Effects of density differences on lateral mixing in open-channel flows

This study investigates lateral mixing of tracer fluids in turbulent open-channel flows when the tracer and ambient fluids have different densities. Longitudinal dispersion in flows with longitudinal density gradients is investigated also.

Lateral mixing was studied in a laboratory flume by introducing fluid tracers at the ambient flow velocity continuously and uniformly across a fraction of the flume width and over the entire depth of the ambient flow. Fluid samples were taken to obtain concentration distributions in cross-sections at various distances, x, downstream from the tracer source. The data were used to calculate variances of the lateral distributions of the depth-averaged concentration. When there was a difference in density between the tracer and the ambient fluids, lateral mixing close to the source was enhanced by density-induced secondary flows; however, far downstream where the density gradients were small, lateral mixing rates were independent of the initial density difference. A dimensional analysis of the problem and the data show that the normalized variance is a function of only three dimensionless numbers, which represent: (1) the x-coordinate, (2) the source width, and (3) the buoyancy flux from the source.

A simplified set of equations of motion for a fluid with a horizontal density gradient was integrated to give an expression for the density-induced velocity distribution. The dispersion coefficient due to this velocity distribution was also obtained. Using this dispersion coefficient in an analysis for predicting lateral mixing rates in the experiments of this investigation gave only qualitative agreement with the data. However, predicted longitudinal salinity distributions in an idealized laboratory estuary agree well with published data.

I. Stokes flow past a thin screen. II. Viscous flows past porous bodies of finite size

Part I

The slow, viscous flow past a thin screen is analyzed based on Stokes equations. The problem is reduced to an associated electric potential problem as introduced by Roscoe. Alternatively, the problem is formulated in terms of a Stokeslet distribution, which turns out to be equivalent to the first approach.

Special interest is directed towards the solution of the Stokes flow past a circular annulus. A "Stokeslet" formulation is used in this analysis. The problem is finally reduced to solving a Fredholm integral equation of the second kind. Numerical data for the drag coefficient and the mean velocity through the hole of the annulus are obtained.

Stokes flow past a circular screen with numerous holes is also attempted by assuming a set of approximate boundary conditions. An "electric potential" formulation is used, and the problem is also reduced to solving a Fredholm integral equation of the second kind. Drag coefficient and mean velocity through the screen are computed.

Part II

The purpose of this investigation is to formulate correctly a set of boundary conditions to be prescribed at the interface between a viscous flow region and a porous medium so that the problem of a viscous flow past a porous body can be solved.

General macroscopic equations of motion for flow through porous media are first derived by averaging Stokes equations over a volume element of the medium. These equations, including viscous stresses for the description, are more general than Darcy's law. They reduce to Darcy's law when the Darcy number becomes extremely small.

The interface boundary conditions of the first kind are then formulated with respect to the general macroscopic equations applied within the porous region. An application of such equations and boundary conditions to a Poiseuille shear flow problem demonstrates that there usually exists a thin interface layer immediately inside the porous medium in which the tangential velocity varies exponentially and Darcy's law does not apply.

With Darcy's law assumed within the porous region, interface boundary conditions of the second kind are established which relate the flow variables across the interface layer. The primary feature is a jump condition on the tangential velocity, which is found to be directly proportional to the normal gradient of the tangential velocity immediately outside the porous medium. This is in agreement with the experimental results of Beavers, et al.

The derived boundary conditions are applied in the solutions of two other problems: (1) Viscous flow between a rotating solid cylinder and a stationary porous cylinder, and (2) Stokes flow past a porous sphere.

On the thermal budget of Pahoehoe lava flows

In this thesis I investigate some aspects of the thermal budget of pahoehoe lava flows. This is done with a combination of general field observations, quantitative modeling, and specific field experiments. The results of this work apply to pahoehoe flows in general, even though the vast bulk of the work has been conducted on the lavas formed by the Pu'u 'O'o - Kupaianaha eruption of Kilauea Volcano on Hawai'i. The field observations rely heavily on discussions with the staff of the United States Geological Survey's Hawaiian Volcano Observatory (HVO), under whom I labored repeatedly in 1991-1993 for a period totaling about 10 months.

