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A diffusion-inertia model (DIM) for calculating of the monodispersed bubble flows in an isothermal flow in a turbulent liquid has been presented. The comparison of the results of the simulation with various experimental data shows that the DIM adequately describes the variety of physical phenomena occurring in bubble flows. The DIM has been incorporated in a three-dimensional computing code that allows one to use this model for engineering calculations.

A statistical model of clustering (accumulation) of monodispersed particles in homogeneous isotropic turbulence taking into account absolutely elastic collisions has been presented. The effect of collisions on clustering of particles has been analyzed in a wide range of variation of their concentration, inertia, and size.

The purpose of the study is to present an explicit self-consistent algebraic Reynolds stress model (ARSM) for two-phase turbulent flows combined with diffusion-inertia model (DIM). The modulation of turbulence by small heavy particles whose diameter is not greater than the Kolmogorov length scale is considered. The volume concentration of dispersed phase is assumed to be sufficiently small so that inter-particle collisions can be neglected; however, the mass concentration is fairly large ( ρ p ≫ ρ f ). Calculations were performed for turbulent flow in a vertical tube and turbulent flow past a backward-facing step. It was shown that the suggested turbulence model correctly predicts anisotropy of velocity fluctuations and the turbulence modulation by particles. In the presence of small particles turbulence of the flow in a vertical tube is attenuated. The level of turbulence attenuation is increased with increasing particles mass loading and decreased with increasing particle response time. In turbulent flow past the backward-facing step regions with attenuating and enhancing turbulence are observed.

A nonlinear model of turbulent viscosity (the explicit algebraic Reynolds stress model) for gas-dispersed flow with small heavy particles has been presented. The calculations have been checked by comparison with the experimental data on the effect of the fine-dispersed admixture on turbulent flow of air in a pipe.

The paper concerns the effects of particle inertia, density, and size on acceleration statistics. A simple analytical model for estimating the acceleration variance of particles suspended in an isotropic homogeneous turbulent flow field is developed. This model is capable of qualitative describing the particle acceleration variance over the entire range of the particle-to-fluid density ratio. Comparisons of model predictions with numerical simulations and experimental data are presented.

The objective of the paper is twofold: (i) to present a new statistical model for predicting the transport and dispersion of the particulate phase (particles and bubbles) in turbulent flows and (ii) to examine the performance of this model in channel flow emphasizing the effect of particle accumulation. The model presented is based on a kinetic equation for the probability density function (PDF) of velocity distribution and covers the entire range of the particle-to-fluid density ratio (from heavy particles in a gas to bubbles in a liquid).

The objective of this paper is twofold: (1) to present a statistical model of particle transport and heat transfer in turbulent flows and (2) to examine the performance of this model in various turbulent flows going from a simple flow to a more complicated one. This model is based on a kinetic equation for the probability density function of the particle velocity and temperature distributions in anisotropic turbulent flow. The model predictions compare reasonable well with numerical simulations and properly reproduce the crucial trends of computations performed in various turbulent flows.

The paper deals with collisions resulting from the interaction between particles (droplets, bubbles) and turbulent eddies of the continuous fluid medium (gas or liquid). A statistical model is developed for predicting the collision rate. This model is valid for arbitrary values of the particle-to-fluid density, the particle inertia parameter, and the ratio between the particle size and the fluid turbulent lengthscale.

A theoretical model is given for describing the inverse effect of small heavy particles on the intensity of turbulence in various flows. Analysis is performed of the effect of particles on isotropic homogeneous shear and wall turbulence.

Comparison with the data of direct numerical calculations in isotropic turbulence with constant temperature gradient is used for verifying the statistical model of heat transfer in gas-dispersed turbulent flows. Analysis is made of the effect of thermal inertia of particles on the correlations between turbulent heat fluxes and intensities of temperature fluctuations in the dispersed and continuous phases.

The objective of the paper is twofold: (i) to present a model (the so-called diffusion-inertia model) for predicting dispersion and deposition of aerosol particles in two-phase turbulent flows and (ii) to examine the performance of this model as applied to the flows in straight ducts and circular bends. The model predictions compare reasonable well with both experimental data and Lagrangian tracking simulations coupled with fluid DNS or LES.

