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.
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 1: Aircraft speed in the year of entry into service.
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.
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).
Figure 2: Compressibility correction at subsonic speeds.
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.
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).
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 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.
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.
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).
Check Full List of Reviewed References
Speed Limits
Extended Constant-Density Flows
Critical Mach Number
Some Transonic Effects
Shock Waves
Flow Separation
Computational Aerodynamics
State of the Art
Related Material
Wave Propagation
Selected References
