A Viscous Velocity Potential/Stream Function

Abstract

The motion of fluids presents interesting phenomena including flow separation, wakes, turbulence etc. The physics of these are enshrined in the continuity equation and the NSE. Therefore, their studies are important in mathematics and physics. They also have engineering applications. These studies can either be carried out experimentally, numerically, or theoretically. Theoretical studies using classical potential theory (CPT) have some gaps when compared to experiments. The present publication is part of a series introducing refined potential (RPT) that bridges these gaps. It leverages experimental observations, physical deductions and the match between CPT and experimentally observed flows in the theoretical development. It analytically imitates the numerical source/vortex panel method to describe how wall bounded eddies in a three-dimensional cylinder crossflow are linked to the detached wake eddies. Unlike discrete and arbitrary vortices/sources on the cylinder surface whose strengths are numerically determined in the panel method, the vortices/sources/sinks in RPT are mutually concentric and continuously distributed on the cylinder surface. Their strengths are analytically determined from CPT using physical deductions starting from Reynolds number dependence. This study results in the incompressible Kwasu function which is a Eulerian velocity potential/stream function that captures vorticity, boundary layer, shed wake vortices, three-dimensional effects, and turbulence. This Eulerian Kwasu function also theorizes streaklines. The Lagrangian form of the function is further exploited to obtain flow pathlines.