Two- and Four-Way Coupled Euler–Lagrangian Large-Eddy Simulation of Turbulent Particle-Laden Channel Flow

Large-eddy simulations (LES) of a vertical turbulent channel flow laden with a very large number of solid particles are performed. The motivation for this research is to get insight into fundamental aspects of co-current turbulent gas-particle flows, as encountered in riser reactors. The particle volume fraction equals about 1.3%, which is relatively high in the context of modern LES of two-phase flows. The channel flow simulations are based on large-eddy approximations of the compressible Navier–Stokes equations in a porous medium. The Euler–Lagrangian method is adopted, which means that for each individual particle an equation of motion is solved. The method incorporates four-way coupling, i.e., both the particle-fluid and particle–particle interactions are taken into account. The results are compared to single-phase channel flow in order to investigate the effect of the particles on turbulent statistics. The present results show that due to particle–fluid interactions the mean fluid profile is flattened and the boundary layer is thinner. Compared to single-phase turbulent flow, the streamwise turbulence intensity of the gas phase is increased, while the normal and spanwise turbulence intensities are reduced. This finding is generally consistent with existing experimental data. The four-way coupled simulations are also compared with two-way coupled simulations, in which the inelastic collisions between particles are neglected. The latter comparison clearly demonstrates that the collisions have a large influence on the main statistics of both phases. In addition, the four-way coupled simulations contain stronger coherent particle structures. It is thus essential to include the particle–particle interactions in numerical simulations of two-phase flow with volume fractions around one percent.     

Direct numerical simulation of stationary particles in homogeneous turbulence decay: Application of the k–ε model

This study focuses on understanding how the presence of particles, in homogeneous turbulence decay, affects the dissipation of dissipation coefficient within the volume averaged dissipation transport equation. In developing this equation, the coefficient for dissipation of dissipation was assumed to be the sum of the single phase coefficient and an additional coefficient that is related to the effects of the dispersed phase. Direct numerical simulation was used to isolate the effect of stationary particles in homogeneous turbulent decay at low Reynolds numbers (Re_L = 3.3 and 12.5). The particles were positioned at each grid point and modeled as point forces and a comparison was made between a 64³ and 128³ domain. The results show that the dissipation of dissipation coefficient correlates well with a dimensionless parameter called the momentum coupling factor.     

Direct numerical simulation of particle interaction with ejections in turbulent channel flows

Direct numerical simulations (DNS) of incompressible turbulent channel flows coupled with Lagrangian particle tracking are performed to study the characteristics of ejections that surround solid particles. The behavior of particles in dilute turbulent channel flows, without particle collisions and without feedback of particles on the carrier fluid, is studied using high Reynolds number DNS (Re = 12,500). The results show that particles moving away from the wall are surrounded by ejections, confirming previous studies on this issue. A threshold value separating ejections with only upward moving particles is established. When normalized by the square root of the Stokes number and the square of the friction velocity, the threshold profiles follow the same qualitative trends, for all the parameters tested in this study, in the range of the experiments. When compared to suspension thresholds proposed by other studies in the Shields diagram, our simulations predict a much larger value because of the measure used to characterize the fluid and the criterion chosen to decide whether particles are influenced by the surrounding fluid. However, for intermediate particle Reynolds numbers, the threshold proposed here is in fair agreement with the theoretical criterion proposed by Bagnold (1966) [Bagnold, R., 1966. Geological Survey Professional Paper, vol. 422-1]. Nevertheless, further studies will be conducted to understand the normalization of the threshold.     

