The Convective Instability of Boundary-layer Flows over Rotating Spheroids

The continuous development of spinning projectiles and other industrial applications has led to the need to understand the laminar boundary-layer flow and subsequent onset of transition over the general family of rotating spheroids. We begin by finding the laminar boundary-layer flow over a general spheroid. In particular, we distinguish between prolate and oblate spheroids and use an appropriate spheroidal coordinate system in each case. The laminar-flow equations are established for each family of spheroid rotating in otherwise still fluid. An eccentricity parameter e is used to distinguish particular bodies within the oblate or prolate families. In each case, setting e = 0 reduces the equations to those already established by Howarth [2] and Banks [4] for the rotating sphere. In this preliminary study the laminar-flow equations at each latitude are solved by extending the original series solutions due to Howarth and Banks for the rotating sphere. The laminar flows obtained are consistent with established results for the rotating sphere as e tends to zero, and tend to the von Kármán [5] solution for the rotating disk as the latitude is reduced close to the nose. Analyses of the convective instability are performed on the rotating prolate family. These extend the linear analyses previously published by Malik, Lingwood and Garrett & Peake [6–10] on related geometries. An investigation into the relative importance of type I (crossflow) and type II (streamline curvature) modes is also presented. At low latitudes increasing eccentricity has negligible effects on the stability characteristics of the ∗Address all correspondence to this author. flow. However as the latitude increases, eccentricity is seen to lower the upper (type I) branch of the neutral curve, reducing the region of instability.