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National Advisory Committee for Aeronautics, Report - On a Solution of the Nonlinear Differential Equation for Transonic Flow past a Wave Shaped Wall

The Prandtl—Busemann small—perturbation method is utilized
to Obtain the flow of a compressible fluid. past an infinitely
long ware—shaped wall. When the essential assumption for
transonicflow (that all Jlach numbers in the region offlow are
nearly unity) is introduced, the expression for the reloeity poten-
tial takes the form of a power series in the transonic similarity
parameter. 0n the basis of this form of the solution, an attempt
is made to solee the nonlinear differential equation for transonic
flow past the wary wall. The analysis utilized exhibits clearly
the difiiculties inherent in nonlinearg‘low problems. The inves-
tigation has nevertheless been rigorously carried to the point
where the question of the existence or nonexistence of a mixed
potentialflowfree of discontinuities can be settled by the behavior
of a single power series in the transonie similarity parameter.
The calculation of the coeficients of this dominant power series
has been reduced to a routine computing problem by means of
recursion. formulas resulting from the solution of the difierential
equation and the boundary condition at the wall. One of the
interesting results of the analysis is the rigorous statement that
the transonic similarity parameter must be less than four-thirds.

The present report considers in detail a notion first ex-
pressed in reference 1, namely, that- the Von Harman transomo
similarity rule is implicitly contained in the potential-flow
solution for high—subsonic flow past a prescribed body. The
calculations are very much simplified by the choice of rec—
tangular Cartesian coordinates as independent variables. For
this reason the solid boundary chosen is a two-dimensional
wavy wall of small amplitude extending to infinity in
both the downstream and upstream directions. The problem
is first treated by means of the Prandtl-Busemann iteration
method for high—subsonic undisturbed speeds. On the basis of
this solution an attempt is made to solve the nonlinear differen—
tial equation for transonic flow (corresponding to the linear
Tricomi equation in the hodograph plane) past a wavy wall.