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The present chapter deals with the FVM as applied to the diffusion equation in one and two dimensions. In the FVM, the domain Ω is divided into a collection of nonoverlapping subdomains, called control volumes and the collection is called a mesh or grid.
In this chapter, we will focus on solving the PDEs governing laminar flows of viscous incompressible fluids using the FVM (this chapter is a counterpart of Chapter 6 on FEM, where velocity–pressure and penalty finite element models of two dimensional flows of viscous incompressible fluids were presented). These equations are expressed in terms of the primitive variable, namely, the velocity field and the pressure. To begin with, we will consider isothermal flows (flows without the presence of the temperature effect), and demonstrate the use of the FVM for two-dimensional laminar flows of viscous incompressible fluids. Then cases of non-isothermal flows with both forced convection and natural convection will be considered in the sequel.
In Chapter 4 we considered finite element analysis of steady state heat transfer. When external stimuli (e.g., boundary conditions and internal heat generation) are independent of time, heat transfer in a medium may attain a steady state; otherwise, the temperature field changes with time (i.e., unsteady state). The governing equations of unsteady heat transfer are obtained using the principle of balance of energy. When unsteady equations are solved the temperature field reaches a steady state if the external stimuli are independent of time (i.e., the time dependence decays with time).
All numerical methods, including the FEM and FVM, ultimately result in a set of linear or nonlinear algebraic equations, relating the values of the dependent variables at the nodal points of the mesh. These algebraic equations can be linear or nonlinear in the nodal values of the primary variables, depending on whether the governing differential equations being solved are linear or nonlinear. When the algebraic equations are nonlinear, we linearize them using certain assumptions and techniques, such as the Picard method or Newton’s method.
The equations governing flows of Newtonian viscous incompressible fluids were reviewed in Chapter 2. The equations are revisited here, in the Cartesian component form, for the two-dimensional case (i.e., set and the derivatives with respect to to zero).
There are several topics that are considered to be “advanced” for this book. We will briefly discuss some (but not all) of these topics to make the readers aware of the fact that the present coverage has precluded them, and then cover three topics in a greater detail.
Most engineering systems can be described, with the aid of the laws of physics and observations, in terms of algebraic, differential, and integral equations. In most problems of practical interest, these equations cannot be solved exactly, mostly because of irregular domains on which the equations are posed, variable coefficients in the equations, complicated boundary conditions, and the presence of nonlinearities. Approximate representation of differential and integral equations to obtain algebraic relations among quantities that characterize the system and implementation of the steps to obtain algebraic equations and their solution using computers constitute a numerical method.
General equations describing transport of momenta and energy by advection–diffusion was given in Chapter 2 (see, also, Example 4.2.3) and will not be repeated here. It is important to note that the entire finite volume formulation is based on local one-dimensional representation in each coordinate direction. The flux crossing a control volume surface is represented using a one-dimensional formulation.
In this chapter we will introduce the FEM as a technique of solving differential equations governing a single variable (or dependent variable). Once we understand how the method works, it can be extended to problems governed by coupled PDEs among several unknowns. In particular, equations governing steady-state heat transfer in one- and two-dimensional problems are used as the “model” equations to introduce the FEM.