On the flow of a compressible fluid by the hodograph method 1: unification and extension of present-day results Page: 1 of 24
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REPORT No. 789
ON THE FLOW OF A COMPRESSIBLE FLUID BY THE HODOGRAPH METHOD
I-UNIFICATION AND EXTENSION OF PRESENT-DAY RESULTS
By I. E. GRRICK and CARL KAPLANSUMMARY
Elementary basic solutions of the equations of motion of a
compressible fluid in the hodograph variables are developed and
used to provide a basis for comparison, in the form of velocity
correction formulas, of corresponding compressible and incom-
pressible flows. The known approximate results of Chaplygin,
von Kdrmdn and Tsien, Temple and Yarwood, and Prandtl
and Clanert are unified by means of the analysis of the
present paper. Two new types of approximations, obtained
from the basic solutions, are introduced; they possess certain
desirable features of the other approximations and appear
preferable as a basis for extrapolation into the range of high
stream Mach numbers and large disturbances to the main
stream. Tables and figures giving velocity and pressure-
coeficient correction factors are included in order to facilitate
the practical application of the results.
INTRODUCTION
The present paper is concerned with a theoretical study
of the hydrodynamical equations of a perfect compressible
fluid in two dimensions, in which the so-called hodograph
variables are used as the independent variables. It is hoped
to achieve herein a unification of the present-day results
obtained in this field and also to provide a working basis for
further developments. The earliest contributors to the
hodograph method for treating compressible fluids were
Molenbroek (reference 1) and Chaplygin (reference 2). The
remarkable work of Chaplygin on gas jets appeared in
Russian in 1904 but remained relatively unnoticed. In
recent years contributions to the hodograph method have
been made chiefly by Demtchenko (reference 3), von Kirmin
(reference 4), Tsien (reference 5), Ringleb (reference 6), and
Temple and Yarwood (reference 7).
The chief reason, and perhaps the only reason, for pre-
ferring the hodograph variables to the physical plane co-
ordinates is that the equations of motion in the hodograph
variables are linear. This simplification is achieved, how-
ever, at the cost of more difficult boundary conditions and
at a loss of physical insight. The great simplification in the
mathematics due to linearity nevertheless makes it desirableto pursue this line of attack as long as it appears profitable
to do so.
The mathematics for handling the flow equations re-
ceived a substantial impetus by the work of Bers and
Gelbart (reference 8), who developed a new function theory
'analogous to ordinary analytic function theory. The
present paper utilizes the methods of this new function
theory to develop certain functions essential to the compres-
sible-flow problem. It is of historical interest that ideas
similar to those of Bers and Gelbart were explored by the
renowned mathematician Hilbert (reference 9) in the early
part of this century but do not appear to have been further
developed at the time.
The material to be treated is conveniently separated into
two parts. In part I, the present paper, basic particular
solutions of the hodograph flow equations are developed and
employed in unifying and extending the results obtained by
Chaplygin, von Kbrman, and Temple and Yarwood. The
results obtained in part I are of immediate practical applica-
tion and are given in the form of tables and graphs of velocity
and pressure-coefficient correction factors. In part II,
general particular solutions of the hodograph flow equations
are developed and discussed. The material in part II, it is
hoped, will lead to a method for handling the actual boundary
problem of the flow of a compressible fluid past a prescribed
body.
ANALYSIS
FLOW EQUATIONS OF AN INCOMPRESSIBLE FLUID
It is well known that the relations between the velocity
potential q and the stream function ' for the steady irrota-
tional two-dimensional motion of a perfect incompressible
fluid arebx by
by bx(1)
These equations are the Cauchy-Riemann equations and
therefore 0+i4 is an analytic function f(z) of the complex
variable z=x+iy.
283
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Garrick, I. E. & Kaplan, Carl. On the flow of a compressible fluid by the hodograph method 1: unification and extension of present-day results, report, January 12, 1944; (https://digital.library.unt.edu/ark:/67531/metadc60081/m1/1/: accessed May 5, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.