Numerical simulation of a new two-dimensional variable-order fractional percolation equation in non-homogeneous porous media

Percolation flow problems are discussed in many research fields, such as seepage hydraulics, groundwater hydraulics, groundwater dynamics and fluid dynamics in porous media. Many physical processes appear to exhibit fractional-order behavior that may vary with time, or space, or space and time. The theory of pseudodifferential operators and equations has been used to deal with this situation. In this paper we use a fractional Darcys law with variable order Riemann-Liouville fractional derivatives, this leads to a new variable-order fractional percolation equation. In this paper, a new two-dimensional variable-order fractional percolation equation is considered. A new implicit numerical method and an alternating direct method for the two-dimensional variable-order fractional model is proposed. Consistency, stability and convergence of the implicit finite difference method are established. Finally, some numerical examples are given. The numerical results demonstrate the effectiveness of the methods. This technique can be used to simulate a three-dimensional variable-order fractional percolation equation.

A Method of Characteristics for Solving Two-dimensional Unsteady Flow Problems

The aim of this investigation is to evolve a method of solving two-dimensional unsteady flow problems by the method of characteristics. This involves the reduction of the given system of equations to an equivalent system where only interior derivatives occur on a characteristic surface. From this system, four special bicharacteristic directional derivatives are chosen. A finite difference scheme is prescribed for solving the equations. General rectangular lattices are also considered. As an example, we investigate the propagation of an initial pressure distribution in a medium at rest.

Modeling, numerical methods, and simulation for particle-fluid two-phase flow problems

In this paper, we study the issues of modeling, numerical methods, and simulation with comparison to experimental data for the particle-fluid two-phase flow problem involving a solid-liquid mixed medium. The physical situation being considered is a pulsed liquid fluidized bed. The mathematical model is based on the assumption of one-dimensional flows, incompressible in both particle and fluid phases, equal particle diameters, and the wall friction force on both phases being ignored. The model consists of a set of coupled differential equations describing the conservation of mass and momentum in both phases with coupling and interaction between the two phases. We demonstrate conditions under which the system is either mathematically well posed or ill posed. We consider the general model with additional physical viscosities and/or additional virtual mass forces, both of which stabilize the system. Two numerical methods, one of them is first-order accurate and the other fifth-order accurate, are used to solve the models. A change of variable technique effectively handles the changing domain and boundary conditions. The numerical methods are demonstrated to be stable and convergent through careful numerical experiments. Simulation results for realistic pulsed liquid fluidized bed are provided and compared with experimental data. (C) 2004 Elsevier Ltd. All rights reserved.

A HYBRID METHOD OF FRACTIONAL STEPS WITH PREDICTOR-CORRECTOR DIFFERENCE-PSEUDOSPECTRUM FOR NUMERICAL-SOLUTION OF THE CONVECTION-DOMINATED FLOW PROBLEMS

A fractional-step method of predictor-corrector difference-pseudospectrum with unconditional L(2)-stability and exponential convergence is presented. The stability and convergence of this method is strictly proved mathematically for a nonlinear convection-dominated flow. The error estimation is given and the superiority of this method is verified by numerical test.

Numerical simulation of incompressible flow problems using an unstructured staggered mesh method

A number of two dimensional staggered unstructured discretisation schemes for the solution of fluid flow and heat transfer problems have been developed. All schemes store and solve velocity vector components at cell faces with scalar variables solved at cell centres. The velocity is resolved into face-normal and face-parallel components and the various schemes investigated differ in the treatment of the parallel component. Steady-state and time-dependent fluid flow and thermal energy equations are solved with the well known pressure correction scheme, SIMPLE, employed to couple continuity and momentum. The numerical methods developed are tested on well known benchmark cases: the Lid-Driven Cavity, Natural Convection in a Cavity and Melting of Gallium in a rectangular domain. The results obtained are shown to be comparable to benchmark, but with accuracy dependent on scheme selection.

Boundary and internal conditions for adjoint fluid-flow problems

The ever-increasing robustness and reliability of flow-simulation methods have consolidated CFD as a major tool in virtually all branches of fluid mechanics. Traditionally, those methods have played a crucial role in the analysis of flow physics. In more recent years, though, the subject has broadened considerably, with the development of optimization and inverse design applications. Since then, the search for efficient ways to evaluate flow-sensitivity gradients has received the attention of numerous researchers. In this scenario, the adjoint method has emerged as, quite possibly, the most powerful tool for the job, which heightens the need for a clear understanding of its conceptual basis. Yet, some of its underlying aspects are still subject to debate in the literature, despite all the research that has been carried out on the method. Such is the case with the adjoint boundary and internal conditions, in particular. The present work aims to shed more light on that topic, with emphasis on the need for an internal shock condition. By following the path of previous authors, the quasi-1D Euler problem is used as a vehicle to explore those concepts. The results clearly indicate that the behavior of the adjoint solution through a shock wave ultimately depends upon the nature of the objective functional.