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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Wall concentration:

t+

cwall

101

102

103

104

5

10

15

20

St=1St=5St=25

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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.