1
|
Saito S, Takada N, Baba S, Someya S, Ito H. Generalized equilibria for color-gradient lattice Boltzmann model based on higher-order Hermite polynomials: A simplified implementation with central moments. Phys Rev E 2023; 108:065305. [PMID: 38243429 DOI: 10.1103/physreve.108.065305] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/15/2023] [Accepted: 11/19/2023] [Indexed: 01/21/2024]
Abstract
We propose generalized equilibria of a three-dimensional color-gradient lattice Boltzmann model for two-component two-phase flows using higher-order Hermite polynomials. Although the resulting equilibrium distribution function, which includes a sixth-order term on the velocity, is computationally cumbersome, its equilibrium central moments (CMs) are velocity-independent and have a simplified form. Numerical experiments show that our approach, as in Wen et al. [Phys. Rev. E 100, 023301 (2019)2470-004510.1103/PhysRevE.100.023301] who consider terms up to third order, improves the Galilean invariance compared to that of the conventional approach. Dynamic problems can be solved with high accuracy at a density ratio of 10; however, the accuracy is still limited to a density ratio of 1000. For lower density ratios, the generalized equilibria benefit from the CM-based multiple-relaxation-time model, especially at very high Reynolds numbers, significantly improving the numerical stability.
Collapse
Affiliation(s)
- Shimpei Saito
- Research Institute for Energy Conservation (iECO), National Institute of Advanced Industrial Science and Technology (AIST), 1-2-1 Namiki, Tsukuba 3058564, Japan
| | - Naoki Takada
- Research Institute for Energy Conservation (iECO), National Institute of Advanced Industrial Science and Technology (AIST), 1-2-1 Namiki, Tsukuba 3058564, Japan
| | - Soumei Baba
- Research Institute for Energy Conservation (iECO), National Institute of Advanced Industrial Science and Technology (AIST), 1-2-1 Namiki, Tsukuba 3058564, Japan
| | - Satoshi Someya
- Research Institute for Energy Conservation (iECO), National Institute of Advanced Industrial Science and Technology (AIST), 1-2-1 Namiki, Tsukuba 3058564, Japan
| | - Hiroshi Ito
- Research Institute for Energy Conservation (iECO), National Institute of Advanced Industrial Science and Technology (AIST), 1-2-1 Namiki, Tsukuba 3058564, Japan
| |
Collapse
|
2
|
Corbetta A, Gabbana A, Gyrya V, Livescu D, Prins J, Toschi F. Toward learning Lattice Boltzmann collision operators. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2023; 46:10. [PMID: 36877295 PMCID: PMC9988764 DOI: 10.1140/epje/s10189-023-00267-w] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 12/13/2022] [Accepted: 02/12/2023] [Indexed: 06/18/2023]
Abstract
In this work, we explore the possibility of learning from data collision operators for the Lattice Boltzmann Method using a deep learning approach. We compare a hierarchy of designs of the neural network (NN) collision operator and evaluate the performance of the resulting LBM method in reproducing time dynamics of several canonical flows. In the current study, as a first attempt to address the learning problem, the data were generated by a single relaxation time BGK operator. We demonstrate that vanilla NN architecture has very limited accuracy. On the other hand, by embedding physical properties, such as conservation laws and symmetries, it is possible to dramatically increase the accuracy by several orders of magnitude and correctly reproduce the short and long time dynamics of standard fluid flows.
Collapse
Affiliation(s)
| | - Alessandro Gabbana
- Eindhoven University of Technology, 5600, Eindhoven, MB, The Netherlands.
