Diffusion-based coarse graining in hybrid continuum–discrete solvers: Theoretical formulation and a priori tests

Coarse graining is an important ingredient in many multi-scale continuum–discrete solvers such as CFD–DEM (computational fluid dynamics–discrete element method) solvers for dense particle-laden flows. Although CFD–DEM solvers have become a mature technique that is widely used in multiphase flow research and industrial flow simulations, a flexible and easy-to-implement coarse graining algorithm that can work with CFD solvers of arbitrary meshes is still lacking. In this work, we proposed a new coarse graining algorithm for continuum–discrete solvers for dense particle-laden flows based on solving a transient diffusion equation. Via theoretical analysis we demonstrated that the proposed method is equivalent to the statistical kernel method with a Gaussian kernel, but the current method is much more straightforward to implement in CFD–DEM solvers. A priori numerical tests were performed to obtain the solid volume fraction fields based on given particle distributions, the results obtained by using the proposed algorithm were compared with those from other coarse graining methods in the literature (e.g., the particle centroid method, the divided particle volume method, and the two-grid formulation). The numerical tests demonstrated that the proposed coarse graining procedure based on solving diffusion equations is theoretically sound, easy to implement and parallelize in general CFD solvers, and has improved mesh-convergence characteristics compared with existing coarse graining methods. The diffusion-based coarse graining method has been implemented into a CFD–DEM solver, the results of which are presented in a separate work.     

Instabilities during batch sedimentation in geometries containing obstacles: A numerical and experimental study

Batch sedimentation of non‐colloidal particle suspensions is studied with nuclear magnetic resonance flow visualization and continuum‐level numerical modelling of particle migration. The experimental method gives particle volume fraction as a function of time and position, which then provides validation data for the numerical model. A finite element method is used to discretize the equations of motion, including an evolution equation for the particle volume fraction and a generalized Newtonian viscosity dependent on local particle concentration. The diffusive‐flux equation is based on the Phillips model (Phys. Fluids A 1992; 4 :30–40) and includes sedimentation terms described by Zhang and Acrivos (Int. J. Multiphase Flow 1994; 20 :579–591). The model and experiments are utilized in three distinct geometries with particles that are heavier and lighter than the suspending fluid, depending on the experiment: (1) sedimentation in a cylinder with a contraction; (2) particle flotation in a horizontal cylinder with a horizontal rod; and (3) flotation around a rectangular inclusion. Secondary flows appear in both the experiments and the simulations when a region of higher density fluid is above a lower density fluid. The secondary flows result in particle inhomogeneities, Rayleigh–Taylor‐like instabilities, and remixing, though the effect in the simulations is more pronounced than in the experiments. Published in 2007 by John Wiley & Sons, Ltd.     

A LBM–DEM solver for fast discrete particle simulation of particle–fluid flows

The lattice Boltzmann method (LBM) for simulating fluid phases was coupled with the discrete element method (DEM) for studying solid phases to formulate a novel solver for fast discrete particle simulation (DPS) of particle–fluid flows. The fluid hydrodynamics was obtained by solving LBM equations instead of solving the Navier–Stokes equation by the finite volume method (FVM). Interparticle and particle–wall collisions were determined by DEM. The new DPS solver was validated by simulating a three-dimensional gas–solid bubbling fluidized bed. The new solver was found to yield results faster than its FVM–DEM counterpart, with the increase in the domain-averaged gas volume fraction. Additionally, the scalability of the LBM–DEM DPS solver was superior to that of the FVM–DEM DPS solver in parallel computing. Thus, the LBM–DEM DPS solver is highly suitable for use in simulating dilute and large-scale particle–fluid flows.     

Estimation of particle dynamics in 2-D fluidized beds using particle tracking velocimetry

The experimental characterization of particle dynamics in fluidized beds is of great importance in fostering an understanding of solid phase motion and its effect on particle properties in granulation processes. Commonly used techniques such as particle image velocimetry rely on the cross-correlation of illumination intensity and averaging procedures. It is not possible to obtain single particle velocities with such techniques. Moreover, the estimated velocities may not accurately represent the local particle velocities in regions with high velocity gradients. Consequently, there is a need for devices and methods that are capable of acquiring individual particle velocities. This paper describes how particle tracking velocimetry can be adapted to dense particulate flows. The approach presented in this paper couples high-speed imaging with an innovative segmentation algorithm for particle detection, and employs the Voronoi method to solve the assignment problem usually encountered in densely seeded flows. Lagrangian particle tracks are obtained as primary information, and these serve as the basis for calculating sophisticated quantities such as the solid-phase flow field, granular temperature, and solid volume fraction. We show that the consistency of individual trajectories is sufficient to recognize collision events.     

