A Survey of Unsteady Hypersonic Flow Problems

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indication of their reliability, even though results of their use under steady
conditions cannot be conclusive evidence of their value when the flow is
unstoaay.
In Figs. 10 and 11 third order piston theory predictions for the
pressure distribution on a blconvex parabolic arc section aerofoil are shown
compared with the distribution from a characteristics solution, which should
be accurate for the inviscid flow. For a two-dimensional section the correction
for boundary layer growth should be straightforward and need not be considered
in this comparison. At M = 3'5, in Fig. 10, the agreement is quite olose,
though third-order piston values are consistently low; at M = IO, in Fig. 11,
the errors are much greater. The departures from the characteristics result are
greatest in the regions of the nose and trailing edge over which MC > 1 (8 is
the surface slope). The error in the prediction of the oentre of pressure for
a single surface would be quite large, but this does not necessarily mean that
the oentre of pressure for the section is similarly in error, since this depends
on the increment of pressure difference with incidence and not the overall distribution.
Calculations by Newtonian impact theory are compared with characteristic
results for the same bioonvex section at M=lO and 02 for zero incidence, and
at M = 10 for an incidence of 19~9~ in Figs. 11, 12 and 13. Results are shown
for equation (6) and for this expression modified to gave the oorreot pressure
at the leading edge. The unmodified equation always gives values of pressure
that are considerably lower than the accurate ,values. The modified equation is
inadequate at M = 10, (1 = O" (Fig. 11); even at M = m it is considerably
in error beyond the one third chord point (Fig. 12). For the section at an
incidence of 19*9O, the theory is being tested under very favourable conditions,
since there should be only a thin shook layer over the lower surface of the
section. Even for this ease (Fig. 13), there are appreolable errors in the
pressure distribution given by the modified expression.
Results for two-dimensional shook-expansion theory are given in
Figs. 14 and 15. In Fig. 14, pressure distributions on the same 1% aerofoil
section as before are given for the characteristics calculation, for the shock
expansion method, and for a simplified version of the shook expansion method
applicable to slender bodies at high Mach number (this slender aerofoil method
corresponds to the expressions used to derive equation (2.49)). There is close
agreement of the methods. In Fig. 15, the effect of real gas thermodynamics
on the pressure distributions are shown. When oalorlc and thermal imperfections
of the gas are fully considered in the shock expansion method, the results differ
very little from the characteristics result, consequently the characteristics
results have not been included in Fig. 15. The results from the slender aerofoil
method use an average value of y throughout the field. Above an incidence of
about IO' it is clear that the departures from perfect gas behaviour become
significant, especially for the, slender aerofoil method.
Figs. 16 and 17 illustrate the use of generalised shock expansion
theory on an ogivalbody of revolution. In Fig. 16 comparzon is made with both
characteristics results and with experiment. At adequate values of M6 = Md/8
agreement with the characteristics results is seen to be gocd, and Fig. 17 shows
that the agreement with experiment is also good if account is taken of the
boundary layer. (In Ref. 29 it is argued on physical grounds that, provided the
boundary layer flow is largely hypersonic, and the conditions governing the
application of two-dimensional shock expansion theory to a three-dimensional
body are satisfied; the boundary layer flow along geodesics can also be
calculated using two-dimensional relationships). In Figs. 19-23 results are
given for the application of shock expansion theory to more OomPlex slender
bodies26 ./