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National Advisory Committee for Aeronautics, Technical Notes - On Possible Similarity Solutions for Three Dimensional Incompressible Laminar Boundary Layer - III - Similarity with Respect to  Stationary Polar Coordinates for Small Angle Variation

Approximate solutions are obtained describing mainstream flows con-
fined to regions of small angle variation over flat surfaces for three—
dimensional, laminar, incompressible, thin boundary-layer flows having
similarity with respect to stationary polar coordinate systems. The solu-
tions, summarized in a table, include accelerating or decelerating flows
and stagnation-point, spiral, or circular flows. An experimental compari-
son of limiting overturning at the wall showed good agreement for the
first 100 of turning of circular mainstream flow.

In addition to providing an insight into secondary—flow behavior
associated with laminar boundary-layer flows, the experimental investiga-
tions of references 1 to 3 demonstrate that the information thus obtained
for laminar flows can be used to interpret and to correlate flow measure-
ments taken in turbomachines at operational conditions. These experimental
investigations thereby provide an important link between applied turbo-
machine research and the similarity—type boundary—layer analyses developed
in references 4 to 12. The link is further strengthened by the combined
theoretical and experimental investigation of reference 15. In reference
13, boundary—layer similarity solutions are obtained for main flows con-
sisting of streamline translates (i.e., the entire streamline pattern can
be obtained by translating any particular streamline parallel to the
leading edge), and the theoretical predictions of boundary-layer over-
turning (more than mainstream turning) near the surface are in close
agreement with experimental results Obtained by tracing the boundary;
layer streamlines with smoke flow—visualization techniques.

Using a generalized_similarity variable n, reference 14 extends
these results analytically to obtain all possible flows with boundary
layers having classical similarity with respect to stationary rectangular
coordinates. 'The dimensionless boundary-layer velocity components in the
plane of the surface are assumed to have similarity with respect to their
respective coordinates. This similarity is expressed by means of two
suitably defined functions of the similarity variable. The boundary-layer
equations are then transformed to equations involving the mainstream flow
components, their derivatives, the similarity functions, and theirmderiva—
tives. All the mainstream flows are then determined for which the trans-
formed boundary-layer equations reduce to ordinary differential equations
in the similarity functions and their derivatives.