Gauge Method for Viscous Incompressible Flows

Abstract

We present a new formulation of the incompressible Navier-Stokes equation in terms of an auxiliary field that diers from the velocity by a gauge transformation. The gauge freedom allows us to assign simple and specific boundary conditions for both the auxiliary field and the gauge field, thus eliminating the issue of pressure boundary condition in the usual primitive variable for- mulation. The resulting dynamic and kinematic equations can then be solved by standard methods for heat and Poisson equations. A normal mode analysis suggests that in contrast to the classi- cal projection method, the gauge method does not suer from the problem of numerical boundary layers. Thus the subtleties in the spatial discretization for the projection method are removed. Con- sequently, the projection step in the gauge method can be accomplished by standard Poisson solves. We demonstrate the eciency and accuracy of the gauge method by several numerical examples, including the flow past cylinder. 1. The gauge formulation In this paper, we introduce a new formulation of the incompressible Navier-Stokes equation and demonstrate that this new formulation is particularly suited for numer- ical purpose. We start with the classical formulation of the Navier-Stokes equation: ‰ u t + (u ¢ r)u + rp = 1 Re 4u