Geometrical interpretations of continuous and complex-lamellar steady flows

Abstract

Three dimensional incompressible steady flows were investigated from a kinematical perspective. It is shown that their geometry, as defined by their streamline curvature, is directly related to the Curl of the associated unit tangent vector t. This relationship reveals that both the complex-lamellar and the continuity flow conditions impose geometrical constraints. These simplify flow analysis considerably, enabling a pure geometric representation of the underlying physics. Specifically, rather than using the traditional physical variables of vorticity, ∇×v and divergence ∇⋅v, for the flow description, the geometric related variables ∇×t and ∇⋅t, representing the streamline curvature KS and the mean curvature of the normal congruence Hnc, respectively, were considered. The systematic implementation of these mathematical findings leads to the appearance of an interesting equation for the flow velocity, as a function of Hnc. Based on this, the concept of “Geometric Lensing” has been introduced. According to this concept, the normal congruence spreads or focuses the flow streamlines through its curvature, thus altering their density and ultimately flow velocity. Geometric Lensing transforms the physical problem of finding the velocity distribution to a purely geometrical one and provides an intuitive explanation of the relieving effect that three dimensional flows exhibit. Finally, the inherent relationship between physical and geometric quantities of irrotational flows is explored. The Geometric Potential Theory, originally developed for planar flows, has been extended in three dimensions. The theoretical findings could provide useful post-processing tools for both experimentalists and CFD engineers, as well as for researchers interested in scientific visualization.