A Spectral Collocation Method for 2D Incompressible Fluids in Vorticity Formulation

A spectral collocation method for viscous incompressible flow in a bounded 2D domain is presented for the vorticity-stream function formulation of the incompressible Navier-Stokes equations, along with its extension to the Boussinesq system. The no-slip boundary condition for velocity is converted into a local boundary formula for the vorticity, which when used in conjunction with an explicit time stepping scheme allows decoupling the computation of the vorticity and stream function time updates. Numerical results are presented for the singular lid-driven caivity problem, a benchmark differentially heated cavity problem, and a Rayleigh-Bernard convection problem for Rayleigh number up to $10^{10}$, demonstrating the efficiency of the method, and in particular that it is well suited for high Reynolds or high Rayleigh number regime simulations.