Direct numerical simulation of particle dispersion in a turbulent jet considering inter-particle collisions

To investigate the behaviour of inter-particle collision and its effects on particle dispersion, direct numerical simulation of a three-dimensional two-phase turbulent jet was conducted. The finite volume method and the fractional-step projection algorithm were used to solve the governing equations of the gas phase fluid and the Lagrangian method was applied to trace the particles. The deterministic hard-sphere model was used to describe the inter-particle collision. In order to allow an analysis of inter-particle collisions independent of the effect of particles on the flow, two-way coupling was neglected. The inter-particle collision occurs frequently in the local regions with higher particle concentration of the flow field. Under the influence of the local accumulation and the turbulent transport effects, the variation of the average inter-particle collision number with the Stokes number takes on a complex non-linear relationship. The particle distribution is more uniform as a result of inter-particle collisions, and the lateral and the spanwise dispersion of the particles considering inter-particle collision also increase. Furthermore, for the case of particles with the Rosin–Rammler distribution (the medial particle size is set d₅₀ = 36.7 μm), the collision number is significantly larger than that of the particles at the Stokes number of 10, and their effects on calculated results are also more significant.     

A Unifying Explanation for the Damping of Turbulence by Additives or External Forces

Recent calculations for the turbulent flow of a suspension of solid spheres in a gas were carried out by solving a Navier Stokes equation for the fluid that recognizes the presence of particles as external point forces. These show that a strong damping of the fluid turbulence can be realized at remarkably small volume fractions. The suggestion is made that the presence of point forces, pseudo-point forces, or added point shear stresses could provide a general explanation for turbulence suppression caused by additives or by the acceleration of a turbulent boundary-layer.     

Effects of vortex pairing on particle dispersion in turbulent shear flows

Particle dispersion in large-scale dominated turbulent shear flow is investigated numerically with special emphasis on the effects of the vortex-pairing phenomenon. The particle dispersion is visualized numerically by following the particle trajectories in a flow consisting of large vortices which are undergoing pairing interaction. The flow field is generated by a discrete vortex method. Important global and local fiow quantities from the numerical simulation compare reasonably well with experimental measurements.

For both cases of point sources with continuous particle release and an initially distributed line source, the particle dispersion results demonstrate that the extent of particle dispersion depends strongly on the Stokes number, the ratio of the particle aerodynamic response time to the characteristic time of the vortex-pairing flow field. Particles with relatively small Stokes numbers disperse laterally at approximately the saine rate as that of the fluid particles and particles with large Stokes numbers disperse much less than the fluid particles. Particles with intermediate Stokes numbers (0.5-5) may be dispersed laterally farther than the fiuid particles and may actually be flung out of the vortex structures. Due to the strong particie entrainment power, the flow during the vortex-pairing process seems to produce higher particle lateral dispersion than the pre-pairing and post-pairing flows.     

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.     

Direct numerical simulation of modulation of isotropic turbulence by poly‐dispersed particles

A direct numerical simulation technique based on two‐way coupling is presented to study a particle‐laden, decaying isotropic turbulent flow. Physical characteristics of turbulence modulation because of the mono‐dispersed (i.e., particles with single Stokes number) and poly‐dispersed particles (i.e., particles with more than one Stokes number) were investigated. A scale dependent effective viscosity that summarizes the aspects of the interaction between the velocity field and particles is defined in the study. Particles of Stokes number (St) 3.2,6.4 and 12.8 were used in performing the simulations. Poly‐dispersed particles were acquired by mixing particles of two different Stokes numbers at a time. As a whole, decay of turbulence because of the poly‐dispersed particles is observed to be larger than that of the decay of turbulence because of the mono‐dispersed particles. Simulations of poly‐dispersed particle indicate nonlinear characteristics in the modification of the temporal evolution of turbulence energy and dissipation. The scale dependent effective viscosity, which correlates with the energy spectrum plot, indicates that the decay of turbulence is mostly observed at the intermediate scales of turbulence. The effective viscosity for the simulations of the poly‐dispersed particles was calculated to be higher than that of the simulations of the mono‐dispersed particles. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical investigation of gas-particle flow in the primary air pipe of a low NOx swirl burner – The DEM-CFD method

