Analysis of two- and three-dimensional flow separation

Abstract

Prandtl (1904) showed that streamlines in a steady flow past a two-dimensional streamlined body separate from the boundary where the skin friction (or wall shear) vanishes and admits a negative gradient. Although commonly thought otherwise, these separation conditions are purely kinematic: They can be derived for any two-dimensional steady vector field that conserves mass (see, e.g. Shariff, Pulliam, and Ottino 1991). Haller (2002) managed to extend the Lagrangian separation theory to compressible two-dimensional velocity fields with general time dependence. Specifically, he defines unsteady flow separation as a material instability induced by an unstable manifold of a distinguished boundary point. In this general context, the unstable manifold is a time-dependent material line that shrinks to the separation point in backward time. In forward time, the unstable manifold attracts and ejects particles from a vicinity of the boundary. Using the above Lagrangian definition, the above kinematic separation theory renders mathematically exact Eulerian criteria for the location of time-dependent unstable manifolds. The theory only assumes local mass conservation and regularity for the unsteady velocity field. After recalling the main points of Haller's theory, we apply it to a specific model: a two-dimensional pitching airfoil. We first analyze the flow around the airfoil, and show how, under certain conditions, separation happens on the upper part of this airfoil. Next we consider the unsteady flow conditions, and determine the shape of the separation profile emanating from the wing. At that point, we also outline a new approach to the control of separation. In the second part of this thesis, we extend Haller's two-dimensional separation theory to

(cont.) three-dimensional flows, treating the case of open and closed separation separately. Next, we use a method developed by Perry and Chong (1986) to derive expansions of the Navier-Stokes equations that we use as models of three-dimensional separation. We verify our theory on those models. Finally we discuss new results on genuinely three-dimensional aspects of flow separation: open and closed separation, separation lines and separation surfaces.