There have actually only been a few airplanes that are truly supersonic (able to cruise at supersonic speeds). Before the F-22, classic “supersonic” fighters used brute force (afterburners) and had extremely limited duration. As an example, consider the two defined supersonic missions for the F-14A:

F-14A Supersonic Missions

• CAP (combat air patrol)
  • 150 miles subsonic cruise to station
  • Loiter
  • Accel, M = 0.7 to 1.35, then dash 25 nautical miles (NM)—4 1/2 minutes and 50 NM total
  • Then, must head home or to a tanker!

• DLI (deck launch intercept)
  • Energy climb to 35K ft, M = 1.5 (4 minutes)
  • 6 minutes at M = 1.5 (out 125-130 NM)
  • 2 minutes combat (slows down fast)
  • After 12 minutes, must head home or to a tanker!

In this chapter, we explain the key supersonic aerodynamics issues facing the configuration aerodynamicist. We start by reviewing the most significant airplanes that had substantial sustained supersonic capability in section 9.1. We examine the key physical underpinnings of supersonic gas dynamics and their implications for configuration design in section 9.2. We present an overview of wave drag in section 9.3 (primarily looking at bodies of revolution), and then we examine supersonic aerodynamics of wings in section 9.4 and aerodynamic center shift in section 9.5. In section 9.6, we discuss the novel oblique-wing concept and aeropropulsion integration issues in section 9.7. We show applications of modern CFD and the application of MDO in section 9.8. This is followed by examples of supersonic airplane design in section 9.9. We will see that developing a practical supersonic airplane is extremely demanding and requires careful integration of the various contributing technologies. We also discuss some examples of efforts to develop new supersonic airplanes.

9.1 Supersonic “Cruise” Airplanes

The supersonic capability is typical of most of the so-called supersonic fighters, but obviously the supersonic performance is limited, as illustrated by the F-14 example above. The following list represents my selection of the previous manned supersonic aircraft. It is a very short list. It is important for aerodynamic designers to be familiar with these airplanes. We cite maximum lift-to-drag (L/D_max) ratio for each design (unless not publicly available) for supersonic flight.

• 1956: The B-58 (L/D_max = 4.5), Convair, 1^st flight Nov. 1956
  • In 1962: Mach 2 for 30 minutes
• 1962: The A-12 (SR-71 in ’64) (L/D_max = 6.6), Lockheed, 1^st flight April 1962
  • 1^st flight as SR-71, Dec. 1964
  • 1^st supersonic flight, May 4, 1962
  • 1^st flight to exceed Mach 3, July 20, 1963
• 1964: The XB-70 (L/D_max = 7.2). North American Aviation, 1^st flight Sept. 1964
  • 1^st Mach 3 flight Oct. 1965
  • In 1966: Mach 3 for 33 minutes
• 1968: The TU-144, Tupolev
  • 1^st flight: Dec. 31, 1968
• 1969: The Concorde (L/D_max = 7.4), Aérospatiale/BAC design
  • 1^st flight, March 2, 1969
  • 1^st Mach 2 flt., Nov. 4, 1970)*
• 1990: The YF-22 and YF-23 (supercruisers)
  • YF-23: Northrop-led team, 1^st flight: Aug. 27, 1990
  • YF-22: Lockheed-led team, 1^st flight: Sept. 29, 1990**
• 1997: The F-22 (supercruiser, supermaneuverable, superstealth)
  • F-22: Lockheed Martin–led team, 1^st flight: Sept. 7, 1997

Note the low L/D values associated with supersonic flight. Nevertheless, these airplanes were all remarkable.

9.1.1 The B-58

The B-58 is shown in figure 9-1. It is a strikingly attractive design. It followed the Convair approach of using a conically cambered delta wing in the tradition of the F-102 (first flight: 1953) and F-106 (first flight: 1956) airplanes from the same company. Today, B-58s can be seen at numerous museums, notably the Museum of the Air Force in Dayton, Ohio, and the Pima Air & Space Museum outside of Tucson, Arizona.

The B-58 had a three-man crew and weighed around 160,000 pounds, of which over 100,000 pounds was fuel! The structural-weight fraction was an amazingly low 14%. To achieve the required aerodynamic performance, the wings were thin—4.08% at the tip and 3.46% at the root. In 1962, the B-58 flew 1,300 mph for 30 minutes and 43 seconds, thus being the first airplane to fly over 2,000 km/h for 30 minutes, enabling its creators to claim the Bleriot prize. It also set a record by flying from Los Angeles to New York in 2 hours and 57 seconds (for comparison, the SR-71 later made the trip in 68 minutes). The required static margin for longitudinal stability was 3%, but greater than 3% was required for directional stability to handle the engine-out case. An aileron-rudder interconnect (ARI) was used to cancel the yawing moment due to aileron deflection. The plane was powered by four pylon-mounted GE J-79 engines, making it an extremely loud airplane.

One of the curious features was the use of a “double bubble” attached pod below the fuselage. The outer pod carried fuel and was jettisoned when empty. The inner pod was in fact the nuclear bomb.

The best paper to read on the B-58 is by Erickson.^[1] The airplane had a poor safety record. In part, this was because the tires were very small to allow the gear to fit in the fuselage. The takeoff and landing speeds were high, and the tires sometimes blew up. It was also very difficult to maintain.

9.1.2 The SR-71 Family

This section covers what is probably the most amazing design achievement ever made in aeronautics. Figure 9-2 shows this familiar airplane, the SR-71. Its predecessor, the A-12, first flew in 1962, while the SR-71’s first flight occurred in 1964. It served for many years after having been developed in complete secrecy. Aerodynamic heating was an important consideration on the design. The airplane was powered by two Pratt J-58 turboramjet engines. The SR-71 had a crew of two. Its wing area was 1,800 square feet, the span was 55 feet 7 inches, and the MTOGW was 172,000 pounds. Its last military flight was in 1990. NASA operated SR-71s sporadically until 1999.

The definitive paper on the aerodynamics of the SR-71 (on “the edge” between supersonic and hypersonic flight) was written by Ben Rich,^[2] who later went on to be a key member of the team that developed the F-117 stealth “fighter.” It is impossible to provide a better description of the plane than the one given by Rich.

A good description of the airplane is available in the AIAA book by Peter Merlin.^[3] This book comes with a DVD that has a wealth of information, including flight manuals, photos, and videos.

9.1.3 The XB-70

The XB-70^[4] is shown in figure 9-3. The airplane was intended to be a Mach 3 intercontinental bomber. However, the successful development of ICBMs meant there was no longer a need for the plane, so instead it became a research airplane. Two were built. It was a large airplane with a wing area of 6,297 square feet, a span of 105 feet, and a MTOGW of 542,000 pounds. In May 1966, the XB-70 flew at Mach 3 for 33 minutes. It had six GE YJ93 engines.

It only reached Mach 3 a few times. This was in part because the second airplane was destroyed in a midair collision with an F-104 in June 1966 during a publicity photo flight for General Electric. Joe Walker, the pilot of the F-104, was killed; Carl Cross, the XB-70 pilot, also died, though his co-pilot Al White survived. To increase directional stability and minimize the aerodynamic center shift, the XB-70 deflected its wingtips down in supersonic flight. This is shown in figure 9-4. Only the second airplane had full wingtip-deflection capability.

Although it was also said to have increased aerodynamic efficiency through the use of compression lift on the lower surface, this could have been achieved by mild wing camber. Note that the canard was essentially a foreplane, acting as a trimmer. It also had a very high base drag associated with the propulsion installation in the transonic flight regime.

The remaining XB-70 is on display at the National Museum of the Air Force in Dayton, Ohio. It was flown to Dayton in February 1969.

9.1.4 The TU-144

The Tupolev TU-144 was a Soviet Union contemporary of the Concorde. It flew before the Concorde, on December 31, 1968. Fifteen were built. It was similar in configuration to the Concorde with the notable exception that it had a retractable “mustache” canard (as well as the drooping nose used by the Concorde). It also had a braking parachute, very unusual for a commercial airplane. The program suffered when a TU-144 crashed during a demonstration flight at the Paris Airshow in June 1973. Although it made a number of operational flights, it wasn’t practical and the program ended. It was resurrected during the US High-Speed Civil Transport (HSCT) program in the 1990s when it was used by NASA as a testbed for supersonic flight research. The TU-144 is cited as having a wing area of 5,450 square feet, a span of 94.4 feet, and a MTOGW of 455,950 pounds.

9.1.5 The Concorde

The development of the Concorde, together with a discussion of the Russian SST effort and the aborted attempt by the US, is described in a wonderful paper by Poisson-Quinton.^[5] His paper includes data showing how the TU-144 used its mustache canard. The AIAA published a case study on the Concorde^[6] that provided a description of its aerodynamic design. Figure 9-5 is a photo from the National Archives website that shows the Concorde planform. The Concorde has a wing area of 3,856 square feet, a span of 84 feet, and a MTOGW of 412,000 pounds. The aerodynamic design of the wing is described in a paper by Wilde and Cormery.^[7] To allow the slender wing configuration to land and takeoff at an acceptable angle of attack, the Concorde took advantage of vortex lift as well as ground effects.

