Hydrodynamic and thermal fields analysis in gas-solid two-phase flow

The present work aims to investigate numerically the flowfield and heat transfer process in gas-solid suspension in a vertical pneumatic conveying pipe. The Eulerian-Lagrangian model is used to simulate the flow of the two-phases. The gas phase is simulated based on Reynolds Average Navier-Stokes equations (RANS) with low Reynolds number k-ε model, while particle tracking procedure is used for the solid phase. An anisotropic model is used to calculate the Reynolds stresses and the turbulent Prandtl number is calculated as a function of the turbulent viscosity. The model takes into account the lift and drag forces and the effect of particle rotation as well as the particles dispersion by turbulence effect. The effects of inter-particles collisions and turbulence modulation by the solid particles, i.e. four-way coupling, are also included in the model. Comparisons between different models for turbulence modulation with experimental data are carried out to select the best model. The model is validated against published experimental data for velocities of the two phases, turbulence intensity, solids concentration, pressure drop, heat transfer rates and Nusselt number distribution. The comparisons indicate that the present model is able to predict the complex interaction between the two phases in non-isothermal gas-solid flow in the tested range. The results indicate that the particle-particle collision, turbulence dispersion and lift force play a key role in the concentration distribution. In addition, the heat transfer rate increases as the mass loading ratio increases and Nusselt number increases as the pipe diameter increases.     

Bubble and liquid turbulence characteristics of bubbly flow in a large diameter vertical pipe

The bubble and liquid turbulence characteristics of air–water bubbly flow in a 200 mm diameter vertical pipe was experimentally investigated. The bubble characteristics were measured using a dual optical probe, while the liquid-phase turbulence was measured using hot-film anemometry. Measurements were performed at six liquid superficial velocities in the range of 0.2–0.68 m/s and gas superficial velocity from 0.005 to 0.18 m/s, corresponding to an area average void fraction from 1.2% to 15.4%. At low void fraction flow, the radial void fraction distribution showed a wall peak which changed to a core peak profile as the void fraction was increased. The liquid average velocity and the turbulence intensities were less uniform in the core region of the pipe as the void fraction profile changed from a wall to a core peak. In general, there is an increase in the turbulence intensities when the bubbles are introduced into the flow. However, a turbulence suppression was observed close to the wall at high liquid superficial velocities for low void fractions up to about 1.6%. The net radial interfacial force on the bubbles was estimated from the momentum equations using the measured profiles. The radial migration of the bubbles in the core region of the pipe, which determines the shape of the void profile, was related to the balance between the turbulent dispersion and the lift forces. The ratio between these forces was characterized by a dimensionless group that includes the area averaged Eötvös number, slip ratio, and the ratio between the apparent added kinetic energy to the actual kinetic energy of the liquid. A non-dimensional map based on this dimensionless group and the force ratio is proposed to distinguish the conditions under which a wall or core peak void profile occurs in bubbly flows.     

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.     

Influence of gravity and lift on particle velocity statistics and transfer rates in turbulent vertical channel flow

The present study reports detailed statistics for velocity and transfer rates of heavy particles dispersed in turbulent boundary layers. Statistics have been extracted from a homogeneous source of data covering a large target parameter space and are used here to analyze the effects of gravity and lift on particle dispersion and deposition in a systematic way. Datasets were obtained performing Direct Numerical Simulation (DNS) of particle-laden turbulent upward/downward flow in a vertical channel. Six values for the particle timescale (the particle Stokes number, St) ranging three orders of magnitude were considered to analyze the deposition process as the controlling mechanism was shifting from turbulent diffusion to inertia-moderated crossing trajectories. For the particle timescales examined, gravity and lift do not influence the qualitative behavior of particles even though velocity profiles and deposition coefficients are modified in a non-monotonic fashion, reaching an optimum for St ? 15. Physical mechanisms for the different behavior are discussed. Raw data and statistics obtained from the present DNS are made available at http://cfd.cineca.it (mirror site: http://158.110.32.35/download/database) and can be used to test simple models and closure equations for multiphase RANS and Large Eddy simulations.     

