Dispersion of Gas Bubbles in Large-Scale Homogeneous Isotropic Turbulence

An approximate analysis is given of the dispersion of gas bubbles that rise at large Reynolds number through large-scale homogeneous, isotropic turbulence, characterized by the Kraichnan energy-spectrum function. A fairly well-established equation of motion of the bubbles, originally proposed by Thomas et al. [16], is used to derive a closed set of equations for the components of the dispersion tensor of the bubbles in a manner analogous to that used by Saffman [12] for fluid particles and by Pismen and Nir [10, 11] for solid particles. The equations are then solved to obtain the diffusivities and the intensities of bubble velocity fluctuations. Analytical solutions are compared with results from simulations of the bubble motion in a Gaussian random velocity field.     

Kinetic theory of disperse systems

A kinetic equation for the motion of solid particles in a liquid or gas is derived on the basis of the Fokker-Planck-Kolmogorov diffusion equation for the N particle distribution function. It is shown that, under appropriate assumptions, Bogolyubov's method can also be applied to equations of diffusion type. The obtained kinetic equation is a generalization of the one proposed earlier in [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 128–132, January–February, 1980.I thank V. P. Myasnikov for suggesting the problem and for helpful discussions.     

Method of outer and inner asymptotic expansions in the theory of Brownian motion of aerosol particles

We examine the Brownian motion of particles in a gaseous medium, complicated by the influence of inertial forces. The equation for the distribution function in phase space describing motion of this type was obtained in [1]. Also presented in [1] are the solutions of this equation for certain simple particular cases. The approximate equations of motion of aerosol particles in coordinate space were first obtained in [2] and solved for certain concrete problems in [3,4]. More exact equations of motion in coordinate space, and also the limits of applicability of the equations of [2], are presented in [5].     

A multiphase DNS approach for handling solid particles motion with heat transfer

In the current work we propose a multiphase DNS method capable of resolving the motion of solid particles coupled with heat transfer effects. The method is based on solving a shared set of momentum and energy balance equations for the carrier phase and the particulate phase. Individual particles are tracked using a number of volume fraction advection equations. The proposed method is in very good agreement with the available data in the literature for the following cases: isothermal particle motion (in the presence of walls and other particles), natural convection around a stationary particle and solid particles motion accompanied with heat transfer effects. In addition, we show that the method is inherently capable of handling deformable particles (i.e. droplets and bubbles) co-existing with solid particles. The method is thus well suited to deal with challenging multiphase systems, such as diesel spray combustion with soot formation, spray drying with particle nucleation, and biological treatment of waste water.     

Calculation of the momentum and heat transfer in turbulent gas-particle jet flows

The equations for the second moments of the dispersed-phase velocity and temperature fluctuations are used for calculating gas-suspension jet flows within the framework of the Euler approach. The advantages of introducing the equations for the second moments of the particle velocity fluctuations has previously been quite convincingly demonstrated with reference to the calculation of two-phase channel boundary flows [9–11]. The flows considered below have a low solid particle volume concentration, so that interparticle collisions can be neglected and, consequently, the stochastic motion of the particles is determined exclusively by their involvement in the fluctuating motion of the carrier flow. In addition to the equations for the turbulent energy of the gas and its dissipation, the calculation scheme includes the equations for the turbulent energy and turbulent heat transfer of the solid phase; however, the model constructed does not contain additional empirical constants associated with the presence of the particles in the flow.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 69–80, May–June, 1992.     

Equations of hydromechanics and compression shock in two-velocity and two-temperature continuum with phase transformations

The equations of motion of multiphase mixtures have been considered in [1–10] and several other studies. In [1] it is proposed that the mixture motion be considered as an interpenetrating motion of several continua when velocity, pressure, mean density, concentration, etc., fields for each phase are introduced in the flowfield. The equations of motion are written separately for each phase, and the force effect of the other components is considered by introducing the interaction forces, which for the entire system are internal. The assumption of component barotropy is used to close the system.The energy equations are used in [2, 3] in place of the component barotropy assumption. Moreover, mixtures without phase transformations are considered. In [4] an analysis is made of the equations of turbulent motion with account for viscous forces for a two-velocity, but single-temperature medium in which equilibrium phase transformations are assumed, i. e., a two-phase medium is considered in which the phase temperatures are the same, the composition is equilibrium, but the phase velocities are different. In [5] the equations are written on the interface in a multicomponent medium consisting of barotropic fluids. A discontinuity classification is also presented here. In the aforementioned work [3] the equations on the shock are written for a continuum with particles without the use of the property of barotropy of the carrier fluid. Various different aspects of the motion of multiphase mixtures are considered in [6–11], for example, the effect of particle collisions with one another, the effect of the volume occupied by the particles on the parameters stream, shock waves, etc. In [7] a study is made of the force effect of an agitated medium on a particle on the basis of the Basset-Boussinesq-Oseen equation.In the following we derive the equations of motion of a two-velocity and two-temperature continuum with drops or particles with nonequilibrium phase transformations, i. e., a medium in which the phase velocities and temperatures are different and the composition may be nonequilibrium. In addition, we study the effect of the presence of particles or drops on the gas parameters behind a shock. Further, the equations obtained here are used to study compression waves, and in particular shock waves.The author wishes to thank Kh. A. Rakhmatulin, S. S. Grigoryan, and Yu. A. Buevich for helpful discussions and valuable comments.     

