A high accuracy hybrid method for two-dimensional Navier–Stokes equations

A dual-mesh hybrid numerical method is proposed for high Reynolds and high Rayleigh number flows. The scheme is of high accuracy because of the use of a fourth-order finite-difference scheme for the time-dependent convection and diffusion equations on a non-uniform mesh and a fast Poisson solver DFPS2H based on the HODIE finite-difference scheme and algorithm HFFT [R.A. Boisvert, Fourth order accurate fast direct method for the Helmholtz equation, in: G. Birkhoff, A. Schoenstadt (Eds.), Elliptic Problem Solvers II, Academic Press, Orlando, FL, 1984, pp. 35–44] for the stream function equation on a uniform mesh. To combine the fast Poisson solver DFPS2H and the high-order upwind-biased finite-difference method on the two different meshes, Chebyshev polynomials have been used to transfer the data between the uniform and non-uniform meshes. Because of the adoption of a hybrid grid system, the proposed numerical model can handle the steep spatial gradients of the dependent variables by using very fine resolutions in the boundary layers at reasonable computational cost. The successful simulation of lid-driven cavity flows and differentially heated cavity flows demonstrates that the proposed numerical model is very stable and accurate within the range of applicability of the governing equations.