A new transformation free generalized (5,5)HOC discretization of transient Navier-Stokes/Boussinesq equations on nonuniform grids

Abstract

This work pertains to the implicit high order compact discretization of the Navier-Stokes (N-S) equations on nonuniform grid. Subsequently, the discretization is used to approximate the Boussinesq equation as well. Contrary to earlier works on nonuniform grids this newly developed scheme is based on a comparatively smaller five-point stencil and leads to an algebraic system of equations with constant coefficients. The scheme carries the flow variable and its gradients as unknown and is seen to report back truncation accuracy of order four for linear flow problems even in a nonuniform mesh. Temporally the scheme is second-order accurate. Both primitive and vorticity-streamfunction formulations have been successfully tackled using the proposed formulation. Verification and validation studies were carried out to establish the efficiency of the formulation in conjunction with both Dirichlet and Neumann boundary conditions. Simulation of interior and exterior flow problems near-critical Hopf bifurcation points using a comparatively lesser number of grid points helps establish the robustness of the scheme. The numerical solution obtained by solving the Boussinesq equation for the problem of natural convection reveals the wider applicability of the scheme involving heat transfer.