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.     

On the necessity of including the turbulence experienced by an inertial particle in Lagrangian random-walk models

Many Lagrangian models have been developed in the literature in order to simulate the dispersion of particles in turbulent gas and liquid flows. The purpose of the present study is to critically analyze the impact of different fluid autocorrelation functions on the behavior of the dispersed phase in homogeneous isotropic turbulence. The “purely Lagrangian” autocorrelation, well-appropriate for turbulent diffusion problems, needs to be modified by other more sophisticated autocorrelation coefficients, including either space–time characteristics or better particle parameters to obtain appropriate numerical dispersion results in concordance with a recent theory.     

Turbulent diffusion in a medium subjected to uniform deformation

Turbulent diffusion in media subjected to uniform deformation caused by the presence of average-velocity gradients which are constant throughout the space is an idealization of real processes, in particular, of processes such as diffusion in channels of variable cross-sectional area [1], in the lowest layers of the atmosphere [2], etc. In this article we formulate the problem of the connection between the statistical characteristics of the transfer of a passive substance in turbulent diffusion in deformed media with the statistical characteristics of the turbulence. The statistical transfer characteristics generally used are the first two moments of the vector of random displacement of a liquid particle under the action of turbulent pulsations in velocity: the average displacement and the components of the dispersion tensor of the displacement of a liquid particle. We obtain connecting relations for the dispersion tensor of a liquid particle in turbulent diffusion of a passive substance in a uniform turbulent medium subjected to uniform deformation caused by average-velocity gradients which are constant throughout the space. These relations are a generalization of known expressions for undeformed media [2, 3]. We investigate the case of rapid deformation when the turbulent characteristics of the medium vary in accordance with the linear theory [4].     

Effect of external turbulence on the development of a turbulent jet

The effect of external turbulent agitation on jet development has been investigated in [1–3]. The difference of the method employed in the present work lies in the assumption that the turbulence scale of the external flow is substantially larger than the turbulence scales in either the jet or the mixing layer. Utilizing this assumption, it becomes possible to solve separately the energy equations for the turbulence of the external flow and of the jet. Solutions obtained on the basis of this assumption are found to be in qualitative agreement with experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 24–29, January–February, 1977.     

细颗粒泥沙净冲刷和输移的大涡模拟研究

在传统水沙输移数值模拟研究中一般采用雷诺时均模拟技术(Reynolds-averaged simulation,RANS).与RANS相比,大涡模拟技术(large eddy simulation,LES)能够更加精确反映细部流动结构,计算机的发展使得采用LES探讨水流和泥沙运动规律成为可能.本文尝试给出净冲刷条件下悬沙计算的边界条件,采用动态亚格子模式对循环槽道和长槽道中的水流运动和泥沙输移进行了三维大涡模拟研究.利用直接数值模拟(direct numerical simulation,DNS)结果对LES模型进行了率定,计算结果符合良好,在此基础上初步探讨了泥沙浓度、湍动强度和湍动通量等的分布特征.结果表明,净冲刷条件下输沙平衡时泥沙浓度符合Rouse公式分布,单向流动中泥沙浓度沿着流向逐渐增大.泥沙浓度湍动强度和湍动通量都在近底部达到最大值,沿着垂向迅速减小.湍动黏性系数和扩散系数基本上在水深中间处达到最大.湍动Schmidt数沿着水深方向不是常数,在近底部和自由水面附近较大,在水深中间处较小.     

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.     

Theory on transverse migration of particles in a turbulent two-phase suspension flow due to turbulent diffusion—I

A new theoretical model has been developed to explain the behavior of transverse particle transport in turbulent flow of a dilute two-phase suspension due to turbulent diffusion. This model is based on the ability of a particle to respond to surrounding fluid motion and depends on particle size and density relative to the carrier fluid, the fractional variation in particle concentration in the transverse direction as well as the existing turbulence structure of the surrounding fluid. The model developed in this investigation has been formulated by dividing the transverse fluid velocity, as seen by a particular particle, into two superimposed components representing, respectively, the transverse turbulent fluid fluctuations and an apparent transverse local fluid drifting velocity due to the effect on the transverse oscillatory component of fluid motion by the transverse concentration distribution of particles. A subsequent paper will show that the theory (together with other new results on the concentration effects on particle drag and lift and fluid turbulence properties) can help to explain the phenomena measured previously.     

