Parametrization of the cumulant lattice Boltzmann method for fourth order accurate diffusion part I: Derivation and validation

Geier, Martin; Pasquali, Andrea; Schönherr, Martin

doi:10.1016/j.jcp.2017.05.040

Cited by 97 publications

(103 citation statements)

References 54 publications

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“…According to Refs. [56,59,60], the K-LBM cannot recover the behavior of the BGK-LBM. This point will be further studied in Sec.…”

Section: F Cumulant Spacementioning

confidence: 99%

“…This is explained by the fact that the only error introduced by the numerical discretization comes from the collision term [58,132]. Hence, one can show that the numerical behavior of LBMs is drastically impacted by the choice of both the moment space and the relaxation parameters [29,56,59,61]. Nevertheless, these errors should not be attributed to a physical problem, such as the Galilean invariance issue, but rather to a purely numerical defect, in order not to mislead the reader on this particularly complex topic.…”

Section: Macroscopic Equationsmentioning

confidence: 99%

“…Nonetheless, it was done in an extremely reduced configuration where the diffusive scaling was adopted, thus reducing the validity of the comparison to flows governed by the incompressible Navier-Stokes equations [57,58]. Hence, no information regarding the generation and propagation of acoustic waves can be deduced from either their work or from those that followed [59][60][61]. More surprisingly, no comparison with central moments was provided for the most complicated case, i.e, the study of the flow around a sphere.…”

Section: Introductionmentioning

confidence: 99%

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Comprehensive comparison of collision models in the lattice Boltzmann framework: Theoretical investigations

2019

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Over the last decades, several types of collision models have been proposed to extend the validity domain of the lattice Boltzmann method (LBM), each of them being introduced in its own formalism. The present article proposes a formalism that describes all these methods within a common mathematical framework, and in this way allows us to draw direct links between them. Here, the focus is put on single and multirelaxation time collision models in either their raw moment, central moment, cumulant or regularized form. In parallel with that, several bases (non orthogonal, orthogonal, Hermite) are considered for the polynomial expansion of populations. General relationships between moments are first derived to understand how moment spaces are related to each other. In addition, a review of collision models further sheds light on collision models that can be rewritten in a linear matrix form. More quantitative mathematical studies are then carried out by comparing explicit expressions for the post collision populations. Thanks to this, it is possible to deduce the impact of both the polynomial basis (raw, Hermite, central, central Hermite, cumulant) and the inclusion of regularization steps on isothermal LBMs. Extensive results are provided for the D1Q3, D2Q9, and D3Q27 lattices, the latter being further extended to the D3Q19 velocity discretization. Links with the most common two and multirelaxation time collision models are also provided for the sake of completeness. The present work ends by emphasizing the importance of an accurate representation of the equilibrium state, independently of the choice of moment space. As an addition to the theoretical purpose of the present article, general instructions are provided to help the reader with the implementation of the most complicated collision models.

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“…According to Refs. [56,59,60], the K-LBM cannot recover the behavior of the BGK-LBM. This point will be further studied in Sec.…”

Section: F Cumulant Spacementioning

confidence: 99%

Section: Macroscopic Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Comprehensive comparison of collision models in the lattice Boltzmann framework: Theoretical investigations

2019

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“…The initial condition for the velocity field for this flow problem is given as

\begin{matrix} lefttrue \\ \frac{u}{u_{0}} & = \frac{L_{0}}{L} \\ \frac{v}{u_{0}} & = \frac{L_{0}}{L} \sin (\frac{2 π x}{L}) \cos (\frac{4 π z}{3 L}) \\ \frac{w}{u_{0}} & = 0, \end{matrix}

and the exact solution is known as

\begin{matrix} lefttrue \\ \frac{u (t)}{u_{0}} & = \frac{L_{0}}{L} \\ \frac{v (t)}{u_{0}} & = \frac{L_{0}}{L} \sin (\frac{2 π ((), x + u_{0} t)}{L}) \cos (\frac{4 π z}{3 L}) e^{- ν italict ({((), \frac{2 π}{L})}^{2} + {((), \frac{4 π}{3 L})}^{2})} \\ \frac{w (t)}{u_{0}} & = 0 . \end{matrix}

