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.     

Coupled RANS–LPT modelling of dilute,particle-laden flow in a duct with a 90° bend

A dilute, particle-laden flow in a square duct with a 90° bend is modelled using a RANS approach, coupled to a second-moment turbulence closure, together with a Lagrangian particle tracking technique, with particle dispersion modelled using a stochastic approach that ensures turbulence anisotropy. Detailed predictions of mean and fluctuating fluid and particle velocities are validated through comparisons of predictions with experimental measurements made for gas–solid flows in a vertical-to-horizontal flow configuration. Reasonable agreement between predicted first and second moments and data is found for both phases, with the consistent application of anisotropic and three-dimensional modelling approaches resulting in predictions that compare favourably with those of other authors, and which provide fluctuating particle velocities in acceptable agreement with data.     

A PDF propagation approach to model turbulent dispersion in swirling flows

A computationally efficient approach that solves for the spatial covariance matrix along the dense particle ensemble-averaged trajectory has been successfully applied to describe turbulent dispersion in swirling flows. The procedure to solve for the spatial covariance matrix is based on turbulence isotropy assumption, and it is analogous to Taylor's approach for turbulent dispersion. Unlike stochastic dispersion models, this approach does not involve computing a large number of individual particle trajectories in order to adequately represent the particle phase; a few representative particle ensembles are sufficient to describe turbulent dispersion. The particle Lagrangian properties required in this method are based on a previous study (Shirolkar and McQuay, 1998). The fluid phase information available from practical turbulence models is sufficient to estimate the time and length scales in the model. In this study, two different turbulence models are used to solve for the fluid phase – the standard k–ε model, and a multiple-time-scale (MTS) model. The models developed here are evaluated with the experiments of Sommerfeld and Qiu (1991). A direct comparison between the dispersion model developed in this study and a stochastic dispersion model based on the eddy lifetime concept is also provided. Estimates for the Reynolds stresses required in the stochastic model are obtained from a set of second-order algebraic relations. The results presented in the study demonstrate the computational efficiency of the present dispersion modeling approach. The results also show that the MTS model provides improved single-phase results in comparison to the k–ε model. The particle statistics, which are computed based on the fundamentals of the present approach, compare favorably with the experimental data. Furthermore, these statistics closely compare to those obtained using a stochastic dispersion model. Finally, the results indicate that the particle predictions are relatively unaffected by whether the Reynolds stresses are based on algebraic relations or on the turbulence isotropy assumption.     

Turbulent particle dispersion in arbitrary wall-bounded geometries: A coupled CFD-Langevin-equation based approach

A Lagrangian continuous random walk (CRW) model is developed to predict turbulent particle dispersion in arbitrary wall-bounded flows with prevailing anisotropic, inhomogeneous turbulence. The particle tracking model uses 3D mean flow data obtained from the Fluent CFD code, as well as Eulerian statistics of instantaneous quantities computed from DNS databases. The turbulent fluid velocities at the current time step are related to those of the previous time step through a Markov chain based on the normalized Langevin equation which takes into account turbulence inhomogeneities. The model includes a drift velocity correction that considerably reduces unphysical features common in random walk models. It is shown that the model satisfies the well-mixed criterion such that tracer particles retain approximately uniform concentrations when introduced uniformly in the domain, while their deposition velocity is vanishingly small, as it should be. To handle arbitrary geometries, it is assumed that the velocity rms values in the boundary layer can locally be approximated by the DNS data of fully developed channel flows. Benchmarks of the model are performed against particle deposition data in turbulent pipe flows, 90° bends, as well as more complex 3D flows inside a mouth-throat geometry. Good agreement with the data is obtained across the range of particle inertia.     

Modeling fluid-particle interaction in dilute-phase turbulent liquid-particle flow simulation

The present work examines the predictive capability of a two-fluid CFD model that is based on the kinetic theory of granular flow in simulating dilute-phase turbulent liquid-particle pipe flows in which the inter-stitial fluid effect on the particle fluctuating motion is significant.The impacts of employing different drag correlations and turbulence closure models to describe the fluid-particle interactions(i.e.drag force and long-range interaction)are examined at both the mean and fluctuating velocity levels.The model pre-dictions are validated using experimental data of turbulent liquid-particle flows in a vertical pipe at different particle Reynolds numbers(ReP > 400 and ReP < 400),which characterize the importance of the vortex shedding phenomenon in the fluid-phase turbulence modulation.The results indicate that(1)the fluctuating velocity level predictions at different ReP are highly sensitive to the drag correlation selec-tion and(2)different turbulence closure models must be employed to accurately describe the long-range fluid-particle interaction in each phase.In general,good agreement is found between the model predic-tions and the experimental data at both the mean and fluctuating velocity levels provided that appropriate combinations of the drag correlation and the turbulence closure model are selected depending on Rep.     

