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Air resistance and aerodynamic heating



 
 
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  #1  
Old March 9th 05, 03:02 PM
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Default Air resistance and aerodynamic heating

I'm trying to get a handle on aerodynamic heating. I'd like to be able
to answer questions like:

How fast could the space shuttle go at sea level to experience the same
heating as reentry (assuming same angle of attack)?

If you used a electromagnetic rail gun as a first stage for a rocket
launch, how fast could you have the rocket going when it left the rails
at sea level....how much faster could it go if the rail gun were at
15000 ft elevation?

Does anyone out there have any equations I can use to answer these
questions? The perfect equation would let me input altitude and it
would give me the fastest that a vehicle could travel at that altitude.
A more complicated but more useful equation would let me input
altitude and velocity and get out leading edge temperature. I assume
that both of these equations are impossible because the geometry and
construction of the vehicle would play a large part in the temperature.
Does anyone have suggestions for how I can proceed?

Here is what I have so far: From
http://www.e31.net/luftwiderstand_e.html I got the equation:

FAir = A/2 × Cd × D × v^2

FAir = force from drag (which I'm assuming is proportional to
aerodynamic heating

A and Cd = constants related to vehicle geometry and construction, so
I'll lump them together into one constant

D = density of air (which I can find from http://www.pdas.com/e2.htm)

v = velocity

Using this information I can't figure out anything about temperature,
but I can compare the performance of the same vehicle at two different
altitudes. For example if I assume my theoretical vehicle can go 1000
miles/hour at sea level without getting too hot, I can figure out how
fast it can go at 15000ft elevation without getting too hot.

Does anyone have any better way of calculating things that can get me
some actual temperatures? Also, I'm concerned with my assumption that
drag is proportional to heating. A pointy reentry vehicle would have
less drag than a blunt reentry vehicle...but they discovered long ago
that pointy things burn up on reentry.

But I'm not really interested in reentry speeds. I'm more interested
in speed around 2000mph, and elevations below regular commercial
airliners.

Thanks for any suggestions on how to proceed!


  #2  
Old March 10th 05, 05:26 PM
David Summers
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Well, to start out: you cannot directly calculate the temperature -
you calculate the amount of energy coming in to the vehicle, you
calculate the energy being radiated by the vehicle, and the rest of the
energy goes into raising the vehicle's temperature. So, if you have a
decellaration caused by air the energy put into the vehicles velocity
(E=0.5*m*v^2) goes into the air. In your case, the energy put into the
air is on the order of constant*Density*velocity^2 (at least that's
what my quick math said, please check by calculating the derivitive of
the kinetic energy of the vehicle being decellarated).

Of course, the trick then is to figure out how much of the energy put
into the air gets put back on the vehicle! At high altitude and
hypersonic, the answer is about half. For pointy things, the answer is
"lots". But for large curves a shockwave sets up in front of the
vehicle, partially shielding it from the heat. So less of the heat
gets to your vehicle (I usually guess 10%, but that is really only a
guess - it depends on a lot of things).

Typical aproach to dealing: Use large curves to lower the heat load,
then use either:

1) things that can get hot enough to radiate the energy
2) things that have enough heat capacity to store the energy (ballastic
trajectories have a low enough total heat load for this, orbit probably
doesn't)
3) you let parts of your craft vaporize and take the energy away
(ablation or boiling water)

Hope that helps!

-David

  #3  
Old March 11th 05, 04:30 PM
David Summers
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I just realized that there is a far easier answer - the energy transfer
rate is approximately Force*velocity. So you can start there, and
guestimate how much energy ends up in the vehicle.

-David

 




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