New Finite Volume Method for rotating channel flows involving boundary layers

Abstract

We investigate in this article the boundary layers appearing in a fluid under moderate rotation when the viscosity is small. The fluid is modeled by the time-dependent rotating Stokes equations also known as the Stokes---Coriolis equations. The equations are considered in an infinite channel with periodicity on the lateral boundary and Dirichlet boundary conditions on the top and bottom of the channel. First, we analytically derive the correctors which describe the sharp variations at large Reynolds number (i.e. small viscosity). Second, thanks to a modified finite volume method (MFVM) we give the numerical solutions of the Stokes---Coriolis system at small viscosity ($$10^{-3}$$10-3---$$10^{-10}$$10-10). We follow the common idea which consists of adding the corrector functions to the Galerkin basis or its analogous for the classical Finite Volume Method, see Gie et al. (Discrete Contin Dyn Syst 36(5):2521---2583, 2016), Gie and Temam (Int J Numer Anal Model 12(3):536---566, 2015), Shih and Bruce (SIAM J Math Anal 18(5):1467---1511, 1987). The MFVM introduced here can be applied to a large class of singular perturbation problems.