The quantitative models I have constructed are based on the physical processes observed by others and myself to be active on pahoehoe lava flows. By building up these models from the basic physical principles involved, this work avoids many of the pitfalls of earlier attempts to fit field observations with "intuitively appropriate" mathematical expressions. Unlike many earlier works, my model results can be analyzed in terms of the interactions between the different physical processes. I constructed models to: (1) describe the initial cooling of small pahoehoe flow lobes and (2) understand the thermal budget of lava tubes.

The field experiments were designed either to validate model results or to constrain key input parameters. In support of the cooling model for pahoehoe flow lobes, attempts were made to measure: (1) the cooling within the flow lobes, (2) the amount of heat transported away from the lava by wind, and (3) the growth of the crust on the lobes. Field data collected by Jones [1992], Hon et al. [1994b], and Denlinger [Keszthelyi and Denlinger, in prep.] were also particularly useful in constraining my cooling model for flow lobes. Most of the field observations I have used to constrain the thermal budget of lava tubes were collected by HVO (geological and geophysical monitoring) and the Jet Propulsion Laboratory (airborne infrared imagery [Realmuto et al., 1992]). I was able to assist HVO for part of their lava tube monitoring program and also to collect helicopterborne and ground-based IR video in collaboration with JPL [Keszthelyi et al., 1993].

The most significant results of this work are (1) the quantitative demonstration that the emplacement of pahoehoe and 'a'a flows are the fundamentally different, (2) confirmation that even the longest lava flows observed in our Solar System could have formed as low effusion rate, tube-fed pahoehoe flows, and (3) the recognition that the atmosphere plays a very important role throughout the cooling of history of pahoehoe lava flows. In addition to answering specific questions about the thermal budget of tube-fed pahoehoe lava flows, this thesis has led to some additional, more general, insights into the emplacement of these lava flows. This general understanding of the tube-fed pahoehoe lava flow as a system has suggested foci for future research in this part of physical volcanology.

Compressible flows at small Reynolds numbers

The problem of the slow viscous flow of a gas past a sphere is considered. The fluid cannot be treated incompressible in the limit when the Reynolds number Re, and the Mach number M, tend to zero in such a way that Re ~ o(M^2 ). In this case, the lowest order approximation to the steady Navier-Stokes equations of motion leads to a paradox discovered by Lagerstrom and Chester. This paradox is resolved within the framework of continuum mechanics using the classical slip condition and an iteration scheme that takes into account certain terms in the full Navier-Stokes equations that drop out in the approximation used by the above authors. It is found however that the drag predicted by the theory does not agree with R. A. Millikan's classic experiments on sphere drag.

The whole question of the applicability of the Navier-Stokes theory when the Knudsen number M/Re is not small is examined. A new slip condition is proposed. The idea that the Navier-Stokes equations coupled with this condition may adequately describe small Reynolds number flows when the Knudsen number is not too large is looked at in some detail. First, a general discussion of asymptotic solutions of the equations for all such flows is given. The theory is then applied to several concrete problems of fluid motion. The deductions from this theory appear to interpret and summarize the results of Millikan over a much wider range of Knudsen numbers (almost up to the free molecular or kinetic limit) than hitherto Believed possible by a purely continuum theory. Further experimental tests are suggested and certain interesting applications to the theory of dilute suspensions in gases are noted. Some of the questions raised in the main body of the work are explored further in the appendices.

Identification of thermally-tagged coherent structures in the zero pressure gradient turbulent boundary layer

A zero pressure gradient boundary layer over a flat plate is subjected to step changes in thermal condition at the wall, causing the formation of internal, heated layers. The resulting temperature fluctuations and their corresponding density variations are associated with turbulent coherent structures. Aero-optical distortion occurs when light passes through the boundary layer, encountering the changing index of refraction resulting from the density variations. Instantaneous measurements of streamwise velocity, temperature and the optical deflection angle experienced by a laser traversing the boundary layer are made using hot and cold wires and a Malley probe, respectively. Correlations of the deflection angle with the temperature and velocity records suggest that the dominant contribution to the deflection angle comes from thermally-tagged structures in the outer boundary layer with a convective velocity of approximately 0.8U∞. An examination of instantaneous temperature and velocity and their temporal gradients conditionally averaged around significant optical deflections shows behavior consistent with the passage of a heated vortex. Strong deflections are associated with strong negative temperature gradients, and strong positive velocity gradients where the sign of the streamwise velocity fluctuation changes. The power density spectrum of the optical deflections reveals associated structure size to be on the order of the boundary layer thickness. A comparison to the temperature and velocity spectra suggests that the responsible structures are smaller vortices in the outer boundary layer as opposed to larger scale motions. Notable differences between the power density spectra of the optical deflections and the temperature remain unresolved due to the low frequency response of the cold wire.

