Asymptotic analysis of the lattice Boltzmann equation

Junk, Michael; Klar, Axel; Luo, Li‐Shi

doi:10.1016/j.jcp.2005.05.003

Cited by 328 publications

(263 citation statements)

References 47 publications

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“…This can be verified with help of a Chapman-Enskog expansion or other asymptotic analyses [19,20]. Note that the arguments ( x n , t s ) were suppressed in (4) for better readability.…”

Section: The Lattice Boltzmann Methodsmentioning

confidence: 80%

Discrete Artificial Boundary Conditions for the Lattice Boltzmann Method in 2D

2015

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Abstract. To confine a spatial domain to a smaller computational domain, one needs artificial boundaries. This work considers the lattice Boltzmann method and deals with boundary conditions for these open boundaries. Ideally, such a condition does not interact with the fluid at all. We present novel two-dimensional discrete artificial boundary conditions to pursue that goal and we discuss four different versions. This type of condition is formulated on the discrete lattice Boltzmann level and does not require a PDE formulation of the fluid. We set a special focus on the D2Q9 model. Our numerical results compare the novel discrete artificial boundary conditions to simulations using the existing non-reflecting characteristic boundary condition and an exit boundary condition.

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“…This can be verified with help of a Chapman-Enskog expansion or other asymptotic analyses [19,20]. Note that the arguments ( x n , t s ) were suppressed in (4) for better readability.…”

Section: The Lattice Boltzmann Methodsmentioning

confidence: 80%

Discrete Artificial Boundary Conditions for the Lattice Boltzmann Method in 2D

2015

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“…The consistency of the LBM with regard to the Navier-Stokes equations (NSE) has been established through various methods in the literature [1,2,3,4,5,6,7,8] and has been applied to many moving boundary simulations [9,10] because of its computational efficiency, simplicity and scalable parallel nature. In this work, the cascaded lattice Boltzmann method (CLBM), recently introduced by Geier et al, [11,12], is used for the fluid flow simulation due to its superior stability properties and higher degree of Galilean invariance over other lattice Boltzmann schemes [13,14,15,16,17,18].…”

Section: Introductionmentioning

confidence: 99%

PROTEUS: A coupled iterative force-correction immersed-boundary cascaded lattice Boltzmann solver for moving and deformable boundary applications

Falagkaris

Ingram

Markakis

et al. 2018

Computers & Mathematics with Applications

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Many realistic fluid flow problems are characterised by high Reynolds numbers and complex moving or deformable geometries. In our previous study, we presented a novel coupling between an iterative force-correction immersed boundary and a multi-domain cascaded lattice Boltzmann method, Falagkaris et al., and investigated flows around rigid bodies at Reynolds numbers up to 10 5 . Here, we extend its application to flows around moving and deformable bodies with prescribed motions. Emphasis is given on the influence of the internal mass on the computation of the aerodynamic forces including deforming boundary applications where the rigid body approximation is no longer valid. Both the rigid body and the internal Lagrangian points approximations are examined. The resulting solver has been applied to viscous flows around an in-line oscillating cylinder, a pitching foil, a plunging SD7003 airfoil and a plunging and flapping NACA-0014 airfoil. Good agreement with experimental results and other numerical schemes has been obtained. It is shown that the internal Lagrangian points approximation accurately captures the internal mass effects in linear and angular motions, as well as in deforming motions, at Reynolds numbers up to 4 · 10 4 . In all cases, the aerodynamic loads are significantly affected by the internal fluid forces.

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“…In [27,28], a Taylor expansion is applied to the LBM scheme to justify it in the case of the 1D convection-diffusion equation and in the case of the 1D wave equation with diffusive term. In [29], a Chapman-Enskog expansion is applied to a LBM scheme in the case of the incompressible Navier-Stokes system. In [30,31], a convergence result in L 2 is proposed for LBM schemes in the case of the incompressible Navier-Stokes system under some assumptions and a stability result in L 2 is obtained by linearizing the LBM schemes.…”

Section: Introductionmentioning

confidence: 99%

“…u = 0), one of these LBM schemes is classical and can be found in [4,6,7,32] (the second one seems to be less classical). As in [29], these LBM schemes are obtained by discretizing a discrete velocity kinetic system whose the fluid limit is (1), this fluid limit being formally obtained with a Chapman-Enskog expansion and with a Hilbert expansion. Then, we prove convergence results in L ∞ and discrete maximum principles when u = 0 satisfied by the proposed LBM schemes with periodic, Neumann or Dirichlet boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

“…Thus, M ε q will be named maxwellian in the sequel, ρ(x) and u(x) being the macroscopic density and velocity associated to M ε q . A similar approach is proposed in [29] in the case of the incompressible Navier-Stokes system. Nevertheless, the integration of (5) is obtained with a second order integration formula instead of a third order integration formula, which obliges the authors in [29] to correct the diffusion coefficient ν to obtain a consistent LBM scheme: this point is a direct consequence of the stiffness of (5) when ε ≪ 1, and is clarified in §3.…”

Section: Introductionmentioning

confidence: 99%

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Construction and Analysis of Lattice Boltzmann Methods Applied to a 1D Convection-Diffusion Equation

Dellacherie

2013

Acta Appl Math

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We construct and we analyze two LBM schemes build on the D1Q2 lattice to solve the 1D (linear) convectiondiffusion equation. We obtain these LBM schemes by showing that the 1D convection-diffusion equation is the fluid limit of a discrete velocity kinetic system. Then, we show in the periodic case that these LBM schemes are equivalent to a finite difference type scheme named LFCCDF scheme. This allows us, firstly, to prove the convergence in L ∞ of these schemes, and to obtain discrete maximum principles for any time step in the case of the 1D diffusion equation with different boundary conditions. Secondly, this allows us to obtain most of these results for the Du Fort-Frankel scheme for a particular choice of the first iterate. We also underline that these LBM schemes can be applied to the (linear) advection equation and we obtain a stability result in L ∞ under a classical CFL condition. Moreover, by proposing a probabilistic interpretation of these LBM schemes, we also obtain Monte-Carlo algorithms which approach the 1D (linear) diffusion equation. At last, we present numerical applications justifying these results.

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Asymptotic analysis of the lattice Boltzmann equation

Cited by 328 publications

References 47 publications

Discrete Artificial Boundary Conditions for the Lattice Boltzmann Method in 2D

Discrete Artificial Boundary Conditions for the Lattice Boltzmann Method in 2D

PROTEUS: A coupled iterative force-correction immersed-boundary cascaded lattice Boltzmann solver for moving and deformable boundary applications

Construction and Analysis of Lattice Boltzmann Methods Applied to a 1D Convection-Diffusion Equation

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