The Supersonic Blunt Body Problem

On November 1, 1952, the United States detonated a 10.4-megaton hydrogen test device on Eniwetok Atoll in the Marshall Islands, the first implementation of physicist Edward Teller’s concept for a "super bomb” and a major milestone toward the development of the American hydrogen bomb. With it came the need for a new entry vehicle beyond the long-range strategic bomber, namely the intercontinental ballistic missile (ICBM). This vehicle would be launched by a rocket booster, go into a suborbital trajectory in space, and then enter Earth’s atmosphere
at hypersonic speeds near orbital velocity. This was a brand-new flight regime, and the design of the entry vehicle was dominated by an emerging design consideration: aerodynamic heating. Knowledge of the existence of aerodynamic heating was not new. Indeed, in 1876, Lord Rayleigh published a paper in which he noted that the compression process that creates a high stagnation pressure on a high-velocity body also results in a correspondingly large increase in temperature. In particular, he com­mented on the flow-field characteristic of a meteor entering Earth’s atmo­sphere, noting: "The resistance to a meteor moving at speeds comparable with 20 miles per second must be enormous, as also the rise of temper­ature due to the compression of the air. In fact it seems quite unneces­sary to appeal to friction in order to explain the phenomena of light and heat attending the entrance of a meteor into the earth’s atmosphere.”[772] We note that 20 miles per second is a Mach number greater than 100. Thus, the concept of aerodynamic heating on very high-speed bodies dates back before the 20th century. However, it was not until the mid­dle of the 20th century that aerodynamic heating suddenly became a showstopper in the design of high-speed vehicles, initiated by the press­ing need to design the nose cones of ICBMs.

In 1952, conventional wisdom dictated that the shape of a missile’s nose cone should be a slender, sharp-nosed configuration. This was a natural extension of good supersonic design in which the supersonic body should be thin and slender with a sharp nose, all designed to reduce the strength of the shock wave at the nose and therefore reduce the supersonic wave drag. (Among airplanes, the Douglas X-3 Stiletto and the Lockheed F-104A Starfighter constituted perfect exemplars of good supersonic vehicle design, with long slender fuselage, sharp noses, and very thin low aspect ratio [that is, stubby] wings having extremely sharp leading edges. This is all to reduce the strength of the shock waves on the vehicle. The X-3 and F-104 were the first jet airplanes designed for flight at Mach 2, hence their design was driven by the desire to reduce wave drag.) With this tradition in mind, early thinking of ICBM nose cones for hypersonic flight was more of the same, only more so. On the other hand, early calculations showed that the aerodynamic heat­ing to such slender bodies would be enormous. This conventional wis­dom was turned on its head in 1951 because of an epiphany by Harry

Julian Allen ("Harvey” Allen to his friends because of Allen’s delight in the rabbit character named Harvey, played by Jimmy Stewart in the movie of the same name). Allen was at that time the Chief of the High­Speed Research Division at the NACA Ames Research Laboratory. One day, Harvey Allen walked into the office and simply stated that hyper­sonic bodies should "look like cannonballs.”

His reasoning was so fundamental and straightforward that it is worth noting here. Imagine a vehicle coming in from space and enter­ing the atmosphere. At the edge of the atmosphere the vehicle velocity is high, hence it has a lot of kinetic energy (one-half the product of its mass and velocity squared). Also, because it is so far above the surface of Earth (the outer edge of the atmosphere is about 400,000 feet), it has a lot of potential energy (its mass times its distance from Earth times the acceleration of gravity). At the outer edge of the atmosphere, the vehicle simply has a lot of energy. By the time it impacts the surface of Earth, its velocity is zero and its height is zero—no kinetic or potential energy remains. Where has all the energy gone? The answer is the only two places it could: the air itself and the body. To reduce aerodynamic heating to the body, you want more of this energy to go into the air and less into the body. Now imagine two bodies of opposite shapes, a very blunt body (like a cannonball) and a very slender body (like a needle), both coming into the atmosphere at hypersonic speeds. In front of the blunt body, there will be a very strong bow shock wave detached from the surface with a very high gas temperature behind the strong shock (typically about 8,000 kelvins). Hence the air is massively heated by the strong shock wave. A lot of energy goes into the air, and therefore, only a moderate amount of energy goes into the body. In contrast, in front of the slender body there will be a much weaker attached shock wave with more moderate gas temperatures behind the shock. Hence the air is only moderately heated, and a massive amount of energy is left to go into the body. As a result, a blunt body shape will reduce the aero­dynamic heating in comparison to a slender body. Indeed, if a slender body would be used, the heating would melt and blunt the nose anyway. This was Allen’s thinking. It led to the use of blunt noses on all modern hypersonic vehicles, and it stands as one of the most important aerody­namic contributions of the NACA over its history.

