Two-Dimensional Irrotational Transonic Flows of a Compressible Fluid Page: 4 of 93
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2 NACA TN No. 1445
in part II. It was found that for finite Mach number the only case in
which the nature of the singularity of the incompressible solution can
remain unchanged is for a ratio of specific heats equal to -1. Part III,
which contains a discussion of the improvement of convergence of the
power series, contains no essentially new material and is included primarily
for the sake of completeness. Detailed proofs are given in appendixes A
Part IV contains the study of two particular flows, one having a
finite circulation and the other having zero circulation. Both are derived
frcm the incompressible flow about an elliptic cylinder of thickness
ratio 0.60. The free-stream Mach number for both cases is taken to be
0.60 in order to avoid the appearance of limiting lines. The pressure
distribution for the flow without circulation has been compared with that
of incompressible flow over approximately the same body. The discrepancies
between the exact results and those predicted by the approximate
Von KArmAn-Tsien and Glauert-Prandtl formulas are so wide as to show
definitely that in this case the effect of geometry cannot be ignored, as
is done in both approximate formulas. In general, it seems that the
effect of geometry cannot be neglected, and the conventional "pressure-
correction" formulas are not valid, even in the subsonic region if the
body is thick, especially if there is a supersonic region in the flow.
The importance of this result cannot be overemphasized, as there is a
widespread tendency in engineering practice to use simple pressure-
correction formulas indiscriminately.
This work was conducted at the Guggenheim Aeronautical Laboratory of
the California Institite of Technology under the sponsorship and with the
financial assistance of the National Advisory Ccmittee for Aeronautics.
I - CONCFP2 AND METHODS
General Consideration of Transonic Flows
The flow of a compressible ideal fluid about an infinite cylindrical
body, unlike that of an incapressible fluid, depends on, among other
conditions, the speed or Mach number at infinity. If the free-stream
Mach number is below a certain value, the flow pattern will be very
similar to that of an incompressible fluid even though part of the flow
may be supersonic. However, as soon as the limiting Mach number is
reached, the situation is entirely different. The phenomenon of major
importance in this new situation is the appearance in the calculations
of limiting lines in the supersonic region, characterized by the fact
that the fluid particles there experience an infinite pressure gradient.
It can be shown that if the assumptions of isotropy and of irrotationality
of the flow are not rejected, it is impossible to continue the solution
beyond these singular lines (reference 1). The failure of potential flow
can be attributed to the effects of viscosity and conductivity of the
fluid. Although the exact relation between the limiting line and shock
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Kuo, Yung-Huai. Two-Dimensional Irrotational Transonic Flows of a Compressible Fluid, report, June 1948; Washington D.C.. (https://digital.library.unt.edu/ark:/67531/metadc172480/m1/4/: accessed April 25, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.