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].     

Numerical modeling of a two-dimensional vertical turbulent two-phase jet

The results of numerically modeling two-dimensional two-phase flow of the “gas-solid particles” type in a vertical turbulent jet are presented for three cases of its configuration, namely, descending, ascending, and without account of gravity. Both flow phases are modeled on the basis of the Navier-Stokes equations averaged within the framework of the Reynolds approximation and closed by an extended k-? turbulence model. The averaged two-phase flow parameters (particle and gas velocities, particle concentration, turbulent kinetic energy, and its dissipation) are described using the model of mutually-penetrating continua. The model developed allows for both the direct effect of turbulence on the motion of disperse-phase particles and the inverse effect of the particles on turbulence leading to either an increase or a decrease in the turbulent kinetic energy of the gas. The model takes account for gravity, viscous drag, and the Saffman lift. The system of equations is solved using a difference method. The calculated results are in good agreement with the corresponding experimental data which confirms the effect of solid particles on the mean and turbulent characteristics of gas jets.     

Energy-balance equation for turbulent pulsations in the theory of the free turbulent boundary layer

Semiempirical expressions are proposed for the coefficient of turbulent viscosity and for the scale of turbulence in the equations for the free turbulent boundary layer in an incompressible fluid, these equations consisting of the equation of continuity, the equations of motion, and the equation for the average energy balance in the turbulent pulsations. The advantage of the expressions over the existing ones is that the two empirical constants in the equations have nearly the same values for circular and plane turbulent streams and also for a turbulent boundary layer at the edge of a semiinfinite homogeneous flow with a stationary fluid. The mean-energy distribution and the mean energy of the turbulent pulsations computed in this paper agree well with the experimental values.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 75–79, November–December, 1970.     

视密度加权的时平均统一二阶矩两相湍流模型

在气粒两相湍流的双流体模型中,颗粒相的视（表观）密度是有脉动的,在时平均的统一二阶矩（USM）模型中出现了和颗粒数密度或视密度脉动有关的项和方程,使模型方程比较复杂。实际上,用LDV或PDPA测量的流体（用小颗粒代表）和颗粒速度都是颗粒数加权平均的结果。因此,在视密度加权平均基础上推导两相湍流模型更为合理。通过推导和封闭了视密度加权平均的统一二阶矩模型（MUSM）方程组,改进了两相速度脉动关联的封闭,并引入了颗粒遇到的气体脉动速度及其输运方程。MUSM模型可以减少所用方程数,节省计算量。视密度加权平均的统一二阶矩两相湍流模型是一种对原有时间平均的统一二阶矩模型和改进和发展。     

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.     

Turbulent cascade modeling of single and bubbly two-phase turbulent flows

The analysis of turbulent two-phase flows requires closure models in order to perform reliable computational multiphase fluid dynamics (CMFD) analyses. A spectral turbulence cascade-transport model, which tracks the evolution of the turbulent kinetic energy from large to small liquid eddies, has been developed for the analysis of the homogeneous decay of isotropic single and bubbly two-phase turbulence. This model has been validated for the decay of homogeneous, isotropic single and two-phase bubbly flow turbulence for data having a 5 mm mean bubble diameter. The Reynolds number of the data based on bubble diameter and relative velocity is approximately 1400.     