Large-eddy simulations of fully developed turbulent channel and pipe flows with smooth and rough walls

Studies in turbulence often focus on two flow conditions, both of which occur frequently in real-world flows and are sought-after for their value in advancing turbulence theory. These are the high Reynolds number regime and the effect of wall surface roughness. In this dissertation, a Large-Eddy Simulation (LES) recreates both conditions over a wide range of Reynolds numbers Re_τ = O(10²)-O(10⁸) and accounts for roughness by locally modeling the statistical effects of near-wall anisotropic fine scales in a thin layer immediately above the rough surface. A subgrid, roughness-corrected wall model is introduced to dynamically transmit this modeled information from the wall to the outer LES, which uses a stretched-vortex subgrid-scale model operating in the bulk of the flow. Of primary interest is the Reynolds number and roughness dependence of these flows in terms of first and second order statistics. The LES is first applied to a fully turbulent uniformly-smooth/rough channel flow to capture the flow dynamics over smooth, transitionally rough and fully rough regimes. Results include a Moody-like diagram for the wall averaged friction factor, believed to be the first of its kind obtained from LES. Confirmation is found for experimentally observed logarithmic behavior in the normalized stream-wise turbulent intensities. Tight logarithmic collapse, scaled on the wall friction velocity, is found for smooth-wall flows when Re_τ ≥ O(10⁶) and in fully rough cases. Since the wall model operates locally and dynamically, the framework is used to investigate non-uniform roughness distribution cases in a channel, where the flow adjustments to sudden surface changes are investigated. Recovery of mean quantities and turbulent statistics after transitions are discussed qualitatively and quantitatively at various roughness and Reynolds number levels. The internal boundary layer, which is defined as the border between the flow affected by the new surface condition and the unaffected part, is computed, and a collapse of the profiles on a length scale containing the logarithm of friction Reynolds number is presented. Finally, we turn to the possibility of expanding the present framework to accommodate more general geometries. As a first step, the whole LES framework is modified for use in the curvilinear geometry of a fully-developed turbulent pipe flow, with implementation carried out in a spectral element solver capable of handling complex wall profiles. The friction factors have shown favorable agreement with the superpipe data, and the LES estimates of the Karman constant and additive constant of the log-law closely match values obtained from experiment.

Towards understanding the mixing characteristics of turbulent buoyant flows

This work proposes a new simulation methodology in which variable density turbulent flows can be studied in the context of a mixing layer with or without the presence of gravity. Specifically, this methodology is developed to probe the nature of non-buoyantly-driven (i.e. isotropically-driven) or buoyantly-driven mixing deep inside a mixing layer. Numerical forcing methods are incorporated into both the velocity and scalar fields, which extends the length of time over which mixing physics can be studied. The simulation framework is designed to allow for independent variation of four non-dimensional parameters, including the Reynolds, Richardson, Atwood, and Schmidt numbers. Additionally, the governing equations are integrated in such a way to allow for the relative magnitude of buoyant energy production and non-buoyant energy production to be varied.

The computational requirements needed to implement the proposed configuration are presented. They are justified in terms of grid resolution, order of accuracy, and transport scheme. Canonical features of turbulent buoyant flows are reproduced as validation of the proposed methodology. These features include the recovery of isotropic Kolmogorov scales under buoyant and non-buoyant conditions, the recovery of anisotropic one-dimensional energy spectra under buoyant conditions, and the preservation of known statistical distributions in the scalar field, as found in other DNS studies.

This simulation methodology is used to perform a parametric study of turbulent buoyant flows to discern the effects of varying the Reynolds, Richardson, and Atwood numbers on the resulting state of mixing. The effects of the Reynolds and Atwood numbers are isolated by looking at two energy dissipation rate conditions under non-buoyant (variable density) and constant density conditions. The effects of Richardson number are isolated by varying the ratio of buoyant energy production to total energy production from zero (non-buoyant) to one (entirely buoyant) under constant Atwood number, Schmidt number, and energy dissipation rate conditions. It is found that the major differences between non-buoyant and buoyant turbulent flows are contained in the transfer spectrum and longitudinal structure functions, while all other metrics are largely similar (e.g. energy spectra, alignment characteristics of the strain-rate tensor). Also, despite the differences noted between fully buoyant and non-buoyant turbulent fields, the scalar field, in all cases, is unchanged by these. The mixing dynamics in the scalar field are found to be insensitive to the source of turbulent kinetic energy production (non-buoyant vs. buoyant).

