Hypersonic vehicles are commonplace. There are many more of them than the supersonic aircraft discussed in the last chapter. Applications include missiles, launch vehicles, and entry bodies. A huge effort has been made in developing hypersonic aerodynamic methods and configurations. This began with missiles, including the intercontinental ballistic missile (ICBM) effort of the 1950s, followed by development work for the Mercury, Gemini, and Apollo manned space flight programs. The next major effort was devoted to the Space Shuttle. There is continued work on hypersonics for future entry vehicles and vehicles that can land on other planets. Finally, there is a perennial effort to develop atmospheric hypersonic vehicles. These efforts have resulted in a massive amount of literature, and we will provide references for further study. In this chapter, we limit our discussion to the key things to know from a configuration aerodynamics viewpoint.

Despite the effort to develop hypersonic configurations, there is no exact consensus on what defines the start of the hypersonic flow regime. Possibilities include:

1. Mach numbers at which supersonic linear theory fails
2. Where the ratio of the constant-pressure and constant-volume specific heats, γ, is no longer constant, and we must consider temperature effects on fluid properties
3. Mach numbers from 3 to 5, where Mach 3 might be required for blunt bodies that cause large disturbances to the flow and Mach 5 might be the starting point for more highly streamlined bodies.

In this section, we will provide a brief outline of the key distinguishing concepts. The books by Bertin and Cummings^[1] and Anderson^[2] provide a starting point for further study.

Essentially, there are five key points to remember:

1. In many cases, surface pressure can be estimated fairly easily.
2. Control and stability issues lead to different shapes.
3. Temperature and aerodynamic heating become critically important.
4. Blunt shapes are commonplace.
5. Engine-airframe integration is critical.

Our discussion will conclude with a summary of the flight vehicles that have been studied extensively and sometimes even flown. Our interest is in the lessons learned from these configurations.

The X-15, shown in figure 10-1, is the only true manned hypersonic airplane flown to date. It was rocket powered and started flight by being dropped from a B-52, so it was purely a research airplane. The first flight was by Scott Crossfield in June of 1959. The X-15 reached 314,750 feet in July 1962, piloted by Joe Walker. An improved version reached a Mach number of 6.7 at an altitude of 102,100 feet in October 1967 with Pete Knight at the controls. The X-15 program flew 199 flights, with the last one being in October 1968. Milt Thompson’s book^[3] describes the X-15 program, including the crackling sounds the airframe made as it heated up!

10.1 Surface Pressure Estimation

In many cases, surface pressures are relatively easy to estimate at hypersonic speeds. At supersonic speed, we have a local relation for two-dimensional flows relating surface slope and pressure, where θ is the surface inclination relative to the freestream:

$$C_p=\frac{2\theta }{\sqrt{M^2-1}}\text{(10-1)}$$

However, this relation is not particularly useful for most cases in actual aircraft configurations. In comparison, hypersonic rules are useful. The most famous relation is based on Newton’s concepts. Although Newton was incorrect about low-speed flow, his ideas happens to apply at hypersonic speeds. The oncoming flow can be thought of as a stream of particles that lose all their momentum normal to a surface when they “hit” the surface. This leads to the relation

$$C_p=2{\sin }^2\theta \text{,}\text{(10-2)}$$

where θ is the angle between the flow vector and the surface. Thus you only need to know the geometry of the body locally to estimate the local surface pressure. Particles only impact the portion of the body facing the flow, as shown in figure 10-2. The rest of the body is in a “shadow,” and the C_p is assumed to be zero.* See Bertin and Cummings^[4] or Anderson^[5] for the derivation of this and other pressure-slope rules.

Two key observations come from the Newtonian pressure rule. First, the Mach number does not appear! Second, the pressure is related to the square of the inclination angle in hypersonic flows and not related linearly as in the supersonic formula. This illustrates how the situation in hypersonic flow is significantly different than the linear flow models at lower speeds.

The Newtonian flow model can be refined to improve agreement with data. This form is known as the modified Newtonian flow formula,

$$C_p=C_{p_{max}}{\sin }^2\theta \text{,}\text{(10-3)}$$

where the stagnation C_pmax is a function of Mach and γ,

$$C_{p_{max}}=\frac{p_{0_2}-p_{\mathrm{\infty }}}{\mathrm{(1}/\mathrm{2)}{\rho }_{\mathrm{\infty }}{V}_{\mathrm{\infty }}^2}\text{(10-4)}$$

and P₀₂ is the stagnation or total pressure behind a normal shock. This expression brings both the Mach number and the ratio of specific heats back into the problem. (Derivation of an expression in terms of Mach number, M, and ratio of specific heats, γ, is left as an exercise for the reader; see exercise 10.2.) The classical Newtonian theory is actually the limit as M → ∞ and γ → 1. These formulas are only valid when θ is positive. There are lots of other local rules (known as surface inclination rules), and Anderson’s book^[6] should be consulted for a more complete discussion. The methods normally heard in hypersonic discussions include the tangent cone, tangent wedge, and shock expansion methods. There is also a modification to the Newtonian pressure rule to include surface curvature effects. This is known as the Newtonian-Busemann rule.

