Shock Interactions and Reflections

Return to the oblique shock wave illustrated in Figure 9.1a. In this picture, we can imagine the shock wave extending unchanged above the comer to infinity. However, in real life this does not happen. In reality, the oblique shock in Figure 9.1a will impinge somewhere on another solid surface and/or will intersect other waves, either shock or expansion waves. Such wave intersections and interactions are important in the practical design and analysis of supersonic airplanes, missiles, wind tunnels, rocket engines, etc. A perfect historical example of this, as well as the consequences that can be caused by not paying suitable attention to wave interactions, is a ramjet flight-test program conducted in the early 1960s. During this period, a ramjet engine was mounted underneath the X-15 hypersonic airplane for a series of flight tests at high Mach numbers, in the range from 4 to 7. (The X-15, shown in Figure 9.16, was an experimental, rocket-powered airplane designed to probe the lower end of hypersonic manned flight.) During the first high-speed tests, the shock wave from the engine cowling impinged on the bottom surface of the X-15, and because of locally high aerodynamic heating in the impingement region, a hole was burned in the X-15 fuselage. Although this problem was later fixed, it is a graphic example of what shock-wave interactions can do to a practical configuration.

The purpose of this section is to present a mainly qualitative discussion of shock­wave interactions. For more details, see Chapter 4 of Reference 21.

First, consider an oblique shock wave generated by a concave corner, as shown in Figure 9.17. The deflection angle at the corner is в, thus generating an oblique shock at point A with a wave angle fi. Assume that a straight, horizontal wall is present above the comer, as also shown in Figure 9.17. The shock wave generated at point A, called the incident shock wave, impinges on the upper wall at point B. Question: Does the shock wave simply disappear at point B1 If not, what happens to it? To answer this question, we appeal to our knowledge of shock-wave properties. Examining Figure 9.17, we see that the flow in region 2 behind the incident shock is inclined upward at the deflection angle в. However, the flow must be tangent everywhere along the upper wall; if the flow in region 2 were to continue unchanged, it would mn into the wall and have no place to go. Hence, the flow in region 2 must eventually be bent downward through the angle в in order to maintain a flow tangent to the upper wall. Nature accomplishes this downward deflection via a second shock wave originating at the impingement point В in Figure 9.17. This second shock is called the reflected shock wave. The purpose of the reflected shock is to deflect the

Figure 9.1 6 The X-15 hypersonic research vehicle. Designed and built during the late 1950s, it served as a test vehicle for the U. S. Air Force and NASA, jCourtesy of Rockwell Inti, North America.)

flow in region 2 so that it is parallel to the upper wall in region 3, thus preserving the wall boundary condition.

The strength of the reflected shock wave is weaker than the incident shock. This is because М2 < Mi, and М2 represents the upstream Mach number for the reflected shock wave. Since the deflection angles are the same, whereas the reflected shock sees a lower upstream Mach number, we know from Section 9.2 that the reflected wave must be weaker. For this reason, the angle the reflected shock makes with the upper wall Ф is not equal to fi (i. e., the wave reflection is not specular). The properties of the reflected shock are uniquely defined by М2 and 0; since М2 is in turn uniquely defined by Mi and в, then the poperties in region 3 behind the reflected shock as well

as the angle Ф are easily determined from the given conditions of M and 9 by using the results of Section 9.2 as follows:

1. Calculate the properties in region 2 from the given M and 9. In particular, this

gives us M2.

2. Calculate the properties in region 3 from the value of M2 calculated above and

the known deflection angle 9.

An interesting situation can arise as follows. Assume that M is only slightly above the minimum Mach number necessary for a straight, attached shock wave at the given deflection angle 9. For this case, the oblique shock theory from Section 9.2 allows a solution for a straight, attached incident shock. However, we know that the Mach number decreases across a shock (i. e., M2 < M ). This decrease may be enough such that M2 is not above the minimum Mach number for the required deflection 9 through the reflected shock. In such a case, our oblique shock theory does not allow a solution for a straight reflected shock wave. The regular reflection as shown in Figure 9.17 is not possible. Nature handles this situation by creating the wave pattern shown in Figure 9.18. Here, the originally straight incident shock becomes curved as it nears the upper wall and becomes a normal shock wave at the upper wall. This allows the streamline at the wall to continue parallel to the wall behind the shock intersection. In addition, a curved reflected shock branches from the normal shock and propagates downstream. This wave pattern, shown in Figure 9.18, is called a Mach reflection. The calculation of the wave pattern and general properties for a Mach reflection requires numerical techniques such as those to be discussed in Chapter 13.

