Page 319 - Aerodynamics for Engineering Students

Compressible flow 301

Since the flow is non-heat conducting the total (or stagnation) temperature remains
constant.

6.4.6 Entropy change across the normal shock
e)
Recalling the basic equation (1.32)
(E)'

= (~~~~-'= equation of state
the
from
which on substituting for the ratios from the sections above may be written as a sum
of the natural logarithms:

These are rearranged in terms of the new variable (M: - 1)

On ex anding these logarithms and collecting like terms, the first and second powers
of (M, !i? - 1) vanish, leaving a converging series commencing with the term

(6.48)

Inspection of this equation shows that: (a) for the second law of thermodynamics to
apply, i.e. AS to be positive, M1 must be greater than unity and an expansion shock
is not possible; (b) for values of M1 close to (but greater than) unity the values of the
change in entropy are small and rise only slowly for increasing MI. Reference to the
appropriate curve in Fig. 6.9 below shows that for quite moderate supersonic Mach
numbers, i.e. up to about M1 = 2, a reasonable approximation to the flow conditions
may be made by assuming an isentropic state.

6.4.7 Mach number change across the normal shock
Multiplying the above pressure (or density) ratio equations together gives the Mach
number relationship directly:

p2 xp' = 2YM; - (7- 1) 2YM; - (7 - 1) =1
P1 P2 Y+l Y+l

Rearrangement gives for the exit Mach number:

(6.49)