How well do these methods work? We look at two cases. First we look at a blunt-body case, and then we will compare results with the pressure on a circular cone at zero angle of attack.

10.1.1 Blunt-Body Case

Figure 10-3 shows the agreement with wind tunnel data for a blunted cone.^[7] This report includes results from a wind tunnel test and has the tabulated data. In this case, the agreement is remarkably good for a Mach number of 1.9. There are two things to note. First, for many hypersonic cases, we plot with a different axis direction than classic subsonic or supersonic aerodynamics; second, it is important to observe the change in slope at s/r of about 0.9. The geometry is not a pure hemisphere cone but has a transition arc. If we had used the standard Newtonian formula, the pressure at the nose would have been 2.0. Clearly the modified formula does an excellent job.

10.1.2 Cones

Next we examine the results of Newtonian and modified Newtonian theory for two cones at zero angle of attack together with predictions from a theory given by DeJarnette et al.^[8] and the exact results from the NASA cone tables.^[9] Figure 10-4 shows the surface pressure results over a Mach number range from 4 to 10 and cone angles of 15° and 25°. The cone table results can be considered to be the exact inviscid values. Again, the cone produces a conical flow field, and the surface pressure variation is constant along the surface for an attached shock. In this case, the Newtonian approximations are not nearly as good.

Since the Newtonian estimates are not particularly good, we will provide the simple approximation from DeJarnette^[10] et al. that works well:

$$C_{p_{\alpha =0}}={\sin }^2{\delta }_v[1+\frac{(\gamma +1)K^2+2}{(\gamma -1)K^2+2}\ln (\frac{\gamma +1}{2}+\frac{1}{K^2})]\text{(10-5)}$$

where

$$K^2=(M^2-1){\sin }^2\delta \text{.}$$

The advantage of the surface inclination rules is that they only need the local geometry. These methods were combined into a program known as the Hypersonic Arbitrary Body Program (HABP), originally developed at Douglas Aircraft and also known as the Gentry code. The program is available for free download from the Carmichael’s Public Domain Aeronautical Software site.^[11] Unfortunately, the user has to be wise enough to choose which rule should be used over various parts of the body. Of course, once you proceed beyond the early stages of configuration design, it is appropriate to use CFD. You essentially trade one set of choices (selecting the right rule) for another (CFD grid and parameters)!

While we are looking at surface pressures, we should look at the change in physics from subsonic to hypersonic flow. This is a key concept in configuration development. What is the maximum (fictitious) lift on a flat plate? I call this the “ultimate” lift. The values are plotted in figure 10-5. To estimate the ultimate lift at a given Mach number, the lower-surface pressure is taken equal to be the stagnation value and the upper-surface pressure is set equal to the vacuum value. We can see that, at low speeds, the lift is generated on the upper surface, while at high speed, the lift is almost completely generated on the lower surface. This is an important consideration when designing hypersonic vehicles.

10.2 Aerodynamic Stability and Control

For stability, many hypersonic vehicles display an unusual geometric feature—thick bases. In this section, we illustrate why they are desirable. Our example relates to the thick trailing edge on the vertical tail of the X-15 shown above in figure 10-1.

The example of the difference in the flow characteristics at hypersonic speed will be used in the hypersonic directional stability problem. To start, we consider the yawing moment contribution from the vertical tail:

$$N_{VT}=q_{VT}{S}_{VT}{l}_{VT}{C}_{Y_{VT}}\text{(10-6)}$$

where q_VT is the dynamic pressure at the vertical tail, S_VT is the vertical tail area, l_VT is the moment arm, and C_YVT is the side force coefficient. The standard definition of the yawing moment coefficient, C_n, is

$$C_n=\frac{N}{q_{ref}{S}_{ref}{b}_{ref}}\text{(10-7)}$$

where b_ref is the reference span. We can write the yawing moment due to the vertical tail as

$$C_{n_{VT}}=\underset{\begin{array}{c}\text{vertical}\\ \text{tail volume}\\ \text{coefficient,}{V}_{\mathrm{V}\mathrm{T}}\end{array}}{\underbrace{\frac{l_{VT}{S}_{VT}}{b_{ref}{S}_{ref}}}}\cdot \underset{\begin{array}{c}\text{ratio of}\\ \text{dynamic pressures,}\\ \text{assume}=1\end{array}}{\underbrace{\frac{q_{VT}}{q_{ref}}}}\cdot C_{Y_{VT}}\text{.}\text{(10-8)}$$

The nomenclature associated with the problem and the equations is illustrated in figure 10-6.

