Symmetries, the current function, and exact solutions for Broadwell’s two-dimensional stationary kinetic model

Ilyin, Oleg

doi:10.1007/s11232-014-0170-1

Cited by 9 publications

(9 citation statements)

References 15 publications

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“…conserving nothing but mass, momentum and energy, can be found in [8]. In two dimensions, special classes of solutions to the Broadwell model are given in [7], [9] and [16]. The Broadwell model, not included in the present results, is a four-velocity model, with v 1 + v 2 = v 3 + v 4 = 0 and v 1 , v 3 orthogonal.…”

Section: Introductionmentioning

confidence: 79%

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Discrete velocity Boltzmann equation in the plane: stationary solutions

Arkeryd¹,

Nouri²

2021

Preprint

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The paper proves existence of stationary mild solutions for normal discrete velocity Boltzmann equations in the plane with no pair of colinear interacting velocities and given ingoing boundary values. An important restriction of all velocities pointing into the same half-space in a previous paper is removed in this paper. A key property is L 1 compactness of integrated collision frequency for a sequence of approximations. This is proven using the Kolmogorov-Riesz theorem, which here replaces the L 1 compactness of velocity averages in the continuous velocity case, not available when the velocities are discrete. .

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Section: Introductionmentioning

confidence: 79%

“…The passage to the limit when ǫ → 0 and ǫ ′ → 0 in (4. 16) results from the monotone convergence theorem, the family (F ǫ ) ǫ∈]0,1[ being non decreasing, with mass uniformly bounded, together with the mass of (χ…”

mentioning

confidence: 99%

Discrete velocity Boltzmann equation in the plane: stationary solutions

Arkeryd¹,

Nouri²

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…conserving nothing but mass, momentum and energy, can be found in [7]. In two dimensions, special classes of solutions to the Broadwell model are given in [8], [6], and [14]. The Broadwell model is a four-velocity model, with v 1 + v 2 = v 3 + v 4 = 0 and v 1 , v 3 orthogonal.…”

Section: Remark 11mentioning

confidence: 99%

Discrete velocity Boltzmann eqations in the plane:stationary solutions for a generic class

Arkeryd¹,

Nouri²

2021

Preprint

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The paper proves existence of renormalized stationary solutions for a dense class of discrete velocity Boltzmann equations in the plane with given ingoing boundary values. The proof is based on the construction of a sequence of approximations with L 1 compactness for the integrated collision frequency and gain term. Compactness is obtained using the Kolmogorov-Riesz theorem, which replaces the L 1 compactness of velocity averages in the continuous velocity case, not available when the velocities are discrete. .

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“…The DV Boltzmann approach attracted the attention of many researchers several decades ago, but, at present, is significantly less popular than the LB method. For instance, the well-known four velocity Broadwell equation in two dimensions has been investigated thoroughly [31][32][33][34][35][36], this model has correct collision invariants, but its discrete velocity set is too small and lacks isotropy [37]; therefore, the correct description of the hydrodynamics is impossible in the framework of this model. In addition, another subtle feature should be mentioned: for the discrete velocity models, the molecular chaos hypothesis can be violated, i.e., the particles can be correlated before the collision [38].…”

Section: Introductionmentioning

confidence: 99%

Discrete Velocity Boltzmann Model for Quasi-Incompressible Hydrodynamics

Ilyin

2021

Mathematics

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In this paper, we consider the development of the two-dimensional discrete velocity Boltzmann model on a nine-velocity lattice. Compared to the conventional lattice Boltzmann approach for the present model, the collision rules for the interacting particles are formulated explicitly. The collisions are tailored in such a way that mass, momentum and energy are conserved and the H-theorem is fulfilled. By applying the Chapman–Enskog expansion, we show that the model recovers quasi-incompressible hydrodynamic equations for small Mach number limit and we derive the closed expression for the viscosity, depending on the collision cross-sections. In addition, the numerical implementation of the model with the on-lattice streaming and local collision step is proposed. As test problems, the shear wave decay and Taylor–Green vortex are considered, and a comparison of the numerical simulations with the analytical solutions is presented.

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Symmetries, the current function, and exact solutions for Broadwell’s two-dimensional stationary kinetic model

Cited by 9 publications

References 15 publications

Discrete velocity Boltzmann equation in the plane: stationary solutions

Discrete velocity Boltzmann equation in the plane: stationary solutions

Discrete velocity Boltzmann eqations in the plane:stationary solutions for a generic class

Discrete Velocity Boltzmann Model for Quasi-Incompressible Hydrodynamics

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