Numerical Analysis of the Lattice Boltzmann Method for the Boussinesq Equations
- PDF / 384,865 Bytes
- 21 Pages / 439.37 x 666.142 pts Page_size
- 28 Downloads / 197 Views
Numerical Analysis of the Lattice Boltzmann Method for the Boussinesq Equations Wen-An Yong1 · Weifeng Zhao2 Received: 6 January 2019 / Revised: 5 June 2020 / Accepted: 26 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This paper is concerned with the lattice Boltzmann method (LBM) with BGK collision models for the two-dimensional Bousinessq equations on periodic domains. We show the numerical stability of the LBM linearized at a quiescent state and conduct a formal asymptotic analysis which indicates the consistency of the method to the Boussinesq equations. With these, we establish the convergence of the LBM for the nonlinear equations. Moreover, our present analysis provides some important hints on how to construct initial data and how to add force terms in the LBM. It can be straightforwardly extended to other LB models or three-dimensional cases. Keywords Lattice Boltzmann method · Boussinesq equations · Stability structure · Asymptotic analysis · Convergence Mathematics Subject Classification 65M12 · 76M28 · 35Q30
1 Introduction During the last two decades, the lattice Boltzmann method (LBM) has been developed into an effective and viable tool for simulating various flow problems. It has been proved to be quite successful in simulating complex fluid flows such as turbulent flows, micro-flows, multi-phase and multi-component flows, particulate suspensions, and interfacial dynamics. Consequently, the LBM is becoming a serious alternative to traditional computational fluid methods. We refer to [1–4] for a comprehensive account of the method and its applications.
B
Weifeng Zhao [email protected] Wen-An Yong [email protected]
1
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, People’s Republic of China
2
Department of Applied Mathematics, University of Science and Technology Beijing, Beijing 100083, People’s Republic of China 0123456789().: V,-vol
123
36
Page 2 of 21
Journal of Scientific Computing
(2020) 84:36
Among various applications, the coupling of flows and diffusion is typical. Examples include electro-osmotic flows [5], multi-phase flows [6] and thermal convective flows [7– 9], which are widely encountered in nature and industrial applications. In these literature, double distributions are adopted to solve the Navier–Stokes equations (for flow fields) and the convection diffusion equations (for diffusions), respectively. In this way, the coupling is easily reflected in the evolution of the two distributions and benchmark quality results agreeing well with those of the conventional numerical methods can be obtained [9]. The goal of this work is to investigate the stability and convergence of the above LBM with double distributions. For the sake of definiteness, we will focus on the method for the thermal convective flow governed by the Boussinesq equations ∇ · u = 0, ∂u + u · ∇u + ∇ p = ν∇ 2 u + F(u, T ), (1.1) ∂t ∂T + u · ∇T = κ∇ 2 T ∂t on the domain [0, T ] × . Here ∇ is the usual gradient operator, u is the velocity
Data Loading...
