A coupled Discrete Element Lattice Boltzmann Method for the simulation of fluid–solid interaction with particles of general shapes
Introduction
It has always been an issue of any numerical modelling effort the fact that greater accuracy usually requires large amounts of details extracted from observations and fed into the model, which is often translated into larger computational requirements. Is not difficult to appreciate how with increasing computational power, the numerical models developed nowadays are including more and more details that were unmanageable years ago.
The computational power that researchers and engineers currently enjoy is enough to solve a broad range of models based on partial differential equations by means of methods such as the Finite Element Method (FEM). Recently it has also been enough to abandon the continuous assumption and start studying the macroscopic response of many systems as a function of the microscopic interactions between the constitutive elements. Examples of such approaches are the Distinct Element Method (DEM) [1] introduced by Cundall for granular materials and the Molecular Dynamic Method (MD) [2] for gases and liquids. In such methods, a macroscopic constitutive equation is not introduced as in the case of FEM modelling. Instead, the modeller introduces the interactions between the particles and obtains the macroscopic response of the complex system. DEM and MD have been very successful tools for research since their results match, qualitatively, the response from controlled experiments.
Recently DEM has started to be used as a predictive tool [3], [4] despite the main difficulty of finding the value of the microscopic parameters from experimental set-ups. It is the author’s opinion that as the computational power increases, DEM will be a viable option for engineering design at the field scale, and the advent of commercial DEM packages seems to follow this trend.
Some research efforts are focused in coupling DEM with other methods. One particular field of interest is the interaction between DEM and Computational Fluid Dynamics (CFD) methods. The need for such tool is evident for the modelling of a broad range of problems such as groundwater transport of colloid and contaminants, erosion and the stability of coastal foundations to name a few.
DEM has been successfully coupled with CFD schemes such as Smooth Particle Hydrodynamics (SPH) [5] and FEM solvers for the Navier Stokes equations [6]. These methods are great for large scale simulations, but lack some details of the fluid at the pore scale. A set of seminal papers by Owen and collaborators [7], [8], [9], [10] showed the coupling between DEM and the Lattice Boltzmann Method (LBM) by means of an immersed boundary method. LBM offers several advantages over other CFD schemes including the locality of the dynamic steps that facilitates the distribution of the domain into massive parallel systems. Such locality also facilitates the formulation of laws involving the interaction of the fluid with moving boundaries which at the ends translates into physically sound coupling schemes with the DEM [9]. It is easy to introduce multiphase and multicomponent systems into the LBM formalism by assigning different lattices to the different components and attractive or repulsive forces to model phase transition and immiscibility [11]. In this way many features at the pore scale such as the menisci and the capillary interaction are reproduced. It is also easy, due to the regular lattice configuration, to deal with obstacles of complex shapes [12] and therefore to cope with the realistic morphology of the porous medium.
For these reasons, the coupling DEM-LBM is a promising tool. It will potentially be able to tackle problems as complex as the mechanical response of unsaturated soils or to model industrial applications such as the hydraulic fracturing. But to be able to accurately address this kind of complexity, the classical DEM spherical element should be abandoned in favour of particles of general shape and with cohesion to model fracture phenomena. The author of the present work has implemented previously the sphero–polyhedra method to model the collision and fragmentation of particles with general shapes [13]. With Voronoi or Delaunay tessellations, solid blocks can be reproduced without voids and the cohesive parameters can be tuned to reproduce any possible elastic moduli, Poisson’s ratios and tensile strengths.
Due to the flexibility of the sphero–polyhedra method, it is a viable choice to expand the DEM-LBM formalism introduced by Owen et al. The presented research work was done with that aim. The sphero–polyhedra method generalises the widely used contact detection between spheres to the collision between polyhedra. In the same way it will be show how the sphero–polyhedra also generalises the contact law between the LBM fluid and the DEM particles introduced before in a simple and efficient way.
The paper is structured as follows: Section 2 describes independently the DEM and LBM and then finalizes with the coupling law between the two. Section 3 presents a series of validation examples for different geometries and compare the results with previous studies. In Section 4 some optimization techniques for the proposed law and their advantages are shown. A preliminary study combining DEM shaped like real animals and multiphase and multicomponent LBM fluids is shown in Section 5. Finally in Section 6 some conclusions and projections of the current work are presented.
Section snippets
The DEM sphero–polyhedra approach
The sphero–polyhedra method was initially introduced by Pourning [14] for the simulation of complex-shaped DEM particles. Later, it was modified by Alonso Marroquin [15], who introduced a multi-contact approach in 2D allowing the modelling of non-convex shapes, and was extended to 3D by the author of this paper and his collaborators [16], [13], [17]. A sphero–polyhedron is a polyhedron that has been eroded and then dilated by a sphere element as seen in Fig. 1. The result is a polyhedron of
Validation for spheres
To validate the method, a comparison with the work carried out by Owen et al. [9] is done to measure the drag coefficient of a sphere immersed into the fluid as a function of the Reynolds number Re. The LBM domain is formed by a lattice of cells. The lattice size step is m. The DEM sphere is placed at the domain’s center and it has a radius of 0.036 m which is equivalent to 9 LBM cells. The fluid density was taken as kg/m3, the kinematic viscosity ν is m2/s and the time
Performance issues of the proposed coupling model
It is easy to identify the step of finding the distance between the LBM cell and the geometric feature of the sphero–polyhedron Eq. 21 as the efficiency bottleneck of the proposed coupling scheme. Finding the volume fraction for each cell and each geometric feature can quickly impair the algorithm performance. A Verlet list algorithm, similar as the one used to speed up DEM simulations, is suggested to address this issue [33]. Basically at the beginning of the simulation a list of pairs
Implementation details of the DEM-LBM coupling method with multiphase and multicomponent fluids
In this section, a less rigorous simulation is going to be performed using a multiphase and multicomponent LBM. In the LBM literature it is common to express the relevant quantities in terms of lattice units (the system in which with the units of density) instead of physical units. For easy comparison and implementation, the parameters of the LBM scheme used in this part are going to be expressed in both unit systems. The conversion between units is easy once the grid size , the
Conclusions and projections of this work
An extension for the Owen et al. [7], [8], [9], [10] coupling law for DEM spheres interacting with LBM fluids is introduced in the present work. The extension generalizes the coupling law for spheres to more complex particles called sphero–polyhedra [13], [33], [17], [16]. The sphero–polyhedra method simplifies the coupling law with the LBM fluid in the same way that it simplifies the collision law with DEM particles.
The method is validated with previous results obtained for spheres by Owen et
Acknowledgements
The author wants to acknowledge the continuous support and encouragement from Alexander Scheuermann and Ling Li during the development of this numerical tool; to Dorival Pedroso for his advice on how to model animal shapes with triangular meshes; and to Alex Jerves for his useful comments which enriched the paper. This work was funded by The University of Queensland Early Career Research Grants Scheme (Grant No. RM2011002323). The algorithm was programmed within the MechSys open source library
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