An integrated design approach to power systems : from power flows to electricity markets

Power system is at the brink of change. Engineering needs, economic forces and environmental factors are the main drivers of this change. The vision is to build a smart electrical grid and a smarter market mechanism around it to fulfill mandates on clean energy. Looking at engineering and economic issues in isolation is no longer an option today; it needs an integrated design approach. In this thesis, I shall revisit some of the classical questions on the engineering operation of power systems that deals with the nonconvexity of power flow equations. Then I shall explore some issues of the interaction of these power flow equations on the electricity markets to address the fundamental issue of market power in a deregulated market environment. Finally, motivated by the emergence of new storage technologies, I present an interesting result on the investment decision problem of placing storage over a power network. The goal of this study is to demonstrate that modern optimization and game theory can provide unique insights into this complex system. Some of the ideas carry over to applications beyond power systems.

Suspension mechanics: I. Inertial and non-Newtonian migration of neutrally buoyant rigid spheres in two-dimensional unidirectional flows; II. The effect of viscoelasticity on the creeping motion of a train of neutrally buoyant Newtonian drops through a circular tube

The lateral migration of neutrally buoyant rigid spheres in two-dimensional unidirectional flows was studied theoretically. The cases of both inertia-induced migration in a Newtonian fluid and normal stress-induced migration in a second-order fluid were considered. Analytical results for the lateral velocities were obtained, and the equilibrium positions and trajectories of the spheres compared favorably with the experimental data available in the literature. The effective viscosity was obtained for a dilute suspension of spheres which were simultaneously undergoing inertia-induced migration and translational Brownian motion in a plane Poiseuille flow. The migration of spheres suspended in a second-order fluid inside a screw extruder was also considered.

The creeping motion of neutrally buoyant concentrically located Newtonian drops through a circular tube was studied experimentally for drops which have an undeformed radius comparable to that of the tube. Both a Newtonian and a viscoelastic suspending fluid were used in order to determine the influence of viscoelasticity. The extra pressure drop due to the presence of the suspended drops, the shape and velocity of the drops, and the streamlines of the flow were obtained for various viscosity ratios, total flow rates, and drop sizes. The results were compared with existing theoretical and experimental data.

Distributed Load Control in Multiphase Radial Networks

The current power grid is on the cusp of modernization due to the emergence of distributed generation and controllable loads, as well as renewable energy. On one hand, distributed and renewable generation is volatile and difficult to dispatch. On the other hand, controllable loads provide significant potential for compensating for the uncertainties. In a future grid where there are thousands or millions of controllable loads and a large portion of the generation comes from volatile sources like wind and solar, distributed control that shifts or reduces the power consumption of electric loads in a reliable and economic way would be highly valuable.

Load control needs to be conducted with network awareness. Otherwise, voltage violations and overloading of circuit devices are likely. To model these effects, network power flows and voltages have to be considered explicitly. However, the physical laws that determine power flows and voltages are nonlinear. Furthermore, while distributed generation and controllable loads are mostly located in distribution networks that are multiphase and radial, most of the power flow studies focus on single-phase networks.

This thesis focuses on distributed load control in multiphase radial distribution networks. In particular, we first study distributed load control without considering network constraints, and then consider network-aware distributed load control.

Distributed implementation of load control is the main challenge if network constraints can be ignored. In this case, we first ignore the uncertainties in renewable generation and load arrivals, and propose a distributed load control algorithm, Algorithm 1, that optimally schedules the deferrable loads to shape the net electricity demand. Deferrable loads refer to loads whose total energy consumption is fixed, but energy usage can be shifted over time in response to network conditions. Algorithm 1 is a distributed gradient decent algorithm, and empirically converges to optimal deferrable load schedules within 15 iterations.

We then extend Algorithm 1 to a real-time setup where deferrable loads arrive over time, and only imprecise predictions about future renewable generation and load are available at the time of decision making. The real-time algorithm Algorithm 2 is based on model-predictive control: Algorithm 2 uses updated predictions on renewable generation as the true values, and computes a pseudo load to simulate future deferrable load. The pseudo load consumes 0 power at the current time step, and its total energy consumption equals the expectation of future deferrable load total energy request.

