Page 191 - Computational Fluid Dynamics for Engineers

Inviscid Flow Equations

for Incompressible Flows

6.1 Introduction

In this chapter we address the solution of the inviscid flow equations for incom-
pressible flows and postpone the discussion on compressible flows to Chapter
10. For incompressible irrotational flows the Euler equations of subsection 2.4.1
simplify further and, in terms of either velocity potential, 0, or stream func-
tion, i\), they reduce to the Laplace equation in <j) or ^, as discussed in Section
6.2. This equation can be solved by finite-difference methods discussed in Sec-
tion 6.3. It can also be solved by superposition of flows, called panel methods,
rather than finite-difference methods. In Section 6.4, a popular and useful panel
method, developed by Hess and Smith, is described for two-dimensional flows.
Section 6.5 presents and describes a computer program based on this method.
Applications of the panel method to several problems are discussed in Section
6.6.

6.2 Laplace Equation and Its Fundamental Solutions

For an incompressible irrotational flow, the inviscid flow equations of subsec-
tion 2.4.1 can be expressed in forms usually more convenient for mathematical
treatment. For example, in terms of the velocity potential 0, which for a two-
dimensional flow is,
9(p d(p
U = V= (6 2 1}
^ Ty ' '
the continuity equation becomes Laplace's equation in 0,
2
V 0 = 0 (6.2.2)
where the operator V 2 is defined by