Questions about the drag equation and aerodynamics

[Bobcent]

Hello!

I have some questions about the drag equation and aerodynamics:

F = \frac{1}{2}ρv^2CA

I'm trying to calculate the atmospheric drag on a streamlined body (the drag coefficient will be a very small number) with a velocity of about 8 km/s at about 38,000 meters altitude, where the atmospheric density is only about 5.4\times10^-3kg/m^3. So my question is; is the drag equation valid even for these extreme values, or is there a better equation that I can use?

Secondly, which is the optimal geometrical shape for \frac{Volume}{Drag}? Is it a streamlined body shape? If it is a streamlined body shape, what is the equation for calculating its volume, and what is the equation for calculating its reference area? Can't find it!

Really appreciate any help on this!

 

[mfb]

Above the speed of sound, and in particular if heating becomes important, that formula will need some corrections.
NASA and other space agencies should know something about atmospheric drag at those velocities, they might have published something.

 

[SteamKing]

Streamlined bodies come in a variety of shapes. In order to calculate something, you would need a description of the particular shape.

 

[rcgldr]

Drag at supersonic speeds gets complicated. In order to use the standard equation for drag, the coefficient of drag becomes a function of speed (usually complicated enough to require a table and interpolation). A related wiki article:

http://en.wikipedia.org/wiki/External_ballistics

 

[Bobcent]

Thanks for your replies!

I have now found out that the optimum shape for \frac{Volume}{Drag} at high hypersonic speeds is the Sears-Haack body, and the equations for calculating the volume and reference area of the Sears-Haack body are on the wikipidea page, so now that bit is solved. See below if interested:

http://en.wikipedia.org/wiki/Sears–Haack_body

However, I still have a big problem. As "rcgldr" points out, in order to use the standard equation for drag (and I still havn't found any better equation), the drag coefficient becomes a function of speed, and the drag coefficient is based on empirical data for drag at different speeds for the specific shape.

This is a big problem since I can't find any empirical data for drag on the Sears-Haack body at around mach 25 (which is about 8 km/s at 38,000 meters altutude). Does anyone know if any experiments even have been conducted at these speeds for the Sears-Haack body, or for any other shape for that matter?

The highest speeds I've found data on for drag on the Sears-Haack body is mach 12 in a scientific article published by NASA in 1967, has no one really conducted experiments for higher speeds since? See article below:

http://ntrs.nasa.gov/archive/nasa/ca...1967030792.pdf

If anyone wonders what this is for, it is for my high school science project where I'm investigating the possibility to fling satellites into orbit around the Earth from the upper atmosphere instead of launching them by rockets. The upper atmosphere would be reached using a huge helium balloon.

I realize of course that it probably won't work, presumably because of too powerful centrifugal forces, crushing the satellite as it rotates during acceleration before it is released in its trajectory to orbit around the earth. But its still a fun project.

Very grateful for help!

 

[mfb]

Those experiments would require a massive rocket, and heating is a really serious problem with those parameters.

If anyone wonders what this is for, it is for my high school science project where I'm investigating the possibility to fling satellites into orbit around the Earth from the upper atmosphere instead of launching them by rockets.

With a circular motion? You would need a cable which is thicker than the actual spacecraft . Attached to the spacecraft with the same strength as within the cable...

 

[Bobcent]

Those experiments would require a massive rocket, and heating is a really serious problem with those parameters.

Hi! I am not so sure heating will be a problem in my case, since it will only be seconds before the spacecraft has reached a high enough altitude for the atmospheric drag to be negligible. But I havn't done the calculations yet, so I don't know. Atmospheric density rapidly decreases with altitude, see graph below:

http://www.wolframalpha.com/input/?i=Atmospheric+density+38000+meters+altitude

 

[mfb]

The graph shows the range of 0...1000km, you can hardly see the relevant range of 30..100km. Sure, atmospheric density drops with height quickly (otherwise concept like the StarTram would be impossible), but the density at 38km is not negligible.

How do you accelerate a spacecraft from 0 to 8km/s in seconds? That would require an extreme acceleration and power output.