A Survey of Unsteady Hypersonic Flow Problems

- 30 -
The cut-of-phase component of Ys is plotted in Fig. 8 and the
cut-of-phase component of &, is plotted in Fig. 9 in comparison with the
quasi-steady Newtonian result. At practical values of kx where 5; = i;/a,
is the non-dimensional amplitude of the cap plunging motion, it is clear that the
cut-of-phase component of the shock wave motion will be negligible over the
range of applicability of the analysis. The cut-of-phase component of the
pressure is small for quasi-steady Newtonian theory; the present theory provides
a correction of the order of .$ at the limit of its application.
The analysis by Hclt 33 can be applied to those parts of the flow
around a body where supersonic conditions exist, and a characteristic analysis
has been carried cut. Starting from an established flow field and known
characteristic directions, Hclt expresses the flow equations in terms of a
cc-ordinate system based on the given characteristic directions and introduces
small perturbations of the flow quantities. If squares and products of the
perturbation quantities are neglected, the equations become linear equations
for the perturbations with coefficients determined by the steady flow solution.
Hclt applied this analysis to the simple case of isentropic flow over an
axisymmetric conical afterbody in Ref. 33. More recently, Kawamura and Tsienx
applied the method of analysis to an axisymmetric body to determine the stability
derivatives, but this is for, effectively, steady state conditions.
The small perturbation methods of solution can be valid only when the
conditions are such that the hypersonic similarity parameter for the body motion,
M6 (where 6 is the change in surface slope due to the motion), is small, so
that disturbances to the flow quantities are sufficiently small. In general,
numerical solutions will be necessary, but this is unlikely to be an important
drawback since a numerical solution of the steady flow held will usually have
been necessary in the conditions for which the methods are best used.
2.2 The Influence of Real Gas Effects and Viscosity
It was pointed cut inAppendix Ithat, in many hypersonic flows,
temperatures will be generated in the gas which are sufficient to cause excitation
of vibrational degrees of freedom of pclyatcmic gas molecules, dissociation,
and icnisaticn; and that these effects can give rise to significant mcdificaticns
to the flow. Fortunately, the characteristic times involved in these reactions
will usually be very short in comparison with the characteristic time of any flow
unsteadiness likely to be met in practice. For example, the relaxsticn time
for dissociation of cqgen for flow in the stagnation region of a blunt body at
M = 15 at 200 000 ft is of the order of 2 x 10e4 seconds, whereas the maximum
frequency for any unsteady motion involving the structure of a vehicle is unlikely
to be as high as 100 cycles per second, and will, usually, be very much less than
this. Consequently, although the effects of these changes in the gas can complicate
analysis, they can usually be dealt with on a quasi-steady basis.
The effects of viscosity are not as clearly defined. The first effects
that must be considered exist already in steady flow and arise from the fact
that boundary layers are in general very much thicker than at lower Mach numbers
because of the rise in the temperature of the gas as a result of its deceleration
in the layer, and the smaller unit Reynolds numbers associated with high altitude
flight. The thickness of the boundary layer can be such that it exerts a
significant influence on the external 'inviscid' flow. A measure of this influence
is usually given by the size of the parameter x defined by:
x = bf(a,JRe,)' . . . pt)
where/