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

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

If you still looking for answer,

You could keep a well-designed wedge, with static holes on the wedge surface, eighther 1. the surface of the wedge is aligned with the flow axis or 2. aligned symmetric line with the flow axis. you will have pitot pressure from Raleigh pitot-tube.

Now you could measure the static pressure directly in case 1. or 2. with Oblique shock relations having known values of wedge angle($\theta$), $P_0$ & $P_{\infty}$ you will get oblique shock angle($\beta$) and Mach number($M$).