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Particle-resolved turbulent flow in a packed bed: RANS, LES, and DNS simulations

AICHE JOURNAL 2023年第1期69卷 e17615-e17615页

作者： Ambekar, Aniket S. Schwarzmeier, Christoph Ruede, Ulrich Buwa, Vivek V. Indian Inst Technol Delhi Dept Chem Engn New Delhi 110016 India Univ Erlangen Nurnberg Dept Comp Sci Chair Syst Simulat Erlangen Germany CERFACS Parallel Algorithms Team Toulouse France

Packed bed reactors are widely used to perform solid-catalyzed gas-phase reactions and local turbulence is known to influence heat and mass transfer characteristics. We have investigated turbulence characteristics in a packed bed of 113 spherical particles by performing particle-resolved Reynolds-averaged Navier-Stokes (RANS) simulations, Large Eddy Simulation (LES), and Direct Numerical Simulation (DNS). The predictions of the RANS and LES simulations are validated with the lattice Boltzmann method (LBM)-based DNS at particle Reynolds number (Re-p) of 600. The RANS and LES simulations can predict the velocity, strain rate, and vorticity with a reasonable accuracy. Due to the dominance of enhanced wall-function treatment, the turbulence characteristics predicted by the epsilon-based models are found to be in a good agreement with the DNS. The omega-based models under-predicted the turbulence quantities by several orders of magnitude due to their inadequacy in handling strongly wall-dominated flows at low Re-p. Using the DNS performed at different Re-p, we also show that the onset of turbulence occurs between 200 <= Rep <= 250.

关键词： Direct Numerical Simulations Large Eddy Simulations Packed bed Particle-resolved simulations Turbulence models

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Particle-resolved simulations of local liquid spreading in packed beds: Effect of wettability at varying particle size

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PHYSICS OF FLUIDS 2022年第6期34卷 1-14页

作者： Ambekar, Aniket S. Ruede, Ulrich Buwa, Vivek V. Indian Inst Technol Delhi Dept Chem Engn Delhi India Univ Erlangen Nurnberg Chair Syst Simulat Nurnberg Germany CERFACS Parallel Algorithms Team Toulouse France

Packed beds are widely used to perform solid-catalyzed gas-liquid reactions, e.g., hydrodesulfurization, oxidation, and hydrogenation. The overall performance of packed beds is often governed by local liquid spreading. In the present work, the dynamics of liquid spreading through a randomly packed three-dimensional bed is investigated using particle-resolved volume-of-fluid simulations. The effect of particle surface-wettability ( theta) at varying particle diameter ( d(p)) on the relative contributions of forces governing the dynamics of liquid spreading is analyzed using the Ohnesorge (Oh(I)), Weber ( We(I)), and AB(I) (proposed in the present work) numbers. With the help of simulated liquid spreading and these numbers, we show that the contribution of inertial force is significant at the beginning of liquid spreading irrespective of theta as well as d(p) and promotes lateral liquid spreading (AB(I) > 1, We(I) > 1). Once the dominance of inertial force diminishes, the capillary force leads to a substantial increase in the lateral spreading (AB(I) > 1, We(I) < 1). In the final stages, the gravitational force dominates restricting the lateral liquid spreading (AB(I) < 1). Furthermore, we have proposed a regime map constructed using AB(I) and We(I), which provides a relationship between different forces and the resultant liquid spreading at breakthrough. We also show that the dominance of capillary force (AB(I) > 1, We(I )< 1) results in the highest lateral spreading, whereas the flow dominated by inertial (AB(I) > 1, We(I) > 1) and gravitational force ( AB(I) (sic) 1) leads to intermediate and least lateral liquid spreading, respectively.

关键词： Packed beds

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Implicitly Extrapolated Geometric Multigrid on Disk-Like Domains for the Gyrokinetic Poisson Equation from Fusion Plasma Applications

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JOURNAL OF SCIENTIFIC COMPUTING 2022年第1期91卷 1-27页

作者： Kuehn, Martin J. Kruse, Carola Ruede, Ulrich German Aerosp Ctr DLR Inst Software Technol Dept High Performance Comp Cologne Germany CERFACS Ctr Europeen Rech & Format Avancee Calcul Parallel Algorithms Team 42 Ave Gaspard Coriolis F-31057 Toulouse 01 France Friedrich Alexander Univ Erlangen Nurnberg Cauerstr 11 D-91058 Erlangen Germany

The gyrokinetic Poisson equation arises as a subproblem of Tokamak fusion reactor simulations. It is often posed on disk-like cross sections of the Tokamak that are represented in generalized polar coordinates. On the resulting curvilinear anisotropic meshes, we discretize the differential equation by finite differences or low order finite elements. Using an implicit extrapolation technique similar to multigrid tau-extrapolation, the approximation order can be increased. This technique can be naturally integrated in a matrix-free geometric multigrid algorithm. Special smoothers are developed to deal with the mesh anisotropy arising from the curvilinear coordinate system and mesh grading.

