Elliptic-blending k - ε model

Elliptic-blending k −ε model
Validation and verification
Sylvain Lardeau

Introduction
Assessing performance of the newly implemented elliptic
blending model from University of Manchester, Billard &
Laurence (2012), in canonical and complex flow
configurations.

Introduction
Assessing performance of the newly implemented elliptic
blending model from University of Manchester, Billard &
Laurence (2012), in canonical and complex flow
configurations.
1. Formulation of the baseline model (comments and
adaptations needed)
2. Wall-treatment for irregular meshes
3. Results

Formulation
Original model, Billard & Laurence (2012)
Dρk
Dt = Pk +E −ρε+ ∂
µ µt ∂k
+
∂xj 2 σk ∂xj
Dρε 1
= Cε1
Dt τ
Pk −ρC ∗ ε2ε + ∂
µ µt ∂ε
+
∂xj 2 σε ∂xj
Dρϕ
Dt = ρ(1−α3 )fw +ρα 3 ϕ
fh −Pk
k +CDϕ + ∂
µ µt ∂
+
∂xj 2 σϕ ∂
L 2 ∂2 α
∂xj∂xj
= α−1

Formulation
Equation-by-equation modifications
Turbulent kinetic energy k:
Modifications:
Dρk
Dt = Pk +Pb +E −ρε+ ∂
◮ Adding buoyancy source term Pb,
∂xj
µ µt ∂k
+
2 σk ∂xj
◮ What alternative formulation for E ? Expensive to compute,
and only active very near the wall.
◮ Definition of time-scale τ and τlim:
k2
τ =
ε
+C 2 t
ν
ε ; τlim
CT
= max τ, √
3CµϕS
with S = 2SijSij. Where is CT = 0.6 coming from?

Formulation
Equation-by-equation modifications
Turbulent dissipation rate ε:
Dρε
Dt
Modifications:
1
= Cε1
τ
(fcPk +Pb)−ρC ∗ ε2ε + ∂
∂xj
◮ Adding buoyancy source term Pb,
µ µt ∂ε
+
2 σε ∂xj
◮ Testing curvature correction fc, as suggested by Durbin for
k −ω models:
◮ Option for C ∗ ε2 :
fc = 1+α1|η3|+3α1|η3|
C ∗ ε2 =
⎧
⎨ Cε2
⎩ Cε2 +α3 (0.4−Cε2 )tanh
∂ νt ∂k
∂xj σk ∂xj
ε−1
3/2

Formulation
Equation-by-equation modifications
Reduced normal stress ϕ:
Dρϕ
Dt = ρ(1−α3 )fw+ρα 3 fh−ρCµτlimϕ 2 S 2 + ∂
with
fw = − ϕε
2k ; fh = − 1
τ
Modifications:
C1 −1+C2
µ µt ∂ϕ
+
∂xj 2 σϕ ∂xj
Pk +Pb
ϕ−
ρε
2
3
◮ Buoyancy source term added here Pb (is it correct?),
◮ Cross-diffusion term neglected: found to pose significant
problems in many cases with coarse mesh.
◮ Dissipation term Pk ϕ
k by ρCµτlimϕ 2 S 2 to avoid division by
zero when k → 0.

Formulation
All-y + wall-treatment
Those terms needs to be defined at the first cell away from the
wall:
Remarks:
Pk =
(U + uk) 2 νt 0.95(1−νt)
2 classical u∗
u4 ∗ dU
ν
+
dy +
1− dU+
dy +
iterative u∗
ε = Γ νk
+(1−Γ)
y2 E = − u2 k
µ 0.0875exp
Cµu∗k
2κy
−20.66
◮ U + is taken from Reichardt’s law:
U + = 1
κ ln(1+κy+ )+C
ln(y + )
ln(10) −1.01
2
1−e
with C = 1
κ ln
E 1 y +
κ and b = mκ 1
2 C +
y + m
−y +
y + m + y+
y + m
.
e −by+