Order and disorder in particle–liquid flows in a pipe

The spherical expanded polystyrene particle–oil two-phase flow in a vertical pipe was used to simulate the dispersed phase distribution in laminar bubbly flows. A three-dimensional particle image tracking technique was used to track the particles in the flow to study the ordered structure of dispersed phase distribution and its transition to disorder. The ordered structures behaved as particle strings aligned in the flow direction as induced by the flow shear. The structures were quite durable in high liquid velocity flows and dispersed gradually as the liquid velocity decreased. In lower velocity flows, the particles tended to form clusters in the horizontal direction, as predicted by potential theory for spherical bubbles rising in a quiescent inviscid liquid and as observed in experiments on non-shear bubbly water flows.     

On the influence of near-wall forces in particle-laden channel flows

Computation of a turbulent dilute gas–solid channel flow has been undertaken to study the influence of using wall-corrected drag coefficients and of the lift force on the dispersed phase characteristics. The incompressible Navier–Stokes equations governing the carrier flow were solved by using a direct numerical simulation approach and coupled with a Lagrangian particle tracking. Calculations were performed at Reynolds number based on the wall-shear velocity and channel half-width, Re_τ ≈ 184, and for three different sets of solid particles. For each particle set, two cases were examined, in the first one the particle motion was governed by both drag and lift wall-corrected forces, whereas in the other one, the standard drag force (not corrected) was solely acting. The lift force model used represents the most accurate available expression since it accounts for weak and strong shear as well as for wall effects. For this study, we considered elastic collisions for particles contacting the walls and that no external forces were acting. Present results indicate that the use of the lift force and of the drag corrections does not lead to significant changes in the statistical properties of the solid phase, at the exception of some statistics for the high inertia particles.     

The effect of two-way coupling and inter-particle collisions on turbulence modulation in a vertical channel flow

Turbulence modulation due to its interaction with dispersed solid particles in a downward fully developed channel flow was studied. The Eulerian framework was used for the gas-phase, whereas the Lagrangian approach was used for the particle-phase. The steady-state equations of conservation of mass and momentum were used for the gas-phase, and the effect of turbulence on the flow-field was included via the standard k–ε model. The particle equation of motion included the drag, the Saffman lift and the gravity forces. Turbulence dispersion effect on the particles was simulated as a continuous Gaussian random field. The effects of particles on the flow were modeled by appropriate source terms in the momentum, k and ε equations. Particle–particle collisions and particle–wall collisions were accounted for in these simulations. Gas-phase velocities and turbulence kinetic energy in the presence of 2–100% mass loadings of two particle classes (50 μm glass and 70 μm copper) were evaluated, and the results were compared with the available experimental data and earlier numerical results. The simulation results showed that when the inter-particle collisions were important and was included in the computational model, the fluid turbulence was attenuated. The level of turbulence attenuation increased with particle mass loading, particle Stokes number, and the distance from the wall. When the inter-particle collisions were negligible and/or was neglected in the model, the fluid turbulence was augmented for the range of particle sizes considered.     

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.     

Investigation of particle-laden turbulent pipe flow at high-Reynolds-number using particle image/tracking velocimetry (PIV/PTV)

An experimental investigation of a high Reynolds number flow (Re = 320 000) of a dilute liquid-solid mixture (<1% by volume) was conducted. The turbulent motion of both the liquid phase (water) and particles (0.5, 1, and 2 mm glass beads) was evaluated in an upward pipe flow using a particle image/tracking velocimetry (PIV/PTV) technique. Results show that the Eulerian mean axial velocity of the glass beads is lower than that of the liquid phase in the central region but higher in the near-wall region. Moreover, the presence of the coarse particles has a negligible effect on the turbulence intensity of the liquid phase. Particles show higher streamwise and radial fluctuations than the liquid-phase at the tested conditions. The profiles of particle concentration across the pipe radius show almost constant concentration in the core of the pipe with a decrease towards the near wall region for 0.5 and 1 mm particles. For the 2 mm particles, a nearly linear concentration gradient from centre to the pipe wall is observed. The results presented here provide new information concerning the effect of a dispersed particulate phase on the turbulence modulation of the liquid carrier phase, especially at high Reynolds numbers. The present study also demonstrates how correlations developed to determine if particles cause turbulence attenuation/augmentation are not applicable for solid-liquid flows at high Reynolds numbers. Finally, the importance of particle-fluid slip velocity on fluid phase turbulence modulation is illustrated.     

