Modern Engineering Thermodynamics

16.9 Shock Waves 675
16.9 SHOCK WAVES
Shock waves can occur only in compressible substances and are often thought of as strong acoustical (sound)
waves. However, they differ from sound waves in two important ways: They travel much faster than normal
sound waves, and there is a large and nearly discontinuous change in pressure, temperature, and density across
a shock wave. The thickness of a shock wave over which these changes occur is typically on the order of 10 –7 m
(4 × 10 –6 in.); consequently, large property gradients occur across the shock wave that make it very dissipative
and irreversible. The amplitude of a large shock wave, such as that created by an explosion or a supersonic aircraft,
decreases nearly with the inverse square of the distance from the source until it weakens sufficiently to
become an ordinary sound wave. The sonic boom heard at the surface of the Earth from a high-altitude supersonic
aircraft is the weak acoustical remnants of its shock wave.
Since strong shock waves are highly irreversible, they cannot be treated even approximately as isentropic
processes. Ordinary sound waves, on the other hand, are very much weaker by comparison and can sometimes
be modeled by isentropic processes.
When a shock wave occurs perpendicular (i.e., normal) to the velocity, it is called a normal shock, and it can be
analyzed with the one-dimensional balance equations. A shock wave that is inclined to the direction of flow is
called an oblique shock and requires a two- or three-dimensional analysis. We limit our analysis to normal shock
waves in this chapter.
The easiest way to generate normal shock waves for laboratory study is to use a supersonic converging-diverging
nozzle. Equation (16.13) gives the pressure profile along the nozzle in terms of the isentropic stagnation pressure
p os and the local Mach number M as
p/p os = 1 + k − 1 k/ð1−kÞ
2 M2
This equation is shown schematically along the nozzle in Figure 16.21. When the back pressure p B is greater
than the throat critical pressure p * given in Eq. (16.19), the Mach number at the throat is less than 1 and the
flow remains subsonic throughout the entire nozzle, with nozzle exit pressure p exit equal to the back pressure p B .
When p B is less than p * , the Mach number at the throat is equal to 1 and the flow becomes supersonic in the
diverging section of the nozzle. If p B /p os is between the points b and c on Figure 16.21, then a normal shock
occurs within the diverging section at the point where the flow can isentropically recover to p exit = p B .Ifwelet
p E be the pressure at the exit of the supersonic nozzle when the flow expands isentropically throughout the
nozzle (see Figure 16.21), then when p B /p os is between the points c and d on Figure 16.21, a normal shock
p os p exit p B
(back
pressure)
1
p*/p os
p/p os
p E /p os
M = 1 at
throat
Shock waves
a
b
c
Decreasing
back pressure
p B /p os
d
Throat
Distance along the nozzle
Exit
FIGURE 16.21
The pressure distribution in a converging-diverging nozzle when the upstream stagnation pressure is held constant and the downstream
back pressure is decreased. Shock waves occur in the diverging section when the flow is supersonic but the back pressure is not low
enough to allow complete expansion to the end of the nozzle.