For a high-speed flow, we will assume that the vertical tail is a two-dimensional surface with a constant pressure on each side, so that C_YVT = Cp_LS - Cp_US. We will consider two cases, one supersonic and the other hypersonic. We compare the results for directional stability at high Mach number using the two-dimensional rule for linearized supersonic flow and Newtonian theory for hypersonic flow, as shown below.

$$\begin{array}{ll}\text{Linear theory}& \text{Newtonian theory}\\ C_p=\frac{2\theta }{\sqrt{M^2-1}}& C_p=2{\sin }^2\theta \end{array}\text{(10-9)}$$

10.2.1 Case 1: Linearized Supersonic Theory

$$C_{Y_{VT}}=\mathrm{\Delta }{C}_p=\frac{2(\theta +\beta )}{\sqrt{M^2-1}}-\frac{2(\theta -\beta )}{\sqrt{M^2-1}}=\frac{4\beta }{\sqrt{M^2-1}}\text{(10-10)}$$

Equation (10-10) above shows that the θs cancel. We use this expression to get C_nβ:

$$C_{n_{{\beta }_{VT}}}=V_{VT}\frac{\mathrm{\partial }{C}_{Y_{VT}}}{\mathrm{\partial }\beta }=V_{VT}\frac{4}{\sqrt{M^2-1}}\text{(10-11)}$$

Note that V_VT is defined in equation (10-8). This expression shows that C_nβ is positive but vanishes for hypersonic Mach numbers.

10.2.2 Case 2: Newtonian Hypersonic Flow Theory

This time, the expression for the side force is

$$C_{Y_{VT}}=\mathrm{\Delta }{C}_p=2{\sin }^2(\theta +\beta )-2{\sin }^2(\theta -\beta )\text{.}\text{(10-12)}$$

Using trig functions, we arrive at

$$C_{Y_{{\beta }_{VT}}}=2[\begin{array}{l}{\sin }^2\theta {\cos }^2\beta +2\cos \theta \sin \beta \sin \theta \cos \beta +{\cos }^2\theta {\sin }^2\beta \\ -{\sin }^2\theta {\cos }^2\beta +2\cos \theta \sin \beta \sin \theta \cos \beta -{\cos }^2\theta {\sin }^2\beta \end{array}\rfloor \text{,}\text{(10-13)}$$

which reduces to

$$\begin{array}{rl}{C}_{Y_{{\beta }_{VT}}}& =8\underset{=1}{\underbrace{\cos \theta }}\underset{=\beta }{\underbrace{\sin \beta }}\underset{=\theta }{\underbrace{\sin \theta }}\underset{=1}{\underbrace{\cos \beta }}\\ & \cong 8\theta \beta \end{array}\text{,}\text{(10-14)}$$

and at β = 0,

$$C_{n_{{\beta }_{VT}}}=8V_{VT}\theta \text{.}\text{(10-15)}$$

If θ is zero, so is C_nβ! But opening up the angle rapidly increases C_nβ. In addition, there is no Mach number dependence. The Case 2 results were verified experimentally, and the wedge vertical tail concept literally saved the X-15 program.^[12] This effect is also the reason for flared “skirts” seen on some launch vehicles. Figure 10-7 shows the X-15 airplane at the Smithsonian Air and Space Museum on the Mall in Washington, DC. This is a photo I took to highlight the wedge vertical tail with the large base area. This airfoil results in significant base pressure drag. However, the airplane was rocket propelled and had enough thrust to ensure the base drag wasn’t critical.

This analysis brings out another key issue. For a flat plate at hypersonic speeds, the use of the classic stability derivative concept is problematic. C_Lα is no longer a constant with angle of attack. This means that the stability and control analysis has to be generalized.

There is, however, more to the story. Let’s take another look at the X-15.

Observe the top and side view of the airplane in figure 10-8.^[13] Though the wedge vertical tail can be seen in the top view, we are really interested in the side view’s depiction of the ventral fin portion of the vertical tail. The lower dashed portion of the ventral tail was designed to be dropped before landing. This turned out to be a fortuitous design feature.