Mean and fluctuating components of drag and lift forces on an isolated finite-sized particle in turbulence

We perform fully resolved direct numerical simulations of an isolated particle subjected to free-stream turbulence in order to investigate the effect of turbulence on the drag and lift forces at the level of a single particle, following Bagchi and Balachandar’s work (Bagchi and Balachandar in Phys Fluids 15:3496–3513, 2003). The particle Reynolds numbers based on the mean relative particle velocity and the particle diameter are Re?=?100, 250 and 350, which covers three different regimes of wake evolution in a uniform flow: steady axisymmetric wake, steady planar symmetric wake, and unsteady planar symmetric vortex shedding. At each particle Reynolds number, the turbulent intensity is 5–10% of the mean relative particle velocity, and the corresponding diameter of the particle is comparable to or larger than the Kolmogorov scale. The simulation results show that standard drag values determined from uniform flow simulations can accurately predict the drag force if the turbulence intensity is sufficiently weak (5% or less compared to the mean relative velocity). However, it is shown that for finite-sized particles, flow non-uniformity, which is usually neglected in the case of the small particles, can play an important role in determining the forces as the relative turbulence intensity becomes large. The influence of flow non-uniformity on drag force could be qualitatively similar to the Faxen correction. In addition, finite-sized particles at sufficient Reynolds number are inherently subjected to stochastic forces arising from their self-induced vortex shedding in addition to lift force arising from the local ambient flow properties (vorticity and strain rate). The effect of rotational and strain rate of the ambient turbulence seen by the particle on the lift force is explored based on the conditional averaging using the generalized representation of the quasi-steady force proposed by Bagchi and Balachandar (J Fluid Mech 481:105–148, 2003). From the present study, it is shown that at Re?=?100, the lift force is mainly influenced by the surrounding turbulence, but at Re = 250 and 350, the lift force is affected by the wake structure as well as the surrounding turbulence. Thus, for a finite-sized particle of sufficient Reynolds number supporting self-induced vortex shedding, the lift force will not be completely correlated with the ambient flow. Therefore, it appears that in order to reliably predict the motion of a finite-sized particle in turbulence, it is important to incorporate both a deterministic component and a stochastic component in the force model. The best deterministic contribution is given by the conditional average. The influence of ambient turbulence at the scale of the particle, which are not accounted for in the deterministic contribution, can be considered in stochastic manner. In the modeling of lift force, additional stochastic contribution arising from self-induced vortex shedding must also be included.     

The Oriented-Eddy Collision Turbulence Model

The Oriented-Eddy Collision (OEC) model treats turbulent flow as a non-Newtonian fluid where the average behavior of turbulence is modeled as a collection of interacting fluid particles which have inherent orientation. The model is derived from the two-point velocity correlation transport equation, and has the form of a collection of Reynolds-stress transport equations, with one set of transport equations for each representative eddy direction. The addition of eddy orientation information adds important physics to the model and allows the model to represent structural (two-point) information about the turbulence. This structural information is sufficient to allow the model to capture the effect of external forces and imposed mean strains (such as rapid distortion theory) exactly. The only physical effects that must be empirically modeled are those that are due to turbulence-turbulence interactions, referred to as eddy collisions. The performance of the model in a number of canonical flow situations is presented.     

Numerical study on flow separation control over NACA0015 aerofoil using electromagnetic fields

In this study, a flow solver was developed based on the governing RANS equations of compressible flows and was further extended to include the effects of electromagnetic forces namely Lorentz forces. Lorentz forces may be added as a source term in the governing fluid flow equations. Numerical studies were carried out for NACA0015 aerofoil at high angles of incidences from 15° to 30° and compared with some available cases of experimental and incompressible numerical solutions. The hydrodynamics performance was improved using a magnetic momentum coefficient of up to 0.048. The size of flow separation zone was decreased or completely eliminated by increasing this coefficient. The overall drag was not changed considerably, however the overall lift was increased up to 80 percent at stall angles.     