Estimation of particle dynamics in 2-D fluidized beds using particle tracking velocimetry

The experimental characterization of particle dynamics in fluidized beds is of great importance in fostering an understanding of solid phase motion and its effect on particle properties in granulation processes.Commonly used techniques such as particle image velocimetry rely on the cross-correlation of illumination intensity and averaging procedures.It is not possible to obtain single particle velocities with such techniques.Moreover,the estimated velocities may not accurately represent the local particle velocities in regions with high velocity gradients.Consequently,there is a need for devices and methods that are capable of acquiring individual particle velocities.This paper describes how particle tracking velocimetry can be adapted to dense particulate flows.The approach presented in this paper couples high-speed imaging with an innovative segmentation algorithm for particle detection,and employs the Voronoi method to solve the assignment problem usually encountered in densely seeded flows.Lagrangian particle tracks are obtained as primary information,and these serve as the basis for calculating sophisticated quantities such as the solid-phase flow field,granular temperature,and solid volume fraction.We show that the consistency of individual trajectories is sufficient to recognize collision events.     

Optimization of unsteady incompressible Navier–Stokes flows using variational level set method

This paper presents the optimization of unsteady Navier–Stokes flows using the variational level set method. The solid–liquid interface is expressed by the level set function implicitly, and the fluid velocity is constrained to be zero in the solid domain. An optimization problem, which is constrained by the Navier–Stokes equations and a fluid volume constraint, is analyzed by the Lagrangian multiplier based adjoint approach. The corresponding continuous adjoint equations and the shape sensitivity are derived. The level set function is evolved by solving the Hamilton–Jacobian equation with the upwind finite difference method. The optimization method can be used to design channels for flows with or without body forces. The numerical examples demonstrate the feasibility and robustness of this optimization method for unsteady Navier–Stokes flows.Copyright © 2012 John Wiley & Sons, Ltd.     

An immersed‐boundary finite‐volume method for direct simulation of flows with suspended paramagnetic particles

In the present study, we have proposed an immersed‐boundary finite‐volume method for the direct numerical simulation of flows with inertialess paramagnetic particles suspended in a nonmagnetic fluid under an external magnetic field without the need for any model such as the dipole–dipole interaction. In the proposed method, the magnetic field (or force) is described by the numerical solution of the Maxwell equation without current, where the smoothed representation technique is employed to tackle the discontinuity of magnetic permeability across the particle–fluid interface. The flow field, on the other hand, is described by the solution of the continuity and momentum equations, where the discrete‐forcing‐based immersed‐boundary method is employed to satisfy the no‐slip condition at the interface. To validate the method, we performed numerical simulations on the two‐dimensional motion of two and three paramagnetic particles in a nonmagnetic fluid subjected to an external uniform magnetic field and then compared the results with the existing finite‐element and semi‐analytical solutions. Comparison shows that the proposed method is robust in the direct simulation of such magnetic particulate flows. This method can be extended to more general flows without difficulty: three‐dimensional particulate flows, flows with a great number of particles, or flows under an arbitrary external magnetic field. Copyright © 2010 John Wiley & Sons, Ltd.     

Efficient simulation of particle-laden turbulent flows with high mass loadings using LES

The paper is concerned with the simulation of particle-laden two-phase flows based on the Euler–Lagrange approach. The methodology developed is driven by two major requirements: (i) the necessity to tackle complex turbulent flows by eddy-resolving schemes such as large-eddy simulation; (ii) the demand to predict dispersed multiphase flows at high mass loadings. First, a highly efficient particle tracking algorithm was developed working on curvilinear, block-structured grids. Second, to allow the prediction of dense two-phase flows, the fluid–particle interaction (two-way coupling) as well as particle–particle collisions (four-way coupling) had to be taken into account. For the latter instead of a stochastic collision model, in the present study a deterministic collision model is considered. Nevertheless, the computational burden is minor owing to the concept of virtual cells, where only adjacent particles are taken into account in the search for potential collision partners. The methodology is applied to different test cases (plane channel flow, combustion chamber flow). The computational results are compared with experimental measurements and good agreement is found.     