A coherent scattering of electromagnetic waves by clusters of inertial Rayleigh particles in atmospheric turbulence is considered. A preliminary estimate based on the Maxwell-Garnett theory and the Rayleigh approximation for single clusters demonstrates an importance of the coherent scattering contribution. It is confirmed by a general solution in a combination with theoretical estimates for the two-point probability density function for low-inertia spherical particles in isotropic turbulence. An approximate analytical expression for the coefficient characterizing effect of coherent scattering by the particle clusters is derived. The calculations for small Stokes numbers typical of water droplets in cumulus clouds yield an estimate of the coherent scattering effect on the microwave radar reflection. The model suggested allows solving the inverse problem to determine the pair correlation function for cloud particles. It is expected to be important for the investigations on particle–turbulence interaction in the atmosphere. The theoretical model developed is true not only in the limit of low-inertia particles and can be potentially used at arbitrary Stokes numbers in other applications.

The development and application of a diffusion-inertia model are shown for computing the propagation and deposition of aerosol particles in turbulent flows. Comparison with experimental and numerical results points to the fact that the developed model adequately describes all the typical peculiarities of the process of the sedimentation of aerosol particles in straight and curved pipes.

Three-dimensional simulation of experiments on aerosol particle deposition in a turbulent flow is carried out. The k — ɛ turbulence model and the diffusion inertia model of particle transport and deposition were used in the simulation. The range of flow velocities and particle sizes is typical for the diffusion and turbophoresis deposition mechanisms. Deposition of particles in a turbulent flow is considered for cases of a direct vertical pipe and for a 90° bend in which the turbophoresis is coupled with centrifugal forces. The calculation results are in good agreement with experimental data. Deviations of results are comparable with those of discrete particle modeling.

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.

The effect of clustering of particles under conditions of homogeneous isotropic turbulence on Rayleigh scattering of radiation is analyzed. It is demonstrated that, similarly to the diagnostics of molecular structure of liquid using X-ray scattering, the radial distribution function of macroscopic particles, which characterizes the clustering phenomenon, may be determined by way of measuring the intensity of scattering of infrared or microwave radiation. An approximate relation for clustering coefficient is obtained, which may be employed for estimating the increase in reflected radiation of microwave radar from cumulus clouds owing to turbulent clustering of water drops in the cloud.

The purpose of this paper is twofold: (i) to present statistical models that describe particle–turbulence interactions as well as particle–particle collisions and (ii) to gain a better understanding of the effect of inter-particle collisions on transport, deposition, and preferential concentration of heavy particles in turbulent channel flows. The models presented are based on a kinetic equation for the probability density function of the particle velocity distribution in anisotropic turbulent flow. The model predictions compare reasonable well with numerical simulations and properly reproduce the crucial trends of computations.

The effect of temperature fluctuations on thermal radiation transfer under conditions of isotropic turbulence is analyzed. Simple analytical dependences are given for estimating the fluctuations of radiation intensity and of radiation flux in a plane layer of nonscattering medium. The solution of model problem is given for microwave radiation of cloud layer in the Earth atmosphere.

The objective of the paper is to advance a code for calculating the behaviour of aerosol-shaped fission products (FP) in the primary circuit of a nuclear reactor. This FP may come into the circuit from the fuel during a severe accident. The paper is focused on development of models for predicting the deposition and coagulation rates of aerosols. Comparisons of model predictions with experimental data and direct numerical simulations are discussed.

A statistical model is given for determining the kernel of coagulation in the presence of turbulent, Brownian, gradient, and gravitational mechanisms of collisions in view of accumulation effect.

The paper concerns the effect of particle inertia on acceleration statistics. A simple analytical model for predicting the acceleration of heavy particles suspended in an isotropic homogeneous turbulent flow field is developed. This model is capable of describing the influence of both Stokes and Reynolds numbers on the particle acceleration variance. Comparisons of model predictions with numerical simulations are presented.

The objective of the paper is to analyse the effects of flow turbulence and inlet moisture on the process of spontaneous condensation of supercooled steam. Calculations of steady and unsteady spontaneously condensing transonic turbulent flows in Laval nozzles are presented. Comparisons of model predictions with experimental data are discussed.

A statistical model is given for determining the kernel of collisions of small aerosol particles in homogeneous isotropic turbulence in view of Brownian motion. The suggested model is based on the kinetic equation for the probability density function of distribution of relative velocity of two particles. Analysis is performed of the effect of Brownian motion on the relative velocity of two particles (transport effect) and on the radial distribution function (accumulation effect).

