Effects of Compressibility

All gases and even solids are compressible. The speed of sound, a, represents the speed of propagation of weak disturbances due to compressibility of the medium. These weak waves often are referred to as acoustic waves. For example, sound waves in water propagate at the acoustic speed—or sound speed—characteristic of that liquid.

Other variables become important when additional physical effects must be accounted for, such as compressibility. For instance, as the speed of a vehicle approaches a significant percentage of the speed of sound, several changes take place in the flow-field characteristics and in the accompanying forces on the vehicle. If the representative aircraft shown in Figs. 1.2-1.12 in Chapter 1 are studied, it becomes apparent that drastic changes in design accompany the increases in speed. Compare the pictures of the high-speed seaplanes used in the Schneider Cup races with the supersonic F-22 and the Concorde transport; quite different shapes are required when the speed is higher than the speed of sound. Similarly, there are important design differences between low-speed aircraft and those that must operate in the transonic range, which begins when the speed is about 80% of the speed of sound. Other families of shapes are demanded when a vehicle must operate at hypersonic speeds; that is, when the velocity is five or more times higher than the acoustic speed.

The dimensionless ratio of the flow speed to the speed of sound represents a measure of the relative importance of compressibility. The thermodynamics and gas – dynamics of this situation are carefully worked out as needed in subsequent chap­ters. It suffices at this point to introduce the speed of sound for isentropic wave propagation in air:

a = ^Jp, (2.5)

where y is the ratio of specific heats. For air, this parameter has a value of у = 1.4. For an ideal gas, it is easy to see that by substituting Eq. 2.1, the speed of sound depends on only the temperature, T. We find:

a = ї/yRT. (2.6)

For sea-level air, the speed of sound is approximately a = 1,116 ft/sec = 340 m/s.

An important parameter is the ratio of vehicle speed to speed of sound—that is, the Mach number:

Table 2.1. Effects of compressibility


Flow Regime

Characteristics of Flow Field



No dependence on compressibility; density is constant everywhere.



Compressibility effects are sometimes negligible.

0.85 to 1.05


Compressibility greatly affects flow. There is a major increase in drag. Shock waves may appear at points on the body where flow is locally higher than sonic speed.

1.05 to 5


The flow field is dominated by shock waves. Major drag increases appear that are related to the shock waves.



Shock waves may lie close to body surfaces and may interact with viscous boundary layers.

Whether M is zero or less than or greater than unity determines which of sev­eral flow “regimes” within which a vehicle operates. Each regime is characterized by distinct flow-field features, which are summarized in Table 2.1. The ramifications of several flow regimes are examined later in great detail because they are important in determining aerodynamic performance and in solving practical vehicle-design problems.

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