To reenter the atmosphere, a high-α recovery was desired. Initial lateral-directional data for the simulator was for the horizontal stabilizer at zero deflection. Once the math model was updated, there was a pilot-induced oscillation (PIO) if the stability augmentation system (SAS) was inoperative above 15° angle of attack. After the aerodynamic characteristics were updated, the roll damper was flight critical. Loss of the SAS at an altitude above 200,000 feet would result in loss of the aircraft. It turns out that although C_nβ was good, the C_lβ was poor. Bob Hoey,^[14] who was one of the flight test engineers, described this story. The problem was the adverse rolling moment created by the large ventral fin. Although the airplane had plenty of directional stability, the dihedral effect was strongly negative at high alpha (with the left sideslip producing left roll). Famous flight test pilot Joe Walker said it was “like a marble rolling on the outside of a barrel.”

The fix proposed by the flight test engineers was to simply leave the detachable portion of the ventral fin off. C_nβ was reduced, but C_lβ was now acceptable at high alpha and Dutch roll was “about the same.” This was counterintuitive since vertical tail size was typically being increased after flight testing at that time (recall the F-100 story in Chapter 8).

It is worthwhile to look at the stability derivatives as presented by Roxanah Yancy.^[15] Figure 10-9 contains the directional data, and figure 10-10 presents the lateral data.

Specifically, look at the α = 15° to 25° areas of figures 10-9 and 10-10. Leaving the lower portion of the ventral off reduces C_nβ compared to when it is on. At a Mach number below 3, C_nβ goes negative. However, at this speed, the angle of attack can be reduced and C_nβ becomes positive. In exchange for sacrificing strongly positive C_nβ (directional stability), figure 10-10 shows that C_lβ is now at least slightly negative. The pilots found this acceptable. These figures should be studied very carefully. This is an important example of configuration aerodynamics understanding that is necessary to achieve a viable configuration.

10.3 Aerodynamic Heating

We are now ready to address what is probably the most challenging consideration in developing hypersonic vehicles. This is illustrated by looking at the relationship between stagnation temperature and static temperature:

$$\frac{T_0}{T}=1+\frac{\gamma -1}{2}{M}_{\mathrm{\infty }}^2\text{(10-16)}$$

For the adiabatic wall temperature,

$$T_{\text{adiabatic}\text{wall}}=(1+r\frac{\gamma -1}{2}{M}_{\mathrm{\infty }}^2)T_e\text{(10-17)}$$

where r is the recovery factor. Equation 10-17 describes a wall temperature with no heat transfer (adiabatic). For many hypersonic vehicles, the surface will have to be cooled and the heating needs to be estimated. For adiabatic wall temperatures, the limit for an aluminum structure is around Mach 2, which was the Concorde’s cruise Mach number. The SR-71 was made of titanium, and temperature limited the speed to slightly over Mach 3. Some sources indicate that this limit was actually the temperature limit on the wiring inside the airplane and that is the condition that limited the speed.

Hypersonic aerodynamic configuration design means that you must always deal with heating. In general, at sustained high speeds, surfaces must be cooled, and since heating is a critical concern, this means that viscous effects are crucial immediately. Unlike normal airplane aerodynamics, hypersonic vehicles fly at very high altitudes and the Reynolds number may be low enough that the flow is laminar, meaning that laminar flows are also often of interest. Recall that the heat transfer is much lower when the flow is laminar. In fact, a critical requirement in hypersonic vehicle design is the ability to estimate the transition location (for these cases, transition occurs over a region and can’t be assumed to occur at a “location”), making this ability the subject of much research.^[16] (Note: ignore the error in the first line of the abstract that has the wrong date and pilot for the Mach 6.7 flight.) A description of the aerodynamic heating on the SR-71 is available in the excellent paper by Ben Rich.^[17] An appendix in his paper provides the equations used to estimate the heat transfer coefficients.

In the 1950s, the problem of aerodynamic heating was a problem of national focus. The ability of ICBMs to reenter the atmosphere and accurately deliver the payload was a critical requirement. Initially it had been assumed that the nose shape should consist of a sharply pointed tip. However, H. Julian Allen and A. J. Eggers at NACA Ames found that a blunt shape would be much better. A blunt nose forces a detached shock wave, and most of the heat goes off the surface and into the flow field, not the vehicle. This insight enabled practical reentry vehicles. Thus, a nose or leading edge radius large enough to prevent the nose from melting had to be used. The analysis by H. Julian Allen and A. J. Eggers convinced the aerodynamic design community that blunt bodies were required to survive entry from orbit.^[18]

The maximum value of the heat transfer, q-dot, is proportional to the leading-edge radius, as shown in this relation:

$${\dot{q}}_{\text{max, laminar}}=\frac{1}{\sqrt{R_{LE}}}\text{(10-18)}$$

where R_LE is the leading-edge radius at the stagnation point. Clearly, a larger R_LE is desirable.

