# Subsonic and Supersonic Airflow Through a Constriction

## Main Question or Discussion Point

Hi everyone!

For the past week I have spend a lot of time thinking about how de Laval nozzles work. But before I convince myself that I have understood it, I want to make sure that the (simpler) scenarios I describe below are correct.

Setup: We have a long pipe with constant radius r = 1 everywhere except at its midpoint, where there is a smooth constriction (smoothly "pinched" to r = 0.5). At one end of the pipe, there is a hypothetical air generator that creates air at p = 1 atm and with a velocity v that matches the pipe direction. At the other end of the pipe, there is a hypothetical air destroyer, where the pressure is also p = 1 atm. (I have assumed that frictional effects are negligible).

My question is: would the velocity profile of the air vary as in the diagram I have drawn? I have included the cases where the air generator expels air at subsonic, sonic and supersonic speeds.

Thanks for the help!
Nat

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cjl
As long as your constriction didn't choke the flow, yes, your velocity profiles are correct for subsonic and supersonic. Your sonic one is not possible though.

russ_watters
As long as your constriction didn't choke the flow, yes, your velocity profiles are correct for subsonic and supersonic. Your sonic one is not possible though.

So if I was to draw an entire set of curves, each one with gradually increasing upstream velocity, what would happen at M=1? I would be surprised if there is some sort of discontinuity in the profile shape as you vary v.

Nat :)

cjl
If you were to draw a set of curves, the problem would arise when the flow hit mach 1 in the throat (this is what is known as "choked flow"), not when it hit mach 1 upstream. Subsonically, once you hit M = 1 at the throat, the only way to increase massflow would be to increase pressure upstream. For your example, with r_initial = 1 and r_throat = 0.5, you have an area ratio of 4. For an area ratio of 4, a flow with an upstream mach number of 0.147 will choke, so you will never be able to achieve higher than that (steady state). Similarly, on the supersonic side, with an area ratio of 4, a flow of mach 2.94 will slow to mach 1 in the throat. This actually makes all 3 of your curves above physically impossible because of the numbers chosen. However, an initial flow above mach 2.94 will (qualitatively) look much like your red curve, while an initial flow below mach 0.147 will look much like your green curve.

All speeds below mach 1 and above mach 0.147 will not be possible, because the backpressure will increase until the flow slows to mach 0.147. You could theoretically have your air generator set between mach 1 and mach 2.9 though, because the pressure from the throat cannot propagate upstream to affect inlet conditions when the inlet is supersonic. In this case, I would guess that a normal shock wave would occur somewhere before the throat, causing the flow to drop subsonic through the rest of the section.

NatanijelVasic
Thanks so much for the explanation! I'm going to have a long think about this now !

So it would look something like this?

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cjl
Yeah, basically. Also, I can't tell on your supersonic ones above mach 2.94 - are those all dipping down to mach 1? If so, that's not quite correct - as you continue to increase mach number, the speed in the throat will increase as well.

Also, this is all very idealized. It's actually very nearly impossible to build a shockless supersonic diffuser, so in reality, the supersonic flow behavior would be a bit different in the constriction (even though the flows as described here are mathematically possible).

NatanijelVasic
Yeah, basically. Also, I can't tell on your supersonic ones above mach 2.94 - are those all dipping down to mach 1? If so, that's not quite correct - as you continue to increase mach number, the speed in the throat will increase as well.

Also, this is all very idealized. It's actually very nearly impossible to build a shockless supersonic diffuser, so in reality, the supersonic flow behavior would be a bit different in the constriction (even though the flows as described here are mathematically possible).
Yeah, they all dip to 1 but i guess they shouldn't. Thanks so much for clearing up my understanding :)

Gold Member
Also note that in any real supersonic flow, that constriction is almost guaranteed to create shocks, which will throw all of this discussion off.

NatanijelVasic and cjl
cjl
Yep, hence my statement above

Also, this is all very idealized. It's actually very nearly impossible to build a shockless supersonic diffuser, so in reality, the supersonic flow behavior would be a bit different in the constriction (even though the flows as described here are mathematically possible).

Gold Member
Yep, hence my statement above
Pfft, you can't honestly expect me to read the whole thread, can you?

cjl
cjl
I usually don't, so that would be rather hypocritical of me...