A vorticity method for the solution of natural convection flows in enclosures

Abstract

Vorticity formulations for the incompressible Navier‐Stokes equations have certain advantages over primitive‐variable formulations including the fact that the number of equations to be solved is reduced through the elimination of the pressure variable, identical satisfaction of the incompressibility constraint and the continuity equation, and an implicitly higher‐order approximation of the velocity components. For the most part, vorticity methods have been used to solve exterior isothermal problems. In this research, a vorticity formulation is used to study the natural convection flows in differentially‐heated enclosures. The numerical algorithm is divided into three steps: two kinematic steps and one kinetic step. The kinematics are governed by the generalized Helmholtz decomposition (GHD) which is solved using a boundary element method (BEM) whereas the kinetics are governed by the vorticity equation which is solved using a finite element method (FEM). In the first kinematic step, vortex sheet strengths are determined from a novel Galerkin implementation of the GHD. These vortex sheet strengths are used to determine Neumann boundary conditions for the vorticity equation. (The thermal boundary conditions are already known.) In the second kinematic step, the interior velocity field is determined using the regular (non‐Galerkin) form of the GHD. This step, in a sense, linearizes the convective acceleration terms in both the vorticity and energy equations. In the third kinetic step, the coupled vorticity and energy equations are solved using a Galerkin FEM to determine the updated values of the vorticity and thermal fields. Two benchmark problems are considered to show the robustness and versatility of this formulation including natural convection in an 8×1 differentially‐heated enclosure at a near critical Rayleigh number.