# Hypersonic Axial Flow about Long Thin Cylinder

## Main Question or Discussion Point

Hi Physics Forums!

I would like to know how to model a thin, infinitely-long circular cylinder moving axially at hypersonic speeds. For something I would expect to be a base case in most hypersonics textbooks, it's surprisingly difficult to find a straight answer.

In particular what is the:
• heat over cylinder surface
• drag force on the cylinder

The closest thing I have been able to find so far is GK Batchelor's "The Skin Friction on Infinite Cylinders Moving Parallel to their Length" (1954) which gives a solution for subsonic speeds only. There, the drag force is:

8 • dynamicViscosity • ∫{ [ e(-k2)(kinematicViscosity)(timePeriod) ] / [ k•BesselJ(cylinderRadius•k)2 + BesselY(cylinderRadius•k)2 ] }dk

The closest thing I can think of to model this is that Rods from God idea about de-orbiting telephone poles to hit targets. Unfortunately I can't find much on their aerodynamics either (just religious stuff).

Many thanks!

-mLuby

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Gold Member
That is absolutely not a "base case" in hypersonic flow. In fact, it is nearly completely useless case, so you probably won't find it anywhere.

The problem is that no hypersonic vehicle would ever be that shape. Typically, you would look at flat plates as a base example, wedges, cones and blunt bodies as the next example and then move to more complicated geometries.

The quickest and dirtiest way to get a rough drag estimation would be modified Newtonian impact theory, which is very nice for back of the envelope calculations. That would not account for drag due to skin friction, but skin friction drag is a fairly small portion of total drag on a blunt body like that. To get the drag due to skin friction, you will need to do some flowfield analysis and get wall shear stress. That is not trivial.

As for temperature, you will need to solve the energy equation. That is also not trivial.

What exactly do you want to solve this for? I don't know how well you know compressible gas dynamics, but if you have never seen it and want to get a good book on it, I would start with Anderson's Modern Compressible Flow and then if you want to read further on hypersonic flow in particular, move on to Anderson's Hypersonic and High-Temperature Gas Dynamics.

I figured it would be basic because flows over infinite flat planes and perpendicular to infinite cylinders are often covered in the introductory texts I've read. Obviously that assumption was incorrect.

My understanding is that at hypersonic speeds most of the form drag & kinetic energy lost to heating is the result of shockwaves caused by the leading edge violently forcing the fluid out of the way of the object. What I want to know is whether once that fluid is out of the way, there is still significant parasitic drag or heating. (Thus the 'infinite' nature of the cylinder.)

Thanks for the book suggestions. I'll check them out and see what I can learn.

Gold Member
There is always going to be significant heating. The hypersonic boundary layer is so unbelievably thin that you have huge velocity gradients and huge shear stress, which drives heating. Heating is one of the single most important factors in hypersonic flight.

Similarly, there will still be plenty of skin friction drag along the surface, it just will be a lot less than the wave drag. If I remember correctly, about 80% of the drag is going to be in the form of wave drag for supersonic bodies. I could go through my stuff and find that figure but the exact number isn't entirely important right now, just the relative dominance.

If the cylinder is moving in the axial direction then won't the only drag be skin friction drag? If this is the case would a good starting point be the drag on a flat plate, because the cylinder could just be thought of a flat plate that has been rolled up. Don't know much about supersonic flows so not sure how useful this would be even as an approximation.

Gold Member
No. The cylinder would still have form drag because it isn't infinitely thin, and wave drag because of the shockwave at the front end.

A flat plate is not a good approximation because while it is still a 2-D flow at 0 AoA, it is a different fundamental geometry governed by te cylindrical Navier-Stokes equations rather than the Cartesian ones.

The OP stated that the cylinder was infinitely long. So there is not front end, so no shockwave. And no rear end means no form drag. Obviously this is unrealistic but that is what the question referred to.

Gold Member
Fair enough, but that doesn't address my last point. Unless the cylinder was of infinite radius a well, it would not be quantitatively similar to a flat plate. Additionally, removing the shock makes the problem entirely trivial and uninteresting in that it serves no purpose and provides no interesting insights into anything. It is a flow with no application to any real situation. I know you already realize that, but the problem doesn't have any practical value. It illustrates how important it is to understand the questions one is trying to answer.

Fair enough, but that doesn't address my last point. Unless the cylinder was of infinite radius a well, it would not be quantitatively similar to a flat plate.
Ok, it was just a suggestion anyways, I didn't really think it would work..

Additionally, removing the shock makes the problem entirely trivial and uninteresting in that it serves no purpose and provides no interesting insights into anything, as it is a flow with no application to any real situation.
Agreed.