Numerical Simulation Of Particulate Flow Research Articles

Heat transfer from an array of resolved particles in turbulent flow

The PHYSALIS method for resolved numerical simulation of particulate flows, recently extended to include particles-fluid heat transfer, is applied to the turbulent flow past a planar particle array perpendicular to the incoming mean flow. The array consists of nine equal spheres. Periodicity boundary conditions are imposed on the boundaries of the computational domain parallel to the mean flow. The Reynolds number based on the particle diameter and mean incident flow is 120, the Taylor-scale Reynolds number is close to 30, and the ratio of particle radius to the Kolmogorov length is about 10. A detailed characterization of the flow and heat transfer is given including probability distribution functions of temperature and streamwise velocity, contour maps of the temperature fluctuations, diagonal Reynolds stresses, turbulent heat flux, and the various contributions to the energy budget. Turbulence moderately increases the heat transfer and considerably shortens the thermal wake of the particles. Temperature and streamwise velocity develop very differently downstream of the spheres in spite of the fact that the Prandtl number equals 1, because of the blockage by the spheres, which has no counterpart for the temperature.

Direct numerical simulation of particulate flows with an overset grid method

We evaluate an efficient overset grid method for two-dimensional and three-dimensional particulate flows for small numbers of particles at finite Reynolds number. The rigid particles are discretised using moving overset grids overlaid on a Cartesian background grid. This allows for strongly-enforced boundary conditions and local grid refinement at particle surfaces, thereby accurately capturing the viscous boundary layer at modest computational cost. The incompressible Navier–Stokes equations are solved with a fractional-step scheme which is second-order-accurate in space and time, while the fluid–solid coupling is achieved with a partitioned approach including multiple sub-iterations to increase stability for light, rigid bodies. Through a series of benchmark studies we demonstrate the accuracy and efficiency of this approach compared to other boundary conformal and static grid methods in the literature. In particular, we find that fully resolving boundary layers at particle surfaces is crucial to obtain accurate solutions to many common test cases. With our approach we are able to compute accurate solutions using as little as one third the number of grid points as uniform grid computations in the literature. A detailed convergence study shows a 13-fold decrease in CPU time over a uniform grid test case whilst maintaining comparable solution accuracy.

Numerical simulation of particulate flows using a hybrid of finite difference and boundary integral methods.

A combination of finite difference (FD) and boundary integral (BI) methods is used to formulate an efficient solver for simulating unsteady Stokes flow around particles. The two-dimensional (2D) unsteady Stokes equation is being solved on a Cartesian grid using a second order FD method, while the 2D steady Stokes equation is being solved near the particle using BI method. The two methods are coupled within the viscous boundary layer, a few FD grid cells away from the particle, where solutions from both FD and BI methods are valid. We demonstrate that this hybrid method can be used to accurately solve for the flow around particles with irregular shapes, even though radius of curvature of the particle surface is not resolved by the FD grid. For dilute particle concentrations, we construct a virtual envelope around each particle and solve the BI problem for the flow field located between the envelope and the particle. The BI solver provides velocity boundary condition to the FD solver at "boundary" nodes located on the FD grid, adjacent to the particles, while the FD solver provides the velocity boundary condition to the BI solver at points located on the envelope. The coupling between FD method and BI method is implicit at every time step. This method allows us to formulate an O(N) scheme for dilute suspensions, where N is the number of particles. For semidilute suspensions, where particles may cluster, an envelope formation method has been formulated and implemented, which enables solving the BI problem for each individual particle cluster, allowing efficient simulation of hydrodynamic interaction between particles even when they are in close proximity. The method has been validated against analytical results for flow around a periodic array of cylinders and for Jeffrey orbit of a moving ellipse in shear flow. Simulation of multiple force-free irregular shaped particles in the presence of shear in a 2D slit flow has been conducted to demonstrate the robustness of the FD-BI scheme. This method can also be used to solve the full Navier-Stokes equations around particles. In this case, it is shown that, below a threshold particle Reynolds number, which in turn depends on the required resolution of the particle surface, the FD-BI scheme will be more computationally efficient than a pure FD scheme solved on a multiblock grid.