During the years that the Concorde was in service, I saw it frequently, flying over Long Island on its way to Kennedy Airport, taxiing around Kennedy, and also at London Heathrow. Compared to the other planes, it was small, and the wing was extremely thin. One of the most notable aspects of its flight was how loud it was. If it had flown over my house more than once a day, it would have been very annoying, even to me (and I lived about 35 miles from the airport).

9.1.6 The F-22

During the late 1970s and the 1980s, the US Air Force studied the requirements for a new fighter. A key requirement was identified as the ability to “supercruise,” wherein the airplane could fly supersonically without the use of the afterburner. This decreased fuel burn and allowed for significantly longer supersonic range than previous fighter aircraft. Stealth was also important, though not publicly discussed. The fighter requirement became official as the Advanced Tactical Fighter (ATF) in 1981. In 1986, teams led by Northrop (YF-23) and Lockheed (YF-22) won awards to build demonstrators. In 1991, after the flight demonstrations were completed, Lockheed was awarded the contract for the F-22. This process is described in detail in the AIAA book by Aronstein, Hirschberg, and Piccirillo.^[8] A description of the F-22 aircraft was given in the 1992 Wright Brothers Lecture by Sherman Mullin.^[9] A photo of the F-22 is shown in figure 9-6.

9.2 The Challenge for Supersonic Airplane Design

From this small number of actual supersonic cruising airplanes, we see that supersonic flight is a challenge. We can get some insight from the range equation:

$$R=\frac{V\left(\frac{L}{D}\right)}{sfc}\ln \left(\frac{W_{\text{initial}}}{W_{\text{final}}}\right)\text{(9-1)}$$

Here we have to counterbalance the reduction in L/D as shown above for these airplanes with an increase in V (recall that subsonic transport L/D values should be between 18 and 20). However, we always incur the extra cost of supersonic wave drag, which reduces the L/D. We also need to be able to fly without afterburner to keep the sfc low. This is difficult because modern transonic transports use high-bypass-ratio engines. The large engine/nacelle diameters are not possible at supersonic speeds; the drag would be unacceptable. Thus we have numerous challenges for economical supersonic flight.

The basic physics of the flow field change between subsonic and supersonic flow is illustrated in figure 9-7. It was originally presented by Von Kármán.^[10]

Poisson-Quinton^[11] has shown how the wave drag leads to a reduction in L/D at supersonic Mach numbers. This is illustrated in figure 9-8. The chief culprit is the C_D0 increase with Mach number due to the volumetric wave drag.

Nicolai and Carichner ^[12] have collected the minimum drag values for a number of supersonic fighters, as shown in figure 9-9. Clearly, the wave drag is large. For modern supersonic designs, the drag increase would be much less. However this figure provides insight into the challenge.

A key overview of the supersonic aerodynamic design issues has been written by Baals et al.^[13] This paper provides a basis for thinking about supersonic airplane design and should be studied for the details of the aerodynamic design thinking used in the US SST program to be described later.

In addition to the pure aerodynamic performance challenge, supersonic airplanes must address the noise problem, both around the airport (“community noise”) and the sonic boom. This proved to be a significant issue during the studies of a possible new supersonic transport in the 1990s, the HSCT.^[14] That airplane was intended to operate at Pacific-Rim ranges with 250 to 300 passengers. This was followed by investigations of the possibility of designing small supersonic business jets with an acceptable level of boom noise.^[15],[16] It is likely that these airplanes will be the next generation of supersonic airplanes, and we will discuss the boom issues below in section 9.9.4.

Next, to understand some of the key aerodynamic ideas, we will be break our discussion into two parts. First, we will present an overview of zero-lift drag in section 9.3 (primarily looking at bodies of revolution), and then we will discuss drag due to lift in section 9.4. By splitting the discussion into two parts, we are implicitly using linear theory to help us understand the main contributors to supersonic aerodynamics.

Before proceeding, we note that the increase in zero-lift drag means that the drag due to lift will also be higher at L/D_max since the drag due to lift should approach the zero-lift drag to maximize L/D. Thus, every effort must be made to reduce the volumetric wave drag. It is unlikely there will ever be enough thrust to cruise at L/D_max.

9.3 Wave Drag

The key idea underlying wave drag is the area rule. We already discussed this in Chapter 3. Basically we want a smooth area distribution. In addition, there are specific shapes that produce the minimum drag. They have been worked out analytically for axisymmetric bodies. The derivation of the wave drag integral is given in Chapter 6 of Ashley and Landahl.^[17] We cite it here because they also provide the derivation of minimum drag bodies of revolution. Equation 9-2 is the slender body wave drag formula:

$$D_{\text{wave}}=-\frac{{\rho }_{\mathrm{\infty }}{U}_{\mathrm{\infty }}^2}{4\pi }{\int }_0^l{\int }_0^l{S}^{\mathrm{\prime }\mathrm{\prime }}\left(x_1\right)S^{\mathrm{\prime }\mathrm{\prime }}\left(x_2\right)\ln |x_1-x_2|d{x}_{1}d{x}_2\text{(9-2)}$$

where S(x) is the cross-sectional area along the body. The integral in equation 9-2 requires that the ends of the body be closed, S = 0, or that S’(l) be zero. This equation is used to analyze different cross-sectional area distributions and also to find shapes with minimum drag. Since the equation is from slender-body theory, the Mach number does not appear (see the derivation in section 9.3 of Ashley and Landahl’s work).

The integral shows that it is actually the second derivative of the area distribution that is required for the computation. Clearly we want to make S’’(x) small. Unless care is taken, the numerical method may result in values that are too high because of artificial noise in the interpolation procedures and the quality of the input data for the area distribution, S. This difficulty was substantially reduced with a rather ingenious scheme due to Evelyn Eminton in Great Britain^[18] and adopted for use in the now-standard Harris wave drag program written by Roy Harris for Boeing at NASA Langley.^[19] Eminton’s approach to finding the value of the integral was to find the interpolating curve passing through the specified input points that minimized the value of the wave drag. Thus she solved an optimization problem to eliminate problems arising artificially from the interpolation procedure. The code that makes this calculation is generally known as the Harris wave drag program and is available from Ralph Carmichael’s PDAS website.^[20]

Minimum wave drag shapes subject to a variety of constraints have been found using the wave drag formual. For an open base (ignoring base pressure), the minimum wave drag is the Von Kármán ogive. Given the base area and length, the minimum wave drag is

$$D_{\text{wave}}=\frac{2{\rho }_{\mathrm{\infty }}{U}_{\mathrm{\infty }}^2}{\pi }\frac{[S(l){]}^2}{l^2}\text{.}\text{(9-3)}$$

Or, if S_ref is the base area, the coefficient of wave drag is given by

$$C_{D_{\text{wave}}}=\frac{4}{\pi }\frac{[S(l)]}{l^2}\text{.}\text{(9-4)}$$

See appendix A, “Geometry for Aerodynamicists,” for the equation of the shape of the Von Kármán ogive. The corresponding radius and area distributions are shown in figures 9-10 and 9-11.

Instead of having an open base, if the body is closed at both ends, then for a given length, l, and volume, V, the wave drag is

$$D_{\text{wave}}=\frac{64V^2}{\pi l^4}{\rho }_{\mathrm{\infty }}{U}_{\mathrm{\infty }}^2\text{.}\text{(9-5)}$$

Or, based on the maximum cross-sectional area, the wave drag coefficient is given by

$$C_{D_{\text{wave}}}=\frac{24\mathrm{V}}{l^3}\text{.}\text{(9-6)}$$

This is known as the Sears–Haack body. See appendix A for the equation of the shape. Figures 9-12 and 9-13 show the radius and cross-sectional area distribution for this body.

Adams^[21] provides derivations of minimum wave drag axisymmetric body shapes for other constraints (i.e., other than the ones for the Von Kármán ogive and Sears–Haack body).

The Sears–Haack body is not the minimum wave drag body for a given max cross-sectional area, which may often be the more relevant constraint. However, using the connection between volume and maximum body radius,

$$V=\pi r_{max}^2\frac{3\pi d}{16}\text{.}\text{(9-7)}$$

we get a form that shows the connection between the drag coefficient and fineness ratio, l/d:

$$C_{D_{\text{wave}}}=\frac{9{\pi }^2}{8}\frac{1}{{\left(l/d_{max}\right)}^2}\text{.}\text{(9-8)}$$

Figure 9-14 shows how important the use of a high fineness ratio is to reduce the wave drag. To make the drag nondimensional, we divide by the dynamic pressure and by the volume raised to the 2/3rd power. Care must always be taken when looking at drag values to make sure you understand the reference area used to define the nondimensional drag coefficient. It is not unheard of that incorrect drag coefficient values were used!