| | - Vitaliy Gyrya
- Los Alamos National Laboratory, Los Alamos, NM, 87545, USA
| | - Daniel Livescu
- Los Alamos National Laboratory, Los Alamos, NM, 87545, USA
| | - Joost Prins
- Eindhoven University of Technology, 5600, Eindhoven, MB, The Netherlands
| | - Federico Toschi
- Eindhoven University of Technology, 5600, Eindhoven, MB, The Netherlands
- Consiglio Nazionale della Ricerche-IAC, Rome, Italy
| |
Collapse
|
3
|
Guo Z, Wang LP, Qi Y. Discrete unified gas kinetic scheme for continuum compressible flows. Phys Rev E 2023; 107:025304. [PMID: 36932506 DOI: 10.1103/physreve.107.025304] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/19/2022] [Accepted: 01/24/2023] [Indexed: 06/18/2023]
Abstract
In this paper, a discrete unified gas kinetic scheme (DUGKS) is proposed for continuum compressible gas flows based on the total energy kinetic model [Guo et al., Phys. Rev. E 75, 036704 (2007)1539-375510.1103/PhysRevE.75.036704]. The proposed DUGKS can be viewed as a special finite-volume lattice Boltzmann method for the compressible Navier-Stokes equations in the double distribution function formulation, in which the mass and momentum transport are described by the kinetic equation for a density distribution function (g), and the energy transport is described by the other one for an energy distribution function (h). To recover the full compressible Navier-Stokes equations exactly, the corresponding equilibrium distribution functions g^{eq} and h^{eq} are expanded as Hermite polynomials up to third and second orders, respectively. The velocity spaces for the kinetic equations are discretized according to the seventh and fifth Gauss-Hermite quadratures. Consequently, the computational efficiency of the present DUGKS can be much improved in comparison with previous versions using more discrete velocities required by the ninth Gauss-Hermite quadrature.
Collapse
Affiliation(s)
- Zhaoli Guo
- Institute of Multidisciplinary Research for Mathematics and Applied Science, Huazhong University of Science and Technology, Wuhan 430074, China
| | - Lian-Ping Wang
- Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Shenzhen 518055, China
| | - Yiming Qi
- State Key Laboratory for Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871, China
| |
Collapse
|
4
|
Shirsat AU, Nayak SG, Patil DV. Simulation of high-Mach-number inviscid flows using a third-order Runge-Kutta and fifth-order WENO-based finite-difference lattice Boltzmann method. Phys Rev E 2022; 106:025314. [PMID: 36109898 DOI: 10.1103/physreve.106.025314] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/27/2022] [Accepted: 08/01/2022] [Indexed: 06/15/2023]
Abstract
A discrete-velocity Boltzmann equation (DVBE) with Bhatnagar-Gross-Krook (BGK) approximation is discretized in time and space using a third-order Runge-Kutta (RK3) and fifth-order weighted essentially nonoscillatory (WENO) finite-difference scheme to simulate benchmark inviscid compressible flows. The implementation of the WENO ensures that solutions behave with minimal or no oscillations, narrowing the gap between the exact and the numerical results. Discrete-velocity sets given by Kataoka and Tsutahara [Phys. Rev. E 69, 056702 (2004)10.1103/PhysRevE.69.056702] are used. The additional dissipation terms as well as artificial viscosity are incorporated in the formulation to solve the compressible flows at high Mach number. Further, the flows which are subjected initially to a high density ratio are effectively simulated. In this article, one-dimensional benchmarks are simulated at initial Mach number up to 30 and density ratio up to 1000, whereas, the benchmarks in two dimensions are simulated with a Mach number up to 10. The algorithm is assessed by simulating numerous benchmarks, namely, (i) one-dimensional Riemann problem for various shock waves combinations [namely (a) shock-shock waves in the case of different Mach numbers, (b) rarefaction-shock waves for various density ratios, (c) sudden contact shock discontinuity, and (d) shock-rarefaction waves for density ratio 5], (ii) isentropic vortex convection test, (iii) regular shock reflection (RR) for Mach numbers 2.9 and 10, (iv) double Mach reflection (DMR) for inflow Mach numbers as 6 and 10, and (v) forward-facing step for inflow Mach numbers 2 to 5. Further, the effect of change in Mach numbers and wedge angles on the flow structures in the case of DMR are detailed. In the case of a forward-facing step, the variations of flow structure (e.g., the Mach stem height, triple points location, and shock standoff distance) are detailed with respect to Mach number, step height, and specific-heat ratios. Finally, the numerical stability of the proposed formulation is carried out to assess the behavior of the free parameters.
