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.     

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.     

Space-time correlations in turbulent flow: A review

This paper reviews some of the principal uses, over almost seven decades, of correlations, in both Eulerian and Lagrangian frames of reference, of properties of turbulent flows at variable spatial locations and variable time instants. Commonly called space—time correlations, they have been fundamental to theories and models of turbulence as well as for the analyses of experimental and direct numerical simulation turbulence data.     

On the orientation probability distribution of flexible fibres in a contracting channel flow

The development of fibre orientation distribution in a plane contracting channel flow is investigated with combining experiments and modelling. A dilute suspension of flexible wood fibres is used in the experiments. The salient feature of the suspension is the flexibility of the fibres. To model the fibre orientation probability distribution (FOPD) a diffusion–convection equation is used. The effect of random motion, in this case turbulence, is considered with translational and rotational diffusion coefficients. In addition to providing the inlet conditions, experiments are used to determine the rotational diffusion coefficient for the model. The work addresses the problems related to the above mentioned modelling method and combines the experiments and modelling in order to understand the mechanisms affecting the development of fibre orientation.     

Acoustic Lagrangian velocity measurement in a turbulent air jet

Measuring Lagrangian velocities in a turbulent flow is of a great interest for turbulence modeling. We report measurements made in an axisymmetric turbulent air jet at Reynolds number R _λ ≃ 320, using acoustical Doppler scattering. Helium-filled soap bubbles are used as Lagrangian tracers. We describe an experimental setup which allows the simultaneous measurement of the full three-component Lagrangian velocity and the longitudinal Eulerian one. Lagrangian velocity probability density functions (PDF) are found Gaussian, close to Eulerian ones. Velocity correlations are analysed as well as the statistical dependence between components.     

A note on the consistency of hybrid Eulerian/Lagrangian approach to multiphase flows

The aim of the present paper is to introduce and to discuss inconsistency errors that may arise when Eulerian and Lagrangian models are coupled for the simulations of turbulent poly-dispersed two-phase flows. In these hybrid models, two turbulence models are implicitly used at the same time and it is important to check that they are consistent, in spite of their apparent different formulations. This issue is best revealed in the case of very small particles, or tracer-limit particles, where it is assessed that coupling inconsistent turbulence models (Eulerian and Lagrangian) can result in non-physical results, notably for second-order fluid velocity moments. Computations for fluid particles in a turbulent channel flow using several coupling strategies are presented to illustrate this question.     

Numerical Study and Lagrangian Modelling of Turbulent Heat Transport

A stochastic Lagrangian model for both fluid velocities and temperature fluctuations is evaluated from Direct Numerical Simulation of heat transport in homogeneous isotropic turbulence submitted to a linear mean temperature gradient. The first stage lies on the study of the Lagrangian fluid turbulence statistics (Lagrangian correlations functions) computed from predictions of DNS. They are crucial for the analysis and the modelling of the fluid turbulent properties along discrete particle trajectories. In the second stage, a velocity-scalar Lagrangian stochastic model is proposed and evaluated from the DNS data. The coefficients of the drift and diffusion terms of the model are determined by only Lagrangian timescales, temperature variance and turbulent flux. The shapes of correlation functions present a good agreement between DNS results and stochastic modelling approach.     

Corrélations de vitesse lagrangienne et échelle intégrale temporelle en simulation des grandes échellesLagrangian velocity correlations and timescales by a large-eddy simulation

Velocity correlations and Lagrangian timescales are studied numerically by means of a direct numerical simulation (DNS) and a large-eddy simulation (LES) coupled with a subgrid Lagrangian stochastic model, in the case of a homogeneous and isotropic turbulence. A Langevin model is used to determine the subgrid component of the velocity of fluid particles. Numerical results of Lagrangian velocity correlations and timescales are presented. These quantities play an important role in turbulent mixing and scalar dispersion. To cite this article: G. Wei et al., C. R. Mecanique 334 (2006).     

A Lagrangian–Eulerian model of particle dispersion in a turbulent plane mixing layer

A Lagrangian–Eulerian model for the dispersion of solid particles in a two‐dimensional, incompressible, turbulent flow is reported and validated. Prediction of the continuous phase is done by solving an Eulerian model using a control‐volume finite element method (CVFEM). A Lagrangian model is also applied, using a Runge–Kutta method to obtain the particle trajectories. The effect of fluid turbulence upon particle dispersion is taken into consideration through a simple stochastic approach. Validation tests are performed by comparing predictions for both phases in a particle‐laden, plane mixing layer airflow with corresponding measurements formerly reported by other authors. Even though some limitations are detected in the calculation of particle dispersion, on the whole the validation results are rather successful. Copyright © 2002 John Wiley & Sons, Ltd.     

