A Survey of Unsteady Hypersonic Flow Problems

-44-
is to reduce further the range of values of K for which divergence occurs, but
to increase the altitude at which divergence starts when It does occur. Fig. 34
shows that, as in the non-iifting case, K = 0 does not show serious divergence,
but K = +2 does.
Confirmation of the qualitative valltity of the simplified analyses of
re-entry which have been discussed is provided in Ref. 53, where the results are
presented of a 6-degree of freedom numerical analysis, using experimentally
determined aerodynamic forces, of the re-entry motion of a blunt uncontrolled
vehxle. This analysis shows; in particular,-that provided
of the body IS positive, the motion converges even for
fk, + cm&).
3.1.3 The effect of aerodynamic non-linearlties
the lift-curve slope
zero value of
In Ref. 51, Kistler and Capalongan give the results of stubes, using
analogue computers, in which they considered the effects of aerodynamic
non-linearitles on the longitudinal dynamx motions of hypervelocity, highaltitude
vehicles. They found that reasonable accuracies could be obtained
using linear aerodynamics if the coeffxients were determined at the trim point
of the vehlole, and provided the perturbaiions were small. For large perturbations
accuracies began to drop rapdly. The study included level flight and shallow
re-entry conditions and one of the conclusions of the report was that artificial
damping of the vehicles would be necessary, and thx m&t well overshadow any
non-linear aerodynamic damping characteristics.
Laitone and Coakley, in Ref. 50, examine the effect of aerodynamic
non-linearities on the pitching oscillations of a vehxle flying in a re-entry
traJect0x-y. The results do not affect the ooncluslons that have been drawn
about small amplitude motions, but they show that a steady llml'c cycle oscillation
can exist and that conditions sre possible III whxh osodlatlons will grow If
the initial disturbance exceeds a certain amplitude.
3.2 The Lateral Behaviour of the Hypersonic Vehicle
In Ref. 52 Nonweiler has also examined, qualitatively, the lateral
dynamic behaviour of a hypersonic vehxcle. The analysis is carried out in the
same way as for the longitudinal behavlour: the approxxnate roots for the
stability equation are found under the assumption that terms involving the
relative density, p, are large. The roots show the usual modes: normally,
two non-oscillatory modes and one oscillatory, the 'dutch roll'. An examination
of the factors governing the modes shows that it should not be difficult to
ensure convergence of the non-oscillatory modes and damping of the dutch roll
even for slender bodies at high incidences. But the period of the lateral
oscillation is likely to be rather shorter than that of the pitching oscillation
for slender bodies at high incidence - typxally 3 seconds for a value of KL
of unity, but since such bodies are not likely to be operating at large incdences
for low values of KL at low altitude, the maximum values of frequency parameter
will probably remain of the same order as those for the longitudinal pitching
oscillation.
The longitudinal behaviour of a hypersonic vehicle flying a ?-e-entry
path has been found to be not essentially different from that of the same vehicle
in level flight and it is reasonable to suppose that the same result would be
fauna for the lateral behaviour. The rate of decay of the lateral OsCilhtiOn
will/