When Allen introduced his blunt body concept in the early 1950s, there were no theoretical solutions of the flow over a blunt body mov­ing at supersonic or hypersonic speeds. In the flow behind the strong
curved bow shock wave, the flow behind the almost vertical portion of the shock near the centerline is subsonic, and that behind the weaker, more inclined part of the shock wave further above the centerline is super­sonic. There were no pure theoretical solutions to this flow. Numerical solutions of this flow were tried in the 1950s, but all without success. Whatever technique worked in the subsonic region of the flow fell apart in the supersonic region, and whatever technique worked in the supersonic region of the flow fell apart in the subsonic region. This was a potential disaster, because the United States was locked in a grim struggle with the Soviet Union to field and employ intercontinental and intermedi­ate-range ballistic missiles, and the design of new missile nose cones desperately needed solutions of the flow over the body were the United States to ever successfully field a strategic missile arsenal.

On the scene now crept CFD. A small ray of hope came from one of the NACA’s and later NASA’s most respected theoreticians, Milton O. Van Dyke. Spurred by the importance of solving the supersonic blunt body problem, Van Dyke developed an early numerical solution for the blunt body flow field using an inverse approach: take a curved shock wave of given shape, calculate the flow behind the shock, and solve for the shape of the body that would generate the assumed shock shape. In turn, the flow over a blunt body of given shape could be approached by repetitive applications of this inverse solution, eventually converg­ing to the shape of interest. If critical, it was nevertheless a potentially tedious task that could have consumed thousands of hours by hand calculation, but by using the early IBM computers at Ames, Van Dyke was able to obtain the first reliable numerical solution of the super­sonic blunt body flow field, publishing his pioneering work in the first NASA Technical Report issued after the establishment of the Agency.[773] Van Dyke’s solution constituted the first important and practical use of CFD but was not without limitations. Although the first major advance­ment toward the solution of the supersonic blunt body problem, it was only half a loaf. His procedure worked well in the subsonic region of the flow field, but it could penetrate only a small distance into the super­sonic region before blowing up. A uniform solution of the whole flow field, including both the subsonic and supersonic regions, was still not obtainable. The supersonic blunt body problem rode into the decade of

the 1960s as daunting as it ever was. Then came the breakthrough, which was both conceptual and numerical.

First the conceptual breakthrough: at this time the flow was being calculated as a steady flow using the Euler equations, i. e., the flow was assumed to be inviscid (frictionless). For this flow, the governing par­tial differential equations of continuity, momentum, and energy (the Euler equations) exhibited one type of mathematical behavior (called elliptic behavior) in the subsonic region of the flow and a completely different type of mathematical behavior (called hyperbolic behavior) in the supersonic region of the flow. The equations themselves remain identical in these two regions, but the actual behavior of the mathemat­ical solutions is different. (This is no real surprise because the physi­cal behavior of the flow is certainly different between a subsonic and a supersonic flow.) This change in the mathematical characteristics of the equations was the root cause of all the problems in obtaining a solu­tion to the supersonic blunt body problem. Hence, any numerical solu­tion appropriate for the elliptic (subsonic) region simply was ill-posed in the supersonic region, and any numerical solution appropriate for the hyperbolic (supersonic) region was ill-posed in the subsonic region. Hence, no unified solutions for the whole flow field could be obtained. Then, in the middle 1960s, the following idea surfaced: the Euler equa­tions written for an unsteady flow (carrying along the time derivatives in the equations) were completely hyperbolic with respect to time no matter whether the flow were locally subsonic or supersonic. Why not solve the blunt body flow field by first arbitrarily assuming flow-field properties at all the grid points, calling this the initial flow field at time zero, and then solving the unsteady Euler equations in steps of time, obtaining new flow-field values at each new step in time? The problem is properly posed because the unsteady equations are hyperbolic with respect to time throughout the whole flow field. After continuing this process over a large number of time steps, eventually the changes in the flow properties from one time step to the next grow smaller, and if one goes out to a sufficiently large number of time steps, the flow con­verges to the steady-state solution. It is this steady-state solution that is desired. The time-marching process is simply a means to the end of obtaining the solution.[774]

The numerical breakthrough was the implementation of this time­marching approach by means of CFD. Indeed, this process can only be carried out in a practical fashion on a high-speed computer using CFD techniques. The time-marching approach revolutionized CFD. Today, this approach is used for the solution of a whole host of different flow problems, but it got its start with the supersonic blunt body problem. The first practical implementation of the time-marching idea to the super­sonic blunt body was carried out by Gino Moretti and Mike Abbett in 1966.[775] Their work transformed the field of CFD. The supersonic blunt body problem in the 1950s and 1960s was worked on by platoons of researchers leading to hundreds of research papers at an untold num­ber of conferences, and it cost millions of dollars. Today, because of the implementation of the time-marching approach by Moretti and Abbett using a finite-difference CFD solution, the blunt body solution is read­ily carried out in many Government and university aerodynamic labo­ratories, and is a staple of those aerospace companies concerned with supersonic and hypersonic flight. Indeed, this approach is so straight­forward that I have assigned the solution of the supersonic blunt body problem as a homework problem in a graduate course in CFD. What bet­ter testimonial of the power of CFD! A problem that used to be unsolv­able and for which much time and money was expended to obtain its solution is now reduced to being a "teachable moment” in a graduate engineering course.