And Hypersonic Flight

Prandtl's student Adolf Busemann developed the ideas thatmade supersonic flight possible. Here Busemann (center)receives an award from the Fusion Energy Foundation inNovember 1981.thinking was quite pervasive even at a point when von Karmanlater on became one of the celebrated people whoallegedly had a lot to do with the development of aerodynamics.I want to emphasize that this collisional and essentiallystatistical model of computing physical events on the basisof certain averages—averaged over particles and groups ofparticles and molecules statistically—is proved one of themost important barriers to a satisfactory development oftheory not only in the areas that we are discussing here,fluid dynamics and hydrodynamics, but also in the equallyimportant areas of quantum theory, plasma physics, and soon, which are essential to the possibility of thermonuclearfusion.These collisional and statistical models do not work, andit is only and precisely to the extent that they were explicitlyrejected by the Gottingen school that these areas can beregarded as possible and developable.Prandtl's MethodThe essential idea that Prandtl had in 1904 is that if onewere to try to directly describe the possibility of flight usingthe very difficult differential equations that govern the flowof so-called viscous fluids (fluids that have internal friction),the so-called Navier-Stokes equations, then one would befaced with an impossible problem. One could experimentallyperhaps define and determine the possibility of flight,but one could never quantitatively explicitly calculate theactual conditions that make flight possible. Rather thanlooking at an airfoil subjected to a stream of air as an airfoilinjected into a viscous fluid, which mathematically is impossibleto handle, Prandtl separated the problem into twoparts. He did this from the standpoint of the geometricaltype of thinking that introduces as an essential characteristicof the geometrical continuum the singularities in thiscontinuum. On the one hand, Prandtl said, we can look atthe flow far away from the airfoil, the so-called free flow,on the basis of the very simple potential equations accordingto Laplace. These are trivial and relatively easy to understanddifferential equations, which have an immediategeometrical interpretation in the context of so-called conformalmapping theory.Prandtl said the only area in which we have to considerflow that has internal friction is in the immediate vicinity ofthe airfoil itself, in the so-called boundary layer. This is thelittle white layer that can be seen in the photographs ofPrandtl's experiments (Figures 1 and 2). In this area, we canno longer ignore viscosity or the internal friction of thefluid, in particular. This is not because we know on the onehand that directly at the surface of the airfoil the flow iszero; that is, the air or the water—or whatever it is—actuallysticks at the surface. A very small distance away fromthis, it is clear that it has already attained a velocity equal tothe free flow velocity. What we must look at is this criticalboundary layer, what Prandtl called the surface of discontinuity,in which, a very, very large difference in velocity isattained over an extremely thin layer—a layer that can, infact, be thought of as arbitrarily thin. If we think of thisboundary layer as a surface of discontinuity, under thesecircumstances we can simplify the Navier-Stokes equationsquite significantly, and are therefore able to give a quantitativesolution to the problems of drag, lift, and all of theother aerodynamical problems that are critical to discussthe possibility of flight.Without the kind of work that Prandtl did—first publishedin 1904 and discussed by him prior to his coming toGottingen, when he was a teacher at the Technical HighSchool in Hannover—without these kinds of discussions ofthe boundary layer problems, it is generally acknowledgedtoday that a quantitative discussion of the possibility offlight would not have been available.One of Prandtl's most important colleagues was Runge,a mathematician who developed many of the mathematicalmethods for calculating the problems in aerodynamics thatPrandtl raised.The Role of Felix KleinFelix Klein was the teacher of many of the students in thelate 19th and early 20th century in Germany in mathematicsand in physics, and at the same time was one of the mostaccomplished organizers of the total scientific, technological,and industrial enterprise in Germany. Klein had earliermade a name for himself by developing some very interestingand significant work in elliptical function theory, and inthe 1890s he came to Gottingen as a professor and made ithis task to try to define a research program for the entiretyof the technical and scientific disciplines at the university.Particularly important, he worked in close collaborationwith Wilamowitz, the senior faculty member in the field ofAltphilologie, ancient languages with specific emphasis onGreek. Klein and Wilamowitz jointly defined an outlook onresearch and education, which I think is uniquely responsible,in terms of its philosophy, for the advances that weremade in Germany in that period. At the same time, Kleinenlisted and in a certain sense forced German industry into42 January-February 1986 FUSION