Fully compact higher-order computation of steady-state natural convection in a square cavity

Abstract

The flow in a thermally driven square cavity with adiabatic top and bottom walls and differentially heated vertical walls for a wide range of Rayleigh numbers ${(10\mathrm{}}^3<~\mathrm{Ra}<~{10}^7)$ has been computed with a fourth-order accurate higher-order compact scheme, which was used earlier only for the stream-function vorticity (ψ-ω) form of the two-dimensional steady-state Navier-Stokes equations. The boundary conditions used are also compact and of identical accuracy. In particular, a compact fourth-order accurate Neumann boundary condition has been developed for temperature at the adiabatic walls. The treatment of the derivative source term is also compact and has been done in such a way as to give fourth-order accuracy and easy assimilation with the solution procedure. As the discretization for the ψ-ω formulation, boundary conditions, and source term treatment are all fourth-order accurate, highly accurate solutions are obtained on relatively coarser grids. Unlike other compact solution procedure in literature for this physical configuration, the present method is fully compact and fully higher-order accurate. Also, use of conjugate gradient and hybrid biconjugate gradient stabilized algorithms to solve the symmetric and nonsymmetric algebraic systems at every outer iteration, ensures good convergence behavior of the method even at higher Rayleigh numbers.