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.     

Comparison of the Eulerian and Lagrangian approaches in studying the flow pattern and heat transfer in a separated axisymmetric turbulent gas-droplet flow

The Eulerian and Lagrangian approaches are used to perform a numerical study of the disperse phase dynamics, turbulence, and heat transfer in a turbulent gas-droplet flow in a tube with sudden expansion with the following ranges of two-phase flow parameters: initial droplet size d ₁ = 0–200 µm and mass fraction of droplets M _L1 = 0–0.1. The main difference between the Eulerian and Lagrangian approaches is the difference in the predictions of the droplet mass fraction: the Eulerian approach predicts a smaller value of M _L both in the recirculation region and in the flow core (the difference reaches 15–20%). It is demonstrated that the disperse phase mass fraction calculated by the Lagrangian approach agrees better with measured data than the corresponding value predicted by the Eulerian approach.     

Two-phase micro- and macro-time scales in particle-laden turbulent channel flows

The micro-and macro-time scales in two-phaseturbulent channel flows are investigated using the direct numerical simulation and the Lagrangian particle trajectorymethods for the fluid-and the particle-phases,respectively.Lagrangian and Eulerian time scales of both phases are calculated using velocity correlation functions.Due to flowanisotropy,micro-time scales are not the same with the theoretical estimations in large Reynolds number(isotropic) turbulence.Lagrangian macro-time scales of particle-phaseand of fluid-phase seen by particles are both dependent onparticle Stokes number.The fluid-phase Lagrangian integral time scales increase with distance from the wall,longerthan those time scales seen by particles.The Eulerian integral macro-time scales increase in near-wall regions but decrease in out-layer regions.The moving Eulerian time scalesare also investigated and compared with Lagrangian integraltime scales,and in good agreement with previous measurements and numerical predictions.For the fluid particles themicro Eulerian time scales are longer than the Lagrangianones in the near wall regions,while away from the walls themicro Lagrangian time scales are longer.The Lagrangianintegral time scales are longer than the Eulerian ones.Theresults are useful for further understanding two-phase flowphysics and especially for constructing accurate predictionmodels of inertial particle dispersion.     

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.     

Analysis of unsteady gas–liquid flows in a rectangular tank: Comparison of Euler–Eulerian and Euler–Lagrangian simulations

We study the dynamics of gas–liquid flows experimentally and computationally in a rectangular bubble column where the gas source is introduced at the corner. The flow in this reactor is complex and inherently unsteady in nature. The two-dimensional liquid phase velocity field is calculated by an Eulerian approach solving the unsteady Reynolds Averaged Navier Stokes equations. The conservation equations are closed using a two parameter turbulence model. The two-way coupling was accounted for by adding source terms in the conservation equations of the continuous phase to take into account the interaction with the dispersed phase. Bubble tracking is achieved through a Lagrangian approach. Here the equations of motion are solved taking into account the drag, pressure, buoyancy and gravity forces. The time-averaged flows along with the variables which characterize turbulence are analyzed for a wide range of gas flow-rates using Euler–Lagrangian simulations. These simulation predictions are validated with Euler–Eulerian simulations where the gas-phase distribution is captured as a void fraction and PIV experiments. The motion of bubbles induces turbulence in the flow. The applicability of two parameter models for turbulence like the standard k–ε model on time-averaged flow properties is addressed. From the results of the time averaged velocity field, turbulence intensity, turbulent viscosity and gas hold-up profiles, it is concluded that the Euler–Lagrangian model is applicable at lower gas flow-rates. The Euler–Eulerian approach was found to be valid at lower as well as higher gas flow-rates.     

The motion of fibres in turbulent flow, stochastic simulation of isotropic homogeneous turbulence

The state of fibres suspended in a turbulent fluid is described in terms of a probability distribution function of fibre orientation and position throughout the suspending fluid. The evolution of the fibre's probability distribution function is governed by a convection–dispersion equation, where the randomizing effect of the turbulence is modelled by rotational and translational dispersion coefficients. To estimate these coefficients a numerical simulation of fibres moving in a turbulent fluid was developed. The trajectory of an ensemble of inertialess, rigid, thin, free-draining fibres was calculated through a stochastic model of homogeneous, isotropic turbulence. The results of the simulation were compared with analytical estimates and were found to provide reasonable agreement over a wide range of fibre length. However, the simulation showed that the Lagrangian integral time scale for rotation was significantly smaller than for translation and the ratio of rotational to translational Lagrangian time scales was smaller than the ratio of Eulerian time scales. The simulation also showed that the Lagrangian velocity correlation increased as fibre length increased and that the temporal correlations approached the analytical estimates of the Eulerian correlations in the limit of long fibres.     

