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The three main approaches in fluid dynamics are actual experiments, numerical simulations, and theoretical solutions. Numerical simulations and theoretical solutions are based on the continuity equation and Navier-Stokes equations (NSE) that govern experimental observations of fluid dynamics.Theoretical solutions can offer huge advantages over numerical solutions and experiments in the understanding of fluid flows and design. These advantages are in terms of cost and time consumption. However, theoretical solutions have been limited by the prized NSE problem that seeks a physically consistent solution than what classical potential theory (CPT) offers. Therefore, the current author refined CPT. He introduced refined potential theory (RPT) that provides a viscous potential/stream function as a physically consistent solution to the NSE problem. This function captures observable unsteady flow features including separation, wake, vortex shedding, compressibility, turbulence, and Reynolds-number-dependence. It appropriately combines the properties of a three-dimensional potential function that satisfy the inertia terms of NSE and the features of a stream function that satisfy the continuity equation, the viscous vorticity equation, and the viscous terms of NSE. RPT has been verified and validated against experimental and numerical results of incompressible unsteady sub-critical Reynolds number flows on stationary finite circular cylinder, sphere, and spheroid.

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The motion of a fluid around a circular cylinder presents interesting phenomena including flow separation, wake and turbulence. The physics of these are enshrined in the continuity equation and the Navier-Stokes Equations (NSE). Therefore, their studies are important in mathematics and physics. They also have engineering applications. These studies can either be carried out experimentally, computationally, or theoretically. Theoretical studies of a cylinder flow using classical potential flow theory (CPT) have some gaps when compared to experiments. Attempting to bridge these gaps, this article introduces refined potential flow theory (RPT) and employs it on a stationary circular cylinder incompressible crossflow at Reynolds number 3,900. It leverages experimental observations, physical deductions and some agreements between CPT and experiments in the theoretical development. This results in the incompressible Eulerian Kwasu function which is a quasi-irrotational stream function that satisfies the governing equations and boundary conditions. It captures vorticity, boundary layer, shed wake vortices, three-dimensional effects, and static unsteadiness. The Lagrangian form of the function is exploited for the flow pathlines that are used to incorporate dynamic unsteadiness. A gravity analogy is used to predict the separation, transition, and reattachment points. The analogy introduces the perifocal frame of fluid motion. The forces are obtained in this frame with a change of variable. The drag prediction is within the error bound of measured data. The RPT pressure distribution, separation point and Strouhal number are also within acceptable ranges. Energy spectra analyses of the wake velocity display Kolmogorov’s Five-Thirds law of homogeneous isotropic turbulence.

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.