The gas-particle flow in the primary air pipe (PAP) of a low NO_x swirl burner was investigated using the computational fluid dynamics (CFD) coupled with the discrete element method (DEM). The mathematical models were validated using the measured values obtained at the outlet of the primary pipe through a phase Doppler anemometer (PDA) system. Particles of different Stokes numbers in the primary air pipe (PAP) were investigated, and the effects of the structure of the primary air pipe and the particle–particle interaction on particle dispersion were analyzed. The results indicate that particles under the combined effects of the Venturi pipe and the spindle body are concentrated into a narrow band area and that the PAP structure can more efficiently concentrate particles with large Stokes numbers. The formed fuel rich/lean jet persists for a long distance out of the burner, thereby favoring of air-staged combustion and NO_x reduction. The particle collision frequency and its fluctuation range increase as the particle Stokes number increases. The collisions among particles result in an increase of the spanwise dispersion of particles. Experimental results indicate that the models that take particle–particle collision into consideration are more able to predict particle concentration.     

Direct numerical simulation of a particle-laden low Reynolds number turbulent round jet

Direct numerical simulation method is used for the investigating of particle-laden turbulent flows in a spatially evolution of low Reynolds number axisymmetric jet, and the Eulerian–Lagrangian point-particle approach is employed in the simulation. The simulation uses an explicit coupling scheme between particles and the fluid, which considers two-way coupling between the particle and the fluid. The DNS results are compared well with experimental data with equal Reynolds number (R_e = 1700). Our objects are: (i) to investigate the correlation between the particle number density and the fluctuating of fluid streamwise velocity; (ii) to examine whether the three-dimensional vortex structures in the particle-laden jet are the same as that in the free-air jet and how the particles modulate the thee-dimensional vortex structures and turbulence properties with different Stokes number particles; (iii) to discover the particle circumferential dispersion with different Stokes number particles. Our findings: (i) all the particles, regardless of their particle size, tend to preferentially accumulate in the region with large-than-mean fluid streamwise velocity; (ii) the small Stokes number particles take an important part in the modulation of three-dimensional vortex structures, but for the intermediate and larger sized particles, this modulation effect seems not so apparent; (iii) the particle circumferential dispersion is more effective for the smaller and intermediate sized particles, especially for the intermediate sized particles.     

Numerical simulation of a supersonic gas–solid flow over a blunt body: The role of inter-particle collisions and two-way coupling effects

A supersonic dusty gas flow over a blunt body is considered. The mathematical model of the two-phase gas–particle flow takes into account the inter-particle collisions and the two-way coupling effects. The carrier gas is treated as a continuum, the averaged flow field of which is described by the complete Navier–Stokes equations with additional source terms modeling the reverse action of the dispersed phase. The dispersed phase is treated as a discrete set of solid particles, and its behavior is described by a kinetic Boltzmann-type equation. Particles impinging on the body surface are assumed to bounce from it. Numerical analysis is carried out for the cross-wise flow over a cylinder. The method of computational simulation represents a combination of a CFD-method for the carrier gas and a Monte Carlo method for the “gas” of particles. The dependence of the fine flow structure of the continuous and dispersed phases upon the free stream particle volume fraction α_p∞ and the particle radius r_p is investigated, particularly in the shock layer and in the boundary layer at the body surface. The particle volume fraction α_p∞ is varied from a negligibly low value to the value α_p∞ = 3 × 10⁵ at which inter-particle collisions and two-way coupling effects are simultaneously essential. Particular attention has been given to the particles of radii close to the critical value r_p1, because in this range of particle size the behavior of the particles and their effect on the carrier gas flow are not yet completely understood. An estimate of the turbulent kinetic energy produced by the particles in the shock layer is obtained.     

Direct numerical simulation of particle-laden plane turbulent wall jet and the influence of Stokes number

