Velocity change across the normal shock

Подпись: or for air Подпись: «2 _ 2 + (7 - 1 )Mj щ (7+1 )M ui 5 + Mj щ= 6 M Velocity change across the normal shock

The velocity ratio is the inverse of the density ratio, since by continuity u2ju = рі/рг. Therefore, directly from Eqns (6.45) and (6.45a):

Velocity change across the normal shock Подпись: Cp7o - I ahead of the shock

Of added interest is the following development. From the energy equations, with cpT replaced by [7/(7 – l)]p/p, pi/pi and pijpi are isolated:

and

Подпись: downstream of the shockP2 7 V 2

Velocity change across the normal shock

The momentum equation (6.37) is rearranged with рщ = piui from the equation of continuity (6.36) to

Disregarding the uniform flow solution of щ = u2 the conservation of mass, motion and energy apply for this flow when

Подпись:2(7-1) m

uu2=——– rr-CpTo

7+1

cpTo = a*

Подпись: 2(7-1) 7+1

i. e. the product of normal velocities through a shock wave is a constant that depends on the stagnation conditions of the flow and is independent of the strength of the shock. Further it will be recalled from Eqn (6.26) that

where a* is the critical speed of sound and an alternative parameter for expressing the gas conditions. Thus, in general across the shock wave:

Подпись: щи2 a(6.52)

This equation indicates that щ > a* > u2 or vice versa and appeal has to be made to the second law of thermodynamics to see that the second alternative is inadmissible.

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