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Compressible Flow

10.1 Introduction

In this chapter we extend the discussion on inviscid flow equations for incom-
pressible flows to compressible flows. As in incompressible flows, inviscid com-
pressible flow equations and their solutions have played a central role in the
development of CFD methods. As discussed in Chapter 6, the incompressible,
irrotational inviscid flow equation can be solved using the Laplace equation,
which is elliptic in form. Removing the incompressible flow assumption leads to
many difficulties. The first is that the panel method of chapter 6 can no longer
be used, except for compressibility corrections, which are limited to small Mach
numbers. One major advantage of a panel method is that it requires only the
generation of a surface mesh. This is no longer the case for compressible flows
and mesh generation can become more complex since the solution of the gov-
erning PDE will now require the entire field to be discretized. This is discussed
in detail in Chapter 9, but some issues pertaining to the discretization of the
flow equations are discussed here.
Another difficulty arising from removing the incompressible assumption is
that the equations are hyperbolic for supersonic flows. Supersonic flows allow
the solution of discontinuous flows, such as shock wave compressions and, while
discontinuous expansions are thermodynamically non-physical, the solution of
the compressible continuity equation has no means to provide the correct phys-
ical behavior. The relation between the upstream and downstream flow proper-
ties across a shock wave are derived in Section 10.2 for the Euler, Full-Potential
and Transonic Small Disturbance equations. Adequate shock capturing methods
ensuring the proper shock jump relations are discussed in Section 10.3.
Adding to the complexity, transonic flows, that is flows that contain both
subsonic and supersonic flow regions, are of mixed elliptic-hyperbolic type and
require careful computation of the density from the velocity fields so that non-
linearity of the flow field is retained [1]. The numerical solutions must therefore
reproduce the properties of the local flow equations and this was a considerable