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Phase-field-based lattice Boltzmann model for immiscible incompressible N-phase flows

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Physical Review E 2020年第6期101卷 063310-063310页

作者： Xiaolei Yuan Hong Liang Zhenhua Chai Baochang Shi School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan 430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan 430074 China Department of Physics Hangzhou Dianzi University Hangzhou 310018 China

In this paper, we develop an efficient and alternative lattice Boltzmann (LB) model for simulating immiscible incompressible N-phase flows (N≥2) based on the Cahn-Hilliard phase field theory. In order to facilitate the design of LB model and reduce the calculation of the gradient term, the governing equations of the N-phase system are reformulated, and they satisfy the conservation of mass, momentum and the second law of thermodynamics. In the present model, (N−1) LB equations are employed to capture the interface, and another LB equation is used to solve the Navier-Stokes (N-S) equations, where a new distribution function for the total force is delicately designed to reduce the calculation of the gradient term. The developed model is first validated by two classical benchmark problems, including the tests of static droplets and the spreading of a liquid lens, the simulation results show that the current LB model is accurate and efficient for simulating incompressible N-phase fluid flows. To further demonstrate the capability of the LB model, two numerical simulations, including dynamics of droplet collision for four fluid phases and dynamics of droplets and interfaces for five fluid phases, are performed to test the developed model. The results show that the present model can successfully handle complex interactions among N (N≥2) immiscible incompressible flows.

关键词： Multiphase flows Lattice-Boltzmann methods

Optimal rates of convergence and error localization of Gegenbauer projections

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arXiv 2020年

作者： Wang, Haiyong School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

Motivated by comparing the convergence behavior of Gegenbauer projections and best approximations, we study the optimal rate of convergence for Gegenbauer projections in the maximum norm. We show that the rate of convergence of Gegenbauer projections is the same as that of best approximations under conditions of the underlying function is either analytic on and within an ellipse and λ ≤ 0 or differentiable and λ ≤ 1, where λ is the parameter in Gegenbauer projections. If the underlying function is analytic and λ > 0 or differentiable and λ > 1, then the rate of convergence of Gegenbauer projections is slower than that of best approximations by factors of nλ and nλ−1, respectively. An exceptional case is functions with endpoint singularities, for which Gegenbauer projections and best approximations converge at the same rate for all λ > −1/2. For functions with interior or endpoint singularities, we provide a theoretical explanation for the error localization phenomenon of Gegenbauer projections and for why the accuracy of Gegenbauer projections is better than that of best approximations except in small neighborhoods of the critical points. Our analysis provides fundamentally new insight into the power of Gegenbauer approximations and related spectral *** Codes 41A10, 41A25 Copyright © 2020, The Authors. All rights reserved.

关键词： Functions

How much faster does the best polynomial approximation converge than Legendre projection?

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arXiv 2020年

We compare the convergence behavior of best polynomial approximations and Legendre and Chebyshev projections and derive optimal rates of convergence of Legendre projections for analytic and differentiable functions in the maximum norm. For analytic functions, we show that the best polynomial approximation of degree n is better than the Legendre projection of the same degree by a factor of n1/2. For differentiable functions such as piecewise analytic functions and functions of fractional smoothness, however, we show that the best approximation is better than the Legendre projection by only some constant factors. Our results provide some new insights into the approximation power of Legendre projections. Copyright © 2020, The Authors. All rights reserved.

关键词： Functions

Discrete unified gas kinetic scheme for nonlinear convection-diffusion equations

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Physical Review E 2020年第2期101卷 023306-023306页

作者： Jinlong Shang Zhenhua Chai Huili Wang Baochang Shi School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan 430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan 430074 China School of Mathematics and Computer Science Wuhan Textile University Wuhan 430073 China

In this paper, we develop a discrete unified gas kinetic scheme (DUGKS) for a general nonlinear convection-diffusion equation (NCDE) and show that the NCDE can be recovered correctly from the present model through the Chapman-Enskog analysis. We then test the present DUGKS through some classic convection-diffusion equations, and we find that the numerical results are in good agreement with analytical solutions and that the DUGKS model has a second-order convergence rate. Finally, as a finite-volume method, the DUGKS can also adopt the nonuniform mesh. Besides, we perform some comparisons among the DUGKS, the finite-volume lattice Boltzmann model (FV-LBM), the single-relaxation-time lattice Boltzmann model (SLBM), and the multiple-relaxation-time lattice Boltzmann model (MRT-LBM). The results show that the present DUGKS is more accurate than the FV-LBM, more stable than the SLBM, and almost has the same accuracy as the MRT-LBM. Moreover, the use of nonuniform mesh may make the DUGKS model more flexible.

