稠密气固两相流各向异性颗粒相矩方法

基于气体分子动力学和颗粒动理学方法,考虑颗粒速度脉动各向异性,建立颗粒相二阶矩模型.应用初等输运理论,对三阶关联项进行模化和封闭.考虑颗粒与壁面之间的能量传递和交换,建立颗粒相边界条件模型.采用Koch等计算方法模拟气固脉动速度关联矩.考虑气体-颗粒间相互作用,建立稠密气体-颗粒流动模型.数值模拟提升管内气固两相流动特性,模拟结果表明提升管内颗粒相湍流脉动具有明显的各向异性.预测颗粒速度、浓度和颗粒脉动速度二阶矩与Tartan等实测结果相吻合.模拟结果表明轴向颗粒速度脉动强度约为平均颗粒相脉动强度的1.5倍,轴向颗粒脉动能大约是径向颗粒脉动能3.0倍.     

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.     

Nonlinear nonequilibrium kinetic model of the Boltzmann equation for a gas with power-law interaction

A model kinetic equation approximating the Boltzmann equation on a wide range of the intensities of nonequilibrium states of gases is derived to describe rarefied gas flows. The kinetic model is based on a distribution function dependent on the absolute velocity of gas particles. Themodel kinetic equation possesses a high computational efficiency and the problem of shock wave structure is solved on its basis. The calculated and experimental data for argon are compared.     

Kinetic description of Brownian movement of heated particles in rarefied gas

Translational and rotational Brownian movement of a spherical particle in a rarefied gas is considered. It is assumed that the particle radius is much less than the free path of the molecules in the gas. The collision integral of the considered particles and the gas molecules is generalized to the case of an arbitrary law of interaction between the molecules and the particle surface, this making it possible to consider the situation when there is no thermodynamic equilibrium between the particles and the gas, in particular, the particle temperature differs from the gas temperature. By expansion with respect to the small parameter — the ratio of the molecule and particle masses — the kinetic equation of the Boltzmann type reduces to the Fokker-Planck equation for the particle distribution function. The coefficients of the equation are calculated in an explicit form for the case of diffuse interaction between the molecules and the particle surface. A dependence of the diffusion coefficients on the ratio of the particle and gas temperatures is obtained.     

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.     

Interaction of molecules with the surface of a solid

The interaction of molecules and atoms with the surface of a solid is considered on the basis of classical mechanics. A two-dimensional square lattice with atoms arranged at the lattice points was taken as the model for describing a solid. It is assumed that only neighboring atoms interact in the solid, while the gas molecules interact with the atoms located in its surface layer.As a result of collisions with the surface, a gas molecule loses a part of its kinetic energy, and this process is characterized by the energy accommodation coefficient. In addition, another coefficient is introduced which takes account of that part of the energy of translational motion converted into energy of internal motion of the molecule (vibration and rotation). The possibility of the occurrence of inelastic losses and some special features of this phenomenon are illustrated by the interaction of a diatomic molecule with an isolated atom.The available experimental data on the interaction of particles of gas with the surface of a solid are essentially associated with the low-energy region (the temperature of the gas is less than or on the order of several hundreds of degrees). One of the objectives of this research was to find the distribution function of particles reflected from the surface; in particular, the hypothesis of diffuse-specular reflection is tested [1–3]. However, the few experimental results provide evidence of the effect of a large number of factors on the nature of the interaction, rather than make it possible to establish strict laws for the process.The theoretical investigations were conducted along the line of improving simple models-instead of modelling a solid by a one-dimensional chain of atoms [4,5], two and three-dimensional lattices are introduced [6–8]. It is noted in [6] that the interaction of gas particles with a one -dimensional chain differs from the interaction with a three-dimensional lattice, and this fact can lead to a considerable divergence in the values of the accommodation coefficient when the mass of the incident molecule is comparable with the mass of an atom of the solid. It also follows from [6] that if we describe the interaction between gas atoms and those of the solid by the Morse potential, then we can select the parameters of the potential so that the calculated data will agree with the experimental data. Moreover, good results are obtained if we make use of parameters of the potential determined on the basis of the combination rule [7].The interaction of atoms of gas with a three-dimensional lattice of finite size is considered in [8]. Forces with the Lennard-Jones potential act between gas atoms and atoms of the solid. The classical equations of motion of all particles were solved numerically on electronic computers.In these references, a gas particle (molecule or atom) is regarded as a whole; however, the problem of the influence of internal degrees of freedom on the coefficients of energy and momentum exchange between the gas particles and the solid is interesting. An attempt is made in this work to take this influence into account on the basis of classical mechanics.     

