A Survey of Unsteady Hypersonic Flow Problems

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The thcorctlcal deduction that @! u a parameter affecting flutter
has not been directly investigated, but some confirmati of it can be found
UI the results of the experiments by Hanson59 and Goetz al. These results are
given in Figs. 42 and 51. They show that, far a given aerofoil at a given Mach
number, the results at widely different values of p and 2 show the same
values of the parameter bwcrda at flutter. Since
b+i "m,J-v Mf
- = . . . (4.13)
a
Vf
the constancy of this parameter is equivalent, for a fsxed Mach number, to the
re1ats.on
vf
found by Chawla (equation (4.8)).
cc -/p . . . (4.14)
Fig. 44, derived from the results of both Hanson and Goetz, for
pointed rleading-edge
sections, shows that the use of the parameter
Vf - '-
correlates results from models of different thickness at different
bwa' PJ'
Mach numbers, but having the same value of M6, provxled M > 2, which is a
normal lower limit for the use of piston theory, m any case. This figure ~11
be dsscussed agaxn III a later sectlon, but It seems to support quite well the
theoretical result for the significance of fl.
(ii) Thxkness parameter M6
The paper by Morgan, Runyan and Huckel
57 .
gives a ccmpar~son between
measurements of the lift and centre of pressure position on a % thick doublewedge
aerofoil m steady flow at M = 6.86, and calculations by linear theory,
which does not include thxkness effects, and by third-order piston theory and
a second-order solution due to Van Dyke for flow round an oscillating
twc-dimensional aerofoil, whxh do include these effects. The results are
reproduced in Fig. 39. There is little difference III the lift coefficient up
to an incidence of about 12 degrees, but there is a considerable error in the
prediction of the centre of pressure position by linear theory. Since the
centre of pressure position is an unportant flutter parameter (see, for example,
Ref. 62, Section 6.5~) it IS to be expected that the thxhess of an aerofoil
may have an important influence on its flutter behaviour at high Mach numbers.
The effect of thickness is shown by the theoretxal flutter boundaries
of Fig. 40, taken from Ref. 57, from which a comparison can be made between
those theories that take account of thickness and linear theory which does not.
For the particular value of bending-pitching frequency ratio the influence of
thictiess is destabilssing. The effect of thickness depends, to some extent,
on frequency ratlo and on the positions of the elastic axes and centre of
gravity positions - this 1s shown in Fig. 41(a)-(d), also from Ref. 57, but, in
general, for a&e the influence of thxkness 1s found to be
destabilising. Ch&lz56'gives similar results.
These theor txal predxtlons are supported by the experimental reSUltS
of Hanson59 and Young %I , which are shown in Fig. 42. The two sets of results are
plotted/