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I am a grader for a fluid mechanics class, and have come across a problem that I don't understand the answer to, and I was hoping someone could explain it to me. I've quoted the question below.

Consider compressible flow through the converging nozzle below, that has a low velocity at the inlet and a sonic velocity (Ma=1) at the exit. the nozzle exit diameter is reduced by hlaf while the inlet temperature and pressure are maintained the same. the nozzle exit velocity will:

  • a) remain the same
  • b) double
  • c) quadruple
  • d) go down by 1/4th
  • e) go down by half

(The included picture is rather useless, it just shows two nozzles one with a smaller opening at one end. I'd take the effort of posting it if it were relevant)

According to the solutions I have, the answer is supposedly a) remain the same but I don't understand why. Based on the continuity equation, shouldn't it be c) quadruple? Or does that assume it is an incompressible fluid?

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user177454
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  • I suspect that the key here is that the original exit velocity is already Mach 1. But i've not studied fluid dynamics enough to know. Though there are some mathematicians who know this area, you would almost certainly have much more help from asking this on the physics forum instead of mathematics. – Paul Sinclair Nov 10 '19 at 03:39
  • Okay, thanks for the advice! I'll post it there and see if someone over there can help me out. – user177454 Nov 12 '19 at 18:45

1 Answers1

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From the area-Mach number relation it follows that when you reduce the area the flow stays choked at M =1. Then, in both cases, the stagnation temperature at exit is the same as the stagnation tmperature at the inlet because it is an adiabatic flow. It follows that the exit static temperature is also the same because in both cases M=1 at exit. Hence a the speed of sound is also the same and since u = Ma, the velocity remains the same.

The velocity would quadruple if the fluid were incompressible - but here the density at exit would increase to keep (rho u A) constant.

user89699
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