Derivation of Hydrodynamics for Multi-Relaxation Time Lattice Boltzmann using the Moment Approach

Kaehler, Goetz; Wagner, A. James

doi:10.4208/cicp.451011.260112s

Cited by 20 publications

(17 citation statements)

References 30 publications

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“…T ai f eq i = 0 if T a is a ghost eigenvector. This emphasizes the physical interpretation of the model, but is otherwise not necessary for a faithful implementation of hydrodynamics [43]. It is often convenient to imagine the f i to be the elements of vector in the space spanned by the velocity vectors c i .…”

Section: Discrete Velocity Boltzmann Equationmentioning

confidence: 99%

Langevin theory of fluctuations in the discrete Boltzmann equation

Groß¹,

Cates²,

Varnik³

et al. 2011

J. Stat. Mech.

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The discrete Boltzmann equation for both the ideal and a non-ideal fluid is extended by adding Langevin noise terms in order to incorporate the effects of thermal fluctuations. After casting the fluctuating discrete Boltzmann equation in a form appropriate to the Onsager-Machlup theory of linear fluctuations, the statistical properties of the noise are determined by invoking a fluctuationdissipation theorem at the kinetic level. By integrating the fluctuating discrete Boltzmann equation, a fluctuating lattice Boltzmann equation is obtained, which provides an efficient way to solve the equations of fluctuating hydrodynamics for ideal and non-ideal fluids. Application of the framework to a generic force-based non-ideal fluid model leads to ideal gas-type thermal noise. Simulation results indicate proper thermalization of all degrees of freedom.fluid at all length scales, not only the largest ones. Notably, the general theory of LB-Langevin equations was also derived in an early account by [17].Recently, the Langevin approach of [15,17], originally devised for the ideal gas, has been generalized to describe thermal fluctuations in the LBE for a class of non-ideal fluids [18]. There, non-ideal fluid models based on a squaregradient (Ginzburg-Landau) free energy functionals were considered, implying that density fluctuations in such a fluid are spatially correlated with a correlation length which is proportional to the theoretical width of the diffuse liquid-vapor interface. This is in contrast to the ideal gas, where equilibrium correlations are generally absent for all degrees of freedom. Clearly, the non-trivial structure factor of a non-ideal fluid has to be faithfully represented by the correlations of the LB population densities. Using results of continuum kinetic theory, a general ansatz for these correlations has been proposed, noticing, however, that certain models might require modifications to account for implementation-specific details [18]. This was, in particular, found to be the case for the model of Swift et al. [19], where, owing to the underlying modified equilibrium distribution, a spatially correlated form of thermal noise arises.In the present paper, we show how thermal noise can be incorporated into both ideal and non-ideal fluid versions of the discrete-velocity Boltzmann equation (DBE). The DBE is a precursor of the LBE that arises from the continuum Boltzmann equation by restricting velocity space to a finite number of velocities, while keeping position space and time continuous [20,21]. The fluctuating DBE (FDBE) is put forward here as a unifying starting point to construct fluctuating LB models and as a theoretical link between two successful frameworks of non-equilibrium physics: the LB method and the theory of linear regression of fluctuations due to Onsager and Machlup [22,23]. The utility of the FDBE is illustrated by re-deriving the fluctuating LBEs of the ideal gas [15] and the modified-equilibrium model [18]. Moreover, based on the model of He, Shan and Doolen [24], we propose a FDBE ...

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Section: Discrete Velocity Boltzmann Equationmentioning

confidence: 99%

Langevin theory of fluctuations in the discrete Boltzmann equation

Groß¹,

Cates²,

Varnik³

et al. 2011

J. Stat. Mech.

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“…The eigenvectors of the collision matrix in these approaches differ from the current ones only by a factor and additions of conserved eigenvectors. Kaehler et al [24] showed that this ensures that the collision terms are equivalent. For the equilibrium distribution in moment space we obtain…”

Section: Lattice Boltzmann Methodsmentioning

confidence: 99%

“…This derivation follows the approach developed by Kaehler et al [24]. We write the lattice Boltzmann eqn.…”

Section: Hydrodynamic Limitmentioning

confidence: 99%

Fluctuating lattice Boltzmann method for the diffusion equation

Wagner

Strand

2016

Phys. Rev. E

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“…The correspondence between the LBE and the macroscopic behavior that it simulates can be shown through different approaches. The standard procedure is the Chapman-Enskog analysis, and one alternative is the recursive substitution developed by Wagner [25] and further developed by Holdych et al [26] and Kaehler and Wagner [27]. Up to second order terms, both procedures result in the same behavior, and it is not known if differences at higher orders will occur.…”

Section: A the Lattice Boltzmann Equationmentioning

confidence: 99%

Shaping the equation of state to improve numerical accuracy and stability of the pseudopotential lattice Boltzmann method

Czelusniak,

Mapelli,

Cabezas-Gómez

et al. 2021

Preprint

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Recently it was discovered that altering the shape of the meta stable and unstable branches of an equation of state (EOS) can greatly improve the numerical accuracy of liquid and gas densities in the pseudopotential method. Inspired by this approach we develop an improved approach that is benchmarked for both equilibrium and non-equilibrium situations. We show here that the original approach reduces the method stability in non-equilibrium situations. Here we propose a new procedure to replace the metastable and unstable regions of these EOS by alternative functions. Our approach does not affects the coexistence densities or the speed of sound of the liquid phase while maintaining continuity of the sound speed in the pressure-density curve. Using this approach we were able to reduce the relative error of the planar interface vapor density compared to the thermodynamic consistent value by increasing the vapor phase sound speed. To allow for the benchmarking of dynamic results we also developed a finite difference method (FD) that solves the same macroscopic conservation equation as the pseudopotential lattice Boltzmann method (LBM). With this FD scheme we are able to perform mesh refinement and obtain reference solutions for the dynamic tests. We observed excellent agreement between the FD solutions and our proposed scheme. We also performed a detailed study of the stability of the methods using simulations of a droplet impacting on a liquid film for reduced temperatures down to 0.35 with Reynolds number of 300. Our approach remains stable for a density ratio up to 3.38 • 10 4 .

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Derivation of Hydrodynamics for Multi-Relaxation Time Lattice Boltzmann using the Moment Approach

Cited by 20 publications

References 30 publications

Langevin theory of fluctuations in the discrete Boltzmann equation

Langevin theory of fluctuations in the discrete Boltzmann equation

Fluctuating lattice Boltzmann method for the diffusion equation

Shaping the equation of state to improve numerical accuracy and stability of the pseudopotential lattice Boltzmann method

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