Separation of three-dimensional laminar boundary layers on a prolate spheroid

Abstract

The laminar flow around a prolate spheroid at 6° angle of attack has been determined by the numerical solution of steady, three-dimensional boundary-layer equations with the external-pressure distribution obtained from an analytic solution of the inviscid-flow equations. The flow is shown to comprise a region of positive crossflow, followed by a substantial region of negative crossflow, a separation line and two terminal lines beyond which solutions of the boundary-layer equations could not be obtained. The separation line defines one boundary of a region of open separation and accords with the argument of Lighthill in that separation of three-dimensional boundary-layer flows is defined by a skin-friction line. A procedure is described that permits the identification of this skin-friction line and requires that it passes through the first location at which the longitudinal component of the wall shear is zero and the circumferential component negative. The numerical tests show that the finite-difference scheme based on the characteristic box allows calculations against the circumferential flow and with an accuracy equal to that of the regular box provided that a stability criterion is used to choose the grid intervals. This stability criterion is shown to be essential for accurate solutions in the vicinity of the separation and terminal lines and implies the need for extremely fine grids. It is evident that similar numerical constraints will apply to calculations performed with an interactive boundary-layer procedure or with higher-order forms of the Navier-Stokes equations.