1212 19 september, 20061 numerical simulation of particle-laden channel flow hans kuerten department...
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19 September, 2006
1
Numerical simulation of particle-laden
channel flow
Hans Kuerten
Department of Mechanical EngineeringTechnische Universiteit Eindhoven
19 September, 2006
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Contents:
1. DNS of particle-laden flow
2. Large-eddy simulation (LES)
3. LES of particle-laden flow
4. Reynolds-averaged Navier-Stokes
5. Conclusions
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1. DNS of particle-laden flow
• Turbulent channel flow
• Particles
• Only drag force:
• Elastic collisions with walls
p
tt
dt
ddt
d
vxuv
vx
)),((
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150Re
xy z
• Spectral method: Fourier-Chebyshev• 128 x 129 x 128 points• Second-order accurate time integration• Fourth-order interpolation for fluid velocity at particle position
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Explanation for turbophoresis:
2yp
pyy u
dy
duv
)1()(
1if
vvv 0p
rmsp
H
u
-1 0 10
0.2
0.4
0.6
y
<uy2>
2yp
pyy u
dy
duv
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Comparison with expansion:
y+
<vy+-u
yp+> St=1
0 50 100 150-0.03
-0.02
-0.01
0
0.01
theoryDNS
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2. Large-eddy Simulation:
• Filter with typical size
• Top-hat filter
yduGu 3)();()( yyxx
x
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Effect on energy spectrum:
100
101
10210
-10
10-5
100
kz
E
resolvedscales
subgridscales
DNSfilteredDNS
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Effect on velocity fluctuations:
-1 -0.5 0 0.5 10
0.2
0.4
0.6DNSfilteredDNS<u
y2>
y
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A priori simulations:
• Filter fluid velocity as calculated in DNS with top-hat filter.
• Solve particle equation of motion with filtered fluid velocity:
p
tt
dt
d
vxuv
)),((
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101
102
103
104
5
10
15
20
St=1St=5St=25
Effect on turbophoresis:
t+
cwall
A priori
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3. Real LES of particle-laden flow:
• Subgrid model in Navier-Stokes– Smagorinsky eddy viscosity– Dynamic eddy viscosity– LES grid 32 x 33 x 64
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Subgrid model in particle equation• Retrieve unfiltered velocity from filtered• Only possible for scales present in LES grid
k
)(ˆ kG
0 10 20 30-0.5
0
0.5
1
0 10 20 30-0.5
0
0.5
1
LESh2
0 10 20 30-0.5
0
0.5
1
LESh
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LES velocity fluctuations:
y
<uy2>
-1 -0.5 0 0.5 10
0.2
0.4
0.6
DNS
filtered DNS
dynamic
Smagorinsky
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Wall concentration:
t+
cwall St=5
102
103
1040
2
4
6
8
10DNSa prioridynamicdynamic inverseSmagorinskySmag. inverse
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1000
20
40
60
80
y+
c
DNSdynamicdynamic inverseSmag.Smag. inverse
Concentration in steady state (St=5):
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Dispersion (St=25):
0 50 100 1500
1
2
3
4
y+
vx,rms
DNSdynamicdynamic inverseSmag.Smag. inverse
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Linear velocity interpolation:
t+
cwall St=5
102
103
1040
2
4
6
8
10DNSa priori4th order4th order inverse2nd order2nd order inverse
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Linear velocity interpolation:
0 50 100 1500
0.2
0.4
0.6
0.8
DNSfourth orderfourth order inverse2nd order2nd order inverse
y+
vz,rms
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First conclusions:• Dynamic model performs better than
Smagorinsky.• Linear interpolation is inaccurate.• Inverse filtering improves results of
dynamic model.• Still discrepancy with DNS results:
– A priori results do not agree well with LES.– Inverse filter is arbitrary.
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Approximate Deconvolution Model (Stolz et al., 2001):
• Approximate unfiltered velocity in LES:
• Add relaxation term for dissipation.
• Deconvolution also in particle equation.
ii
N
i
j
ji
j
ji
uuu
x
uu
x
uu
1
0
*
**
)(
GGI
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Dispersion (St=25):
y+
vx,rms
0 50 100 1500
1
2
3
4DNSdynamicdynamic inverse
ADMADM inverse
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Concentration (St=5):
1000
20
40
60
80
y+
c
DNSdynamicdynamic inverse
ADMADM inverse
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Drift velocity (St=1):
0 50 100 150-20
-15
-10
-5
0
5 x 10-3
y+
<vy-u
yp>
DNSdynamicdynamic inverse
ADMADM inverseSmag.Smag. inverse
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High Reynolds number simulations:
• No DNS of particle-laden flow.
• DNS data of channel flow is available (Moser, Kim & Mansour) at Re=590.
• Particle velocity rms should be close to fluid velocity rms at low Stokes number.
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Dispersion (Re=590, St=1):
y+
vx,rms
0 200 400 6000
1
2
3
4DNS (fluid)dynamicdynamic inverseADMADM inverse
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y+
vy,rms
0 200 400 6000
0.5
1
1.5
DNS (fluid)dynamicdynamic inverseADMADM inverse
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y+
vz,rms
0 200 400 6000
0.5
1
1.5
DNS (fluid)dynamicdynamic inverseADMADM inverse
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4. Reynolds-averaged Navier-Stokes
• Often used in CFD packages
•
• Only mean velocity is known and some information about turbulence
pp
ttttt
dt
d
vxuxuvxuv
)),(('))(()),((
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k-ε model• k and ε are known• • isotropic
Reynolds-stress model• all Reynolds stresses and ε
are known• anisotropic
wku 3/2'
For both models:
• w is constant during time interval
• eddy-turnover time, te=ck/ ε
• crossing trajectories, tc depends on τp
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Results:
• a priori: obtain RANS quantities from DNS
• a posteriori: real RANS simulations performed with fluent on fine grid
• same test case as in DNS and LES
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Velocity fluctuations (St=1):
0 50 100 1500
0.5
1
1.5
2DNSreal k-real RSM
a priori k- a priori RSM
y+
vy,rms
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Velocity fluctuations (St=1):
y+
vx,rms
0 50 100 1500
1
2
3DNSreal k-real RSM
a priori k- a priori RSM
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Particle concentration (St=1):
0 0.5 1 1.5 2
x 104
0
5
10
15DNS
real k-real RSMa priori k- a priori RSM
t+
cwall
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5. Conclusions (LES):
• A priori: turbophoresis is changed if eqs of motion are solved with .
• Real LES confirms this.
• Inverse filtering improves results.
• Similar results for particle dispersion.
• Inverse ADM gives best results for concentration and dispersion.
• Also applicable at higher Reynolds number.
u
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Conclusions (cont.)
• Linear interpolation of fluid velocity is inaccurate.
• Smagorinsky model is inaccurate.
• Inverse filtering hardly improves Smagorinsky results.
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Conclusions (RANS)• Reynolds-stress model gives accurate
results for particle dispersion if stress tensor is accurately predicted.
• k-ε model is not accurate because of isotropy of velocity fluctuations.
• Turbophoresis is not well predicted since preferential concentration cannot be taken into account.