Howard H. Hu

Direct simulation of flows of solid-liquid mixtures

Abstract A finite element technique based on moving unstructured grids is developed to simulate the motion of a large number of solid particles in a flowing liquid. A generalized Galerkin finite element formulation which incorporates both the fluid and particle equations of motion into a single variational equation is developed for Newtonian fluids. The hydrodynamic forces and moments acting on the solid particles are eliminated in the formulation, so need not be computed explicitly. An arbitrary Lagrangian-Eulerian (ALE) technique is adopted to deal with the motion of the particles. In the implementation, the nodes on the particle surface are assumed to move with the particle. The nodes in the interior of the fluid are computed using Laplaces equation, to guarantee a smoothly varying distribution of the nodes. At each time step, the grid is updated according to the motion of the particles and checked for element degeneration. If unacceptable element distortion is detected, a new finite element grid is generated and the flow fields are projected from the old grid to the new grid. This generalized ALE Galerkin finite element approach gives rise to a set of non-linear algebraic equations which is solved via a quasi-Newton scheme. The corresponding linearized system is solved with an iterative solver using a preconditioned generalized minimal residual algorithm. Initially, the particles are positioned randomly in the fluid, with zero velocity. The particles are then released and the motion of the combined fluid-particle system is simulated using a procedure in which the positions of the particles and of the mesh grids are updated explicitly, while the velocities of the fluid and the solid particles are determined implicitly. Using the developed numerical procedure, we study the Poiseuille flow of solid-liquid mixtures in a vertical channel. The computation is performed within a unit cell which is periodic in the direction along the channel. The gravity is directed along the channel walls, and a pressure gradient is applied against the gravity and drives the flow. The solid particles are slightly heavier than the liquid. The effects of the applied pressure gradient, the particle Reynolds number and the fraction of the solid loading on the flow pattern of the solid-liquid mixture are studied. It was found that when the applied pressure gradient is large enough to overcome the gravity, the particles migrate away from the channel walls and there is a clear liquid layer next to the wall which lubricates the flow. As the particle Reynolds number is increased, particles interact more strongly and large clusters of particles are formed in the flow.

Numerical Simulation of Electroosmotic Flow

We have developed a numerical scheme to simulate electroosmotic flows in complicated geometries. We studied the electroosmotic injection characteristics of a cross-channel device for capillary electrophoresis. We found that the desired rectangular shape of the sample plug at the intersection of the cross-channel can be obtained when the injection is carried out at high electric field intensities. The shape of the sample plug can also be controlled by applying an electric potential or a pressure at the side reservoirs. Flow induced from the side channels into the injection channel squeezes the streamlines at the intersection, thus giving a less distorted sample plug. Results of our simulations agree qualitatively with experimental observations.

Direct Simulation of Initial Value Problems for the Motion of Solid Bodies in a Newtonian Fluid Part 1. Sedimentation

This paper reports the result of direct simulations of fluid–particle motions in two dimensions. We solve the initial value problem for the sedimentation of circular and elliptical particles in a vertical channel. The fluid motion is computed from the Navier–Stokes equations for moderate Reynolds numbers in the hundreds. The particles are moved according to the equations of motion of a rigid body under the action of gravity and hydrodynamic forces arising from the motion of the fluid. The solutions are as exact as our finite-element calculations will allow. As the Reynolds number is increased to 600, a circular particle can be said to experience five different regimes of motion: steady motion with and without overshoot and weak, strong and irregular oscillations. An elliptic particle always turn its long axis perpendicular to the fall, and drifts to the centreline of the channel during sedimentation. Steady drift, damped oscillation and periodic oscillation of the particle are observed for different ranges of the Reynolds number. For two particles which interact while settling, a steady staggered structure, a periodic wake-action regime and an active drafting–kissing–tumbling scenario are realized at increasing Reynolds numbers. The non-linear effects of particle–fluid, particle–wall and interparticle interactions are analysed, and the mechanisms controlling the simulated flows are shown to be lubrication, turning couples on long bodies, steady and unsteady wakes and wake interactions. The results are compared to experimental and theoretical results previously published.

Direct simulation of initial value problems for the motion of solid bodies in a Newtonian fluid. Part 2. Couette and Poiseuille flows

This paper reports the results of a two-dimensional finite element simulation of the motion of a circular particle in a Couette and a Poiseuille flow. The size of the particle and the Reynolds number are large enough to include fully nonlinear inertial effects and wall effects. Both neutrally buoyant and non-neutrally buoyant particles are studied, and the results are compared with pertinent experimental data and perturbation theories. A neutrally buoyant particle is shown to migrate to the centreline in a Couette flow, and exhibits the Segre-Silberberg effect in a Poiseuille flow. Non-neutrally buoyant particles have more complicated patterns of migration, depending upon the density difference between the fluid and the particle. The driving forces of the migration have been identified as a wall repulsion due to lubrication, an inertial lift related to shear slip, a lift due to particle rotation and, in the case of Poiseuille flow, a lift caused by the velocity profile curvature. These forces are analysed by examining the distributions of pressure and shear stress on the particle. The stagnation pressure on the particle surface are particularly important in determining the direction of migration.