Collapse
Affiliation(s)
- Akshay U Shirsat
- Department of Mechanical, Materials, and Aerospace Engineering, Indian Institute of Technology Dharwad, Dharwad, Karnataka 580011, India
| | - Sagar G Nayak
- Department of Mechanical, Materials, and Aerospace Engineering, Indian Institute of Technology Dharwad, Dharwad, Karnataka 580011, India
| | - Dhiraj V Patil
- Department of Mechanical, Materials, and Aerospace Engineering, Indian Institute of Technology Dharwad, Dharwad, Karnataka 580011, India
| |
Collapse
|
5
|
Lehmann M, Krause MJ, Amati G, Sega M, Harting J, Gekle S. Accuracy and performance of the lattice Boltzmann method with 64-bit, 32-bit, and customized 16-bit number formats. Phys Rev E 2022; 106:015308. [PMID: 35974647 DOI: 10.1103/physreve.106.015308] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/12/2022] [Accepted: 06/07/2022] [Indexed: 06/15/2023]
Abstract
Fluid dynamics simulations with the lattice Boltzmann method (LBM) are very memory intensive. Alongside reduction in memory footprint, significant performance benefits can be achieved by using FP32 (single) precision compared to FP64 (double) precision, especially on GPUs. Here we evaluate the possibility to use even FP16 and posit16 (half) precision for storing fluid populations, while still carrying arithmetic operations in FP32. For this, we first show that the commonly occurring number range in the LBM is a lot smaller than the FP16 number range. Based on this observation, we develop customized 16-bit formats-based on a modified IEEE-754 and on a modified posit standard-that are specifically tailored to the needs of the LBM. We then carry out an in-depth characterization of LBM accuracy for six different test systems with increasing complexity: Poiseuille flow, Taylor-Green vortices, Karman vortex streets, lid-driven cavity, a microcapsule in shear flow (utilizing the immersed-boundary method), and, finally, the impact of a raindrop (based on a volume-of-fluid approach). We find that the difference in accuracy between FP64 and FP32 is negligible in almost all cases, and that for a large number of cases even 16-bit is sufficient. Finally, we provide a detailed performance analysis of all precision levels on a large number of hardware microarchitectures and show that significant speedup is achieved with mixed FP32/16-bit.
Collapse
Affiliation(s)
- Moritz Lehmann
- Biofluid Simulation and Modeling-Theoretische Physik VI, University of Bayreuth, Bayreuth, Germany
| | - Mathias J Krause
- Institute of Mechanical Process Engineering and Mechanics, Karlsruhe Institute of Technology, Karlsruhe, Germany
| | - Giorgio Amati
- CINECA, SCAI-SuperComputing Applications and Innovation Department, Rome Branch, Italy
| | - Marcello Sega
- Helmholtz Institute Erlangen-Nürnberg for Renewable Energy, Erlangen, Germany
| | - Jens Harting
- Helmholtz Institute Erlangen-Nürnberg for Renewable Energy, Erlangen, Germany
- Department of Chemical and Biological Engineering and Department of Physics, Friedrich-Alexander-Universität, Erlangen, Germany
| | - Stephan Gekle
- Biofluid Simulation and Modeling-Theoretische Physik VI, University of Bayreuth, Bayreuth, Germany
| |
Collapse
|
6
|
Ilyin O. Discrete-velocity Boltzmann model: Regularization and linear stability. Phys Rev E 2022; 105:045312. [PMID: 35590549 DOI: 10.1103/physreve.105.045312] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/22/2021] [Accepted: 03/22/2022] [Indexed: 06/15/2023]
Abstract
A discrete-velocity Boltzmann model for a nine-velocity lattice is considered. Compared to the conventional lattice Boltzmann (LB) schemes the collisions for the model are defined explicitly. Space and time discretization of the model is based on the collide and stream method; in addition, the regularization of the collision term is proposed. It is demonstrated that the regularized model can be represented as a two-relaxation-time LB model of a special type. The scheme is compared to the Onsager regularized (a specific filtered Galilean invariant model) and recursively regularized LB equations in terms of stability and dissipation properties, and linear stability analysis is performed. Several numerical experiments are carried out: double shear layer, lid-driven cavity flow, and propagation of acoustic and shear waves are considered for different grid resolutions, Mach and Reynolds numbers. It is shown that free parameters in the model corresponding to collision cross sections can be adjusted in such a way that the dissipation and stability properties can be controlled.
