An improved time-splitting method for simulating natural convection heat transfer in a square cavity by Legendre spectral element approximation

A Legendre spectral element formulation of an improved time-splitting method is developed for the natural convection heat transfer problem in a square cavity. The convection term, Stokes term and temperature diffusion term are considered separately. The standard Galerkin variation is applied for the convection and temperature diffusion term, and the least-squares variation is adopted to the Stokes term due to the inf-sup constraint. Therefore, assumptions about the intermediate velocity and pressure in the original time-splitting method is eliminated, and the final velocity is divergence free. The numerical test verifies the validation and the h−p convergence characteristics of this method. Besides, the natural convection heat transfer in a square cavity with the Rayleigh number ranging in 103-106 is simulated, and the numerical solutions reach an satisfactory agreement with existing both numerical and experimental results. Furthermore, the natural convection in a square cavity with four heat sources is investigated, depicting a further application of this method.