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National Advisory Committee for Aeronautics, Technical Notes - On Possible Similarity Solutions for Three Dimensional Incompressible Laminar Boundary Layers - II - Similarity with Respect to Stationary Polar Coordinates

Solutions of mainstream flow patterns for three-dimensional, laminar,
incompressible thin—boundary—layer flows (over flat or slightly curved
surfaces) having similarity with respect to stationary polar coordinates
in the plane of the surface are derived. The solutions are summarized in
a table.

Considerable attention has been devoted in laminar-boundary—layer
research to theoretical solutions of the two— and three-dimensional
incompressible—boundary—layer equations using the "similarity" approach.
In this method, the partial differential boundary-layer equations are
transformed by means of a similarity parameter n and rewritten in terms '
of functions of n, their derivatives, the mainstream velocity components,
and their derivatives. Solutions are then sought for the mainstream flow
conditions for which the transformed equations reduce to ordinary differ-
ential equations for the functions of n (refs. 1 to 10). Some experi-
mental evidence is presented in reference 10 in support of this kind of
theoretical development for laminar flows. Reference ll presents a sys-
tematic approach to similarity-type solutions using a generalized simi-
larity parameter. As a result, reference ll has Obtained solutions for
the permissible mainstream flows for all the boundary-layer flows having
classical similarity with respect to stationary rectangular coordinates.
The present report is an extension of the work of reference ll. Solutions
are sought for the mainstream flows in stationary cylindrical coordinates
for all the boundary—layer flows having classical similarity with_fespect
to the polar coordinates in the plane of the surface.