Abstract

This dissertation concerns the development of a loosely coupled, finite element method for the numerical simulation of 2-D hypersonic, thermo-chemical equilibrium and nonequilibrium flows, with an emphasis on resolving directional flow features, such as shocks, by an anisotropic mesh adaptation procedure. Since the flow field of such problems is chemically reacting and molecular species are vibrationally excited, numerical analyses based on an ideal gas assumption result in inaccurate if not erroneous solutions. Instead, hypersonic flows must be computed by solving the gasdynamic equations in conjunction with species transport and vibrational energy equations. The number of species transport equations could be very high but is drastically reduced by neglecting the ionization, thus leaving one to represent the air by only five neutral species: O, N, NO, O$\sb2$ and N$\sb2.$ This system of equations is further simplified by considering an algebraic equation for conservation of the fixed nitrogen to oxygen ratio in air. The chemical source terms are computed according to kinetic models, with reaction rate coefficients given by Park's reaction models. All molecular species are characterized by a single vibrational temperature, yielding the well-known two-temperature thermal model which requires the solution of a single conservation equation for the total vibrational energy. In this thesis, the governing equations are decoupled into three systems of PDEs--gasdynamic, chemical and vibrational systems--which are integrated by an implicit time-marching technique and discretized in space by a Galerkin-finite element method. This loosely-coupled formulation maintains the robustness of implicit techniques, while keeping the memory requirements to a manageable level. It also allows each system of PDEs to be integrated by the most appropriate algorithm to achieve the best global convergence. This particular feature makes a partially-decoupled formulation attractive for the extension of existing gasdynamic codes to hypersonic nonequilibrium flow problems, as well as for other applications having stiff source terms. The hypersonic shocks are resolved in a cost-effective manner by coupling the flow solver to a directionally mesh adaptive scheme using an edge-based error estimate and an efficient mesh movement strategy. The accuracy of the numerical solution is continuously evaluated using a bound available from finite element theory. The Hessian (matrix of second derivatives) of a selected variable is numerically computed and then modified by taking the absolute value of its eigenvalues to finally produce a Riemannian metric. Using elementary differential geometry, the edge-based error estimate is thus defined as the length of the element edges in this Riemannian metric. This error is then equidistributed over the mesh edges by applying a mesh movement scheme made efficient by removing the usual constraints on grid orthogonality. The construction of an anisotropic mesh may thus be interpreted as seeking a uniform mesh in the defined metric. The overall methodology is validated on various relevant benchmarks, ranging from supersonic frozen flows to hypersonic thermo-chemical nonequilibrium flows, and the results are compared against experimental data and, when not possible, to other computational approaches.