Numerical simulation of particulate flows in CFB riser with drag corrections based on particle distribution characterization

A heterogeneous drag force model was developed using a Eulerian-Lagrangian scheme for multiphase flow modeling. The model is based on an idea called particle distribution characterization (PDC), in which transient distribution of particle position is analyzed with an intuitive method to quantify the local heterogeneity of gas-solid flow for drag force correction. The model was implemented into the multiphase particle-in-cell (MP-PIC) method and applied to particulate flow simulation in circulating fluidized bed (CFB) risers. Whereas traditional homogeneous drag models tend to over-estimate gas-particle interactions and present homogeneous flow fields, the developed PDC model successfully predicts the heterogeneous gas-solid flow structures. The PDC predictions on time-averaged distributions and solid fluxes also correspond better with experimental measurements. Effects of the statistical parcel weight, an important parameter in computational parcel assumption in MP-PIC method (wherein parcels are employed to represent real particles), were investigated with both traditional and the PDC drag models. Results show defects of the assumption that agree with literature. It is proved that the PDC drag model is applicable to particulate flow modeling in CFB riser, and considerable for simulation of other industrial processes.

Forcing homogeneous turbulence in direct numerical simulation of particulate flow with interface resolution and gravity

We consider the case of finite-size spherical particles which are settling under gravity in a homogeneous turbulent background flow. Turbulence is forced with the aid of the random forcing method of Eswaran and Pope [“An examination of forcing in direct numerical simulations of turbulence,” Comput. Fluids 16(3), 257-278 (1988)], while the solid particles are represented with an immersed-boundary method. The forcing scheme is used to generate isotropic turbulence in vertically elongated boxes in order to warrant better decorrelation of the Lagrangian signals in the direction of gravity. Since only a limited number of Fourier modes are forced, it is possible to evaluate the forcing field directly in physical space, thereby avoiding full-size transforms. The budget of box-averaged kinetic energy is derived from the forced momentum equations. Medium-sized simulations for dilute suspensions at low Taylor-scale Reynolds number Reλ = 65, small density ratio ρp/ρf = 1.5, and for two Galileo numbers Ga = 0 and 120 are carried out over long time intervals in order to exclude the possibility of slow divergence. It is shown that the results at zero gravity are fully consistent with previous experimental measurements and available numerical reference data. Specific features of the finite-gravity case are discussed with respect to a reduction of the average settling velocity, the acceleration statistics, and the Lagrangian auto-correlations.

Fully resolved simulations of the flow through a packed bed of cylinders: Effect of size distribution

Fully resolved simulations of the flow through a fixed bed of pellets are performed to better understand the effect of the local microstructure on the energy loss, i.e., pressure drop through the bed. Both mono-disperse and poly-disperse systems as well as spherical and cylindrical pellets (solid particles) are investigated. Using a DEM-DLM/FD simulation method inspired by “Wachs, A. (2009). A DEM-DLM/FD method for direct numerical simulation of particulate flows: sedimentation of polygonal isometric particles in a Newtonian fluid with collisions. Comput. Fluids 38(8), 1608–1628” and implemented here in a Finite Volume context with second order reconstruction of the particle boundary as in “Rahmani, M., Wachs, A. (2014). Free falling and rising of spherical and angular particles. Phys. Fluids 26, 083301”, we evidence that the computed solution converges nicely with mesh refinement and provide guidelines on the grid size to guarantee a satisfactory level of accuracy. Based on these trustworthy simulation results, we investigate the impact of the particle shape as well as the degree of poly-dispersity in the system on the pressure drop over the fixed bed in the viscous regime. Unprecedented simulation results on the flow through a bed of poly-disperse cylinders indicate that the correlation for poly-disperse spheres suggested in “Van der Hoef, M.A., Beetstra, R., Kuipers, J.A.M. (2005). Lattice-Boltzmann simulations of low-Reynolds-number flow past mono- and bidisperse arrays of spheres: results for the permeability and drag force. J. Fluid Mech. 528, 233–254” may still be valid for cylinders of moderate aspect ratio.

