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Atif M, Kolluru PK, Ansumali S. Essentially entropic lattice Boltzmann model: Theory and simulations. Phys Rev E 2022; 106:055307. [PMID: 36559488 DOI: 10.1103/physreve.106.055307] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/05/2022] [Accepted: 10/26/2022] [Indexed: 11/12/2022]
Abstract
We present a detailed description of the essentially entropic lattice Boltzmann model. The entropic lattice Boltzmann model guarantees unconditional numerical stability by iteratively solving the nonlinear entropy evolution equation. In this paper we explain the construction of closed-form analytic solutions to this equation. We demonstrate that near equilibrium this analytic solution reduces to the standard lattice Boltzmann model. We consider a few test cases to show that the analytic solution does not exhibit any significant deviation from the iterative solution. We also extend the analytical solution for the Ellipsoidal Statistical (ES)-Bhatnagar-Gross-Krook model to remove the limitation on the Prandtl number for heat transfer problems. The simplicity of the analytic solution removes the computational overhead and algorithmic complexity associated with the entropic lattice Boltzmann models.
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Affiliation(s)
- Mohammad Atif
- Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore 560064, India
| | - Praveen Kumar Kolluru
- Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore 560064, India
| | - Santosh Ansumali
- Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore 560064, India.,SankhyaSutra Labs Limited, Bangalore 560045, India
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Tang X, Yu Y, Oztekin A. Asymptotic method for entropic multiple relaxation time model in lattice Boltzmann method. Phys Rev E 2022; 106:015303. [PMID: 35974631 DOI: 10.1103/physreve.106.015303] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/29/2021] [Accepted: 06/09/2022] [Indexed: 06/15/2023]
Abstract
To improve the numerical stability of the lattice Boltzmann method, Karlin et al. [Phys. Rev. E 90, 031302(R) (2014)10.1103/PhysRevE.90.031302] proposed the entropic multiple relaxation time (EMRT) collision model. The idea behind EMRT is to construct an optimal postcollision state by maximizing its local entropy value. The critical step of the EMRT model is to solve the entropy maximization problem under certain constraints, which is often computationally expensive and even not feasible. In this paper, we propose to employ perturbation theory and obtain an asymptotic solution to the maximum entropy state. With mathematical analysis of particular cases under relaxed constraints, we obtain the unperturbed form of the original problem and derive the asymptotic solution. We show that the asymptotic solution well approximates the optimal states; thus, our approach provides an efficient way to solve the constrained maximum entropy problem in the EMRT model. Also, we use the same idea of the EMRT model for the initial condition of the distribution function and propose to leave the entropy function to determine the missing information at the initial nodes. Finally, we numerically verify that the simulation results of the EMRT model obtained via the perturbation theory agree well with the exact solution to the Taylor-Green vortex problem. Furthermore, we also demonstrate that the EMRT model exhibits excellent stability performance for under-resolved simulations in the doubly periodic shear layer flow problem.
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Affiliation(s)
- Xiangshuo Tang
- Department of Mechanical Engineering and Mechanics, Lehigh University, Bethlehem, Pennsylvania 18015, USA
| | - Yue Yu
- Department of Mathematics, Lehigh University, Bethlehem, Pennsylvania 18015, USA
| | - Alparslan Oztekin
- Department of Mechanical Engineering and Mechanics, Lehigh University, Bethlehem, Pennsylvania 18015, USA
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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.
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Affiliation(s)
- Oleg Ilyin
- Federal Research Center "Computer Science and Control" of the Russian Academy of Sciences, Vavilova - 44,2, Moscow 119333, Russia
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Buzzicotti M, Tauzin G. Inertial range statistics of the entropic lattice Boltzmann method in three-dimensional turbulence. Phys Rev E 2021; 104:015302. [PMID: 34412200 DOI: 10.1103/physreve.104.015302] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/01/2021] [Accepted: 06/10/2021] [Indexed: 11/06/2022]
Abstract
We present a quantitative analysis of the inertial range statistics produced by entropic lattice Boltzmann method (ELBM) in the context of three-dimensional homogeneous and isotropic turbulence. ELBM is a promising mesoscopic model particularly interesting for the study of fully developed turbulent flows because of its intrinsic scalability and its unconditional stability. In the hydrodynamic limit, the ELBM is equivalent to the Navier-Stokes equations with an extra eddy viscosity term. From this macroscopic formulation, we have derived a new hydrodynamical model that can be implemented as a large-eddy simulation closure. This model is not positive definite, hence, able to reproduce backscatter events of energy transferred from the subgrid to the resolved scales. A statistical comparison of both mesoscopic and macroscopic entropic models based on the ELBM approach is presented and validated against fully resolved direct numerical simulations. Besides, we provide a second comparison of the ELBM with respect to the well-known Smagorinsky closure. We found that ELBM is able to extend the energy spectrum scaling range preserving at the same time the simulation stability. Concerning the statistics of higher order, inertial range observables, ELBM accuracy is shown to be comparable with other approaches such as Smagorinsky model.
