Entropy change across the normal shock

Recalling the basic equation (1.32)

eAS/cv _ 02^ (jf} = (jr) from the equation of state

which on substituting for the ratios from the sections above may be written as a sum of the natural logarithms:

— = In 2lMl-~ ^ ~ Ц + 7 in (2 + (7 – 1 )M) – 7 In M

Cy 7+1

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

On expanding these logarithms and collecting like terms, the first and second powers of (M[ — 1) vanish, leaving a converging series commencing with the term

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