Solutions of Buoyancy-driven Flow Problems by the Local Collocation Method That Uses Multiquadrics as the Radial Basis Function

Abstract

The local multiquadric collocation method is a meshless method that uses radial basis functionsknown as multiquadrics to approximate functions and their derivatives. In this paper, buoyancy-drivenflow problems in a square cavity and a horizontal concentric annulus are solved by this method. The streamfunction-vorticity formulation is used because there are only two unknowns in this formulation. However,since the vorticity boundary condition is required, but not explicitly given, it must be determined by usingthe definition of vorticity. A scheme for computing boundary vorticity that is appropriate to the localmultiquadric collocation method is presented. Results from the buoyancy-driven flow problem in a squarecavity show that the accuracy of solutions obtained by using this scheme is comparable with the accuracyof solutions obtained by using a more accurate scheme. Furthermore, it is also shown that numerical resultsof the buoyancy-driven flow problem in a horizontal concentric annulus for cases of D_o/D_i = 1.5 and 2.0, Pr= 0.7, and RaDi between 105 and 106 by the local multiquadric collocation method agree with experimentalresults