Approximation of the perturbation equations of a quasi-linear hyperbolic system in the neighborhood of a bicharacteristic

When we look at photographs of real transonic flows which are predicted to be shockless, we find a very large number of weak shocks almost perpendicular to the streamlines. These are no more than almost-trapped upstream-propagating nonlinear waves. In this paper we try to obtain a simple approximate equation which gives their complete history and takes into account both their turning effect, owing to a non-zero gradient of the fluid velocity in a direction normal to the streamlines, and also the finite radius of curvature of the wave front. We first give a brief discussion of a few results which can be easily obtained from the solution of the approximate equation and then compute the history of two nonlinear pulses by numerically integrating the equation.

Nonlinear wave propagation in a two-dimensional steady transonic flow

It is my assignment to provide a background of the history and development of transonic aerodynamics as a preamble to the remainder of the papers in this book presenting critical reviews of recent advances and suggestions of prospective avenues for further progress. Such assignments are a sure sign of passing years, but I do have the advantage of getting an early start in transonic aerodynamics when I was assigned to the Flight Research Branch at NACA/Ames Aeronautical Laboratory upon arrival as a young engineer in mid-1943. This group, under the direction of Larry dousing, who was also a leading test pilot, was just beginning research to determine the causes of catastrophic accidents that were occurring when fighter planes were put into steep dives at high altitude, and hopefully to determine ways to cure or avoid the problems. In 1947, I transferred to the Theoretical Aerodynamics Branch under the direction^of Dr. Max A. Heaslet, and was soon at work trying to develop theories for dealing with transonic flows. During the following years, it fell my lot to provide summary papers on the state of transonic aerodynamics on a number of occasions. I will draw rather heavily on these papers, inasmuch as they were designed to be comprehensive summaries within imposed limits of length of the state of knowledge of transonic aerodynamics, with somewhat of a bias toward theory and its evaluation in wind-tunnel experiments. While this may seem a bit parochial, since corresponding research was also under way at a number of other places, the work at Ames was both at the forefront and in

Chapter I. Transonic Aerodynamics—History and Statement of the Problem

Efforts to replace Sobieczky's complicated analog computations of solutions to the hodograph equations by a fast elliptic solver in order to generate shock-free airfoil designs more effectively are described. The indirect design of airfoil and wing shapes that are free from shock waves even though the local flow velocity exceeds the speed of sound is described. The problem of finding an airfoil in two dimensional, irrotational flow that has a prescribed pressure distribution is as addressed. Sobieczky's suggestion to use a fictitious gas for finding shock-free airfoils directly in the physical plane was the basis for a more efficient procedure for achieving the same end.

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Shock-Free Configurations in Two- and Three- Dimensional Transonic Flow,

The question of the stability of steady compressible two-dimensional potential flows with embedded regions of supersonic flow adjacent to a body is examined. The behaviour of unsteady disturbances is investigated and an essential difference between the stability of transonic one-dimensional and two-dimensional flow is shown to exist. It is shown that two-dimensional transonic potential flows are stable. An experimental investigation on quasi-elliptical aerofoil sections is discussed. The results of this investigation suggest that the theoretical potential flows can be approached arbitrarily close. It is concluded that the use of potential flow theory in the transonic region is equally acceptable as anywhere else in aerodynamics.

Bernardus Maria Spee

Papers

Investigations on the transonic flow around aerosoils