Network constraints, e.g., transformer loading constraints and voltage regulation constraints, bring significant challenge to the load control problem since power flows and voltages are governed by nonlinear physical laws. Remarkably, distribution networks are usually multiphase and radial. Two approaches are explored to overcome this challenge: one based on convex relaxation and the other that seeks a locally optimal load schedule.

To explore the convex relaxation approach, a novel but equivalent power flow model, the branch flow model, is developed, and a semidefinite programming relaxation, called BFM-SDP, is obtained using the branch flow model. BFM-SDP is mathematically equivalent to a standard convex relaxation proposed in the literature, but numerically is much more stable. Empirical studies show that BFM-SDP is numerically exact for the IEEE 13-, 34-, 37-, 123-bus networks and a real-world 2065-bus network, while the standard convex relaxation is numerically exact for only two of these networks.

Theoretical guarantees on the exactness of convex relaxations are provided for two types of networks: single-phase radial alternative-current (AC) networks, and single-phase mesh direct-current (DC) networks. In particular, for single-phase radial AC networks, we prove that a second-order cone program (SOCP) relaxation is exact if voltage upper bounds are not binding; we also modify the optimal load control problem so that its SOCP relaxation is always exact. For single-phase mesh DC networks, we prove that an SOCP relaxation is exact if 1) voltage upper bounds are not binding, or 2) voltage upper bounds are uniform and power injection lower bounds are strictly negative; we also modify the optimal load control problem so that its SOCP relaxation is always exact.

To seek a locally optimal load schedule, a distributed gradient-decent algorithm, Algorithm 9, is proposed. The suboptimality gap of the algorithm is rigorously characterized and close to 0 for practical networks. Furthermore, unlike the convex relaxation approach, Algorithm 9 ensures a feasible solution. The gradients used in Algorithm 9 are estimated based on a linear approximation of the power flow, which is derived with the following assumptions: 1) line losses are negligible; and 2) voltages are reasonably balanced. Both assumptions are satisfied in practical distribution networks. Empirical results show that Algorithm 9 obtains 70+ times speed up over the convex relaxation approach, at the cost of a suboptimality within numerical precision.

Experimental study on inertial effects in liquid-solid flows

This thesis presents experimental measurements of the rheological behavior of liquid-solid mixtures at moderate Reynolds (defined by the shear rate and particle diameter) and Stokes numbers, ranging from 3 ≤ Re ≤ 1.6 × 10³ and 0.4 ≤ St ≤ 195. The experiments use a specifically designed Couette cylindrical rheometer that allows for probing the transition from transporting a pure liquid to transporting a dense suspension of particles. Measurements of the shear stress are presented for a wide range of particle concentration (10 to 60% in volume) and for particle to fluid density ratio between 1 and 1.05. The effective relative viscosity exhibits a strong dependence on the solid fraction for all density ratios tested. For density ratio of 1 the effective viscosity increases with Stokes number (St) for volume fractions (φ) lower than 40% and becomes constant for higher φ. When the particles are denser than the liquid, the effective viscosity shows a stronger dependance on St. An analysis of the particle resuspension for the case with a density ratio of 1.05 is presented and used to predict the local volume fraction where the shear stress measurements take place. When the local volume fraction is considered, the effective viscosity for settling and no settling particles is consistent, indicating that the effective viscosity is independent of differences in density between the solid and liquid phase. Shear stress measurements of pure fluids (no particles) were performed using the same rheometer, and a deviation from laminar behavior is observed for gap Reynolds numbers above 4× 10³, indicating the presence of hydrodynamic instabilities associated with the rotation of the outer cylinder. The increase on the effective viscosity with Stokes numbers observed for mixtures with φ ≤ 30% appears to be affected by such hydrodynamic instabilities. The effective viscosity for the current experiments is considerably higher than the one reported in non-inertial suspensions.