Numerical simulation of particulate flow in interconnected porous media for central particle-heating receiver applications

The use of central solar particle-heating receivers (SPR) in concentrated solar power (CSP) systems offers numerous advantages over other central receiver concepts. The two main advantages are: (1) ordinary particulate minerals can allow collection temperatures approaching 1000°C compared with conventional molten salts which are limited to about 650°C, and (2) the low-cost high temperature particulate material can also be used as the storage medium in a highly cost effective thermal energy storage system. An innovative SPR design invented by researchers at King Saud University and Georgia Institute of Technology allows the particulate material to flow downward through a stationary porous structure where the concentrated solar energy is absorbed. The porous structure reduces the speed of the falling particulate material, i.e. increases its residence time within the receiver, thereby allowing a large temperature rise to be achieved in a single pass. The design increases absorption of the incident solar radiation (i.e. receiver efficiency) and reduces convective heat loss and particle loss (Al-Ansary et al., 2013). A numerical two-fluid solid–gas Eulerian–Eulerian flow model has been used to evaluate the effect of the porous structure on particulate flow through the receiver. The stationary porous structure was modeled using the packed bed concept in the FLUENT commercial CFD code. Two benchmarking experiments were conducted to assess the validity of the two-fluid flow and packed bed model for flow through the porous structure. This paper describes the two benchmarking experiments, and provides a comparison between the model predictions and the measured data. In the first experiment, particle flow rates through variable-width slits at the bottom of a constant head plenum were measured. In the second experiment, a porous layer was placed above the opening at the bottom of the constant head plenum to limit the particle discharge flow. Experiments with two different porous layer heights were conducted using two particle materials with different diameters and densities. The simulation results were in reasonable agreement with the test data.

Direct numerical simulation of particulate flow with heat transfer

The Immersed Boundary (IB) method proposed by Uhlmann for Direct Numerical Simulation (DNS) of fluid flow through dense fluid-particle systems is extended to systems with interphase heat transport. A fixed Eulerian grid is employed to solve the momentum and energy equations by traditional computational fluid dynamics methods. Our numerical method treats the particulate phase by introducing momentum and heat source terms at the boundary of the solid particle, which represent the momentum and thermal interactions between fluid and particle. Forced convection heat transfer was simulated for a single sphere and an in-line array of 3 spheres to assess the accuracy of the present method. Non-isothermal flows past stationary random arrays of spheres are investigated to assess the capability of our simulation method for dense particulate systems. All results are in satisfactory agreement with reported experimental and numerical results.

Combination of the fictitious domain method and the sharp interface method for direct numerical simulation of particulate flows with heat transfer

In this paper, the fictitious domain (FD) method and the sharp interface (SI) method are combined for the direct numerical simulations of particulate flows with heat transfer in three dimensions. The flow field and the motion of particles are solved with the FD method. The temperature field is solved in both fluid and solid media with the SI method. The accuracy of the proposed FD/SI method is validated via two problems: the natural convection in a two- dimensional cavity with fixed solid particles, and the flow over a cold sphere. The method is then applied to the natural convection in a three dimensional cavity with a fixed sphere, the motion of a spherical particle in a non-isothermal fluid, and the rising of spherical catalyst particles in an enclosure. The effects of the thermal conductivity ratio are examined in the first and third problems, respectively, and the significant effects of the thermal expansion coefficient ratio on the particle motion are demonstrated in the second problem.

