Daniel D. Joseph

Boundary conditions at a naturally permeable wall

Experiments giving the mass efflux of a Poiseuille flow over a naturally permeable block are reported. The efflux is greatly enhanced over the value it would have if the block were impermeable, indicating the presence of a boundary layer in the block. The velocity presumably changes across this layer from its (statistically average) Darcy value to some slip value immediately outside the permeable block. A simple theory based on replacing the effect of the boundary layer with a slip velocity proportional to the exterior velocity gradient is proposed and shown to be in reasonable agreement with experimental results.

A distributed Lagrange multiplier/fictitious domain method for particulate flows

Abstract A new Lagrange-multiplier based fictitious-domain method is presented for the direct numerical simulation of viscous incompressible flow with suspended solid particles. The method uses a finite-element discretization in space and an operator-splitting technique for discretization in time. The linearly constrained quadratic minimization problems which arise from this splitting are solved using conjugate-gradient algorithms. A key feature of the method is that the fluid–particle motion is treated implicitly via a combined weak formulation in which the mutual forces cancel—explicit calculation of the hydrodynamic forces and torques on the particles is not required. The fluid flow equations are enforced inside, as well as outside, the particle boundaries. The flow inside, and on, each particle boundary is constrained to be a rigid-body motion using a distributed Lagrange multiplier. This multiplier represents the additional body force per unit volume needed to maintain the rigid-body motion inside the particle boundary, and is analogous to the pressure in incompressible fluid flow, whose gradient is the force required to maintain the constraint of incompressibility. The method is validated using the sedimentation of two circular particles in a two-dimensional channel as the test problem, and is then applied to the sedimentation of 504 circular particles in a closed two-dimensional box. The resulting suspension is fairly dense, and the computation could not be carried out without an effective strategy for preventing particles from penetrating each other or the solid outer walls; in the method described herein, this is achieved by activating a repelling force on close approach, such as might occur as a consequence of roughness elements on the particle. The development of physically based mathematical methods for avoiding particle–particle and particle–wall penetration is a new problem posed by the direct simulation of fluidized suspensions. The simulation starts with the particles packed densely at the top of the sedimentation column. In the course of their fall to the bottom of the box, a fingering motion of the particles, which are heavier than the surrounding fluid, develops in a way reminiscent of the familiar dynamics associated with the Rayleigh–Taylor instability of heavy fluid above light. We also present here the results of a three-dimensional simulation of the sedimentation of two spherical particles. The simulation reproduces the familiar dynamics of drafting, kissing and tumbling to side-by-side motion with the line between centers across the flow at Reynolds numbers in the hundreds.

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.

The lattice Boltzmann equation method: theoretical interpretation, numerics and implications

Abstract During the last ten years the lattice Boltzmann equation (LBE) method has been developed as an alternative numerical approach in computational fluid dynamics (CFD). Originated from the discrete kinetic theory, the LBE method has emerged with the promise to become a superior modeling platform, both computationally and conceptually, compared to the existing arsenal of the continuum-based CFD methods. The LBE method has been applied for simulation of various kinds of fluid flows under different conditions. The number of papers on the LBE method and its applications continues to grow rapidly, especially in the direction of complex and multiphase media. The purpose of the present paper is to provide a comprehensive, self-contained and consistent tutorial on the LBE method, aiming to clarify misunderstandings and eliminate some confusion that seems to persist in the LBE-related CFD literature. The focus is placed on the fundamental principles of the LBE approach. An excursion into the history, physical background and details of the theory and numerical implementation is made. Special attention is paid to advantages and limitations of the method, and its perspectives to be a useful framework for description of complex flows and interfacial (and multiphase) phenomena. The computational performance of the LBE method is examined, comparing it to other CFD methods, which directly solve for the transport equations of the macroscopic variables.

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.

Direct simulation of fluid particle motions

Continuum models of two-phase flows of solids and liquids use constitutive assumptions to close the equations. A more fundamental approach is a “molecular dynamic” simulation of flowing “big” particles based on reliable macroscopic equations for both solid and liquid. We developed a package that simulates the unsteady two-dimensional solid-liquid two-phase flows using the Navier-Stokes equations for the liquid and Newtons equations of motion for the solid particles. The Navier-Stokes equations are solved using a finite-element formulation and Newtons equations of motion are solved using an explicit-implicit scheme. We show that the simplest fully explicit scheme to update the particle motion using Newtons equations is unstable. To correct this instability we propose and implement and Explicit-Implicit Scheme in which, at each time step, the positions of the particles are updated explicitly, the computational domain is remeshed, the solution at the previous time is mapped onto the new mesh, and finally the nonlinear Navier-Stokes equation and the implicitly discretized Newtons equations for particle velocities are solved on the new mesh iteratively. The numerical simulation reveals the effect of vortex shedding on the motion of the cylinders and reproduces the drafting, kissing, and tumbling scenario which is the dominant rearrangement mechanism in two-phase flow of solids and liquids in beds of spheres which are constrained to move in only two dimensions.

