Theory on transverse migration of particles in a turbulent two-phase suspension flow due to turbulent diffusion—I

A new theoretical model has been developed to explain the behavior of transverse particle transport in turbulent flow of a dilute two-phase suspension due to turbulent diffusion. This model is based on the ability of a particle to respond to surrounding fluid motion and depends on particle size and density relative to the carrier fluid, the fractional variation in particle concentration in the transverse direction as well as the existing turbulence structure of the surrounding fluid. The model developed in this investigation has been formulated by dividing the transverse fluid velocity, as seen by a particular particle, into two superimposed components representing, respectively, the transverse turbulent fluid fluctuations and an apparent transverse local fluid drifting velocity due to the effect on the transverse oscillatory component of fluid motion by the transverse concentration distribution of particles. A subsequent paper will show that the theory (together with other new results on the concentration effects on particle drag and lift and fluid turbulence properties) can help to explain the phenomena measured previously.     

LDV measurements of a turbulent air-solid two-phase flow in a 90° bend

Simultaneous measurements of the mean streamwise and radial velocities and the associated Reynolds stresses were made in an air-solid two-phase flow in a square sectioned (10×10 cm) 90° vertical to horizontal bend using laser Doppler velocimetry. The gas phase measurements were performed in the absence of solid particles. The radius ratio of the bend was 1.76. The results are presented for two different Reynolds numbers, 2.2×10⁵ and 3.47×10⁵, corresponding to mass ratios of 1.5×10^–4 and 9.5×10^–5, respectively. Glass spheres 50 and 100 m in diameter were employed to represent the solid phase. The measurements of the gas and solid phase were performed separately. The streamwise velocity profiles for the gas and the solids crossed over near the outer wall with the solids having the higher speed near the wall. The solid velocity profiles were quite flat. Higher negative slip velocities are observed for the 100 m particles than those for the 50 gm particles. At angular displacement =0°, the radial velocity is directed towards the inner wall for both the 50 and 100 m particles. At =30° and 45°, particle wall collisions cause a clear change in the radial velocity of the solids in the region close to the outer wall. The 100 m particle trajectories are very close to being straight lines. Most of the particle wall collisions occur between the =30° and 60° stations. The level of turbulence of the solids was higher than that of the air.List of symbols D hydraulic diameter (100 mm) - De Dean number,De = - mass flow rate - number of particles per second (detected by the probe volume) - r radial coordinate direction - r _i radius of curvature of the inner wall - r ₀ radius of curvature of the outer wall - r ^* normalized radial coordinate, - R mean radius of curvature - Re Reynolds number, - R _r radius ratio, - U ,U _z mean streamwise velocity - U _r ,U _y mean radial velocity - U _b bulk velocity - , _z rms fluctuating streamwise velocity - _r , _y rms fluctuating radial velocity - -r shear stress component - z-y shear stress component - x spanwise coordinate direction - x ^* normalized spanwise coordinate, - y radial coordinate direction in straight ducts - y ^* normalized radial coordinate in straight ducts, - z streamwise coordinate direction in straight ducts - z ^* normalized streamwise coordinate in straight ducts, Greek symbols streamwise coordinate direction - kinematic viscosity of air     

Drag models for simulating gas–solid flow in the turbulent fluidization of FCC particles

This paper examines the suitability of various drag models for predicting the hydrodynamics of the turbulent fluidization of FCC particles on the Fluent V6.2 platform.The drag models included those of Syamlal–O’Brien,Gidaspow,modified Syamlal–O’Brien,and McKeen.Comparison between experimental data and simulated results showed that the Syamlal–O’Brien,Gidaspow,and modified Syamlal–O’Brien drag models highly overestimated gas–solid momentum exchange and could not predict the formation of dense phase in the fl...     

