Eulerian-Lagrangian (EL) models are developed that account for stochasticity and randomness in tracers of inertial particles forced by a carrier flow phase. Central to the novelty of the models is a forcing formulation that uses a series expansion with random coefficients to account for epistemic and aleatoric uncertainties, in lieu of commonly used stochastic, random-walk processes.

Starting from randomly forced ordinary differential equations that govern the Lagrangian inertial point-particle tracer dynamics, Lagrangian cloud and Liouville models are derived. Both cloud and Liouville models are closed and are shown to more accurately and computationally efficiently predict the propagation of the forcing randomness into confidence intervals of the particle phase solution as compared to Monte Carlo sampling methods.

The closed and predictive particle cloud tracer models the mean motion and deformation of a cloud of inertial particles at a singular point in space and along its Lagrangian trajectory in time. The tracer builds upon the Subgrid Particle-Averaged Reynolds Stress Equivalent (SPARSE) formulation first introduced in Davis et al. (2017) for the tracing of particle clouds. Using a combination of the forcing models, averaging and a truncated Taylor series expansion to estimate the statistical correlations in the cloud region, the SPARSE model is closed and achieves a third convergence for the confidence interval with respect the number of samples.

The Liouville models are rigorously derived with the method of distributions and do not require truncation or ad-hoc assumptions. The deterministic PDF models are described by hyperbolic partial differential equations (PDEs). In Eulerian form, the PDEs are solved with grid-based spectral methods. To recover the Lagrangian character of the disperse phase, the method of characteristics is employed to derive a PDF formulation based on the computation of flow maps, circumventing difficulties of solving high-dimensional PDE equations. This formulation is local, does not require grid based methods nor sampling, and offers a complete statistical description. It is shown that the Liouville PDF models may generalize Langevin and Fokker-Planck descriptions of particle statistics to non-Gaussian noise of the random walk.

An inverse model to infer stochastic descriptions of particle forcings from noisy trajectory data using an adjoint formulation is also introduced using a point-particle approach.

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.

This dissertation is a comprehensive account of the low-Reynolds number (Re) flow over a cambered airfoil for a wide range of angles of attack with a focus on the dynamics of boundary layer separation and transition. The unsteady and complex phenomena of the transitional flow are analyzed through a combination of direct numerical simulations (DNS), large-eddy simulations (LES), experiments, and development of Lagrangian theory and methods.

A discontinuous Galerkin spectral element method (DGSEM) is used to model the compressible Navier-Stokes equations in two and three dimensions. The DGSEM generates high-order accurate results with low dispersion and diffusion errors and has been developed to include kinetic-energy conserving volume fluxes, tools to efficiently track Lagrangian fluid tracers, and computation of higher wall-normal velocity derivatives. The code is benchmarked through a series of Navier-Stokes flows using different DG variants and polynomial orders.

High-fidelity DNS in three dimensions show that the transitional flow over a cambered NACA 65(1)-412 airfoil at Re = 20,000 swiftly changes from a state of laminar separation at mid-chord without reattachment to a laminar separation bubble (LSB) at the leading edge with a turbulent boundary layer. The bifurcation occurs within an angle-of-attack change of two degrees and is accompanied by a rapid increase of the lift and decrease of the drag force, which is observed in computations and experiments likewise. Each flow regime is governed by different dynamics, instabilities, and wake structures that change with the transition location of the separated shear layer. The kinematic aspects of flow separation are further investigated in the Lagrangian frame, where the initial motion of upwelling fluid material from the wall is related to the long-term attracting manifolds in the flow field.

An objective finite-time diagnostic for instabilities in shear flows based on the curvature of Lagrangian material lines is introduced. By defining a flow instability in the Lagrangian frame as the increased folding of lines of fluid particles, subtle perturbations and unstable growth thereof are detected early based solely on the curvature change of material lines over finite time.