The distribution and motion of inertial particles in plane turbulent wall jet are investigated using direct numerical simulation, under the assumption of one-way coupling. To our knowledge, this appears to be the first direct numerical simulation of a particle-laden plane turbulent wall jet. It is shown that, in outer part of the wall jet, the behaviour of particles closely resembles that of a free plane jet. Due to the streamwise decay of particle Stokes number, the particle streaks formed in the near wall region of the wall jet are characterized by their intensity variation, which differs significantly from those in the channel flow. The streamwise growth of the particle velocity half-width is approximately equal to that of the fluid velocity half-width and the maximum velocity of particles decays slower than that of fluid due to inertia. The outer scaling can collapse the mean particle velocity in both the inner and outer region for heavier particles. In the buffer region, the particle–fluid velocity difference can be negative or positive depending on the Stokes number since there are two competing effects, namely the memory effect and turbophoresis. In the viscous region, the larger particles are on average faster than fluid and the velocity difference is found to be self-similar depending on outer Stokes number. The near-wall distribution of velocity difference is significantly correlated with the presence of high-momentum particles which are entrained by vortical structures generated in the outer region of the wall jet. These results are useful for environmental and engineering applications.     

Linear stability of particle laden flows: the influence of added mass,fluid acceleration and Basset history force

Both modal and non-modal linear stability analysis of a channel flow laden with particles is presented. The particles are assumed spherical and solid and their presence modelled using two-way coupling, with Stokes drag, added mass and fluid acceleration as coupling terms. When the particles considered have a density ratio of order one, all three terms become important. To account for the volume and mass of the particles, a modified Reynolds number is defined. Particles lighter than the fluid decrease the critical Reynolds number for modal stability, whereas heavier particles may increase the critical Reynolds number. Most effect is found when the Stokes number defined with the instability time scale is of order one. Non-modal analysis shows that the generation of streamwise streaks is the most dominant disturbance-growth mechanism also in flows laden with particles: the transient growth of the total system is enhanced proportionally to the particle mass fraction, as observed previously in flows laden with heavy particles. When studying the fluid disturbance energy alone, the optimal growth hardly changes. We also show that the Basset history force has a negligible effect on stability. The inclusion of the extra interaction terms does not show any large modifications of the subcritical instabilities in wall-bounded shear flows.     

Dispersion of particles in wall-bounded particle-laden turbulent flows with high wall permeability

A series of numerical simulations were performed to investigate the distribution and deposition properties of particles in turbulent flows bounded by permeable walls using the Large Eddy Simulation (LES) with a Lagrangian trajectory approach. The wall permeation speeds were taken from 10⁻⁴ to 10⁻² of the bulk velocity. The directions of the permeation speed were the same at both walls, and they were inward on one wall but outward on the other wall to reserve the fluid mass. Particles with Stokes number (respecting viscous time scale) around 0.1, 1 and 10 were released in the fully developed turbulent channel flow. The particle–particle interaction and the retroaction from particles to the fluid were neglected. The fluid-phase turbulence statistical properties and particle's transport characteristics by vortexes were then analyzed in details. If the wall permeation exists, the turbulence intensities will be depressed close to the outward permeable wall but increased near the inward permeable wall. Not influenced by the wall permeation, the suspended particles with St⁺ ∼O(1) tend to accumulate in the less vortical zones away from the wall, while those particles in the flow regions near the outward permeable wall will distribute disregarding of the vorticity. The turbulence structures near the outward permeable wall are found to exert promotional effects on the particle deposition rate, but such effects are different for particles with various Stokes number. A distribution tendency of streamwise streaks for the deposited particles is also found on the wall imposed by the high outward permeation speed and the clustering deposition pattern is more obvious with increasing particle Stokes number.     

Global Existence of Strong Solutions to the Cucker–Smale–Stokes System

A coupled kinetic–fluid model describing the interactions between Cucker–Smale flocking particles and a Stokes fluid is presented. We demonstrate the global existence and uniqueness of strong solutions to this coupled system in a three-dimensional spatially periodic domain for initial data that are sufficiently regular, but not necessarily small.     

Generation and direct viscous dissipation of turbulent energy

Existing information about the generation and viscous dissipation of turbulent energy is based, as a rule, on the Laufer test data obtained for fluid flow in circular tubes at two Reynolds numbers (5 · 10⁵ and 5 · 10⁴). Computational dependences are presented herein for the generation and viscous dissipation of turbulent energy, common over the whole stream section and for the whole range of variation of the Reynolds number. The equation of the average energy balance during fluid flow in a circular tube and a flat channel is solved taking account of the equation of motion and the turbulent friction profile obtained by the author [1]. The computational dependences satisfy all the evident boundary conditions, agree with the Laufer test results [2] and yield a well-founded passage to the limit modes of average turbulent motion.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 30–36, November–December, 1973.     

