Introduction to Compressible Fluid Flow, Second Edition, 2nd Edition

87

5

Normal Shock Waves

Shock Waves

It has been found experimentally that, under some circumstances, it is possible for an

almost spontaneous change to occur in a ow, with the velocity decreasing and the pres-

sure increasing through this region of sharp change. The possibility that such a change

can actually occur follows from the analysis given below. It has been found experimen-

tally, and it also follows from the analysis given below, that such regions of sharp change

can only occur if the initial ow is supersonic. The extremely thin region in which the

transition from the initial supersonic velocity, relatively low-pressure state to the state that

involves a relatively low velocity and high pressure is termed a shock wave. The changes

that occur through a normal shock wave, i.e., a shock wave that is straight and at right

angles to the ow direction, are shown in Figure 5.1. A photograph of a normal shock wave

is shown in Figure 5.2.

A shock wave is extremely thin, usually only a few mean free paths thick. A shock wave

is analogous in many ways to a “hydraulic jump,” which occurs in free-surface liquid

ows (shown schematically in Figure 5.3). A hydraulic jump occurs, for example, in the

ow downstream of a weir.

A shock wave is, in general, curved. However, many shock waves that occur in practi-

cal situations are straight, being either at right angles (i.e., normal) to or at an angle to the

upstream ow (see Figure 5.4). A straight shock wave that is at right angles to the upstream

ow is, as noted above, termed a normal shock wave, whereas a straight shock wave that

is at an angle to the upstream ow is termed an oblique shock wave.

In the case of a normal shock wave, the velocities both ahead (i.e., upstream) of the shock

and after (i.e., downstream) of the shock wave are at right angles to the shock wave. In the

case of an oblique shock wave, there is a change in ow direction across the shock. This is

illustrated in Figure 5.5.

A complete shock wave may be effectively normal in part of the ow, curved in other

parts of the ow, and effectively oblique in other parts of the ow, as shown in Figure 5.6.

Because of its own importance and because, as will be shown later, the oblique shock

relations can be deduced from those for a normal shock wave, the normal shock wave will

be considered rst, such waves being the subject of this chapter. Oblique shock waves will

then be discussed in the next chapter. Curved shock waves are relatively difcult to ana-

lyze and they will not be discussed in detail in this book.

Normal shock waves occur in a number of practical situations such as, for example, in

the intakes to the engines in some supersonic aircraft, in the exhaust system of recipro-

cating engines, in long distance gas pipelines, and in mine shafts as a result of the use of

explosives.

88 Introduction to Compressible Fluid Flow

Shock

wave

p

V

T

FIGURE 5.1

Changes through a normal shock wave.

Normal

shock wave

FIGURE 5.2

Photograph of a normal shock wave. (Reprinted with permission from W. Bleakney, D. K. Weimer, and

C.H.Fletcher, The Shock Tube: A Facility for Investigations in Fluid Dynamics, Rev. Sci. Instr., 20(11), pp. 807–

815. Copyright 1949, American Institute of Physics.)

Hydraulic

jump

Free

surface

Liqui

d

flow

V

2

< V

1

H

1

V

1

H

2

FIGURE 5.3

Hydraulic jump.

89Normal Shock Waves

Normal shock

wave

Oblique shock

wave

Curved shock

wave

FIGURE 5.4

Curved, normal, and oblique shock waves.

Oblique shock wave

V

1

V

2

V

1

V

2

Normal sh

ock wave

FIGURE 5.5

Velocity changes across normal and oblique shock waves.

Expansion

wave

Body

Oblique

shock

wave

M > 1

Normal

shock

wave

FIGURE 5.6

Shock wave with changing shape along wave.