关键词： Lattice-Boltzmann methods

GLOBAL-IN-TIME SOLVABILITY AND BLOW-UP FOR A NON-ISOSPECTRAL TWO-COMPONENT CUBIC CAMASSA-HOLM SYSTEM IN A CRITICAL BESOV SPACE

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arXiv 2020年

作者： Zhang, Lei Qiao, Zhijun SCHOOL OF MATHEMATICS AND STATISTICS HUBEI KEY LABORATORY OF ENGINEERING MODELING AND SCIENTIFIC COMPUTING HUAZHONG UNIVERSITY OF SCIENCE AND TECHNOLOGY WUHANHUBEI430074 China SCHOOL OF MATHEMATICAL AND STATISTICAL SCIENCES THE UNIVERSITY OF TEXAS RIO GRANDE VALLEY EDINBURGTX78539 United States

In this paper, we prove the global Hadamard well-posedness of strong solutions to anon-isospectral two-component cubic Camassa-Holm system in the critical Besov space B22,1(T). 1 Our results shows that in comparison with the well-known work for classic Camassa-Holm-type equations, the existence of global solution only relies on the L1-integrability of the variable coefficients α(t) and γ(t), but nothing to do with the shape or smoothness of the initial data. The key ingredient of the proof hinges on the careful analysis of the mutual effect among two component forms, the uniform bound of approximate solutions, and several crucial estimates of cubic nonlinearities in low-regularity Besov spaces via the Littlewood-Paley decomposition theory. A reduced case in our results yields the global existence of solutions in a Besov space for two kinds of well-known isospectral peakon system with weakly dissipative terms. Moreover, we derive two kinds of precise blow-up criteria for a strong solution in both critical and non-critical Besov spaces, as well as providing specific characterization for the lower bound of the blow-up time, which implies the global existence with additional conditions on the time-dependent parameters α(t) an γ(t). Copyright © 2020, The Authors. All rights reserved.

关键词： Banach spaces

Topological structures on saturated sets, optimal orbits and equilibrium states

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arXiv 2020年

作者： Hou, Xiaobo Tian, Xueting Zhang, Yiwei School of Mathematical Sciences Fudan University Shanghai200433 China School of Mathematics and Statistics Center for Mathematical Sciences Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Sciences and Technology Wuhan430074 China

Pfister and Sullivan proved that if a topological dynamical system (X, T) satisfies almost product property and uniform separation property, then for each nonempty compact subset K of invariant measures, the entropy of saturated set GK satisfies (0.1) hBtop(T, GK) = inf{h(T, µ): µ ∈ K}, where hBtop(T, GK) is Bowen's topological entropy of T on GK, and h(T, µ) is the Kolmogorov-Sinai entropy of µ. In this paper, we investigate topological complexity of GK by replacing Bowen's topological entropy with upper capacity entropy and packing entropy in (0.1) and obtain the following formulas: hUCtop (T, GK) = htop(T, X) and hPtop(T, GK) = sup{h(T, µ): µ ∈ K}, where hUCtop (T, GK) is the upper capacity entropy of T on GK and hPtop(T, GK) is the packing entropy of T on GK. In the proof of these two formulas, uniform separation property is unnecessary. As applications, when (X, T) is a transitive Anosov diffeomorphism on a compact manifold, we show that (1) for any continuous function f, hUCtop (T, Sfop) = htop(T, X) where Sfop is the set of initial values of f-optimal orbits;(2) there exists a Baire generic subset F in the space of continuous functions and an open and dense subset G in the space of Hölder continuous or C1 smooth functions such that for any f ∈ F or f ∈ G, hBtop(T, Sfop) = hPtop(T, Sfop) = 0;(3) for any f ∈ G, hUCtop (T, SfMR) = 0 where SfMR is the set of initial values of measure-recurrent optimal orbits;(4) there exists a dense subset in the space of continuous functions such that for any f in the subset, hBtop(T, Sfop) = hPtop(T, Sfop) > 0. For equilibrium states, we prove that (1) for any continuous function f and equilibrium state µ of f, hUCtop (T, Gµ) = htop(T, X);(2) if f is Hölder continuous and is not cohomologous to a constant, then hBtop(T, Gµ) = hPtop(T, Gµ) fopMSC Codes 37C50, 37B20, 37B05, 37D45, 37C45 © 2020, CC BY.