It is worth investigating the minimum wave drag shapes in a little more detail. Since we’ve seen the key role that the fineness ratio plays in the drag, we can write the wave drag coefficient for three different cases: (i) the Sears–Haack minimum wave drag for a given volume and length, equation 9-9; (ii) the minimum wave drag for a specified maximum cross-sectional area, equation 9-10; and (iii) the minimum wave drag for the Von Kármán ogive, equation 9-11

$$C_{D_{S-H}}{\left(\frac{l}{d}\right)}^2=\frac{9{\pi }^2}{8}=11.1\text{(9-9)}$$

$$C_{D_{S-H}}{\left(\frac{l}{d}\right)}^2={\pi }^2=9.87\text{(9-10)}$$

$$C_{D_{S-H}}{\left(\frac{l}{d}\right)}^2=1\text{(9-11)}$$

Thus the minimum wave drag body for a given max cross-sectional area is 11% less than the minimum wave drag body of a given volume with the same max cross-sectional area. This also shows that ignoring base drag (reasonable if the base is filled with a jet exhaust), the drag is much lower when the base is open. In fact, the wave drag is very nearly a linear function of the ratio of the base area to the max cross-sectional area (see figure 3 in NACA TN-2550 by Adams):

$$C_{D_w}{\left(\frac{l}{d}\right)}^2={\pi }^2-({\pi }^2-1)\frac{S_n}{S_{max}}\text{(9-12)}$$

The minimum drag for a body of revolution with a constant cross section placed between the front and back of a body (with a fairing to close the shape in the front and back) has been given by Heaslet and Lomax.^[22] To be a little more complete, we note that Harder and Rennemann^[23] found minimum drag shapes that have slightly less drag than Adams found. Their work presents results for the case of a given volume.

For comparison, Krasnov^[24] points out that using a simple cone forebody has a drag of about twice the value of the optimum forebody value for a fineness ratio of 3. However, a tangent ogive only has a drag about 7% higher than the optimum value.

A general minimum wave drag axisymmetric shape can also be found using the analysis by Lord and Eminton.^[25] I have implemented this analysis in the interactive program MinDrag. It computes the minimum value of the supersonic wave drag for an axisymmetric body as well as the area distribution required to attain this value for a given length, volume, nose area, base area, and another area specified at a given location along the body. A screenshot of the program is shown in figure 9-15, and the program can be interactively executed by selecting Mindrag.exe in section E.5.2.4.

Often, the wave drag of supersonic airplanes is given in terms of a multiple of the wave drag of a minimum drag body of revolution. Typically the Sears–Haack body is chosen because of its well known drag expression. In the early stages of design, it’s worth collecting data from previous designs. One rule of thumb was to use 19/(l/d)² as opposed to the Sears–Haack value of 11.1. Jobe has collected data in an AIAA book^[26] based on his Air Force Report.^[27] It is also worth investigating examples of cross-sectional area distributions for the B-58 and XB-70 given by Tinetti, Maglieri, and Bobbitt.^[28]

Because the surface area can change as the fineness ratio changes, it would be interesting to find the minimum of the sum of the friction and wave drag. Although I worked this out many years ago, I discovered that it had already been done by W. Bailey Oswald.^[29] Figure 9-16 shows his result for two cases. The first is the minimum drag fineness ratio when the maximum frontal cross sectional area is specified, and the second case is the result when the volume is specified. For the first case, the minimum fineness ratio occurs for a value slightly above 15 (this would be the usual case). When the volume is specified, the minimum drag fineness ratio is a little less than 25 (this is much too high for a practical manned airplane). In both cases, the minimum is fairly shallow, so you can select values that are significantly lower than the minimum shown in his results without incurring a large penalty.

Once we get past the “smooth area,” the next step is to make the area distribution for the entire airplane close to the axisymmetric minimum wave drag shapes described above. We will see examples below.

9.3.1 A Curious Story

Both the Von Kármán ogive and Sears–Haack bodies have a curious feature. They are slightly blunted. Intuitively we expect them to have sharp noses. Yet, the slope at the nose is 90°, and the leading-edge radius is zero, which is hard to visualize. The explanation is that it’s better to have a high slope, and thus high pressure, at the nose, where dS is small, rather than further aft, where r is larger and dS is much larger (dS = 2πrdr). These shapes are geometrically blunt while being aerodynamically sharp. More details associated with this feature of the geometry are available.^[30]

In addition, our analysis revealed that the nose shape of the Von Kármán ogive and Sears–Haack bodies are essentially equivalent to a power law body with an exponent of 0.75. This means that the minimum drag supersonic and hypersonic bodies are related. Figure 9-17 shows an extreme “blow-up” of the power law body at the nose.

9.3.2 Multiple Bodies to Reduce Wave Drag and Favorable Interference

Another possibility for reducing wave drag occurs when multiple bodies of revolution are arranged for that purpose. There can be both a favorable and an unfavorable interaction between bodies that are located in close proximity. The idea of favorable interaction is the reason for concepts like the Grumman Tribody of 1978. Unfortunately, the interaction can be sensitive to the Mach number. One of the ideas has been to stagger stores based on flight Mach number to reduce drag, which was the basis for the work of Jack Nielsen.^[31] Previously, Friedman ^[32] had also done an analysis that showed the possibility of favorable interaction. Nielsen says that, compared to the sum of the drag of the individual bodies, “the drag of a pair of bodies can be either double or nearly halved, depending upon the lateral and longitudinal spacings of the bodies.” According to Friedman, a “three-body configuration is found for which the total wave drag is about 35 percent less than the sum of the individual wave drags of the three bodies.” Figure 9-18 from Nielsen^[33] shows the sensitivity of the interference drag for one Mach number. The potential advantage is available, but it must be used very carefully.

Multiple axisymmetric bodies are just one possibility for favorable interference. A number of other possibilities exist. Kulfan has examined many of them for applications in supersonic airplanes.^[34] That paper includes the pertinent ideas and an extensive reference list.

9.3.3 Planar Wing Wave Drag

We generally think of axisymmetric bodies as the shapes that minimize wave drag. However, this isn’t the case. An example of a potential reduction has been shown in Küchemann’s book.^[35] Planar wings with the same volume as a Sears–Haack body can actually have less drag than the axisymmetric body. The data is plotted as a fraction of the Sears–Haack drag values in figure 9-19. A significant reduction in wave drag can be obtained with these flattened-out volumes.

9.4 Wings: Lift and Drag Due to Lift

Now we consider the lift and drag for traditional supersonic wing planforms. One of the fundamental notions in supersonic aerodynamics is the distinction between a subsonic edge and a supersonic edge. The concept is illustrated in figure 9-20.

Essentially all trailing edges are supersonic. This disconnects the upper- and lower-surface pressure distributions at the trailing edge, so there is no need for Kutta condition. The issue then becomes whether the leading edge is subsonic or supersonic. If the leading edge is subsonic, the flow about the leading edge is similar to the subsonic case in two dimensions. If the leading edge is supersonic, the flow at the leading edge behaves as if it were in a locally two-dimensional supersonic flow. This leads to a significant difference in the physics of the flow on the wing.

We will use a delta wing planform as an example. Figure 9-21 shows the difference in the planform and shock wave locations. It also introduces the concept of conical flows as important in thinking about supersonic wing performance. In a conical flow, both the geometry and the flow properties are constant along rays through the apex.

For the subsonic leading edge case, the upper and lower surfaces can still communicate with each other, as shown in the figure 9-22. Here, a point on the lower surface can create a disturbance that propagates in front of the wing, where a point on the upper surface aft of the lower surface disturbance point can be influenced.

Let’s examine the linear theory spanwise pressure distributions for the two different cases, with the first case being subsonic leading edge and the second being supersonic. Figure 9-23 is for the case of a subsonic leading edge, showing the singularity at the leading edge, just as we expect in subsonic flow. The equation for the loading is given by

$$\frac{\mathrm{\Delta }{C}_p}{C_L}=\frac{2}{\pi }\frac{1}{\sqrt{1-{\left(\frac{\tau }{m}\right)}^2}}\text{(9-13)}$$

where m = βcotΛ, and τ = (βy/x).

Figure 9-24 is based on a supersonic leading edge, and here we see that the pressures outside of the Mach cone are constant. Thus, outside the Mach cone, the constant value of the loading is given by

$$\frac{\mathrm{\Delta }{C}_p}{C_L}=\frac{1}{\sqrt{1-n^2}}\text{(9-14)}$$

where n = tan Λ/β = 1/(βcotΛ) = 1/m, m>1. Inside the Mach cone,

$$\frac{\mathrm{\Delta }{C}_{\rho }}{C_L}=\frac{1}{\sqrt{1-n^2}}(1-\frac{2}{\pi }{\sin }^{-1}\sqrt{\frac{n^2-{\sigma }^2}{1-{\sigma }^2}})\text{(9-15)}$$

where σ = ytanΛ/x, 0<σ<1.

The solutions reflect the significant difference in the physics of the two cases.

9.4.1 Arrow Wings and Conical Camber

To think about the supersonic aerodynamics of wings, it is useful to consider a class of planforms for which exact linear theory solutions are available, namely the arrow wing. The flow is still conical and departs from a delta wing by adding a trailing-edge cutout, usually described as the notch ratio. Figure 9-25 provides the nomenclature.

The linear theory lift is given in figure 9-26 for two different notch ratios, ζ. The calculations were made using program arrow.f (see section E.8.3). Note the change in character when passing from the subsonic-edge case to the supersonic-edge case. The parameters are presented in the figure to “scale out” the Mach number, so the results are applicable for all Mach numbers.