An Eulerian–Lagrangian hybrid model for the coarse-grid simulation of turbulent liquid jet breakup

In this paper we present a numerical model for the coarse-grid simulation of turbulent liquid jet breakup using an Eulerian–Lagrangian coupling. To picture the unresolved droplet formation near the liquid jet interface in the case of coarse grids we considered a theoretical model to describe the unresolved flow instabilities leading to turbulent breakup. These entrained droplets are then represented by an Eulerian–Lagrangian hybrid concept. On the one hand, we used a volume of fluid method (VOF) to characterize the global spreading and the initiation of droplet formation; one the other hand, Lagrangian droplets are released at the liquid–gas interface according to the theoretical model balancing consolidating and disruptive energies. Here, a numerical coupling was required between Eulerian liquid core and Lagrangian droplets using mass and momentum source terms. The presented methodology was tested for different liquid jets in Rayleigh, wind-induced and atomization regimes and validated against literature data. This comparison reveals fairly good qualitative agreement in the cases of jet spreading, jet instability and jet breakup as well as relatively accurate size distribution and Sauter mean diameter (SMD) of the droplets. Furthermore, the model was able to capture the regime transitions from Rayleigh instability to atomization appropriately. Finally, the presented sub-grid model predicts the effect of the gas-phase pressure on the droplet sizes very well.     

Role of nuclei density as a “hidden” parameter in the formation of anomalous zones in a heavy cavitating magma

The formation of zones with anomalously high values of the basic flow characteristics in decompression waves in a heavy cavitating magma with an intense increase in the density of cavitation nuclei is numerically studied within the framework of a mathematical model of multiphase media with a system of kinetic equations. The basic effects leading to the formation of the anomalous zone are identified.     

Transport modeling of sedimenting particles in a turbulent pipe flow using Euler–Lagrange large eddy simulation

A volume-filtered Euler–Lagrange large eddy simulation methodology is used to predict the physics of turbulent liquid–solid slurry flow through a horizontal periodic pipe. A dynamic Smagorinsky model based on Lagrangian averaging is employed to account for the sub-filter scale effects in the liquid phase. A fully conservative immersed boundary method is used to account for the pipe geometry on a uniform cartesian grid. The liquid and solid phases are coupled through volume fraction and momentum exchange terms. Particle–particle and particle–wall collisions are modeled using a soft-sphere approach. Three simulations are performed by varying the superficial liquid velocity to be consistent with the experimental data by Dahl et al. (2003). Depending on the liquid flow rate, a particle bed can form and develop different patterns, which are discussed in light of regime diagrams proposed in the literature. The fluctuation in the height of the liquid-bed interface is characterized to understand the space and time evolution of these patterns. Statistics of engineering interest such as mean velocity, mean concentration, and mean streamwise pressure gradient driving the flow are extracted from the numerical simulations and presented. Sand hold-up calculated from the simulation results suggest that this computational strategy is capable of predicting critical deposition velocity.     

Numerical study of gas–solid two-phase flow and erosion in a cavity with a slope

Gate valve is mainly used to turn on or turn off the pipeline in pneumatic conveying. When the gate valve is fully open, the particles are easy to collide with the cavity rear wall and enter into the cavity, resulting in particles’ accumulation in the cavity. The particles in cavity will accumulate between the cavity bottom and the flashboard bottom wall and prevent the gate from turning off normally. Meanwhile, the particles’ collision with cavity rear wall will cause serious erosion. Both the particles’ accumulation and erosion will cause the poor sealing of the gate valve, further resulting in the leakage of the pipeline system. To reduce the particles’ accumulation in cavity and erosion on cavity when the gate valve is fully open, we simplify the gate valve into a cavity structure and study it. We find that adding a slope upstream the cavity can effectively reduce the particles’ accumulation in the cavity and the erosion on the cavity rear wall. In this work, Eulerian–Lagrangian method in commercial code (FLUENT) was used to study the gas–solid two-phase flow and erosion characteristics of a cavity with a slope. The particle distribution shows that the particles with Stokes number St = 1.3 and St = 13 cannot enter the cavity due to the slope, but the particles with St = 0.13 enter the cavity following the gas. For St = 13, the particles collide with the wall many times in the ideal cavity. Erosion results show that the slope can transfer the erosion on cavity rear wall to the slope and reduce the maximum erosion rate of the wall near the cavity to some degrees.     

