The discrete Green's function paradigm for two-way coupled Euler–Lagrange simulation

We outline a methodology for the simulation of two-way coupled particle-laden flows. The drag force that couples fluid and particle momentum depends on the undisturbed fluid velocity at the particle location, and this latter quantity requires modelling. We demonstrate that the undisturbed fluid velocity, in the low particle Reynolds number limit, can be related exactly to the discrete Green's function of the discrete Stokes equations. In addition to hydrodynamics, the method can be extended to other physics present in particle-laden flows such as heat transfer and electromagnetism. The discrete Green's functions for the Navier–Stokes equations are obtained at low particle Reynolds number in a two-plane channel geometry. We perform verification at different Reynolds numbers for a particle settling under gravity parallel to a plane wall, for different wall-normal separations. Compared with other point-particle schemes, the Stokesian discrete Green's function approach is the most robust at low particle Reynolds number, accurate at all wall-normal separations. To account for degradation in accuracy away from the wall at finite Reynolds number, we extend the present methodology to an Oseen-like discrete Green's function. The extended discrete Green's function method is found to be accurate within $6\,\%$ at all wall-normal separations for particle Reynolds numbers up to 24. The discrete Green's function approach is well suited to dilute systems with significant mass loading and this is highlighted by comparison against other Euler–Lagrange as well as particle-resolved simulations of gas–solid turbulent channel flow. Strong particle–turbulence coupling is observed in the form of turbulence modification and turbophoresis suppression, and these observations are placed in context of the different methods.

Competing flow and collision effects in a monodispersed liquid–solid fluidized bed at a moderate Archimedes number

We study the effects of fluid–particle and particle–particle interactions in a three-dimensional monodispersed reactor with unstable fluidization. Simulations were conducted using the immersed boundary method for particle Reynolds numbers of 20–70 with an Archimedes number of 23 600. Two different flow regimes were identified as a function of the particle Reynolds number. For low particle Reynolds numbers ( $20 < Re_p < 40$ ), the porosity is relatively low and the particle dynamics are dominated by interparticle collisions that produce anisotropic particle velocity fluctuations. The relative importance of hydrodynamic effects increases with increasing particle Reynolds number, leading to a minimized anisotropy in the particle velocity fluctuations at an intermediate particle Reynolds number. For high particle Reynolds numbers ( $Re_p > 40$ ), the particle dynamics are dominated by hydrodynamic effects, leading to decreasing and more anisotropic particle velocity fluctuations. A sharp increase in the anisotropy occurs when the particle Reynolds number increases from 40 to 50, corresponding to a transition from a regime in which collision and hydrodynamic effects are equally important (regime 1) to a hydrodynamic-dominated regime (regime 2). The results imply an optimum particle Reynolds number of roughly 40 for the investigated Archimedes number of 23 600 at which mixing in the reactor is expected to peak, which is consistent with reactor studies showing peak performance at a similar particle Reynolds number and with a similar Archimedes number. Results also show that maximum effective collisions are attained at intermediate particle Reynolds number. Future work is required to relate optimum particle Reynolds number to Archimedes number.

The slip velocity of nearly neutrally buoyant tracers for large-scale PIV

The behaviour of nearly neutrally buoyant tracers is studied by means of experiments with helium-filled soap bubbles and numerical simulations. The current models used for estimating the slip velocity of heavy micro particles and neutrally buoyant particles are reviewed and extended to include the effect of unsteady forces and particle Reynolds number. The particle motion is analysed via numerical simulations of a rectilinear oscillatory flow and in the flow around an airfoil within a particle flow parameter space that is typical of large-scale PIV experiments. An empirical relation is obtained that estimates the particle slip velocity, depending on the particle-to-fluid density ratio, the particle Reynolds number and frequency of the local flow fluctuations. The model developed is applied to assess the slip velocity of helium-filled soap bubbles in a large-scale experiment conducted at the German–Dutch wind (DNW) tunnels in the flow around an airfoil, with chord Reynolds numbers up to three millions. Furthermore, a procedure is proposed that can be used to retrieve the bubbles mean density and dispersion from measurements of mean velocity and fluctuations, respectively. Graphic abstract

Settling Behavior of Particles in Bubble Containing Newtonian Fluids: Experimental Study and Model Development