Calculation of a turbulent two-phase isobaric jet

A numerical calculation is carried out by the finite-difference method based on proposed equations for a turbulent submerged jet containing an admixture of solid particles. The relative longitudinal particle velocity and the influence of particles on the turbulence intensity are taken into account. The calculated results adequately agree with available experimental data. A turbulent two-phase jet is examined in [1] on the basis of the theory for a variable density jet, assuming equal mean velocities for the gas and particles and not considering the influence of particles on the turbulence intensity. Particles are analogously taken into account by a noninertial gas mixture in [2, 3], and a particle Schmidt number of 1.1 is assumed in [4]. A model is proposed in [5] which takes into account the influence of particles on the turbulence intensity of the gas phase. Problems concerning the initial and main sections of a submerged jet were solved in [6] by the integral method on the basis of this model and the assumed equality of the mean velocities of the gas and particles. Turbulent mixing of homogeneous two-phase flows with allowance made for dynamic nonequilibrium of the phases is considered in [7]. However, the neglect of turbulent transfer of particle mass and momentum led to a physically unrealistic solution for the particle concentration in the far field of the mixture. A two-phase jet is considered in the present work on the basis of the theory of a two-velocity continuous medium [8, 9] with allowance made for turbulent transfer of particle mass and momentum. The influence of particles on the turbulence intensity of the gas phase is taken into account with the model of [5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 57–63, September–October, 1976.The author acknowledges useful comments and discussion.of the work by G. N. Abramovich and participants of his seminar. The author sincerely thanks I. N. Murzinov for scientific supervision of the work.     

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.     

网格湍流微结构的实验研究

本文试验研究了网格湍流从前期到后期整个连续衰变过程即湍能和Taylor微尺度随时间的变化规律,以及高阶速度相关系数的变化规律,试验结果是在一个低湍流度、低速风洞内,用TSI热线风速仪测得的,而拟均匀各向同性湍流是用在风洞试验段入口处加网格产生的,本文的试验结果与文献[1]提出的涡旋结构理论的计算结果做了比较,发现理论计算的和曲线与本文实测值非常吻合,本文的实测结果与Townsendt早年的试验以及Beanett七十年代末的试验也做了比较。结果表明,这些试验结果彼此也很一致,因而,所有这些试验结果与理论计算值都相互获得了验证。     

Filtered particle tracking in isotropic turbulence and stochastic modeling of subgrid-scale dispersion

A numerical study based on the Eulerian–Lagrangian formulation is performed for dispersed phase motion in a turbulent flow. The effect of spatial filtering, commonly employed in large-eddy simulations, and the role of the subgrid scale turbulence on the statistics of heavy particles, including preferential concentration, are studied through a priori analysis of DNS of particle-laden forced isotropic turbulence. In simulations where the subgrid scale kinetic energy attains 30–35% of the total we observe the impact of residual fluid motions on particles of a smaller inertia. It is shown that neglecting the influence of subgrid scale fluctuations has a significant effect on the preferential concentration of those particles. A stochastic Langevin model is proposed to reconstruct the residual (or subgrid scale) fluid velocity along particle trajectories. The computation results for a selection of particle inertia parameters are performed to appraise the model through comparisons of particle turbulent kinetic energy and the statistics of preferential concentrations.     

分层流体中栅格湍流的特性

本文考察了分层流体中栅格湍流衰减和演化过程的细节,通过对空间场的信息二维图象处理,获得了湍流动能、耗散率、功率谱及多种湍流尺度等主要湍流特征量。结果显示层结加速了湍流垂直动能的衰减,而水平动能的衰减几乎很少受影响。流场各特征量与前人结果也符合得较好。     

Statistical models for predicting the effect of bidisperse particle collisions on particle velocities and stresses in homogeneous anisotropic turbulent flows

The purpose of this paper is to present and compare two statistical models for predicting the effect of collisions on particle velocities and stresses in bidisperse turbulent flows. These models start from a kinetic equation for the probability density function (PDF) of the particle velocity distribution in a homogeneous anisotropic turbulent flow. The kinetic equation describes simultaneously particle–turbulence and particle–particle interactions. The paper is focused on deriving the collision terms in the governing equations of the PDF moments. One of the collision models is based on a Grad-like expansion for the PDF of the velocity distributions of two particles. The other model stems from a Grad-like expansion for the joint fluid–particle PDF. The validity of these models is explored by comparing with Lagrangian simulations of particle tracking in uniformly sheared and isotropic turbulent flows generated by LES. Notwithstanding the fact that the fluid turbulence may be isotropic, the particle velocity fluctuations are anisotropic due to the impact of gravitational settling. Comparisons of the model predictions and the numerical simulations show encouraging agreement.     