Using conical flow theory, the analytic solution programmed in arrow.f for a subsonic leading edge is

$$\beta C_{L_0}=\frac{4m}{E^{\mathrm{\prime }}(m)}[\frac{\zeta }{1+\zeta }+\frac{1-\zeta }{{(1-{\zeta }^2)}^{3/2}}{\cos }^{-1}(-\zeta )],m\le 1\text{(9-16)}$$

where m = βcotΛ_LE and E’(m) = E(k), where k = √(1-m²) and E(k) is the complete elliptic integral of the second kind.^[36]

In the case of the supersonic leading edge^[37]

$$\beta C_{L_0}=\frac{8m}{\pi (1+\zeta )}[\frac{1}{\sqrt{m^2-{\zeta }^2}}{\cos }^{-1}(-\frac{\zeta }{m})+\frac{\zeta }{\sqrt{m^2-1}}{\cos }^{-1}\left(\frac{1}{m}\right)],m\ge 1\text{.}\text{(9-17)}$$

The drag due to lift is also available from the theory. It is computed based on the 0% leading-edge suction drag, and then it is reduced if the leading edge is subsonic by the leading-edge suction; the component of the suction parallel to the velocity vector is called the leading-edge thrust, T. Thus the drag due to lift is

$$D=\alpha L-T\text{(9-18)}$$

where T is zero if the leading edge is supersonic. This can be put in terms of the force coefficients and written in a form that eliminates both Mach number and C_L dependence using the equation given above. The second term inside the bracket is zero for the supersonic leading-edge case:

$$\frac{\mathrm{\Delta }{C}_D}{\beta C_L^2}=\frac{1}{\beta C_{L_0}}[1-\frac{\pi m}{\beta C_{L_0}}\frac{1}{(1-\zeta )}\frac{\sqrt{1-m^2}}{E^2}]\text{(9-19)}$$

The results are shown in figure 9-27. This time, the figure is more complicated. For subsonic edges, two branches of the drag are shown. One assumes that the full leading-edge suction can be realized, while the other assumes that no leading-edge suction is achieved. If full leading-edge suction can be obtained, subsonic edges are desirable to reduce the drag. However, if no suction can be obtained, then the drag is reduced when the edges are supersonic. Although we have presented results for planar wings, the effect of leading suction is attained in practice by cambering the wing. Essentially, the linear theory says that by cambering the wing, a benefit equivalent to the full leading-edge suction can be realized. Experience shows that only a portion of the leading-edge suction predicted by linear theory can be achieved, possibly explaining why many of the supersonic airplanes listed above have had supersonic leading edges at the design Mach number. Many NASA designs were more aggressive, and selected an m of about 0.75. They were expected to attain the full value of leading-edge suction. In general, the calculation of the leading-edge suction is difficult. However, progress has been made.^[38]

The conical flow theory used to obtain the solutions presented here was also the basis for a special type of camber that was widely applied to many delta-wing supersonic aircraft to reduce the drag due to lift. Known as conical camber, the surface was cambered so that the geometry was straight along rays through the apex of the wing.^[39] Conical camber was used on the F-102, F-106, and B-58. It was also used on the F-15.

Figure 9-28 shows the conical camber on an F-102. The photo was taken by W. H. Mason at the Pima Air & Space Museum outside of Tucson, Arizona. It is taken from behind the airplane looking forward along the leading edge, showing the large amount of camber. The camber is pronounced on this plane because it was added after the plane was built and was limited to the wing outboard of the 85% spanwise cuts to maintain the basic structure. The F-15 has conical camber, but it is distributed across the span and difficult to see.

Figure 9-29 shows one of the inventors of conical camber, NACA/NASA research scientist Charles F. Hall, looking at a model in a wind tunnel at the Ames Research Center.

Figure 9-30, from a survey paper by Von Kármán,^[40] shows how the concept works. With a small leading-edge radius required for supersonic flight, the effect of leading edge suction can be achieved by providing a forward-facing surface with a low pressure. The resulting drag polar is shifted upward, approaching the value of a flat surface with 100% leading suction.

9.4.2 Modified Arrow Wings

Once the arrow wing is used to establish a basis for thinking about the aerodynamics of supersonic wings, one proceeds to think in terms of modifying it for practical application. The literature frequently refers to a “modified arrow wing” configuration that to me never appeared at all like the planform shown above.^[41]

Figure 9-31 traces the evolution of the arrow wing to a configuration applicable to actual aircraft design.^[42] The sequence of modifications shown in the figure can be described as follows. For the basic arrow planform, the tip is “clipped.” The structural span is decreased, and a portion of the wing that doesn’t actually contribute aerodynamically is eliminated. Next, for planform A, the outboard portion of the wing is unswept slightly. This improves the subsonic efficiency with little penalty for supersonic efficiency. It also reduces the aerodynamic center (ac) shift. Finally, for planform B, the trailing edge is filled in. This helps the subsonic pitchup, makes trailing-edge flaps more effective, and also further reduces the ac shift. For a specified thickness, the extra chord means that the root t/c is reduced. This explains why a planform that doesn’t look anything like an arrow wing becomes a “modified” arrow wing.

A related issue for modified arrow wings with the outboard portion unswept is the potential for pitchup at subsonic speeds and high angle of attack (described previously in Chapter 5). A review of this effect and an approximate method for estimating for pitchup for these types of planforms is available in a paper by Benoliel and Mason.^[43]

9.5 The Aerodynamic Center Shift

Another important consideration in supersonic configuration design is the shift of the aerodynamic center (AC). In the classical 2D case illustrated in figure 9-32, the aerodynamic center shifts from the 25% chord to the 50% chord. The arrow.f program also provides the aerodynamic center location as well as the lift and drag (but only at supersonic conditions). We need to see how modifications to the classic delta wing shape can be used to reduce the AC shift.

Figure 9-33 is taken from the Concorde case study,^[44] and it shows that the Concorde planform shaping is critical in reducing the shift. However, the shift is still significant and fuel transfer must be used to maintain an acceptable relation between the center of gravity and aerodynamic center.

The allowable CG range shown in figure 9-34 for the Concorde shows how important the CG control was.^[45] The Concorde used a root chord of 90.75 feet as the reference chord. At takeoff and landing, the CG was required to be between 52.5% and 53.5% of the root chord. At supersonic cruise, the AC shifts about 5.5%, or 5 feet. The allowable CG limits are between 58% and 59% of the root chord. Thus for an airplane of slightly over 101 feet in length, the CG must be maintained within about 11 inches. This is why configuration designers consider airplane balance immediately upon examining any design drawing.

All supersonic airplanes use fuel transfer to control the CG location and maintain a relation between the CG and AC that allows the airplane to be adequately controlled and also fly with minimum control surface deflections to minimize trim drag. The XB-70 was unusual in that the wingtips folded down in supersonic flight. Although usually described as providing extra aerodynamic efficiency and additional directional stability, perhaps the most important consequence of folding the tips was to control the AC shift.

Additional examples of planform shaping to reduce the AC shift were presented by Lamar and Alford.^[46] They used a slightly different parameter to correlate AC shift, showing the benefit of using a double-delta planform, which would be a rough approximation of the Concorde planform. Figure 9-35 shows the result of their parametric study of the aerodynamic center shift. They show that a pure delta wing has a much larger AC shift then a double delta. They found that the inboard delta meeting the aft delta at about 45% of the semispan produces the smallest AC shift. When it comes to the extent of the more highly swept inboard delta, the larger the better.

Why does this work? The idea is that the inboard wing, being more highly swept and having a lower aspect ratio, has a lower C_Lα, that is insensitive to Mach number. The outboard section, with less sweep, has a higher aspect ratio and a C_Lα that decreases with Mach number. Thus the composite planform has a smaller AC shift. How this was discovered is a little mysterious. The NASA paper has a date of 1966. Ben Rich provided a similar explanation is his paper on the SR-71.^[47] The first version of the SR-71, the A-12, was flown in 1962. The Lockheed’s proposed commercial supersonic transport (SST) design used a double delta, and the configuration was apparently chosen in 1963.*

A final example requiring attention to aerodynamic center shift arises for airplanes with variable-swept wings. Although no longer fashionable, variable-sweep wings have the attraction of performing well at both subsonic and supersonic speeds. When the wings are swept back, the low aspect ratio also reduces the gust-load response for high subsonic-speed on-the-deck penetration missions. The problem of the airplane becoming too stable when the AC shifts aft is aggravated when the variable-sweep wing is swept back at supersonic speeds. This adds more (stabilizing) planform area aft of the CG. Figure 9-36(a) from a paper by Kress^[48] illustrates the problem by comparing the static margin shift with Mach number for F-111 and F-14. Clearly the F-111 supersonic static margin is excessive. The F-14 has a lower static margin, and it has another feature that decreases the static margin even more: the glove vane.

Kress invented the glove-vane for the F-14, as shown in figure 9-36(b), to further reduce the static margin. When the wings were swept back, the glove vane came out, adding planform area forward of the center of gravity. This resulted in a reduction in AC shift.

The main reason that the F-14 had a reduced static margin shift was the use of an outboard pivot location, which differed in comparison to the F-111, as shown in figure 9-36(c). The use of the outboard pivot reduced the AC shift. The effect of pivot location had been the subject of numerous wind tunnel tests at NASA Langley. Good background information on variable-sweep aircraft development can be found in the report by Polhamus and Toll.^[49] Their paper also discusses the skewed-wing concept that became known as an oblique wing, as detailed in the next section.

