Global Matrix-Free Finite-Element Scheme for Natural Convection in a Square Cavity with Step Blockage

Abstract

A global matrix-free finite-element scheme is proposed for the solution of two-dimensional Navier-Stokes equations in velocity–vorticity form. By including the boundary conditions of the field variables at the element level itself, the global assembly of matrices is completely eliminated, thus resulting in a significant saving in computer memory. Only the global vectors obtained as a result of matrix–vector products are assembled at the time of solution of the simultaneous equations, using a conjugate gradient iterative solver. The method is validated by solving a benchmark problem on natural convection in a square cavity. The incompressibility constraint of the flow field is satisfied by imposing the vorticity definition at the boundaries using a second-order-accurate Taylor series expansion scheme. Results obtained for natural convection in a square cavity for Rayleigh number 103 < Ra < 106 indicate excellent agreement with benchmark solutions. The capability of the method to treat complex flow geometries is demonstrated by extending the study to natural convection in a square cavity with an internal square step block positioned on the left wall. Comparison of the memory storage with the compact vector storage scheme is also discussed.