Modelling of the diffusion of fluid particles in turbulent flows approaching a circular cylinder by random Fourier modes and random flight

To investigate the diffusion of fluid particles around a cylinder in a turbulent flow, we have developed two new types of model for simulating the trajectory of particles:(1) a model combining random Fourier modes and random flight (RF); (2) a pure kinematic simulation (KS) by random Fourier modes. In model 1 the large-scale turbulence is simulated by a sum of random Fourier modes varying in space and time, and the small-scale random motion of particles is simply modelled by an Itô type of stochastic differential equation with a memory time comparable to the Lagrangian time scaleT _s ^L of the small-scale motion. In model 2, both large- and small-scale turbulence is simulated using random Fourier modes. The change of turbulence around the cylinder is modelled by rapid distortion theory (RDT), although the small-scale motion of particles in the RF model is simply assumed to keep the homogeneous random behaviour. These models give very similar and realistic trajectories showing rapid changes of direction due to the small-scale motion.     

Theory of early onset of turbulence in a dispersion

The effects of phase interaction are considered for the hydrodynamic instability of a flowing dispersion, and a model is proposed for premature onset of turbulence.A disperse phase can stabilize the laminar flow of a suspension or aerosol by impeding the growth of neutral perturbations and the onset of turbulent conditions. This stabilizing action is seen also in the shift of the curves for neutral stability towards higher Reynolds number R [1,2] and in the occurrence of additional dissipation of pulsation energy as a resutt of phase interaction [3,4].These conclusions from the theory are closely confirmed by tests on the motion of finely divided or neatly equal-density systems, where turbulence starts at R. higher than those for the homogeneous dispersion medium, However, systems with coarser particles and substantial density differences show the converse effect (lower R,). This has been observed [5] for a liquid containing sand in vertical pipes, where R. was only t60-200 instead of 2300. Similar results have been obtained with oil in sands [6]. Loss of laminar flow at low R. is characteristic also of emulsions having disperse material less dense than the dispersion medium, e.g., water-oil emulsions of very low concentration [7]. The quantity R. is not an invariant characteristic of the type of motion but is dependent on the physical parameters of the phases and the concentration.This indicates that a different mechanism accompanies the usual mechanism for toss of hydrodynamic stability (from inertial effects, undamped shear waves, and secondary flows); energy is transferred to pulsations and causes premature onset of turbulence. This energy transfer must be due to interaction between phases, which produces differences between the local speeds of liquid and particles; it does not require shear in order to arise. A preliminary analysis of early turbulence thus can be based on the stability of a one-dimensional homogeneous two-phase flow.     

Turbulent collision of inertial particles: Point-particle based,hybrid simulations and beyond

Point-particle based direct numerical simulation (PPDNS) has been a productive research tool for studying both single-particle and particle-pair statistics of inertial particles suspended in a turbulent carrier flow. Here we focus on its use in addressing particle-pair statistics relevant to the quantification of turbulent collision rate of inertial particles. PPDNS is particularly useful as the interaction of particles with small-scale (dissipative) turbulent motion of the carrier flow is mostly relevant. Furthermore, since the particle size may be much smaller than the Kolmogorov length of the background fluid turbulence, a large number of particles are needed to accumulate meaningful pair statistics. Starting from the relative simple Lagrangian tracking of so-called ghost particles, PPDNS has significantly advanced our theoretical understanding of the kinematic formulation of the turbulent geometric collision kernel by providing essential data on dynamic collision kernel, radial relative velocity, and radial distribution function. A recent extension of PPDNS is a hybrid direct numerical simulation (HDNS) approach in which the effect of local hydrodynamic interactions of particles is considered, allowing quantitative assessment of the enhancement of collision efficiency by fluid turbulence. Limitations and open issues in PPDNS and HDNS are discussed. Finally, on-going studies of turbulent collision of inertial particles using large-eddy simulations and particle-resolved simulations are briefly discussed.     