Collapse
Affiliation(s)
- Oleg Ilyin
- Federal Research Center "Computer Science and Control" of the Russian Academy of Sciences, Vavilova - 44,2, Moscow 119333, Russia
| |
Collapse
|
7
|
Gounley J, Vardhan M, Draeger EW, Valero-Lara P, Moore SV, Randles A. Propagation pattern for moment representation of the lattice Boltzmann method. IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS : A PUBLICATION OF THE IEEE COMPUTER SOCIETY 2022; 33:642-653. [PMID: 35498162 PMCID: PMC9053389 DOI: 10.1109/tpds.2021.3098456] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/14/2023]
Abstract
A propagation pattern for the moment representation of the regularized lattice Boltzmann method (LBM) in three dimensions is presented. Using effectively lossless compression, the simulation state is stored as a set of moments of the lattice Boltzmann distribution function, instead of the distribution function itself. An efficient cache-aware propagation pattern for this moment representation has the effect of substantially reducing both the storage and memory bandwidth required for LBM simulations. This paper extends recent work with the moment representation by expanding the performance analysis on central processing unit (CPU) architectures, considering how boundary conditions are implemented, and demonstrating the effectiveness of the moment representation on a graphics processing unit (GPU) architecture.
Collapse
Affiliation(s)
- John Gounley
- Computational Sciences and Engineering Division at Oak Ridge National Laboratory
| | | | - Erik W Draeger
- Center for Applied Scientific Computing at Lawrence Livermore National Laboratory
| | - Pedro Valero-Lara
- Computer Science and Mathematics Division at Oak Ridge National Laboratory
| | - Shirley V Moore
- Department of Computer Science at the University of Texas at El Paso
| | | |
Collapse
|
8
|
Wilde D, Krämer A, Reith D, Foysi H. High-order semi-Lagrangian kinetic scheme for compressible turbulence. Phys Rev E 2021; 104:025301. [PMID: 34525552 DOI: 10.1103/physreve.104.025301] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/13/2020] [Accepted: 07/12/2021] [Indexed: 11/07/2022]
Abstract
Turbulent compressible flows are traditionally simulated using explicit time integrators applied to discretized versions of the Navier-Stokes equations. However, the associated Courant-Friedrichs-Lewy condition severely restricts the maximum time-step size. Exploiting the Lagrangian nature of the Boltzmann equation's material derivative, we now introduce a feasible three-dimensional semi-Lagrangian lattice Boltzmann method (SLLBM), which circumvents this restriction. While many lattice Boltzmann methods for compressible flows were restricted to two dimensions due to the enormous number of discrete velocities in three dimensions, the SLLBM uses only 45 discrete velocities. Based on compressible Taylor-Green vortex simulations we show that the new method accurately captures shocks or shocklets as well as turbulence in 3D without utilizing additional filtering or stabilizing techniques other than the filtering introduced by the interpolation, even when the time-step sizes are up to two orders of magnitude larger compared to simulations in the literature. Our new method therefore enables researchers to study compressible turbulent flows by a fully explicit scheme, whose range of admissible time-step sizes is dictated by physics rather than spatial discretization.
Collapse
Affiliation(s)
- Dominik Wilde
- Department of Mechanical Engineering, University of Siegen, Paul-Bonatz-Straße 9-11, 57076 Siegen-Weidenau, Germany.,Institute of Technology, Resource and Energy-efficient Engineering (TREE), Bonn-Rhein-Sieg University of Applied Sciences, Grantham-Allee 20, 53757 Sankt Augustin, Germany
| | - Andreas Krämer
- Department of Mathematics and Computer Science, Freie Universität Berlin, Arnimallee 6, 14195 Berlin, Germany
| | - Dirk Reith
- Institute of Technology, Resource and Energy-efficient Engineering (TREE), Bonn-Rhein-Sieg University of Applied Sciences, Grantham-Allee 20, 53757 Sankt Augustin, Germany.,Fraunhofer Institute for Algorithms and Scientific Computing (SCAI), Schloss Birlinghoven, 53754 Sankt Augustin, Germany
| | - Holger Foysi
- Department of Mechanical Engineering, University of Siegen, Paul-Bonatz-Straße 9-11, 57076 Siegen-Weidenau, Germany
| |
Collapse
|
9
|
Qiu R, Zhou T, Bao Y, Zhou K, Che H, You Y. Mesoscopic kinetic approach for studying nonequilibrium hydrodynamic and thermodynamic effects of shock wave, contact discontinuity, and rarefaction wave in the unsteady shock tube. Phys Rev E 2021; 103:053113. [PMID: 34134242 DOI: 10.1103/physreve.103.053113] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/20/2021] [Accepted: 05/12/2021] [Indexed: 06/12/2023]
Abstract
This paper presents a detailed description of a molecular velocity distribution-based mesoscopic kinetic approach that enables a better understanding of various nonequilibrium hydrodynamic and thermodynamic effects in shock waves, contact discontinuities, and rarefaction waves. This builds on the mesoscopic kinetic approach in a previous investigation into regular reflection shocks by further addressing the mesoscopic physical meaning of kinetic moments from the view of kinetics and the implications of the magnitude and sign of nonequilibrium kinetic moments. To deepen understanding of nonequilibrium effects, this work focuses on the one-dimensional unsteady shock tube problem, which contains the typical and essential features of the discontinuous flows, and has no interference of two-dimensional flow direction. The approach uses a lattice Boltzmann method to solve the flow field, and describes nonequilibrium effects through the nonequilibrium kinetic moments of molecular velocity distribution functions. The mechanism of nonequilibrium effect in discontinuous flows is further probed. This work develops the mesoscopic kinetic approach and clarifies the mesoscopic physics of shock waves, contact discontinuities, and rarefaction waves.
