eⁿ Transition Prediction in Three-Dimensional Boundary Layers on Inclined Prolate Spheroids

Abstract

The capability of the eⁿ method to predict transition in complex three-dimensional, laminar boundary layers is demonstrated. Flows around inclined prolate spheroids are ideally suited for investigation because they exhibit highly divergent and convergent three-dimensional viscous flows on curved surfaces. The invicid flowfield is described by the theory and the viscous layer by a three-dimensional laminar boundary-layer method. This approach is applied in weak viscous/inviscid interaction regions only, that is, in attached laminar flow regions, and validated by comparison with measured wall pressure and skin friction in magnitude and direction. For the determination of the transition location, the laminar boundary layer is analyzed by the two N factor eⁿ method. Excited Tollmien-Schlichting waves of constant frequencies and stationary crossflow waves of constant wavelength are computed by the local, linear stability theory. Both N factor integrations are executed along 21 streamlines, which regularly cover the surface of the prolate spheroid. First, the values of both N factors at the measured transition locations are calculated, which deliver the stability limit of the prolate spheroid in the considered wind tunnel. Second, based on the knowledge of the stability limit, the transition locations are evaluated. The prolate spheroid with an aspect ratio of 6:1 was tested in two wind tunnels at angles of attack from 0 to 30 deg and Reynolds numbers from 1.5 x 10⁶ to 43 x 10⁶. The measured transition locations compare remarkably well with computations for all investigated flow cases.