DRBEM applications in fluid dynamics problems and DQM solutions of hyperbolic equations

Pekmen, Bengisen
In this thesis, problems of fluid dynamics defined by the two-dimensional convection-diffusion type partial differential equations (PDEs) are solved using the dual reciprocity boundary element method (DRBEM). The terms other than the Laplacian are treated as inhomogeneous terms in the DRBEM application. Once the both sides are multiplied by the fundamental solution of Laplace equation, and then integrated over the domain, all the domain integrals are transformed to boundary integrals using the Green's identities. The inhomogeneous terms are approximated with radial basis functions, and the space derivatives in convective terms are easily handled by using the DRBEM coordinate matrix constructed from the radial basis functions. The discretization of the boundary is achieved with linear elements. For the solution of unsteady problems, first order Backward-Euler and third order Houbolt time integration schemes are used. The boundary only nature of DRBEM provides one to obtain the results in a small computational cost compared to the domain discretization methods. Incompressible fluid flow in cavities, natural and mixed convection flow in enclosures are simulated when the medium is porous or non-porous, and with or without magnetic effect. The numerical results are visualized for different non-dimensional physical parameters in terms of streamlines, isotherms, vorticity, induced magnetic field lines and current density contours. In the thesis, the differential quadrature method (DQM) is also used for solving especially problems defined by hyperbolic equations and nonlinear in nature. DQM is made use of discretizing both time and space domains, and the solution is obtained at one stroke or blockwise without the need of an iteration. The nonlinearities are handled using an iteration procedure. Accurate results are obtained using considerably small number of Gauss-Chebyshev-Lobatto discretization points at very small expense. Test problems include Klein-Gordon, sine-Gordon equations, hyperbolic telegraph equations, and viscous Burgers' equation.
Citation Formats
B. Pekmen, “DRBEM applications in fluid dynamics problems and DQM solutions of hyperbolic equations,” Ph.D. - Doctoral Program, Middle East Technical University, 2014.