Compressible flow

In previous chapters the study of aerodynamics has been almost exclusively restricted to incompressible flow. This theoretical model is really only suitable for the aero­dynamics of low-speed flight and similar applications. For incompressible flow the air density and temperature are assumed to be invariant throughout the flow Field. But as flight speeds rise, greater pressure changes are generated, leading to the compression of fluid elements, causing in turn a rise in internal energy and, in consequence, temperature. The resulting variation of these flow variables throughout the flow Field makes the results obtained from incompressible flow theory less and less accurate as flow speeds rise. For example, in Section 2.3.4 we showed how use of the incompressibility assumption led to errors in estimating the stagnation-pressure coefFicient of 2% at M = 0.3, rising to 6% at M = 0.5, and 28% at M = l.

But these errors in estimating pressures and other flow variables are not the most important disadvantage of using the incompressible flow model. Far more signiFicant is the marked qualitative changes to the flow Field that take place when the local flow speed exceeds the speed of sound. The formation of shock waves is a particularly important phenomenon and is a consequence of the propagation of sound through the air. In incompressible flow the fluid elements are not permitted to change in volume as they pass through the flow Field. And, since sound waves propagate by alternately compressing and expanding the medium (see Section 1.2.7), this is tanta­mount to assuming an inFinite speed of sound. This has important consequences when a body like a wing moves through the air otherwise at rest (or, equivalently, a uniform flow of air approaches the body). The presence of the body is signalled by sound waves propagating in all directions. If the speed of sound is inFinite the presence of the body is instantly propagated to the farthest extent of the flow Field and the flow instantly begins to adjust to the presence of the body.

The consequences of a Finite speed of sound for the flow Field are illustrated in Fig. 6.11(p.308). Figure 6.11b depicts the wave pattern generated when a source of disturbances (e. g. part of a wing) moves at subsonic speed into still air. It can be seen that the wave fronts are closer together in the direction of flight. But, otherwise, the flow field is qualitatively little different from the one (analogous to incompressible flow) correspond­ing to the stationary source shown in Fig. 6.1 la. In both cases the sound waves eventually reach all parts of the flow field (instantly in the case of incompressible flow). Contrast this with the case, depicted in Fig. 6.11c, of a source moving at supersonic speed. Now the waves propagating in the forward direction line up to make planar wave fronts. The flow Field remains undisturbed outside the regions reached by these planar wave fronts, and waves no longer propagate to all parts of the flow field. These planar wave fronts are formed from a superposition of many sound waves and are therefore much stronger than an individual sound wave. In many cases they correspond to shock waves, across which the flow variables change almost discontinuously. At supersonic speeds the flow Field is fundamentally wavelike in character, meaning that information is propagated from one part of the flow Field to another along wave fronts. Whereas in subsonic flow fields, which are not wavelike in character, information is propagated to all parts of the flow Field.

This wavelike character of supersonic flow fields makes them qualitiatively different from the low-speed flow Fields studied in previous chapters. Furthermore, the existence of shock waves brings about additional drag and many other undesirable changes from the viewpoint of wing aerodynamics. As a consequence, the effects of flow compressi­bility has a strong influence on wing design for high-speed flight even at subsonic flight speeds. It might at First be assumed that shock waves only affect wing aerodynamics at
supersonic flight speeds. This is not so. It should be recalled that the local flow speeds near the point of minimum pressure over a wing are substantially greater than the free – stream flow speed. The local flow speed first reaches the speed of sound at a free-stream flow speed termed the critical flow speed. So, at flight speeds above critical, regions of supersonic flow appear over the wing, and shock waves are generated. This leads to wave drag and other undesirable effects. It is to postpone the onset of these effects that swept-back wings are used for high-speed subsonic aircraft. It is also worth pointing out that typically for such aircraft, wave drag contributes 20 to 30% of the total.

In recent decades great advances have been made in obtaining computational solutions of the equations of motion for compressible flow. This gives the design engineer much greater freedom to explore a wide range of possible configurations. It might also be thought that the ready availability of such computational solutions makes a knowledge of approximate analytical solutions unnecessary. Up to a point there is some truth in this view. There is certainly no longer any need to learn complex and involved methods of approximation. Nevertheless, approximate analytical methods will continue to be of great value. First and foremost, the study of relatively simple model flows, such as the one-dimensional flows described in Sections 6.2 and 6.3, enables the essential flow physics to be properly understood. In addition, these relatively simple approaches offer approxi­mate methods that can be used to give reasonable estimates within a few minutes. They also offer a valuable way of checking the reliability of computer-generated solutions.

Preamble

Hitherto in this volume the study of aerodynamics has almost exclusively been restricted to incompressible flow. This is really only suitable for the aero­dynamics of low-speed flight and similar applications. For incompressible flow the density and temperature of the fluid are assumed invariant throughout the flow field. As the flow speeds rise the changes in pressure become greater, leading to the compression of fluid elements, causing in turn the internal energy, and therefore the temperature, to rise. The generation and transfer of heat due to viscous effects and heat conduction are also significant in the boundary layer. But these and other viscous effects are not considered in this chapter.

The chapter begins with the study of (quasi-) one-dimensional flow, This is an approximate approach that is suitable for flows through ducts and nozzles when the changes in the cross-sectional area are gradual. Under this circumstance the flow variables can be assumed uniform across a cross-section so that they only vary in the streamwise direction. Despite its apparently restrictive nature one­dimensional flow theory is applicable to a wide range of practical problems. It also serves as a good introduction to the concepts and phenomena of compressible flow, such as the development of shock waves when the air is accelerated through and beyond the speed of sound.

The chapter continues with a description of the formation of Mach and shock waves in two-dimensional flow. An important application of this theory is the study of wing aerodynamics. The nature of the flow around wings is greatly affected when the local flow speed exceeds the speed of sound. The flight speed at which this first occurs is called the critical Mach number and methods of estimating this quantity for specified wing sections are demonstrated. The (inviscid) equations of motion governing high-speed flows change their character so that their solutions become wavelike when the local Mach number exceeds unity. The behaviour of the Mach and shock waves in two-dimensional flow is described in some detail. In general, the equations of motion are non­linear in form and not amenable to analytical solution. Special approximate approaches exist for pure subsonic or supersonic flows. For example, the assumption of small perturbations to the freestream flow can be exploited to obtain approximate analytical solutions for both subsonic and supersonic flows around wings. Other approximate methods are also explored. The chapter closes with a short description of compressible flow around wings of finite span.