Turbulence energy balance in reacting turbulent flows

The turbulence accompanying combustion and the propagation of detonation waves in gases has been studied theoretically and experimentally in many papers [1–8]. The attention of researchers has been concentrated on essential questions like how the turbulent flow field interacts with the kinetics of the chemical reaction and to what extent the process of chemical change is intensified, and how the turbulence itself is deformed by the heat released and the accompanying expansion of the gases. The various mechanisms proposed for these phenomena are based on various hypotheses concerning the structure of the combusion zone and the determinative stage of the interaction of the turbulence with the chemical-reaction kinetics. The mechanism of turbulence generation by combustion proposed in a number of papers [3–6] is based on the observation in turbulent flow of a weakly curved flickering laminar flame. This gives rise to a nonuniform flow field of the gas, part of the energy of which goes over into the energy of turbulent fluctuations. Other authors [7, 8] considered the turbulence field to interact with the chemical-reaction kinetics via a volume mechanism and suggested a criterion of turbulence intensification based on certain physical considerations, e.g., the condition for the intensification of thermogaskinetic oscillations proposed by Rayleigh [9]. In the present paper the problem is analyzed by introducing Kolmogorov's general equation for the turbulence energy balance in reacting turbulent flows [10]. In accordance with, this equation the turbulence energy can vary due to energy exchange between the turbulent motion and the mean gas flow as a result of the work on turbulent mass transport in the acceleration field of the mean flow, and due to the effect of pressure fluctuations on the rate of thermal expansion from the chemical reaction. Each of these effects is considered and analyzed.     

Two-scale semiempirical theory of turbulent boundary layers and jets

To characterize the turbulence of boundary layers in the energy-bearing interval of wave numbers several turbulence scales are sometimes used (for example, [1, 2]). In particular, the universality of the semiempirical model of turbulence [2] can be extended in this way. A turbulence model with one equation (energy balance of the turbulence) has been constructed and used [3–6] and it has been established that the number of problems that can be solved for a universal choice of the values of the empirical coefficients increases appreciably if not one but two turbulent scales are used. In the present paper, it is shown that the introduction of a second scale makes it possible to take into account the interaction of shear layers in flows with two shear layers (for example, a channel or jet), and also to take into account the influence of turbulence of an external flow on a boundary layer. The interaction of shear layers is taken into account in theories containing a transport equation for the turbulent frictional stress _t (for example, [7]), in which the essence of the interaction reduces to diffusion of _t from layer to layer. In the present paper, a predominant volume interaction effect is assumed. It takes the form of a difference between the interaction of large-scale vortices with a shear deformation motion in flows with one and two shear layers, and also in the presence of turbulence in an external flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 8, pp. 17–25, November–December, 1982.     

A generalized mass transfer law unifying various particle transport mechanisms in dilute dispersions

A generalized mass transfer law for dilute dispersion of particles (or droplets) of any sizes suspended in a fluid has been described, which can be applied to turbulent or laminar flow. The generalized law reduces to the Fick’s law of diffusion in the limit of very small particles. Thus the study shows how the well-known and much-used Fick’s law of diffusion fits into the broader context of particle transport. The general expression for particle flux comprises a diffusive flux due to Brownian motion and turbulent fluctuation, a diffusive flux due to temperature gradient (thermophoresis plus stressphoresis) and a convective flux that arises primarily due to the interaction of particle inertia and the inhomogeneity of the fluid turbulence field (turbophoresis). Shear-induced lift force, electrical force, gravity, etc. also contribute to the convective flux. The present study includes the effects of surface roughness, and the calculations show that the presence of small surface roughness even in the hydraulically smooth regime significantly enhances deposition especially of small particles. Thermophoresis can have equally strong effects, even with a modest temperature difference between the wall and the bulk fluid. For particles of the intermediate size range, turbophoresis, thermophoresis and roughness are all important contributors to the overall deposition rate. The paper includes a parametric study of the effects of electrostatic forces due to mirror charging. The present work provides a unified framework to determine the combined effect of various particle transport mechanisms on mass transfer rate and the inclusion of other mechanisms not considered in this paper is possible.     