Vapor Flow with Evaporation–Condensation on Solid Particles

Strongly nonequilibrium vapor (gas) flows in a region filled by solid particles are considered with allowance for particlesize variation due to evaporation–condensation on the particle surface. The study is performed by directly solving the kinetic Boltzmann equation with allowance for the transformation of the distribution function of gas molecules due to their interaction with dust particles.     

高浓度固-液两相流紊流的动理学模型

采用分子动理学方法,基于固-液两相流液相分子或颗粒相颗粒的Boltzmann方程,对Boltzmann方程分别取零矩和一次矩,则得到高浓度固-液两相流紊流的连续方程和动量方程,再和较成熟的低浓度两相流连续方程和动量方程比较,取低浓度两相流控制方程中较成熟合理的有关项和高浓度时由动理学方法推导出的颗粒间碰撞项,则得到高浓度固-液两相流紊流的最终控制方程:连续方程和动量方程.     

Effect of solid mass flux on anisotropic gas–solid flow in risers determined with an LES-SOM model

Within the framework of the two-fluid approach, gas was treated with a large-eddy simulation and a sub-grid-scale (SGS) turbulent kinetic energy model while particles were treated with a second-order-moment method to describe the anisotropy of the fluctuating velocity. A modified Simonin model was derived for the gas–solid interphase fluctuating energy transfer. The anisotropic gas–solid flow in a circulating fluidized bed was investigated. Predictions were in good agreement with experimental data. The distributions of the second-order moment of particles and SGS-turbulent kinetic energy of gas were simulated at different solid mass fluxes. The effects of the solid mass flux on the particle second-order moment, particle anisotropic behavior, gas SGS-turbulent kinetic energy and gas SGS energy dissipation were analyzed for the circulating fluidized bed.     

Kinetic model of a collisional admixture in dusty gas and its application to calculating flow past bodies

Using the methods of statistical physics, the basic kinetic equation describing the dynamics of a polydisperse admixture of solid particles in a dilute dusty-gas flow is derived. Particle rotation, inelastic collisions, and interaction with the carrier gas are taken into account. The basic kinetic equation is used to obtain a Boltzmann-type equation for the one-particle distribution function, for which the boundary conditions for the problem of dusty-gas flow past a body are formulated. On the basis of the kinetic model developed, using direct statistical modeling, the flow patterns and the fields of the dispersed-phase macroparameters in a uniform crosswise dusty-gas flow past a cylinder are obtained for various free-stream particle sizes and concentrations. Sankt-Peterburg. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 81–97, May–June, 2000. The work received financial support from the Russian Foundation for Basic Research (projects 96-01-01467 and 99-01-00674).     

Stagnation region of a blunt body in two-phase hypersonic flow

The fluid-mechanics equations of a two-velocity, two-temperature medium are used to investigate flow near the stagnation point of a blunt body washed by a hypersonic stream of gas containing solid or liquid deformed particles. The effect of particles of the gasdynamic flow parameters is analyzed. A relaxation layer was found to occur near the body, with marked changes in the gas parameters. It is shown that the presence of particles in the flow reduces the shock stand-off distance. The results of computations on the dynamics and heating of particles in the shock layer are discussed. A solution in finite form is obtained in the limiting case of fine particles by the method of asymptotic expansions. The motion of solid or liquid particles in hypersonic shock layers has been the subject of several papers [1–6], in which particle dynamics was examined, assuming that the particles have a negligible influence on the gasdynamic flow parameters. The solutions obtained are therefore limited to the case of low mass particle concentration in the incident flow. A numerical solution not subject to this limitation was obtained in [7] for supersonic two-phase flow over a wedge.     