This result led to the choice of the manned space capsule shapes for the Mercury, Gemini, and Apollo programs. The Space Shuttle entered the atmosphere at a very high angle of attack so that it was in effect a blunt body. Figure 10-11 show a photo of Harvey Allen demonstrating the concept.

Figure 10-12 shows an example of the flow field over a sphere at a Mach number of 7.6. The shock standoff distance was critically important and its estimation was one of the most important efforts at the time.

Even when using a blunt-body reentry shape, the heating problem is severe. The Mercury, Gemini, and Apollo vehicles used an ablative heat shield, where portions of the shield actually burn off. The Space Shuttle, which reentered from a relatively low Earth orbit, used special heat-resistant tiles, a number of which had to be replaced after each flight. Details of the Apollo capsule thermal protection system can be found in the NASA report by Pavlosky and Leger.^[19]

This was the first great challenge problem for CFD (though this was in the 1960s, and the computational solution of the flow field wasn’t called CFD until the early 1970s). This challenge, known as the blunt-body problem, was particularly difficult because the flow is subsonic behind the strong normal, or nearly normal, shock wave. The flow then accelerates quickly to supersonic/hypersonic speed. Thus, this is the reverse of the transonic flow problem. In this case, the freestream is supersonic/hypersonic rather than subsonic. We want to know the shock standoff distance, the shock shape, and the flow properties at the nose, where the aerodynamic heating is the highest. The shape of the shock determines the distribution of flow properties such as entropy, which vary as the shock slope changes.

The problem was solved by computing the unsteady flow field, which is always mathematically hyperbolic. If the solution is steady, the computation will converge to the steady state result while overcoming the difficulty of the mixed elliptic-hyperbolic equation type that describes the steady-state problem. The successful approach invented to obtain a practical blunt-body calculation method is generally attributed to Moretti.^[20]

In case the issue of aerodynamic heating seems academic, consider the M = 6.7 flight of the X-15. It turned out to be the last flight of that airplane. A dummy scramjet^∗ installation was tested by placing the scramjet mockup below the airplane. The shocks from the front of the scramjet inlet impinged on the pylon supporting the scramjet. The shock interference heating was so severe that the shocks acted as a blowtorch, cutting through the structure and effectively slicing off the scramjet. The internal damage to the airplane from the heating led to the scrapping of this high-speed version of the airplane and to the termination of the program. Figure 10-13 below shows the scramjet hanging below the airplane.

Figure 10-14 shows a schematic representation of the scramjet installation. Apparently, this was done in an ad hoc fashion without serious analysis. Figure 10-15 shows the result. This was an indication of the seriousness of the aerodynamic heating problem. Both of these figures are contained in the paper by Iliff and Shafer,^[21] who had taken the figures directly from NASA TM X-1669.^[22]

The message from this “incident,” as it was described, is that shock impingement on a surface at hypersonic speed leads to extreme heating. Special care must be taken when developing a hypersonic configuration to avoid shock impingement heating.

Recall that the Space Shuttle Columbia was destroyed by the breakdown of the thermal protection system on February 1, 2003. In that case, a piece of insulating foam from the external tank broke off during ascent and damaged the leading edge of the Space Shuttle, exposing the internal structure to the heating during reentry. There was essentially nothing left to see when debris was found, in contrast to the X-15 case we’ve shown. Some of the recent flight test vehicles have also been lost due to adverse effects of aerodynamic heating.

10.4 Additional Considerations for Gas Dynamics

Because of the extreme conditions in hypersonic flows, many somewhat unique and distinctive gas dynamics effects become important. These effects are the result of a difference in the viscous effects at hypersonic speeds. Specifically, flight at high altitudes leads to a significant extent of laminar flow. In addition, Mach number effects (heating in particular) result in a thicker boundary layer. Figure 10-16 from Hayes and Probstein^[23] illustrates the situation.

The thicker hypersonic boundary layer means that it immediately affects the pressure distribution compared to the low-speed case where the flow over a flat plate surface in line with the freestream flow will not change the pressure distribution significantly. An example of this effect on pressures is shown in figure 10-17, where pressures were measured on a plate at α = 0° and M = 6.86.^[24] At low speeds, we would expect p₂ – p₁ to be zero (here, p₂ is the static pressure at some point on the surface and p₁ is the freestream static pressure). Especially near the leading edge, there is an effect of the boundary layer on the pressures. This effect is known as viscous-inviscid interaction and can be characterized as either “strong” or “weak.” An extensive description is available in Hayes and Probstein.^[25] Once again, modern CFD methods need to be used in design, and this flow feature should arise without having to be deeply involved in the theory.

