Implementation of a flexible fiber model in a general purpose CFD code

Transcription

1 Implementation of a flexible fiber model in a general purpose CFD code Jelena Andrić Supervisor: Håkan Nilsson Co-supervisors: Srdjan Sasic and Alf-Erik Almstedt Department of Applied Mechanics Chalmers University of Technology Göteborg, Sweden, 2012

2 Industrial Application Process of making pulp mats in absorbent hygiene product industry (diapers, sanitary napkins ) Flexible fibers transported in flowing air and forming mat Achieve desired fiber spatial distribution in forming head Deposition of fibers on forming mat

3 Aim of the Present Work Implement fiber model with relevant physical properties Study fiber motion in channel Air-fiber interaction Fiber-fiber interaction Formation and break-up of fiber flocks Current stage of the work Air-fiber interaction One-way coupling (air flow is not affected by presence of fibers)

4 Numerical Methods Different numerical methods to study flows with fibers Eulerian-Eulerian approach Both phases treated as continuum Eulerian-Lagrangian approach Particles are moving points in continuum fluid Microhydrodynamics approach (many particles combined in multibody systems) Direct Numerical Simulation (geometry resolved)

5 Particle-level Simulation Technique Yamamoto and Matsuoka Chain of spheres Two-dimensional shear flows Ross and Klingenberg Chain of prolate spheroids Two-dimensional shear flows Schmid et al. Chain of massless rigid cylindrical segments Three-dimensional sheared suspensions One-way coupling Lindström and Uesaka Chain of cylindrical segments (inertia is taken into account) Two-way coupling Hydrodynamic interactions fiber-fluid

6 Present Work Implementation of fiber model in OpenFOAM Software choice Open-source general purpose CFD code Implemented solvers for fluid phase Interpolation and tracking routines available Good platform for building fiber model Fibers modelled as chains of rigid cylindrical segments Each segment tracked individually - LPT Segment inertia taken into account Connectivity forces ensure fiber integrity Bending and twisting torques and collisional forces are not considered

7 Fiber Geometry Geometrical properties defined for each segment diameter length start point unit direction vectors position vector

8 Equations of Motion Direct application of Newton s second law for each fiber segment Linear momentum equation - hydrodynamic force - sum of body forces - connectivity force exerted by segment i-1 on segment i - for end segments Angular momentum equation - inertia tensor for cylindrical segment - hydrodynamic torque

9 Equations of Motion (cont.) Connectivity constraint - end points of adjacent segments coincide Connectivity equation - time derivative of connectivity constraint

10 Hydrodynamic Forces and Torques Approximations for hydrodynamic forces and torques Segment Reynolds number Hydrodynamic effects - sum of viscous and inertia components Viscous effects - dominant for small segment Reynolds numbers Inertia effects dominant for large segment Reynolds numbers Assumption numerically investigated by Lindström and Uesaka Good agreement in viscous and inertia regime, some discrepancy in intermediate regime

11 Viscous Hydrodynamic Forces and Torques Formula of Cox - circular cylinder is hydrodynamically equivalent to prolate spheroid if - spheroid minor axis Orbiting behaviours of cylinder and prolate spheroid in shear flow are the same Valid for isolated particles when slender body theory applies Kim and Karilla - analytical solution for viscous drag force and torque for isolated spheroid and laminar conditions

12 Viscous Hydrodynamic Forces and Torques Expressions for hydrodynamic force and torque (Kim and Karilla) Hydrodynamic resistance tensors Hydrodynamic coefficients ( depend on spheroid ecentricity )

13 Dynamic Drag Force and Torque Expressions for dynamic drag force and torque Dynamic drag resistance tensors cross-flow velocity of fluid relative to segment Total hydrodynamic force and torque

14 Discretized Governing Equations Time discretization (implicit numerical scheme) Discretized linear momentum equation Discretized angular momentum equation Discretized connectivity equation Equations made dimensionless to improve numerical stability

15 Connectivity Force Linear System Linear system for connectivity forces, for 7-segment fiber Tikhonov regularization applied to reach numerical stability

16 Numerical Algorithm Solve for dimensionless connectivity forces Calculate dimensionless velocities and angular velocities Scale to dimensional velocities and angular velocities Update fiber segment position and orientation Correct for segment position

17 Case Descriptions Implemented model tested in different types of flows Imposed flow fields Only fiber motion is solved for Shear flows (different shear rates) Simultaneously predicted flow field Both flow and fiber motion are simulated Lid-driven cavity Validation test case physical pendulum

18 Fiber and Fluid Properties Air properties Density Kinematic viscosity Fiber properties Diameter Length Density

19 Case Descriptions Imposed Flow Fields Computational domain box of side 0.1 m Cartesian mesh with10 cells in each direction Each case includes 50 fibers Initial fiber positions and shapes (horizontal, vertical, zig-zag) Fiber positions in box Fiber shapes in z plane