关键词： Fusion plasma Disk-like Cross section Multigrid Extrapolation

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Forces governing the dynamics of liquid spreading in packed beds

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JOURNAL OF FLUID MECHANICS 2022年 948卷 A13-A13页

作者： Ambekar, Aniket S. Ruede, Ulrich Buwa, Vivek V. Indian Inst Technol Delhi Dept Chem Engn New Delhi 110016 India Univ Erlangen Nurnberg Chair Syst Simulat D-91058 Erlangen Germany CERFACS Parallel Algorithms Team 42 Av Gaspard Coriolis F-31100 Toulouse France

Liquid spreading through a randomly packed particle-resolved bed influenced by capillary or inertial (AB(S) similar to 1), and gravitational force (moderately (AB(S) similar to 0.1) and strongly (AB(S) similar to 0.01)) is investigated using the volume-of-fluid simulations. The relative contribution of governing forces at different stages of spreading is analysed using the time evolution of Weber (We(I)) and AB(I) numbers. We show that the dynamics of liquid spreading at AB S N 1 is primarily governed by the inertial force in the beginning (AB(I) > 1, We(I) > 1) followed by the capillary force at t/t* similar to 1. This interplay of governing forces leads to inertia- and capillary-induced bubble entrapments at the void scale and promote lateral liquid spreading. When the AB(S) similar to 0.1, the t/t* for which the flow is governed by inertial (AB(I) > 1, Wei > 1) and capillary forces (AB(I) > 1, Wei < 1) decreases and the relative contribution of gravitational force is substantial at large t/t* (AB(I) < 1). This force balance leads to unified-void filling characterised by negligible bubble trapping and results in a decrease in the lateral spreading. Further decrease in the AB(S) to similar to 0.01 results in liquid spreading primarily governed by gravitational force (AB(I)< 1) with small contribution of inertial and capillary forces at the very beginning leading to trickling flow and a further decrease in lateral spreading. Finally, a regime map is proposed, which provides the relationship between different forces, void-scale events, and the resultant liquid spreading at t/t* similar to 1.

关键词： multiphase flow gas/liquid flow capillary flows

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EXTENSIONS OF THE AUGMENTED BLOCK CIMMINO METHOD TO THE SOLUTION OF FULL RANK RECTANGULAR SYSTEMS

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2021年第5期43卷 S516-S539页

作者： Dumitrasc, Andrei Leleux, Philippe Popa, Constantin Ruede, Ulrich Ruiz, Daniel FAU Erlangen Nurnberg Chair Comp Sci Syst Simulat 10 D-91058 Erlangen Germany CERFACS Parallel Algorithms Team F-31057 Toulouse 01 France Ovidius Univ Constanta Gheorghe Mihoc Caius Iacob Inst Stat Math & Appl Bucharest Romania Acad Romanian Scientists Bucharest Romania Ecole Natl Super Electrotech Elect Informat & Hyd Inst Rech Informat Toulouse F-31071 Toulouse France

For the solution of large sparse unsymmetric systems, Duff et al. [SIAM J. Sci. Comput., 37 (2015), pp. A1248-A1269] proposed an approach based on the block Cimmino iterations [Numer. Math., 35 (1980), pp. 1-12], in which the solution is computed in a single iteration, so we call it a pseudodirect solver. In this approach, matrices are augmented with additional variables and constraints so that a partitioning of the matrix in blocks of rows defines mutually orthogonal subspaces. The augmented system can then be solved efficiently with a sum of projections onto these orthogonal subspaces. The purpose of this manuscript is to extend this method to the minimum norm solution of underdetermined systems and to the solution of least-squares problems. In the latter case, a partitioning of the matrix in blocks of columns rather than rows is used, and the system must be suitably augmented to define mutually orthogonal subspaces again to recover the least-squares solution of the original problem. This article proves the equivalence between the solution of the original and the augmented system. In order to complete the extension to overdetermined systems, we also propose an iterative block conjugate gradient acceleration [SIAM J. Sci. Comput., 16 (1995), pp. 1478-1511] for the solution of least-squares problems. The efficiency of both the iterative and the augmented pseudodirect approaches, as implemented in the ABCD-Solver, is illustrated on large rectangular matrices from the SuiteSparse Matrix Collection.

