Eight UK Conference on Boundary Integral Methods, Leeds, United Kingdom, 4 - 05 July 2011, pp.65-72
Abstract: In this paper, the dual reciprocity boundary element method (DRBEM) is applied to solve the two-dimensional unsteady Navier-Stokes equations in a lid-driven square cavity, and natural convection flow equations in a cavity. Stream function-vorticity and temperature variables are used, and vorticity transport and energy equations are transformed to modified Helmholtz equations by utilizing forward difference with relaxation parameters for the time derivatives. The resulting modified Helmholtz equations are solved by DRBEM using the fundamental solution (1/2π)K0(x) whereas in the stream function Poisson's equation (1/2π)\ln(x) is made use of. This procedure eliminates the need of another time integration scheme in vorticity transport and energy equations, and has the advantage of using large time increments. The inhomogeneities are approximated by using coordinate functions f=1+r and f=r2ln(r) in the stream function and vorticity-energy equations, respectively, and the missing vorticity boundary conditions are also obtained with the help of coordinate matrix F. The solutions are obtained for Reynolds number up to 2000 and Rayleigh number values between 102 and 106 by using constant boundary elements. The solution procedure needs considerably small number of iterations and large time increments with suitable values of relaxation parameters which occur in the argument of Bessel function K0(x). The results are in good agreement with the results available in the literature.