9.6 The Oblique-Wing Concept

This section describes the supersonic airplane concept based on the oblique wing. When considering both lift and volumetric drag, a somewhat radical wing concept with good potential for a supersonic transport application is the oblique-wing idea advocated by R. T. Jones for many years.^[50],[51] The idea is that both the longitudinal as well as spanwise distribution of lift should be elliptical. The natural way to do this is with an oblique wing planform. Jones made the argument that there is no reason that an airplane has to be symmetrical.

Originated for its good drag-due-to-lift characteristics, this shape is also good for volumetric wave drag because the area diagram can be very good, especially for a flying wing configuration. Figure 9-37 from the Wing Theory book by Jones^[52] illustrates the advantage. It compares the wave drag for an oblique elliptic wing to a swept wing having the same aspect ratio and thickness-to-chord ratio. There is a clear advantage for the oblique wing. Note that the advantage occurs for Mach numbers around 1.4 to 1.5. Earlier in this chapter, we saw how important length was in reducing the drag of axisymmetric bodies, and that is important here too. Admittedly, the speed range where the concept shows most benefit is less than the usual notion of a Mach 2 airplane attributed to the Concorde. However, the large decrease in drag makes this concept compelling for a supersonic configuration. Because Jones worked at NACA/NASA and had made many contributions, he was able to convince NASA to conduct numerous wind tunnel tests. Many of the wind tunnel tests were nominally “transonic” tests, but they actually emphasized the upper-transonic speed regime, which might actually be called lower supersonic.

Many other aerodynamicists have examined the concept. It has always been found to be attractive. We only cite one among many, the paper by Li, Seebass, and Sobieczky.^[53] They include a good reference list to other work and advocated the design of an oblique flying wing at a Mach number of 1.4. Aeroelasticity was also an issue, and Weisshaar and his colleagues have presented results of their work in this area in numerous papers; we reference only one here.^[54] Although aeroealsticity needs to be addressed, today’s modern design and active control technology should be capable of handling any serious issues.

As an unconventional configuration, an oblique wing was built and flown by NASA. The AD-1 (Ames-Dryden-1) oblique-wing research aircraft was successful; figure 9-38 shows it in flight. It was strictly a subsonic airplane, intended to investigate the flying qualities of an oblique wing. The wing could sweep from 0 to 60 degrees in flight. Its first flight was in December 1979, and it finished the flight test program in August 1982. The demonstrator was very basic; there was no automatic flight control system. The plane suffered from some issues with handling qualities, as well as adverse aeroelastic effects. With the use of a modern flight control system, however, it seems that the oblique wing would be a viable concept. A description of the program and outcomes are documented in two AIAA Papers.^[55],[56]

After it finished its demonstration, it was displayed at the visitor’s center at NASA Ames Research Center, located at Moffett Field in California, as shown in figure 9-39. It was so small that we include a photo with the pilot standing in front of it (figure 9-40) to appreciate its size. Today, it’s on display indoors at Hiller Aviation Museum in San Carlos, California.

Numerous studies of the oblique wing as a new supersonic transport have been made,^[57],[58] but it was not considered in the HSCT studies conducted by NASA with the industry in the 1990s.

A comprehensive survey of oblique-wing work has been compiled by Hirschberg, Hart, and Beutner.^[59] Subsequent to this paper, a major optimization study was carried out at NASA Ames showing that advanced computational methods could further improve oblique-wing aerodynamic designs.^[60] The case in favor of the oblique-wing concept for modest supersonic speeds remains compelling.

9.7 Aeropropulsion Integration

Efficiently integrating the propulsion system with the airframe requires much more effort for supersonic airplane design than for subsonic airplanes. Covering the entire subject here is beyond the scope of this course, but we point out a few of the issues and provide an example showing how to include the propulsion system efficiently. The design of the inlet and exhaust nozzle often requires as much computational and testing effort as the rest of the airframe.

The inlet has to be carefully designed to handle the required mass flow at supersonic cruise while also operating efficiently at low speed. A variable geometry inlet is often required if the cruise Mach number is more than 1.6. A stable shock system providing the air to the engines is hard to maintain. The XB-70 and SR-71 had notorious problems with unstarts (breakdowns of the supersonic airflow). Eventually the SR-71 had an automated system to recover from an unstart. Many fighters that have inlets close to the side of the fuselage are offset and use boundary layer diverters to ensure a uniform flow into the inlet. The F-35 uses a diverterless inlet, which features a “bump” on the fuselage side in front of the inlet to force most of the boundary layer to flow around the inlet. This became possible when CFD could be used to design the fuselage/inlet.

Figure 9-41 from Don Baals et al.^[61] shows how integrating a podded-engine installation under the wing can lead to favorable interference. The wing is reflexed around the nacelle. The drag of the aircraft is very sensitive to the location of the pod.^[62]

The exhaust nozzle also requires careful design. Because of the afterburner, the nozzle has to have a variable geometry. Thrust vectoring may also be used. Each of these features will require careful attention. The external contour typically has what is known as a boattail shape, which has to be carefully shaped to avoid extra drag. Boattail analysis was one of the first intense CFD efforts.

The best place to learn about propulsion system installation in supersonic aircraft configuration work is in the book by Nicolai and Carichner,^[63] which describes inlet design in Chapter 15 and nozzles in Chapter 16. Studies continue to be carried out for the aeropropulsion integration of proposed supersonic business jets.^[64]

9.8 Computational Methods and Supersonic Aerodynamic Design

Although powerful CFD methods are available today, we need to provide a context for their use. We’ve described the fundamental aerodynamic ideas that can be used to develop a supersonic configuration. These came from slender-body and linear-theory analysis. For most supersonic flight, the flow is necessarily attached to keep drag low, ensuring that the inviscid flow field combined with skin friction estimated from boundary layer theory can provide good estimates for drag. However, supersonic flight is demanding and very sensitive to the accuracy of the estimates. We will describe methods to use as a starting point, but complete CFD will have to be used for accurate drag values.

The drawbacks to using CFD during the initial stages of design are threefold. First, CFD requires a high-quality grid to be generated on a detailed geometry. We need the grid to perform computational aerodynamic simulations. The process is time consuming, and it is highly desirable to have a good starting point for the design when starting the work of creating a detailed geometry. The second issue is the need to represent the geometry with a flexible parametric representation that is capable of being readily changed to improve the design. The third issue is that, to obtain the design shape using CFD, some sort of formal optimization method is needed because the “cut-and-try” approach is inefficient. Formulating the optimization problem to achieve a good design requires skill and experience. The optimizer is excellent at finding any weaknesses in the problem formulation or the analysis! It’s harder than it appears in typical student textbooks. Nevertheless numerical optimization methods are very powerful and will be increasingly used to develop future configurations. More importantly, serious aerodynamic design is only of interest when the entire system is included. This means using the techniques that fall under the umbrella of multidisciplinary design optimization (MDO) to do the aerodynamic design together with the other key disciplines.

With all this in mind, we’ll lay out a way to get to a good supersonic design. Details on CFD can be found in the companion volume to this work.^[65]

9.8.1 The Linear Theory Starting Point

Linear theory was used to investigate many configurations at NASA and in the aviation industry during the 1960s, used in conjunction with an active aerodynamic testing program. This was done in anticipation of the development of the US SST (supersonic transport). The program produced many insights into supersonic aerodynamics for application to aircraft configurations. The basic tools are the Harris wave drag code,^[66] a supersonic panel method^[67] with corrections for attainable leading-edge thrust,^[68] and formulas for estimating skin friction drag using the Van Driest method.^[69] We will review how well these methods work.

Let us look at the wave drag estimation first. Figure 9-42 is one of the famous figures from the NASA wave drag report by Harris.^[70] The Harris wave drag code uses the input geometry to find the cross-sectional areas to compute the wave drag integral; see equation 9-2 and Chapter 3. Like most NASA codes of that era, it used the Craidon input, as described in NASA TM X-2074.^[71] It was much more efficient to use essentially the same input geometry in numerous codes.

9.8.1.1 Body of Revolution

Examples of the accuracy of the Harris code for a Haack–Adams body of revolution of fineness ratio (l/d) of 10 are shown in figure 9-43. The geometry is shown in figure 9-43(a) in terms of normalized radius distribution along the length of the body, and the computed zero-lift wave drag coefficients at different Mach numbers are shown in figure 9-43(b). Wind tunnel data is from a NASA Langley Research Center Test.^[72] The solution labeled GASP: Space Marched is an inviscid Euler solution from the GASP CFD code.^[73],[74] At moderate Mach numbers, the Harris code is reasonably good. These results are from work at Virginia Tech.^[75] If you are interested in more details of the analysis performed to generate the results presented here, you should read the cited MAD Center report.

9.8.1.2 Wing

We next consider the zero-lift wave drag prediction for a wing shape. The geometry of the so-called Squire wing^[76] is shown in figure 9-44(a). The wing has a 9-percent-thick biconvex airfoil section on the centerline, and it features elliptic spanwise sections that are picked to match the centerline airfoil. The wing may be a little thick for wave drag application but has wind tunnel data available for comparison in the same report (ARC R&M 3818).

