A Closed-Form Analytical Solution to the Complete Three-Dimensional Unsteady Compressible Navier-Stokes Equation

Abstract

The three main approaches to exploring 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 embarked on a doctoral research on the refinement of CPT. He introduced the Refined Potential Theory (RPT) that provides the Kwasu function as a physically consistent solution to the NSE problem. The Kwasu function is a viscous scalar potential function that captures known and observable unsteady features of experimentally observed wall bounded flows including flow separation, wake formation, vortex shedding, compressibility effects, turbulence and Reynolds-number-dependence. It is appropriately defined to combine the properties of a three-dimensional potential function to satisfy the inertia terms of the NSE and the features of a stream function to satisfy the continuity equation, the viscous vorticity equation and the viscous terms of the 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. It is concluded that the Kwasu function is a physically consistent and closed-form analytical solution to the NSE problem.