Finite surface discretization for incompressible Navier-Stokes equations and coupled conservation laws

Abstract

© 2020 Elsevier Inc. We present a new Finite Surface Discretization (FSD) aiming at the incompressible Navier-Stokes Equations (NSE) and other coupled conservation laws. This discretization defines the velocities as surfaced-averaged values living on the faces of the pressure volumes in which the mass is set to be conserved. Consequently, the calculation of the mass balance on these control volumes is exact which allows more accurate information to be kept in the velocity field and produces a very accurate prediction of the pressure in the next time step. The proposed discretization reduces the stencil size of the Poisson equation in the projection method compared to the finite volume and finite difference discretizations. Due to highly accurate mass conservation, the compact sixth-order approximation of FSD can be used with an explicit fourth-order pressure treatment. This property greatly reduces the cost and complexity of the implementation. We present the discrete evolution equation of the surface-averaged velocities together with the enforcement of mass-conservation and the solution procedure for the pressure. The approximation of the NSE under this new discretization uses a combination of finite-difference and finite-volume methods. The proposed method is validated using standard laminar test cases. We identify the conditions under which a fourth-order pressure treatment can support the sixth-order and eighth-order approximations of the convection term using Fourier analysis. The performance of the method is evaluated on turbulent channel flows up to friction Reynolds number of 950. The quantitative relationships between the accuracy of the solution and grid size are identified. We present two performance indices for comparison with other methods. At the error level of 1 per mille, the proposed method is 28-times faster than the classic second-order scheme.