I Subsonic and Supersonic Airflow Through a Constriction

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

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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.
 
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.
Thank you for the reply!

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

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

cjl

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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, 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 :)
 

boneh3ad

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

cjl

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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).
 

cjl

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I usually don't, so that would be rather hypocritical of me...
 
I recall something similar to this question from a Thermodynamics homework assignment back in the day. Gas flowing in a tube will gain heat from frictiom causing it to accelerate until it reaches mach 1. At that point it will shock and return to subsonic flow. The abrupt jump is defined by the Fanno and Raleigh lines. Once subsonic, it will begin to heat further and repeat the process. Supersonic flow does not occur in a pipe of constant diameter. A speed of mach 1 can occur at the throat of a nozzle and then accelerate into supersonic speeds in the divergent part of the nozzle. If the end of the nozzle transitions into a straight pipe, the shock will occur and back to subsonic flow. It has been a long time since I worked on nozzles and I am certainly no expert but maybe this is useful. If there is an expert out there, please correct me.
 

cjl

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At mach 1, you can't have a shock, because it wouldn't decelerate the flow. Also, as far as I know, there's no way for the flow to continue accelerating in a pipe with friction after mach 1. As a result, what will happen will simply be (for a long enough pipe) that the flow reaches mach 1 exactly once: at the exit of the pipe, and this fact will significantly impact the upstream conditions.
 

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