A Survey of Unsteady Hypersonic Flow Problems

- 14 -
In practice, equation (2.1) is replaced by the approximation given
by the leating terms in the binomial expansion of the bracket. Expressed in
non-dimensional terms, this expansion is
where c- =
P - pm
SYPe#dS
w
2 -iv0 = -, and 6 is the measure of
U6 maximum surface slope defined
above.
When equation (2.2) is used to calculate the lift distribution on an
aerofoil it shows the effects both of thickness and mean incidence. Fig. 39,
from Ref. 57, shows that the predicted thickness effect agrees well with that
found by experiment, and by the more exact theory of Van Byke.
Appendix IV shows that changes in lift distribution due to thickness and
mean incidence can have an important influence on the flutter of an aerofoil
section.
From its derivation, equation (2.1) only applies for isentrcpio
conditions, and, for a compr $fl sion, this requires that W/G CC 1 (or M6 1, Miles'5, Raymond
that
wave
16 and East 17 have suggested
piston theory be extended on a semi-empirical basis by substituting shockrelationships
for equation (2.1) for compression surfaces. The methods
suggested are based on the expressions for one-dimensional-shock flow in front
of a piston moving at a speed w > G. These expressions are:
1
where
P-RX?
- = p ,
PC.2
P
-=
PO3
1 + [(Y+~wYlcI
1 + r(Y-fwYlP
p = y(,)[~(,)+[(~y(,.j +I]&]
. . . (2.3)
Milesi suggests two ways in which these relationships could be used:
(1) Where the shock wave movement due to the unsteady disturbances is
small, equations (2.3) should be used to calculate the local flow conditions
due to the steady state surface slope, and the values of p and a from this
calculation should then be used in (2.1) with the unsteady disturbance w' to
calculate the unsteady pressure disturbance.
(PI/