Conventional fluid dynamics models such as the Navier-Stokes equations are derived for prediction of fluid motion at or near equilibrium, classic examples being the motion of fluids for which inter-molecular collisions are dominant. Flows at equilibrium permit simplifications such as the introduction of viscosity and also lead to solutions that are single-valued. However, many other regimes of interest include "fluids"' far from equilibrium; for example, rarefied gases or particle-laden flows in which the dispersed phase can be comprised of granular solids, droplets, or bubbles. Particle motion in these flows is not typically dominated by collisions and may exhibit significant memory effects; therefore, is often poorly described using continuum, field-based (Eulerian) approaches. Non-equilibrium flows generally lack a straightforward counterpart to viscosity and their multi-valued solutions cannot be represented by most Eulerian methods. This strongly motivates different strategies to address current shortcomings and the novel approach adopted in this work is based on the Conditional Quadrature Method of Moments (CQMOM). In CQMOM, moment equations are derived from the Boltzmann equation using a quadrature approximation of the velocity probability density function (PDF). CQMOM circumvents the drawbacks of current methods and leads to multivariate and multidimensional solutions in an Eulerian frame of reference. In the present work, the discretized PDF is resolved using an adaptive two-point quadrature in three-dimensional velocity space. The method is applied to computation of a series of non-equilibrium flows, ranging from simple two-dimensional test cases to fully-turbulent three-dimensional wall-bounded particle-laden flows. The primary contribution of the present effort is on development, application, and assessment of CQMOM for predicting the key features of dilute particle-laden flows. Statistical descriptors such as mean concentration and mean velocity are in good agreement with previous results, for both collision-less and collisional flows at varying particle Stokes numbers. Turbulent statistics and measures of local accumulation agree less favorably with prior results and identify areas for improvement in the modeling strategy.

Rotorcraft operation in austere environments can result in difficult operating conditions, particularly in the vicinity of sandy areas. The uplift of sediment by rotorcraft downwash, a phenomenon known as brownout, hinders pilot visual cues and may result in a potentially dangerous situation. Brownout is a complex multiphase flow problem that is not unique and depends on both the characteristics of the rotorcraft and the sediment. The lack of fundamental understanding constrains models and limits development of technologies that could mitigate the adverse effects of brownout. This provides the over-arching motivation of the current work focusing on models of particle-laden sediment beds. The particular focus of the current investigations is numerical modeling of near-surface fluid-particle interactions in turbulent boundary layers with and without coherent vortices superimposed on the background flow, that model rotorcraft downwash. The simulations are performed with two groups of particles having different densities both of which display strong vortex-particle interaction close to the source location. The simulations include cases with inter-particle collisions and gravitational settling. Particle effects on the fluid are ignored. The numerical simulations are performed using an Euler- Lagrange method in which a fractional-step approach is used for the fluid and with the particulate phase advanced using Discrete Particle Simulation. The objectives are to gain insight into the fluid-particle dynamics that influence transport near the bed by analyzing the competing effects of the vortices, inter-particle collisions, and gravity. Following the introduction of coherent vortices into the domain, the structures convect downstream, dissipate, and then recover to an equilibrium state with the boundary layer. The particle phase displays an analogous return to an equilibrium state as the vortices dissipate and the boundary layer recovers, though this recovery is slower than for the fluid and is sensitive to the particle response time. The effects of inter-particle collisions are relatively strong and apparent throughout the flow, being most effective in the boundary layer. Gravitational settling increases the particle concentration near the wall and consequently increase inter-particle collisions.