In Equations and , u 0 and L 0 are the reference velocity and length scales, respectively, and ν is the kinematic viscosity. Geier et al have studied this test case to demonstrate the accuracy and efficiency of different LBMs, and in particular, they have used the fourth‐order LBM based on a standard local stencil. Herein, the decay rate of the double shear wave is studied for the two values of the kinematic viscosities ν = 10 −2 and 10 −4 with u 0 = 0.01 and L 0 = 1.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…, u 0 and L 0 are the reference velocity and length scales, respectively, and is the kinematic viscosity. Geier et al have studied this test case10,[30][31][32] to demonstrate the accuracy and efficiency of different LBMs, and in particular, they have used the fourth-order LBM based on a standard local stencil. Herein, the decay rate of the double shear wave is studied for the two values of the kinematic viscosities =10 −2 and 10 −4 with u 0 = 0.01 and L 0 = 1.…”

mentioning

confidence: 99%

Simulation of three‐dimensional incompressible flows in generalized curvilinear coordinates using a high‐order compact finite‐difference lattice Boltzmann method

Ezzatneshan

Hejranfar

2018

Numerical Methods in Fluids

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Summary In the present study, a high‐order compact finite‐difference lattice Boltzmann method is applied for accurately computing 3‐D incompressible flows in the generalized curvilinear coordinates to handle practical and realistic geometries with curved boundaries and nonuniform grids. The incompressible form of the 3‐D nineteen discrete velocity lattice Boltzmann method is transformed into the generalized curvilinear coordinates. Herein, a fourth‐order compact finite‐difference scheme and a fourth‐order Runge‐Kutta scheme are used for the discretization of the spatial derivatives and the temporal term, respectively, in the resulting 3‐D nineteen discrete velocity lattice Boltzmann equation to provide an accurate 3‐D incompressible flow solver. A high‐order spectral‐type low‐pass compact filtering technique is applied to have a stable solution. All boundary conditions are implemented based on the solution of the governing equations in the 3‐D generalized curvilinear coordinates. Numerical solutions of different 3‐D benchmark and practical incompressible flow problems are performed to demonstrate the accuracy and performance of the solution methodology presented. Herein, the 2‐D cylindrical Couette flow, the decay of a 3‐D double shear wave, the cubic lid‐driven cavity flow with nonuniform grids, the flow through a square duct with 90° bend and the flow past a sphere at different flow conditions are considered for validating the present computations. Numerical results obtained show the accuracy and robustness of the present solution methodology based on the implementation of the high‐order compact finite‐difference lattice Boltzman method in the generalized curvilinear coordinates for solving 3‐D incompressible flows over practical and realistic geometries.

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Periodic hill flow simulations with a parameterized cumulant lattice Boltzmann method

Gehrke

Rung

2022

Numerical Methods in Fluids

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The article is concerned with the assessment of a cumulant lattice Boltzmann method in wall‐bounded, separated turbulent shear flows. The approach is of interest for its resolution‐spanning success in turbulent channel flows without using a specific turbulence treatment. The assessment focuses upon the flow over a periodic hill, which offers a rich basis of numerical and experimental data, for Reynolds numbers within 700≤Re≤37,000. The analysis involves the mean flow field, second moments and their invariants, as well as spectral data obtained for a wide range of resolutions with 2≲Δxi+≲100. With the emphasis on a recently published parameterized cumulant collision operator, the universality of the value assigned to a stability preserving regularization parameter is assessed. Reported results guide resolution‐dependent optimized values and indicate a required minimum resolution of Δxi+≈30. Analog to the findings for attached turbulent shear flows, the approach appears to adequately resolve complex turbulent flows without the need for ad hoc modeling for a range of scale resolving resolutions.

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Parametrization of the cumulant lattice Boltzmann method for fourth order accurate diffusion part I: Derivation and validation

Cited by 97 publications

References 54 publications

Comprehensive comparison of collision models in the lattice Boltzmann framework: Theoretical investigations

Comprehensive comparison of collision models in the lattice Boltzmann framework: Theoretical investigations

Simulation of three‐dimensional incompressible flows in generalized curvilinear coordinates using a high‐order compact finite‐difference lattice Boltzmann method

Periodic hill flow simulations with a parameterized cumulant lattice Boltzmann method

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