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Affiliation(s)
- Michele Buzzicotti
- Department of Physics and INFN, University of Rome Tor Vergata, via della Ricerca Scientifica 1, 00133, Rome, Italy
| | - Guillaume Tauzin
- Department of Physics and INFN, University of Rome Tor Vergata, via della Ricerca Scientifica 1, 00133, Rome, Italy.,Chair of Applied Mathematics and Numerical Analysis, Bergische Universität Wuppertal, Gaußstrasse 20, 42119 Wuppertal, Germany
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Wang L. Enhanced multi-relaxation-time lattice Boltzmann model by entropic stabilizers. Phys Rev E 2020; 102:023307. [PMID: 32942451 DOI: 10.1103/physreve.102.023307] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/08/2019] [Accepted: 07/23/2020] [Indexed: 11/07/2022]
Abstract
The difficulty of choice of relaxation rates in multi-relaxation-time lattice Boltzmann model (MRT-LBM) is surmounted by solution of least-square problem of entropic stabilizer equations. Relaxation rates in the enhanced MRT-LBM are evolving with time rather than remain constants. To derive entropic stabilizer equations, nonequilibrium population is split into different modes in terms of column vectors in the inverse transform matrix. The entropic stabilizer equations are achieved by minimization of H-function. Different moment representations in MRT-LBM, such as Gram-Schmidt orthogonal moment, natural moment, and central moment, are tested for double periodic shear flow, shock tube problem, and lid-driven cavity flow, which demonstrates the potential of enhanced MRT-LBM.
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Affiliation(s)
- Long Wang
- School of Aeronautics, Northwestern Polytechnical University, Xi'an 710072, China
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Krämer A, Wilde D, Küllmer K, Reith D, Foysi H. Pseudoentropic derivation of the regularized lattice Boltzmann method. Phys Rev E 2019; 100:023302. [PMID: 31574640 DOI: 10.1103/physreve.100.023302] [Citation(s) in RCA: 15] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/08/2019] [Indexed: 11/07/2022]
Abstract
The lattice Boltzmann method (LBM) facilitates efficient simulations of fluid turbulence based on advection and collision of local particle distribution functions. To ensure stable simulations on underresolved grids, the collision operator must prevent drastic deviations from local equilibrium. This can be achieved by various methods, such as the multirelaxation time, entropic, quasiequilibrium, regularized, and cumulant schemes. Complementing a part of a unified theoretical framework of these schemes, the present work presents a derivation of the regularized lattice Boltzmann method (RLBM), which follows a recently introduced entropic multirelaxation time LBM by Karlin, Bösch, and Chikatamarla (KBC). It is shown that both methods can be derived by locally maximizing a quadratic Taylor expansion of the entropy function. While KBC expands around the local equilibrium distribution, the RLBM is recovered by expanding entropy around a global equilibrium. Numerical tests were performed to elucidate the role of pseudoentropy maximization in these models. Simulations of a two-dimensional shear layer show that the RLBM successfully reproduces the largest eddies even on a 16×16 grid, while the conventional LBM becomes unstable for grid resolutions of 128×128 and lower. The RLBM suppresses spurious vortices more effectively than KBC. In contrast, simulations of the three-dimensional Taylor-Green and Kida vortices show that KBC performs better in resolving small scale vortices, outperforming the RLBM by a factor of 1.8 in terms of the effective Reynolds number.
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Affiliation(s)
- Andreas Krämer
- Institute of Technology, Resource and Energy-efficient Engineering (TREE), Bonn-Rhein-Sieg University of Applied Sciences, Grantham-Allee 20, 53757 Sankt Augustin, Germany.,National Heart, Lung, and Blood Institute, National Institutes of Health, Bethesda, Maryland 20892, United States
| | - Dominik Wilde
- Institute of Technology, Resource and Energy-efficient Engineering (TREE), Bonn-Rhein-Sieg University of Applied Sciences, Grantham-Allee 20, 53757 Sankt Augustin, Germany.,Department of Mechanical Engineering, University of Siegen, Paul-Bonatz-Straße 9-11, 57076 Siegen-Weidenau, Germany
| | - Knut Küllmer
- Institute of Technology, Resource and Energy-efficient Engineering (TREE), Bonn-Rhein-Sieg University of Applied Sciences, Grantham-Allee 20, 53757 Sankt Augustin, Germany.,Department of Mechanical Engineering, University of Siegen, Paul-Bonatz-Straße 9-11, 57076 Siegen-Weidenau, 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
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