A Cartesian cut cell based two-way strong fluid–solid coupling algorithm for 2D floating bodies

In a recent paper Kelly et al. (2015) [PICIN: A Particle-In-Cell solver for incompressible free surface flows with two-way fluid–solid coupling. SIAM Journal on Scientific Computing 37 (3), B403–24.] detailed the PICIN full particle Particle-In-Cell (PIC) solver for incompressible free-surface flows. The model described in that paper employed a tailored version of the Distributed Lagrange Multiplier (DLM) method for the strong coupling of fluid–solid interaction. In this paper we propose an alternative strong fluid–solid coupling algorithm based on a modification to the cut cell methodology that is informed by the variational approach. The solid velocity flux/integral on the boundary is expressed purely in terms of pressure leading to a revised pressure Poisson equation that is discretised in a finite volume sense. This approach allows the PICIN model to simulate the motion of floating bodies of arbitrary configuration. 2D test cases involving floating bodies with one or more degrees of freedom (DoF) are used to validate the modified PICIN model. The results presented show that the modified PICIN model is able to both efficiently and robustly predict the motions of surface-piercing floating structures under either regular or extreme wave action.     

A numerical scheme for Euler–Lagrange simulation of bubbly flows in complex systems

An Eulerian–Lagrangian approach is developed for the simulation of turbulent bubbly flows in complex systems. The liquid phase is treated as a continuum and the Navier–Stokes equations are solved in an unstructured grid, finite volume framework for turbulent flows. The dynamics of the disperse phase is modeled in a Lagrangian frame and includes models for the motion of each individual bubble, bubble size variations due to the local pressure changes, and interactions among the bubbles and with boundaries. The bubble growth/collapse is modeled by the Rayleigh–Plesset (RP) equation. Three modeling approaches are considered: (a) one‐way coupling, where the influence of the bubble on the fluid flow is neglected, (b) two‐way coupling, where the momentum‐exchange between the fluid and the bubbles is modeled, and (c) volumetric coupling, where the volumetric displacement of the fluid by the bubble motion and the momentum‐exchange are modeled. A novel adaptive time‐stepping scheme based on stability‐analysis of the non‐linear bubble dynamics equations is developed. The numerical approach is verified for various single bubble test cases to show second‐order accuracy. Interactions of multiple bubbles with vortical flows are simulated to study the effectiveness of the volumetric coupling approach in predicting the flow features observed experimentally. Finally, the numerical approach is used to perform a large‐eddy simulation in two configurations: (i) flow over a cavity to predict small‐scale cavitation and inception and (ii) a rising dense bubble plume in a stationary water column. The results show good predictive capability of the numerical algorithm in capturing complex flow features. Copyright © 2010 John Wiley & Sons, Ltd.     

An SPH model for free surface flows with moving rigid objects

This paper presents a computational model for free surface flows interacting with moving rigid bodies. The model is based on the SPH method, which is a popular meshfree, Lagrangian particle method and can naturally treat large flow deformation and moving features without any interface/surface capture or tracking algorithm. Fluid particles are used to model the free surface flows which are governed by Navier–Stokes equations, and solid particles are used to model the dynamic movement (translation and rotation) of moving rigid objects. The interaction of the neighboring fluid and solid particles renders the fluid–solid interaction and the non‐slip solid boundary conditions. The SPH method is improved with corrections on the SPH kernel and kernel gradients, enhancement of solid boundary condition, and implementation of Reynolds‐averaged Navier–Stokes turbulence model. Three numerical examples including the water exit of a cylinder, the sinking of a submerged cylinder and the complicated motion of an elliptical cylinder near free surface are provided. The obtained numerical results show good agreement with results from other sources and clearly demonstrate the effectiveness of the presented meshfree particle model in modeling free surface flows with moving objects. Copyright © 2013 John Wiley & Sons, Ltd.     

A PDF propagation approach to model turbulent dispersion in swirling flows

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

Research on the effect of cylinder particles on the turbulent properties in particulate flows

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Numerical study of concentrated fluid–particle suspension flow in a wavy channel

The particle migration effects and fluid–particle interactions occurring in the flow of highly concentrated fluid–particle suspension in a spatially modulated channel have been investigated numerically using a finite volume method. The mathematical model is based on the momentum and continuity equations for the suspension flow and a constitutive equation accounting for the effects of shear‐induced particle migration in concentrated suspensions. The model couples a Newtonian stress/shear rate relationship with a shear‐induced migration model of the suspended particles in which the local effective viscosity is dependent on the local volume fraction of solids. The numerical procedure employs finite volume method and the formulation is based on diffuse‐flux model. Semi‐implicit method for pressure linked equations has been used to solve the resulting governing equations along with appropriate boundary conditions. The numerical results are validated with the analytical expressions for concentrated suspension flow in a plane channel. The results demonstrate strong particle migration towards the centre of the channel and an increasing blunting of velocity profiles with increase in initial particle concentration. In the case of a stenosed channel, the particle concentration is lowest at the site of maximum constriction, whereas a strong accumulation of particles is observed in the recirculation zone downstream of the stenosis. The numerical procedure applied to investigate the effects of concentrated suspension flow in a wavy passage shows that the solid particles migrate from regions of high shear rate to low shear rate with low velocities and this phenomenon is strongly influenced by Reynolds numbers and initial particle concentration. Copyright © 2008 John Wiley & Sons, Ltd.     

