Hypervelocity Drag

Based on Trajectory Optimization for an Apollo-type Vehicle under Entry Conditions Encountered During Lunar Returm by John W. Young (famous astronaut) and Robert E. Smith Jr., May 1967, NASA TR-R-258, Langley Research Center.

Equations on Page 5 in Foot-second-slug-BTU :

Density in slugs/ft3: multiply kg/m3 by 1.9403203e-3

Power in Btu/ft2-s: multiply by 11350.54 to get W/m2

Velocity in ft/s: divide m/s by 0.3048

Metric equations:

These are for a 1 foot diameter nose, and scale by {r_n}^{-1/2} according to equation 4B-4 on page 520 of Part 4B (Entry Heat Transfer) of the SAE Aerospace Applied Thermodynamics Manual. That sites reference 1, A study of the motion and aerodynamic heating of missiles entering the earth's atmosphere at high supersonic speeds, H. Julian Allen and A. J. Eggers, Jr, NACA TN 4047, 1957. If r_n is in meters, scale by 0.552 {r_n}^{-1/2} .

If we scale these for a half-spherical nose, area \pi {r_n}^2 , we get:

Effective time:

Assume constant acceleration for the vehicle, v = a t , to a maximum velocity V = a T .

define t_{eff} = {\Large { T \over { n+1 } } } = { \Large { V \over { a ( n+1) } } }

If the drag power \dot Q = k v^n = k a^n t^n , then the time integrated power:

Q=k a^n{\Large {T^{n+1}\over {n+1}}}=k a^n T^n{\Large {T\over{n+1}}} = k V^n t_{eff} = \dot Q_{max} t_{eff}

There will also be additional exit or climb-out time for the launch loop added to t_eff , TBD.

The drag losses are much higher; most of the lost energy ends up heating the upper atmosphere (where it radiates efficiently into space, not to the ground). The drag power is P = C_D \rho Area V^3 and the drag loss is P = C_D \rho Area V^3 T/4

Examples:

For a 1 meter diameter nose, V=11 km/s, a=3*9.8m/s, T=374 s, CD = 2.0 and density at 80, 100, and 120 km:

altitude km

80

100

120

density kg/m3

1.85e-5

5.60e-7

2.22e-8

\dot Q_c W

3.49e-6

6.08e+5

1.21e+5

\dot Q_r W

1.17e+5

6.21e+2

4.90e+0

t_{eff}

Q_c J

94

3.26e+8

5.69e+7

1.13e+7

exponent n = 3

Q_r J

18

2.09e+6

1.11e+4

8.73e+1

exponent n = 20

Q_total J

3.28e+8

5.69e+7

1.13e+7

heat fraction

1.09e-3

1.88e-4

3.74e-5

drag loss J

1.44e+10

4.38e+8

1.74e+7

drag fraction

4.77e-2

1.45e-3

5.75e-5

heat/drag

0.023

0.130

0.651