20 Imposed Flow Fields - Case u=(y,0,0) Time = 0.1s Time = 0.5s Time = 1s Time = 1s; zoom-up at vertical and zig-zag fibers Horizontal fibers with higher y position move faster Vertical and zig-zag fibers move and rotate acording to flow direction and rotation Zig-zag fibers stretch out in flow direction

21 Imposed Flow Fields - Case u=(10y,0,0) Time = 0.01s Time = 0.1s Time = 0.1s; zoom-up at vertical and zig-zag fibers Fibers respond faster to flow Have the same pattern as in previous case

22 Imposed flow fields - Case u=(100y,0,0) Time = 0.01s Time = 0.01s; zoom-up at vertical and zig-zag fibers Fibers reach right end of computational domain in 0.1 s Horizontal and vertical fibers have similar pattern to previous cases Zig-zag fibers do not completely aligh with flow end segment more bent Fiber shape evolution depend on shear rate

23 Imposed Flow Fields - Case u=(0,x,0) Time = 0.5s Time = 1s; zoom-up at fibers Time = 1s Vertical fibers with higher x position move faster Horizontal fibers move upwards and rotate Zig-zag fibers are stretching out in flow direction

24 Imposed Flow Fields - Case u=(y,x,0) Time = 0.3s Time = 0.7s Time = 0.7s; zoom-up at fibers Test case for flow with two non-zero components Motion in xy plane to right and vertically upwards Horizontal and vertical fibers keep their shape Zig-zag stretched out

25 Imposed Flow Fields - Case u=(y,z,x) Time = 0.25s Time = 0.5s Test case for flow with three non-zero components Time = 0.5s; zoom-up at fibers Horizontal and vertical fibers keep their shape Zig-zag fibers stretched out

26 0.1m Simultaneously Predicted Flow Field Lid-driven cavity case modified standard OpenFOAM tutorial 0.1m/s Box of sides 0.1 m Top wall moves in x direction at velocity 0.1 m/s Flow is assumed to be laminar Modified icofoam solver Graded Cartesian mesh 50x50x5 cells

27 Simultaneously Predicted Flow Field Lid-driven Cavity Time = 0s Fibers at initial positions Time = 10s Fibers straighten out and align with flow field

28 Simultaneously Predicted Flow Field Channel Flow Domain size: m x 0.1m x m Fibers at initial position Reference paper: DNS in a plane vertical channel with and without buoyancy; LarsDavidson, Dalibor Cuturic and Shia-Hui Peng

29 Simultaneously Predicted Flow Field Channel Flow test and fromback Fibers in flow field after 0.3s Fibers in flow field after 0.7s

30 Validation of Energy Conservation and Response Time Test Case Description Fiber segment motion composed of translation and rotation Generic test case - check energy conservation and response time Simple mechanical system of two thin rods Initial positions and orientations are set Body forces applied at centers of gravity Each rod resembles physical pendulum fixed at pivot point 0

31 Validation of Energy Conservation and Response Time Equations of Motion Simplified version of fiber model (connectivity force, constant body force) Linear momentum equations Angular momentum equations -constant scalar for infinitely thin rods Connectivity constraint as in fiber model

32 Validation of Energy Conservation and Response Time Results m=1kg; l=1m System oscillates around connectivity point Two-rod system motion for different time steps; from left to right t=0s, 0.23s, 0.47s, 0.70s, 0.94s, 1.17s, 1.41s, 1.64s Comparisson with analytical period for physical pendulum

33 Validation of Energy Conservation and Response Time Energy Conservation Total rod energy = potential energy + total kinetic energy Total kinetic energy = translational k.e. + rotational k.e Total energy transformed between potential and kinetic and conserved over time Total energy over time Translational and rotational energy over time Greatest part of kinetic enegy comes form translational motion In real pendulum case only rotational kinetic energy

34 Conclusions Flexible fiber model has been implemented in OpenFOAM Translational and rotational degrees of freedom Fiber integrity Coupling with imposed and simultaneously predicted flows Fibers follow flow and deform according to flow gradients Connectivity forces ensure fiber integrity Fiber motion concept validation Energy conservation Response time

35 Future Work On-going work: Add bending and twisting torques Add interaction forces Simulations in realistic geometries Fiber-wall interactions Parallelizing

36 Acknowledgments Supervisor Håkan Nilsson Co-supervisors Srdjan Sasic and Alf-Erik Almstedt Henrik Rusche, Wikki GmbH and Hrvoje Jasak, Wikki Ltd and FSB Zagreb Djordje S. Čantrak and Milan Lečić, University of Belgrade Stefan Lindström, Linköping University Financially supported by Bo Rydin Foundation and SCA Hygiene Products AB

37 Thank you for your attention