# Boundary element method solution of initial and boundary value problems in fluid dynamics and magnetohydrodynamics

2008
Bozkaya, Canan
In this thesis, the two-dimensional initial and boundary value problems invol\-ving convection and diffusion terms are solved using the boundary element method (BEM). The fundamental solution of steady magnetohydrodynamic (MHD) flow equations in the original coupled form which are convection-diffusion type is established in order to apply the BEM directly to these coupled equations with the most general form of wall conductivities. Thus, the solutions of MHD flow in rectangular ducts and in infinite regions with mixed boundary conditions are obtained for high values of Hartmann number, M. For the solution of transient convection-diffusion type equations the dual reciprocity boundary element method (DRBEM) in space is combined with the differential quadrature method (DQM) in time. The DRBEM is applied with the fundamental solution of Laplace equation treating all the other terms in the equation as nonhomogeneity. The use of DQM eliminates the need of iteration and very small time increments since it is unconditionally stable. Applications include unsteady MHD duct flow and elastodynamic problems. The transient Navier-Stokes equations which are nonlinear in nature are also solved with the DRBEM in space - DQM in time procedure iteratively in terms of stream function and vorticity. The procedure is applied to the lid-driven cavity flow for moderate values of Reynolds number. The natural convection cavity flow problem is also solved for high values of Rayleigh number when the energy equation is added.

# Suggestions

 Development of an incompressible navier-stokes solver with alternating cell direction implicit method on structured and unstructured quadrilateral grids Baş, Onur; Tuncer, İsmail Hakkı; Department of Aerospace Engineering (2007) In this research, the Alternating Cell Direction Implicit method is used in temporal discretisation of the incompressible Navier-Stokes equations and compared with the well known and widely used Point Gauss Seidel scheme on structured and quadrilateral unstructured meshes. A two dimensional, laminar and incompressible Navier-Stokes solver is developed for this purpose using the artificial compressibility formulation. The developed solver is used to obtain steady-state solutions with implicit time stepping m...
 Error analysis for the numerical evaluation of the diagonal forms of the scalar spherical addition theorem Koc, S; Song, JM; Chew, WC (Society for Industrial & Applied Mathematics (SIAM), 1999-04-29) The numerical solution of wave scattering from large objects or from a large cluster of scatterers requires excessive computational resources and it becomes necessary to use approximate-but fast-methods such as the fast multipole method; however, since these methods are only approximate, it is important to have an estimate for the error introduced in such calculations. An analysis of the error for the fast multipole method is presented and estimates for truncation and numerical integration errors are obtain...
 Numerical solution of nonlinear reaction-diffusion and wave equations Meral, Gülnihal; Tezer, Münevver; Department of Mathematics (2009) In this thesis, the two-dimensional initial and boundary value problems (IBVPs) and the one-dimensional Cauchy problems defined by the nonlinear reaction- diffusion and wave equations are numerically solved. The dual reciprocity boundary element method (DRBEM) is used to discretize the IBVPs defined by single and system of nonlinear reaction-diffusion equations and nonlinear wave equation, spatially. The advantage of DRBEM for the exterior regions is made use of for the latter problem. The differential quad...
 FINITE-ELEMENT METHOD FOR SOLVING MHD FLOW IN A RECTANGULAR DUCT Tezer, Münevver (Wiley, 1989-02-01) A finite element method is given to obtain the solution in terms of velocity and induced magnetic field for the steady MHD (magnetohydrodynamic) flow through a rectangular pipe having arbitrarily conducting walls. Linear and then quadratic approximations have been taken for both velocity and magnetic field for comparison and it is found that with the quadratic approximation it is possible to increase the conductivity and Hartmann number M (M ≤ 100). A special solution procedure has been used for the resulti...
 The dual reciprocity boundary element method solution of fluid flow problems Gümgüm, Sevin; Tezer, Münevver; Department of Scientific Computing (2010) In this thesis, the two-dimensional, transient, laminar flow of viscous and incompressible fluids is solved by using the dual reciprocity boundary element method (DRBEM). Natural convection and mixed convection flows are also solved with the addition of energy equation. Solutions of natural convection flow of nanofluids and micropolar fluids in enclosures are obtained for highly large values of Rayleigh number. The fundamental solution of Laplace equation is used for obtaining boundary element method (BEM) ...
Citation Formats
C. Bozkaya, “Boundary element method solution of initial and boundary value problems in fluid dynamics and magnetohydrodynamics,” Ph.D. - Doctoral Program, Middle East Technical University, 2008. 