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.

Modern gas turbines operate at high mainstream gas temperatures and pressures, which requires high durability materials. A method of preventing these hot gases from leaking into the turbine cavities is essential for improved reliability and cost reduction. Utilizing bleed-off air from the compressor to cool internal components has been a common solution, but at the cost of decreasing turbine performance. The present work thoroughly describes the complex flow field between the mainstream gas and a single rotor-stator disk cavity, and mechanisms of mainstream gas ingestion. A combined approach of experimental measurement and numerical simulation are performed on the flow in a single-stage model gas turbine. Mainstream gas ingestion into the cavity is further reduced by utilizing two axially overlapping seal rings, one on the rotor disk and the other on the stator wall. Secondary purge air is injected into the rotor-stator cavity pre-swirled through the stator radially inboard of the two seal rings. Flow field predictions from the simulations are compared against experimental measurements of static pressure, velocity, and tracer gas concentration acquired in a nearly identical model configuration. Operational conditions were performed with a main airflow Reynolds number of 7.86e4 and a rotor disk speed of 3000rpm. Additionally the rotational Reynolds number was 8.74e5 with a purge air nondimensional flow rate cw=4806. The simulation models a 1/14 rotationally periodic sector of the turbine rig, consisting of four rotor blades and four stator vanes. Gambit was used to generate the three-dimensional unstructured grids ranging from 10 to 20 million cells. Effects of turbulence were modeled using the single-equation Spalart-Allmaras as well as the realizable k-epsilon models. Computations were performed using FLUENT for both a simplified steady-state and subsequent time-dependent formulation. Simulation results show larger scale structures across the entire sector angle inside the cavity and certain unsteady mainstream ingestion mechanisms are realized from the tracer gas. Simulated velocity distributions were scrutinized against Particle Image Velocimetry plots in the rotor-stator cavity and are in reasonable agreement with all of the measurements.