A Survey of Unsteady Hypersonic Flow Problems

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minimum are that its variations for small variations in the values of p, q
and $ should be zero. These conditions can be shown to lead to the equations
of continuity, irrotationality and momentum for the flow, so that the problem
of finding the flow by solving the equations of motion for the fluid can be
change& into that of finding a function for p which makes the integral in
equation (2.42) a minimum.
For isentropic conditions, p can be expressed in terms of the
velocity potential, 6, in the form
p = r, (y-l) ( f! + $o(y-')
. . . ( 2 . 4 3 )
Ibo L
where s, is the free-stream
can be written in the form
ta
PC.3 I
t; ‘J(t) ti lb (t) [
& Iat /-I
speed of sound. Using this expression, equation (2.42)
-!I$(;+ $$)~y-r)w a, (2.44)
and the problem becomes that of finding a function for $ such that the variation
in I is sero for small variations in +. This problem canbe solved in an
approximate manner by assuming a finite series for 4 which satisfies the boundary
conditions (including the known conditions at k and I+) and in which the
coefficients are determined by the condition that the variation in I for small
variations in each coefficient must be sero.
When this general method is applied to the case of flow around a
two-dimensional piston in Ref. 19 it is shown that the statement of the variational
principle must be modified slightly to take account of the fact that the conditions
at time te are not known. The modified statement has the form
A][(; dv at +~t2;pA~lt~t2~~ = 0 . . . (2.45)
+a
where A p dV dt is the small variation in the integral for a small variation
h u(t)
in P (or +), and A+ is the corresponding small variation in +. The variations
must be taken so that A$ = 0 at the outer wave from the piston, the boundary
conditions at the piston are unaltered, and at time b the disturbed volume is
zero or the flm is known everywhere and the variations are correspondingly
restrained. When both p and p are expressed in terms of the velocity potential
equation (2.45) can be written as:
where S(t) is the disturbed area in the two-dimensional problem.
. . . (2.46)
Zartari*n19/