T.-W. Pan

Applications of operator-splitting methods to the direct numerical simulation of particulate and free-surface flows and to the numerical solution of the two-dimensional elliptic Monge-Ampère equation

The main goal of this article is to review some recent applications of operator-splitting methods. We will show that these methods are well-suited to the numerical solution of outstanding problems from various areas in Mechanics, Physics and Differential Geometry, such as the direct numerical simulation of particulate flow, free boundary problems with surface tension for incompressible viscous fluids, and the elliptic real Monge-Ampère equation. The results of numerical experiments will illustrate the capabilities of these methods.

A Fictitious-Domain Method with Distributed Multiplier for the Stokes Problem

This article is devoted to the numerical analysis of a fictitious domain method for the Stokes problem, where the boundary condition is enforced weakly by means of a multiplier defined in a portion of the domain. In practice, this is applied for example to the sedimentation of many particles in a fluid. It is found that the multiplier is divergence-free. We present here sufficient conditions on the relative mesh sizes for convergence of the discrete method. Also, we show how the constraint on the divergence of the discrete multiplier can be relaxed when such a sedimentation problem is discretized.

A fictitious domain method with distributed Lagrange multipliers for the numerical simulation of particulate flow and its parallel implementation

The main goal of this article, which generalizes [4] considerably, is to discuss the numerical simulation of particulate flow for mixtures of incompressible viscous fluids and rigid particles. Such flow occurs in liquid/solid fluidized beds, sedimentation, and other applications in Science and Engineering. Assuming that the number of particles is sufficiently large, those simulations are useful to adjust parameters in the homogenized models approximating the above two-phase flow. From a computational point of view, the methodology to be discussed in this article combines distributed Lagrange multipliers based fictitious domain methods, which allow the use of fixed structured finite element grids for the fluid flow computations, with time discretizations by operator splitting à la Marchuk-Yanenko to decouple the various computational difficulties associated to the simulation; these difficulties include collisions between particles, which are treated by penalty type methods. After validating the numerical methodology discussed here by comparison with some well documented two particle fluid flow interactions, we shall present the results of two and three dimensional particulate flow simulations, with the number of particles in the range 10− 10; these results include the simulation of a Rayleigh-Taylor instability occurring when a sufficiently large number of particles, initially at rest, are positioned regularly over a fluid of smaller density, in the presence of gravity. The methods described in this article will be discussed with more details (of computational and physical natures) in [6]. Actually, ref. [6] will contain, also, many references to the work of several investigators, showing that the most popular methodology to simulate particulate flow has been so far the one based on ALE (Arbitrary Lagrange-Euler) techniques; these methods are clearly more complicated to implement than those described in this article (particularly on parallel platforms).

Wavelet Methods in Computational Fluid Dynamics

We discuss in this paper the numerical solution of boundary value problems for partial differential equations by methods relying on compactly supported wavelet approximations. After defining compactly supported wavelets and stating their main properties we discuss their application to boundary value problems for partial differential equations, giving a particular attention to the treatment of the boundary conditions. Finally, we discuss application of wavelets to the solution of the Navier-Stokes equations for incompressible viscous fluids.

A Fictitious Domain Method for the Incompressible Navier-Stokes Equations

In this article we briefly discuss a fictitious domain method for the numerical solution of the Navier-Stokes equations modelling incompressible viscous flow. The methodology described here takes a systematic advantage of time discretization by operator splitting in order to treat separately advection and incompressibility; it seems well suited to moving boundary flow problems. Indeed, 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 non regular geometry. Preliminary numerical results show that this new method looks quite promising.

Parallel Solution of Multibody Store Separation Problems by a Fictitious Domain Method

The numerical simulation of interaction between fluid and complex geometries, for example,, multi-body store separation, is computationally expensive and parallelism often appears as the only way toward large scale simulations, even if one has a fast Navier-Stokes solver. The method that is presented in this chapter is a combination of a distributed Lagrange multiplier based fictitious domain method and operator splitting schemes. This method allows the use of a fixed structured finite element grid on a simple shape auxiliary domain containing the actual one for the entire fluid flow simulation. It can be easily parallelized and there is no need to generate a new mesh at each time step right after finding the new position of the rigid bodies. Numerical results of multi-body store separation in an incompressible viscous fluid on an SGI Origin 2000 are presented further in the chapter.