Page 293 - Aerodynamics for Engineering Students

Compressible flow 275

supersonic flight speeds. This is not so. It should be recalled that the local flow speeds
near the point of minimum pressure over a wing are substantially greater than the free-
stream flow speed. The local flow speed first reaches the speed of sound at a free-stream
flow speed termed the critical flow speed. So, at flight speeds above critical, regions of
supersonic flow appear over the wing, and shock waves are generated. This leads to
wave drag and other undesirable effects. It is to postpone the onset of these effects that
swept-back wings are used for high-speed subsonic aircraft. It is also worth pointing out
that typically for such aircraft, wave drag contributes 20 to 30% of the total.
In recent decades great advances have been made in obtaining computational solutions
of the equations of motion for compressible flow. This gives the design engineer much
greater freedom to explore a wide range of possible configurations. It might also be
thought that the ready availability of such computational solutions makes a knowledge
of approximate analytical solutions unnecessary. Up to a point there is some truth in this
view. There is certainly no longer any need to learn complex and involved methods of
approximation. Nevertheless, approximate analytical methods will continue to be of
great value. First and foremost, the study of relatively simple model flows, such as the
one-dimensional flows described in Sections 6.2 and 6.3, enables the essential flow physics
to be properly understood. In addition, these relatively simple approaches offer approxi-
mate methods that can be used to give reasonable estimates within a few minutes. They
also offer a valuable way of checking the reliability of computer-generated solutions.

6.2 Isentropic one-dimensional flow

For many applications in aeronautics the viscous effects can be neglected to a good
approximation and, moreover, no significant heat transfer occurs. Under these circum-
stances the thermodynamic processes are termed adiabatic. Provided no other irrever-
sible processes occur we can also assume that the entropy will remain unchanged, such
processes are termed isentropic. We can, therefore, refer to isentropic flow. At this
point it is convenient to recall the special relationships between the main thermo-
dynamic and flow variables that hold when the flow processes are isentropic.
In Section 1.2.8 it was shown that for isentropic processes p = kpy (Eqn (1.24)),
where k is a constant. When this relationship is combined with the equation of state
for a perfect gas (see Eqn (1.12)), namely p/(pT) = R, where R is the gas constant, we
can write the following relationships linking the variables at two different states (or
stations) of an isentropic flow:
P1 - P2 P1 -P2
---
PIT1 P27-2' p: p;
From these it follows that

(6.2)

A useful, special, simplifed model flow is one-dimensional, or more precisely quasi-
one-dimensional flow. This is an internal flow through ducts or passages having
slowly varying cross-sections so that to a good approximation the flow is uniform
at each cross-section and the flow variables only vary with x in the streamwise
direction. Despite the seemingly restrictive nature of these assumptions this is a very
useful model flow with several important applications. It also provides a good way to
learn about the fundamental features of compressible flow.