关键词： full rank linear systems extended system augmented block Cimmino hybrid methods least-squares problems minimum norm solution

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Block low-rank single precision coarse grid solvers for extreme scale multigrid methods

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NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS 2022年第1期29卷

作者： Buttari, Alfredo Huber, Markus Leleux, Philippe Mary, Theo Ruede, Ulrich Wohlmuth, Barbara CNRS IRIT Toulouse France Tech Univ Munich Inst Numer Math Garching Germany CERFACS Parallel Algorithms Team Toulouse France Sorbonne Univ CNRS LIP6 Paris France Friedrich Alexander Univ Lehrstuhl Syst Simulat Erlangen Germany

Extreme scale simulation requires fast and scalable algorithms, such as multigrid methods. To achieve asymptotically optimal complexity, it is essential to employ a hierarchy of grids. The cost to solve the coarsest grid system can often be neglected in sequential computings, but cannot be ignored in massively parallel executions. In this case, the coarsest grid can be large and its efficient solution becomes a challenging task. We propose solving the coarse grid system using modern, approximate sparse direct methods and investigate the expected gains compared with traditional iterative methods. Since the coarse grid system only requires an approximate solution, we show that we can leverage block low-rank techniques, combined with the use of single precision arithmetic, to significantly reduce the computational requirements of the direct solver. In the case of extreme scale computing, the coarse grid system is too large for a sequential solution, but too small to permit massively parallel efficiency. We show that the agglomeration of the coarse grid system to a subset of processors is necessary for the sparse direct solver to achieve performance. We demonstrate the efficiency of the proposed method on a Stokes-type saddle point system solved with a monolithic Uzawa multigrid method. In particular, we show that the use of an approximate sparse direct solver for the coarse grid system can outperform that of a preconditioned minimal residual iterative method. This is demonstrated for the multigrid solution of systems of order up to 1011 degrees of freedom on a petascale supercomputer using 43,200 processes.

关键词： block low-rank efficient coarse level solver hierarchical hybrid grids high-performance computing MUMPS sparse direct solver

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parallel adaptive FETI-DP using lightweight asynchronous dynamic load balancing

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING 2020年第4期121卷 621-643页

作者： Klawonn, Axel Kuhn, Martin J. Rheinbach, Oliver Univ Cologne Dept Math & Comp Sci Cologne Germany Univ Cologne Ctr Data & Simulat Sci Cologne Germany CERFACS Parallel Algorithms Team F-31057 Toulouse France Tech Univ Bergakad Freiberg Inst Numer Math & Optimierung Freiberg Germany Tech Univ Bergakad Freiberg Univ Rechenzentrum URZ D-09596 Freiberg Germany

A parallel FETI-DP domain decomposition method using an adaptive coarse space is presented. The implementation builds on a recently introduced adaptive FETI-DP approach for elliptic problems in three dimensions and uses small, local eigenvalue problems for faces and, additionally, for a small number of edges. The condition number of the preconditioned operator then satisfies a bound that is independent of coefficient heterogeneities in the problem. The computational cost of the local eigenvalue problems is not negligible, and also a significant load imbalance can be introduced. As a remedy, certain eigenvalue problems are discarded by a theory-guided heuristic strategy, based on the diagonal entries of the stiffness matrices. Additionally, a lightweight pairwise dynamic load balancing strategy is implemented for the eigenvalue problems. The load balancing is supervised by an orchestrating rank using asynchronous point-to-point communication. The resulting method shows good weak and strong scalability up to thousands of cores while fast convergence is obtained even for heterogeneous problems.

关键词： adaptive coarse spaces domain decomposition methods highly heterogeneous high performance computing parallelization partial differential equations

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Solver comparison for Poisson-like equations on tokamak geometries

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JOURNAL OF COMPUTATIONAL PHYSICS 2023年第1期488卷

作者： Bourne, Emily Leleux, Philippe Kormann, Katharina Kruse, Carola Grandgirard, Virginie Guclu, Yaman Kuhn, Martin J. Rude, Ulrich Sonnendrucker, Eric Zoni, Edoardo CEA IRFM F-13108 St Paul Les Durance France IRITCRCT Team INSERM Toulouse France Ctr Europeen Rech & Format Avancee Calcul Sci Parallel Algorithms Team CERFACS 42 Ave Gaspard Coriolis F-31057 Toulouse France Ruhr Univ Bochum Fak Math Univ Str 150 D-44801 Bochum Germany Max Planck Inst Plasma Phys Garching Germany Inst Software Technol German Aerosp Ctr Dept High Performance Comp Cologne Germany Friedrich Alexander Univ Erlangen Nurnberg Cauerstr 11 D-91058 Erlangen Germany Tech Univ Munich Garching Germany Lawrence Berkeley Natl Lab Berkeley CA USA

