Tsorng-Whay Pan

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.

A fictitious domain method for Dirichlet problem and applications

Abstract In this article we discuss the solution of the Dirichlet problem for a class of elliptic operators by a Lagrange multiplier/fictitious domain method. This approach allows the use of regular grids and therefore of fast specialized solvers for problems on complicated geometries; the resulting saddle-point system can be solved by an Uzawa/conjugate gradient algorithm. In the case of two-dimensional problems, a quasi-optimal preconditioner has been found by Fourier analysis and numerical experiments confirm its nice scaling properties. The resulting methodology is applied to a nonlinear time dependent problem, namely the flow of a viscous-plastic medium in a cylindrical pipe showing the potential of this methodology for some classes of nonlinear problems.

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.

A fictitious domain method for external incompressible viscous flow modeled by Navier-Stokes equations

In this article we discuss a fictitious domain method for the numerical solutions of three-dimensional elliptic problems with Dirichlet boundary conditions and also of the Navier-Stokes equations modeling incompressible viscous flow. The methodology for the Navier-Stokes equations described here takes a systematic advantage of time discretization by operator splitting in order to treat separately advection, imbedding and incompressibility. Due to the decoupling, fast elliptic solvers can be used to treat the incompressibility condition even if the original problem is taking place on a nonregular geometry. The resulting methodology is applied to two-dimensional unsteady external incompressible viscous flow problems and three-dimensional Stokes problems.

Fluidization of 1204 spheres: simulation and experiment

In this paper we study the fluidization of 1204 spheres at Reynolds numbers in the thousands using the method of distributed Lagrange multipliers. The results of the simulation are compared with a real experiment. This is the first direct numerical simulation of a real fluidized bed at the finite Reynolds number encountered in the applications. The simulations are processed like real experiments for straight lines in lot-log plots leading to power laws as in celebrated correlations of Richardson and Zaki [1954]. The numerical method allows for the first ever direct calculation of the slip velocity and other averaged values used in two-fluid continuum models. The computation and the experiment show that a single particle may be in balance under weight and drag for an interval of fluidizing velocities; the expectation that the fluidizing velocity is unique is not realized. The numerical method reveals that the dynamic pressure actually decreases slowly with the fluidizing velocity. Tentative interpretations of these new results are discussed.

A distributed Lagrange multiplier/fictitious domain method for the simulation of flow around moving rigid bodies : Application to particulate flow

In this article we discuss the application of a Lagrange multiplier based fictitious domain method to the numerical simulation of incompressible viscous flow modeled by the Navier‐Stokes equations around moving rigid bodies; the rigid body motion is due to hydrodynamical forces and gravity. The solution method combines finite element approximations, time discretization by operators splitting and conjugate gradient algorithms for the solution of the linearly constrained quadratic minimization problems coming from the splitting method. We conclude this article by the presentation of numerical results concerning the simulation of an incompressible viscous flow around a NACA0012 airfoil with a fixed center, but free to rotate, then the sedimentation of circular cylinders in 2-D channels, and finally the sedimentation of spherical balls in cylinders with square cross-sections. ” 2000 Elsevier Science S.A. All rights reserved.

Distributed Lagrange multiplier methods for incompressible viscous flow around moving rigid bodies

Abstract In this article we discuss the application of a distributed Lagrange multiplier based fictitious domain method to the numerical simulation of incompressible viscous flow modelled by the Navier-Stokes equations around moving bodies, we suppose the rigid bodies motion known a priori. The solution method combines finite element approximations, time discretization by operator splitting and conjugate gradient algorithms for the solution of the linearly constrained quadratic minimization problems coming from the splitting method. Numerical experiment results for two-dimensional flow around a moving disk are presented.

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.

A distributed Lagrange multiplier/fictitious domain method for viscoelastic particulate flows

Abstract A distributed Lagrange multiplier/fictitious domain method (DLM) is developed for simulating the motion of rigid particles suspended in the Oldroyd-B fluid. This method is a generalization of the one described in [R. Glowinski, T.W. Pan, T.I. Hesla, D.D. Joseph, Int. J. Multiphase Flows 25 (1998) 755–794] where the motion of particles suspended in a Newtonian fluid was simulated. In our implementation of the DLM method, the fluid–particle system is treated implicitly using a combined weak formulation in which the forces and moments between the particles and fluid cancel. The governing equations for the Oldroyd-B liquid are solved everywhere, including inside the particles. The flow inside the particles is forced to be a rigid body motion by a distribution of Lagrange multipliers. We use the Marchuk–Yanenko operator-splitting technique to decouple the difficulties associated with the incompressibility constraint, nonlinear convection and viscoelastic terms. The constitutive equation is solved using a scheme that guarantees the positive definiteness of the configuration tensor, while the convection term in the constitutive equation is discretized using a third-order upwinding scheme. The nonlinear convection problem is solved using a least square conjugate gradient algorithm, and the Stokes-like problem is solved using a conjugate gradient algorithm. The code is verified performing a convergence study to show that the results are independent of the mesh and time step sizes. Our simulations show that, when particles are dropped in a channel, and the viscoelastic Mach number (M) is less than 1 and the elasticity number (E) is greater than 1, the particles chain along the flow direction; this agrees with the results presented in [P.Y. Huang, H.H. Hu, D.D. Joseph, J. Fluid Mech. 362 (1998) 297–325]. In our simulations of the fluidization of 102 particles in a two-dimensional bed, we find that the particles near the channel walls form chains that are parallel to the walls, but the distribution of particles away from the walls is relatively random.

A distributed Lagrange multiplier/fictitious domain method for flows around moving rigid bodies: Application to particulate flow

This article discusses the application of a Lagrange multiplier-based fictitious domain method to the numerical simulation of incompressible viscous flow modeled by the Navier–Stokes equations around moving rigid bodies; the rigid body motions are due to hydrodynamical forces and gravity. The solution method combines finite element approximations, time discretization by operator splitting and conjugate gradient algorithms for the solution of the linearly constrained quadratic minimization problems coming from the splitting method. The study concludes with the presentation of numerical results concerning four test problems, namely the simulation of an incompressible viscous flow around a NACA0012 airfoil with a fixed center but free to rotate, then the sedimentation of 200 and 1008 cylinders in a two-dimensional channel, and finally the sedimentation of two spherical balls in a rectangular cylinder. Copyright