Modeling of dynamics, heat transfer, and combustion in two-phase turbulent flows: 1. Isothermal flows

This paper presents a review of authors' collective works in the field of two-phase flow modeling done in the past few decades. The paper is aimed at the construction of mathematical models for simulation of particle-laden turbulent flows. A kinetic equation was obtained for the probability density function (PDF) of the particle velocity distribution in turbulent flows. The proposed kinetic equation describes both the interaction of particles with turbulent eddies of the carrier phase and particle-particle collisions. This PDF equation is used for the derivation of different schemes describing turbulent momentum transfer in the dispersed particle phase. The turbulent characteristics of the gaseous phase are calculated on the basis of the k - turbulence model with a modulation effect of particles on the turbulence.

The constructed models have been applied to the calculation of various two-phase gas-particle turbulent flows in jets and channels as well as particle deposition in tubes and separators. For validating the theoretical and numerical results, a wide range of comparisons with experimental data from Russian and foreign sources has been done.     

Calculation of the momentum and heat transfer in turbulent gas-particle jet flows

The equations for the second moments of the dispersed-phase velocity and temperature fluctuations are used for calculating gas-suspension jet flows within the framework of the Euler approach. The advantages of introducing the equations for the second moments of the particle velocity fluctuations has previously been quite convincingly demonstrated with reference to the calculation of two-phase channel boundary flows [9–11]. The flows considered below have a low solid particle volume concentration, so that interparticle collisions can be neglected and, consequently, the stochastic motion of the particles is determined exclusively by their involvement in the fluctuating motion of the carrier flow. In addition to the equations for the turbulent energy of the gas and its dissipation, the calculation scheme includes the equations for the turbulent energy and turbulent heat transfer of the solid phase; however, the model constructed does not contain additional empirical constants associated with the presence of the particles in the flow.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 69–80, May–June, 1992.     

Turbulent mass transport and attenuation in Stokes waves

The motion of turbulent Stokes waves on a finite constant depth fluid with a rough bed is considered. First and second order turbulent boundary layer equations are solved numerically for a range of roughness parameters, and from the solutions are calculated the mass transport velocity profiles and attenuation coefficients. A new mechanism of turbulent mass transport is found which predicts a reduction and reversal of drift velocity in shallow water in agreement with experimental observations under turbulent conditions. This transpires because the second order Stokes wave motion, in a turbulent boundary layer, can directly influence the mass transport velocity by mode coupling interactions between different second order Fourier modes of oscillation. It is also found that the Euler contribution due to the radiation stress of the first order motion is reduced to half of it's corresponding laminar value as a consequence of the velocity squared stress law. The attenuation is found to be of inverse algebraic type with the reciprocal wave height varying linearly with either distance or time. The severe wave height restriction applicable to the Longuet-Higgins [4] solution is shown not to apply to progressive waves on a finite constant depth of fluid. The existence of sand bars on sloping beaches exposed to turbulent waves is predicted.     

Turbulence modulation for gas - particle flow in vertical tube and horizontal channel using four-way Eulerian-Lagrangian approach

Turbulence modulation of gas-solid flow in vertical tube and horizontal channel in dilute and moderately dense suspensions is investigated numerically using a four way Eulerian-Lagrangian approach. Low Reynolds number k-l model is used for analyzing the fluid phase motion. A new model is presented based on a source-term formulation, which can predict fluid phase turbulence augmentation due to the presence of large particles and damping of turbulence due to small particles in the core of the channel and tube. Particle-particle and particle-wall collisions are simulated based on a deterministic approach, and coupling terms representing the fluid-particle interactions are also taken into account. The predicted fluid mean velocity and turbulence intensity profiles are in good agreement with the available experimental data. Additional numerical simulation results for variation of the eddy viscosity, turbulence production and dissipation are also presented for different values of loading ratios.