Another type of shock interaction is shown in Figure 9.19. Here, a shock wave is generated by the concave corner at point G and propagates upward. Denote this wave as shock A. Shock A is a left-running wave, so-called because if you stand on top of the wave and look downstream, you see the shock wave running in front of you

Figure 9.19 Intersection of right – and left-running shock waves.

toward the left. Another shock wave is generated by the concave comer at point H, and propagates downward. Denote this wave as shock B. Shock В is a right-running wave, so-called because if you stand on top of the wave and look downstream, you see the shock mnning in front of you toward the right. The picture shown in Figure 9.19 is the intersection of right – and left-running shock waves. The intersection occurs at point E. At the intersection, wave A is refracted and continues as wave D. Similarly, wave В is refracted and continues as wave C. The flow behind the refracted shock D is denoted by region 4; the flow behind the refracted shock C is denoted by region 4′. These two regions are divided by a slip line EF. Across the slip line, the pressures are constant (i. e„ p = p4<), and the direction (but not necessarily the magnitude) of velocity is the same, namely, parallel to the slip line. All other properties in regions 4 and 4′ are different, most notably the entropy (s4 / ,v4′). The conditions which must hold across the slip line, along with the known Mi, 6, and 6L, uniquely determine

the shock-wave interaction shown in Figure 9.19. (See Chapter 4 of Reference 21 for details concerning the calculation of this interaction.)

Figure 9.20 illustrates the intersection of two left-running shocks generated at comers A and B. The intersection occurs at point C, at which the two shocks merge and propagate as the stronger shock CD, usually along with a weak reflected wave CE. This reflected wave is necessary to adjust the flow so that the velocities in regions 4 and 5 are in the same direction. Again, a slip line CF trails downstream of the intersection point.

The above cases are by no means all the possible wave interactions in a supersonic flow. However, they represent some of the more common situations encountered frequently in practice.

Consider an oblique shock wave generated by a compression corner with a 10° deflection angle. The Mach number of the flow ahead of the corner is 3.6; the flow pressure and temperature are standard sea level conditions. The oblique shock wave subsequently impinges on a straight wall opposite the compression comer. The geometry for this flow is given in Figure 9.17. Calculate the angle of the reflected shock wave Ф relative to the straight wall. Also, obtain the pressure, temperature, and Mach number behind the reflected wave.

Solution

From the 6-fi-M diagram, Figure 9.7, for M = 3.6 and в = 10°, i = 24°. Hence,

Mn ] = M] sin /1] = 3.6 sin 24° = 1.464

From Appendix B,

Also,

These are the conditions behind the incident shock wave. They constitute the upstream flow properties for the reflected shock wave. We know that the flow must be deflected again by в = 10° in passing through the reflected shock. Thus, from the 6-fl-M diagram, for M2 = 2.96 and в = 10°, we have the wave angle for the reflected shock, fi2 = 27.3°. Note that fi2 is not the angle the reflected shock makes with respect to the upper wall; rather, by definition of the wave angle, f}2 is the angle between the reflected shock and the direction of the flow in region 2. The shock angle relative to the wall is, from the geometry shown in Figure 9.17,

Ф = p2 – в = 27.3 – 10

Also, the normal component of the upstream Mach number relative to the reflected shock is M2 sin /Т = (2.96) sin 27.3° = 1.358. From Appendix B,

— = 1.991 — = 1.229 M„,3 = 0.7572

Pi T2

M„ з 0.7572 ГТ7Т

kf3 =——– —– =———————- = 2.55

sin(f}2 — 0) sin(27.3 — 10) _____

For standard sea level conditions, p =2116 lb/ft3 and T, = 519°R. Thus,

Ръ = — — Px= (1 -991)(2.32)(2116) =

Pi Px

h =T^Ti = (1.229)(1.294)(519) ;

12 l

Note that the reflected shock is weaker than the incident shock, as indicated by the smaller pressure ratio for the reflected shock, p2/p2 = 1.991 as compared to p2/p = 2.32 for the incident shock.