High-fidelity numerical tools based on high-order Discontinuous-Galerkin (DG) methods and Lagrangian Coherent Structure (LCS) theory are developed and validated for the study of separated, vortex-dominated flows over complex geometry. The numerical framework couples prediction of separated turbulent flows using DG with time-dependent analysis of the flow through LCS and is intended for the development of separation control strategies for aerodynamic surfaces. The compressible viscous flow over a NACA 65-(1)412 airfoil is solved with a DG based Navier-Stokes solver in two and three dimensions. A method is presented in which high-order polynomial element edges adjacent to curved boundaries are matched to boundaries defined by non-smooth splines. Artificial surface roughness introduced by the piecewise-linear boundary approximation of straight-sided meshes results in the simulation of incorrect physics, including wake instabilities and spurious time-dependent modes. Spectral accuracy in the boundary approximation is not achieved for non-analytic boundary functions, particularly in high curvature regions. An algorithm is developed for the high-order computation of Finite-Time Lyapunov Exponent (FTLE) fields simultaneously and efficiently with two and three dimensional DG-based flow solvers. Fluid tracers are initialized at Gauss-Lobatto quadrature nodes within an element and form the high-order basis for a flow map at later time. Gradients of the flow map and FTLE are evaluated with DG operators. Multiple flow maps are determined from a single particle trace by remapping the flow map to the quadrature nodes on deformed mesh elements. For large integration times, excessive subdomain deformation deteriorates the interpolating conditioning. The conditioning provides information on the fluid deformation and identifies subdomains that contain LCS. An exponential filter smooths the flow map in highly deformed areas. The algorithm is tested on several benchmarks and is shown to have spectral convergence. The two and three-dimensional LCS field are analyzed for the unsteady flow over a NACA 65-(1)412 airfoil at a free-stream Reynolds number of Re=20,000 based on the chord length and a Mach number of 0.3. In two-dimensions, a Karman vortex street forms at the trailing edge. The three-dimensional vortex street breaks down to turbulence at the trailing edge.