As shown above, viscous compressibility effects can be important. We illustrate this by looking at the change in the skin friction drag coefficient on a flat plate with increasing Mach number. The code FRICTION can be used to study this effect; see section E.5.1. As posted on the website, the code assumes that the wall is at the adiabatic wall temperature (no heat transfer). This is the usual assumption for typical aircraft aerodynamics, up to around a Mach number of 2. However, the hardwired value of the ratio of the wall temperature to the adiabatic wall temperature, TWTAW, can be easily changed. Figure 10-18 shows the variation of skin friction drag coefficient for an adiabatic wall in laminar flow for values of the wall-temperature-to-freestream-temperature ratios (T_w/T_e) of 0.25, 1.0, and 4.0. These laminar flow results use the Eckert reference temperature method to incorporate Mach number and wall temperature effects.^[26]

Figure 10-19 shows the compressibility effects for turbulent flows for a flap plate using the Van Driest II method.^[27] The values are given normalized by the zero Mach number adiabatic wall value, T_aw. In general, the skin friction coefficient decreases with increasing Mach number and wall temperature.

At one time, theoretical aerodynamicists devoted a large part of their efforts to developing prediction methods to predict viscous-inviscid interaction effects. Today, we can use CFD to find the hypersonic flow field. A good overview of CFD for hypersonic flow appeared in a special issue of the Journal of Spacecraft and Rockets.^[28] This paper is an overview of all the papers contained in this issue of the journal. Finally, an excellent chart shown in figure 10-20 was sent to me by Chris Johnston at NASA Langley. It illustrates the range of considerations that might be important when making hypersonic aerothermodynamic predictions.^[29]

For a much more thorough description of these effects, study the books by Bertin^[30] and Hirschel and Weiland.^[31] An excellent discussion of aerothermodynamics is available in the survey paper by Hollis and Borrelli.^[32] This paper includes a discussion of radiation effects. These also have to be included when the entry velocities become extremely high.

10.5 Considerations for High-Temperature Gas Dynamics

We now briefly discuss the effects of high temperatures on gas dynamic properties. From the classic subsonic viewpoint through the supersonic aerodynamic viewpoint, we almost always assume a calorically and thermally perfect gas. This means that the ratio of specific heats, γ, is a constant. At high temperatures, this is no longer the case. According to Anderson,^[33] γ is not constant above about 980°F (800 K). Oxygen starts to dissociate above about 3,140°F (2000 K) and is completed at 6,740°F (4000 K). Nitrogen dissociation begins at 15,740°F (9000 K). Above 15,740°F (9000 K), gas starts to ionize and become a plasma. Clearly these are temperatures more closely connected to reentry vehicles than any atmospheric flight vehicles. Anderson also makes the point that this effect should be called “high-temperature gas dynamics.” It is very common in the community and in the literature to describe this as a “real gas” effect, which is not precisely correct. In addition, the gas can be in equilibrium or “reacting” in time and space. If the flow composition varies in time, additional equations must be added to the governing equation set. This situation involves finite-rate chemistry. Here we will present the difference between our classical constant-γ gas dynamics and air that is in equilibrium, using the values from tables generated at the Cornell Aeronautical Lab.^[34] An engineering applet is available to compute the same information,^[35] making use of that is known as the Tannehill curve fits.^[36]

Figure 10-21 shows the difference between calorically perfect gas and equilibrium air values for the temperature behind a normal shock wave. In this case, we present results for two different altitudes as a function of freestream Mach number. The equilibrium air results also depend on the freestream value of pressure. It is clear that the Mach number is not necessarily an appropriate reference value for equilibrium and finite-rate chemistry flows, and you will not necessarily see results presented in this fashion. However, from a low-speed perspective, it is useful in trying to understand the effects. We see that there is an extremely large difference between the calorically perfect and equilibrium gas values. For flow behind the shock, a significant amount of energy goes into dissociation of the gas instead of raising the temperature.

We next present the values for the change in pressure and density across the normal shock. The density jump is presented in figure 10-22, and pressure jump is presented in figure 10-23. The density is strongly affected by the strong shock. Conversely, there is just a small pressure difference between equilibrium air and the calorically perfect gas case. Anderson^[37] shows that the pressure is more closely connected to the fluid mechanics of the shock jump, while the temperature and density are the result of the thermodynamics of the shock jump.

To illustrate the importance of including high-temperature effects, we describe a situation where the space shuttle was almost lost during the first reentry, apparently because the high-temperature gas effects were not included in the predictions. The difference between the perfect gas and equilibrium air simulation for the pitching moment is shown in figure 10-24.^[38] There is a significant difference between the two.