Modulation of homogeneous turbulence seeded with finite size bubbles or particles

The dynamics of homogeneous, isotropic turbulence seeded with finite sized particles or bubbles is investigated in a series of numerical simulations, using the force-coupling method for the particle phase and low wavenumber forcing of the flow to sustain the turbulence. Results are given on the modulation of the turbulence due to massless bubbles, neutrally buoyant particles and inertial particles of specific density 1.4 at volumetric concentrations of 6%. Buoyancy forces due to gravity are excluded to emphasize finite size and inertial effects for the bubbles or particles and their interactions with the turbulence. Besides observing the classical entrapment of bubbles and the expulsion of inertial particles by vortex structures, we analyze the Lagrangian statistics for the velocity and acceleration of the dispersed phase. The turbulent fluctuations are damped at mid-range wavenumbers by the bubbles or particles while the small-scale kinetic energy is significantly enhanced. Unexpectedly, the modulation of turbulence depends only slightly on the dispersion characteristics (bubble entrapment in vortices or inertial sweeping of the solid particles) but is closely related to the stresslet component (finite size effect) of the flow disturbances. The pivoting wavenumber characterizing the transition from damped to enhanced energy content is shown to vary with the size of the bubbles or particles. The spectrum for the energy transfer by the particle phase is examined and the possibility of representing this, at large scales, through an additional effective viscosity is discussed.     

A closed model of fluctuating particle motion in a turbulent flow

Random particle motion in a turbulent and molecular velocity fluctuation field is considered. Using a spectral representation of the carrier-phase Eulerian velocity fluctuation correlations, a closed system of integral equations for calculating the carrier-phase velocity correlation along the particle trajectory and the particle Lagrangian velocity fluctuation correlation is obtained. Based on this system, the fluctuations of the particle parameters are analyzed. In the limiting case of a passive admixture, an estimate is found for the ratio of the integral Lagrangian and Eulerian time scales and the Kolmogorov constant for the Lagrangian structure function of the carrier-phase velocity fluctuations.     

Estimate of the correlation between pressure pulsations and the divergence of the velocity in subsonic flows of variable density

In flows with variable density, the turbulence energy equation contains a large number of correlations, about which little is at present known [1]. One of the least studied is the correlation between the pressure and the divergence of the velocity. Usually, this correlation is ignored [2, 3]. The aim of the present paper is to estimate the pulsations of the divergence of the velocity and the correlation with the pressure pulsations in a subsonic turbulent flow with variable density. Three cases are considered: 1) mixing of gases having different densities, 2) diffusion combustion, 3) combustion of a homogeneous mixture. It is assumed that the Mach number is small, the Reynolds number large, and the coefficients of molecular diffusion and thermal diffusivity equal; external forces are absent.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 4–11, May–June, 1979.     

Lagrangian particle dispersion in turbulent flow over a wall mounted obstacle

Large-eddy simulations (LES) of particle-laden turbulent flows are presented in order to investigate the effects of particle response time on the dispersion patterns of a space developing flow with an obstruction, where solid particles are injected inside the wake of an obstacle [Vincont, J.Y., Simoens, S., Ayrault M., Wallace, J.M., 2000. Passive scalar dispersion in a turbulent boundary layer from a line source at the wall and downstream of an obstacle. J. Fluid Mech. 424, 127–167]. The numerical method is based on a fully explicit fractional step approach and finite-differences on Cartesian grids, using the immersed boundary method (IBM) to represent the existence of solid obstacles. Two different turbulence models have been tested, the classical Smagorinsky turbulence model and the filtered structure function model. The dispersed phase was modelled either by an Eulerian approach or a Lagrangian particle tracking scheme of solid particles with Stokes numbers in the range St = 0–25, assuming one-way coupling between the two phases. A very good agreement was observed between the Lagrangian and Eulerian approaches. The effect of particle size was found to significantly differentiate the dispersion pattern for the inhomogeneous flow over the obstacle. Although in homogeneous flows like particle-laden turbulent channels near-wall particle clustering increases monotonically with particle size, for the examined flow over an obstacle, preferential concentration effects were stronger only for an intermediate range of Stokes numbers.     