Investigation of the microstructure a turbulent jet in a wake

The study of the characteristics of the turbulence in the boundary layer and in free jets is one of the most important problems of the aerodynamics of viscous fluids. The accumulation of information on the pulsation characteristics of jet flows and the establishment of the corresponding governing laws may serve to verify the basic hypotheses of the semiempirical theories of turbulence, and also for the development of more advanced computational methods. In many cases the measurement of the pulsation characteristics of turbulent jets is of practical interest.The studies made up till now [1–5] of the microstructure of turbulent flow in the primary region of submerged axisymmetric jets have made it possible to obtain several interesting results. In particular, in addition to the average velocity profiles, hot-wire anemometric equipment has been used to measure the normal and tangential Reynolds stresses and also the intermittency factor in cross sections of the jet, the distribution of the intensity of the longitudinal and lateral velocity pulsations along the axis, the correlation coefficients and the corresponding integral turbulence scales, etc. These measurements have made it possible to draw several important conclusions on the mechanism of turbulent exchange, on the order of the terms omitted in the equation of motion, and on the semiempirical theories of turbulence [6–9].The common deficiency of the studies mentioned above is that near the boundary of a submerged jet, where the average velocity is practically equal to zero, the intensity of the pulsations is so great that it makes the reliability of the results obtained by means of the hotwire anemometer questionable. In this connection Townsend [6] indicated the advisability of studying the microstructure of a turbulent jet issuing into a low-velocity ambient flow.The present study had as its objective the investigation of the microstructure of the primary region of an axisymmetric jet in a wake flow over quite a broad range of the flow ratio parameter m=u/u₀;here u₀ is the average velocity at the nozzle exit, u is the velocity of the ambient stream. For various values of the parameter m in the primary region of the jet measurements were made of the profiles of the three components of the pulsation velocity and the Reynolds shear stresses, and also the values of the average velocity and two components of the pulsation velocity at a large number of points on the jet axis. The measured profiles of the Reynolds shear stresses were compared with the corresponding profiles calculated on the basis of the boundary layer equations from the experimentally determined average velocity profiles. For two values of the parameter m, in one of the sections of the jet measurements were made of the correlation coefficients of the longitudinal components of the pulsation velocity and the variation across the jet of the integral turbulence scale was determined.The results obtained give an idea of the influence of the parameter m on the characteristics of the turbulent jet in an ambient stream.     

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.     

Turbulent thermal diffusion in a multi-fan turbulence generator with imposed mean temperature gradient

We studied experimentally the effect of turbulent thermal diffusion in a multi-fan turbulence generator which produces a nearly homogeneous and isotropic flow with a small mean velocity. Using particle image velocimetry and image processing techniques, we showed that in a turbulent flow with an imposed mean vertical temperature gradient (stably stratified flow) particles accumulate in the regions with the mean temperature minimum. These experiments detected the effect of turbulent thermal diffusion in a multi-fan turbulence generator for relatively high Reynolds numbers. The experimental results are in compliance with the results of the previous experimental studies of turbulent thermal diffusion in oscillating grid turbulence (Buchholz et al. 2004; Eidelman et al. 2004). We demonstrated that the turbulent thermal diffusion is an universal phenomenon. It occurs independently of the method of turbulence generation, and the qualitative behavior of particle spatial distribution in these very different turbulent flows is similar. Competition between turbulent fluxes caused by turbulent thermal diffusion and turbulent diffusion determines the formation of particle inhomogeneities.     

Calculation of principal characteristics of turbulent streams in a state of structural equilibrium

A review of articles on the study of turbulent streams having transverse displacement, in which a turbulent energy balance equation is used, is contained in [1]. Levin [2] proposed a certain development of Rotta's method [3] making it possible to determine the characteristics of the average flow and the radial distribution of pulsation magnitudes. However, in this article the scale of the turbulence (the quantityl) was given as an empirical function of the coordinates. At the same time it is clear that the distribution of the turbulence scale depends on the conditions of the problem. A special differential equation proposed in [4,5] describing the variation in time and space of the quantityl has the drawback that in deriving this equation it is necessary to invoke additional hypotheses which are difficult to test experimentally. In the present article, along with the velocity of the average flow, the pressure, and the pulsation magnitudes, the scale of the turbulence is considered as an important characteristic of the stream, determined by the reference system which consists of the Reynolds equations, continuity equations, and equations for the component of the Reynolds stress tensor. Rotta's approximate semiempirical relations and an experimental relation for the single-point correlation coefficient between the turbulent pulsations in velocity are used for closure of the system obtained. An approximate calculation is given for the principal average and pulsation characteristics of the flow for the region of the stream where the turbulence is in a state of structural equilibrium [6]. A comparison of the calculated and experimental data is presented.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 95–99, January–February, 1973.     

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.