关键词： Entropy

Numerical methods for stochastic Volterra integral equations with weakly singular kernels

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arXiv 2020年

作者： Li, Min Huang, Chengming Hu, Yaozhong School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modelling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China Department of Mathematical and Statistical Sciences University of Alberta EdmontonT6G 2G1 Canada

In this paper, we first establish the existence, uniqueness and Hölder continuity of the solution to stochastic Volterra integral equations with weakly singular kernels. Then, we propose a θ-Euler-Maruyama scheme and a Milstein scheme to solve the equations numerically and we obtain the strong rates of convergence for both schemes in Lp norm for any p ≥ 1. For the θ-Euler-Maruyama scheme the rate is min{1 − α, 12 − β} and for the Milstein scheme the rate is min{1 − α, 1 − 2β} when α, 12, where (0 12 ). These results on the rates of convergence are significantly different from that of the similar schemes for the stochastic Volterra integral equations with regular kernels. The difficulty to obtain our results is the lack of Itô formula for the equations. To get around of this difficulty we use instead the Taylor formula and then carry a sophisticated analysis on the equation the solution satisfies. Copyright © 2020, The Authors. All rights reserved.

关键词： Integral equations

A diffuse-interface lattice Boltzmann method for thermal particulate flows with Dirichlet boundary conditions

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Physics of Fluids 2025年第6期37卷

作者： Liu, Jiao Liu, Xi Chai, Zhenhua School of Mathematics and Computer Science Nanchang University Jiangxi Nanchang330031 China School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Institute of Interdisciplinary Research for Mathematics and Applied Science Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China The State Key Laboratory of Intelligent Manufacturing Equipment and Technology Huazhong University of Science and Technology Wuhan430074 China

The lattice Boltzmann method has been widely used to study thermal particulate flows that are usually observed in both nature and engineering, while how to efficiently and accurately treat the fluid-solid boundary conditions is still an important and not well-resolved issue. In this paper, a diffuse-interface lattice Boltzmann method (DI-LBM) is developed for complex thermal particulate flows with Dirichlet boundary conditions. Unlike the previous works where the fluid-solid boundary is treated as a sharp interface, the fluid-solid boundary in the DI-LBM is implemented as a diffuse interface such that the physical variables and transport coefficients in the fluid and solid regions are varied smoothly. To depict the diffuse interface, a smooth function is introduced, and simultaneously, an additional source term is added in the evolution equation to reflect the effect of Dirichlet boundary condition imposed on the fluid-solid boundary. The present method is also tested by several classical problems, and the numerical results show that the DI-LBM is not only of second-order convergence rate in space, but also efficient in the study of complex thermal particulate flows. © 2025 Author(s).

关键词： Kinetic theory of gases

The influence of stochastic forcing on strong solutions to the Incompressible Slice Model in 2D bounded domain

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arXiv 2020年

作者： Zhang, Lei Shi, Yu Hu, Chaozhu Wang, Weifeng Liu, Bin SCHOOL OF MATHEMATICS AND STATISTICS HUBEI KEY LABORATORY OF ENGINEERING MODELING AND SCIENTIFIC COMPUTING HUAZHONG UNIVERSITY OF SCIENCE AND TECHNOLOGY HUBEI WUHAN430074 China SCHOOL OF SCIENCE WUHAN UNIVERSITY OF TECHNOLOGY HUBEI WUHAN 430070 China SCHOOL OF SCIENCE HUBEI UNIVERSITY OF TECHNOLOGY HUBEI WUHAN 430068 China SCHOOL OF MATHEMATICS AND STATISTICS SOUTH-CENTRAL UNIVERSITY FOR NATIONALITIES WUHAN430074 China

The Cotter-Holm Slice Model (CHSM) was introduced to study the behavior of whether and specifically the formulation of atmospheric fronts, whose prediction is fundamental in meteorology. Considered herein is the influence of stochastic forcing on the Incompressible Slice Model (ISM) in a smooth 2D bounded domain, which can be derived by adapting the Lagrangian function in Hamilton’s principle for CHSM to the Euler-Boussinesq Eady incompressible case. First, we establish the existence and uniqueness of local pathwise solution (probability strong solution) to the ISM perturbed by nonlinear multiplicative stochastic forcing in Banach spaces Wk, p(D) with k > 1 + 1/ p and p ≥ 2. The solution is obtained by introducing suitable cut-off operators applied to the W1,∞-norm of the velocity and temperature fields, using the stochastic compactness method and the Yamada-Watanabe type argument based on the Gyöngy-Krylov characterization of convergence in probability. Then, when the ISM is perturbed by linear multiplicative stochastic forcing and the potential temperature does not vary linearly on the y-direction, we prove that the associated Cauchy problem admits a unique global-in-time pathwise solution with high probability, provided that the initial data is sufficiently small or the diffusion parameter is large enough. The results partially answer the problems left open in Alonso-Orán et al. (Physica D 392:99–118, 2019, pp. 117). Copyright © 2020, The Authors. All rights reserved.

关键词： Banach spaces