Different mechanisms of the possible effect of surfactant on the condensation of supercooled steam in transonic nozzles are considered.

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A transonic turbulent flow of spontaneously condensing steam in Laval nozzles is treated. Analysis is made of the effect of flow turbulence at the nozzle inlet on steady and unsteady spontaneous condensation.

A new concept of an in-vessel corium melt catcher is proposed. The lower part of an elongated reactor vessel, which is filled with a sacrificial material of a proper composition, porosity, and arrangement, is used as such a catcher. The concept accounts of the scientific and design experience with the development of the ex-vessel corium catcher for the Tyan’van NPP with VVER-1000 reactors.

Operation of filters of postaccident decontamination of pressurized rooms of a nuclear power plant with a type-VVER-440 reactor is analyzed. The distribution of radioactive nuclides over filter stages, the time variation of the thermal state of filter, and the characteristic features of the processes of sorption in the section of fine cleaning are considered.

A statistical model is given for describing the clustering of small heavy particles in homogeneous isotropic turbulence. The suggested model is based on the kinetic equation for the probability density function of distribution of relative velocity of two particles. Analysis is performed of the phenomenon of clustering of low-inertia particles at high values of the Reynolds number.

The objective of the paper is to present a statistical model for predicting collision rates of inertial particles immersed in turbulent flow. This model is valid over the entire range of particle inertia (from the zero-inertia to the high-inertia limit) and accounts for two mechanisms influencing the collision rate, namely, the particle relative motion induced by turbulence and the accumulation effect that leads to an additional enhancement to the collision rate. The model is applicable to near-homogeneous two-phase turbulent flows with colliding or coalescing particles.

A statistical model is given for determining the kernel of collisions of inertial particles (droplets, bubbles) in the entire range of variation of the density ratio between the dispersed and continuous phases. The model is based on the assumption that the joint probability density function of velocities of continuous medium and particles is a Gaussian distribution. The ranges of variation of bubble sizes are determined, in which the decisive part is played by the “inertial” and “spatial” collision mechanisms.

The effects of particle inertia and particle-to-fluid density on the time scales that measure the dispersion of particles by isotropic turbulence are investigated. Numerical calculations are performed by means of a kinematic simulation method coupled with Lagrangian tracking of discrete particles. Moreover, a simple semi-empirical model for predicting the time scales of the fluid velocity seen by a particle as well as of the fluid–particle velocity covariance and the particle velocity variance is proposed. Comparisons between simulations and model predictions are presented.

A two-point statistical model based on the kinetic equation for the probability density function (PDF) of relative velocity is used to perform analysis of relative dispersion of inertial particles in isotropic turbulence. It is demonstrated that, in the case of highly inertial particles, the Richardson law for dispersion may be valid only in the case of very high values of the Reynolds number.

Quasi-steady-state thermal conditions of the corium bath found on a reactor vessel bottom is considered. Using approximate heat transfer models proposed earlier, a conjugate problem is solved with taking into account radiation heat transfer in the space above the bath surface. The results of calculations of heat flow-rates from the corium bath to the VVER-440 reactor vessel are presented. It is shown that water delivery to the bath surface leads to an essential decrease in the heat flowrate towards the wall in the most dangerous area, where a layer of melted steel contacts the reactor vessel, and retains a possibility of keeping the melt inside the reactor vessel under external cooling.

The objective of the paper is to present a statistical approach to modelling dispersion and preferential concentration of inertial particles suspended in the turbulent fluid. This approach departs from kinetic equations for the one-point and two-point probability density functions of particle velocity distributions in turbulent Gaussian fluid flow fields. Preferential concentration of particles in sheared as well as in isotropic turbulent flows is analysed from a unified viewpoint, and an analogy between both phenomena is discussed.

Two statistical models are given for determining the collision rate of bidisperse particles exhibiting different inertia in a homogeneous isotropic turbulence. One of these models is based on the assumption that the joint probability density function of gas and particle velocities is a Gaussian distribution. The second model follows from the kinetic equation for the probability density of the velocities of two particles. These two models are compared with each other and with data of direct numerical simulation.

