Compression and expansion waves

Imagine the source of a pressure pulse to be the pointed nose of a projectile (although any other point would do, in fact a sharp leading edge is made up of a line of such points). At very low speed the disturbance of the air by the moving point would spread out spherically through the surrounding air, like the much larger waves from a chiming bell. The waves all travel away from the source at the local speed of sound. As the source accelerates to higher speeds, however, the pressure pattern begins to change as each pulse is made further and further ahead of the previous one. These two cases are shown in Fig. 5.13(a) and (b), in which the source is initially almost stationary and is then seen moving at a speed half that of sound, M = 0.5.

Fig. 5.13 The generation of the compression wave and Mach cone (after T. von Karman).

As long as pulses precede the source the air particles lying in its path receive warning of its approach and can begin to adjust themselves beforehand for the coming displacement. The acceleration of each particle is therefore smaller than if it received no warning, and the force required to cause displacement is less.

When the source moves at the speed of sound it moves forward with the advancing pressure pulses, as shown in Fig. 5.13(c). Ahead of the source lies a zone of silence and behind is a zone of action, while between them is a sharp pressure wave formed by the piled-up pulses. When the source moves faster than sound successive pulses are sent out from points ahead of the preceding ones and the pressure wave generated by the source takes on a conical form. The resulting Mach cone is unique in that the semi-vertex angle, θ, is related to the Mach number by

θ = = cosec a V M (5-12)

Now consider the airspeed relative to the source. If the source is considered to be stationary the air ahead of it is moving towards it with speed V, while behind the cone the relative airspeed is less than V. This is the situation in a wind-tunnel where the formation of shock waves (the Mach cone is a conical shock wave — shock because of the sharp pressure change through it) marks a deceleration in the flow relative to the source. Clearly, for a relative deceleration to have taken place, the air in the zone of action must have been swept along to a certain extent by the moving source.

It follows that if the air behind the Mach cone has been swept along by the source, then such a wave must be a cause of drag. Applying Bernoulli’s theorem, the static pressure must be higher in the air behind the cone than in the silent region ahead. If the source is part of an aeroplane, then there must be another process of changing the pressure of the air back again to the undisturbed value as the aeroplane passes. The process of decreasing the pressure is by accelerating the air again, relative to the aeroplane, through an expansion wave.

Consider a supersonic body passing through a mass of air contained within an imaginary cylinder, as shown in Fig. 5.14.

Fig. 5.14 Pressure distribution along solid body moving at zero angle of attack through a compressible fluid. The cylinder defines the limit of the undisturbed air during the interval of time between the nose and tail passing a datum point. Beyond the cylinder the air is only disturbed after the body has passed. Inside the cylinder the air is disturbed during the actual passage. The cylinder has a cross-sectional area, A, that is reduced from a maximum to a minimum at the mid-point of the body, increasing to a maximum again at the tail. As the body is moving along the cylinder of air faster than pressure pulses can be transmitted, the air is compressed by the forward surface of the body, with a corresponding decrease in velocity relative to the body

and an increase in pressure. As the slope of the forward-facing surface decreases, however, the compression is reduced until, as the mid-point of the body passes, the area opens out, allowing the compressed particles of air to expand again. The air is therefore accelerated, relative to the body (in reality only trying to move back to where it was pushed away from), and the pressure in that region is decreased to a suction over the

rearward-facing surface. This is the opposite of the case at low speed, when the air is sensibly incompressible. A shock wave occurs at the tail where the rearward movement of the air is terminated and the pressure finally readjusted back to the ambient value.

The frontispiece, of fighters no longer in front-line service, nevertheless shows very clearly the

formation of shock and expansion waves experimentally and in practice. The expansion waves are marked by water vapor condensing as mist.

The drag caused by compressibility of the air is termed wave drag, which has two components. The first is due to the distribution of volume along the length of a body, and this is independent of the lift generated. The second is due to the lift generated, as may be seen from Fig. 5.15, in which the slope of the pressure diagram is similar to that shown in Fig. 5.14, except that the values have been altered by the angle of attack and, hence, the inclination of the surfaces to the air.

Fig. 5.15 Supersonic biconvex aerofoil section with angle of attack equal to leading-edge surface angle, showing lifting pressure distribution.

The double-wedge aerofoil section is a simpler section to make than the biconvex version, although a biconvex section gives a more flexible performance throughout the flight envelope. When an aircraft is designed for one performance mode (e.g. an anti-aircraft missile) then a wedge is more convenient. The experimental North American X-15 employed single-wedge fin surfaces, as shown in Fig. 5.16(f), and in this way eliminated the rear-facing wedge surfaces which, in experiencing a suction, contributed to the drag as much as a flat base. In this way the weight of the fin surfaces was reduced without loss of control

Fig. 5.16 Typical aerofoil sections.

The wave drag is minimized by selecting the optimum area, or volume distribution along the length of the body. A Sears-Haack profile, rather like Fig. 5.14, is such an example. Area and volume distribution is an important design technique that will be discussed under area-ruling, in the next chapter.