A stochastic Langevin model of turbulent particle dispersion in the presence of thermophoresis

Direct numerical simulation (DNS) and experimental data have shown that inertial particles exhibit concentration peaks in isothermal turbulent boundary layers, whereas tracer-like particles remain well mixed in the domain. It is therefore expected that the interactions between turbulence and thermophoresis will be strong in particle-laden flows where walls and carrier fluid are at significantly different temperatures. To capture turbulent particle dispersion with active thermophoresis, a coupled CFD-Lagrangian continuous random walk (CRW) model is developed. The model uses 3D mean flow velocities obtained from the Fluent 6.3 CFD code, to which are added turbulent fluid velocities derived from the normalized Langevin equation which accounts for turbulence inhomogeneities. The mean thermophoretic force is included as a body force on the particle following the Talbot formulation. Validation of the model is performed against recent integral thermophoretic deposition data in long pipes as well as the TUBA TT28 test with its detailed local deposition measurements. In all cases, the agreement with the data is very good. In separate parametric studies in a hypothetical cooled channel flow, it is found that turbulence strongly enhances thermophoretic deposition of particles with dimensionless relaxation times τ⁺ of order 1 or more. On the other hand, the thermophoretic deposition of very small inertia particles (τ⁺ < 0.2) in the asymptotic region far from the injection point tends to that which characterizes stagnant flow conditions, in agreement with the DNS results of Thakurta et al.     

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

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

Characterization of particle-laden, confined swirling flows by phase-doppler anemometry and numerical calculation

The particle dispersion characteristics in a confined swirling flow with a swirl number of approx. 0.5 were studied in detail by performing measurements using phase-Doppler anemometry (PDA) and numerical predictions. A mixture of gas and particles was injected without swirl into the test section, while the swirling airstream was provided through a co-flowing annular inlet. Two cases with different primary jet exit velocities were considered. For these flow conditions, a closed central recirculation bubble was established just downstream of the inlet.

The PDA measurements allowed the correlation between particle size and velocity to be obtained and also the spatial change in the particle size distribution throughout the flow field. For these results, the behaviour of different size classes in the entire particle size spectrum, ranging from about 15 to 80 μm, could be studied, and the response of the particles to the mean flow and the gas turbulence could be characterized. Due to the response characteristics of particles with different diameters to the mean flow and the flow turbulence, a considerable separation of the particles was observed which resulted in a streamwise increase in the particle mean number diameter in the core region of the central recirculation bubble. For the lower particle inlet velocity (i.e. low primary jet exit velocity), this effect is more pronounced, since here the particles have more time to respond to the flow reversal and the swirl velocity component. This also gave a higher mass of recirculating particle material.

The numerical predictions of the gas flow were performed by solving the time-averaged Navier-Stokes equations in connection with the well known kε turbulence model. Although this turbulence model is based on the assumption of isotropic turbulence, the agreement of the calculated mean velocity profiles compared to the measured gas velocities is very good. The gas-phase turbulent kinetic energy, however, is considerably underpredicted in the initial mixing region. The particle dispersion characteristics were calculated by using the Lagrangian approach, where the influence of the particulate phase on the gas flow could be neglected, since only very low mass loadings were considered. The calculated results for the particle mean velocity and the mass flux are also in good agreement with the experiments. Furthermore, the change in the particle mean diameter throughout the flow field was predicted approximately, which shows that the applied simple stochastic dispersion model also gives good results for such very complex flows. The variation of the gas and particle velocity in the primary inlet had a considerable impact on the particle dispersion behaviour in the swirling flow and the particle residence time in the central recirculation bubble, which could be determined from the numerical calculations. For the lower particle inlet velocity, the maximum particle size-dependence residence time within the recirculation region was considerably shifted towards larger particles.     

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.     

