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Development of discontinuous galerkin method 2 dimensional flow solver

Güngör, Osman
In this work, 2 dimensional flow solutions of Euler equations are presented from the developed discontinuous Galerkin method finite element method (DGFEM) solver on unstructured grids. Euler equations govern the inviscid and adiabatic flows with a set of hyperbolic equations. The discretization of governing equations for DGFEM is given in detail. The DGFEM discretization provides high order solutions on an element-compact stencil hence only elements having common boundary are coupled. The required elementwise operations and mathematical operations are revisited and derivations are provided when necessary. Among the two major approaches, modal and nodal, nodal DGFEM is employed. Gaussian quadrature is utilized in the evaluation of volume and surface integrals. The flux through the cell boundaries are calculated through flux functions and several flux functions are implemented and compared. Proper boundary conditions are employed on the solution space boundaries. Severaltestcasesinliteratureareusedforverificationandvalidationpurposes. The high order accuracy is easily achieved in problems with smooth solutions. On the other hand, problems with shocks requires stabilization techniques which may limit the order of accuracy or degrade solution success. The satisfactory results are obtained with comparison of experimental results which are carefully selected considering the fidelity of governing equations. Moreover, importance of curved wall boundary representations in high order methods are experienced. Furthermore, effect of grid adaptation around shocks or discontinues is pointed out.