As shown in figure 9-44(b), for Mach numbers less than 2.2, the Harris code correctly predicts the trend of the zero-lift wave drag coefficient with increasing Mach number. However, the Harris code overpredicts the drag, as in the body of revolution case examined above.

9.8.1.3 HSCT Configuration

We complete our examples of what to expect from the Harris code and a linear theory panel code by comparing results for a hypothetical high-speed civil transport (HSCT) study from the same MAD Center report (MAD 96-12-01, Dec. 1996) that we used to compare wave drag above. Figure 9-45 shows the configuration geometry and the grid for GASP code analysis. In much of our work, we used the space marching option in GASP to reduce computer time. As long as the flow is supersonic, we can replace iterations in time with iterations in space, starting at the nose and marching downstream. This scheme was key to our optimization work that required thousands of analysis runs. We were fortunate to be able to use an automatic grid-generation code written by Ray Barger at NASA Langley.^[77],[78]

Figure 9-46 compares the computed lift and pitching moment coefficients from CFD methods and linear theory at M = 2.4 for the HSCT configuration. Both the linear theory and Euler results include skin friction (SF) estimates, although they don’t influence the lift and moment estimates. The figure also contains the predictions from GASP using the parabolized Navier–Stokes (PNS) option in GASP using the Baldwin–Lomax turbulence model, which can be expected to be accurate for attached flow. The linear theory overpredicts the lift slightly, as well as being slightly different for the pitching moment. In general, however, it appears that the simpler methods can provide valuable information.

The drag polar comparison is presented in figure 9-47. This is the most important comparison for design. Let us look at the values at the design C_L of 0.082. The value of the drag coefficient, C_D, is 79.2 counts for PNS analysis, 78.9 counts for Euler plus skin friction (SF), and 77.1 counts for linear theory plus skin friction. We see that the linear theory predicts a drag that is consistently low. The Harris wave drag estimates are within two counts of the Euler CFD values. The skin friction estimate is about one count higher than that of the PNS prediction. So at first glance, it appears that combining data from the Harris code, linear theory, and skin friction code provides an excellent estimate. The drawback for this case is that a two-count drag underprediction results in a 120-NM overestimation of the range. This illustrates how sensitive the HSCT design was to drag. Eventually the HSCT program was cancelled, and presumably it was in part because of the extreme requirements, as reflected in the sensitivity to drag.

9.8.1.4 Wing-Body Configuration

Figure 9-48 is the final example of basic linear theory. The model, shown in the top-left corner of the figure, is a mid-mounted 45° swept wing on an ogive cylinder body. The wing has an NACA 64A005 airfoil section. The correlation between wing pressures from a wind tunnel test^[79] and predictions from the panel method code Woodward II^[80] is shown at three stations on the wing and the agreement is reasonably good. But the main reason I’m including this figure is to reiterate that, for a supersonic trailing edge, there is no Kutta condition and the upper- and lower-surface pressures don’t come together. For highly swept wings, we often look at spanwise pressure distributions and this feature of the trailing-edge behavior isn’t readily apparent. Students should be aware of this difference between subsonic and supersonic flow.

The linear theory tools available can be used very effectively to understand supersonic configurations. Before undertaking your own studies, there are a few aspects of supersonic aerodynamics that are worth understanding.

9.8.1.5 Wave Drag

It is important to remember to subtract the capture area from the cross-sectional area values. Although the wave drag code does this automatically, students should be aware that this is how it’s done. In addition, although we often talk about the Sears–Haack body as a minimum wave drag value, opening up the base results in an even lower value of drag. I suggest “playing” with the simple interactive MinDrag code on the software site. Of course, base drag would have to be added to the wave drag. Often we assume that the base is the exhaust nozzle from the propulsion system, meaning there is no base drag. The Harris code is available from the PDAS site.^[81]

9.8.1.6 Wing Camber to Reduce Drag Due to Lift

At supersonic speeds, the minimum drag design for a typical cruise lift coefficient (often 0.1 or less) has very little camber and twist compared to the camber and twist used in subsonic and transonic wings. The camber and twist tend to achieve a drag level essentially the same as a flat surface achieving 100% suction (of course this is zero for supersonic leading edges). This was surprising to me when I observed this.

Linear theory methods normally provide optimization options that allow the drag due to lift to be minimized due to constraints on the specified lift coefficient and trim. This is an important advantage for these methods. For a supersonic planform, we often think of the wing camber and twist as being combined, designating the combination “warp.”

Three panel method codes are available at PDAS:^[82] (1) TEA201, the Boeing wing design methods originating from their US SST work,^[83] (2) the original Carmichael–Woodward code,^[84] and (3) the combination of Woodward versions with other capabilities, W12SC3.^[85]

Several survey papers should be read or reread to review key aspects of supersonic configuration design before designing a new configuration.^{[86],[87],[88]}

9.8.2 Modifications to Linear Theory: Attainable Thrust

“Normal” linear theory supersonic methods are weak with respect to the amount of leading-edge suction predicted. Over many years, Harry Carlson at NASA Langley developed a modified linear theory to address the problem. After studying massive amounts of experimental results, he developed the concept of attainable thrust, which supplements linear theory with an empirically based expression for the degree of leading-edge suction that should be expected in practice. Work was summarized together with a discussion of wing design in NASA TP-3202.^[89] A final version of the attainable thrust formulation was published in NASA TP-3557.^[90] This led to development of a design method to determine the wing camber and twist required to get the best possible wing.^[91] The code is known as WNGDES; we will describe an application below.

9.8.3 Nonlinear Aerodynamics of Supersonic Wings

To do wing design in conditions where linear theory may no longer be applicable, we need to use higher fidelity methods. One way to include better physics involves using the nonlinear full-potential flow theory (see section 2.5) and the conical-flow theory, not the linear theory approximation. It provides an excellent framework for thinking about wing design at supersonic speeds.

For highly swept wings used at supersonic speeds, the equivalent of an airfoil on a moderately-swept high-aspect-ratio transonic transport wing is the spanwise section. To analyze spanwise sections on supersonic wings, the first modern computational method used the full-potential equation. The method solved the flow in the crossplane between the bow shock and the spanwise geometry on the wing (actually a spherical cut).^[92] In this plane, the crossflow can become “supercritical,” and the result is that a crossflow shock wave may arise for high lift coefficients, just as the shock wave occurs in transonic flow over a two-dimensional airfoil. The full potential flow model is a good approximation under conditions where the shocks are not too stong. Extensive comparison with theory was done to validate the method for these applications.

The code developed to make this calculation is known as COREL (for COnical RELaxation), and several supersonic maneuver wings were designed and tested at supersonic speed. The idea was to shape the spanwise section to control the crossflow such that the crossflow shock was weak. This concept was known as SC3, or supercritical conical camber. Wings tested in the NASA Langley Unitary Plan Wind Tunnel demonstrated that the full-potential method could be used to design wings with superior performance.^[93] SC3 will be discussed in more detail in section 9.9.2. The conical-flow code was extended to handle nonconical flows and was known as NCOREL.^[94]

Today, a Euler code would be used based on solving the Euler equations given by equation (2-65) without the visous terms. The advantage of the approach used in COREL and NCOREL was that the grid generation was done as part of the calculation, removing that often time-consuming step from the computational process.

9.9 Supersonic Airplane Configuration Design Examples

We conclude this chapter with examples of supersonic airplane and wing design experiences.

9.9.1 The US Supersonic Transport (SST) Story

In the 1950s and 1960s, the public assumed that rapid advances in aircraft performance would continue and that the next step would be supersonic commercial passenger travel. The experts recognized that a supersonic transport would be difficult to achieve. Essentially no aircraft company could do this alone. By 1962, the British and French had merged their individual work into a single program and had agreed to build Concorde.

In response to the Anglo-French program, and also because the Soviet Union was developing a supersonic transport, President Kennedy made a national program for the development of US SST in 1963. This was very much in the spirit of the notion that “the US has to be the leader.” The following timeline recounts what turned out to be a poor decision. Since the program was too big for one company, a national program was established, funded in large part by the government and administered mainly by the FAA (did they have experience with aircraft design?). What follows is a very brief description of events.

• Aug. 15, 1963: FAA issued an RFP.
• May 15, 1964: The FAA selected Boeing and Lockheed to propose the airframes, and General Electric and Pratt & Whitney to propose the engines.
  • Lockheed proposed a double delta.
  • Boeing proposed a variable-sweep wing.
• Dec. 31, 1966: Boeing and General Electric were selected to build the SST.
• Oct. 21, 1968: Boeing abandoned the variable-sweep concept.
• Mar. 24, 1971: The program was cancelled by congress.*

A little more information may be helpful. The US SST was to be much bigger than the Concorde and would fly faster. This necessitated using titanium instead of aluminum, a very difficult and expensive difference between the airplanes. During the time the design was being developed, the government ruled that there would be no supersonic flight over the US (a restriction imposed by many governments). This is due, in part, to the fact that the much-heavier US SST would have generated a stronger sonic boom than the Concorde. All the while, Boeing was discovering a multitude of problems with their variable-sweep concept, as we’ll see below. The US design was becoming much more expensive. The US Congress was asking why the government should be expected to support this program. To add to the confusion, the noise, both around the airport and from the sonic boom, was not the only issue. Environmental concerns had emerged everywhere, and the SST was predicted to destroy the ozone layer. The result of all these issues was that the program was cancelled.