Direct numerical simulation of particulate flows with heat transfer in a rotating cylindrical cavity

The purpose of this work was the direct numerical simulation of heat and fluid flow by granular mixing in a horizontal rotating kiln. To model particle behaviour and the heat and fluid flow in the drum, we solve the mass conservation, momentum and energy conservation equations directly on a fixed Eulerian grid for the whole domain including particles. At the same time the particle dynamics and their collisions are solved on a Lagrangian grid for each particle. To calculate the heat transfer inside the particles we use two models: the first is the direct solution of the energy conservation equation in the Lagrangian and Eulerian space, and the second is our so-called linear model that assumes homogeneous distribution of the temperature inside each particle. Numerical simulations showed that, if the thermal diffusivity of the gas phase significantly exceeds the same parameter of the particles, the linear model overpredicts the heating rate of the particles. The influence of the particle size and the angular velocity of the drum on the heating rates of particles is studied and discussed.

Direct Numerical Simulation of particulate flow with heat transfer

Numerical simulations of heat transfer in non-isothermal particulate flows are important to better understand the flow pattern. The complexity of numerical algorithms coupling the heat and mass transfer and the considerable computational resources required limit the number of such direct simulations that can be reasonably performed. We suggest a Distributed Lagrange Multiplier/Fictitious Domain (DLM/FD) method to compute the temperature distribution and the heat exchange between the fluid and solid phases. The Boussinesq approximation is considered for the flow/temperature fields coupling. We employ a Finite Element Method (FEM) to solve the fluid flow conservation equations for mass, momentum and energy. The motion of particles is computed by a Discrete Element Method (DEM). On each particle, heat transfer is solved using a FEM. For each class of particles, we generate a single FEM grid and translate/rotate it at each time step to match the physical configuration of each particle. Distributed Lagrange multipliers for both the velocity and temperature fields are introduced to treat the fluid/solid interaction. This work is an extension of the method we proposed in Yu et al. (2006). Two two-dimensional (2D) test cases are proposed to validate the implementation by comparing our computational results with those reported in the literature. Finally, the sedimentation of a single sphere in a semi-infinite channel is presented and the results are discussed.

SMOOTHED PROFILE METHOD TO SIMULATE COLLOIDAL PARTICLES IN COMPLEX FLUIDS

A new direct numerical simulation scheme, called "Smoothed Profile (SP) method," is presented. The SP method, as a direct numerical simulation of particulate flow, provides a way to couple continuum fluid dynamics with rigid-body dynamics through smoothed profile of colloidal particle. Our formulation includes extensions to colloids in multicomponent solvents such as charged colloids in electrolyte solutions. This method enables us to compute the time evolutions of colloidal particles, ions, and host fluids simultaneously by solving Newton, advection-diffusion, and Navier–Stokes equations so that the electro-hydrodynamic couplings can be fully taken into account. The electrophoretic mobilities of charged spherical particles are calculated in several situations. The comparisons with approximation theories show quantitative agreements for dilute dispersions without any empirical parameters.

Short-time motion of Brownian particles in a shear flow

The short-time motion of Brownian particles in an incompressible Newtonian fluid under shear, in which the fluid inertia becomes important, was investigated by direct numerical simulation of particulate flows. Three-dimensional simulations were performed, wherein external forces were introduced to approximately form Couette flows throughout the entire system with periodic boundary conditions. In order to examine the validity of the method, the mean-square displacement of a single spherical particle in a simple shear flow was calculated, and these results were compared with a hydrodynamic analytical solution that includes the effects of the fluid inertia. Finally, the dynamical behavior of a monodisperse dispersion composed of repulsive spherical particles was examined on short-time scales, and the shear-induced diffusion coefficients were measured for several volume fractions up to 0.50.