Computational Fluid Dynamics (CFD) modelling of transfer chutes: Assessment of viscosity,drag and turbulence models

Computational Fluid Dynamics (CFD) has been successfully applied to evaluate potential dust emissions from bulk material transfer chutes. The implementation of appropriate models and modelling parameters is shown to be critical to the overall accuracy of the simulation results. This paper presents the influence of different models on the CFD simulation of transfer chutes and follows from an earlier study that details the influence of model parameters. The aim of this paper is to offer guidance to select models, and provide a better understanding of their influence in order to evaluate the most appropriate viscosity model, drag model and turbulence model for this application. A two-phase three-dimensional Euler–Euler model in commercial CFD software ANSYS FLUENT has been selected to model the granular and air flow in the transfer chute. The simulated air velocity profiles are discussed by comparing with each other and against experimental data obtained from Particle Image Velocimetry (PIV) results. The simulated particle velocity distributions were compared with results obtained using a well-established continuum method which was developed by Roberts. Furthermore, the air mass flow rates were analysed to evaluate the influence of different models. The results show that the granular viscosity model had a strong influence on the predictions of both air and particle flow. It was observed that both the drag model and turbulence model had limited influence on the outlet air velocities. The results also indicate that the Di-Felice drag model and SST k–ω turbulence model provide solutions closer to the experimental values than the other models investigated.     

Numerical prediction of particle slip velocity in turbulence by CFD-DEM simulation

Turbulent environment improves the flotation recovery of fine particles by promoting the particle–bubble collision rate, which directly depends on the particle slip velocity. However, the existing slip velocity models are not applicable to fine particles in turbulence. The mechanism of turbulence characteristics and particle properties on the slip velocity of fine particles in turbulence was unclear. In this study, a coupled ANSYS FLUENT and EDEM based on computational fluid dynamics (CFD) and discrete element method (DEM) were used to simulate the slip velocity of fine particles in the approximately homogenous isotropic turbulence, which was excited by the grid. The reliability of the used CFD-DEM simulation method was validated against the slip velocity measured by the particle image velocimetry (PIV) experiments. In particular, the effects of the particle shapes, particle densities, and turbulence intensities on the slip velocity have been investigated with this numerical method. Numerical results show that particle shapes have no significant effect on fine particles between 37 and 225 μm. The slip velocity of the spherical particles increases with the turbulence intensity and particle density. Based on the simulated data, a model which has a correlation coefficient of 0.95 is built by using nonlinear fitting.     

Eulerian prediction of the fluid/particle correlated motion in turbulent two-phase flows

An approximate equation governing the turbulent fluid velocity encountered along discrete particle path is used to derive the fluid/particle turbulent moments required for dispersed two-phase flows modelling. Then, closure model predictions are compared with results obtained from large-eddy simulation of particle fluctuating motion in forced isotropic fluid turbulence.     

CFD-DEM simulation of the hole cleaning process in a deviated well drilling: The effects of particle shape

We investigate the effect of particle shape on the transportation mechanism in well-drilling using a three-dimensional model that couples computational fluid dynamics (CFD) with the discrete element method (DEM). This numerical method allows us to incorporate the fluid–particle interactions (drag force, contact force, Saffman lift force, Magnus lift force, buoyancy force) using momentum exchange and the non-Newtonian behavior of the fluid. The interactions of particle−particle, particle−wall, and particle−drill pipe are taken into account with the Hertz–Mindlin model. We compare the transport of spheres with non-spherical particles (non-smooth sphere, disc, and cubic) constructed via the multi-sphere method for a range of fluid inlet velocities and drill pipe inclination angles. The simulations are carried out for laboratory-scale drilling configurations. Our results demonstrate good agreement with published experimental data. We evaluate the fluid–particle flow patterns, the particle velocities, and the particle concentration profiles. The results reveal that particle sphericity plays a major role in the fluid–solid interaction. The traditional assumption of an ideal spherical particle may cause inaccurate results.     

Influence of geometry on separation efficiency in a hydrocyclone

A numerical study of the gas–liquid–solid multiphase flow in hydrocyclones is presented. Three models of turbulence, the RNG k–ε model, the Reynolds stress model and Large eddy simulation with the volume of fluid model (VOF) multiphase model for simulating air core are compared in order to predict axial and tangential velocity distributions. This presentation is mainly aimed at identifying an optimal method, used to study effective parameters, based on which, eventually, effect of inlet flow rate variations and body dimension variations such as underflow diameter, overflow diameter and cone angle on the separation performance and pressure drop are investigated. The results are then used in the simulation of particle flow described by the stochastic Lagrangian model. The results suggest that the predicted size classifications are approximately similar to those of RSM and LES methods. Predictions using the RSM model are found in agreement with experimental results with a marginal error within the range of 4 to 8%. Proceeding model validation, parametric studies have been carried out concerning the influence of velocity inlet, particle size and body dimension such as underflow and overflow diameter and cone angle. The predictions demonstrate that the flow fields in the hydrocyclones with different sizes and lengths are different, which yields different performances.     