Mechanics of sediment transport and bedrock erosion in steep landscapes

Erosion is concentrated in steep landscapes such that, despite accounting for only a small fraction of Earth’s total surface area, these areas regulate the flux of sediment to downstream basins, and their rugged morphology records transient changes (or lack thereof) in geologic and climatic forcing. Steep landscapes are geomorphically active; large sediment fluxes and rapid landscape evolution rates can create or destroy habitat for humans and wildlife alike, and landslides, debris flows, and floods common in mountainous areas represent a persistent natural and structural hazard. Despite the central role that steep landscapes play in the geosciences and in landscape management, the processes controlling their evolution have been poorly studied compared to lower-gradient areas. This thesis focuses on the basic mechanics of sediment transport and bedrock incision in steep landscapes, as these are the fundamental processes which set the pace and style of landscape evolution. Chapter 1 examines the spatial distribution of slow-moving landslides; these landslides can dominate sediment fluxes to river networks, but the controls on their occurrence are poorly understood. Using a case-study along the San Andreas Fault, California, I show that slow-moving landslides preferentially occur near the fault, suggesting a rock-strength control on landslide distribution. Chapter 2 provides the first field-measurements of incipient sediment motion in streams steeper than 14% and shows a large influence of slope-dependent flow hydraulics and grain-scale roughness on particle motion. Chapter 3 presents experimental evidence for bedrock erosion by suspended sediment, suggesting that, in contrast to prevailing theoretical predictions, suspension-regime transport in steep streams can be the dominant erosion agent. Steep streams are often characterized by the presence of waterfalls and bedrock steps which can have locally high rates of erosion; Chapters 4 and 5 present newly developed, experimentally validated theory on sediment transport through and bedrock erosion in waterfall plunge pools. Finally, Chapter 6 explores the formation of a bedrock slot canyon where interactions between sediment transport and bedrock incision lead to the formation of upstream-propagating bedrock step-pools and waterfalls.

Entrainment of fine sediments by turbulent flows

A study was made of the means by which turbulent flows entrain sediment grains from alluvial stream beds. Entrainment was considered to include both the initiation of sediment motion and the suspension of grains by the flow. Observations of grain motion induced by turbulent flows led to the formulation of an entrainment hypothesis. It was based on the concept of turbulent eddies disrupting the viscous sublayer and impinging directly onto the grain surface. It is suggested that entrainment results from the interaction between fluid elements within an eddy and the sediment grains.

A pulsating jet was used to simulate the flow conditions in a turbulent boundary layer. Evidence is presented to establish the validity of this representation. Experiments were made to determine the dependence of jet strength, defined below, upon sediment and fluid properties. For a given sediment and fluid, and fixed jet geometry there were two critical values of jet strength: one at which grains started to roll across the bed, and one at which grains were projected up from the bed. The jet strength K, is a function of the pulse frequency, ω, and the pulse amplitude, A, defined by

K = Aω^-s

Where s is the slope of a plot of log A against log ω. Pulse amplitude is equal to the volume of fluid ejected at each pulse divided by the cross sectional area of the jet tube.

Dimensional analysis was used to determine the parameters by which the data from the experiments could be correlated. Based on this, a method was devised for computing the pulse amplitude and frequency necessary either to move or project grains from the bed for any specified fluid and sediment combination.

Experiments made in a laboratory flume with a turbulent flow over a sediment bed are described. Dye injection was used to show the presence, in a turbulent boundary layer, of two important aspects of the pulsating jet model and the impinging eddy hypothesis. These were the intermittent nature of the sublayer and the presence of velocities with vertical components adjacent to the sediment bed.

A discussion of flow conditions, and the resultant grain motion, that occurred over sediment beds of different form is given. The observed effects of the sediment and fluid interaction are explained, in each case, in terms of the entrainment hypothesis.

The study does not suggest that the proposed entrainment mechanism is the only one by which grains can be entrained. However, in the writer’s opinion, the evidence presented strongly suggests that the impingement of turbulent eddies onto a sediment bed plays a dominant role in the process.

Particle fluid mechanics in shear flows, acoustic waves, and shock waves

Three different categories of flow problems of a fluid containing small particles are being considered here. They are: (i) a fluid containing small, non-reacting particles (Parts I and II); (ii) a fluid containing reacting particles (Parts III and IV); and (iii) a fluid containing particles of two distinct sizes with collisions between two groups of particles (Part V).

Part I

A numerical solution is obtained for a fluid containing small particles flowing over an infinite disc rotating at a constant angular velocity. It is a boundary layer type flow, and the boundary layer thickness for the mixture is estimated. For large Reynolds number, the solution suggests the boundary layer approximation of a fluid-particle mixture by assuming W = W_p. The error introduced is consistent with the Prandtl’s boundary layer approximation. Outside the boundary layer, the flow field has to satisfy the “inviscid equation” in which the viscous stress terms are absent while the drag force between the particle cloud and the fluid is still important. Increase of particle concentration reduces the boundary layer thickness and the amount of mixture being transported outwardly is reduced. A new parameter, β = 1/Ω τ_v, is introduced which is also proportional to μ. The secondary flow of the particle cloud depends very much on β. For small values of β, the particle cloud velocity attains its maximum value on the surface of the disc, and for infinitely large values of β, both the radial and axial particle velocity components vanish on the surface of the disc.

