An algebraic factorisation scheme for spectral element solution of incompressible flow and scalar transport

A spectral element algorithm for solution of the unsteady incompressible Navier–Stokes and scalar (species/heat) transport equations is developed using the algebraic factorisation scheme. The new algorithm utilises Nth order Gauss–Lobatto–Legendre points for velocity and the scalar, while (N-2)th order Gauss–Legendre points are used for pressure. As a result, the algorithm does not require inter-element continuity for pressure and pressure boundary conditions on solid surfaces. Implementations of the algorithm are performed for conforming and non-conforming grids. The latter is accomplished using both the point-wise matching and integral projection methods, and applied for grids with both polynomial and geometric non-conformities. Code validation cases include the unsteady scalar convection equation, and Kovasznay flow in two- and three-dimensional domains. Using cases with analytical solutions, the algorithm is shown to achieve spectral accuracy in space and second-order accuracy in time. The results for the Boussinesq approximation for buoyancy-driven flows, and the species mixing in a continuous flow micro-mixer are also included as examples of applications that require long-time integration of the scalar transport equations.