Hypersonics and Computational Fluid Dynamics

The design of hypersonic vehicles today is greatly dependent on the use of computa­tional fluid dynamics, much more so than the design of vehicles for any other flight regime. The primary reason for this is the lack of experimental ground test facilities that can simultaneously simulate the Mach numbers, Reynolds numbers, and high – temperature levels associated with hypersonic flight. For such simulation, CFD is the primary tool. Reflecting once again on the philosophy illustrated in Figure 2.44, in the realm of hypersonic flow the three partners are not quite equal. Pure experimental work in hypersonics usually involves tests at either the desired Mach number, the desired Reynolds number, or the desired temperature level, but not all at the same time nor in the same test facilities. As a result, experimental data for the design of hypersonic vehicles is a patchwork of different data taken in different facilities under different conditions. Moreover, the data are usually incomplete, especially for the high-temperature effects, which are difficult to simulate in a wind tunnel. The de­signer must then do his or her best to piece together the information for the specified design conditions. The next partner shown in Figure 2.44, pure theory, is greatly hampered by the nonlinear nature of hypersonic flow, hence making mathematical solutions intractable. In addition, the proper inclusion of high-temperature chemi­cally reacting flows in any pure theory is extremely difficult. For these reasons, the third partner shown in Figure 2.44, computational fluid dynamics, takes on a domi-

nant role. The numerical calculation of both inviscid and viscous hypersonic flows, including all the high-temperature effects discussed in Section 14.2, has been a major thrust of CFD research and design application since the 1960s. Indeed, hypersonics has paced the development of CFD since its beginning.

As an example of CFD applied to a hypersonic flight vehicle appropriate to this chapter, consider the space shuttle shown in Figure 14.15. A numerical solution of the three-dimensional inviscid flow field around the shuttle was carried out by Maus et al. in Reference 78. They made two sets of calculations, one for a perfect gas with у — 1.4, and one assuming chemically reacting air in local chemical equilibrium. The freestream Mach number was 23 in both cases. The CFD technique used for these calculations involved a time-dependent solution of the flow in the blunt nose region, patterned after our discussion in Section 13.5, and starting beyond the sonic line a downstream marching approach patterned after our discussion in Section 13.4. The calculated surface pressure distributions along the windward centerline of the space shuttle for both the perfect gas case (the circles) and the chemically reacting case (the triangles) are shown in Figure 14.16. The expansion around the nose, the pressure plateau over the relatively flat bottom surface, and the further expansion over the slightly inclined back portion of the body, are all quite evident. Also note that there is little difference in the pressure distributions between the two cases; this is an example of the more general result that pressure is usually the flow variable least affected by chemically reacting effects.

It is interesting to note, however, that a flight characteristic as mundane as the vehicle pitching moment coefficient is affected by chemically reacting flow effects. Close examination of Figure 14.16 shows that, for the chemically reacting flow, the

Figure 14.15 Space shuttle geometry.

pressures are slightly higher on the forward part of the shuttle, and slightly lower on the rearward part. This results in a more positive pitching moment. Since the moment is the integral of the pressure through a moment arm, a slight change in pressure can substantially affect the moment. This is indeed the case here, as shown in Figure 14.17, which is a plot of the resulting calculated pitching moment as a function of angle of attack for the space shuttle. Clearly, the pitching moment is substantially greater for the chemically reacting case. The work by Maus et al. was the first to point out this effect on pitching moment, and it serves to reinforce the importance of high-temperature flows on hypersonic aerodynamics. It also serves to reinforce the importance of CFD in the analysis of hypersonic flows. The predicted pitching mo­ment used for the space shuttle design came from “cold-flow” wind tunnel tests which did not simulate the high-temperature effects, that is, the designers used data for a perfect gas with у = 1.4 obtained in the wind tunnel. This is represented by the lower curve in Figure 14.7. The early flight experience with the shuttle indicated a much higher pitching moment at hypersonic speeds than predicted, which required that the body flap deflection for trim to be more than twice that predicted—an alarming situa­tion. The reason for this is now known; the actual flight environment encountered by the shuttle at high Mach numbers was that of a high-temperature chemically reacting flow—the situation reflected in the upper curve in Figure 14.17. The difference in the pitching moment between the two curves in Figure 14.17 is enough to account for the unexpected extra body flap deflection required to trim the shuttle. Although these CFD results were obtained well after the design of the shuttle, they serve to underscore the importance of CFD to present and future hypersonic vehicle designs.

Figure 1 4.1 7 Predicted pitching moment coefficient for the space shuttle; comparison between a calorically perfect gas and equilibrium air calculations. {Source: Maus et al., Reference 78.)

14.8 Summary

Only a few of the basic elements of hypersonic flow are presented here, with special emphasis on newtonian flow results. Useful information on hypersonic flows can be extracted from such results. We have derived the basic newtonian sine-squared law;

Cp = 2 sin2 9 [14.4]

and used this result to treat the case of a hypersonic flat plate in Section 14.4. We also obtained the limiting form of the oblique shock relations as Mx —»■ oo, that is, the hypersonic shock relations. From these relations, we were able to examine the significance of newtonian theory more thoroughly, namely, Equation (14.4) becomes an exact relation for a hypersonic flow in the combined limit of Мж —»• oo and у —у 1. Finally, these hypersonic shock relations illustrate the existence of the Mach number independence principle.

Problems

1. Repeat Problem 9.13 using (ia) Newtonian theory (b) Modified newtonian theory

Compare these results with those obtained from exact shock-expansion theory (Problem 9.13). From this comparison, what comments can you make about the

accuracy of newtonian and modified newtonian theories at low supersonic Mach numbers?

2. Consider a flat plate at a = 20° in a Mach 20 freestream. Using straight newto­nian theory, calculate the lift – and wave-drag coefficients. Compare these results with exact shock-expansion theory.

3. Consider a hypersonic vehicle with a spherical nose flying at Mach 20 at a standard altitude of 150,000 ft, where the ambient temperature and pressure are 500°R and 3.06 lb/ft2, respectively. At the point on the surface of the nose located 20° away from the stagnation point, estimate the: (a) pressure, (b) temperature,

(c) Mach number, and (d) velocity of the flow.

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