One of the controls on the space shuttle is the body flap used to trim the shuttle. It is barely visible in figure 10-24 in the side view of the shuttle, located below the rocket nozzle. Figure 10-25 shows the flap in more detail. It has a limited deflection range. At a Hypersonics Short Course given at the State University of New York at Buffalo in August 1986, a speaker said that the predicted body-flap deflection required to trim was 11°. It turned out they needed a 16° deflection to trim in flight, which was nearly all that was available! The pressure distributions didn’t look very different between the two simulations, but the integrated effect was extremely important.

Figure 10-26 is a photo of the body flap on the Space Shuttle Discovery, as displayed at the Smithsonian National Air and Space Museum at the Udvar-Hazy Center. Discovery made thirty-nine flights, with the last one on February 24, 2011. It is worth visiting the museum to see Discovery. If you do, you’ll notice that the surface is not at all smooth. This is in contrast to the previous shuttle on display, Enterprise. That shuttle never flew into space and therefore never reentered the atmosphere. Its surface is very smooth. (Note: Enterprise is currently on display on the Intrepid Sea, Air and Space Museum in the New York City Harbor.)

Although this explanation of the pitching moment discrepancy seems completely plausible, there have been other explanations. More details of the problem are contained on pages 141 to 147 of the book by Bertin.^[39] This variety of opinions illustrates the importance of studying problems independently and developing judgment as a configuration aerodynamicist.

10.6 Hypersonic Vehicle Design

Hypersonic flight vehicles encompass a wide variety of applications. Rockets and missiles have become routine. Above, we described the evolution of the shape required to survive the aerodynamic heating environment for entry where the use of blunt shapes enabled success. Although hypersonic transports have been the dream of aerodynamicists for many years, we are still a long way from achieving them. Good histories of hypersonic efforts have been written by Hallion^[40] and Heppenheimer;^[41] the latter is available as a free PDF file that can be downloaded. As histories, they focus on vehicle efforts as well as the associated technology development.

10.6.1 Minimum Drag Axisymmetric Shapes at Hypersonic Speeds

Before discussing flight vehicles, it is worth reviewing the shapes of minimum drag bodies of revolution at hypersonic speeds. Using the Newtonian pressure formula, equation (10-2), the minimum forebody drag can be found for a variety of constraints using the calculus of variations. Eggers et al. presented this analysis together with wind tunnel test verification results.^[42] They considered five cases: given (i) forebody length and base diameter, (ii) length and volume, (iii) length and wetted surface area, (iv) diameter and wetted surface area, and (v) diameter and volume. Surprisingly, they found that when the body length is fixed, the body has a blunt nose. If the length is not fixed, the body has a sharp nose. They also found that, when the diameter and wetted surface area are specified, the minimum drag forebody shape is a cone. When the length and diameter are given, the minimum drag forebody shape has “as much as” 20% less forebody drag than a cone of the same fineness ratio. Although the theoretical results lead to a blunt nose, the radius at the nose is actually very small. Furthermore, the forebody shape is very closely approximated by the power law shape:

$$\frac{r}{r_{\text{base}}}={\left(\frac{x}{l}\right)}^n\text{(10-19)}$$

where the value of n is 0.75 for the minimum drag body of given length and diameter. Many other variations of minimum drag bodies have been found and collected in a book edited by Angelo Miele.^[43]

Because of the interest in minimum drag shapes, work has been done using CFD to verify the results found from Newtonian theory. Mason and Lee^[44] studied a power law body with various values of n. The CFD results found the minimum drag exponent have an n of very nearly 0.69, and the drag was indeed about 20% less than a cone with the same length and diameter. The results of the computational study are shown in figure 10-27.

Power law bodies with an n greater than 0.50 have a peculiar property. The slope at the nose is 90°, but there is no longer a leading-edge radius. So the shape could be described as “blunt,” but only weakly!^[45]

10.6.2 Brief Review of Hypersonic Flight Vehicles

NASP. A description of “recent” efforts should probably start with NASP. This was the acronym for the National Aero-Space Plane concept, although considerable work had been done previously.^{[46],[47],[48]} Based on hopes and dreams, this program was announced publicly by President Ronald Reagan in his 1986 State of the Union address. It was to be a single-stage-to-orbit (SSTO) vehicle and a passenger plane capable of a two-hour flight from Washington to Tokyo. Anyone who has taken a propulsion course and studied the staging equations understands the difficulty of developing a successful SSTO. Nevertheless, a large government program was started. An artist’s conception of the plane is given in figure 10-28. Recall that rockets have to carry both the oxidizer and the fuel.