Formulation
All-y + wall-treatment
Those terms needs to be defined at the first cell away from the
wall:
ϕ + ⎧
⎪⎨ 0.0015y
=
⎪⎩
+2
if y + < y + 1
Fi(t) if y +
i ≤ y+ < y +
i+1
0.30766 log(y+ )
log(10) −0.2775 if y+ ≥ y + 5
Fi(t) = (2t 3 −3t 2 +1)f +
i +(t3 −2t 2 +t)m +
i ∆y+ +(3t 2 −2t 3 )f +
i+1
⎧
˜y
⎨
˜y+ 1 if y
˜y
+ > 17
α =
⎩
with uk = νy + /y.
What about Pb?
−y +
1−exp
24.52
if y + < 17

Results: convergence

Validation
Convergence
Best convergence rate obtained when initial flow condition from:
Turbulence Specification:
turbulence intensity + turbulent viscosity ratio
with Tu = 5% and TVR = 200.
All cases computed in Double Precision.
Residual
1
0.01
1E−4
1E−6
1E−8
1E−10
1E−12
1E−14
1E−16
1E−18
0 1000 2000 3000 4000 5000
Iteration
Continuity
X−momentum
Y−momentum
Tke
Tdr
Phi
Alpha
1E−12
0 200 400 600 800
Iteration
1000 1200 1400
Channel flow backward-facing step
Residual
10
1
0.1
0.01
0.001
1E−4
1E−5
1E−6
1E−7
1E−8
1E−9
1E−10
1E−11
Continuity
X−momentum
Y−momentum
Tke
Tdr
Phi
Alpha

Validation
Convergence
For all cases shown below, the best convergence rate was obtained
when the initial flow condition were computed from: Turbulence
Specification: turbulence intensity + turbulent
viscosity ratio with Tu = 5% and TVR = 200. The following
cases were also run in Double Precision to validate the linearization
method.
Residual
10
0.001
1
0.1
0.01
1E−4
1E−5
1E−6
1E−7
1E−8
1E−9
1E−10
1E−11
1E−12
1E−13
1E−14
1E−15
1E−16
1E−17
1E−18
1E−19
0 2000 4000 6000 8000 10000
Iteration
12000 14000 16000 18000 20000
1E−17
0 400 800 1200 1600 2000 2400 2800
Iteration
Diffuser 2d naca 4412
Residual
10
0.1
0.001
1E−5
1E−7
1E−9
1E−11
1E−13
1E−15
Continuity
X−momentum
Y−momentum
Tke
Tdr
Phi
Alpha

Results: validation of all-y + wall-treatment

Validation — Wall-treatment
Channel flow
7 different meshes, with different y + values but also different grid
ratio above the first cell,
for Reτ =550 but similar conclusion where obtained for Reτ = 2000.
Mesh1 and LowRe Mesh2 Mesh3 Mesh4
y + =1 y + =14 y + =28 y + =34
Mesh5 Mesh6 Mesh7
y + =66 y + =40 y + =14

Validation — Wall-treatment
Channel flow
Realizable k −ε two-layer model:
U/uτ
20
15
10
5
0
DNS
Mesh 1
Mesh 2
Mesh 6
1 10 100
y +

Validation — Wall-treatment
Channel flow
SST k −ω model:
U/uτ
20
15
10
5
0
DNS
Mesh 1
Mesh 2
Mesh 6
1 10 100
y +

Validation — Wall-treatment
Channel flow
Elliptic-blending k −ε model:
U/uτ
20
15
10
5
0
DNS
Mesh 1
Mesh 2
Mesh 6
1 10 100
y +

Validation — Wall-treatment
Channel flow
Realizable k −ε two-layer model:
k/u 2 τ
5
4
3
2
1
0
1 10 100
y +

Validation — Wall-treatment
Channel flow
SST k −ω model:
k/u 2 τ
5
4
3
2
1
0
1 10 100
y +

Validation — Wall-treatment
Channel flow
Elliptic-blending k −ε model:
k/u 2 τ
5
4
3
2
1
0
1 10 100
y +

