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 differs 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 suffer 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 efficiency 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:{ ut + (u · ∇)u+∇p = 1Re4u ∇ · u = 0 (1.1) on Ω, where u = (u, v) is the velocity and p is the pressure, with the simplest physical boundary condition: u = 0 (1.2) at Γ = ∂Ω. By taking the divergence of the momentum equation one can obtain a Poisson equation for pressure: −4p = ∇ · (u · ∇u) (1.3) This can be thought of as a replacement for the incompressibility condition. The task of solving (1.1) would become much easier if we could attach a simple boundary condition to this equation. Unfortunately, the most natural candidate obtained from extending (1.1) to the boundary