Duality based boundary conditions and dual consistent finite difference discretizations of the Navier-Stokes and Euler equations

In this paper we derive new far-field boundary conditions for the time-dependent Navier-Stokes and Euler equations in two space dimensions. The new boundary conditions are derived by simultaneously considering well-posedness of both the primal and dual problems. We moreover require that the boundary conditions for the primal and dual Navier-Stokes equations converge to well-posed boundary conditions for the primal and dual Euler equations.We perform computations with a high-order finite difference scheme on summation-by-parts form with the new boundary conditions imposed weakly by the simultaneous approximation term. We prove that the scheme is both energy stable and dual consistent and show numerically that both linear and non-linear integral functionals become superconvergent.