Collapse
Affiliation(s)
- Ruofan Qiu
- School of Aerospace Engineering, Xiamen University, Xiamen 361005, People's Republic of China
| | - Tao Zhou
- School of Aerospace Engineering, Xiamen University, Xiamen 361005, People's Republic of China
| | - Yue Bao
- School of Aerospace Engineering, Xiamen University, Xiamen 361005, People's Republic of China
| | - Kang Zhou
- School of Aerospace Engineering, Xiamen University, Xiamen 361005, People's Republic of China
| | - Huanhuan Che
- School of Aerospace Engineering, Xiamen University, Xiamen 361005, People's Republic of China
| | - Yancheng You
- School of Aerospace Engineering, Xiamen University, Xiamen 361005, People's Republic of China
| |
Collapse
|
10
|
Hosseini SA, Darabiha N, Thévenin D. Compressibility in lattice Boltzmann on standard stencils: effects of deviation from reference temperature. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2020; 378:20190399. [PMID: 32564724 PMCID: PMC7333953 DOI: 10.1098/rsta.2019.0399] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 03/04/2020] [Indexed: 05/05/2023]
Abstract
With growing interest in the simulation of compressible flows using the lattice Boltzmann (LB) method, a number of different approaches have been developed. These methods can be classified as pertaining to one of two major categories: (i) solvers relying on high-order stencils recovering the Navier-Stokes-Fourier equations, and (ii) approaches relying on classical first-neighbour stencils for the compressible Navier-Stokes equations coupled to an additional (LB-based or classical) solver for the energy balance equation. In most cases, the latter relies on a thermal Hermite expansion of the continuous equilibrium distribution function (EDF) to allow for compressibility. Even though recovering the correct equation of state at the Euler level, it has been observed that deviations of local flow temperature from the reference can result in instabilities and/or over-dissipation. The aim of the present study is to evaluate the stability domain of different EDFs, different collision models, with and without the correction terms for the third-order moments. The study is first based on a linear von Neumann analysis. The correction term for the space- and time-discretized equations is derived via a Chapman-Enskog analysis and further corroborated through spectral dispersion-dissipation curves. Finally, a number of numerical simulations are performed to illustrate the proposed theoretical study. This article is part of the theme issue 'Fluid dynamics, soft matter and complex systems: recent results and new methods'.
Collapse
Affiliation(s)
- S. A. Hosseini
- Laboratory of Fluid Dynamics and Technical Flows, University of Magdeburg ‘Otto von Guericke’, 39106 Magdeburg, Germany
- Laboratoire EM2C, CNRS, CentraleSupélec, Université Paris-Saclay, 3 rue Joliot Curie, 91192 Gif-sur-Yvette Cedex, France
- International Max Planck Research School (IMPRS) for Advanced Methods in Process and Systems Engineering, Magdeburg, Germany
| | - N. Darabiha
- Laboratoire EM2C, CNRS, CentraleSupélec, Université Paris-Saclay, 3 rue Joliot Curie, 91192 Gif-sur-Yvette Cedex, France
| | - D. Thévenin
- Laboratory of Fluid Dynamics and Technical Flows, University of Magdeburg ‘Otto von Guericke’, 39106 Magdeburg, Germany
| |
Collapse
|