3. Finite Difference Methods
3.1 Introduction
As was mentioned in Chap. 1, all conservation equations have similar structure and may be regarded as special cases of a generic transport equation,
Eq. (1.26), (1.27) or (1.28). For this reason, we shall treat only a single,
generic conservation equation in this and the following chapters. It will be
used to demonstrate discretization methods for the terms which are common
to all conservation equations (convection, diffusion, and sources). The special
features of the Navier-Stokes equations and techniques for solving coupled
non-linear problems will be introduced later. Also, for the time being, the
unsteady term will be dropped so we consider only time-independent problems.
For simplicity, we shall use only Cartesian grids at this point. The equation
we shall deal with is:
We shall assume that p, u j , r and q4 are known. This may not be the case
because the velocity may not have been computed yet and the properties of
the fluid may depend on the temperature and, if turbulence models are used,
on the velocity field as well. As we shall see, the iterative schemes used to
solve these equations treat 4 as the only unknown; all other variables are
fixed at their values determined on the previous iteration so regarding these
as known is a reasonable approach.
The special features of non-orthogonal and unstructured grids will be
discussed in Chap. 8. Furthermore, of the many possible discretization techniques, only a selected few which illustrate the main ideas will be described;
others may be found in the literature cited.
3.2 Basic Concept
The first step in obtaining a numerical solution is to discretize the geometric
domain - i.e. a numerical grid must be defined. In finite difference (FD)
3.1 Introduction
As was mentioned in Chap. 1, all conservation equations have similar structure and may be regarded as special cases of a generic transport equation,
Eq. (1.26), (1.27) or (1.28). For this reason, we shall treat only a single,
generic conservation equation in this and the following chapters. It will be
used to demonstrate discretization methods for the terms which are common
to all conservation equations (convection, diffusion, and sources). The special
features of the Navier-Stokes equations and techniques for solving coupled
non-linear problems will be introduced later. Also, for the time being, the
unsteady term will be dropped so we consider only time-independent problems.
For simplicity, we shall use only Cartesian grids at this point. The equation
we shall deal with is:
We shall assume that p, u j , r and q4 are known. This may not be the case
because the velocity may not have been computed yet and the properties of
the fluid may depend on the temperature and, if turbulence models are used,
on the velocity field as well. As we shall see, the iterative schemes used to
solve these equations treat 4 as the only unknown; all other variables are
fixed at their values determined on the previous iteration so regarding these
as known is a reasonable approach.
The special features of non-orthogonal and unstructured grids will be
discussed in Chap. 8. Furthermore, of the many possible discretization techniques, only a selected few which illustrate the main ideas will be described;
others may be found in the literature cited.
3.2 Basic Concept
The first step in obtaining a numerical solution is to discretize the geometric
domain - i.e. a numerical grid must be defined. In finite difference (FD)