)
V (m/s)
0.2
0.1
0
-0.1
-0.2
0.2
0.1
0
-0.1
-0.2
0.2
0.1
0
-0.1
Exp y/H = 0.50
ϕ - f
LDM
k-ε L-S
Exp y/H = 0.70
Exp y/H = 0.95
-0.2
0 0.02 0.04 0.06 0.08
x(m)
a) Ra = 0.86x10 6 b) Ra = 1.43x10 6
b)
Elliptic relaxation for near wall turbulence models – p. 17/22

Cp
0.8
0.6
0.4
0.2
0
Test cases
Separated flows: Asymmetric plane diffuser (Exp: Buice
and Eaton, 1997)
Exp
k-ε
k - ω
ϕ model
-20 0 20 40 60
x/h
y/H
y/H
4
3
2
1
0
4
3
2
1
Profiles at x/H = 16.14
0 0.1 0.2 0.3 0.4 0.5
U/Ub
0
0 0.005 0.01
vv/Ub
0.015 0.02
y/H
y/H
4
3
2
1
ϕ model
k-ε
k-ω
0
0 0.005 0.01
uu/Ub
0.015 0.02
4
3
2
1
0
-0.004 -0.002 0 0.002 0.004
uv/Ub
Elliptic relaxation for near wall turbulence models – p. 18/22

Test cases
Flow over periodic hills (Fine LES, Temmerman and
Leschziner, 2001)
Streamlines: LES, k − ε, ϕ
Y/h
Y/h
3
2.5
2
1.5
1
0.5
3
2.5
2
1.5
1
0.5
LES
ϕ model
k - ε
1 2 3 4
4 5 6 7 8
U+x/h
Elliptic relaxation for near wall turbulence models – p. 19/22
a)
b)

Test cases
Three dimensional symmetric bump: Work in progress.
(Simpson, 2002)
Elliptic relaxation for near wall turbulence models – p. 20/22

Test cases
Multiple impinging jets. Work in progress.
(Geers et. al, 2003)
Top: Exp, Middle: k − ε, Bottom: ϕ − f
Elliptic relaxation for near wall turbulence models – p. 21/22

Conclusions
The elliptic relaxation approach reproduces the
near-wall effects.
No need for damping functions or distance from the
wall.
With full Reynolds stress model too expensive (13
equations)
v 2 − f with EVM, cheaper but stiff due to boundary
conditions.
ϕ − f better. Same effects but allows for uncoupling of
equations, can be used in industrial codes.
Tested in separated, impinging and buoyant flows.
It is still more expensive than two-equation models.
Elliptic relaxation for near wall turbulence models – p. 22/22