Part II

The “inviscid” equation for a gas-particle mixture is linearized to describe the flow over a wavy wall. Corresponding to the Prandtl-Glauert equation for pure gas, a fourth order partial differential equation in terms of the velocity potential ϕ is obtained for the mixture. The solution is obtained for the flow over a periodic wavy wall. For equilibrium flows where λ_v and λ_T approach zero and frozen flows in which λ_v and λ_T become infinitely large, the flow problem is basically similar to that obtained by Ackeret for a pure gas. For finite values of λ_v and λ_T, all quantities except v are not in phase with the wavy wall. Thus the drag coefficient C_D is present even in the subsonic case, and similarly, all quantities decay exponentially for supersonic flows. The phase shift and the attenuation factor increase for increasing particle concentration.

Part III

Using the boundary layer approximation, the initial development of the combustion zone between the laminar mixing of two parallel streams of oxidizing agent and small, solid, combustible particles suspended in an inert gas is investigated. For the special case when the two streams are moving at the same speed, a Green’s function exists for the differential equations describing first order gas temperature and oxidizer concentration. Solutions in terms of error functions and exponential integrals are obtained. Reactions occur within a relatively thin region of the order of λ_D. Thus, it seems advantageous in the general study of two-dimensional laminar flame problems to introduce a chemical boundary layer of thickness λ_D within which reactions take place. Outside this chemical boundary layer, the flow field corresponds to the ordinary fluid dynamics without chemical reaction.

Part IV

The shock wave structure in a condensing medium of small liquid droplets suspended in a homogeneous gas-vapor mixture consists of the conventional compressive wave followed by a relaxation region in which the particle cloud and gas mixture attain momentum and thermal equilibrium. Immediately following the compressive wave, the partial pressure corresponding to the vapor concentration in the gas mixture is higher than the vapor pressure of the liquid droplets and condensation sets in. Farther downstream of the shock, evaporation appears when the particle temperature is raised by the hot surrounding gas mixture. The thickness of the condensation region depends very much on the latent heat. For relatively high latent heat, the condensation zone is small compared with Ʌ_D.

For solid particles suspended initially in an inert gas, the relaxation zone immediately following the compression wave consists of a region where the particle temperature is first being raised to its melting point. When the particles are totally melted as the particle temperature is further increased, evaporation of the particles also plays a role.

The equilibrium condition downstream of the shock can be calculated and is independent of the model of the particle-gas mixture interaction.

Part V

For a gas containing particles of two distinct sizes and satisfying certain conditions, momentum transfer due to collisions between the two groups of particles can be taken into consideration using the classical elastic spherical ball model. Both in the relatively simple problem of normal shock wave and the perturbation solutions for the nozzle flow, the transfer of momentum due to collisions which decreases the velocity difference between the two groups of particles is clearly demonstrated. The difference in temperature as compared with the collisionless case is quite negligible.

A kinetic theory description for external spherical flows with arbitrary Knudsen number by a moment method

The Maxwell integral equations of transfer are applied to a series of problems involving flows of arbitrary density gases about spheres. As suggested by Lees a two sided Maxwellian-like weighting function containing a number of free parameters is utilized and a sufficient number of partial differential moment equations is used to determine these parameters. Maxwell's inverse fifth-power force law is used to simplify the evaluation of the collision integrals appearing in the moment equations. All flow quantities are then determined by integration of the weighting function which results from the solution of the differential moment system. Three problems are treated: the heat-flux from a slightly heated sphere at rest in an infinite gas; the velocity field and drag of a slowly moving sphere in an unbounded space; the velocity field and drag torque on a slowly rotating sphere. Solutions to the third problem are found to both first and second-order in surface Mach number with the secondary centrifugal fan motion being of particular interest. Singular aspects of the moment method are encountered in the last two problems and an asymptotic study of these difficulties leads to a formal criterion for a "well posed" moment system. The previously unanswered question of just how many moments must be used in a specific problem is now clarified to a great extent.