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Cost Action FP1005 meeting
Oct. 13th-14th, 2011, Nancy, France
Simulations of inertial point-particles
at NTNU
Lihao Zhao, Christopher Nilsen, Mustafa Barri and
Helge Andersson
Dept. of Energy and Process Engineering, NTNU, Norway
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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
Dec. 2009 – Dec. 2012
Postdoc
– Numerical simulations of particle suspensions
• Christopher Nilsen
PhD-student
Jul. 2011 – Jul. 2014
– Numerical simulations of particle suspensions
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• 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 wallbounded 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: Re 
u h

• Governing equations (non-dimensional):
– Mass balance
u  0
 u

 u  u  p  2u  f p
 t

– Momentum balance  
• Direct numerical simulations (DNS), i.e. turbulence from first principles
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Lagrangian approach - Characteristic parameters
•
b
Inertial prolate ellipsoidal particles  
a
- 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
Z’
Y’
• Point-particle assumption
– Smaller than Kolmogorov length scale
X’
• Particle Reynolds number
ui  vi a
Re p 
1

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Two-way coupled method (Force-coupling)
• Drag force is the point-force on the particle.
dv
m
 Fi ( x p )
dt
Fi ( x p ) : drag force and lift force on the particle i
• Body force f p in the momentum equation of fluid:
 u

   u  u  p  2u  f p
 t

n
1 p
fp  
Fi ( x p )

Vcell i 1
z
fp
Fi
y
Staggered grid system
x
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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:
Dui T ji


  fp
Dt x j
– where Tji is a stress tensor and fp is a body force. For a Newtonian fluid,
the stress tensor is:
 u i u j
Tij   p ij   

 x
 j xi




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Two-way coupled method (Torque-coupling)
• Tji is a stress tensor in a fluid-particle mixture:
 u i u j
Tij   p ij   

 x
 j xi




Micropolar fluids Eringen (1966)
 u i u j
Tij   p ij   

 x
 j xi
Angular velocity of the fluid
1
ui
m   mji
2
x j
 u i u j
Tij   p ij   

 x
 j xi

 u
   r  j  u i

 x

 i x j

  2  r  mij  m


 u j ui 
  2 r m mij
r 


 xi x j 

  2  r  mij  m   m 
 



P
T
ij
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Two-way coupled method (Torque-coupling)
 ui u j
Tij   p ij   

 x
 j xi

  2  r  mij  m  m 
 

TP
ij
•
p
Tij - 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)
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Two-way coupled method (Torque-coupling)
From tensor analysis: any vector -Nm corresponds an anti-symmetric tensor
p
Tij 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
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 N m    mijTijP 
2
TijP  -TjiP
In practice, the torque should be the sum of torques over all particles inside
the grid cell under consideration.
TijP 
np
np
0
1
1

N

 N3


mij
m
  1
  1
 N2
 N3
 N2
0
 N1
 N1
0
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Two-way coupled method (Force- and torque-coupling)
• Momentum equation with force- and torque-coupling:

T ji
T ji
p
Du i


  fp
Dt
x j
x j
where :
 ui u j 

Tij  T ji   p ij   

 x

 j xi 
np
1
T  T     mij N m
 1
n
1 p
f p    Fi ( x p )
 i 1
P
ji
P
ij
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Results
• Re‫=ז‬360
• 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 (With fiber)
Streamwise velocity contour in YZ plane (Without fiber)
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Instantaneous contours
Streamwise velocity contour in YZ plane (With fiber)
Streamwise velocity contour in YZ plane (Without fiber)
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Instantaneous contours
0.4
0.4
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20
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16
14
12
10
8
6
4
2
0
0.1
0.2
0.4
0.6
0.3
Z
0.2
0.2
0.1
0.8
0.2
Y
0.4
0.6
Y
Streamwise force (on fluid)
Streamwise velocity contour
0.4
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13
9
5
2
-2
-5
-9
-13
-16
-20
0.3
Z
Z
0.3
20
16
13
9
5
2
-2
-5
-9
-13
-16
-20
0.2
0.1
0.2
0.4
0.6
Y
Streamwise torque-force (on fluid)
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0.8
T ji
x j
p
0.8
fp
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Statistical results
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Statistical results
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Ellipsoidal particle in laminar Couette flow
uw
z
h
y
x
Re b 
uw h

 200
 p 4 b2  p
St  Re 


f

f
St = 100, 1000, 10000
AR = 1.01, 10, 50
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Ellipsoidal particle in laminar Couette flow
St=100, AR=10
St=10000, AR=10
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Ellipsoidal particle in laminar Couette flow
St=10000, AR=10
St=10000, AR=50
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Ellipsoidal particle in laminar Couette flow
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.
2.
3.
4.
5.
6.
Initial positions and orientations (Euler angles) of ellipsoids and initial velocity conditions for
particle velocities and angular velocities are specified.
Euler parameters are evaluated.
Obtain transformation matrix A.
Calculate resistance tensor.
Solve equations for translational and rotational motion for calculating the new particle positions
and Euler’s parameters.
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
e0  cos
e1  cos
e2  sin
e3  sin





2

2

2

2
cos
e  e  e  e 1
2
0
2
1
2
2
2
3
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2

cos
• Constraint
2

sin
sin

2

2
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x '  Ax' '
A1 x '  A1 Ax' '
x ' '  A1 x '
 a11 a12

A   a21 a22
a
 31 a32
a13 

a23 
a33 
AT  A1
x ' '  AT x '
•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.
2 Sa
 
9 Re

p
– Density ratio:
2

ln    2  1
 2 1
p
S
f
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
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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
Torque components for an ellipsoid
subjected to linear shear under
creeping flow by Jeffery 1922
Euler equations
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Description of rotational motion
• Time evolution of Euler parameters
• Constraint
e02  e12  e22  e32  1
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Hydrodynamic drag force (Brenner)- Translational motion
• The translational motion in the laboratory frame is governed by
dvi
m
 fi ( x p )
dt
• The drag force in creeping flow conditions is given by:
 K ' xx

'
Kij   0
 0

fi ( x p )  A1Kij A(u j  v j )
'
0
K ' yy
0
0 

0 
K ' zz 
– 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
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