Subsonic and Supersonic Airflow Through a Constriction

In summary: This means that the massflow rate cannot be increased anymore by decreasing the backpressure. You can increase the massflow by raising the upstream pressure though. This is how a de Laval nozzle works.In summary, the conversation was about the behavior of air flow in a long pipe with a constriction at the midpoint. The velocity profiles of the air were discussed for subsonic, sonic, and supersonic speeds. It was determined that the sonic velocity profile was not possible, and that a shock wave would occur at the throat of the nozzle for supersonic flow. The conversation also touched on the idealized nature of the discussion and the effects of shocks on real supersonic flow. The concept of
  • #1
NatanijelVasic
19
5
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
 

Attachments

  • pipeFlow.png
    pipeFlow.png
    56.5 KB · Views: 495
Physics news on Phys.org
  • #2
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.
 
  • Like
Likes russ_watters
  • #3
cjl said:
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 :)
 
  • #4
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.
 
  • Like
Likes NatanijelVasic
  • #5
Thanks so much for the explanation! I'm going to have a long think about this now !
 
  • #6
So it would look something like this?
 

Attachments

  • 28AB4808-0AA1-4000-8FAF-4A1955A6A609.jpeg
    28AB4808-0AA1-4000-8FAF-4A1955A6A609.jpeg
    103.9 KB · Views: 436
  • #7
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).
 
  • Like
Likes NatanijelVasic
  • #8
cjl said:
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 :)
 
  • #9
Also note that in any real supersonic flow, that constriction is almost guaranteed to create shocks, which will throw all of this discussion off.
 
  • Like
Likes NatanijelVasic and cjl
  • #10
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).
 
  • #11
cjl said:
Yep, hence my statement above

Pfft, you can't honestly expect me to read the whole thread, can you?
 
  • Like
Likes cjl
  • #12
I usually don't, so that would be rather hypocritical of me...
 
  • Like
Likes boneh3ad
  • #13
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.
 
  • Like
Likes NatanijelVasic
  • #14
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.
 

1. What is the difference between subsonic and supersonic airflow?

Subsonic airflow refers to air moving at a speed below the speed of sound, while supersonic airflow refers to air moving at a speed above the speed of sound.

2. How does a constriction affect airflow?

A constriction in a fluid flow system, such as an airplane's wing or a wind tunnel, causes a decrease in the cross-sectional area through which the fluid can flow. This results in an increase in the velocity of the fluid, as described by the continuity equation.

3. What is the difference between a subsonic and supersonic constriction?

A subsonic constriction occurs when the fluid velocity is below the speed of sound, while a supersonic constriction occurs when the fluid velocity is above the speed of sound. This difference affects the behavior of the airflow and the resulting pressure distribution.

4. How does the Mach number affect airflow through a constriction?

The Mach number, which is the ratio of the fluid velocity to the speed of sound, plays a crucial role in determining the behavior of airflow through a constriction. At low Mach numbers, the airflow behaves similarly to incompressible flow, while at high Mach numbers, the airflow becomes compressible and shock waves can form.

5. What factors affect the critical pressure ratio in a supersonic constriction?

The critical pressure ratio, which is the ratio of the downstream pressure to the upstream pressure at which the flow becomes sonic, is affected by several factors, including the shape of the constriction, the Mach number, and the specific heat ratio of the fluid. A higher critical pressure ratio indicates a more efficient supersonic flow through the constriction.

Similar threads

  • Aerospace Engineering
Replies
10
Views
601
  • General Engineering
Replies
14
Views
538
  • Mechanical Engineering
Replies
15
Views
651
Replies
11
Views
2K
Replies
3
Views
863
Replies
39
Views
4K
  • Other Physics Topics
Replies
14
Views
2K
Replies
11
Views
1K
Replies
5
Views
2K
  • Mechanics
Replies
20
Views
2K
Back
Top