Despite controversy over noise and environmental pollution, the Concorde entered service in 1976 and flew its last flights in 2003.

Figure 9-49 compares the different planform for SSTs. The figure is taken from a 1969 paper by Swan.^[95] The Boeing 2707-300—the final proposed design—is clearly much larger than the others.

The cited paper by Swan is well worth reading for any supersonic airplane designer. He describes the evolution of the configuration and includes a description of design criteria used in developing the final configuration. The issues he describes are just as valid today as they were back then. Swan explains why the Boeing 2707-300 had a horizontal tail, while the other designs were tailless. Although the horizontal tail contributed little to the stability at low alpha (recall that 1 – dε/dα can be very small or negative for a tail behind a huge wing), it helped prevent the configuration from pitching up at high alpha.

It is worth taking a look at the initial design. The original variable-sweep wing had become what can only be described as a monstrosity. Figure 9-50 shows the Boeing 2707-200 design. The figure is from a review by Leroy Spearman.^[96] Since Spearman worked at NASA Langley, his review also covers the Langley work. The Boeing 2707-200 had engines that were mounted on the horizontal tail. The inboard engines were mounted behind the landing gear (not shown in Spearman’s figure). To avoid ingesting the landing gear wake and any spray from a wet runway, a ramp was used to provide air for the inboard engines from the wing upper surface. Clearly the design was out of control! The features evident in figure 9-50 do not address the problems faced by the designers.

Figure 9-51 shows a plastic model that was sold at the time. Figure 9-51 shows the upper and lower surfaces for both the cruise and low-speed configurations (the landing gear have been broken off for many years). The cruise configuration does indeed look terrific. However, look at the landing configuration. Instead of the single drooped nose on the Concorde, the plane has a double-droop design. Additionally, its high-lift system and airflow ramps are much more complicated than the impression given in figure 9-50. Taken all together, it’s easy to see why the design was abandoned. I can’t imagine the internal chaos surrounding the project.

There’s another key issue. We have neglected aeroelasticity in most of our discussions, but this was also a major problem. Figure 9-52 is from a paper by Kumar Bhatia,^[97] written when the government resurrected SST design, this time calling it High-Speed Civil Transport (HSCT). Clearly aeroelasticity needs to be considered early on in the design process.

Once the Boeing engineers realized that the variable-sweep wing concept had to be abandoned, they selected a fixed delta wing with a horizontal tail and a slight inboard glove. Many of the Lockheed engineers are of the opinion that Boeing decided to build the Lockheed design! Looking at the designs, that position seems to be a little bit of a stretch. At the time, there was a discussion of recompeting, but it was decided to continue with Boeing as the airframer. The resulting redesign was known as the Boeing 2707-300, shown in figure 9-53. It was obviously much simpler. Although the plane was still supposed to fly at Mach 2.7, most of the wing was swept at 50° and thus had a supersonic leading edge. This must have been hard for the NASA Langley engineers to accept. It points to the importance of making the design with all the considerations, not just aerodynamics. We’ll see this below when we discuss the HSCT and MDO. Erik Conway’s book* is highly recommended for anyone interested in the history and technical issues of supersonic transport designs.

9.9.2 Supersonic Maneuver Wing—Transonic Aerodynamics at Supersonic Speed

During the early stages of the advanced tactical fighter (ATF) program in the late 1970s, there was interest in the design of a wing capable of maneuvering efficiently at supersonic speeds. As the supercruise requirement emerged, it was clear that efficient supersonic maneuvering could be a requirement also. As previously discussed, linear theory was adequate for cruise design. However, it was not adequate for maneuvering C_L’s. This deficiency was shown in work from NASA Langley, as illustrated in figure 9-54.^[99] Although linear theory predicted good performance, wings designed and tested at higher C_L’s fell short of the predictions. The requirement for supersonic maneuvering meant that C_L’s well above the linear theory limit would be required. The figure also shows the expected maneuver C_L’s. A detailed discussion of the relationship between linear theory breakdown and supercrtical crossflow is given in a paper by Mason and Miller.^[100]

We have included this discussion on supersonic maneuver wing design because it shows how an understanding of flow physics can be used to develop an aerodynamic concept. Rudy Meyer at Grumman realized that there was a close correspondence between the 2D full-potential equation that was then being solved numerically at transonic speeds and the full-potential conical flow equations for supersonic flow.^[101]* He knew that the same techniques could be used for supersonic conical flow. He had also read a paper by Clint Brown that attributed the performance shortfall of wings designed by linear theory to the presence of crossflow shocks on the wing.^[102] Rudy was sure that a numerical method that included crossflow shocks could be used to shape the spanwise wing section to reduce the strength of the crossflow shock. The resulting concept was called supercritical conical camber, SC3.

Figure 9-55 from the final report on this work^[103] shows an illustration of the flow field physics and the associated spanwise pressure distribution. Part A (on the left side of the figure) is a schematic that shows a conical geometry and the spanwise section pressure distribution that will be used to do the design. Part B shows the predicted spanwise pressure distribution for an uncambered conical wing computed using the full-potential flow model that can include the supercritical crossflow shock and the prediction from linear theory (which obviously can’t include the key physics of the flow field). Clearly the nonlinear effect has to be included on the calculation.

The program to design a wing demonstrating controlled supercritical crossflow was carried out in a series of steps. The following history illustrates a physics-based approach to aerodynamic development:

• SC3 was conceived by Rudy Meyer in 1977.
• The program concept was developed by Gianky DaForno in 1977 and funded by NASA.
• The computational method COREL was developed by Bernie Grossman in 1978.
• Aerodynamic design was done by W. H. Mason from 1978 to 1982 in cooperation with NASA Langley, primarily Dave Miller.

The SC3 program included three different prototypes in the NASA Langley Unitary Wind Tunnel. To start, two purely conical wings were designed, built, and tested. One was flat, and the other was cambered based on the computational design work. This wing was termed the “conceptual wing” and was designed using iterative analysis with COREL. Initially, the wing used a simple cambered spanwise shape. The thickness distribution used a superellipse with the leading edge slightly “rounder” than a typical parabolic nose shape. The crossflow shock was considerably weaker than the flat wing. Next, local shaping was used in the vicinity of the crossflow shock to completely eliminate the crossflow shock. This was done by reducing the spanwise curvature around the crossflow shock in the same way that Whitcomb had reduced the shock strength for transonic supercritical airfoils. Figure 9-56 shows both the initial spanwise camber design and the final shockless supercritical crossflow design.^[104]

The dimensioned planform of the wind tunnel test model is shown in figure 9-57(a), which also shows the location of pressure taps. In figure 9-57(b), the final conical cambered wing cross section with shockless supercritical flow is compared to that of a flat uncambered wing model. A photo of the model installed in the wind tunnel is shown in figure 9-57(c).

Figure 9-58 shows the resulting pressure distribution at several angles of attack. The results were considered highly successful.^[105]

The next step was to design and test a three-dimensional wing using the SC3 concept to create what was called the “demonstration wing.” Figure 9-59 shows the demonstration wing installed in the wind tunnel.

The drag polar from the wind tunnel test is shown in figure 9-60. At the design lift coefficient of 0.4, a 21% reduction in drag due to lift was achieved compared to the 0% and 100% leading-edge suction envelopes. The work is described in the paper by Mason et al.^[106]

The work produced a number of papers, including AIAA papers and NASA reports. The following NASA reports are available online at no charge and contain details of the work and wind tunnel results with tabulated data: NASA TP-1759, April 1981; NASA TP-2249, February 1984; and NASA TP-2336, August 1984.

9.9.3 HSCT and MDO

Supersonic research continued as a low-level effort ever since the US SST was cancelled. In the 1980s, NASA began to revisit the possibility of a supersonic commercial transport. The initial NASA study was for a tailless Mach 3 design exploiting the ideas of attainable leading-edge thrust.^[107] This design was wind tunnel tested.^[108] Both McDonnell Douglas and Boeing were involved in the program,^[109],[110] though Boeing had not bought McDonnell Douglas yet. After these studies, a Mach number of 2.4 was chosen as the HSCT design Mach number. The companies also investigated control issues for an HSCT.^[111],[112] Although the program was not classified as such, access was restricted. Eventually the results of the work were made available, and many papers were given at AIAA Meetings. See the book by Conway for details of the evolution of the program.^[113] The public planform was known as Reference H and is shown in figure 9-61.^[114] The design Mach number was 2.4, and it was required to have a range of 5,000 to 6,000 NM. It was intended to carry 250 to 300 passengers. The notional TOGW was 700,000 lbs.