Phase-field simulations of interfacial dynamics in viscoelastic fluids using finite elements with adaptive meshing

This paper describes a novel numerical algorithm for simulating interfacial dynamics of non-Newtonian fluids. The interface between two immiscible fluids is treated as a thin mixing layer across which physical properties vary steeply but continuously. The property and evolution of the interfacial layer is governed by a phase-field variable ? that obeys a Cahn-Hilliard type of convection-diffusion equation. This circumvents the task of directly tracking the interface, and produces the correct interfacial tension from the free energy stored in the mixing layer. Viscoelasticity and other types of constitutive equations can be incorporated easily into the variational phase-field framework. The greatest challenge of this approach is in resolving the sharp gradients at the interface. This is achieved by using an efficient adaptive meshing scheme governed by the phase-field variable. The finite-element scheme easily accommodates complex flow geometry and the adaptive meshing makes it possible to simulate large-scale two-phase systems of complex fluids. In two-dimensional and axisymmetric three-dimensional implementations, the numerical toolkit is applied here to drop deformation in shear and elongational flows, rise of drops and retraction of drops and torii. Some of these solutions serve as validation of the method and illustrate its key features, while others explore novel physics of viscoelasticity in the deformation and evolution of interfaces.

Migration of a sphere in tube flow

e–Silberberg radius. The slip angular velocity discrepancy Ωs − Ωse is the circulation for the free particle and it changes sign with the lift. A method of constrained simulation is used to generate data which is processed for correlation formulas for the lift force, slip velocity, and equilibrium position. Our formulae predict the change of sign of the lift force which is necessary in the Segr´ e–Silberberg effect. Our correlation formula is compared with analytical lift formulae in the literature and with the results of two-dimensional simulations. Our work establishes a general procedure for obtaining correlation formulae from numerical experiments. This procedure forms a link between numerical simulation and engineering practice.

Direct Simulation of the Sedimentation of Elliptic Particles in Oldroyd-B Fluids

Cross-stream migration and stable orientations of elliptic particles falling in an Oldroyd-B fluid in a channel are studied. We show that the normal component of the extra stress on a rigid body vanishes; lateral forces and torques are determined by the pressure. Inertia turns the long side of the ellipse across the stream and elasticity turns it along the stream; tilted off-centre falling is unstable. There are two critical numbers: the elasticity and Mach numbers. When the elasticity number is smaller than critical the fluid is essentially Newtonian with broadside-on falling at the centreline of the channel. For larger elasticity numbers the settling turns the long side of the particle along the stream in the channel centre for all velocities below a critical one, identified with a critical Mach number of order one. For larger Mach numbers the ellipse flips into broadside-on falling again. The critical numbers are functions of the channel blockage ratio, the particle aspect ratio and the retardation/relaxation time ratio of the fluid. Two ellipses falling near to each other, attract, line-up vertically and straighten-out with long sides vertical. Stable, off-centre tilting is found for ellipses falling in shear-thinning fluids and for cylinders with flat ends in which particles tend to align their longest diameter with gravity.

Direct simulation of the motion of solid particles in Couette and Poiseuille flows of viscoelastic fluids

This paper reports the results of direct numerical simulation of the motion of a twodimensional circular cylinder in Couette flow and in Poiseuille flow of an Oldroyd-B fluid. Both neutrally buoyant and non-neutrally buoyant cylinders are considered. The cylinder’s motion and the mechanisms which cause the cylinders to migrate are studied. The stable equilibrium position of neutrally buoyant particles varies with inertia, elasticity, shear thinning and the blockage ratio of the channel in both shear flows. Shear thinning promotes the migration of the cylinder to the wall while inertia causes the cylinder to migrate away from the wall. The cylinder moves closer to the wall in a narrower channel. In a Poiseuille flow, the eect of elastic normal stresses is manifested by an attraction toward the nearby wall if the blockage is strong. If the blockage is weak, the normal stresses act through the curvature of the inflow velocity profile and generate a lateral force that points to the centreline. In both cases, the migration of particles is controlled by elastic normal stresses which in the limit of slow flow in two dimensions are compressive and proportional to the square of the shear rate on the body. A slightly buoyant cylinder in Couette flow migrates to an equilibrium position nearer the centreline of the channel in a viscoelastic fluid than in a Newtonian fluid. On the other hand, the same slightly buoyant cylinder in Poiseuille flow moves to a stable position farther away from the centreline of the channel in a viscoelastic fluid than in a Newtonian fluid. Marked eects of shear thinning are documented and discussed.

Direct numerical simulation of the sedimentation of solid particles with thermal convection

Based on the study of isothermal inert particle sedimentation, the Arbitrary Lagrangian-Eulerian technique was used to solve the problem of sedimentation of two solid particles with thermal convection, including the energy equation. The results show that because of the dynamic wake caused by thermal convection, the trajectory of particles are distinct from each other when settling in isothermal, cool and hot fluid; there is vortex shedding in hot fluid, while in cool fluid a strong upward thermal plume forms.

Sedimentation of an ellipsoid inside an infinitely long tube at low and intermediate Reynolds numbers

The motion of a heavy rigid ellipsoidal particle settling in an infinitely long circular tube filled with an incompressible Newtonian fluid has been studied numerically for three categories of problems, namely, when both fluid and particle inertia are negligible, when fluid inertia is negligible but particle inertia is present, and when both fluid and particle inertia are present. The governing equations for both the fluid and the solid particle have been solved using an arbitrary Lagrangian-Eulerian based finite-element method. Under Stokes flow conditions, an ellipsoid without inertia is observed to follow a perfectly periodic orbit in which the particle rotates and moves from side to side in the tube as it settles. The amplitude and the period of this oscillatory motion depend on the initial orientation and the aspect ratio of the ellipsoid. An ellipsoid with inertia is found to follow initially a similar oscillatory motion with increasing amplitude. Its orientation tends towards a flatter configuration, and the rate of change of its orientation is found to be a function of the particle Stokes number which characterizes the particle inertia. The ellipsoid eventually collides with the tube wall, and settles into a different periodic orbit. For cases with non-zero Reynolds numbers, an ellipsoid is seen to attain a steady-state configuration wherein it falls vertically. The location and configuration of this steady equilibrium varies with the Reynolds number.