An Analytical Solution to the Complete Navier-Stokes Equation

What is it about?

From the subsonic `incompressible' low Reynolds number flow over a finite circular cylinder to the transonic flow over transport aircraft with their wake vortices and beyond to the hypersonic high Reynolds number flow over an atmospheric re-entry vehicle, it is almost a fact of existence that the complete Navier-Stokes equations (NSE) which were formulated about two centuries ago with a yet unknown general analytical solution govern the physics of the complete spectrum of continuum fluid dynamics. However, it is known and easily verifiable that the NSE can lend themselves to a solution in the form of a viscous potential function which must exhibit the features of the flow around a finite circular cylinder. Therefore, as a work in-progress, an attempt is made in this paper to arrive at such a solution starting from the classical potential theory.

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Why is it important?

The attempted refinement of the classical potential theory of the flow over a circular cylinder is achieved by introducing a viscous sink-source-vortex sheet on the surface of the cylinder. These singularities introduced into the flow are modeled as concentric at every location. The vortices are modeled as variations of Lamb-Oseen, Batchelor and Burgers vortices and analytic expressions for their strengths and those of the sinks/sources are obtained from the classical theory. Preliminary results of the work show that the theory presented captures important features of a bluff body flow including flow separation, wake formation, vortex shedding as well as compressibilty effects. The condition at a viscous wall is shown to be transient from slip towards a complete no-slip for a steady freestream flow. It is the hope that the present theory will shed more light on the important phenomenon of turbulence in on-going work in which quantitative analysis of the theory is being carried out.

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