Diffusion of suspended particles in an isotropic turbulence field

A model for turbulent motion is proposed which makes it possible to evaluate the pulsation characteristics and the diffusion coefficients of the dispersed phase and also makes it possible to describe the effect of the suspended particles on the turbulence of the dispersing medium. Specific calculations are made for the situation when the undisturbed turbulent field is isotropic.The diffusion of an admisture having inertia in a turbulent stream has been studied previously on the assumption that the three-dimensional turbulence characteristics have practically no effect on the behavior of the suspended particles, so that the random motion of the latter is described by ordinary differential equations containing the natural independent variable the motion travel time [1–4]. In many cases this assumption is incorrect and the corresponding theory is obviously deficient. For example, a fundamental result of this theory, asserting that the turbulent diffusion coefficients of the particles and of the fluid moles are equal for a long diffusion time, is obviously incorrect if the relative motion of the particles is significant [5].     

Kinetic model of heat transfer processes in a fluidized bed

In the present paper equations are obtained for determining the temperature field in a fluidized layer. The heat and mass transfer processes in a fluidized bed depend significantly on the motion of the solid particles which form the bed. In any small volume of a fluidized bed with nonuniform thermal conditions there are particles with different average temperatures. Therefore it is natural to resort to the statistical representation of such a system, developed previously in [1, 2], for the study of the heat transfer processes. The expression obtained here for the heat conductivity coefficient of the bed is in good qualitative agreement with the experimental data.The author wishes to thank V. G. Levich for his interest and valuable discussions.     

Boundary conditions on a rigid surface in a two-phase flow

A study is made of the boundary conditions on a rigid surface in a two-component disperse flow. Appropriate boundary conditions are obtained for the kinetic equation and macroscopic equations of a pseudogas of solid particles proposed in [1–3]. The reasons for the occurrence of bubbles in two-phase systems are discussed. On the basis of the similitude parameters of the kinetic equation of the pseudogas, disperse systems are classified generally on the basis of the concentration of solid particles and their diameters.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 46–51, May–June, 1980.I thank V. P. Myasnikov for suggesting the problem and for a helpful discussion.     

Effect of particles suspended in a fluid flow on the decay of isotropic turbulence

Dynamic equations have been obtained for the two-point double correlations of the fluctuation velocities of a fluid and the particles suspended in it at low volume concentrations of the solid phase. In the case of uniform isotropic turbulence these equations can be considerably simplified. The final period of decay of isotropic turbulence has been studied in detail. At this stage in the case of high-inertia particles the inhomogeneous-fluid turbulence is similar to the turbulence of a homogeneous fluid (without particles) in the sense that the presence of the particles affects only the fluctuation energy but leaves unchanged the spatial scales of turbulence and the spatial energy spectrum function. The suspended particles lead to exponential damping of the turbulent pulsations.Little theoretical information is available on the hydrodynamics of a suspension of fine particles in a turbulent liquid or gas. Research has been mainly confined to the behavior of the individual particles in a given turbulence field [1]. The problem of the turbulent motion of the mixture as a whole has been examined by Barenblatt [2], who derived the equations of motion of the mixture, using Kolmogorov's hypothesis to close them. Hinze [3] has also attempted to derive equations for turbulent pulsations of the mixture. However, as Murray showed [4], Hinze' s equations contradict Newton' s third law.The effect of suspended particles on the turbulence of a two-phase flow is governed by the noncorrespondence of the local velocities of the particles and the medium. The forces of resistance to the motion of the particles relative to the fluid lead to additional dissipation of fluctuation energy and decay of turbulence [2]. On the other hand, if the averaged velocities of particles and medium do not correspond, the suspended particles may also have a destabilizing effect [5, 6], causing energy transfer from the averaged to the pulsating motion. Below we shall consider the case where the averaged velocities of the two phases coincide, i.e., we shall deal only with the first of the two above-mentioned effects.The authors thank G.I. Barenblatt for his useful advice.     

Macroscopic equations of motion of disperse systems

The macroscopic equations of motion of a two-component system consisting of a continuous phase and a large number of solid particles are considered. The generalized kinetic equation of a pseudogas obtained earlier by the author is expressed in a form more convenient for calculations. The Chapman-Enskog method is used to solve the kinetic equation at small Knudsen numbers and dimensionless number characterizing the transfer of momentum between the phases of order unity. Because of the influence of the continuous phase, the stress tensor in the macroscopic conservation equations of the pseudogas is anisotropic. The obtained macroscopic equations of the pseudogas are more general than the ones proposed earlier by Myasnikov, this being due to the anisotropy of the time constants which occur in the operator of the hydrodynamic interaction.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 39–44, March–April, 1980.I thank V. P. Myasnikov for posing the problem and for helpful discussions.     

