In Chapters 3 and 4, we studied major changes in the flow thermophysical properties as it traverses a turbine or compressor stage. The analysis, then, was one-dimensional, with the underlying assumption that average flow properties will prevail midway between the endwalls. Categorized as a pitch-line flow model, this “bulk-flow” analysis proceeds along the “master” streamline (or pitch line), with no attention given to any lateral flow property gradients.

The primary assumptions and formulations for single-phase flow regimes are reviewed in this chapter. This includes the governing partial differential equations for general fluid dynamics (mass, momentum, energy, and species), equations of state and associated flow regimes, rotational effects and the stream function for incompressible flow, and viscous effects with the Reynolds number, including flow instability mechanisms.

This chapter develops the point-force Equations of Motion for a single spherical particle moving in an unbounded fluid. This includes the particle Equations of Motion, which are considered as a sum of pointwise forces. The drag force is described for solid and fluid particles for Reynolds numbers ranging from creeping flow to turbulent flow. The three acceleration forces of added-mass, fluid-stress, and history forces are explained, and all the forces are combined to provide various Equations of Motion. Finally, heat and mass transfer effects on the particle are discussed.

This chapter provides an overview of the key elements of turbulent flow. First, the basic averaging approach and examples of turbulent flow decompositions are discussed. Using these techniques, the average transport equations for mass, momentum, and species with closure models are given, followed by advanced numerical techniques for turbulent flows. Turbulent time and length scales as well as the kinetic energy cascade are overviewed, and theoretical turbulent species diffusion is treated.

This chapter considers the drag force for velocity gradients in the surrounding fluid, particle Mach number and Knudsen number, temperature gradients in the surrounding fluid, particle spin and fluid vorticity, flow turbulence and particle roughness, shape for a solid particle, surface contamination and internal recirculation for a spherical fluid particle, and deformation and drag for a fluid particle. This includes theory, experimental results, and numerical prediction of the drag coefficient for point-force models.

This chapter identifies systems where dispersed multiphase flow is important as well as the key fluid physics via important engineered and natural systems. This includes energy systems and propulsion systems, manufacturing, processing and transport systems, as well as environmental and biological systems. In addition, this chapter sets forth key terminology and assumptions for dispersed multiphase flow, the key velocity reference frames used for multiphase flow, and the assumption of continuum conditions.

This chapter includes a broad survey of numerical approaches for multiphase flow using classifications based on particle reference frame, relative velocity magnitude, and relative particle size to the grid size. These approaches are then considered for Brownian motion and turbulent diffusion. Throughout this chapter, comments are made on selecting the numerical approach for a given flow based on the approach’’s ability to capture physics and computational requirements (best tools for the job and costs).

This chapter discusses modification of the fluid dynamic point forces due to proximity to the wall and due to neighboring particles, where the latter focuses on the Richardson–Zaki exponent. In addition, particle collision with other particles and with walls is discussed for normal and tangential restitution. This includes effects of viscoelasticity, spin, plasticity, fluid viscosity, and adhesion for solid particles as well as deformation and wetting for fluid particles.

This chapter first overviews the types of coupling and particle concentration descriptors, and then considers aspects of one-way coupling, with the special case of Brownian motion. The remainder of the chapter considers two-way coupling, three-way coupling, and four-way coupling.

This chapter discusses lift, added-mass, and history forces for an isolated particle. Shear-induced and spin-induced lift is considered, along with angular particle torque. The lift for fixed torque and equilibrium spin are discussed. In addition, added-mass and history force are considered for solid particles and bubbles.

This chapter considers size distributions and nonspherical particles and trajectories. Clouds of particles with sizes that vary significantly are described using effective averages. Nonspherical particles shapes are characterized along with their motion in free fall. Nonsphericity effects for drops in free fall and for bubbles in free rise are discussed via Weber number. Finally, shape deformation due to shear and due to deformation dynamics is considered for fluid particles.