Page 337 - Computational Fluid Dynamics for Engineers

Incompressible

Navier—Stokes Equations

11.1 Introduction

The solution of the incompressible Navier-Stokes equations is discussed in this
chapter and that of the compressible form postponed to Chapter 12. It may
appear logical to consider the two together and this can be done readily to
the boundary layer equations, where the equation representing conservation of
energy has to be added to complete the compressible form. The differences be-
tween the incompressible and compressible form of the Navier-Stokes equations
are greater, the continuity equation for the former is elliptic but is hyperbolic
in space and parabolic in time for the latter. These differences are discussed
in more detail in Section 11.2, and the influence of the equation type on the
boundary conditions are discussed in Section 11.3.
Once the equations and boundary conditions have been formulated, a solu-
tion algorithm is required. There are several numerical methods with which to
solve the incompressible Navier-Stokes (INS) equations, and they can be divided
in two classes. The first class uses the stream function variable to ensure that
the conservation of mass is satisfied and this approach is well-established with
two-dimensional equations, but extension to three-dimensions is not straightfor-
ward since the stream function scalar has to be replaced by a three-dimensional
vector potential, which typically is the vorticity vector. Thus, the modified sys-
tem of equations contains six variables as opposed to the four variables in in-
compressible Navier-Stokes equations expressed in primitive variables, which
corresponds to the second class of solution methods.
The Marker and Cell (MAC) algorithm developed by Harlow and Welch [1]
is an early method with which to solve unsteady equations. It uses a staggered
grid with the Poisson equation to obtain the pressure at every time step. The
method is described in [2] and is not discussed here. Another technique, this
time mainly for steady flows, was developed by Patankar and Spalding [3] and