Transient Fanno Flow

[butelja]

I'm attempting to model the fluid dynamics of a 2 stage gas gun, which is essentially an oversized BB gun. When examining the internal ballistics of any gun, the projectile can never go faster than the speed of sound of the propelling medium. By causing air (or another inert gas) to undergo rapid adiabatic compression, the temperature and speed of sound in the gas can be raised significantly.

So, what am I talking about? Here's a brief description.

A (relatively) massive piston is accelerated to a low to moderate velocity (100-200 ft/s) by either a metallic or gas spring. This piston then uses its kinetic energy to compress air from atmospheric pressure to a very high (2,000-4,000 PSIG) pressure and stagnation temperature. At a predetermined pressure, the pressurized resevoir is vented to a tube that has a relatively much lighter weight projectile in it. The projectile is propelled to velocities much higher than would be possible using a static high pressure air resevoir. NASA and others use a similar setup to achieve orbital velocities for small projectiles, only using hydrogen instead of air, and using gunpowder to drive the primary piston.

When examining the gas dynamics in the small projectile barrel, it is ASSUMED that steady state fanno flow relations can be used for each time step in a numerical simulation. Does anyone with relevant experience know how valid this is (or not)? Bear in mind that the stagnation temperature of the high pressure resevoir could be >2,000 °F and the mach number of the flow behind the projectile could be > 0.8, so it is very much compressible flow. Its the transient nature that makes the problem so difficult.

 

[Haf]

Butelja,

I'm not sure how well I understand your problem, so maybe I'm giving an oversimplified answer. But here goes: You mentioned that the transient nature of the problem makes it difficult. For transient problems, numerical approaches are often used, and you yourself are using one. For any transient numerical problem, it is usually assumed that properties are constant and processes are steady over any given timestep. For this to be valid, the time step must be sufficiently small. In other words, if you're worried about using steady vs. transient fanno flow relations, the answer lies in your time step. If steady fanno flow relations are not valid, your timestep is too large. So, my simple answer is this: if your timestep is small enough, you can assume steady state fanno flow relations over that time step.

One note of caution that you are undoubtedly aware of: with any numerical model, it is essential to demonstrate both grid (spatial) and timestep (temporal) independence. Of course, this must be done while adhering to the stability criterion, which inevitably comes up in transient problems. Demonstrating grid and timestep independence obviously tells you that your grid/timestep is refined enough to validate whatever numerical approach you are using.

Haf