Download Methods for Analysis of Preliminary Spacecraft Designs AAE450

Purdue University, AAE450 Spacecraft Design, August 18, 2003
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This type of skin friction analysis is much simpler, in concept and use, than integral boundary
layer methods. Results obtained using the reference temperature method were within 10% of
results obtained using a more complex integral boundary layer method (Ref. cited in paper)
even at high hypersonic Mach numbers. Also, the computation time required by the reference
temperature method is very small when compared to an integral boundary layer method.
Boundary layer transition is predicted using a correlation of the local transition Reynolds
number, Rex,t , with the local edge Mach number, Me , as follows:
log10 (Rex,t ) = 6.421 exp (1.209 × 10−4 Me2.641 ).
(72)
This correlation is based on the experimental data of DiCristina (ref. cited in paper) for transition
on sharp cones at zero angle of attack. This correlation is used due to the lack of better methods
of transition prediction in hypersonic flows. It is also presented in Anderson, HHTG, p. 280,
eq. 6.140. For 450, we will go ahead and use this even at angle of attack. Gustafson suggests
estimating Me for your angle of attack using Me = M∞ cos α. This crude formula neglects the bow
shock, but it will have to do. Note that for Me larger than about 35, Rex,t is greater than about
1030 . For this case, the flow will certainly be laminar, so computation of this very large number is
not needed.
7.8.2
Low-Speed Skin Friction Coefficient and Drag
The following information for low-speed computations is given for background only, to help
you in understanding and applying the high-speed methods.
At low speed, the local flat-plate laminar skin friction coefficient is
Cf (x) =
τ
= 0.664(Rex )−1/2 .
0.5ρ∞ V∞2
(73)
Here, τ is the local shear stress. The drag on one surface of the flat plate is
D=b
Z
l
0
τ (x)dx =
0.332bρ∞ V∞2
Z
l
0
(Rex )−1/2 dx,
(74)
where b is the width of the flat plate, and l is the length. We can make a drag coefficient for a
length l of this plate, using bl for the reference area, as follows:
CD,l =
D
0.664 Z l
=
(Rex )−1/2 dx.
2
0.5ρ∞ V∞ bl
l
0
(75)
Simplifying,
CD,l
0.664
=
l
Note that
Z
so
CD,l
l
0
s
µ∞ Z l −1/2
x
dx.
ρ∞ V ∞ 0
x−1/2 dx = 2l1/2 ,
0.664
=
l
"s
#
µ∞
2l−1/2 ,
ρ∞ V ∞
(76)
(77)
(78)