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.     

Dynamic simulation of the hydrodynamic interaction among immersed particles in Stokes flow

A numerical method for the dynamic simulation of the hydrodynamic interaction among particles in Stokes flow is developed. The method couples the quasi-static Stokes equations for the fluid with the equilibrium equations for the particles. The boundary element method is used to represent the velocity at a general field point in terms of surface velocities and stresses. However, neither the stresses nor the velocities are assumed to be known on the surface of the particles. Kinematic equations relating the linear and angular velocities at the centroids of the particles to the surface velocities are combined with the discretized boundary element equations and the equilibrium equations to generate a system of linear equations. The associated coefficient matrix is correspondent to the grand resistance matrix which relates the velocities of the particles to a given geometry.     

Lie symmetry solutions for two-phase mass flows

We apply Lie symmetry method to a set of non-linear partial differential equations, which describes a two-phase rapid gravity mass flow as a mixture of solid particles and viscous fluid down a slope (Pudasaini, J. Geophys. Res. 117 (2012) F03010, 28 pp [1]). In order to systematically explore the mathematical structure and underlying physics of the two-phase mixture flow, we generate several similarity forms in general form and construct self-similar solutions. Our analysis generalizes the results, obtained by applying the Lie symmetry method to relatively simple single-phase pressure-driven gravity mass flows, to the two-phase mass flows that include several dominant driving forces and strong phase-interactions. Analytical and numerical solutions are presented for the symmetry-reduced homogeneous and non-homogeneous systems of equations. Analytical and numerical results show that the new models presented here can adequately describe the dynamics of two-phase debris flows, and produce observable phenomena that are consistent with the physics of the flow. The solutions are strongly dependent on the choice of the symmetry-reduced model, as characterized by the group parameters, and the physical parameters of the flows. These solutions reveal strong non-linear and distinct dynamic evolutions, and phase-interactions between the solid and fluid phases, namely the phase-heights and phase-velocities.     

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.     

Pipe flow with solid particles fixed in space

A group of solid particles were hung by slender rods in a pipe to make a model of two-phase flow of coarse particles. Pressure gradient and velocities were measured for different types of the models. The drag on the particles (spheres) were obtained from measurements of pressure gradient with some assumptions. The results are summarized as follows. (1) Mean velocities of fluid are lower in the central part of the pipe than in the circumferential part. Turbulence is remarkably increased by particles. The spectrum distribution of turbulent velocity becomes flatter. These results are similar to the gas-solid flow of coarse particles in a vertical pipe. (2) At a large Reynolds number, the drag coefficient per one sphere in the group is larger than that of a single isolated sphere in a uniform flow. When the spheres are arranged along the same line in the longitudinal direction, the drag coefficient becomes smaller as the longitudinal distance between the spheres is shortened.     

垂直湍流液-固流中大颗粒的相对速度

通过量纲分析和实验测量,对于垂直、局部均匀的湍流稀态液一固流中,大颗粒的相对速度,建立了无量纲参数表达式．用分析和实验相结合的方法,确定了表达式中无量纲参数的幂次及有关系数．实验中用激光多普勒分相测量技术,分别测出流体和颗粒的时均速度结果表明,大颗粒相对速度强烈依赖于流体雷诺数,当流体雷诺数较高时,其幂次渐近于1.5。     

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.     

Investigating the diameter of solid particles effects on a laminar nanofluid flow in a curved tube using a two phase approach

In this paper, we report the results of our numerical studies on laminar mixed convection heat transfer in a circular Curved tube with a nanofluid consisting of water and 1 vol.% Al₂O₃. Three dimensional elliptic governing equations have been used. Two phase mixture model and control volume technique have been implemented to study flow field. Effects of the diameter of particles on the hydrodynamic and thermal parameters are investigated and discussed. Increasing the solid particles diameter decreases the Nusselt number and secondary flow, while the axial velocity augments. When the particles are in order of nano meter, increasing the diameter of particles, do not change the flow behaviors. The distribution of solid nanoparticles is uniform and constant in curved tube.     

Comparison of the RANS and PDF methods for air-particle flows

Two simulation methods, namely Reynolds-Averaged Navier–Stokes (RANS) equations, and Probability Distribution Function (PDF) are currently widely used for the modeling of multiphase flows. These two approaches are supplemented with appropriate closure equations that take into account all the pertinent forces and interaction effects on the solid particles, such as: particle–turbulence interactions; turbulence modulation; particle–particle interactions; particle–wall interactions; gravitation, drag and lift forces. The two methods have been used in order to simulate the turbulent particulate flow in upward pipes. The flow domain in all cases was a cylindrical pipe and the computations were carried for upward pipe flow. Monodisperse as well as polydisperse mixtures of particles have been considered. In general, the average velocity results obtained from the two methods are in close agreement, because the methods predict well the average velocity distribution of the carrier fluid as well as the solids. Thus, the differences in the average axial velocities predicted by the methods are not substantial. Differences in the turbulence intensity are more significant. A comparison of the numerical results obtained shows the relative importance of retaining the diffusion terms in both the axial and radial directions in the RANS method. Also the comparisons of the results show the relative effect of the lift forces in the distribution of solid particles.     

