Accurate and efficient numerical algorithms for solving the three dimensional Navier Stokes equations with a generalized thermodynamic and chemistry model and a one equation turbulence model on structured and unstructured mesh topologies are presented. In the thermo-chemical modeling, particular attention is paid to the modeling of the chemical source terms, modeling of equilibrium thermodynamics, and the modeling of the non-equilibrium vibrational energy source terms. In this work, nonequilibrium thermo-chemical models are applied in the unstructured environment for the first time. A three-dimensional, second-order accurate k-exact reconstruction algorithm for the inviscid and viscous fluxes is presented. Several new methods for determining the stencil required for the inviscid and viscous k-exact reconstruction are discussed. A new simplified method for the computation of the viscous fluxes is also presented. Implementation of the one equation Spalart and Allmaras turbulence model is discussed. In particular, an new integral formulation is developed for this model which allows flux splitting to be applied to the resulting convective flux. Solutions for several test cases are presented to verify the solution algorithms discussed. For the thermo-chemical modeling, inviscid solutions to the three dimensional Aeroassist Flight Experiment vehicle and viscous solutions for the axi-symmetric Ram-II C are presented and compared to experimental data and/or published results. For the hypersonic AFE and Ram-II C solutions, focus is placed on the effects of the chemistry model in flows where ionization and dissociation are dominant characteristics of the flow field. Laminar and turbulent solutions over a flat plate are presented and compared to exact solutions and experimental data. Three dimensional higher order solutions using the k-exact reconstruction technique are presented for an analytic forebody. / Ph. D.