Cross Correlators and Galilean Invariance in Fluctuating Ideal Gas Lattice Boltzmann Simulations

Kaehler, Goetz; Wagner, A. James

doi:10.4208/cicp.151109.161110s

Cited by 4 publications

(5 citation statements)

References 15 publications

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“…However, we are not aware of any studies investigating these issues in detail for the FDBE or FLBE at the present time. We remark that violation of Galilean invariance appears to limit the application of the FLBE to systems with rather small mean flow velocities [54]. In this work, we will be only concerned with fluctuations around equilibrium states.…”

Section: Fluctuating 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: Fluctuating 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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“…( 22) therefore qualifies the requirements on the transforms in addition to the necessity that they preserve hydrodynamics. An extensive study on the second condition has been published in [9]. There we found that the Hermite norm of Eq.…”

Section: Lattice Boltzmann Simulation Of a Fluctuating Ideal Gasmentioning

confidence: 76%

“…The result is that the MRT orthogonality condition employed is identical to what is typically known as the Hermite norm [8]. This approximation, as we first discussed in [9] and show later in this paper, formally introduces non-Galilean invariant terms.…”

Section: Introductionmentioning

confidence: 88%

“…Of course one has also to show that this is also consistent with identifying the M a with the hydrodynamic moments. For a discussion of this see [9]. Now that it has been established that the moment fluctuations can be decoupled according to Eq.…”

Section: Lattice Boltzmann Simulation Of a Fluctuating Ideal Gasmentioning

confidence: 99%

“…This new process does not alter the hydrodynamic limit of the lattice Boltzmann method because we only alter the moments multiples of u(r) with the conserved quantity eigenvectors of the density 1 i and momentum modes v iα . If we interpret the local velocity u(r) as an arbitrary constant we do not alter the hydrodynamic equations at all by virtue of our discussion in [9]. We will refer to Eq.…”

Section: Local Velocity Dependent Transformsmentioning

confidence: 99%

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A fluctuating lattice-Boltzmann method with improved Galilean invariance

Kaehler,

Wagner

2012

Preprint

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In this paper we show that standard implementations of fluctuating Lattice Boltzmann methods do not obey Galilean invariance at a fundamental level. In trying to remedy this we are led to a novel kind of multi-relaxation time lattice Boltzmann methods where the collision matrix depends on the local velocity. This new method is conceptually elegant but numerically inefficient. With a small numerical trick, however, this method recovers nearly the original efficiency and allows the practical implementation of fluctuating lattice Boltzmann methods with significantly improved Galilean invariance. This will be important for applications of fluctuating lattice Boltzmann for non-equilibrium systems involving strong flow fields.

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