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High Speed Aerodynamics

In this Chapter


High-Speed in aerodynamics is not a question of highway driving schedules. There are strong physical reasons attached to the definition of high speed, and they involve fundamental changes in the flow compressibility and the nature of propagation of disturbances in the flow field. These facts have major consequences on the solution of the characteristic equations.

Speed Limits

As it turns out, also aircraft have speed limits. The figure below reports a summary compiled from a number of sources. It is shown the cruise speed of some notable aircraft in the year of entry into service (some data are not labelled for clarity). After the year 1960 the growth of speed has virtually stalled.

Exceptional performances have been achieved with research and military aircraft. For example, the North American XB-70A Valkyrie reached M = 3 (1965). The USAF Lockheed SR-71A Blackbird of the 1970s was capable of reaching routinely M = 3.5.

The progress made earlier was due to increasingly more powerful engines (turboprops). With the jet revolution the speed reached the upper transonic. The new limit is not due to lack of enough propulsion power, rather to the unacceptable wave drag that must be overcome at supersonic speeds. [figure unavailable in this version]

Transonic flows are flows at speeds below the speed of sound that feature pockets of supersonic flow, as well as flows slightly above the speed of sound. The Mach number range is M=0.6÷1.2.

In this section we will be concerned only with transonic flows past airfoils, wings and wing-bodies, although the problem is more general, since it includes wedges, slender bodies of various type, helicopter rotor blades, aeronautic propellers, nozzle flows, inlets, and other internal flows.

Extended Constant-Density Flows

From an applied point of view flows with a free stream Mach number below 0.6 are treated as low subsonic flows (provided no shock waves occur) and compressibility corrections to speeds and pressure are performed using some well known theories: Prandlt-Glauert-Gothert and Karman-Tzien for external flows; Lieblein-Stockmann for internal flows.

The effect of compressibility on the Cp is shown in the figure below (results obtained by expanding in series the energy equation).

Compressible Cp

Figure 2: Compressibility correction at subsonic speeds.

Critical Mach Number

The critical Mach number is the free stream Mach number at which sonic flow first appears on the airfoil/wing. Critical Mach numbers can be estimated in several ways, including the methods of Relf-Jacobs, Von Karman, Prandtl-Glauert, and Temple-Yarwood.

Some Transonic Effects

Above certain speeds airfoils and wings experience a phenomenon of transonic drag rise that ultimately sets a limit to the aircraft speed. The drag rise is due to the presence of shock waves and shock-induced separation. Shock waves also have a large effect on the lift, on the structural response of the wing, and on the noise emission.

In the early days of high speed research (started in the 1930s) it was feared that supersonic speeds, or even high transonic speeds, could not be reached by any practical airplane, since the drag rise would make it difficult to cross the sound barrier.

In a constant size stream tube (channel of constant area), the resistance would be effectively very large, due to a singularity in the stream-wise momentum equation. However, high subsonic flows could be easily accelerated past the speed of sound in convergent- divergent nozzles (Rankine-Hugoniot).

Shock Waves

A shock wave is a strong perturbation propagating at supersonic speeds. At transonis speeds the shock wave is described as a discontinuity (or nearly so) of the aero- thermodynamic variables (pressure, density, velocity, entropy) in the flow. The actual shock thickness is of the order of one micron.

The air ahead of the aircraft (or other high speed system), cannot be informed of the approaching and it reacts abruptly by a strong reaction (the shock wave): within a few molecular lengths the air is accelerated to follow the perturbation, which is subsequently radiated in the field. The propagation characteristics depend on the specific problem.

Flow Separation

Flow separation at transonic speeds is generally related to the presence of the shock. One well known aspect is the boundary layer separation, others include trailing edge separation and three-dimensional separation induced by strakes.

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Computational Aerodynamics

The first methods for transonic flows were linearized and small perturbation theories. Linearized theories work well at speeds away from the speed of sound (for example, Ackeret’s theory of supersonic lifting line, 1925; Gothert’s compressibility correction for thin wings, 1940; von Karman’s transonic similary law, 1947; Lighthill’s slender body theory, 1948). Non linearities appear strong just below and above the sonic speed.

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State of the Art

State-of-the-art transonic aerodynamics is the realm of computational aerodynamics (CFD). There is extensive technical literature on full potential, Euler and Navier-Stokes calculations on a wide range of configurations. The small disturbance methods are no longer popular (see a review in Ashley-Landhal, 1965).

Methods for transonic flows require hard numerical work because of the mixed nature of the governing equations (elliptic-hyperbolic). Breakthrough in this field were non-linear numerical schemes able to capture the shock (Murman-Cole, 1971; Beam-Warming, 1976, and many others).

Related Material (available on CD-ROM)

  • Shock Waves
  • Shock Stall
  • Buffeting
  • Noise Emission (aeroacoustics, sonic boom)
  • Supersonic Nozzles
  • Wave Propagation
  • Supercritical Airfoils
  • Swept back Wings
  • Oblique Flying Wing

Selected References

  • Ashley H, Landhal M. Aerodynamics of Wings and Bodies, Addison-Wesley, 1965.

  • Jones RT. Wing Theory, Princeton Univ. Press, Princeton, NJ, 1990.

  • Moulden TH. Fundamentals of Transonic Flow, John Wiley, 1984.

  • Becker JV. The High Speed Frontier, NASA SP-445, 1980

  • Gulderley KG. The Theory of Transonic Flow, Pergamon Press, Oxford, 1962.

  • Bers L. Mathematical Aspects of Subsonic and Transonic Gas Dynamics, John Wiley, 1958.

Check Full List of Reviewed References


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Copyright © A. Filippone (1999-2005). All Rights Reserved.