A Survey of Unsteady Hypersonic Flow Problems

- 20 -
of a disturbance is of order sd(= a@). Consequently, the second limitation
can be expressed by
03..
x >> 66
where- X is the wavelength of the disturbance.
h/d >> 6 . . . (2.22)
Sychev has shown, in Ref. 25, that a fdrm of small disturbance theory
can be developed for bodies at large incidence in hypersonic flow, provided that
the flow on the leeward side of the bcdy can be neglected. From this development
of the theory it can be shown that on the windward side of the body the flow in
a lamina of fluid normal to the axis of the body can be considered as independent
of the flow in adJacent laminae. The form that the equations take will be
illustrated, as before, by considering the general equations of motion of the
fluid. The equations are presented this time in non-dimensional form as, far
this case, the development can be seen more clearly.
The axes &, '1, C., are taken with origin at the nose of the body and
the &-axis in the mean direction of the principal body axis and the z-axis in
the plane of the &-axis and the flow direction (Fig. 6). The independent
variables are made non-dimensIona by the transformations:
g = -, & 11 T = tu co9 a
;i = --&,
. . . (2.23)
b
where b is the body length, 6 is a ratio representing the maximum surface
slope, a is the angle of incidence of the body and U is the flow velocity
of the free stream.
The dependent variables are made non-dimensional by the transformations:
u cos 01 + u v W
= l+ii, 7 = f w =
u 00s a U sin a U sin a
P P
P = , and p = - . . .
pm ua sFna a
PC0
where u, v, w are the velocity perturbations in the &, i, G directions.
Then the general equations of fluid motion become
pm u cos a a;; pm u cos OL as; poou sina a(3 pmU sina a(3
-+ -+ -+
b ax b E b6 a? b& a4
b
(2.24)
poou co.9 a a(3
= - -, . . . (2.25)
b E
u2cos2a
b
I