Validation — Wall-treatment
Channel flow
Realizable k −ε two-layer model:
νt/ν
100
10
1
0.1
0.01
0.001
1 10 100
y +

Validation — Wall-treatment
Channel flow
SST k −ω model:
νt/ν
100
10
1
0.1
0.01
0.001
1 10 100
y +

Validation — Wall-treatment
Channel flow
Elliptic-blending k −ε model:
νt/ν
100
10
1
0.1
0.01
0.001
1 10 100
y +

Results: performance compare to current default
models
(Realizable k −ε two-layer and SST k −ω)

Validation — Performance
Channel flow - Comparison with defaults models
Simulation on fine mesh (y + = 1), forced dp/dx, C ∗ ε2 =cst:
U/uτ
20
15
10
5
0
DNS
Rke 2-layer
SST k −ω
EB k −ε
1 10 100
y +
k/u 2 τ
5
4
3
2
1
0
1 10 100
y +

Validation — Performance
Channel flow - Comparison with defaults models
Simulation on fine mesh (y + = 1), forced dp/dx, C ∗ ε2 =cst:
εν/u 4 τ
0.3
0.2
0.1
0
1 10 100
y +
νt/ν
90
80
70
60
50
40
30
20
10
0
1 10 100
y +

Validation — Performance
Backward facing step - Comparison with defaults models
Set-up:
◮ Periodic boundary conditions,
◮ Variable C ∗ ε2 ,
◮ Iterative u ∗

Validation — Performance
Backward facing step - Comparison with defaults models
Cf
0.003
0.002
0.001
0
-0.001
-0.002
-0.003
-0.004
expe.
Rke 2-layer
SST k −ω
EB k −ε
0 5 10
x/H
15 20
Skin friction coefficient

Validation — Performance
Backward facing step - Comparison with defaults models
y/H
2
1
0
-4 -2 0 2 4 6
x/H
8 10 12 14
Streamwise velocity

Validation — Performance
Backward facing step - Comparison with defaults models
y/H
2
1
0
-4 -2 0 2 4 6
x/H
8 10 12 14
Turbulent kinetic energy

Validation — Performance
Periodic 2d hill - Comparison with defaults models
Comparison with case from Fröhlhich et al. Two meshes used:
fine, wall-resolved, mesh, and coarse mesh.
Set-up:
Coarse mesh
◮ Periodic boundary conditions,
◮ Variable C ∗ ε2 ,
◮ Iterative u ∗

Validation — Performance
Periodic 2d hill - Comparison with defaults models
LES Realizable k −ε two-layer
SST k −ω EB k −ε

Validation — Performance
Periodic 2d hill - Comparison with defaults models
Cf
Comparison with Fröhlhich et al. (2001) LES data:
0.01
0
LES
Rke 2-layer
SST k −ω
EB k −ε
0 1 2 3 4 5 6 7 8 9
x/H
Skin friction coefficient, bottom wall

Validation — Performance
Periodic 2d hill - Comparison with defaults models
y/H
Comparison with Fröhlhich et al. (2001) LES data:
1
0
0 1 2 3 4 5 6 7 8 9
x/H
Streamwise velocity profiles

Validation — Performance
Periodic 2d hill - Comparison with defaults models
y/H
Comparison with Fröhlhich et al. (2001) LES data:
1
0
0 1 2 3 4 5 6 7 8 9
x/H
Turbulent kinetic energy

Validation — Performance
2d diffuser - Comparison with defaults models

Validation — Performance
2d diffuser - Comparison with defaults models
Realizable k −ε two-layer
SST k −ω
Elliptic-blending k −ε

Cf
Validation — Performance
2d diffuser - Comparison with defaults models
0.002
0.001
0
-0.001
Exp.
Rke 2-layer
SST k −ω
EB k −ε
0 10 20 30
x/H
40 50 60
Skin friction coefficient