Accurate description of cuttings slippage in the gas-liquid phase is of great significance for wellbore cleaning and the control accuracy of bottom hole pressure during MPD. In this study, the wellbore bubble flow environment was simulated by a constant pressure air pump and the transparent wellbore, and the settling characteristics of spherical particles under different gas volume concentrations were recorded and analyzed by highspeed photography. A total of 225 tests were conducted to analyze the influence of particle diameter (1–12mm), particle density (2700–7860kg/m^3), liquid viscosity and bubble volume concentration on particle settling velocity. Gas drag force is defined to quantitatively evaluate the bubble’s resistance to particle slippage. The relationship between bubble drag coefficient and particle Reynolds number is obtained by fitting the experimental results. An explicit settling velocity equation is established by introducing Archimedes number. This explicit equation with an average relative error of only 8.09% can directly predict the terminal settling velocity of the sphere in bubble containing Newtonian fluids. The models for predicting bubble drag coefficient and the terminal settling velocity are valid with particle Reynolds number ranging from 0.05 to 167 and bubble volume concentration ranging from 3.0% to 20.0%. Besides, a trial-and-error procedure and an illustrative example are presented to show how to calculate bubble drag coefficient and settling velocity in bubble containing fluids. The results of this study will provide the theoretical basis for wellbore cleaning and accurate downhole pressure to further improve the performance of MPD in treating gas influx.

Analysis on the Characteristics of Bed Materials in the Jinghong Reservoir on the Lancang River

Nine samples of bed materials along the Jinghong reservoir on the Lancang River were obtained using a gravity sampler. The grain size characteristics of the samples were analyzed by the laser diffraction particle size analyzer. The results show that the median grain size of bed materials is in the range of 6.7 to 18.9 µm. From the upstream to the front of the dam, the overall grain size of the bed materials tends to decrease significantly; the sorting is poor but becomes better along the way; the skewness gradually changes from negative to near symmetrical; the kurtosis is platykurtic and mesokurtic with little change along the way. Based on the measured data, an exponential model is constructed to describe the distribution of representative grain size of bed materials along the way. Furthermore, the concept of representative particle Reynolds number is proposed. The relationship between representative particle Reynolds number and flow parameters with boundary conditions is established, and the coefficient and exponents in the equation are determined based on the measured data of the Jinghong reservoir. This study provides valuable first-hand information for reservoir sediment research and new ideas for sediment sorting and deposition studies.

A new model for settling velocity of non-spherical particles

The settlement of non-spherical particles, such as propagules of plants and natural sediments, are commonly observed in riverine ecosystems. The settling process is influenced by both particle properties (size, density and shape) and fluid properties (density and viscosity). Therefore, the drag law of non-spherical particles is a function of both particle Reynolds number and particle shape. Herein, a total of 828 settling data are collected from the literatures, which cover a wide range of particle Reynolds number (0.008–10000). To characterize the influence of particle shapes, sphericity is adopted as the general shape factor, which varies from 0.421 to 1.0. By comparing the measured drag with the standard drag curve of spheres, we modify the spherical drag law with three shape-dependent functions to develop a new drag law for non-spherical particles. Combined with an iterative procedure, a new model is thus obtained to predict the settling velocity of non-spherical particles of various shapes and materials. Further applications in hydrochorous propagule dispersal and sediment transport are projected based on deeper understanding of the settling process.

Drag on Janus Sphere in a Channel: Effect of Particle Position

Potential use of Janus spheres in novel engineering applications is being explored actively in recent years. Hydrodynamics around Janus spheres is different from that around homogeneous sticky or slippery spheres. Instantaneous motion of a sphere in channel flow is governed by hydrodynamic force experienced by the sphere, which in turn depends on the particle to channel size ratio, its instantaneous position, hydrophobicity of its surface and the particle Reynolds number. We investigate numerically the drag experienced by a Janus sphere located at different off-centre positions in a square channel. Two orientations of Janus sphere consisting of a sticky and a slippery hemisphere with the boundary between them parallel to the channel mid-plane are studied: (1) slippery hemisphere facing the channel centreline and (2) sticky hemisphere facing the channel centreline. The flow field around Janus sphere is found to be steady (for Re ≤ 50 investigated in this work) and asymmetric. Based on the data obtained, a correlation for drag coefficient as a function of particle Reynolds number and dimensionless particle position is also proposed.

Effect of inertial lift on a spherical particle suspended in flow through a curved duct

We develop a model of the forces on a spherical particle suspended in flow through a curved duct under the assumption that the particle Reynolds number is small. This extends an asymptotic model of inertial lift force previously developed to study inertial migration in straight ducts. Of particular interest is the existence and location of stable equilibria within the cross-sectional plane towards which particles migrate. The Navier–Stokes equations determine the hydrodynamic forces acting on a particle. A leading-order model of the forces within the cross-sectional plane is obtained through the use of a rotating coordinate system and a perturbation expansion in the particle Reynolds number of the disturbance flow. We predict the behaviour of neutrally buoyant particles at low flow rates and examine the variation in focusing position with respect to particle size and bend radius, independent of the flow rate. In this regime, the lateral focusing position of particles approximately collapses with respect to a dimensionless parameter dependent on three length scales: specifically, the particle radius, duct height and duct bend radius. Additionally, a trapezoidal-shaped cross-section is considered in order to demonstrate how changes in the cross-section design influence the dynamics of particles.