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.     

Local gas and liquid phase velocity measurement in a miniature stirred vessel using PIV combined with a new image processing algorithm

A method which combines standard two-dimensional particle image velocimetry (PIV) with a new image processing algorithm has been developed to measure the average local gas bubble velocities, as well as the local velocities of the liquid phase, within small stirred vessel reactors. The technique was applied to measurements in a gas–liquid high throughput experimentation (HTE) vessel of 45 mm diameter, but it is equally suited to measurements in larger scale reactors. For the measurement of liquid velocities, 3 μm latex seeding particles were used. For gas velocity measurements, a separate experiment was conducted which involved doping the liquid phase with fluorescent Rhodamine dye to allow the gas–liquid interfaces to be identified. The analysis of raw PIV images enabled the detection of bubbles within the laser plane, their differentiation from obscuring bubbles in front of the laser plane, and their use in lieu of tracer particles for gas velocity analysis using cross-correlation methods. The accuracy of the technique was verified by measuring the velocity of a bubble rising in a vertical glass column. The new method enabled detailed velocity fields of both phases to be obtained in an air–water system. The overall flow patterns obtained showed a good qualitative agreement with previous work in large scale vessels. The downward liquid velocities above the impeller were greatly reduced by the addition of the gas, and significant differences between the flow patterns of the two-phases were observed.     

Stability of the plane Couette flow of a disperse medium with a finite volume fraction of the particles

The linear hydrodynamic stability of the plane Couette flow of a suspension with a finite volume fraction of the particles is considered. The two-phase medium flow is described within the framework of the model of mutually penetrating continua which allows for the finiteness of the volume occupied by the particles. In the main flow the phase velocities are the same, while gravity is not taken into account. The stability of disperse flows with both uniform and nonuniform particle distributions is studied. The linearized system of the equations of suspension motion with the no-slip boundary conditions imposed on solid walls is reduced to the eigenvalue problem for an ordinary differential fourth-order equation in the stream function. The eigenvalues are sought using the orthogonolization method. The parametric investigation of the stability characteristics of the disperse flow is performed. It is shown that in the case of the uniform spatial distribution of the particles in the main flow, the presence of an admixture in the flow leads to a slight variation in the wave decay rates, while the flow remains stable for any permissible combinations of the dimensionless governing parameters. In the case of nonuniform distribution of inclusions the flow loses stability already for low Reynolds numbers on a wide range of the dimensionless governing parameters.     

Effect of Slow and Fast Moving Liquid Zones on Solute Transport in Porous Media

A conceptual model is developed in this article that accounts for the effect of slow and fast moving liquid zones on solute transport in porous media. The liquid phase within the porous media is divided into three zones—immobile, slow moving, and fast moving. Slow moving liquids surround the solid particles in thin layers and have lower velocity in flow. Fast moving liquids have higher velocity and are not in contact with the solid particles. Solute mass transfer occurs between the slow and fast liquids, and the slow and immobile liquids. The immobile and slow moving liquids interact with the solid matrix in the media through the mechanism of sorption and desorption. Implicit finite-difference methods are used to solve the partial differential equations that describe the slow and fast movement of solute in the porous medium. The model was validated for a laboratory column experimental data. Sensitivity analyses were conducted to ascertain the effects of the model parameters on solute movement. The effect of each parameter on retardation of the solute movement was analyzed. It was observed that the maximum retardation of solute occurs when there is high adsorption coefficient, high mass transfer rates, and high volume of slow moving liquid in the porous media.     

Two‐dimensional modeling for stability analysis of two‐phase stratified flow

The effect of wavelength and relative velocity on the disturbed interface of two‐phase stratified regime is modeled and discussed. To analyze the stability, a small perturbation is imposed on the interface. Growth or decline of the disturbed wave, relative velocity, and surface tension with respect to time will be discussed numerically. Newly developed scheme applied to a two‐dimensional flow field and the governing Navier–Stokes equations in laminar regime are solved. Finite volume method together with non‐staggered curvilinear grid is a very effective approach to capture interface shape with time. Because of the interface shape, for any time advancement, a new grid is performed separately on each stratified field, liquid, and gas regime. The results are compared with the analytical characteristics method and one‐dimensional modeling. This comparison shows that solving the momentum equation including viscosity term leads to physically more realistic results. In addition, the newly developed method is capable of predicting two‐phase stratified flow behavior more precisely than one‐dimensional modeling. It was perceived that the surface tension has an inevitable role in dissipation of interface instability and convergence of the two‐phase flow model. Copyright © 2009 John Wiley & Sons, Ltd.     

