I was working with orifices that were restricting flow and had to do a few calculation to verify some measurments (and to see if stuff was within tolerances).

So I stumbled upon the calculations our supplier provides

$$\frac{K f_T P_1}{Q}$$ (Sonic condition i.e. $P_1/P_2 \ge 1.9$)

and $$\frac{2K f_T \sqrt{\Delta P P_2}}{Q}$$ (Subsonic condition i.e. $P_1/P_2 \lt 1.9$)

(The whole description on how to use the formulas is in the link).

What got me thinking was that in the Sonic condition the downstream pressure $P_2$ does not matter anymore. My thinking is, that this is because at sonic and supersonic speeds, the downstream pressure shockwave can not propagate upstream anymore because it can't move faster than sonic speed.

Assuming this is true, what happens if I increase $P_2$ continiously until $P_1/P_2 \lt 1.9$. The gas is still moving supersonic and in my naive understanding, the pressure can not propagate upstream to slow down the flow...

How does the transition from (super)sonic flow to subsonic happen?

  • $\begingroup$ I am rather confused by this. Is this a rarefaction shock? How is the downstream pressure lower than the upstream pressure? $\endgroup$ Sep 19, 2022 at 12:32
  • $\begingroup$ In my case, we just apply higher pressure on the upstream side with a pump... The nice thing is, that it toes not matter if we reduce pressure on the downstream side or increase pressure on the upstream side, it's physically the same... $\endgroup$
    – kruemi
    Sep 20, 2022 at 4:42
  • $\begingroup$ I think you misunderstand. In a shock wave, there is some kind of piston/driver that plows through a compressible medium faster than the speed of communication. That piston/driver increases the pressure substantially in doing so. The region of increased pressure over ambient is called the downstream, not the upstream. Do you see why I am a bit confused now? $\endgroup$ Sep 20, 2022 at 12:06
  • $\begingroup$ @honeste_vivere sorry for taking so long to answer. I get the feeling that we're talking about different things. I see the difference in nomenclature. But we're not talking about an object moving trough a compressible medium but a compressible medium going trough a nozzle / orifice. $\endgroup$
    – kruemi
    Nov 7, 2022 at 13:32

1 Answer 1


Yes, pressure waves cannot travel upstream in (super)sonic flow, but the density of the gas also affects the speed of sound (SoS). If you increase the pressure P2, eventually you're going to create a compressive shock at the exit of the restrictor that would increase the density such that the gas becomes slower than the SoS, even if the gas is faster than the SoS at STP, thus the pressure wave would eventually be able to move back up the restrictor.


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