1. Simulations of inertial point-particles at NTNU
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

2. Part I. Overview of recent activities. Part II. Some recent examples
Content
Part I. Overview of recent activities.
Part II. Some recent examples
Two-way coupling of fiber suspensions - Lihao Zhao
Rotation of fiber in laminar Couette flow - Christopher Nilsen

3. 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”

4. Current Lihao Zhao PhD-student Aug. 2008 – Feb. 2012
Numerical simulations of turbulent flow modulations by particles
Mustafa Barri Postdoc Dec – Dec. 2012
Numerical simulations of particle suspensions
Christopher Nilsen PhD-student Jul – Jul. 2014
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Afshin Abbasi-Hoseini PhD-student Jan – Jan. 2013
Experiment of fiber suspensions in channel flow

5. 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)

6. 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)

7. 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

8. 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

9. 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

10. Two-way coupled method (Force-coupling)
Drag force is the point-force on the particle.
Body force in the momentum equation of fluid:
: drag force and lift force on the particle i
Staggered grid system z y x

11. 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:

12. Two-way coupled method (Torque-coupling)
Tji is a stress tensor in a fluid-particle mixture:
Micropolar fluids Eringen (1966)
Angular velocity of the fluid

13. Two-way coupled method (Torque-coupling)
- 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)

14. Two-way coupled method (Torque-coupling)
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.

15. Two-way coupled method (Force- and torque-coupling)
Momentum equation with force- and torque-coupling:
where :

16. Results Reז=360 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

17. Instantaneous contours
Streamwise velocity contour in YZ plane (With fiber)
Streamwise velocity contour in YZ plane (Without fiber)

18. Instantaneous contours
Streamwise velocity contour in YZ plane (With fiber)
Streamwise velocity contour in YZ plane (Without fiber)

19. Instantaneous contours
Streamwise force (on fluid)
Streamwise velocity contour
Streamwise torque-force (on fluid)
Streamwise torque-force (on fluid)

20. Statistical results

21. Statistical results

22. Ellipsoidal particle in laminar Couette flowuw x y z h
St = 100, 1000, 10000
AR = 1.01, 10, 50

23. Ellipsoidal particle in laminar Couette flow
St=100, AR=10
St=10000, AR=10

24. Ellipsoidal particle in laminar Couette flow
St=10000, AR=10
St=10000, AR=50

25. Ellipsoidal particle in laminar Couette flow
St=100, AR=10
St=100, AR=1.01

26. 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

27. Thank you!

28. Simulation procedure
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.

30. Rotation description of fibre
Relation between Euler angles and Euler parameters
Constraint

31. A is the orthogonal transformation matrix comprised the direction cosines

32. 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:

33. Lagrangian approach - Kinematics
Three different coordinate systems:
Laboratory system: [x, y, z]
Co-moving frame: [x”, y”, z”]
Particle frame: [x’, y’, z’]

34. 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

35. Description of rotational motion
Constraint
Time evolution of Euler parameters

36. 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