This paper deals with the effects of flow turbulence and inlet wetness on the process of spontaneous condensation in Laval nozzles. Two-dimensional computations are presented and compared with experimental data for transonic flows in flat and round nozzles in the case of both the absence and the presence of initial wetness supplied to the nozzle inlet.

A statistical model is given of the formation of ordered structures (clusters) in a homogeneous isotropic turbulent field. The suggested model is based on the kinetic equation for the probability density function of relative velocity of a pair of particles. The solution of the problem is treated for a steady-state rarefied system of dispersed particles in view of pair interaction.

A three-fluid model of the dispersed-annular regime of two-phase flow is suggested. The model is based on the conservation equations of mass, momentum, and energy for the gas phase, the dispersed phase (droplets), and the film. Additionally, this model includes the equation for the number density of particles of the dispersed phase, which is used to determine the mean particle size. Calculations are compared with experimental data on the entrainment coefficient, film and droplet flow rates, film thickness, pressure drop, and droplet size.

As intermediate quantities available from various existing numerical computations, the fluid Lagrangian time scales are of primary importance in the development of probability density function models for turbulent flows. Similarly, the time scales of the fluid seen by discrete particles in two-phase flows are essential for the development of dispersion models based on stochastic differential equations. Such time scales are obviously depending not only on the particle properties but also on the fluid Lagrangian and Eulerian time scales. A model is proposed here to estimate the directional dependence of the fluid Lagrangian time scales and drift coefficients in one-directional equilibrium turbulent shear flows, based on a local homogeneity assumption in the frame of the generalized Langevin model. Through comparison with available direct and large eddy simulation predictions in channel flows and in a homogeneous shear flow, the model is shown to lead to significant improvements in the streamwise and spanwis...

The purposes of the paper are twofold: (i) to present a rational approach to the modeling of inertial particle transport in anisotropic turbulent flows and (ii) to show how the anisotropy of fluid turbulence timescales affects the particle fluctuating velocities in homogeneous shear flow. For these purposes, the directional dependence of the Lagrangian autocorrelations of fluid velocities is incorporated into the statistical probability density function (PDF) model proposed previously. The anisotropic timescale (ATS) model is evaluated against numerical simulations for homogeneous shear turbulent flows and is compared to results predicted by the isotropic timescale (ITS) model. The new ATS model for the PDF of the particle velocity distribution in turbulent flow appears to yield slightly improved results over the ITS model.

Statistical models are given of the dispersion of inertial particles in inhomogeneous wall turbulence and homogeneous isotropic turbulence. The models are based on the kinetic equations for the one-point and two-point probability density functions of particle velocities. Comparison is made with the data of direct numerical calculations for channel and isotropic flows, known from the literature.

A diffusion-inertia model is suggested for the description of the transport of low-inertia particles of arbitrary density in turbulent flows. Comparison with experimental data is used to demonstrate the validity of the suggested model for the calculation of the motion of bubble in a vertical pipe under different conditions of the effect of the gravity force.

This paper is a continuation of the papers [1, 2] dealing with the development of a three-fluid model of a two-phase medium for the dispersed-annular mode of flow. Calculations and experiments are performed to determine the liquid film thickness and the pressure drop.

Two models are given for determining the collision rate of inertial particles in a homogeneous isotropic turbulence. The advantages and disadvantages of the known models of particle collisions are discussed. One of the models suggested by us is based on the assumption that the joint probability density function of gas and particle velocities is a Gaussian distribution. The second model follows from the kinetic equation for the probability density of the relative velocity of two particles. These two models are compared with each other and with the available data of direct numerical simulation.

A transonic turbulent flow of supercooled steam under conditions of spontaneous condensation is treated. Analysis is made of the effect of flow turbulence at the nozzle inlet on the process of spontaneous condensation.

This paper is a continuation of the paper [1] dealing with the development of a three-fluid model of a two-phase medium for a dispersed-annular mode of flow. The processes of deposition of droplets from the flow core and of entrainment of droplets from liquid film are treated.

A three-fluid one-dimensional model of two-phase medium is suggested which is based on the equations of conservation of mass, momentum, and energy for the gas phase, dispersed phase (droplets), and film. The model includes the equation for the number density of particles of the dispersed phase, which is used to determine the average particle size. The calculation results are compared with the experimental data on the size of droplets in dispersed flow.