<DOC>
<DOCNO>
1248
</DOCNO>
<TITLE>
an analytic extension of the shock-expansion method .
</TITLE>
<AUTHOR>
waldman,g.d. and probstein,r.f.
</AUTHOR>
<BIBLIO>
j. ae. scs. 1961, 119.
</BIBLIO>
<TEXT>
  the problem is considered of calculating approximately the
inviscid rotational flow field and pressure distribution about a
smooth two-dimensional airfoil with sharp leading and trailing
edges in a uniform supersonic or hypersonic stream .  the assumption
of a perfect gas is made, and the basic flow pattern for
the analysis is taken to be given by the simple isentropic shockexpansion
method with straight characteristics .  an elementary
characteristics treatment is discussed to show when the simple
shock-expansion method should be satisfactory for computing
the surface pressure distribution, and under what circumstances
it may be expected to break down .  by utilizing characteristic
variables the isentropic shock-expansion method is then formulated
analytically, and an analytic result is obtained for the shock
shape corresponding to this zero-order approximation .  in the
special case where hypersonic similitude is applicable, that is,
for slender bodies and high mach numbers, the shock-shape
expression for large distances is found to reduce to the result
previously given by mahony, which for weak shocks and slender
bodies in turn reduces to the simple-wave result first given by
friedrichs .
  employing the analytic form of the isentropic shock-expansion
method as a zero-order approximation, an analytically consistent
perturbation method is developed by expanding the dependent
flow variables in the exact partial differential equations in powers
of the reflection coefficient for simple waves interacting with an
oblique shock .  the scheme by its nature helps to define those
regions in which shock expansion can be used, in addition to
taking into account in a perturbation sense the factors neglected
in simple shock-expansion theory, namely, the curvature and
reflection of the mach waves and the correct boundary conditions
at the shock wave .  analytic solutions are obtained for
the first-order corrections, including the surface pressure distribution
.  the necessary numerical computation of the integrals
involved is considerably simpler than a direct application of the
method of characteristics .  to illustrate the method and its
accuracy, the zero-order shock shape and first-order pressure distribution
are calculated for a family of parabolic arc airfoils at
an infinite free-stream mach number .  these results are compared
with rotational characteristic solutions where available,
and the present method is found to be in excellent agreement .
</TEXT>
</DOC>
