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National Advisory Committee for Aeronautics, Technical Notes - Displacement Effect of a Three Dimensional Boundary Layer

A method is described for determining the "displacement surface"
of a known three—dimensional compressible boundary—layer flow in terms
of the mass—flow defects associated with the profiles of the two
velocity components parallel to the surface.

The height of the displacement surface above the body surface for
flow about a yawed infinite cylinder is shown to be equal to the height
characterizing the mass-flow defect of the chordwise velocity profile.
The displacement surface height is shown to differ, in general, from
that associated with the resultant mass-flow defect, even at stagnation
points of the secondary flow. Nmnerical values are found for the known
three~dimensional bmmdary—layer flow about a cone at a small angle of
the boundary layer established in the flow of a sligitly viscous
fluid about a body is normally considered an isolated region wherein
the effects of viscosity predominate, and outside of which the motion
of the fluid is governed by the laws of’nonviscous motion. For large
Reynolds number, the bomadary layer is assumed to be so thin that the
nonviscous portion of the flow occurs as though there were no boundary
layer. This assumption is strictly correct in the limit of infinite
Reynolds mnhher. For large but finite Reynolds numbers, the growth of
the boundary layer causes the stream to be deflected away from the body
surface.

This displacement effect of the boundary layer on the nonviscous
flow may properly be determined from the behavior of the boundary layer
itself, as established either by experiment or by solution of the
Prandtl boundary-layer equations for laminar flow.

It does not follow, however, that this revised outer flow may be
used. to improve the solution for the boundary layer, still using the
Prandtl equations. A new set of equations must be used for this pur—
pose taking into account the variation of pressure across the boundary
layer, which is neglected in the Prandtl equations. . (See Alden's iter-
ative solution for incompressible flat-plate flow, reference 1.)