Development of Gas-Particle Euler-Euler LES Approach: A Priori Analysis of Particle Sub-Grid Models in Homogeneous Isotropic Turbulence

A new large eddy simulation (LES) approach for particle-laden turbulent flows in the framework of the Eulerian formalism for inertial particle statistical modelling is developed. Local instantaneous Eulerian equations for the particle cloud are first written using the mesoscopic Eulerian formalism (MEF) proposed by Février et al. (J Fluid Mech 533:1–46, 2005), which accounts for the contribution of an uncorrelated velocity component for inertial particles with relaxation time larger than the Kolmogorov time scale. Second, particle LES equations are obtained by volume filtering the mesoscopic Eulerian ones. In such an approach, the particulate flow at larger scales than the filter width is recovered while sub-grid effects need to be modelled. Particle eddy-viscosity, scale similarity and mixed sub-grid stress (SGS) models derived from fluid compressible turbulence SGS models are presented. Evaluation of such models is performed using three sets of particle Lagrangian results computed from discrete particle simulation (DPS) coupled with fluid direct numerical simulation (DNS) of homogeneous isotropic decaying turbulence. The two phase flow regime corresponds to the dilute one where two-way coupling and inter-particle collisions are not considered. The different particle Stokes number (based on Kolmogorov time scale) are initially equal to 1, 2.2 and 5.1. The mesoscopic field properties are analysed in detail by considering the particle velocity probability function (PDF), correlated velocity power spectra and random uncorrelated velocity moments. The mesoscopic fields measured from DPS+DNS are then filtered to obtain large scale fields. A priori evaluation of particle sub-grid stress models gives comparable agreement than for fluid compressible turbulence models. It has been found that the standard Smagorinsky eddy-viscosity model exhibits the smaller correlation coefficients, the scale similarity model shows very good correlation coefficient but strongly underestimates the sub-grid dissipation and the mixed model is on the whole superior to pure eddy-viscosity model.     

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.     

Evaluation of erosion in a two‐way coupled fluid–particle system

In this study, the effects of flow turbulence intensity, temperature, particle sizes and impinging velocity on erosion by particle impact are demonstrated numerically. Underlying turbulent flow on an Eulerian frame is described by the compressible Reynolds averaged Navier–Stokes equations with a RNG k–ε turbulence model. The particle trajectories and particle–wall interactions are evaluated by a Eulerian–Lagrangian approach in a two‐way coupling system. An erosion model considering material weight removal from surfaces is used to predict erosive wear. Computational validation against measured data is demonstrated satisfactorily. The analysis of erosion shows that the prevention of erosion is enhanced by increasing the effects of flow temperature and turbulence intensity and reducing particle inertial momentum. Copyright © 2001 John Wiley & Sons, Ltd.     

Boundary layer in the vicinity of the stagnation point of a body of axial symmetry in a turbulent stream

The flow in the boundary layer in the vicinity of the stagnation point of a flat plate is examined. The outer stream consists of turbulent flow of the jet type, directed normally to the plate. Assumptions concerning the connection between the pulsations in velocity and temperature in the boundary layer and the average parameters chosen on the basis of experimental data made it possible to obtain an isomorphic solution of the boundary layer equations. Equations are obtained for the friction and heat transfer at the wall in the region of gradient flow taking into account the effect of the turbulence of the impinging stream. It is shown that the friction at the wall is insensitive to the turbulence of the impinging stream, while the heat transfer is significantly increased with an increase in the pulsations of the outer flow. These properties are confirmed by the results of experimental studies [1–4].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 83–87, September–October, 1973.     

Accounting for Polydispersion in the Eulerian Large Eddy Simulation of the Two-Phase Flow in an Aeronautical-type Burner

A major issue for the simulation of two-phase flows in engines concerns the modeling of the liquid disperse phase, either in the Lagrangian or the Eulerian approach. In the perspective of massively parallel computing, the Eulerian approach seems to be more suitable, as it uses the same algorithms as the gaseous phase solver. However taking into account the whole physics of a turbulent spray, especially in terms of polydispersity, requires an additional modeling effort. The Mesoscopic Eulerian Formalism (MEF) [13] accounts for the effect of turbulence on the disperse phase, and was extended to the Large Eddy Simulation framework [41], but is limited to monodisperse flows. In [38], the influence of polydispersity on resolved and unresolved turbulent motions of the disperse phase was highlighted, and a first model was proposed, based on size-conditioned statistics. Starting from this idea, a coupling between the MEF and the Multifluid Approach (MA) [30] is proposed. The MA decomposes the Eulerian phase into several fluid classes called sections, and corresponding to size intervals. Each section uses then size-conditioned closures. The original idea of this work is to use the MEF closures independently in each section, taking into account the mean droplet size of this section. This new approach, called Multifluid Mesoscopic Eulerian Formalism (MMEF), is then able to capture polydispersion with associated size-conditioned turbulent dynamics. First, the importance of polydispersity and the ability of MMEF to capture it are highlighted with a 0D evaporation case and a 2D vortex case, showing its impact on dynamics in both size and physical spaces. Then, the MMEF is applied to the MERCATO configuration of ONERA [18]. Results are compared to monodisperse Eulerian, Lagrangian and experimental results.