Simultaneous velocity and concentration measurements in the near field of a turbulent low-pressure jet by digital particle image velocimetry–planar laser-induced fluorescence

The main purpose of this work is to develop a method for simultaneous measurement of velocity and passive scalar concentration by means of digital particle image velocimetry and planar laser-induced fluorescence. Details of the implementation of the method are given, and the technique is applied to measurements of concentration and velocity in the centre-plane of a liquid jet with a Reynolds number of 6,000. The measurements are compared with large eddy simulations. Mean velocities and concentrations, fluctuating velocities and concentrations, and correlation between fluctuating velocities and concentrations are analysed for the first six diameters downstream of the jet exit. The general agreement between measured and simulated results was found to be good, in particular for mean quantities. Mean profiles are also found to be in good agreement with other experimental work on jets reported in the literature. The “whole-plane” measurement method was found to be very useful for detailed comparisons of turbulent statistics with simulated data. The inadequacy of models for turbulent mass transport based on the standard gradient diffusion concept is demonstrated through the experimental data. Received: 4 October 2000/Accepted: 27 November 2000     

Existence of a Solution “in the Large” for Ocean Dynamics Equations

For the system of equations describing the large-scale ocean dynamics, an existence and uniqueness theorem is proved “in the large”. This system is obtained from the 3D Navier–Stokes equations by changing the equation for the vertical velocity component u ₃ under the assumption of smallness of a domain in z-direction, and a nonlinear equation for the density function ρ is added. More precisely, it is proved that for an arbitrary time interval [0, T], any viscosity coefficients and any initial conditions

Experimental study of a liquid jet flowing into another immiscible liquid “a local analysis of the interface”

This paper describes an experimental study of a liquid jet leaving a cylindrical nozzle under gravity. A special optical system was used to study the spatial and temporal interface variations between two liquids. A photoelectric cell was used to measure the light intensity and to obtain the physical parameters of the jet. Spatial analysis revealed a continual contraction of the jet from the nozzle exit to the break-up zone. Fluctuations of the interface over time are characteristic of a random signal with a narrow bandpass. The Fourier transform of the different samples shows a bandpass of finite width centered around a characteristic frequency. The distribution of interface amplitude fluctuations was symmetrical to the average diameter, except in the zone in which the jet breaks up. By systematically tracing the main parameters of the jet diameter, we observed three zones with different jet behavior. The characteristic frequency of interface fluctuations increases as a linear function of the distance from the nozzle. The amplitude of interface fluctuations was an exponential function of the distance at which jet diameter fluctuations were measured.     

Thermo-fluid-dynamic analysis of the flow in a rotating channel with a sharp “U” turn

Infrared thermography has been employed to carry out a detailed convective heat transfer measurements at Re?=?20,000 in a two-pass square channel both for the static case (absence of channel rotation) and for the rotating case (Ro?=?0.3). At the same time, the main and secondary flow fields have been measured by means of particle image velocimetry with the aim to investigate how the flow behavior affects the local distributions of the convective heat transfer coefficient for the two cases. The normal-to-wall velocity component (w) and the turbulent kinetic energy, both measured close to the heat exchanging wall, have been used to formulate an empirical heat transfer correlation within an attempt to identify the role performed by these two quantities on the convective heat transfer coefficient distributions. The latter ones have been reported in terms of normalized Nusselt number (Nu/Nu*) maps, where Nu* is the Nusselt number evaluated with the classical Dittus-B?lter correlation.     

Global stability of periodic solutions for a discrete predator–prey system with functional response

The purpose of this paper is to study the existence and global stability of a periodic solution for a discrete predator–prey system with the Beddington–DeAngelis functional response and predator cannibalism. By using the continuation theorem, the existence conditions of at least one periodic solution are obtained, and the sufficient conditions, which ensure the global stability of the positive periodic solution, are derived by constructing a special Lyapunov function.     