The solution of Poisson-like equations defined on a complex geometry is required for gyrokinetic simulations, which are important for the modelling of plasma turbulence in nuclear fusion devices such as the ITER tokamak. In this paper, we compare three existing solvers finely tuned to solve this problem, in terms of the accuracy of the solution, and their computational efficiency. We also consider practical implementation aspects, including the parallel efficiency of the code, potentially enabling an integration of the solvers in a state-of-the-art first-principle gyrokinetic simulation framework. The first, the Spline FEM solver, uses C1 polar splines to construct a finite elements method which solves the equation on curvilinear coordinates. The resulting linear system is solved using a conjugate gradient method. The second, the GMGPolar solver, uses a symmetric finite difference method to discretise the differential equation. The resulting linear system is solved using a tailored geometric multigrid scheme, with a combination of zebra circle and radial line smoothers, together with an implicit extrapolation scheme. The third, the Embedded Boundary solver, uses a finite volumes method on Cartesian coordinates with an embedded boundary scheme. The resulting linear system is solved using a multigrid scheme. The Spline FEM solver is shown to be the most accurate. The GMGPolar solver is shown to use the least memory. The Embedded Boundary solver is shown to be the fastest in most cases. All three solvers are shown to be capable of solving the equation on a realistic non-analytical geometry. The Embedded Boundary solver is additionally used to attempt to solve an X-point geometry.(C) 2023 Elsevier Inc. All rights reserved.

关键词： Tokamak plasma simulation Poisson -like equations Polar splines finite elements Multigrid solver Embedded boundary scheme

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ENERGY-MINIMIZING, SYMMETRIC DISCRETIZATIONS FOR ANISOTROPIC MESHES AND ENERGY FUNCTIONAL EXTRAPOLATION

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2021年第4期43卷 A2448-A2473页

作者： Kuhn, Martin Joachim Kruse, Carola Rude, Ulrich CERFACS Ctr Europeen Rech & Format Avancee Calcul Parallel Algorithms Team 42 Ave Gaspard Coriolis F-31057 Toulouse 01 France German Aerosp Ctr DLR Dept High Performance Comp Inst Software Technol D-51147 Cologne Germany Friedrich Alexander Univ Erlangen Nurnberg Dept Comp Sci 10 Cauerstr 11 D-91058 Erlangen Germany

Self-adjoint differential operators often arise from variational calculus on energy functionals. In this case, a direct discretization of the energy functional induces a discretization of the differential operator. Following this approach, the discrete equations are naturally symmetric if the energy functional is self-adjoint, a property that may be lost when using standard difference formulas on nonuniform meshes or when the differential operator has varying coefficients. Low order finite difference or finite element systems can be derived by this approach in a systematic way and on logically structured meshes they become compact difference formulas. Extrapolation formulas used on the discrete energy can then lead to higher oder approximations of the differential operator. A rigorous analysis is presented for extrapolation used in combination with nonstandard integration rules for finite elements. Extrapolation can likewise be applied on matrix-free finite difference stencils. In our applications, both schemes show up to quartic order of convergence.

关键词： partial differential equation energy functional symmetry anisotropy extrapolation finite differences finite elements

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Block partitioning of sparse rectangular matrices

Block partitioning of sparse rectangular matrices

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90th Annual Meeting of the International Association of Applied Mathematics and Mechanics, GAMM 2019

作者： Dumitrasc, Andrei Leleux, Philippe Rüde, Ulrich Friedrich-Alexander-Universität Erlangen-Nürnberg Cauerstraße 11 Erlangen91058 Germany Centre Européen de Recherche et de Formation Avancée en Calcul Scientifique Parallel Algorithms Team 42 Avenue Gaspard Coriolis Toulouse Cedex 131057 France

We present a means of reordering large, sparse rectangular matrices such that their nonzeros are closer to the diagonal. This enables a block-partitioning which is useful in parallel contexts. We use the Reverse Cuthill-McKee (RCM) algorithm on the adjacency matrix of the associated bipartite graph. The resulting, reordered matrix has a block bidiagonal structure. © 2019 The Authors Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim.

关键词： Matrix algebra

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