Connection between coefficient of diffusion and Eulerian turbulence characteristics in various flows

The turbulent velocity pulsation time scales and turbulent diffusion coefficients in airstreams of various types are determined experimentally. The connection between the integral time scales in Eulerian and Lagrangian coordinate systems is found. The validity of various relations for calculating the turbulent diffusion coefficient from the results of the determination of the scales in the Eulerian coordinate system are discussed.The authors thank G. B. Krayushkin for assistance in conducting the experiments and express their sincere gratitude to V. R. Kuznetsov and O. V. Yakovlevskii for their interest in the study and participation in discussions of the results of the study.     

Direct numerical simulation of turbulent flow with evaporating droplets at high temperature

Direct numerical simulation (DNS) is used to investigate turbulent flows with evaporating fuel droplets. For the solution of the carrier gas fluid, the Eulerian method is employed, while for fuel droplets, the Lagrangian method is used. The two-way coupling interactions between the carrier fluid and the fuel droplets are described by the mass, momentum and energy transfers. Direct numerical simulation is performed by a compressible code, named S3D. In this paper, the effects of evaporating and non-evaporating droplets on isotropic turbulent flows are investigated. From the simulations it is found that for the case without evaporation, the inclusion of small droplets suppresses the turbulence, while evaporation usually enhances turbulence at later times for higher mass-loading ratios.     

On the use of Csanady’s formulae in a turbulent gas–solid channel flow

The paper examines the use of expressions proposed by Csanady to predict the influence of the crossing trajectory and continuity effects on the decorrelation time scales of the fluid along solid particle trajectories in horizontal and downward vertical channel flows. The model is evaluated using data provided by a direct numerical simulation (DNS) of the carrier phase combined with a Lagrangian simulation of discrete particle (LS). Two particle relaxation times and two values of the gravity acceleration are considered. The results show the possibility of using Csanady’s expressions in a turbulent channel flow provided that the spatial and temporal correlations anisotropy is included in the model. As in isotropic homogeneous turbulence, a decrease of the decorrelation time scales is found to be more important in the directions perpendicular to the mean relative velocity.     

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.     

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.     

Numerical Experiments in Scalar Diffusion under Stochastic Convection

We study scalar diffusion, both from Eulerian and Lagrangian perspectives, advected by two dimensional flows. Emphasis is devoted to the problem of scalar diffusion under a synthetic turbulent flow. We present numerical and analytical results for the turbulent diffusion coefficient either from the influence of the turbulent synthetic field and a periodic array of eddies. Preliminary results concerning Lagrangian dispersion are also reported.     

Dynamics of fibres in a turbulent flow field – A particle-level simulation technique

A particle-level simulation technique has been developed for modelling the flow of fibres in a turbulent flow field. A single fibre is conceived here as a chain of segments, thus enabling the model fibre to have all the degrees of freedom (translation, rotation, bending and twisting) needed to realistically reproduce the dynamics of real fibres. Equations of motion are solved for each segment, accounting for the interaction forces with the fluid, the contact forces with other fibres and the forces that maintain integrity of the fibre.The motion of the fluid is resolved as a combination of 3D mean flow velocities obtained from a CFD code and fluctuating turbulent velocities derived from the Langevin equation. A case of homogeneous turbulence is treated in this paper.The results obtained show that fibre flocs in air-fibre flows can be created even when attractive forces are not present. In such a case, contacts between fibres, properties of an individual fibre (such as flexibility and equilibrium shapes) and properties of the flow of the carrying fluid are shown to govern the physics behind formation and breaking up of fibre flocs. Highly irregular fibre shapes and stiff fibres lead to strong flocculation.The modelling framework applied in this work aims at making possible a numerical model applicable for designing processes involving transport of fibres by air at industrial scale.     

Systematic tuning of dispersion models for simulation of evaporating sprays

In this study, via an Eulerian–Lagrangian framework, the performance of two recent dispersion models, i.e. a first-order autoregressive process and the PDF model, is compared. The appropriate relations for the turbulence scales and the drift correction term are suggested and the tuned values for the constants of the models are proposed in a systematic approach by starting with the simplest case, i.e. particle-laden stationary isotropic turbulence and adding more complexities in the subsequent cases, including the homogeneous anisotropic shear flow, decaying grid turbulence, and inhomogeneous gas–solid spray. Also, the isotropic relation for the effect of inertia in the Lagrangian turbulence time scale seen by particles is extended to the anisotropic case while it remains consistent in the isotropic limit. Finally, the performance of the tuned models is evaluated for the simulation of an evaporating spray. It is observed that, the tuned constants for the evaporating spray are close to the ones obtained for the homogeneous shear flow.