Flow structures inside a large-scale turbulent fluidized bed of FCC particles： Eulerian simulation with an EMMS-based sub-grid scale model

Turbulent fluidized bed reactors are widely used in industry. However, CFD simulations of the hydrody- namic characteristics of these reactors are relatively sparse, despite the urgent demand from industry. To address this problem, Eulerian simulations with an EMMS-based sub-grid scale model, accounting for the effect of sub-grid scale structures on the inter-phase friction, are performed to study the hydrodynamics inside a large-scale turbulent fluidized bed of FCC particles. It is shown that the simulated axial and radial solid concentration profiles, entrained solid fluxes and standard deviation of the solid concentration fluc- tuation agreed well with experimental data available in the literature. In-depth analysis of time-averaged particle velocity and solid concentration shows that a dense-suspension upflow regime coexists with fast fluidization regime in this bed, which is reminiscent of the hydrodynamic characteristics in high-density circulating fluidized bed （CFB） risers, even though they are operated in different fluidization regimes. The Reynolds stresses in turbulent fluidized beds are anisotropic, but the degree of anisotropy is far less pro- nounced than the reported values in CFB risers. It was also found that the solid concentration fluctuation and axial particle velocity fluctuation are strongly correlated. 2009 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.     

Influences of the wall thickness on the granular dispersion in a dense gas–solid coaxial jet

The granular dispersion for different wall thicknesses of the inner channel in a dense gas–solid coaxial jet is investigated by morphology. In the wave dispersion, the Kelvin–Helmholtz instability has a strong influence on the granular stream. However, increasing the wall thickness damps the K–H instability, namely weakens the wave dispersion. Consequently, the non-dimensional wavelength increases and the non-dimensional amplitude decreases as the wall thickness increases for the same annular gas velocity. The instability analysis combining with the modification of the vorticity thickness of the annular gas stream indicates that the non-dimensional wavelength increases linearly with the non-dimensional modified vorticity thickness. The relation between the amplitude of the granular dispersion and the non-dimensional modified vorticity thickness is reliable for the wave dispersion as well. In addition, the granular dispersion angle is proportional to the annular gas velocity, which slightly depends on the wall thickness.     

A coupled model for simulation of the gas–liquid two-phase flow with complex flow patterns

A new model coupling two basic models, the model based on interface tracking method and the two-fluid model, for simulating gas–liquid two-phase flow is presented. The new model can be used to simulate complex multiphase flow in which both large-length-scale interface and small-length-scale gas–liquid interface coexist. By the physical state and the length scale of interface, three phases are divided, including the liquid phase, the large-length-scale-interface phase (LSI phase) and the small-length-scale-interface phase (SSI phase). A unified solution framework shared by the two basic models is built, which makes it convenient to perform the solution process. Based on the unified solution framework, the modified MCBA–SIMPLE algorithm is employed to solve the Navier–Stokes equations for the proposed model. A special treatment called “volume fraction redistribution” is adopted for the special grids containing all three phases. Another treatment is proposed for the advection of large-length-scale interface when some portion of SSI phase coalesces into LSI phase. The movement of the large-length-scale interface is evaluated using VOF/PLIC method. The proposed model is equivalent to the two-fluid model in the zone where only the liquid phase and the SSI phase are present and to the model based on interface tracking method in the zone where only the liquid phase and the LSI phase are present. The characteristics of the proposed model are shown by four problems.     

On wall pressure fluctuations and their coupling with vortex dynamics in a separated–reattached turbulent flow over a blunt flat plate