A New Approach in Modeling Phase Distribution in Fully Developed Bubbly Pipe Flow

Turbulent two-phase flow equations are derived and solved for fully developed pipe flow using a composite eddy-viscosity model and a new void-fraction equation. The void fraction profile is first specified from experiments and the velocity field is calculated to validate the eddy-viscosity model. Consequently, a new equation is presented for calculation of the void fraction. This void-fraction equation incorporates the gradient of turbulent normal stresses in the radial direction, the conventional lift force, and a contribution from the unsteady drag force. The implications of this new equation, for the bubbly flow regime, are investigated by calculating the void-fraction distribution for a given velocity field. Inclusion of the normal turbulent stresses in the radial direction is shown to simulate correctly the experimentally observed trends of the phase distribution, both for upward and downward bubbly flow, without the need for a fictitious term such as the so called ``lubrication force'. This revised version was published online in July 2006 with corrections to the Cover Date.     

Turbulence modification in vertical upward annular flow passing through a throat section

Experimental studies on the turbulence modification in annular two-phase flow passing through a throat section were carried out. The turbulence modification in multi-phase flow due to the interactions between two-phases is one of the most interesting scientific issues and has attracted research attention. In this study, the gas-phase turbulence modification in annular flow due to the gas–liquid phase interaction is experimentally investigated. The annular flow passing through a throat section is under the transient state due to the changing cross sectional area of the channel and resultantly the superficial velocities of both phases are changed compared with a fully developed flow in a straight pipe. The measurements for the gas-phase turbulence were precisely performed by using a constant temperature hot-wire anemometer, and made clear the turbulence structure such as velocity profiles, fluctuation velocity profiles. The behavior of the interfacial waves in the liquid film flow such as the ripple or disturbance waves was also observed. The measurements for the liquid film thickness by the electrode needle method were also performed to measure the base film thickness, mean film thickness, maximum film thickness and wave height of the ripple or the disturbance waves.     

Symmetries, Invariance and Scaling-Laws in Inhomogeneous Turbulent Shear Flows

An approach to derive turbulent scaling laws based on symmetry analysis is presented. It unifies a large set of scaling laws for the mean velocity of stationary parallel turbulent shear flows. The approach is derived from the Reynolds averaged Navier–Stokes equations, the fluctuation equations, and the velocity product equations, which are the dyad product of the velocity fluctuations with the equations for the velocity fluctuations. For the plane case the results include the logarithmic law of the wall, an algebraic law, the viscous sublayer, the linear region in the centre of a Couette flow and in the centre of a rotating channel flow, and a new exponential mean velocity profile that is found in the mid-wake region of high Reynolds number flat-plate boundary layers. The algebraic scaling law is confirmed in both the centre and the near wall regions in both experimental and DNS data of turbulent channel flows. For a non-rotating and a moderately rotating pipe about its axis an algebraic law was found for the axial and the azimuthal velocity near the pipe-axis with both laws having equal scaling exponents. In case of a rapidly rotating pipe, a new logarithmic scaling law for the axial velocity is developed. The key elements of the entire analysis are two scaling symmetries and Galilean invariance. Combining the scaling symmetries leads to the variety of different scaling laws. Galilean invariance is crucial for all of them. It has been demonstrated that two-equation models such as the k– model are not consistent with most of the new turbulent scaling laws.     

