# At what altitude does transonic compressibility become a non-issue?

I know that on the forthcoming SpaceX attempt at sea landing the first stage, they have added hypersonic fins to the first stage. I was curious as to what altitude transonic compressibility is an issue, and whether the first stage upon reentry would be affected by such, and whether these fins could become a standard feature for reentry craft, given their potential to create drag at hypersonic speeds, thus possibly lowering requirements for heat shielding.

• Could this get a better answer over on the Aviation stack? – Everyone Dec 21 '14 at 15:26
• Not sure. Since it's a condition of re-entry, I would think it would be with spacey things, but it also is a condition of supersonic flight as well. I wasn't sure of the answer, but the altitude of the space shuttles sonic booms upon re-entry is probably the right answer. I was just curious, as at those level of forces and stress, would the spacex Merlin engines be damaged by the sheer forces involved in transonic transition, making reusability an even loftier goal than all of us think. – TomSwift Dec 21 '14 at 15:48

This will heavily depend on the re-entry speed. The typical solutions of fluid mechanics available to us via the Navier-Stokes equation are only applicable in the continuum flow regime (where the molecules of gas are dense enough such that their mean free path is quite short, i.e. they collide with each other rather than just ping off into the distance).

We can compute whether the flow around a spacecraft is in the continuum flow regime by the Knudsen number:

$$Kn = \lambda / L = Ma / Re \sqrt{\gamma\pi / 2}$$

where lambda is the mean free path and L is a characteristic length scale (length of your spacecraft, or your fin).

Alternatively it can be defined in terms of the Mach number, the Reynolds number, and the ratio of specific heats. So, you've got to know how fast they are entering the atmosphere.

When $Kn$ is greater than 10, it is safe to say you're now in the land of rarefied gas dynamics. Each molecule must now be treated individually, with its own speed, energy, temperature, etc. instead of as a continuous mass of fluid.

Does this mean that compressible gas effects go away when your gas is rarefied? No, it doesn't. Using DSMC methods, this research paper (on the characteristics of a Brazilian satellite during re-entry) shows that there's still a significant drag coefficient when you're in rarefied flow: http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0103-97332003000200044 (Hypersonic flow of rarefied gas near the Brazilian satellite during its reentry into atmosphere, Felix Sharipov). Shock waves are still quite possible but they have a much smaller standoff distance.

I hate to be glib, but the question is a difficult one. When we say "non-issue", what do we mean? Can we quantify the strength of the shock that will form at a Mach number close to 1? Can we say what drag force the shock will cause, and whether this matters as to the controllability of the vehicle? The altitude at which rarefied flow exists will also change depending on the atmospheric density - and that can vary with the energy output of the sun, getting you into the whole category of space weather.

This is the sort of question that good instrumentation around your spacecraft helps to answer, as well as thoughtful analysis and simulation - and it's still hard then. You'd need the CAD of the vehicle, the flight profile, a powerful particle solver, a good computer cluster, and a month or two of your time to get a reasonably good answer.

It may also just be that the vehicle quickly passes through transonic to subsonic (where the grid fins again will behave predictably). If you're not in that regime for too long, you can perhaps just ride out the transient effects of the shock - like your car tires slipping on ice for just a fraction of a second. This is hard to know without seeing the flight profile with Mach numbers vs. time (and I'd note that the speed at which the vehicle is moving is will not necessarily give you a correct idea of the Mach number, which is influenced by temperature).

given their potential to create drag at hyper-sonic speeds, thus possibly lowering requirements for heat shielding

The Falcon 9 booster does not use a heat shield.

Grid fins are designed to be in the Q zone AFTER that which require a heat shield.

For reference, the titanium used by SpaceX for the grid fins melt at 1650°C.

A Soyuz heat shield encounter temperatures of up to 3000°C. (Close to the Ebullition point of titanium).