Symmetry Plane Viscous Layer on a Sharp Cone

Abstract

Numerical solutions for the wind and lee symmetry planes on sharp cones at incidence are obtained for laminar boundary layers. Comparisons are made between solutions obtained with the similarity boundary-layer equations, the nonsimilar boundary-layer equations, and the nonsimilar boundary region equations. For moderate incidence angles, boundary-layer theory fails at the leeplane and a boundary region develops. For large incidence, after crossflow reversal has occurred, nonsimilar boundary-layer solutions are obtained at the leeplane but are in poor agreement with experimental data. The boundary region system provides a significant improvement. 16~18 § analyses. With approximations 1 and 2, it has been shownl that the symmetry plane equations uncouple from the interior system so that for weak viscous-inviscid interaction the wind- and leeplane flow histories are apparently independent of the boundary-layer behavior around the cone. Solutions for the leeplane boundary layer therefore can be obtained by direct integration of the uncoupled symmetry plane equations16 (ordinary differential equations), or, alternatively, by a wind- to-leeplane marching procedure715 using the partial dif- ferential system governing the three-dimensional similarity boundary layer. The windward boundary-layer solution, as obtained from the uncoupled system, serves as the initial con- dition for the marching integration. For group 3 problems this uncoupling is not possible as diffusion effects are important not only normal to the surface, as with boundary layers, but also normal to the symmetry or separation planes. These domains, where a boundary layer forms within a boundary layer, are nonsimilar and termed boundary regions. Solutions of the nonsimilar boundary region equations can be obtained only by a streamwise integration procedure. Boundary-layer profiles are not specified at either of the symmetry planes. For unseparated flows, i.e., small to moderate angles of in- cidence, the boundary-layer methods of categories 1 and 2 have failed to provide leeplane solutions for all but the smallest yaw angles. This difficulty is most apparent with the uncoupled symmetry plane studies, as the calculations have failed to converge, 1-6 but even with the complete boundary- layer integration starting at the windplane, anomalous leeplane behavior has been observed.713 Moreover, the similarity symmetry plane equations exhibit nonunique windward plane solutions for all incidence angles. This nonuniqueness prevails at the leeplane in the limited range where solutions have been obtained.