A DEM-DLM/FD method for direct numerical simulation of particulate flows: Sedimentation of polygonal isometric particles in a Newtonian fluid with collisions

An efficient direct numerical simulation method to tackle the problem of particulate flows at moderate to high concentration and finite Reynolds number is presented. Our method is built on the framework established by Glowinski and his co-workers [Glowinski R, Pan TW, Hesla TI, Joseph DD. A distributed lagrange multiplier/fictitious domain method for particulate flow. Int J Multiphase Flow 1999;25:755–94] in the sense that we use their Distributed Lagrange Multiplier/Fictitious Domain (DLM/FD) formulation and their operator-splitting idea but differs in the treatment of particle collisions. Compared to our previous works [Yu Z, Wachs A, Peysson Y. Numerical simulation of particle sedimentation in shear-thinning fluids with a fictitious domain method. J Non Newtonian Fluid Mech 2006;136:126–139; Yu Z, Shao X, Wachs A. A fictitious domain method for particulate flow with heat transfer. J Comput Phys 2006;217:424–52; Yu Z, Wachs A. A fictitious domain method for dynamic simulation of particle sedimentation in Bingham fluids. J Non Newtonian Fluid Mech 2007;145:78–91], the novelty of our present contribution relies on replacing the simple artificial repulsive force based collision model usually employed in the literature by an efficient Discrete Element Method (DEM) granular solver. The use of our DEM solver enables us to consider particles of arbitrary shape (at least convex) and to account for actual contacts, in the sense that particles actually touch each other, in contrast with the repulsive force based collision model. We validate GRIFF, 1 ▪ GRIFF stands for “GRains In Fluid Flow”. 1 our numerical code, against benchmark problems and compare our predictions with those available in the literature. Results, which, to the best of our knowledge, have never been reported elsewhere, on the 2D sedimentation of isometric polygonal particles with collisions are presented.

Simulating (electro)hydrodynamic effects in colloidal dispersions: Smoothed profile method

Previously, we have proposed a direct simulation scheme for colloidal dispersions in a Newtonian solvent (Phys. Rev. E 71, 036707 (2005)). An improved formulation called the "Smoothed Profile (SP) method" is presented here in which simultaneous time-marching is used for the host fluid and colloids. The SP method is a direct numerical simulation of particulate flows and provides a coupling scheme between the continuum fluid dynamics and rigid-body dynamics through utilization of a smoothed profile for the colloidal particles. Moreover, the improved formulation includes an extension to incorporate multicomponent fluids, allowing systems such as charged colloids in electrolyte solutions to be studied. The dynamics of the colloidal dispersions are solved with the same computational cost as required for solving non-particulate flows. Numerical results which assess the hydrodynamic interactions of colloidal dispersions are presented to validate the SP method. The SP method is not restricted to particular constitutive models of the host fluids and can hence be applied to colloidal dispersions in complex fluids.

Collision modeling between two non-Brownian particles in multiphase flow

In this paper, we develop a collision model for particle–particle and particle–wall interaction inside fluid medium based on stereomechanical impact model. One of the important parameters identified in literature that characterizes the interaction is the Stokes number ( St), which is the ratio of the particles inertia and fluid viscosity. In dry air collisions, the coefficient of restitution ( e) and coefficient of friction ( f) are used to describe post-collision velocities on the basis of pre-collision velocities. In the case of collisions in fluid medium, we use the experimental results that provide the co-relation between the values of e and f as a function of the Stokes number. The collision models developed here are used in our direct numerical simulation of particulate flows to verify the model with some of the well-known behavior of particle interaction in fluids.

Applications of operator-splitting methods to the direct numerical simulation of particulate and free-surface flows and to the numerical solution of the two-dimensional elliptic Monge-Ampère equation

The main goal of this article is to review some recent applications of operator-splitting methods. We will show that these methods are well-suited to the numerical solution of outstanding problems from various areas in Mechanics, Physics and Differential Geometry, such as the direct numerical simulation of particulate flow, free boundary problems with surface tension for incompressible viscous fluids, and the elliptic real Monge-Ampère equation. The results of numerical experiments will illustrate the capabilities of these methods.