Numerical analysis of swirling turbulent droplet-laden flow and heat transfer in a sudden pipe expansion

The effect of swirling intensity on the structure and heat transfer of a turbulent gas–droplet flow after a sudden pipe expansion has been numerically simulated. Air is used as the carrier phase, and water, ethanol, and acetone are used as the dispersed phase. The Eulerian approach is applied to simulate the dynamics and heat transfer in the dispersed phase. The gas phase is described by a system of Reynolds-averaged Navier-Stokes (RANS) equations, taking into account the effect of droplets on mean transport and turbulent characteristics in the carrier phase. Gas phase turbulence is predicted using the second-moment closure. A swirling droplet-laden flow is characterized by an increase in the number of small particles on the pipe axis due to their accumulation in the zone of flow recirculation and the action of the turbulent migration (turbophoresis) force. A rapid dispersion of fine droplets over the pipe cross-section is observed without swirling. With an increase in swirling intensity, a significant reduction in the length of the separation region occurs. The swirling of a two-phase flow with liquid droplets leads to an increase in the level of turbulence for all three types of liquid droplets investigated in this work due to their intensive evaporation. It is shown that the addition of droplets leads to a significant increase in heat transfer in comparison with a single-phase swirling flow. The greatest effect of flow swirling on heat transfer intensification in a two-phase gas-droplet flow is obtained for the droplets of ethanol and water and smallest effect is for the acetone droplets.     

Simulation of turbulent flows around a circular cylinder using nonlinear eddy-viscosity modelling: steady and oscillatory ambient flows

Steady incident flow past a circular cylinder for sub- to supercritical Reynolds number has been simulated as an unsteady Reynolds-averaged Navier–Stokes (RANS) equation problem using nonlinear eddy-viscosity modelling assuming two-dimensional flow. The model of Craft et al. (Int. J. Heat Fluid Flow 17 (1996) 108), with adjustment of the coefficients of the ‘cubic’ terms, predicts the drag crisis at a Reynolds number of about 2×10⁵ due to the onset of turbulence upstream of separation and associated changes in Strouhal number and separation positions. Slightly above this value, at critical Reynolds numbers, drag is overestimated because attached separation bubbles are not simulated. These do not occur at supercritical Reynolds numbers and drag coefficient, Strouhal number and separation positions are in approximate agreement with experimental measurements (which show considerable scatter). Fluctuating lift predictions are similar to sectional values measured experimentally for subcritical Reynolds numbers but corresponding measurements have not been made at supercritical Reynolds numbers. For oscillatory ambient flow, in-line forces, as defined by drag and inertia coefficients, have been compared with the experimental values of Sarpkaya (J. Fluid Mech. 165 (1986) 61) for values of the frequency parameter, β=D²/νT, equal to 1035 and 11240 and Keulegan–Carpenter numbers, KC=U₀T/D, between 0.2 and 15 (D is cylinder diameter, ν is kinematic viscosity, T is oscillation period, and U₀ is the amplitude of oscillating velocity). Variations with KC are qualitatively reproduced and magnitudes show best agreement when there is separation with a large-scale wake, for which the turbulence model is intended. Lift coefficients, frequency and transverse vortex shedding patterns for β=1035 are consistent with available experimental information for β≈250−500. For β=11240, it is predicted that separation is delayed due to more prominent turbulence effects, reducing drag and lift coefficients and causing the wake to be more in line with the flow direction than transverse to it. While these oscillatory flows are highly complex, attached separation bubbles are unlikely and the flows probably two dimensional.     

Fluid flow in a helical pipe

Without simplifying the N-S equations of Germano's^[5], we study the flow in a helical circular pipe employing perturbation method. A third perturbation solution is fully presented. The first- second- and third-order effects of curvature κ and torsion τ on the secondary flow and axial velocity are discussed in detail. The first-order effect of curvature is to form two counter-rotating cells of the secondary flow and to push the maximum axial velocity to the outer bend. The two cells are pushed to the outer bend by the pure second-order effect of curvature. The combined higher-order (second-, third-) effects of curvature and torsion, are found to be an enlargement of the lower vortex of the secondary flow at expense of the upper one and a clockwise shift of the centers of the secondary vortices and the location of maximum axial velocity. When the axial pressure gradient is small enough or the torsion is sufficiently larger than the curvature, the location of the maximal axial velocity is near the inner bend. The equation of the volume flux is obtained from integrating the perturbation solutions of axial velocity. From the equation the validity range of the perturbation solutions in this paper can be obtained and the conclusion that the three terms of torsion have no effect on the volume flux can easily be drawn. When the axial pressure gradient is less than 22.67, the volume flux in a helical pipe is larger than that in a straight pipe.     

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.