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

An analysis is presented for obtaining approximate solutions of the
similarity eQuations for three-dimensional laminaréboundary-layer flows
over a flat surface under main-flow streamlines that are translates and
representable as infinite series expansions. For the particular case of
streamline shapes described by a power of the distance along the surface
from the leading edge, relativeLy simple expressions are Obtained for
flow deflection at the boundary surface, limiting streamline shape, and
shear stress at the surface.

In recent years, a great deal of attention has been focused on theo-
retical investigations of three-dimensional incompressible boundary—layer
flows. One class of investigations, having application to internal flow
problems in turbomachines and flow over wings at high altitudes, has been
concerned with finding exact solutions of the boundary-layer equations
when the boundary layer develops over a flat surface (e.g., refs. 1 to 8).
To date, all exact solutions have been based on similarity-type boundary-
layer analyses. The essence of this technique involves the reduction of
the partial differential equations for the boundary layer to a system of
ordinary differential equations. The solutions for the actual boundary-
layer flow are then obtained from the solutions of the ordinary differ-
ential equations. It is generally necessary, however, to employ numerical
methods and high-speed computing equipment to Obtain accurate solutions
to the ordinary differential equations because of their complex nature.

As this process is time-consuming and often laborious, it is of interest
to determine whether or not approximate solutions to such equations might
be readily obtained which would encompass a wide variety of boundary-
layer flows.