Validation — Performance
2d diffuser - Comparison with defaults models
y/H
5
4
3
2
1
0
y/H
0
0 5 10 15 20 25 30
x/H
5
4
3
2
1
0 5 10 15 20 25 30
x/H
Streamwise velocity Turbulent kinetic energy

Validation — Performance
3d diffuser - Comparison with defaults models

Validation — Performance
3d diffuser - Comparison with defaults models
Realizable k −ε two-layer SST k −ω
Elliptic-blending k −ε

Where model should be improved:
1. Rotating and strong curvature
⇒ frame-rotation independent solution,
curvature neglected
2. Buoyancy-driven flows
⇒ laminar solution, poor convergence,
algebraic-heat flux

Part 1: Rotation and strong curvature

Failure — need for improvement
Rotating channel flow
Rotating channel flow DNS study from Kristofferson & Andersson
(1993), for Rossby number varying as 0 < Ro < 0.5:
U/U0
45
40
35
30
25
20
15
10
5
0
Ro = 0.1
Ro = 0
DNS
Baseline
CC
Ro = 0.5
-1 -0.5 0
y/H
0.5 1
k/U 2 0
10
5
0
5
0
5
Ro = 0.5
Ro = 0.1
Ro = 0
0
-1 -0.5 0
y/H
0.5 1
Streamwise velocity Turbulent kinetic energy

Failure — need for improvement
NACA 0012 at 10 ◦ incidence
Experimental results from Chow, Zilliac and Bradshaw (1997)
Conclusions from previous studies:
1. T.E. vortex diffuses too quickly with all classical 2-eq. models,
2. lag-model and RSM improves on results,
3. CC very beneficial in predicting correct position/intensity of
vortex

Failure — need for improvement
NACA 0012 at 10 ◦ incidence
Axial velocity:
Without CC
With CC

Failure — need for improvement
NACA 0012 at 10 ◦ incidence
Axial velocity:
Without CC
With CC

Failure — need for improvement
NACA 0012 at 10 ◦ incidence
U/Uin
2
1.8
1.6
1.4
1.2
1
Exp.
Baseline
CC
-0.4-0.2 0 0.20.40.60.8 1
x/C
Axial velocity in the core of the
vortex
Comments:
◮ Without Curvature
correction, vortex bursts
too quickly → not good
for F1,
◮ Current curvature
correction probably not
ideal: needs bound to
get current results, and
changes results for some
of the baseline case → it
is a new model.

Part 2: Buoyancy-driven flows

Failure — need for improvement
Buoyancy driven flow — Tall cavity
Case from Betts and Bokhari (2001)
◮ Symbols: Exp.
◮ Yellow line: v 2 −f
◮ Red line: EB k −ε
Comments:
Too large decay of k close to
the wall (laminar-state like)
⇒ over-prediction of velocity
and over-prediction of
temperature gradient
y/H
Turbulent kinetic energy
1
0.8
0.6
0.4
0.2
0
0 0.01 0.02 0.03
x/H

Failure — need for improvement
Buoyancy driven flow — Tall cavity
Case from Betts and Bokhari (2001)
Temperature Velocity
y/H
308
306
304
302
300
298
296
294
292
290
0 0.01 0.02 0.03
x/H
y/H
1
0.8
0.6
0.4
0.2
0
0 0.01 0.02 0.03
x/H

Failure — need for improvement
Vertical channel — Kasagi & al
Normalized velocity profiles (local scaling):
Hot aiding wall Cold opposing wall
U/U0
20
18
16
14
12
10
8
6
4
2
0
DNS
Std k −ε
v2−f
EB k −ε
1 10
x/H
100
U/U0
20
18
16
14
12
10
8
6
4
2
0
DNS
Std k −ε
v2−f
EB k −ε
1 10
x/H
100

Failure — need for improvement
Vertical channel — Kasagi & al
U/U0
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Velocity Turbulent kinetic energy
DNS
Std k −ε
v2−f
EB k −ε
-1 -0.5 0
x/H
0.5 1
U/U0
6
5
4
3
2
1
0
-1 -0.5 0
x/H
0.5 1