Development of a hybrid particle‐mesh method for two‐phase flow simulations

A hybrid particle‐mesh method was developed for efficient and accurate simulations of two‐phase flows. In this method, the main component of the flow is solved using the constrained interpolated profile/multi‐moment finite volumemethod; the two‐phase interface is rendered using the finite volume particle (FVP) method. The effect of surface tension is evaluated using the continuum surface force model. Numerical particles in the FVP method are distributed only on the surface of the liquid in simulating the interface between liquid and gas; these particles are used to determine the density of each mesh grid. An artificial term was also introduced to mitigate particle clustering in the direction of maximum compression and sparse discretization errors in the stretched direction. This enables accurate interface tracking without diminishing numerical efficiency. Two benchmark simulations are used to demonstrate the validity of the method developed and its numerical stability. Copyright © 2016 John Wiley & Sons, Ltd.     

Gas-Dynamic Processes in Two-Phase Flows in MHD Generators

A plane problem of a two-phase monodisperse flow of combustion products of plasma-forming composite solid propellants in the duct of a Faraday's MHD generator with continuous electrodes, including an accelerating nozzle, MHD channel, and diffuser, is considered. An algorithm based on the pseudo-transient method is developed to solve the system of equations describing the two-phase flow. Gas-dynamic processes in the channels of the Pamir-1 setup are numerically studied. It is shown that shock-free deceleration of a supersonic flow to velocities close to the equilibrium velocity of sound in a two-phase mixture and significantly lower than the velocity of sound in the gas is possible in two-phase flows.     

Interactions of particles and microbubbles with turbulence

Dilute, dispersed two-phase flows arise in many contexts ranging from solid particles or droplets in gas flows to bubbles in liquids. Many of the flows of interest are turbulent, which presents a complex problem to analyze or to determine the dominant physical processes contributing to the observed phenomena. Advances in experimental techniques have made it possible to measure directly turbulent and particle velocity fluctuations in dilute systems. This has provided a counterpart to advances in computational and analytical models and a basis on which to test these models. Three specific areas are considered: the fluctuating forces on an individual particle in an unsteady flow, the response of a solid particle to a turbulent air flow, and the corresponding response of a small bubble in turbulent liquid flows. Results from direct numerical simulations are presented for each of these, including the nonuniform distribution of particles generated by local instantaneous features of the flow. The issue of turbulence modulation at low to moderate void fractions is discussed.     

Relations Between Seepage Velocities in Immiscible,Incompressible Two-Phase Flow in Porous Media

Based on thermodynamic considerations, we derive a set of equations relating the seepage velocities of the fluid components in immiscible and incompressible two-phase flow in porous media. They necessitate the introduction of a new velocity function, the co-moving velocity. This velocity function is a characteristic of the porous medium. Together with a constitutive relation between the velocities and the driving forces, such as the pressure gradient, these equations form a closed set. We solve four versions of the capillary tube model analytically using this theory. We test the theory numerically on a network model.     

A compressible single‐temperature conservative two‐phase model with phase transitions

A model for multidimensional compressible two‐phase flow with pressure and velocity relaxations based on the theory of thermodynamically compatible system is extended to study liquid–gas flows with cavitation. The model assumes for each phase its own pressure and velocity, while a common temperature is considered. The governing equations form a hyperbolic system in conservative form and are derived through the theory of a thermodynamically compatible system. The phase pressure‐equalizing process and the interfacial friction are taken into account in the balance laws for the volume fractions of one phase and for the relative velocity by adding two relaxation source terms, while the phase transition is introduced into the model considering in the balance equation for the mass of one phase the relaxation of the Gibbs free energies of the two phases. A modification of the central finite‐volume Kurganov–Noelle–Petrova method is adopted in this work to solve the homogeneous hyperbolic part, while the relaxation source terms are treated implicitly. In order to investigate the effect of the mass transfer in the solution, a 1D cavitation tube problem is presented. In addition, two 2D numerical simulations regarding cavitation problem are also studied: a cavitating Richtmyer–Meshkov instability and a laser‐induced cavitation problem. Copyright © 2014 John Wiley & Sons, Ltd.     

Charging of disperse particles in two-phase media with unipolar charge

Charging of disperse particles with good conduction in two-phase media with unipolar charge is considered in the case when the volume concentration of the particles is low. For this, in the framework of electrohydro-dynamics [1, 2], a study is made of the charge of one perfectly conducting liquid particle in a gas (or liquid) with unipolar charge in a fairly strong electric field. The influence of the inertial and electric forces on the motion of the gas is ignored, and the velocities are found by solving the Hadamard—Rybczynski problem. We consider the axisymmetric case when the gas velocity and electric field intensity far from the particle are parallel to a straight line. The analogous problem for a solid spherical particle was solved in [3–6] (in [3], the relative motion of the gas was ignored, while in [4–6] Stokes flow around the particle was considered). The two-dimensional problem of the charge of a solid circular, perfectly conducting cylinder in an irrotational flow of gas with unipolar charge was studied in [7].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 108–115, November–December, 1980.We thank L. I. Sedov and V. V. Gogosov for a helpful discussion of the present work.