It is widely accepted that the accuracy and efficiency of computational fluid dynamics (CFD) simulations is heavily influenced by the quality of the mesh upon which the solution is computed. Unfortunately, the computational tools available for assessing mesh quality remain rather limited. This report describes a methodology for rigorously investigating the interaction between a flow solver and a variety of mesh configurations for the purposes of deducing which mesh properties produce the best results from the solver. The techniques described herein permit a more detailed exploration of what constitutes a quality mesh in the context of a given solver and a desired flow regime. In the present work, these newly developed tools are used to investigate mesh quality as it pertains to a high-order accurate discontinuous Galerkin solver when it is used to compute inviscid and high-Reynolds number flows in domains possessing smoothly curving boundaries. For this purpose, two flow models have been generated and used to conduct parametric studies of mesh configurations involving curved elements. The results of these studies allow us to make some observations regarding mesh quality when the discontinuous Galerkin method is used to solve these types of problems. Briefly, we have found that for inviscid problems, the mesh elements used to resolve curved boundaries should be at least third order accurate. For viscous problems, the domain boundaries must be approximated by mesh elements that are of the same order as the polynomial approximation of the solution if the theoretical order of accuracy of the scheme is to be maintained. Increasing the accuracy of the boundary elements to at least one order higher than the solution approximation typically results in a noticeable improvement in the computed error norms. It is also noted that C1-continuity of the mesh is not required at element interfaces along the boundary.