Comparison of Kinetic and Hybrid-Equilibrium Models Simulating Colloid-Facilitated Contaminant Transport in Porous Media

The presence of colloidal particles in groundwater can enhance contaminant transport by reducing retardation effects and carrying them to distances further than predicted by a conventional advective/dispersive equation with normal retardation values. When colloids exist in porous media and affect contaminant migration, the system can best be simulated as a three-phase medium. Mechanisms of mass transfer from one phase to another by colloids and contaminants can be kinetic or equilibrium-based, depending on the sorption–desorption reaction rate between the aqueous and solid phases. When the rate of sorption between the water phase and the solid phase(s) is not much greater than the rate of change in contaminant concentration in the water phase, kinetic sorption models may better describe the phenomenon. In some cases of modeling one or more mass transfer processes, a useful simplification may be to introduce the local equilibrium assumption. In this study, the local equilibrium assumption for sorption processes on colloidal surfaces (hybrid equilibrium model) was compared with kinetic-based models. Sensitivity analyses were conducted to deduce the effect of major parameters on contaminant transport. The results obtained from the hybrid equilibrium model in predicting the transport of colloid-facilitated groundwater contaminant are very similar to those of the kinetic model, when the point of interest is not at contaminant and colloid source vicinities and the time of interest is sufficiently long for imposed sources.     

A Kinetic Approach to the Plane Poiseuille Flow Over a Porous Matrix

The Poiseuille–Couette gas flow in a channel and the gas flow through an adjacent porous medium are considered when the governing equations are obtained via a molecular kinetic approach based on the Boltzmann equation. The mass continuity, momentum balance and energy conservation are written for the gas in the contiguous regions, whereas the behavior of the solid matrix obeys to the heat diffusion equation. Two different space scalings lead to different forms of the equations for the steady flow through the fully saturated matrix. The boundary conditions at the interface between the two domains are investigated via a matching procedure.     

Calculation of the transport coefficients and slip velocities for molecules in the form of solid spheres

A solution of the Boltzmann equation is carried out by the Monte Carlo method for problems of rarefied gasdynamics in a linear formulation. The problems are solved by calculating the transport coefficients and slip velocities on a solid wall for molecules in the form of solid spheres. The accuracy of the method due to various parameters of the computational scheme in the solution of the problem is investigated by calculating the transport coefficients for pseudo-Maxwellian molecules.The Boltzmann kinetic equation is a complex integro-differential equation which is very difficult to solve and analyze. Hence, the solution of even one-dimensional problems and for the linearized Boltzmann equation turns out to be quite difficult, and such problems are solved by approximate methods (the expansion in Knudsen numbers, the method of moments, the expansion in series, etc. [1]). A method of solving the linearized Boltzmann equation by the Monte Carlo method is proposed in [2]. An exact solution of a number of problems of rarefied gas dynamics has been obtained by this method [3, 4]. However, the method was applied for pseudo-Maxwellian molecules, for which the collision cross section is inversely proportional to the relative velocity of the colliding particles =₀/g.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 155–158, March–April, 1971.In conclusion, the author is grateful to M. N. Kogan for formulating the problem and for great assistance provided during the research, and also to V. I. Vlasov, S. L. Gorelov and V. A. Perepukhov for assistance in compiling the program.     

Boltzmann equation and moment equations in curvilinear coordinates

Various forms of writing the Boltzmann equation in an arbitrary orthogonal curvilinear coordinate system are discussed. The derivation is presented of a general transport equation and moment equations containing moments of the distribution function no higher than the fourth. For a gas of Maxwellian molecules it is shown that the system of moment equations for flows which differ little from equilibrium flows transforms into the system of hydrodynamic equations. The resulting equations may be useful in solving problems on motions of a rarefied gas by the moment methods. The results are valid for both the Boltzmann equation and model kinetic equations.The author wishes to thank A. A. Nikol'skii for discussions and helpful comments.     

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.