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.

Development of roll waves in open channels

This study is concerned with some of the properties of roll waves that develop naturally from a turbulent uniform flow in a wide rectangular channel on a constant steep slope . The wave properties considered were depth at the wave crest, depth at the wave trough, wave period, and wave velocity . The primary focus was on the mean values and standard deviations of the crest depths and wave periods at a given station and how these quantities varied with distance along the channel.

The wave properties were measured in a laboratory channel in which roll waves developed naturally from a uniform flow . The Froude number F (F = u_n/√gh_n, u_n = normal velocity , h_n = normal depth, g =acceleration of gravity) ranged from 3. 4 to 6. 0 for channel slopes S_o of . 05 and . 12 respectively . In the initial phase of their development the roll waves appeared as small amplitude waves with a continuous water surface profile . These small amplitude waves subsequently developed into large amplitude shock waves. Shock waves were found to overtake and combine with other shock waves with the result that the crest depth of the combined wave was larger than the crest depths before the overtake. Once roll waves began to develop, the mean value of the crest depths h_nmax increased with distance . Once the shock waves began to overtake, the mean wave period T_av increased approximately linearly with distance.

For a given Froude number and channel slope the observed quantities h^-_max/h_n , T' (T' = S_o T_av √g/h_n), and the standard deviations of h^-_max/h_n and T', could be expressed as unique functions of l/h_n (l = distance from beginning of channel) for the two-fold change in h_n occurring in the observed flows . A given value of h^-_max/h_n occurred at smaller values of l/h_n as the Froude number was increased. For a given value of h /hh^-_max/h_n the growth rate of δh^-_max/h^-_maxδl of the shock waves increased as the Froude number was increased.

A laboratory channel was also used to measure the wave properties of periodic permanent roll waves. For a given Froude number and channel slope the h^-_max/h_n vs. T' relation did not agree with a theory in which the weight of the shock front was neglected. After the theory was modified to include this weight, the observed values of h^-_max/h_n were within an average of 6.5 percent of the predicted values, and the maximum discrepancy was 13.5 percent.

For h^-_max/h_n sufficiently large (h^-_max/h_n > approximately 1.5) it was found that the h^-_max/h_n vs. T' relation for natural roll waves was practically identical to the h^-_max/h_n vs. T' relation for periodic permanent roll waves at the same Froude number and slope. As a result of this correspondence between periodic and natural roll waves, the growth rate δh^-_max/h^-_maxδl of shock waves was predicted to depend on the channel slope, and this slope dependence was observed in the experiments.

On concentrated loads and Green's functions in elastostatics

This investigation is concerned with the notion of concentrated loads in classical elastostatics and related issues. Following a limit treatment of problems involving concentrated internal and surface loads, the orders of the ensuing displacements and stress singularities, as well as the stress resultants of the latter, are determined. These conclusions are taken as a basis for an alternative direct formulation of concentrated-load problems, the completeness of which is established through an appropriate uniqueness theorem. In addition, the present work supplies a reciprocal theorem and an integral representation-theorem applicable to singular problems of the type under consideration. Finally, in the course of the analysis presented here, the theory of Green's functions in elastostatics is extended.

Organization, functions, and achievements of the Inter-American Tropical Tuna Commission

ENGLISH: The Inter-American Tropical Tuna Commission (IATTC) operates under the authority and direction of a Convention originally entered into by the governments of Costa Rica and the United States. The Convention, which came into force in 1950, is open to the adherence by other governments whose nationals participate in the fisheries for tropical tunas in the eastern Pacific Ocean. The member nations of the Commission now are t in addition to Costa Rica and the United States, Canada, France, Japan, Mexico, Nicaragua, and Panama.This report is a description of the organization, functions, and achievements of the Commission. It has been prepared to provide in a convenient format answers to requests for information concerning the Commission. It replaces a similar, earlier report (Carroz, 1965), which is now largely outdated. SPANISH: La Comisión Interamericana del Atún Tropical (CIAT) funciona bajo la autoridad y dirección de un Convenio firmado originalmente por los gobiernos de Costa Rica y los Estados Unidos de America. El Convenio, que entro en vigencia en 1950, se encuentra libre para que otros gobiernos cuyos ciudadanos participen en la pesca de atunes tropicales en el Océano Pacifico oriental se afilien a el. Las naciones miembros de la Comisión, además de Costa Rica y los Estados Unidos, son Cañada, Francia, Japón, México, Nicaragua y Panamá. Este informe es una descripción de la organización, funciones y resultados de la Comisión. Ha sido preparado para suministrar en forma conveniente respuestas a preguntas sobre la Comisión. Reemplaza un informe anterior similar (Carroz 1965), que ya es anticuado en su mayor parte.

Frame presentations: variants of the reals, rings of functions, their Dedekind completions, and the unit circle

176 p.

Organization, functions, and achievements of the Inter-American Tropical Tuna Commission

ENGLISH: The Inter-American Tropical Tuna Commission (IATTC) operates under the authority and direction of a Convention originally entered into by the governments of Costa Rica and the United states. The Convention, which came into force in 1950, is open to the adherence by other governments whose nationals participate in the fisheries for tropical tunas in the eastern Pacific Ocean. The member nations of the Commission now are France. Japan, Nicaragua. Panama, and the United States. This report is a description of the organization, functions. and achievements of the Commission. It has been prepared to provide in a convenient format answers to requests for information concerning the Commission. It replaces similar, earlier reports (Carroz, 1965; Spec. Rep., 1), which are now largely outdated. SPANISH: La Comisión Interamericana del Atún Tropical (CIAT) funciona bajo la autoridad y dirección de un Convenio firmado originalmente por los gobiernos de Costa Rica y los Estados Unidos de America. El Convenio, Que entre en vigencia en 1950, se encuentra libre para Que otros gobiernos cuyos ciudadanos participan en la pesca de los atunes tropicales en el Océano Pacifico oriental se afilien a 61. Las naciones actuales que son miembros de la Comisión son: Francia, Japón, Nicaragua, Panamá y los Estados Unidos. Este informe es una descripci6n de la organización, funciones y resultados de la Comisión. Se ha preparado para suministrar en forma conveniente, informaci6n sobre la Comisión. Este informe renueva otros anteriores similares (Carroz, 1965; CIAT, Inf. Esp., 1) que en su mayor parte determinados.