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National Advisory Committee for Aeronautics, Technical Notes - Some Numerical Solutions of Similarity Equations for Three-Dimensional Laminar Incompressible Boundary-Layer Flows

Numerical solutions are presented for two classes of similarity
equations corresponding to a range of three—dimensional boundary—layer
flows. Equations for limiting—flow deflection and equations for the
calculation.of boundary-layer streamlines are also presented.

Recent years have seen an increase in research activity aimed at
understanding three-dimensional boundary-layer behavior. At present,
basic research is continuing in both the theoretical and experimental
phases of the problem. In the theoretical phase, one particular approach
has been to seek exact solutions of the laminar incompressible boundary-
layer equations as was done earlier for two-dimensional flows. The
search for exact solutions has been based on the so—called similarity
method in which the partial differential equations of the boundary layer
are reduced to a two—equation system of ordinary differential equations.
This research can in turn be divided into two categories. One category
consists of solutions of specific prdblems. The other consists of gen—
eral investigations for determining the circumstances under which simi—
larity solutions exist. Research in this last category (e.g., refs. 1
to 6) has generally been carried to the point of obtaining the similarity
equations but has not taken the next logical step, namely, solving the
systems. In no small measure, this is a consequence of the complexity
of the equations and the great variety of possible cases. Such a program
is probably best accomplished by numerical analyses on high—speed com—
puting equipment.

The value of carrying out a program for determining solutions of
systems of sindlarity equations is twofold. First, the solutions can
give qualitative and quantitative information on boundary—layer behavior
(see ref. 7). Secondly, the solutions provide a basis for developing so—
called approximate methods (e.g., momentum integral techniques). In view
of these considerations, certain specific systems of equations were
chosen from reference 1 and analyzed on high—speed computers. The sys—
tems chosen correspond to mainstream flows which were felt to be of
greatest practical interest from those given in reference 1.