Regimes of Compressible Flow

For steady, inviscid and adiabatic flows, Bernoulli’s law is

У 2

H = h + — = C (12.1)

where C is constant along a streamline and h = CpT for a perfect gas.

The above relation can be rewritten in terms of the speed of sound (a2 = y RT) as

y 2 + 2 a2 = const = y2ax (12.2)

Y – 1

where ymax is the maximum possible velocity in the fluid (where the absolute temperature is zero) corresponding to the escape velocity when the fluid is expanded to vacuum (Fig. 12.1).

© Springer Science+Business Media Dordrecht 2015 399

J. J. Chattot and M. M. Hafez, Theoretical and Applied Aerodynamics,

DOI 10.1007/978-94-017-9825-9_12

Fig. 12.1 Reentry capsule flow (from history. nasa. gov)

In the following sketch, the different regimes of compressible flow are identified and hence their physical characteristics are discussed, see Fig. 12.2.

(i) For incompressible flow M ^ 1, V ^ a and Aa ^ AV

Notice that

2

V AV + aAa = 0 (12.3)

7 – 1

Fig. 12.2 Regimes of compressible flow

(ii) For subsonic flow M < 1, V, a and |Aa| < IA V |

(iii) For supersonic flow M > 1, V > a and |Aa| > |AV|

(iv) For transonic flow M ~ 1, V ~ a and |Aa| ~ |AV|

(v) For hypersonic flow M > 1, V > a and |Aa| > |A V|

In hypersonic flow, changes in velocity are very small and the variations in Mach number are mainly due to changes in a.