Lattice Boltzmann model for simulation of magnetohydrodynamics

The lattice Boltzmann equation describes the evolution of the velocity distribution function on a lattice in a manner that macroscopic fluid dynamical behavior is recovered. Although the equation is a derivative of lattice gas automata, it may be interpreted as a Lagrangian finite-difference method for the numerical simulation of the discrete-velocity Boltzmann equation that makes use of a BGK collision operator. As a result, it is not surprising that numerical instability of lattice Boltzmann methods have been frequently encountered by researchers. We present an analysis of the stability of perturbations of the particle populations linearized about equilibrium values corresponding to a constantdensity uniform mean flow. The linear stability depends on the following parameters: the distribution of the mass at a site between the different discrete speeds, the BGK relaxation time, the mean velocity, and the wavenumber of the perturbations. This parameter space is too large to compute the complete stability characteristics. We report some stability results for a subset of the parameter space for a 7-velocity hexagonal lattice, a 9-velocity square lattice and a 15-velocity cubic lattice. Results common to all three lattices are 1) the BGK relaxation time τ must be greater than 1 2 corresponding to positive shear viscosity, 2) there exists a maximum stable mean velocity for fixed values of the other parameters and 3) as τ is increased from 1 2 the maximum stable velocity increases monotonically until some fixed velocity is reached which does not change for larger τ .

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“…where α is a constant that determines the distribution of mass between the moving and nonmoving populations [23].…”

Section: Review Of Chapman-enskog Methodsmentioning

confidence: 99%

Stability Analysis of Lattice Boltzmann Methods

Sterling

Chen

1996

Journal of Computational Physics

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383

232

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“…The LBM is a kinetic-based computational fluid dynamics method that can be used to numerically model a vast array of physical systems including single-phase , multi-phase (Shan and Chen, 1993) and multi-component fluids (Arcidiacono et al, 2007), magnetohydrodynamic systems (Chen et al, 1991), oil-watersurfactant systems Nekovee et al, 2000) and reactive flow systems (Frouzakis, 2011), (see Chen and Doolen, 1998, for a comprehensive review of the LBM). It derives from the non-equilibrium Boltzmann equation:…”

Section: The Chemical Lattice Boltzmann Model Fluid Dynamicsmentioning

confidence: 99%

Emergence of Competition between Different Dissipative Structures for the Same Free Energy Source

Bartlett¹,

Bullock²

2015

07/20/2015-07/24/2015

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In this paper, we explore the emergence and direct interaction of two different types of dissipative structure in a single system: self-replicating chemical spot patterns and buoyancyinduced convection rolls. A new Lattice Boltzmann Model is developed, capable of simulating fluid flow, heat transport, and thermal chemical reactions, all within a simple, efficient framework. We report on a first set of simulations using this new model, wherein the Gray-Scott reaction diffusion system is embedded within a non isothermal fluid undergoing natural convection due to temperature gradients. The non-linear reaction which characterises the Gray-Scott system is given a temperature-dependent rate constant of the form of the Arrhenius equation. The enthalpy change (exothermic heat release or endothermic heat absorption) of the reaction can also be adjusted, allowing a direct coupling between the dynamics of the reaction and the thermal fluid flow. The simulations show positive feedback effects when the reaction is exothermic, but an intriguing, competitive and unstable behaviour occurs when the reaction is sufficiently endothermic. In fact when convection plumes emerge and grow, the reaction diffusion spots immediately surround them, since they require a source of heat for the reaction to proceed. Then however, the proliferation of spot patterns dampens the local temperature, eventually eliminating the initial convection plume and reducing the ability of the spots to persist. This behaviour appears almost ecological, similar as it is, to competitive interactions between organisms competing for the same nutrient source.

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“…as the Mach number Ma = |u|/c s tends to zero. Most recent work with lattice Boltzmann equations follows Chen et al [8] and Qian et al [29] in employing the Bhatnagar-Gross-Krook (BGK) collision operator [5], for which every variable relaxes towards equilibrium with the same timescale τ . The BGK approximation was originally seen as a simplification over previous lattice Boltzmann equations using first linearized forms of binary collision operators originating in lattice gas cellular automata, and then general linear operators constrained by symmetry and conservation properties [4,23,24].…”

Section: Introductionmentioning

confidence: 99%

Incompressible limits of lattice Boltzmann equations using multiple relaxation times

Dellar¹

2003

Journal of Computational Physics

159

150

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Lattice Boltzmann equations using multiple relaxation times are intended to be more stable than those using a single relaxation time. The additional relaxation times may be adjusted to suppress non-hydrodynamic modes that do not appear directly in the continuum equations, but may contribute to instabilities on the grid scale. If these relaxation times are fixed in lattice units, as in previous work, solutions computed on a given lattice are found to diverge in the incompressible (small Mach number) limit. This non-existence of an incompressible limit is analysed for an inclined one dimensional jet. An incompressible limit does exist if the nonhydrodynamic relaxation times are not fixed, but scaled by the Mach number in the same way as the hydrodynamic relaxation time that determines the viscosity.

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Lattice Boltzmann model for simulation of magnetohydrodynamics

Cited by 648 publications

References 11 publications

Stability Analysis of Lattice Boltzmann Methods

Stability Analysis of Lattice Boltzmann Methods

Emergence of Competition between Different Dissipative Structures for the Same Free Energy Source

Incompressible limits of lattice Boltzmann equations using multiple relaxation times

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