A Survey of Unsteady Hypersonic Flow Problems

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This simple form of the method ignores the
disturbances frcm the body surface at the shock wave
gradient in the flow, and. the picture of the flow 33
effect of the reflection of
ana from regions of entropy
consisting of a straight
shock wave from the nose followed by a simple expansion is not in general
adequate for the whole field. The significance of these reflections has been
examined. in Ref. 28 and it is found that, at the body surface, the reflected
disturbances tend to cancel each other for the condition y > l-3. Since this
condition holds for most flows of practical interest, the simple theory is
adequate to give surface pressure information.
For thin two-dimensional sections undergoing small-amplitude sinusozdal
distortions about zero mean incidence in flow3 of sufficiently high Mach numbers,
the expressions for the nose shock conditions and for the Prandtl-Meyer relation
can be simplified and, as a result, a closed-form expression for the lift
distribution can be developed. For a symmetrical section the expression is 19 :
Pe‘PU = ze ikt
PC0
c &[(8)"-II+1
(y-1 1
1 +- Ei.J 'ii(x)
+ 2 3
- k(O)
X? - + ikg(0) 1 emilm +
CL ax
+ yi& [miq x) -1 J
2y/(v-1)
h(O)
- + ikg(O)] emikx +[z + ikg]] . . . (2.49)
ax
where g(x) is the complex amplitude of the time-dependent motion; n and m
are the rates-of change of pressure and Mach number at the nose with change of
nose angle; "N is the value of the Mach number at the nose for sero distortion;
@ is the value of the shock inclination angle for zero distortion; and z(x)
is the turning angle of the flow from the nose due to thichess alone.
For conditions where the simplified expressions for the nose conditions
and the expansion could not be applied, where the amplitude was not small, or
when real gas effect3 became important, numerical methods would have tL be
employed.
The shock-expansion method can be applied to slender three-dimensional
bodies for which MS is greater than a limit around unity (the limit is not
rigid; errors become greater as unity is approached) because it can be shown
for such bodies that the flow on the body surface is locally two-d' TJym; :",
planes normal to the body surface and tangential to the streamlines
as a consequence of this, the Prandtl-Meyer relation can be applied along the
surface streamlines. The surface streamlines can be shown to follow closely the
surface geodesics through the nose. A geodesic is a line on a surface such that
at any point its projection on the tangent plane at that point has zero curvature;
it is determined by the geometry of the surface so that once the initial direction
of the surface streamlines is known the expansion conbtiona can, in principle,
be determined from the geometry of the surface. The condition3 at the nose of
the body must, of course, be found from the flow over a cone hating the 3ame
cross-section as the body at the nose. This presents a limitation for the
application of the method since the flow is known only for cones with certain
simple cross-sections at small angles of yaw.
x