Modeling particle sedimentation in viscous fluids with a coupled immersed boundary method and discrete element method

Numerical techniques have increasingly been used to model fluid–particle two-phase flows. Coupling the immersed boundary method (IBM) and discrete element method (DEM) is one promising approach for modeling particulate flows. In this study, IBM was coupled with DEM to improve the reliability and accuracy of IBM for determining the positions of particles during the sedimentation process within viscous fluids. The required ratio of the particle diameter to the grid size (D/dx) was determined by comparing the simulation results with the analytical solution and experimental data. A dynamic mesh refinement model was utilised in the IBM model to refine the computational fluid dynamics grid near the particles. In addition, an optimum coupling interval between the IBM and DEM models was determined based on the experimental results of a single particle sedimentation within silicon oil at a Reynolds number of 1.5. The experimental results and the analytical solution were then utilised to validate the IBM–DEM model at Reynolds numbers of 4.1, 11.6, and 31.9. Finally, the validated model was utilised to investigate the sedimentation process for more than one particle by modeling the drafting-kissing-tumbling process and the Boycott phenomenon. Benchmark tests showed that the IBM–DEM technique preserves the advantages of DEM for tracking a group of particles, while the IBM provides a reliable and accurate approach for modeling the particle–fluid interaction.     

Nonhomogeneous Incompressible Herschel–Bulkley Fluid Flows Between Two Eccentric Cylinders

The equations for the nonhomogeneous incompressible Herschel–Bulkley fluid are considered and existence of a weak solution is proved for a boundary-value problem which describes three-dimensional flows between two eccentric cylinders when in each two-dimensional cross-section annulus the flow characteristics are the same. The rheology of such a fluid is defined by a yield stress τ* and a discontinuous stress-strain law. A fluid volume stiffens if its local stresses do not exceed τ*, and a fluid behaves like a nonlinear fluid otherwise. The flow equations are formulated in the stress–velocity–density–pressure setting. Our approach is different from that of Duvaut–Lions developed for the classical Bingham viscoplastic fluids. We do not apply the variational inequality but make use of an approximation of the generalized Bingham fluid by a non-Newtonian fluid with a continuous constitutive law.     

Simulating surface tension with smoothed particle hydrodynamics

A method for simulating two‐phase flows including surface tension is presented. The approach is based upon smoothed particle hydrodynamics (SPH). The fully Lagrangian nature of SPH maintains sharp fluid–fluid interfaces without employing high‐order advection schemes or explicit interface reconstruction. Several possible implementations of surface tension force are suggested and compared. The numerical stability of the method is investigated and optimal choices for numerical parameters are identified. Comparisons with a grid‐based volume of fluid method for two‐dimensional flows are excellent. The methods presented here apply to problems involving interfaces of arbitrary shape undergoing fragmentation and coalescence within a two‐phase system and readily extend to three‐dimensional problems. Boundary conditions at a solid surface, high viscosity and density ratios, and the simulation of free‐surface flows are not addressed. Copyright © 2000 John Wiley & Sons, Ltd.     

Highly elastic solutions for Oldroyd-B and Phan-Thien/Tanner fluids with a finite volume/element method: planar contraction flows

A hybrid finite volume/element method is analysed through the computation of creeping flows of viscoelastic fluids in plane 4:1 sharp and rounded-corner contraction geometries. Simulations are presented for three models: a constant viscosity Oldroyd-B fluid, and Phan-Thien/Tanner (PTT) shear thinning fluids of exponential and linear approximation form. A Taylor–Galerkin/pressure-correction scheme is implemented as the base time-stepping framework. The momentum equations are solved by a finite element method, whilst the constitutive equations are solved by a finite volume approach. Mesh convergence is analysed via refinement around the contraction to capture boundary layers and flow structure. Pressure drop is shown to increase with flow rate for a fixed fluid. For the Oldroyd-B model, singular behaviour is reported in the main stress component as one approaches the corner in the rounded, as with the sharp geometry. Velocity components display an asymptotic trend with a positive slope. Higher values of Weissenberg numbers (We) are reached with these finite volume schemes compared to their finite element counterparts, attributing this to superior accuracy properties.