A new formulation of the distributed Lagrange multiplier/fictitious domain method for particulate flows

A Lagrange-multiplier-based fictitious-domain method (DLM) for the direct numerical simulation of rigid particulate flows in a Newtonian fluid was presented previously. An important feature of this finite element based method is that the flow in the particle domain is constrained to be a rigid body motion by using a well-chosen field of Lagrange multipliers. The constraint of rigid body motion is represented by ua Ua o r; u being the velocity of the fluid at a point in the particle domain; U and o are the translational and angular velocities of the particle, respectively; and r is the position vector of the point with respect to the center of mass of the particle. The fluid‐particle motion is treated implicitly using a combined weak formulation in which the mutual forces cancel. This formulation together with the above equation of constraint gives an algorithm that requires extra conditions on the space of the distributed Lagrange multipliers when the density of the fluid and the particles match. In view of the above issue a new formulation of the DLM for particulate flow is presented in this paper. In this approach the deformation rate tensor within the particle domain is constrained to be zero at points in the fluid occupied by rigid solids. This formulation shows that the state of stress inside a rigid body depends on the velocity field similar to pressure in an incompressible fluid. The new formulation is implemented by modifying the DLM code for two-dimensional particulate flows developed by others. The code is verified by comparing results with other simulations and experiments. 7 2000 Elsevier Science Ltd. All rights reserved.

Nonlinear mechanics of fluidization of beds of spherical particles

Experiments on fluidization with water of spherical particles falling against gravity in columns of rectangular cross-section are described. All of them are dominated by inertial effects associated with wakes. Two local mechanisms are involved: drafting and kissing and tumbling into stable cross-stream arrays. Drafting, kissing and tumbling are rearrangement mechanisms in which one sphere is captured in the wake of the other. The kissing spheres are aligned with the stream. The streamwise alignment is massively unstable and the kissing spheres tumble into more stable cross-stream pairs of doublets which can aggregate into larger relatively stable horizontal arrays. Cross-stream arrays in beds of spheres constrained to move in two dimensions are remarkable. These arrays may even coalesce into aggregations of close-packed spheres separated by regions of clear water. A somewhat weaker form of cooperative motion of cross-stream arrays of rising spheres is found in beds of square cross-section where the spheres may move freely in three dimensions. Horizontal arrays rise where drafting spheres fall because of greater drag. Aggregation of spheres seems to be associated with relatively stable cooperative motions of horizontal arrays of spheres rising in their own wakes.

Hyperbolicity and Change of Type in the Flow of Viscoelastic Fluids.

The equations governing the flow of viscoelastic liquids are classified according to the symbol of their differential operators. Propagation of singularities is discussed and conditions for a change of type are investigated. The vorticity equation for steady flow can change type when a critical condition involving speed and stresses is satisfied. This leads to a partitioning of the field of flow into subcritical and supercritical regions, as in the problem of transonic flow.

Breakup of a liquid drop suddenly exposed to a high-speed airstream

The breakup of viscous and viscoelastic drops in the high speed airstream behind a shock wave in a shock tube was photographed with a rotating drum camera giving one photograph every 5 ms. From these photographs we created movies of the fragmentation history of viscous drops of widely varying viscosity, and viscoelastic drops, at very high Weber and Reynolds numbers. Drops of the order of one millimeter are reduced to droplet clouds and possibly to vapor in times less than 500 ms. The movies may be viewed at http://www.aem.umn.edu /research/Aerodynamic_Breakup. They reveal sequences of breakup events which were previously unavailable for study. Bag and bag-and-stamen breakup can be seen at very high Weber numbers, in the regime of breakup previously called ‘catastrophic’. The movies allow us to generate precise displacement‐time graphs from which accurate values of acceleration (of orders 10 4 to 10 5 times the acceleration of gravity) are computed. These large accelerations from gas to liquid put the flattened drops at high risk to Rayleigh‐Taylor instabilities. The most unstable Rayleigh‐ Taylor wave fits nearly perfectly with waves measured on enhanced images of drops from the movies, but the eAects of viscosity cannot be neglected. Other features of drop breakup under extreme conditions, not treated here, are available on our Web site. # 1999 Elsevier Science Ltd. All rights reserved.