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Static Pressure Measurement in Supersonic Flow

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Old   May 20, 2020, 15:31
Default Static Pressure Measurement in Supersonic Flow
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Carlos Gonzalez
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Hi all,

I'm doing some reading about supersonic pressure measurements. The key formula I've found for calculating the Mach number of a super sonic vehicle is the Rayleigh Pitot Tube formula. I understand the derivation of the formula, it's straight forward. What I don't understand is why we care about the ratio of P_{02} / P_{infinity}. If the pitot tube (as well as the static pressure ports) is downstream of a bow shock, then isn't the static pressure being measured by the instrument P2? I don't understand how we could possible measure P_{infinity}.

Here is a link of what I'm looking at.

I would really appreciate any help understanding this. I feel like I'm missing something obvious.
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Old   July 27, 2020, 23:50
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Will Macleod
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I'm surely too late in replying, since you posted the question over 2 months ago, but I thought I would have a good stab at it anyhow as I enjoy the subject and these sorts of questions are good fun!

Initially I considered the that perhaps the only reason the relation is described as P02/p1 is for the plot on the final page - so that they can directly compare P01/p1 and P02/p2 in the graph, to clearly show the energy loss over the shock, and the incorrect aircraft speed that would be calculated if the right equation is not used. I quickly dismissed this however as it names the equation the Reighleigh formula and so I assume P02/p1 is used universally to calculate supersonic aircraft speed. I could be wrong about this though, I didn't look very hard for confirmation of the Rayleigh formula!

Then another thought occured to me. I thought that maybe, because the fluid flow that goes past the static pressure tapping may not go through the bow shock directly, that means it doesnt go through a normal shock, and hence it is indeed p1 not p2. Even if it goes through the edge of the bow shock, oblique shocks are much weaker than normal shocks, like the one right at the centre of the bow shock. This could be the answer, but if you trace a streamline from the location of the static tapping forwards along its path, it would have to avoid going through a strong shock at any point. The pressure read at a pressure tapping is the pressure of the inviscid part of the flow, just outside of the boundary layer. If you imagine an aircraft cruising, at incidence, with a pressure tapping underneath its fuselage, the forward trace of a streamline taken from just outside of the fuselage's boundary layer may indeed avoid all shocks and just be pure freestream flow, with a static pressure of - p1.

I'm not claiming this to be correct - I am just writing my thoughts down with the hopes that it will invoke a discussion where someone more knowledgeable then me can give us the answer! I'm not sure how readily I believe that a pressure tapping from a supersonic body has flow going past it that hasn't seen a shock. On top of this, even if this is achieved, there would surely be a number of flight conditions (incidences, roll, Mach number) at which the path of the air does go directly through a strong shock, which would throw the instruments totally out of control! Or maybe these particular conditions are simply well documented and managed, and control systems are built to be robust to this behaviour! Or perhaps there are a huge number of tappings and the average is taken with anomalous entries removed? I am actually really interested to hear the answer to all of these thoughts!

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Old   July 30, 2020, 06:59
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Will Macleod
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Update to my post:

I have found some lecture slides which say that the static pressure orifaces in supersonic flow pitot tubes are assumed to be unaffected by normal shock waves! So my guess above was correct

The lecture slides can be found at this link:

And the page in question where it confirms my guess is page 32 in writing (or page 42 of the actual pdf file)
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