Quick flow math for new probe nose

Per my previous post, I have been thinking about getting rid of the static pressure tube just using an extra hole in the front "ball" nose. The following is some simple back-of-the-envelope math to see if that is likely to work.

First let's describe our setting. We plan to add a 6th hole 90 degrees from the centerline, at the bottom of the probe, as follows:

You can see that we measure pressures dpα and dpβ as usual, but dp0 is now between the center hole and our new 6th hole in the 90-degree position.

Using the standard potential flow formula for a sphere, we came up with the expected pressure ratios, to see if we are likely to get good data or if we'll have nasty singularities. The good news is that all seems promising.

We will plot various values for a range of α and β of ± 25 degrees. The color in the following plots is a measure of total angular distance from the centerline, to help with visualizing the surfaces.

First, let us plot the actual pressure differences measured by the sensors, in relation to the free stream dynamic pressure q. We plot (dp0 / q), (dpα / q) and (dpβ / q):

The ratios are all reasonable and there are no surprises. (dp0 / q) is around [1.0, 2.5], and (dpα / q) and (dpβ / q) are both in the range [-1.5, 1.5]. In particular, (dp0 / q) is a reasonably solid value and will not be close to a divide-by-zero

Now we plot our pressure ratios that we will use to calculate α and β, (dpα / dp0) and (dpβ / dp0), to see if there is a clear signal:

The range in both cases is about [-1.5, 1.5] and approximately linear. That is very good.

We conclude that this probe nose design is well worth pursuing. The pesky static pressure rod is a pain in the neck to mount, and it makes our probes a lot more bulky. It would be awesome to get rid of it!