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

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

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”

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 *************************************************************************** Afshin Abbasi-Hoseini PhD-student Jan. 2010 – Jan. 2013 Experiment of fiber suspensions in channel flow

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)

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)

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

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

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

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

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:

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

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)

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.

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

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

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

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

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

Statistical results

Statistical results

Ellipsoidal particle in laminar Couette flow uw x y z h St = 100, 1000, 10000 AR = 1.01, 10, 50

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

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

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

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

Thank you!

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.

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

A is the orthogonal transformation matrix comprised the direction cosines

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:

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

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

Description of rotational motion Constraint Time evolution of Euler parameters

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