Stochastic simulations of particle-laden isotropic turbulent flow

Numerical simulations are performed of dispersion and polydispersity of particles in isotropic incompressible turbulence. The mass loading of the particles is assumed to be small; thus the effects of particles on turbulence is neglected (one-way coupling). A stochastic model is employed to simulate the carrier phase. The results of the simulations are compared with direct numerical simulation (DNS) data and theoretical results. The stochastic model predicts most of the trends as portrayed by DNS and theory. However, the continuity effect associated with the crossing trajectories effect is not captured. Also, the peaking in the variation of the particle asymptotic diffusivity coefficient with the particle time constant is not observed. For evaporating particles, the stochastic model predicts thinner probability density functions (pdfs) for the particle diameter as compared with DNS generated pdfs. The model is implemented to investigate the effects of gravity on evaporation. It is shown that the depletion rate increases with increase of the drift velocity at short and intermediate times, but an opposite trend is observed at long times. The standard deviation and skewness of the particle diameter indicate peak values in their variations with the drift velocity. Dispersion of evaporating particles decreases with respect to that of non-evaporating particles at small drift velocities; an opposite trend is observed at large drift velocities. The effects of the initial evaporation rate and the particle Schmidt number on the evaporation in the gravity environment are also studied.     

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.     

Modeling of the phase lag causing fluidelastic instability in a parallel triangular tube array

Fluidelastic instability is considered a critical flow induced vibration mechanism in tube and shell heat exchangers. It is believed that a finite time lag between tube vibration and fluid response is essential to predict the phenomenon. However, the physical nature of this time lag is not fully understood. This paper presents a fundamental study of this time delay using a parallel triangular tube array with a pitch ratio of 1.54. A computational fluid dynamics (CFD) model was developed and validated experimentally in an attempt to investigate the interaction between tube vibrations and flow perturbations at lower reduced velocities U_r=1–6 and Reynolds numbers Re=2000–12 000. The numerical predictions of the phase lag are in reasonable agreement with the experimental measurements for the range of reduced velocities U_g/fd=6–7. It was found that there are two propagation mechanisms; the first is associated with the acoustic wave propagation at low reduced velocities, U_r<2, and the second mechanism for higher reduced velocities is associated with the vorticity shedding and convection. An empirical model of the two mechanisms is developed and the phase lag predictions are in reasonable agreement with the experimental and numerical measurements. The developed phase lag model is then coupled with the semi-analytical model of Lever and Weaver to predict the fluidelastic stability threshold. Improved predictions of the stability boundaries for the parallel triangular array were achieved. In addition, the present study has explained why fluidelastic instability does not occur below some threshold reduced velocity.     

SECOND-ORDER MOMENT MODEL FOR DENSE TWO-PHASE TURBULENT FLOW OF BINGHAM FLUID WITH PARTICLES

The USM-θmodel of Bingham fluid for dense two-phase turbulent flow was developed, which combines the second-order moment model for two-phase turbulence with the particle kinetic theory for the inter-particle collision. In this model, phases interaction and the extra term of Bingham fluid yield stress are taken into account. An algorithm for USM-θmodel in dense two-phase flow was proposed, in which the influence of particle volume fraction is accounted for. This model was used to simulate turbulent flow of Bingham fluid single-phase and dense liquid-particle two-phase in pipe. It is shown USM-θmodel has better prediction result than the five-equation model, in which the particle-particle collision is modeled by the particle kinetic theory, while the turbulence of both phase is simulated by the two-equation turbulence model. The USM-θmodel was then used to simulate the dense two-phase turbulent up flow of Bingham fluid with particles. With the increasing of the yield stress, the velocities of Bingham and particle decrease near the pipe centre. Comparing the two-phase flow of Bingham-particle with that of liquid-particle, it is found the source term of yield stress has significant effect on flow.     

Dispersion de particules solides en mouvement de saltation dans un écoulement turbulentDispersion of solid saltating particles in a turbulent flow

The solid particle dispersion in saltating motion is studied in an homogeneous turbulence and in a turbulent boundary layer. The fluid velocity along the particle trajectory is estimated using a continuous stochastic differential equation in which the correlation integral time takes into account gravity and inertia effects. As far as the boundary layer is concerned, the aerodynamic entrainment of particles and the rebound are modelised as random variables with Gaussian probability density functions. Compared with experimental results, the numerical results show good agreement for dispersion, although velocity fluctuations are slightly under evaluated. To cite this article: C. Aguirre et al., C. R. Mecanique 332 (2004).