Numerical Methods and Its Solution
The conservation equations for fluid flow (i.e. mass, momentum and energy) derived in Chap. 5 are partial differential equations (PDEs) that are non-linear and cannot be solved analytically. The equations for particle flows derived in Chap. 6 can be in
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Numerical Methods and Its Solution
7.1
Introduction
The conservation equations for fluid flow (i.e. mass, momentum and energy) derived in Chap. 5 are partial differential equations (PDEs) that are non-linear and cannot be solved analytically. The equations for particle flows derived in Chap. 6 can be in Eulerian form which are in the same form as the fluid flow equations, or in Lagrangian form which is an ordinary differential equation (ODE). The single ODE for the particle equation in Lagrangian form, is simpler to solve compared to the coupled non-linear PDEs. In this chapter we first present the discretisation and numerical solution for the set of PDEs, followed by numerical integration techniques for the ODE. The general transport equation for the flow variables that govern fluid flow must be converted into a set of algebraic equations and solved numerically. There are two processes involved in obtaining the computational solution. Figure 7.1 presents these processes where the first process is the conversion of governing equations (which are partial differential equations) derived in Chap. 5 into algebraic equations and applied to the flow domain that has been divided into smaller cells, i.e. the mesh generation. This process is called discretisation. As its name suggests, applying a discretisation on the governing equations means that the solutions obtained are at discrete points in the domain, namely the mesh points. This differs from analytical solutions which provide a continuous solution throughout the domain. For the discretisation we present two approaches. The first is the finite difference method which is a natural introduction to the discretisation features, and the second approach is the finite volume method which has been readily adopted by many CFD codes. The application of these methods is shown for simplified CFD problems such as a one-dimensional steady state diffusion and a steady state convection-diffusion problem. The discretisation forms the numerical framework that is ready to be solved, and this process is known as its numerical solution. Direction solution methods and iterative solutions are presented which are basically in the form of matrix solvers. A key ingredient in the numerical solution of convective flow problems is to solve the
J. Tu et al., Computational Fluid and Particle Dynamics in the Human Respiratory System, Biological and Medical Physics, Biomedical Engineering, DOI 10.1007/978-94-007-4488-2_7, © Springer Science+Business Media Dordrecht 2013
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7 Numerical Methods and Its Solution
Fig. 7.1 Overview process of the computational solution procedure
velocity field based on a coupling between the pressure and velocity fields. The well known SIMPLE algorithm is described. Finally the approximate solutions of the flow variables need to be analysed using advanced visualisation techniques, i.e. through post-processing. The data provided from a numerical simulation in raw form is simply representative numbers stored at locations within the computational domain. The
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