The first step was to create the X-30 demonstrator. It was hoped that the X-30 would be viable because it would employ a propulsion system that used atmospheric air for most of the oxidizer. This would result in a large weight savings. The propulsion system envisioned is known as a scramjet (remember the X-15 fiasco with the dummy scramjet engine described previously). This is a ramjet where the incoming flow is only slowed to supersonic speed in the combustor. Details of scramjet propulsion systems can be found in a book by Segal.^[49] When the NASP program was initiated, no scramjet had been demonstrated! The development of such a system is ongoing, clearly very slowly. The NASP program was cancelled in 1993.

10.6.3 Engine-Airframe Integration and Modern Vehicle Development

X-43. Efforts since the NASP have been much more modest. Clearly the first step had to be a successful demonstration of a scramjet propulsion system. This requires flight demonstration because no ground-based facilities have the capability to simulate the flight environment. This requirement led to a concept known as Hyper-X, which became the X-43. A description of the evolution of this concept is described in a highly readable book by Curtis Peebles.^[50] As we saw above in figure 10-5, all of the important forces are on the lower surface of a hypersonic vehicle. Therefore, it had become clear that the airframe and engine would rely on controlling the lower surface flow field. In effect, the forebody became the inlet and the aft body became the nozzle. Figure 10-29 illustrates the concept.

The X-43 was mounted on the Orbital Sciences rocket-powered Pegasus vehicle that was dropped from a B-52. It boosted the X-43 to the speed where it could be separated from the Pegasus and undergo scramjet-powered propulsive flight. The first attempt resulted in a Pegasus failure on June 2, 2001. Subsequently there were two successful flights of the X-43. The second flight occurred on March 27, 2004, achieving a Mach number of 6.83 after 10 seconds of powered flight (the q was 980psf at 110,000 feet of altitude). The third flight, on November 16, 2004, reached a Mach number of 9.68 with 11 seconds of powered flight (also at 110,000 feet of altitude). In each case, the vehicle came down in the Pacific Ocean and was never recovered. All the results were obtained from onboard telemetry. Figure 10-30 provides an idea of what the X-43 actually looked like.

The November-December 2001 issue of the Journal of Spacecraft and Rockets had a special section devoted to the Hyper-X, and the 2006 AIAA Dryden Lecture was given on the X-43 by McClinton.^[51] The McClinton paper shows that a surface temperature as high as 2,000°F was measured.

X-51. The next step in the evolution of hypersonic air-breathing vehicles is the X-51. This is known as a Waverider concept. Waveriders provide efficient hypersonic flight. The designs can be thought of as a shape placed in the streamline of a body-generated flow field so that the bottom surface is “resting” on the pressure field generated by the shock wave of this flow field. The arrangement is designed to obtain lift with very low drag. See Bertin and Cummings^[52] for a good, detailed description of waveriders. At one time, waveriders had a precise definition, but the term now appears to refer to any concept that exploits lower surface lift. The application of the X-51 type vehicle is likely a missile. The X-51 concept is shown in figure 10-31. It is 25 feet long and weighs 4,000 pounds.

The first flight of the X-51 took place on May 26, 2010, achieving 200 seconds of powered flight and reaching a speed of Mach 5 at 70,000 feet. The second flight had a problem, as the scramjet had an “unstart” when they tried to switch from ethylene to JP-7 as fuel. On the third flight, a control fin locked up and the vehicle went out of control. On the fourth flight, May 1, 2013, they had success with 210 seconds of flight at a Mach number of 5.21.

Clearly, significant progress was made between the X-43 and X-51. This also illustrates the importance of maturing a technology before undertaking a vehicle development program. Any air-breathing hypersonic vehicle will have a highly integrated engine and airframe. To reiterate, in these concepts, the hypersonic propulsion is provided by a scramjet engine, which obtains thrust with a combustion chamber in which the flow is supersonic. This can be seen in figure 10-29, which is based on a presentation to the class by Walt Engelund of NASA Langley. The entire forebody of the vehicle underside is used as an external inlet to provide flow at just the right conditions to the engine. The entire underside afterbody is the exhaust nozzle. The successful design of an air-breathing hypersonic vehicle is an excellent example of the need for multidisciplinary optimization methods. This is a very hard problem, and the structure, aerodynamics, aerothermodynamics, aeroservoelasticity, and propulsion systems are very tightly integrated. Two reviews are available with more details.^[53],[54]

With further maturation of scramjet technology, we can expect to see hypersonic vehicle designs proposed using this technology. However, the vehicle must still be boosted to a high enough Mach number for the scramjet propulsion system to start. It will be interesting to see how these concepts are developed in the future.

Figure References