BEM solution of unsteady convection-diffusion type fluid flow problems

Fendoğlu, Hande
The time-dependent convection-diffusion-reaction (CDR) type equations with constant and variable convective coefficients are solved by using two different boundary element methods (BEM), namely dual reciprocity BEM (DRBEM) and domain BEM (DBEM), in the spatial discretization while an implicit backward finite difference scheme is used in time. In the applications of DRBEM and DBEM, the fundamental solutions of both CDR equation and the modified Helmholtz (mH) equation are made use of. This results in some leftover terms (e.g. time derivative of the unknown) in the equations; and consequently some leftover domain integrals after the weighting process of the differential equations with each aforementioned fundamental solutions. The treatment of these leftover domain integrals generates different BEM formulations. That is, the DRBEM arises following the transformation of these domain integrals into equivalent boundary integrals by using radial basis functions, while keeping these domain integrals and computing them numerically, produce the DBEM. The physical applications of the present techniques are mainly on the solutions of some fluid dynamics problems which are governed by time-dependent CDR type equations. In this respect, first the time-dependent magnetohydrodynamic (MHD) flow equations which are actually convection-diffusion type equations with constant convective coefficients, are solved in ducts with straight and perturbed walls of variable electrical conductivities in the presence of an inclined magnetic field. It is found that for MHD duct flow problems, the DBEM results are almost invariant to the use of the fundamental solutions of either convection-diffusion (CD) or mH equations, while DRBEM with the fundamental solution of CD equation gives reasonably good results. Both methods capture good the well-known MHD flow characteristics for increasing values of Hartmann number. Secondly, the problems governed by Navier-Stokes and/or energy equations are considered in order to extend the application of the present method to the non-linear CD type equations with variable convective coefficients. Thus, the DBEM with the fundamental solution of CD equation is employed for the solution of the benchmark problems of fluid dynamics and heat transfer such as lid-driven cavity, natural and MHD-natural convection flow in cavities and channels. It is observed that, the obtained numerical findings are quite compatible with the physics of the fluid flow and the temperature distribution for moderate values of Reynolds, Rayleigh and Hartmann numbers.