Fictitious boundary and moving mesh methods for the numerical simulation of rigid particulate flows

In this paper, we investigate the numerical simulation of particulate flows using a new moving mesh method combined with the multigrid fictitious boundary method (FBM) [S. Turek, D.C. Wan, L.S. Rivkind, The fictitious boundary method for the implicit treatment of Dirichlet boundary conditions with applications to incompressible flow simulations. Challenges in Scientific Computing, Lecture Notes in Computational Science and Engineering, vol. 35, Springer, Berlin, 2003, pp. 37–68; D.C. Wan, S. Turek, L.S. Rivkind, An efficient multigrid FEM solution technique for incompressible flow with moving rigid bodies. Numerical Mathematics and Advanced Applications, ENUMATH 2003, Springer, Berlin, 2004, pp. 844–853; D.C. Wan, S. Turek, Direct numerical simulation of particulate flow via multigrid FEM techniques and the fictitious boundary method, Int. J. Numer. Method Fluids 51 (2006) 531–566]. With this approach, the mesh is dynamically relocated through a (linear) partial differential equation to capture the surface of the moving particles with a relatively small number of grid points. The complete system is realized by solving the mesh movement and the partial differential equations of the flow problem alternately via an operator-splitting approach. The flow is computed by a special ALE formulation with a multigrid finite element solver, and the solid particles are allowed to move freely through the computational mesh which is adaptively aligned by the moving mesh method in every time step. One important aspect is that the data structure of the undeformed initial mesh, in many cases a tensor-product mesh or a semi-structured grid consisting of many tensor-product meshes, is preserved, while only the spacing between the grid points is adapted in each time step so that the high efficiency of structured meshes can be exploited. Numerical results demonstrate that the interaction between the fluid and the particles can be accurately and efficiently handled by the presented method. It is also shown that the presented method significantly improves the accuracy of the previous multigrid FBM to simulate particulate flows with many moving rigid particles.

Direct numerical simulation of particulate flow via multigrid FEM techniques and the fictitious boundary method

AbstractDirect numerical simulation techniques for particulate flow by the fictitious boundary method (FBM) are presented. The flow is computed by a multigrid finite element solver and the solid particles are allowed to move freely through the computational mesh which can be chosen independently from the particles of arbitrary shape, size and number. The interaction between the fluid and the particles is taken into account by the FBM in which an explicit volume based calculation for the hydrodynamic forces is integrated. A new collision model based on papers by Glowinski, Joseph, Singh and coauthors is examined to handle particle–particle and particle–wall interactions. Numerical tests show that the present method provides a very efficient approach to directly simulate particulate flows with many particles. Copyright © 2005 John Wiley & Sons, Ltd.

Numerical simulation of particulate flow by the Eulerian–Lagrangian and the Eulerian–Eulerian approach with application to a fluidized bed

This work presents a computational study of the flow behavior of a lab-scale fluidized bed. The results obtained from a ‘discrete particle method’ (DPM) are qualitatively compared to the results obtained from a multi-fluid computational fluid dynamic (CFD) model. In the DEM, also referred as Eulerian–Lagrangian (EL) model, the two-dimensional motion of each individual spherical particle is directed calculated from the forces acting on them, accounting for the interaction between the particle and the gas-phase. The implemented collision model is based on the conservation laws for linear and angular momentum and requires, apart from geometrical factors, two empirical parameters: a restitution coefficient and a friction coefficient. The fluidynamic model of the gas is based on the volume-averaged Navier–Stokes equations. In the multi-fluid CFD model, also referred as Eulerian–Eulerian (EE), the gas and the solid phases are considered to be continuous and fully inter-penetrating. Both phases are described in terms of separate sets of conservation equations with appropriate interaction terms representing the coupling between the phases. Experiments results of a two-dimensional lab-scale bubbling fluidized bed are furthermore compared to the computational results obtained by the two approaches. A discussion about the obtained results and their discrepancies is presented.