Expansion Waves

Let us consider the two supersonic flows pictured in Figure 5.15a and 5.15b. When the flow is turned by a surface concave to the flow, as in Figure 5.15a, we have seen that an oblique shock originating from the bend in the surface will compress the flow and turn it though the angle, 8. The question then posed is, how is the flow turned around a bend convex to the flow, as shown in Figure 15b. As suggested by the figure, this is accomplished through a continuous ensemble of weak expansion waves, known as an expansion fan.

In order to examine the flow relationships in this case, we take an approach similar to that for oblique shock waves. Consider supersonic flow through a single, weak wave, known as a Mach wave, as illustrated in Figure 5.16. The wave represents a limiting case of zero entropy gain across the wave. Hence the turning and velocity changes are shown as differentials instead of as finite changes. Since the wave is a weak wave, it propagates

Figure 5.15a Deflection of a supersonic flow by an oblique shock wave (com­pression).

Expansion fan

Figure 5.15b Deflection of a supersonic flow by a series of Mach waves (expan­sion).

normal to itself at the acoustic velocity, a, which added vectorially to the free-stream velocity, V, defines the angle of the wave p,.

■ – i a M = sm —

= sin-1 – гг (5.45)

Applying momentum principles across the wave, as done previously for the oblique shock wave, results in

dVt = 0

and

-dp = pa[(V + dV) sinGu. + dS)~ a]

This reduces to

-dp = pa2|^pr +VM2- 1dS j

Since the tangential velocity component is unchanged across the wave, it follows that

—^— = (V + d V) cos (p + d8) tan ju.

Expanding this and substituting Equation 5.45 results in

dV V dS л/м2-1

dp __ ypM2

dS ~ VM!- 1

Thus, this weak wave, deflecting the flow in the direction shown in Figure

5.16, results in an expansion of the flow, since dp/dS is negative. It is also possible for small deflections in the opposite direction to produce a com­pression with a Mach wave. This fepresents a limiting case of an oblique shock wave.

[vVMr-l L* v

The expansion fan shown in Figure 5.15b represents a continuous dis­tribution of Mach waves. Each wave deflects the flow a small amount, so that the integrated effect produces the total deflection, 8. The changes in the flow can be related to the total deflection by integrating Equation 5.46. The energy equation is used to relate the local sonic velocity to V. It is convenient in so doing to let 8 = 0 at M = 1.0. This corresponds to V = a* for a given set of reservoir conditions. Therefore,

The details of performing this integration will not be presented here. They can be found in several texts and in Reference 5.7. The final expression for 8 becomes

This relationship is presented graphically in Figure 5.17 and is referred to as Prandtl-Meyer flow. To use this graph one relates a given flow state back to the M = 1 condition. For example, suppose the local Mach is equal to 3.0. This means that, relative to M = 1, the flow has already been deflected

Figure 5.17

through an angle of approximately 50°. Suppose the flow is turned an additional 50°. Relative to M = 1, this gives a total deflection of 100°. Thus, one enters Figure 5.17 with this value of 8 to determine a final Mach number slightly in excess of 9.0. Since the Prandtl-Meyer flow is isentropic, the flow state is determined completely by the reservoir conditions and the local Mach number (Equation 5.17).