Exact Solutions to the Three-Dimensional Navier–Stokes Equations Using the Extended Beltrami Method

In a 1966 publication, Chi-Yi Wang used the streamfunction in concert with the vorticity equations to develop a methodology for obtaining exact solutions to the incompressible Navier–Stokes equations, now known as the extended Beltrami method. In Wang's approach, the vorticity is represented by the sum of a linear function of the streamfunction and an assumed auxiliary function, such that the vorticity equation can be reduced to a quasi-linear partial differential equation, and exact solutions are obtainable for many choices of the auxiliary function. In the present work, a natural extension of Wang's formulation to three-dimensional flows in arbitrary orthogonal curvilinear coordinates has been derived, wherein two auxiliary functions are formed at the outset, with the caveat that the pressure and velocity components may vary in two spatial dimensions. As is the case with two-dimensional extended Beltrami flows, exact solutions are only obtainable when the forms of the auxiliary functions are “simple enough” to render the governing equations solvable. To demonstrate the solutions which may be obtained using the extended formulation, the well-known Kovasznay flow is generalized to a three-dimensional flow. A unique solution in plane polar coordinates is found. An extension to the solution to Burgers vortex has been derived and discussed in the context of existing literature. Finally, a new 3D swirling flow solution which is the angular analogue to Kovasznay flow has been developed.