An Eulerian model for the motion of granular material with a large Stokes number in fluid flow

This study introduced a novel Euler–Euler approach for modeling granular multiphase flow. The motion of particles with a large Stokes number was investigated assuming that granular material has unilateral compressibility. Solid pressure in the momentum equations for granular multiphase flow was determined so that the unilateral incompressibility condition was satisfied. Using the continuity condition of the granular phase, the equation was rewritten in the optimal form to calculate the solid pressure. A discrete formulation of smoothed particle hydrodynamics was applied for the convective terms so that the discrete matrix was positive semidefinite for the convergence and the discretization for an unstructured mesh was allowed. Frictional stress was then determined from solid pressure and, by using the solid pressure and frictional stress, momentum equations for the granular phase were solved. The method was incorporated into ANSYS FLUENT by a UDF (user defined function). Model validation was performed through a comparison with two previous results, and efficacy of the proposed model was confirmed.     

Influence of the hall effect on development of rotational MHD flow in a flat channel

In an experimental study of the heat transfer from a partially ionized gas it was found that the heat flux to the wall for flow of an electrically conducting gas in a circular tube located in a magnetic field of a solenoid depends not only on the magnitude of the magnetic field but also on the field orientation [1]; with the magnetic field parallel to the velocity the heat transfer is reduced by 15%, with antiparallel orientation it is reduced by only 1% in comparison with the heat transfer without the magnetic field. No explanation for this was given either in [1] or in the subsequent discussion [2]; moreover, on the basis of the constructed equations [1] this effect cannot be obtained at all, since the solution of the equations clearly is not changed by a change of the field sign. In the following we attempt to explain this effect by the processes which take place during the development of rotational flow of an anisotropically conducting medium. The idea of the possibility of such an explanation for this effect was proposed in general form in the survey paper [3].The detailed calculation of the development of MHD flows has been made previously only for the case of a transverse magnetic field and very simple channel geometry (see, for example, the survey [3]).In all the considered problems the components of the electrical field which appeared in the motion equations were known with an accuracy to constants from symmetry considerations. Therefore, under the assumption of smallness of the induced magnetic field these problems reduced simply to the solution of the equation of motion with additional terms which are linear in the velocity. In the present paper we construct an approximate simultaneous solution of a system consisting of the motion equations and the equation for the electrical potential.     

Steady two-phase flow in a vertical tube subject to combined free and forced convection

Using the two-velocity, two-temperature model of a continuous medium, the viscousgravitational flow of a mixture of incompressible liquid and solid particles in a vertical round tube is considered. The free-convection equations are written down on the basis of the general equation of motion and the energy equation of a two-phase medium [1, 2]. Using a finite Hankel integral transformation, a solution is constructed for the case of a linear wall-temperature distribution along the tube. The results of some practical calculations of the velocity and temperature fields over the cross section of the tube are presented, together with the dimensionless heat-transfer coefficient expressed as a function of the Rayleigh number and phase concentration. Here it is assumed that the dynamic and thermal-interaction coefficients between the phases correspond to the Stokes mode of flow for each particle, as a result of which the velocity and thermal phase lag is very small [3].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 132–136, July–August, 1975.     

Approximate statistical theory of a fluidized bed

Under the conditions of developed fluidization there are intensive fluctuations both in the fluidizing medium and in the dispersed solid phase. These motions have a decisive effect on the rheologlcal properties of the fluidized bed, and on the chemical reactions and transport processes taking place in it [1], Thus, for example, in the experiments of Wicke and Fetting [2], who investigated the heat transfer between a fluidized bed and the walls of a heated container, the effective heat transfer coefficient was found to be higher by an order of magnitude than the corresponding result for a fluidized bed held down by a wire grid so that the random motion of the solid phase was reduced. It is clear that the initial stage of any study of the structure of the fluidized bed as a whole, and of the subsequent development of any model, must involve an investigation of local structural properties, including the above fluctuations.The time variation of the individual particle velocities is due to two different causes. First, there is the interaction between the particles both through direct collisions and through the medium of the liquid phase, and, secondly, there is the interaction with the viscous fluid. These two factors are not independent, so that the set of fluidized particles has certain features characteristic for both a dense gas, with a potential intramolecular interaction, and a set of particles executing Brownian motion in a continuous medium.Any detailed statistical theory of a system of fluidized particles must be based on a representation of the random particle motions in the medium by a stochastic process with some definite properties (see, for example, [3–4]). Ideally, this theory should lead to the formulation of a transport equation which, in view of the above properties of the system, should have some of the features of both the usual Boltzman transport equation and the Fokker-Planck equation. The solution of this final equation is, of course, more difficult than the solution of the Boltzman or Fokker-Planck equations. Moreover, there is also the problem of applying this equation to different special cases. An alternative approach is to develop an approximate, but still sufficiently effective, theory of the local properties of the fluidized bed, which would combine relative simplicity in application with sufficient rigor and generality. This kind of theory is put forward in the present paper. The conclusions to which it leads are in good qualitative agreement with experiment.The author wishes to thank G. I. Barenblatt and the participants of his seminar for useful discussions.