Mechanical realization of the medium imitating electromagnetic fields and particles

A turbulent fluid exhibits elastic properties. Turbulence may generate in the medium a body force. Small perturbations of averaged ideal turbulence reproduce the electromagnetic field. The averaged fluid velocity corresponds to the magnetic vector-potential, the perturbation of the averaged pressure to the scalar potential, and the body force due to nonuniformity of Reynolds stresses corresponds to the electric field. Discontinuities of the medium model particles and electric charges. A vapor bubble can be taken as a model of the neutron. Under the action of turbulent fluctuations the bubble turns into the inhomogeneity of the elastic medium. The region of rarefaction of the medium thus formed produces in the turbulent fluid the field of positive perturbation of the turbulence energy. This medium defect may serve as a model of the proton. In order to maintain the energy balance the respective field of negative perturbation of the turbulence energy should be formed. This field can be viewed as generated by an isle of quiescent fluid. The latter singularity models the electron. Electromagnetic interactions may be concerned with the entrainment of the pressure center in the turbulent aether. The magnetic force is due to the entrainment by the fluid stream, and the electric force is due to the entrainment by the turbulence.     

Use of an equation describing the transport of the energy of turbulent pulsations to calculate two-phase jets

A finite-difference calculation of an axisymmetric two-phase jet is made on the basis of a turbulence model and an equation describing the transport of the energy of the turbulent pulsations of the carrier phase. The influence of the particles of the disperse phase on the energy of the turbulent pulsations is taken into account.     

湍流边界层中固体颗粒运动的数值模拟

根据Ｌａｇｒａｎｇｅ颗粒运动微分方程及不可压缩湍流边界层中流体的壁面速度分布规律，数值求解了颗粒在湍流边界层中的运动，考虑了Ｓａｆｆｍａｎ升为对颗粒运动的影响，壁面对运动阻力的影响，给出了固体颗粒沉积边壁，在边界层外缘上所需的最小速度和最小入射角，计算结果还表明边界层对固体颗粒撞击边壁的速度和入射角有较大影响，从数值结果可可以发现一个重要现象。     

Numerical calculations of two-phase turbulent round jet

The investigation deals with the effect of suspended particles on the dissipation of turbulence energy.Additional dissipation is hypothesized as caused by the relative velocity between the particles and the fluid, and by structural changes of turbulence.An extended model for the turbulence energy equation is derived and applied to the case of an axially symmetrical free jet. The governing equations are solved numerically, and the results are compared with experimental data. Reasonably good agreement is obtained.     

Transport, Mixing and Agglomeration of Particles in Turbulent Flows

This paper describes methods and approaches that have been used to simulate and model the transport, mixing and agglomeration of small particles in a flowing turbulent gas. The transported particles because of their inertia are assumed not to follow the motion of the large scales of the turbulence and or the motion of the small dissipating scales of the turbulence. We show how both these behaviours can be represented by a PDF approach analogous to that used in classical kinetic theory. For large scale dispersion the focus is on transport in simple generic flows like statistically stationary homogeneous and isotropic turbulence and simple shear flows. Special consideration is given to the transport and deposition of particles in turbulent boundary layers. For small scale transport the focus is on how the small scales of turbulence together with the particle inertial response enhance collision processes like particle agglomeration. In this case the importance of segregation and the formation of caustics, singularities and random uncorrelated motion is highlighted and discussed.     

Turbulence in multiphase flow

We establish in this paper the foundations of a two-field turbulent flow model that includes two turbulent fields. The case of dispersed particles in an incompressible carrier fluid is treated here, but the very presence of these two fields allows for the generalization of the model to the instability-induced turbulent mixing of two materials. This model describes both cases of turbulent mass diffusion and small drag regime, “wave-like” interpenetration of the two components. It also includes the damping of the turbulence due to the presence of the particles. In addition, a theoretical derivation of the drag-induced decay of the large-scale turbulence kinetic energy is proposed as another mechanism specific to turbulent multiphase flow.