Qualitative Aspects of Hypersonic Flow
Consider a 15° half-angle wedge flying at Mx — 36. From Figure 9.7, we see that the wave angle of the oblique shock is only 18°; that is, the oblique shock wave is very close to the surface of the body. This situation is sketched in Figure 14.1. Clearly, the shock layer between the shock wave and the body is very thin. Such thin shock layers are one characteristic of hypersonic flow. A practical consequence of a thin shock layer is that a major interaction frequently occurs between the inviscid flow behind the shock and the viscous boundary layer on the surface. Indeed, hypersonic vehicles generally fly at high altitudes where the density, hence Reynolds number, is low, and therefore the boundary layers are thick. Moreover, at hypersonic speeds, the boundary-layer thickness on slender bodies is approximately proportional to hence, the high Mach numbers further contribute to a thickening of the boundary layer. In many cases, the boundary-layer thickness is of the same magnitude as the shock – layer thickness, such as sketched in the insert at the top of Figure 14.1. Here, the shock layer is fully viscous, and the shock-wave shape and surface pressure distribution are affected by such viscous effects. These phenomena are called viscous interaction phenomena—where the viscous flow greatly affects the external inviscid flow, and, of course, the external inviscid flow affects the boundary layer. A graphic example of such viscous interaction occurs on a flat plate at hypersonic speeds, as sketched in Figure 14.2. If the flow were completely inviscid, then we would have the case shown
Figure 14.1 For hypersonic flow, the shock layers are thin and viscous.
in Figure 14.2a, where a Mach wave trails downstream from the leading edge. Since there is no deflection of the flow, the pressure distribution over the surface of the plate is constant and equal to p^. In contrast, in real life there is a boundary layer over the flat plate, and at hypersonic conditions this boundary layer can be thick, as sketched in Figure 14.2b. The thick boundary layer deflects the external, inviscid flow, creating a comparably strong, curved shock wave which trails downstream from the leading edge. In turn, the surface pressure from the leading edge is considerably higher than poo, and only approaches px far downstream of the leading edge, as shown in Figure 14.2b. In addition to influencing the aerodynamic force, such high pressures increase the aerodynamic heating at the leading edge. Therefore, hypersonic viscous interaction can be important, and this has been one of the major areas of modem hypersonic aerodynamic research.
There is a second and frequently more dominant aspect of hypersonic flow, namely, high temperatures in the shock layer, along with large aerodynamic heating of the vehicle. For example, consider a blunt body reentering the atmosphere at Mach 36, as sketched in Figure 14.3. Let us calculate the temperature in the shock layer immediately behind the normal portion of the bow shock wave. From Appendix B, we find that the static temperature ratio across a normal shock wave with Moo = 36 is 252.9; this is denoted by Ts/Tx in Figure 14.3. Moreover, at a standard altitude of 59 km, Тж = 258 К. Hence, we obtain Ts = 65,248 К—an incredibly high temperature, which is more than six times hotter than the surface of the sun! This is, in reality, an incorrect value, because we have used Appendix В which is good only for a calorically perfect gas with у = 1.4. However, at high temperatures, the gas will become chemically reacting; у will no longer equal 1.4 and will no longer be constant. Nevertheless, we get the impression from this calculation that the temperature in the shock layer will be very high, albeit something less than 65,248 K. Indeed, if a proper calculation of Ts is made taking into account the chemically
reacting gas, we would find that Ts & 11,000 К— still a very high value. Clearly, high-temperature effects are very important in hypersonic flow.
Let us examine these high-temperature effects in more detail. If we consider air at p — 1 atm and T = 288 К (standard sea level), the chemical composition is essentially 20 percent O2 and 80 percent N2 by volume. The temperature is too low for any significant chemical reaction to take place. However, if we were to increase T to 2000 K, we would observe that the O2 begins to dissociate; that is,
2 -* 20 2000 К < T < 4000 К
If the temperature were increased to 4000 K, most of the O2 would be dissociated, and N2 dissociation would commence:
N2 -* 2N 4000 К < T < 9000 К
If the temperature were increased to 9000 K, most of the N2 would be dissociated, and ionization would commence:
N -* N+ + e_ O -* 0+ + e~
Hence, returning to Figure 14.3, the shock layer in the nose region of the body is a partially ionized plasma, consisting of the atoms N and O, the ions N+ and 0+, and electrons, e~. Indeed, the presence of these free electrons in the shock layer is responsible for the “communications blackout” experienced over portions of the trajectory of a reentry vehicle.
One consequence of these high-temperature effects is that all our equations and tables obtained in Chapters 7 to 13 which depended on a constant у = 1.4 are no longer valid. Indeed, the governing equations for the high-temperature, chemically reacting shock layer in Figure 14.3 must be solved numerically, taking into account the proper physics and chemistry of the gas itself. The analysis of aerodynamic flows with such real physical effects is discussed in detail in Chapters 16 and 17 of Reference 21; such matters are beyond the scope of this book.
Associated with the high-temperature shock layers is a large amount of heat transfer to the surface of a hypersonic vehicle. Indeed, for reentry velocities, aerodynamic
heating dominates the design of the vehicle, as explained at the end of Section 1.1. (Recall that the third historical example discussed in Section 1.1 was the evolution of the blunt-body concept to reduce aerodynamic heating; review this material before progressing further.) The usual mode of aerodynamic heating is the transfer of energy from the hot shock layer to the surface by means of thermal conduction at the surface; that is, if 3T/3n represents the temperature gradient in the gas normal to the surface, then qc — —к(дТ/дп) is the heat transfer into the surface. Because ЗT/Эи is a flow-field property generated by the flow of the gas over the body, qc is called convective heating. For reentry velocities associated with ICBMs (about 28,000 ft/s), this is the only meaningful mode of heat transfer to the body. However, at higher velocities, the shock-layer temperature becomes even hotter. From experience, we know that all bodies emit thermal radiation, and from physics you know that blackbody radiation varies as 7’4: hence, radiation becomes a dominant mode of heat transfer at high temperatures. (For example, the heat you feel by standing beside a fire in a fireplace is radiative heating from the flames and the hot walls.) When the shock layer reaches temperatures on the order of 11,000 K, as for the case given in Figure 14.3, thermal radiation from the hot gas becomes a substantial portion of the total heat transfer to the body surface. Denoting radiative heating by qr, we can express the total aerodynamic heating q as the sum of convective and radiative heating; q = qc + q, . For Apollo reentry, qr/q ~ 0.3, and hence radiative heating was an important consideration in the design of the Apollo heat shield. For the entry of a space probe into the atmosphere of Jupiter, the velocities will be so high and the shock-layer temperatures so large that the convective heating is negligible, and in this case q ~ qr. For such a vehicle, radiative heating becomes the dominant aspect in its design. Figure 14.4 illustrates the relative importance of qc and q, for a typical manned reentry vehicle in the earth’s atmosphere; note how rapidly qr dominates the aerodynamic heating of the body as velocities increase above 36,000 ft/s. The
Nose radius = 15 ft Altitude = 200,000 ft
heating rates of a blunt reentry vehicle as a function of flight velocity. (Source: Anderson, Reference 36.j
details of shock-layer radiative heating are interesting and important; however, they are beyond the scope of this book. For a thorough survey of the engineering aspects of shock-layer radiative heat transfer, see Reference 36.
In summary, the aspects of thin shock-layer viscous interaction and high – temperature, chemically reacting and radiative effects distinguish hypersonic flow from the more moderate supersonic regime. Hypersonic flow has been the subject of several complete books; see, for example, References 37 to 41. In particular, see Reference 55 for a modem textbook on the subject.