Kinetic models in $n$-dimensional Euclidean spaces: From the Maxwellian to the Poisson kernel

In this work, minimal kinetic theories based on unconventional entropy functions, $H\sim \text{ln}f$ (Burg entropy) for 2D and $H\sim f^{1-\frac{2}{n}}$ (Tsallis entropy) for $n\text{D}$ with $n\ge 3$, are studied. These entropy functions were originally derived by Boghosian et al. [Phys. Rev. E 68, 025103 (2003)] as a basis for discrete-velocity and lattice Boltzmann models for incompressible fluid dynamics. The present paper extends the entropic models of Boghosian et al. and shows that the explicit form of the equilibrium distribution function (EDF) of their models, in the continuous-velocity limit, can be identified with the Poisson kernel of the Poisson integral formula. The conservation and Navier-Stokes equations are recovered at low Mach numbers, and it is shown that rest particles can be used to rectify the speed of sound of the extended models. Fourier series expansion of the EDF is used to evaluate the discretization errors of the model. It is shown that the expansion coefficients of the Fourier series coincide with the velocity moments of the model. Employing two-, three-, and four-dimensional (2D, 3D, and 4D) complex systems, the real velocity space is mapped into the hypercomplex spaces and it is shown that the velocity moments can be evaluated, using the Poisson integral formula, in the hypercomplex space. For the practical applications, a 3D projection of the 4D model is presented, and the existence of an $H$ theorem for the discrete model is investigated. The theoretical results have been verified by simulating the following benchmark problems: (1) the Kelvin-Helmholtz instability of thin shear layers in a doubly periodic domain and (2) the 3D flow of incompressible fluid in a lid-driven cubic cavity. The present results are in agreement with the previous works, while they show better stability of the proposed kinetic model, as compared with the BGK type (with single relaxation time) lattice Boltzmann models.

Figure 1

Building a rectangular doubly periodic region using a hexagonal lattice base.

Figure 2

Vorticity fields using the standard $d2q9$ on $G_2$ and for $\text{Mach}=0.04$. (Left) $T=1$ and $\text{Re}=10000$; (right) $T=0.45$ and $\text{Re}=30000$.

Figure 3

Vorticity fields at $T=1$ using the present $d2z6$ on $G_1$ and for $\text{Mach}=0.04$. (Left) $\text{Re}=10,000$; (right) $\text{Re}=30000$.

Figure 4

Vorticity fields at $T=1$ using the present $d2z7$ on $G_1$ and for $\text{Mach}=0.04$. (Left) $\text{Re}=10000$; (right) $\text{Re}=30000$.

Figure 5

The geometry of the problem.

Figure 6

Convergence rate (in space) of the results: $\text{Re}=100$; $\text{Ma}=0.05$; single precision; using $G_1\text{−}{G}_4$. (Left) Basic model ($d3z18$); (right) modified model ($d3z19$).

Figure 7

Centreline profiles: using $G_3$; $\text{Re}=1000$, $\text{Ma}=0.05$; double precision; simulations with the basic and modified models ($d3z18$ and $d3z19$).

Figure 8

Pseudostreamlines at midsections parallel to the $XY$ and $YZ$ planes, at $\text{Re}=1000$, $\text{Ma}=0.05$, and using double precision on grid $G_3$.