Page 109 - Computational Fluid Dynamics for Engineers

Numerical Methods
for Model Parabolic

and Elliptic Equations

4.1 Introduction

The equations of Chapters 2 and 3 are of little value unless they can be solved.
Since known analytical methods are limited to some special flows such as fully
developed laminar flows in ducts, numerical methods are required. These meth-
ods are invariably implemented on digital computers, so this chapter provides
an introduction to computational fluid dynamics and to the numerical methods
to be discussed in the subsequent chapters.
The numerical solution of the Navier-Stokes and Euler equations can be
obtained by using finite-difference and finite-volume methods. Their numeri-
cal solution can also be obtained by finite-element methods. However, these
methods are mainly used in computational structural mechanics and are yet
to gain popularity in computational fluid dynamics. Thus, this book considers
only finite-difference and finite-volume methods. The finite-difference methods
are mainly used with the differential form of the conservation equations, and
the finite-volume methods are used in conjunction with the integral form of
the conservation equations. For reduced forms of the Navier-Stokes equations,
such as the boundary-layer equations and the Orr-Sommerfeld equation which
involve the numerical solution of ordinary and partial-differential equations,
only finite-difference methods are considered. The finite-volume methods will
be considered in Chapter 12 in connection with the numerical solution of the
Navier-Stokes and Euler equations together with the finite-difference methods.
Model equations for the conservation equations are considered in Section 4.2.
These equations are useful to "model" the behavior of the more complete and
complicated partial-differential equations considered in subsequent chapters. In
this chapter they are used to describe the numerical solution of parabolic and
elliptic equations and in the following chapter the numerical solution of hyper-
bolic equations.