Studying the natural convection problem in a square cavity by a new vorticity-stream-function approach

In this study, a benchmark natural convection problem is studied under a Gay'Lussac type approximation incorporating centrifugal effects in the context of a new vorticity-stream-function approach. This approximation differs from the classic Boussinesq approximation in that density variations are considered in the advection term as well as the gravity term in the momentum equations. Such a treatment invokes Froude number as a non-Boussinesq parameter deviating results from the classic Boussinesq approximation. Numerical simulations of the natural convection in square cavity are performed up to Rα = 10⁶ and ε = 0.3 at Pr = 0.71 via proposed formulation and results are compared against the conventional Boussinesq approximation in terms of the average and local Nusselt number and entropy generation. Comparing results indicate that, up to Rα = 10⁵, mentioned approaches are showing almost identical performance, but as the Rayleigh number exceeds 10⁵, formed thermal boundary layer under Gay'Lussac type approximation is slightly thicker compared to the Boussinesq approximation accompanied by a stronger velocity gradient.