simulations of inertial point-particles at ntnu

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Simulations of inertial point-particles at NTNU

Lihao Zhao, Christopher Nilsen, Mustafa Barri and Helge Andersson

Dept. of Energy and Process Engineering, NTNU, Norway

Cost Action FP1005 meetingOct. 13th-14th, 2011, Nancy, France

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Content

Part I. Overview of recent activities.

Part II. Some recent examples

1. Two-way coupling of fiber suspensions - Lihao Zhao

2. Rotation of fiber in laminar Couette flow - Christopher Nilsen

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History

• Joint Industry Project (Gassco, Statoil, Norsk Hydro) 2003 – 2007

• Collaborative research with TU Delft, Professor B.J. Boersma, Laboratory for Aero and Hydrodynamics

• Pål H. Mortensen, PhD (NTNU Jan. 2008)– ”Particle dynamics in wall-bounded turbulence”

• Jurriaan J.J. Gillissen, PhD (TU Delft, Sep. 2008)– ”Numerical simulation of fiber-induced drag reduction in turbulent channel

flow”

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Current

• Lihao Zhao PhD-student Aug. 2008 – Feb. 2012– Numerical simulations of turbulent flow modulations by particles

• Mustafa Barri Postdoc Dec. 2009 – Dec. 2012– Numerical simulations of particle suspensions

• Christopher Nilsen PhD-student Jul. 2011 – Jul. 2014– Numerical simulations of particle suspensions

***************************************************************************• Afshin Abbasi-Hoseini PhD-student Jan. 2010 – Jan. 2013

– Experiment of fiber suspensions in channel flow

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Activities in numerical simulations

• Past– Numerical simulations of spherical and prolate ellipsoidal particle

suspensions with one-way coupling (Dr. Mortensen)• Spherical particle spin• Dynamics and orientations of prolate ellipsoidal particle suspensions in wall-

bounded turbulence– One-way coupling of spherical particle suspensions (Lihao)

• Reynolds number effect on the particles suspensions• Stokes number effect on the fluid-particle slip velocity

– Two-way coupling of spherical particle suspensions (Lihao)• Modulations on turbulent flow by presence of particle• Particle spin in two-way coupled simulation

• Current– Two-way coupling of fiber suspensions (Lihao)

• Force-coupling• Torque-coupling

– Fiber suspensions in laminar Couette flow (Christopher)

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Activities in numerical simulations

• Future– Fully coupling simulation of fiber suspensions (Lihao)

• Add the stress coupling– Simulation of microbubbles in the channe flow (Lihao)– Particle aggregation (Christopher)– Particle-particle interaction (Christopher)– Spherical and non-spherical particle suspensions in channel flow with

roughness and in orifice flow (Mustafa)

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Outline

• Eulerian-Lagrangian point-particle approach – Eulerian fluid representation– Lagrangian particle representation

• Two-way coupling scheme– Force-coupling– Torque-coupling

• Recent examples– Some new results of ellpsoidal particle suspensions with two-way coupling– Ellpsoidal particle suspensions in laminar Couette flow

• Summary

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Eulerian fluid representation

• Incompressible and isothermal Newtonian fluid.

• Frictional Reynolds number:

• Governing equations (non-dimensional):

– Mass balance

– Momentum balance

• Direct numerical simulations (DNS), i.e. turbulence from first principles

Reu h

0 u

pfupuutu

2

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Lagrangian approach - Characteristic parameters

• Inertial prolate ellipsoidal particles- a: radius in minor-axis- b: radius in major-axis

• Lagrangian approach - Translational and rotational motions of each individual particle

• Finite number of particles ~106

• Point-particle assumption – Smaller than Kolmogorov length scale

• Particle Reynolds number

Re 1i ip

u v a

ba

Z’Y’

X’

Z’Y’

X’

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Two-way coupled method (Force-coupling)• Drag force is the point-force on the particle.

• Body force in the momentum equation of fluid:

pn

ipi

cellp xF

Vf

1

)(1

iF

x

yz

pf

Staggered grid system

pf

pfupuutu

2

)( pi xF)( pi xFdtdvm : drag force and lift force on the particle i

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Two-way coupled method (Torque-coupling)

• Cauchy’s equation of motion, i.e. the principle of conservation of linear momentum, can be expressed in Cartesian tensor notation as:

– where Tji is a stress tensor and fp is a body force. For a Newtonian fluid, the stress tensor is:

pj

jii fxT

DtDu

i

j

j

iijij x

uxu

pT

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12


• Tji is a stress tensor in a fluid-particle mixture:

mmijrj

i

i

jr

i

j

j

iijij x

uxu

xu

xu

pT 2

i

j

j

iijij x

uxu

pT

Micropolar fluids Eringen (1966)

j

imjim x

u

21

mijmrj

i

i

jr x

uxu

2

PijT

mmmijri

j

j

iijij x

uxu

pT

2

Angular velocity of the fluid

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PijT

mmmijri

j

j

iijij x

uxupT

2

• - the effect of the particles on the motion of the fluid-particlemixture

• ωm - the field of micro-rotation and represent the angular velocity of the rotation of the solid particles.