At the cruise Mach number, the major portion of the leading edge of the Reference H configuration is subsonic. The planform is much closer to the ideas we described in section 9.4.2 for the modified arrow wing. This design was pushing technology to the limit, with the sensitivity of the takeoff gross weight to drag quoted at 10,400 pounds per count! From work we will describe below, we found a value of 14,000 pounds per count. From a commercial point of view, this was too sensitive to be practical. Some of my peers at NASA turned this around to say that the design would be a success if the drag could only be reduced by a few more counts. Nevertheless, concerns over the environment—in the form of noise (community and sonic boom) and emissions (concerns about high-altitude atmospheric chemistry)—remained formidable challenges. These concerns, together with the uncertainty about economic viability, resulted in the program being cancelled. The story is well told by Conway.^[115]

9.9.3.1 Multidisciplinary Design Optimization (MDO)

As part of the HSCT program, various universities, including Virginia Tech, conducted a nearly decade-long program developing MDO methods for HSCT design. The work explored a number of approaches to MDO. The key objective was to find ways to include high-fidelity disciplinary analysis methods in conceptual design, where the high-fidelity methods could have the most impact. The primary disciplines included aerodynamics and structures. Numerous constraints were specified. Some of these were practical geometric constraints, but many also addressed trim and static stability and control requirements. Figure 9-62 shows a nominal statement of the optimization problem. One of the key issues was the development of a general parametric geometry model, where the parameters could be used as design variable in the optimization, as described in the figure. Numerous variations of this problem were investigated.

There were a variety of outcomes from this work. Perhaps the key understanding was that it was impractical to include high-fidelity analysis methods into a “giant” program, which we called disaggregation.^[116] Instead, it was necessary to represent the disciplines through models that could be used during the optimization. These models are called response surface models (RSMs) by the statisticians. We used these RSMs for several reasons. First, we were interested primarily in using gradient-based optimization. Just about any analysis method will produce slightly “noisy” results as geometry changes are made (shocks bouncing between grid lines, etc.). Gradient-based optimizers are extremely sensitive to this artificial noise. In addition, the experts can use the latest versions of their software to provide data for the RSMs to be built for the optimization. Finally, in this approach, the many analysis runs needed from an analysis code can be carried out simultaneously rather than sequentially. This allows the use of coarse-grained parallel computing to reduce the time required to do a design. The RSMs representing the disciplines can be used with a variety of optimization schemes. Because we found local minima in the design space, the models could be employed with global optimization schemes. The models also allowed parameter studies to be made very easily. We’ve repeated some of the discussion given previously in Chapter 4 because this was the work that led to our preferred aerodynamic design and optimization process.

An example result from the HSCT optimization work described above is shown in figure 9-63.^[117] The figure contains the results from two different approaches. The first is labeled VCM Optimum, in which VCM stands for “variable complexity model.” In this approach, low-fidelity models for the aerodynamics are used together with a few high-fidelity aerodynamic models. The other approach is labeled Drag RS Optimum, and it uses response surface models. Both produce similar results.

In the optimization process, we used numerous different starting designs or starting points. With the high-dimensional design space used here, there are many local minima in which a gradient-based optimization can get “stuck.” Figure 9-64 shows the convergence history of the optimization. An example of design space visualization illustrating the local minima issue is discussed in the paper by Knill et al.^[118]

A great deal of work on this problem has been carried out by other researchers, and we cite two that are of interest.^[119],[120]

9.9.4 Design to Reduce the Strength of the Sonic Boom

As stated earlier in this chapter, government regulations do not allow supersonic flight over the United States as well as most other countries. This is a major limiting factor preventing the development of a commercial supersonic airplane. Thus, attention in the research community turned to an effort to reduce the strength of the sonic boom enough that a new noise criteria could be established allowing low-strength/no-boom supersonic flight over land. On August 27, 2003, a modified F-5 flown at Edwards AFB demonstrated sonic boom shaping, proving that the strength of the sonic boom hitting the ground could be reduced. The modified F-5 used for the demonstration is shown in figure 9-65.

Quieting the Boom: The Shaped Sonic Boom Demonstrator and the Quest for Quiet Supersonic Flight, by Lawrence R. Benson, part of the NASA Aeronautics Book Series, is an excellent source of complete story.^[121] Examples of the theory that can be used to design low-boom concepts have been published by Li and Rallabhandi^[122] and Rallabhandi et al.^[123] This remains an area of active research.

The goal behind research into shaped booms is to change the shape of the classic N-wave associated with the sonic boom. Figure 9-66 shows the N-wave and a design from a paper by Aronstein and Schueler^[124] that reduces the sudden pressure jump.

Another idea is to use an extensible nose spike to make the length of an airplane longer. Figure 9-67 shows an F-15 modified with what they call a Quiet Spike to extend a nose boom in flight. This was a program conducted by Gulfstream and NASA from 2004 to 2007.^[125]

9.9.5 Additional Efforts

In this section, we describe two projects that may lead to new supersonic airplanes. After having tried to pursue a large commercial transport without success, the future appears to be in more modest designs for executive jets. One has been underway for a number of years, and the other is a new NASA effort. Both are much smaller than the HSCT program discussed above. The NASA design has the purpose of demonstrating a supersonic airplane with a low sonic boom strength. Hopefully the NASA work will establish the basis for relaxing the restriction on supersonic flight over land in the US.

9.9.5.1 Aerion

The Aerion Corporation* has been working on a small supersonic airplane design since 2003. Figure 9-68 shows one of the preferred concepts, which is a three-engine design. Nominally they expect to be able to have a boomless cruise at Mach 1.1 to 1.2, with a long-range cruise Mach number of 1.4. The AS2 is designed to carry 8 to 11 passengers and is expected to operate at runways from 6,000 to 7,500 feet depending on the airplane weight. The range is expected to be 7,500 NM.

The Aerion concept depends on a fundamental property of fluid mechanics. Note the essentially unswept wing. The concept arose from the idea that, for an unswept wing at supersonic speed, the pressure decreases on the wing falls continuously from the leading edge to the trailing edge. This means that the pressure gradient is favorable over the entire wing and for a modest Reynolds number the flow should be laminar. The resulting low skin friction drag is an enabling technology. Here we have an example where two-dimensional supersonic airfoil theory is useful.^[126] We discuss the relevant aspects of the supersonic airfoil theory in the next section below.

2D: The supersonic airfoil story

Supersonic airfoil theory has not been particularly useful in the swept-wing concepts discussed above. Now we have a case where supersonic airfoil characteristics can be exploited, as seen in the example provided below. Generally the textbooks have students work the diamond airfoil problem. In this section, we’ll look at a biconvex airfoil. Let’s examine the pressure distribution on the 5-percent-thick biconvex airfoil shown in figure 9-69.

The linear theory pressure distribution on the 5-percent-thick biconvex airfoil is given in figure 9-70 for a Mach 2 case at 5° angle of attack. Note that the pressures vary linearly so that there is a constant favorable pressure gradient all the way from the leading edge to the trailing edge. And of course since the trailing edge is supersonic, the pressures don’t need to come together at the trailing edge as they would for a subsonic case.

This is the idea behind using the unswept wing on the Aerion. The hope is that, without an adverse pressure gradient, the wing will achieve a significant amount of natural laminar flow. Once we’ve started this discussion, we use the opportunity to make a few other points.

Although figure 9-70 shows the idea for the Aerion concept, we are left to wonder how well linear theory agrees with data. Looking at the literature, almost all data used for comparison with airfoils comes from the wind tunnel tests conducted by Antonio Ferri at Guidonia in Italy.^[127] Figure 9-71 shows a comparison of the predictions from linear theory, Ferri’s wind tunnel data, and predictions from second-order supersonic airfoil theory. The model was a 10-percent-thick biconvex airfoil at 10° angle of attack and Mach 2.13. This is a fairly thick airfoil at a fairly high angle of attack, so we may be asking too much from linear theory. The second-order airfoil theory formulas are available in NACA TN-1428.^[128] The figure shows that the second-order theory is in generally good agreement with the wind tunnel data, and the idea of a favorable pressure gradient is still essentially valid. Because first- and second-order supersonic airfoil theories are relatively simple analytically, numerous studies have been conducted seeking optimum aerodynamic characteristics. We will not repeat them here, instead referring to an entire book that covers the subject.^[129]

The other interesting feature of supersonic airfoil theory is the effect of camber. Linear theory only considers the leading- and trailing-edge values; camber and thickness don’t affect lift. This is not true for second-order supersonic airfoil theory. For a flat lower surface and a curved upper surface, generally considered positive camber, the lift at zero angle of attack is negative. Figure 9-72 shows this effect for one of the airfoils tested by Ferri—a one-sided wedge 6.3% thick. Although the camber effect is not huge, it is noticeable. This is particularly interesting in that it explains the F-16 device deflection schedule. At supersonic speed, the F-16 automatic device schedule includes leading and trailing edges with a negative deflection, as shown in figure 9-73. The F-16 has a NACA 64A204 airfoil, and the leading and trailing edges use an automatic -2° deflection at supersonic speeds.^[130]

9.9.5.2 Sonic QueSST

In 2016, NASA started work on a low-boom flight demonstrator. The Lockheed Martin’s Skunk Works is responsible for designing and building the vehicle. The project is known as the QueSST, and the US Air Force assigned it the X-59 designation in 2018. Figure 9-74 shows the final configuration.^[131] It is a very long (nearly 100 feet), lightweight design. The design Mach number is 1.42.

Figure References