| Περίληψη: | The development and application of the high-order accurate finite element Discontinuous Galerkin method (DG) for the numerical solution of three-dimensional compressible fluid flows, electromagnetic fields, and the flow of fully or partially
ionized gases under the influence of electromagnetic fields is pursued. The DG method is used for the spatial discretization of the Euler/Navier-Stokes and the full Maxwell’s equations, and subsequently, the gas-dynamics and electrodynamics equations are coupled through the source terms for the solution of plasma flows. As opposed to the classic magnetohydrodynamic theory (MHD), where the displacement current is ignored, the proposed coupling approach allows imposed time-varying electrical fields in the solution domain and the computation of the induced electrical fields. The discrete coupled system of equations is solved
numerically first for fully ionized gas flows to validate the method and then for partially ionized gas flows. The unsteady Euler and Navier-Stokes equations are solved for subsonic, transonic and supersonic flows at high Mach numbers, using second and higher-order solution approximations. For advancing the solution in time, high-order explicit and diagonally implicit Runge-Kutta (RK ) schemes are used. New approaches well-suited for shock and contact discontinuity capturing in three-dimensional
flows and mixed type meshes have been developed. Elements in the physical space are transformed into the standard cubical element of the computational space, where the TVB limiting procedure is performed for all the element types in a unified manner. Better resolution of the flow structures of interest was achieved by using h- and p- adaptive techniques for the local adaptive mesh refinement and increase of solution approximation, respectively. It was found that the use of the proposed TVB limiter in conjunction with the adaptive mesh refinement at regions with solution discontinuities constitutes a quite efficient and highly accurate approach to solve three-dimensional compressible flows at high Mach numbers. In the same spirit a dissipative filter for discontinuity capturing has been developed.Validation of the discretization method and shock-capturing approaches has been carried out.
The full Maxwell’s equations are also solved numerically with the DG method. Three-dimensional divergence-free vector bases were constructed and tested for approximating the magnetic vector field in order to satisfy the divergence constraint for the magnetic field. It was verified that the divergence-free bases ensure divergence-free solution in the interior of the elements whereas the solution jumps accross the element interfaces cause errors to the divergence-free condition of the magnetic field that diminish with the order of expansion. Furthermore, the perfectly hyperbolic Maxwell’s equations formulation was employed to ensure preservation of the constraints for the magnetic and electric field. In addition, the so-called Perfectly Matched Layers (PML), widely used in the finite differences, have been adapted to the DG framework for the representation of the radiation
condition and elimination of reflections at the boundaries of the solution domain. Validation of the method for the electromagnetic field has been carried out.
These developments led to the fulfillment of the final aim of this thesis in achieving high-order accurate, three-dimensional simulations with the DG method of fully or partially ionized gas flows under the influence of electromagnetic fields. It was found that the coupled system including nonlinear and stiff source terms of the gas-dynamics/electrodynamics equations must be advanced simultaneously in an implicit fashion. Implicit time marching is used for the fully coupled system to avoid wrong wave shapes and propagation speeds that are obtained when the coupling source terms are lagged in time or by using splitting iterative schemes. Validation of fully ionized gas flow simulations was carried out, using benchmark problems. A two-temperature model for the partially ionized gas flows consisting of three different species (electrons, positive ions and neutral particles) was used. All the required transport properties for such flows were computed by the use of relations of statistical mechanics. It was found that the partially ionized gas formulation converges to the fully ionized gas formulation as the degree of ionization increases. Application of the partially ionized gas formulation is shown for control of supersonic flow shock standoff distance with electromagnetic field.
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