This study deals with the numerical predictions through Large-Eddy Simulation (LES) of the separated–reattached turbulent flow over a blunt flat plate for analyzing main coherent structure features and their relation to the unsteady pressure field. A compressible approach that inherently includes acoustic propagation is here followed to describe the relationship between pressure fluctuations and vortex dynamics around the separation bubble. The objective of the present work is then to contribute to a better understanding of the coupling between the vortex dynamics and the wall pressure fluctuations. The filtered compressible Navier–Stokes equations are then solved with a numerical method that follows a Lax–Wendroff approach to recover a high accuracy in both time and space. For validations, the present numerical results are compared to experimental measurements, coming from both the Pprime laboratory (Sicot el al., 2012) and the literature (Cherry et al., 1984; Kiya and Sasaki, 1985; Tafti and Vanka,1991; Sicot et al., 2012). Our numerical results very well predict mean and fluctuating pressure and velocity fields. Flapping, shedding as well as Kelvin–Helmholtz characteristic frequencies educed by present simulations are in very good agreement with the experimental values generally admitted. These characteristic modes are also visible on unsteady pressure signatures even far away from the separation. Spectral, POD and EPOD (extended POD) analyses are then applied to these numerical data to enhance the salient features of the pressure and velocity fields, especially the unsteady wall pressure in connection with either the vortex shedding or the low frequency shear-layer flapping. A contribution to the understanding of the coupling between wall pressure fluctuations and eddy vortices is finally proposed.     

“Non-free” interaction of plane shock waves and the boundary layer in the neighborhood of the leading edge of a plate with slip

The interaction between a normally impinging shock wave and the boundary layer on a plate with slip is studied in the neighborhood of the leading edge using various experimental methods, including special laser technology, to visualize the supersonic conical gas flows. It is found that in the “non-free” interaction, when the leading edge impedes the propagation of the boundary layer separation line upstream, the structure of the disturbed flow is largely identical to that in the developed “free” interaction, but with higher parameter values and gradients in the leading part of the separation zone. The fundamental property of developed separation flows, namely, coincidence of the values of the pressure “plateau” in the separation zone and the pressure behind the oblique shock above the separation zone of the turbulent boundary layer, is conserved. Moscow. e-mail: ostap@inmech.msu.su. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 57–69, May–June, 2000. The work was carried out with financial support from the Russian Foundation for Basic Research (project No. 97-01-00099).     

“Simple” solutions of the equations of dynamics for a polytropic gas

The notion of a “simple” solution of a system of differential equations that admit a local Lie group G of transformations of the basic space is considered as an invariant H-solution of type (0, 0) with respect to the subgroup HυG. Such solutions are attractive since they are described by explicit formulas that provide a clear physical interpretation for them. For gas-dynamic equations with a polytropic gas law, all simple solutions that are not related to special forms of gas flow are listed. Examples of simple solutions are given and the collapse phenomenon, which has been previously studied for barochronic flows, is described. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 5–12, March–April, 1999.     

The flow pattern map of a two-phase non-Newtonian liquid–gas flow in the vertical pipe

A map for the determination of flow pattern for two-phase flow of gas and non-Newtonian liquid in the vertical pipe has been presented. Our own experimental data confirm applicability of such a map.     

On a Theory of the Infusion Method of Measuring the “Opening Pressure” in the Urethra

At rest the muscles which control the urethra (urine duct) are contracted and its lumen is practically equal to zero over its entire length. To open the urethra, a mechanical effort (due, for example, to a pressure rise in the bladder) must be applied. Reduced contractile activity of the muscles may be one of the reasons for incontinence (enuresis). A widespread method of estimating the blocking capability of the urethra consists in inserting a catheter with lateral perforations near the end. The catheter enters the bladder and is then removed at a constant velocity while a fluid is constantly pumped (infused) into it by a syringe pump at a steady rate and then flows out through the gap between the catheter and the urethral wall. The pumping pressure is considered to be a local measure of the blocking capability, and its dependence on the location of the catheter is regarded as an important diagnostic characteristic.Below, we will consider the simple, longitudinally homogeneous model system formed by an elastic tube pulled over a catheter segment when the initial stresses in the tube are constant over its length. An incompressible viscous fluid flows out of the perforations and percolates in a thin layer along the catheter. In solving the model problem, we will use the lubricating layer approximation under the assumption of small layer curvature. On the basis of an analysis of the results and a comparison of the model with a practical intraurethral measurement procedure, we discuss, firstly, the relationship between the measured quantities and the real characteristics of the urethra and, secondly, the possible formulation of a more realistic model problem.