A Survey of Unsteady Hypersonic Flow Problems

- 16 -
nsteady Newtonian theory is di ussed by Hayes and Probstein', by
and by Zsrtarian and Saurwein 3s . The theorv can be shown to anulv . . "
in the limits M +m, Y +I, when E +t, where E
:
is the density ratio
aoross the shook at the body surface. The theory assumes that there are no
interactions between fluid particles, that the only change in the velocity of a
particle impinging on a body surface takes place normal to the surface, and
that, after impact, the particle follows the surface of the body within an
infinitesimally thin shock layer. The pressure at a point at the surface of
the shock layer can be found directly from the change in the component of the
fluid momentum normal to the body surface, so that the pressure coefficient at
the surface of the shook layer is given by:
P-&Z
C e
P &@" = 2( "yqn]
where qn is the normal velocity of the body surface point arising from any
motion of the body, and any time-dependent distortion of the surface, and u,
is the component of the free-stream velocity normal to the surface. It is clear
from the assumptions in the theory that it can be applied only where the flow
impinges directly on the surface: it can give no information about surfaces
shielded from the flow and the pressures on such surfaces are assumed to be
negligible.
A fully rational theory'920 requires a calculation of the nressure
differences aoross the shock iayer necessary to account for the accelerations
of the fluid partioles following the body surface. This would involve an analysis
of the structure of the shock layer, and a consideration of the complications that
can occur in the behaviour of the layer. But, since corrections for these effects
wolld introduce very great complications, the theory, as it has been empirically
applied, assumes that the pressure at the body surface is the same as that at the
surface of the shock layer and, in this form, it is sometimes known as Newtonian
impact theory.
Under steady conditions the simple theory has been found to give
reasonably satisfactory results on convex surfaces where the Mach number is high
enough for the shook to be close to the surface, provided that the expression (2.6)
is factored to give the correct value of the Cp‘ at the stagnation point or the
leading edge of the body. The accuracy of impact theory in these circumstances
is, apparently, due in part to the oancelling of opposing errors: the pressure
behind the shook is higher than the impact theory value, but the pressure
difference across the shock.layer due to the centrifugal effect compensates for
this (Ref. 2).
Examples of the use of Newtonian impact theory for unsteady flow
analysis are to be found in Befs. 21, 22 and 23. In Ref. 21 Tobak and Wehrend
compare the results of impact theory analysis for a cone with first and second
order potential flow solutions and, for the stiffness derivative, with exact
values over the Mach number range from 3-O to 5-O. The impact theory values
appear to be a limit which the exact values approach with increasing Mach
number, but the cone is an especially favourable case for the application of
Newtonian theory. East'7 has shown that, in the limits M +co, y + 1, the
expressions from Newtonian impact theory and f-ran strong shook piston theory are
the same.