•The micro-rotation fieldis obtained from a transport equationfor angular momentum.(see Eringen1966 & Lukaszewicz1999)

pijT

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From tensor analysis: any vector -Nm corresponds an anti-symmetric tensor

of second order that contains the same information as the vector (see e.g. Irgens 2008).

The torque vector -Nm can thus be obtained from a particle stress tensor

In practice, the torque should be the sum of torques over all particles inside the grid cell under consideration.

Pijmijm TN

21

pp nn

mmijPij

NNNNNN

NT1

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23

1 00

011

pijT

Pi

Pij TT j-

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Two-way coupled method (Force- and torque-coupling)

• Momentum equation with force- and torque-coupling:

where :

pj

pji

j

jii fxT

xT

DtDu

i

j

j

iijjiij x

uxupTT

pn

mmijPP NTT

1ijji

1

pn

ipip xFf

1

)(1

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Results

• Re 360=ז• Channel model

– 6h*3h*h (x*y*z)----------------------------------• Particle number 2.5 million• Minor axis radius=0.001• Aspect ratio 5• Translational response time 30

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Instantaneous contours

Streamwise velocity contour in YZ plane (Without fiber)

Streamwise velocity contour in YZ plane (With fiber)

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Streamwise velocity contour in YZ plane (Without fiber)

Streamwise velocity contour in YZ plane (With fiber)

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Y

Z

0.2 0.4 0.6 0.8

0.1

0.2

0.3

0.42220181614121086420

Y

Z

0.2 0.4 0.6 0.8

0.1

0.2

0.3

0.4201613952

-2-5-9-13-16-20

Y

Z

0.2 0.4 0.6 0.8

0.1

0.2

0.3

0.4201613952

-2-5-9-13-16-20

Streamwise velocity contour

Streamwise force (on fluid)

Streamwise torque-force (on fluid)

Streamwise torque-force (on fluid)

j

pji

xT

pf

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Statistical results

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Statistical results

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Ellipsoidal particle in laminar Couette flow

uw

x

y

zh

200Re huw

b

St = 100, 1000, 10000

AR = 1.01, 10, 50

f

p

f

p bSt

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Re

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St=100, AR=10 St=10000, AR=10


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St=10000, AR=10 St=10000, AR=50

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St=100, AR=10 St=100, AR=1.01

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Summary

• Two-way coupling scheme for ellipsoids suspensions is introduced and some new results are shown

• The effect of Stokes number and aspect ratio is studied for ellipsoidal particles in laminar Couette flow

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Thank you!

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Simulation procedure

1. Initial positions and orientations (Euler angles) of ellipsoids and initial velocity conditions for particle velocities and angular velocities are specified.

2. Euler parameters are evaluated.3. Obtain transformation matrix A.4. Calculate resistance tensor.5. Solve equations for translational and rotational motion for calculating the new particle positions

and Euler’s parameters.6. Return to step 3 and continue procedure until desired time period is reached.

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Rotation description of fibre

• Relation between Euler angles and Euler parameters

• Constraint

0

1

2

3

cos cos2 2

cos sin2 2

sin sin2 2

sin cos2 2

e

e

e

e

2 2 2 20 1 2 3 1e e e e

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333231

232221

131211

aaaaaaaaa

A

'''

'''

'''

'''

1

1

11

xAx

AA

xAx

AxAxA

Axx

T

T

•A is the orthogonal transformation matrix comprised the direction cosines

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Particle response time

• Ellipsoidal particle response time (by Shapiro and Goldenberg): – the time required by the particles to respond to changes in the flow field.

– Density ratio:

2 2

2

ln 129Re 1

pSa

p

f

S

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Lagrangian approach - Kinematics

Three different coordinate systems:

1. Laboratory system: [x, y, z]

2. Co-moving frame: [x”, y”, z”]

3. Particle frame: [x’, y’, z’]

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Hydrodynamic torque - Rotational motions

• The rotational motion in the particle frame is governed by

Euler equations Torque components for an ellipsoid subjected to linear shear under creeping flow by Jeffery 1922

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Description of rotational motion

• Time evolution of Euler parameters

• Constraint2 2 2 20 1 2 3 1e e e e

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Hydrodynamic drag force (Brenner)- Translational motion

• The translational motion in the laboratory frame is governed by

• The drag force in creeping flow conditions is given by:

– Resistance tensor K ’ is an intrinsic property of the particle. Particle orientation is absorbed in the resistance tensor.

– A is the orthogonal transformation matrix comprised the direction cosines

)()( '1jjijpi vuAKAxf

)( pii xfdtdvm

zz

yy

xx

ij

KK

KK

'

'

'

'

000000

simulations of inertial point-particles at ntnu

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