## How to know whether the flow is supersonic in a nozzle?

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For a project I had built a convergent divergent nozzle designed for Mach number = 3. In that project, I could know the flow has gone supersonic by seeing the manometer fixed between the throat and the divergent section (drop in pressure, as the divergent section acts like a nozzle for the supersonic flow).

However, this got me thinking, If I am to build a nozzle for the propulsion purpose (or any practical purpose), it is not desirable to have holes in it for the manometer in order to maintain the uniform strength. My theoretical calculations tell me that the flow should go supersonic and no shock in the nozzle, but while building, the surface finish, geometric tolerances and supply pressure might not be what I expect. In that case, how do I know if the flow has gone supersonic?

I thought about following ways. So far I haven't tried any of them.

1. Using a Pitot tube might not be useful since there will be a bow shock in front of the tube if in case the flow is indeed supersonic (as shown in the figure), which will increase the total pressure. We can use Reyleigh pitot tube formula, but how to compute static free stream pressure without affecting the flow / nozzle?

2. Schlieren Photography: If we see oblique shocks / shock diamonds, then the inference will be: 'flow is supersonic'. This will work only when the shock features are super clear.

2I think it would be fine to ask the 2 parts of this question as separate questions. In case an answerer only knows the answer to one part.dcorking 2015-01-26T19:11:13.247

Counterargument: the two are pretty intertwined, and an answerer that knows one is very likely to have an answer to the other. I voted up.Rick Teachey 2015-01-26T19:18:17.407

Fun (+1), I know nothing about this, but can you measure the mass flow up stream (or down stream) and then use the geometry? Re: sensor changing measurement, maybe try bigger and smaller sizes, make a plot and draw a line to zero. Is there some probe calibration?George Herold 2015-01-27T03:29:59.670

1@GeorgeHerold On the first issue, measuring the mass doesn't work well because the fluid is compressible, so setting up a control volume isn't a trivial matter. On the pitot tube, it's not a matter of size, it's the actual physics behind it. A pitot tube brings flow to a stop, and in order for the supersonic flow to stop, it goes through a shock wave first, which prevents anything from before the shock wave from being reasonably measured after it.Trevor Archibald 2015-01-27T04:11:20.987

OK, then probe it, sound waves? light waves? Maybe put some pieces/ markers in the stream that you can image and measure.George Herold 2015-01-27T04:17:18.130

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Subodh, would you be willing to edit this question to focus on Part A and ask a new question about Part B? You can link to this one from the Part B question. Anyone with opinions on this can join the discussion in the main chat, beginning here.

Paul Gessler 2015-01-27T15:04:17.950

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Sure!, I will make part B a new question.

Subodh 2015-01-27T16:25:06.123

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From my brief involvement in shocks, I think the most likely solution would be to image the exhaust, probably optically, but maybe using interferometry or something depending on what the exhaust is. The most obvious indication that you have supersonic flow is if you can see a shock diamond. I think you could probably also work it out from the length of the exhaust but I cannot remember how.

Alternatively you could also look at the thrust generated. You should be able to calculate the expected thrust. This is what they do when testing rockets/jet engines as they don't actually care if the flow is supersonic, just that it generates enough power.

The simple way for pipes is to just measure the exit flow. Its a pipe so flow should be constant. However, in practice I suspect long pipes also have regular inspection hatches/areas where they measure the flow somehow to check for leaks/faults.

dcorking 2015-01-27T21:13:44.547

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That is a really nice thought experiment! In general I would argue that you only need to know:

• Total-Pressure ( $p_t$ of your supply)
• Static/Ambient Pressure ( $p_\infty$into which you are expanding)

Taking into account the uncertainties (or tolerances) you will know whether your flow will have reached $M=1$ in the throat (or how likely it is that it has not).

$$\frac{p_\infty}{p_t}\leq 0.528 \quad, \text{assuming two-atomic-gas with } \gamma=1.4 \text{ in } \frac{p^*}{p_t}=\left(\frac{2}{\gamma+1}\right)^{\gamma/(\gamma-1)}$$

Looking at the equations for 1D steady compressible flow there is only one solution so the only way for the flow not to reach sonic speed would be a large total pressure loss so that the critical ratio is never reached.

As far as thrust is concerned the answer to your question is a bit more complicated since different set-ups (over/under expanded) or geometries (e.g. dual-bell).

As far as measurement is concerned you might want to have a look on acoustic air speed measurement systems.