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. \textit{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 (R. Sun and H. Xiao, Diffusion-based coarse graining in hybrid continuum-discrete solvers: Application in CFD-DEM solvers for particle laden flows)

Diffusion-Based Coarse Graining in Hybrid Continuum-Discrete Solvers: Applications in CFD-DEM

In this work, a coarse-graining method previously proposed by the authors in a companion paper based on solving diffusion equations is applied to CFD-DEM simulations, where coarse graining is used to obtain solid volume fraction, particle phase velocity, and fluid-particle interaction forces. By examining the conservation requirements, the variables to solve diffusion equations for in CFD-DEM simulations are identified. The algorithm is then implemented into a CFD-DEM solver based on OpenFOAM and LAMMPS, the former being a general-purpose, three-dimensional CFD solver based on unstructured meshes. Numerical simulations are performed for a fluidized bed by using the CFD-DEM solver with the diffusion-based coarse-graining algorithm. Converged results are obtained on successively refined meshes, even for meshes with cell sizes comparable to or smaller than the particle diameter. This is a critical advantage of the proposed method over many existing coarse-graining methods, and would be particularly valuable when small cells are required in part of the CFD mesh to resolve certain flow features such as boundary layers in wall bounded flows and shear layers in jets and wakes. Moreover, we demonstrate that the overhead computational costs incurred by the proposed coarse-graining procedure are a small portion of the total costs in typical CFD-DEM simulations as long as the number of particles per cell is reasonably large, although admittedly the computational overhead of the coarse graining often exceeds that of the CFD solver. Other advantages of the present algorithm include more robust and physically realistic results, flexibility and easy implementation in almost any CFD solvers, and clear physical interpretation of the computational parameter needed in the algorithm. In summary, the diffusion-based method is a theoretically elegant and practically viable option for CFD-DEM simulations

M5 branes and Theta Functions

We propose quantum states for Little String Theories (LSTs) arising from M5 branes probing A- and D-type singularities. This extends Witten's picture of M5 brane partition functions as theta functions to this more general setup. Compactifying the world-volume of the five-branes on a two-torus, we find that the corresponding theta functions are sections of line bundles over complex 4-tori. This formalism allows us to derive Seiberg-Witten curves for the resulting four-dimensional theories. Along the way, we prove a duality for LSTs observed by Iqbal, Hohenegger and Rey.Comment: 27 pages, 1 Figure. v3: included further explanations and corrected typos. This the version published in JHE

T-duality, Non-geometry and Lie Algebroids in Heterotic Double Field Theory

A number of issues in heterotic double field theory are studied. This includes the analysis of the T-dual configurations of a flat constant gauge flux background, which turn out to be non-geometric. Performing a field redefinition to a non-geometric frame, these T-duals take a very simple form reminiscent of the constant Q- and R-flux backgrounds. In addition, it is shown how the analysis of arXiv:1304.2784 generalizes to heterotic generalized geometry. For every field redefinition specified by an O(D,D+n) transformation, the structure of the resulting supergravity action is governed by the differential geometry of a corresponding Lie algebroid.Comment: 27 pages, 1 figure, v2: refs adde