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We present a kinetic model for nonideal fluids, where the local thermodynamic pressure is imposed through appropriate rescaling of the particle's velocities, accounting for both long- and short-range effects and hence full thermodynamic consistency. The model features full Galilean invariance together with mass, momentum, and energy conservation and enables simulations ranging from subcritical to supercritical flows, which is illustrated on various benchmark flows such as anomalous shock waves or shock droplet interaction. © 2020 American Physical Society. ISSN:1539-3755 ISSN:1063-651X ISSN:1095-3787 ISSN:1550-2376

Compressible lattice Boltzmann model on standard lattices [M. H. Saadat, F. Bösch, and I. V. Karlin, Phys. Rev. E 99, 013306 (2019).] is extended to deal with complex flows on unstructured grid. Semi-Lagrangian propagation [A. Krämer et al., Phys. Rev. E 95, 023305 (2017).] is performed on an unstructured second-order accurate finite-element mesh and a consistent wall boundary condition is implemented which makes it possible to simulate compressible flows over complex geometries. The model is validated through simulation of Sod shock tube, subsonic and supersonic flow over NACA0012 airfoil and shock-vortex interaction in Schardin's problem. Numerical results demonstrate that the present model on standard lattices is able to simulate compressible flows involving shock waves on unstructured meshes with good accuracy and without using any artificial dissipation or limiter., Physical Review E, 101 (2), ISSN:1539-3755, ISSN:1063-651X, ISSN:1095-3787, ISSN:1550-2376

We present a lattice Boltzmann model (LBM) that covers the entire range of fluid flows, from low Mach weakly compressible to transonic and supersonic flows. One of the most restrictive limitations of the lattice Boltzmann method, the low Mach number limit, is overcome here by three fundamental changes to the LBM scheme: use of an appropriately chosen multispeed lattice, accurate evaluation of the equilibrium, and the entropic relaxation for the collision. The range of applications is demonstrated through the simulation of a bow shock in front of an airfoil and the simulation of decaying compressible turbulence with shocklets.

An energy-conserving lattice Boltzmann (LB) model based on the entropic theory of admissible higher-order lattice is presented in detail. The entropy supporting 'zero-one-three" lattice is used to construct a model capable of reproducing the full Fourier-Navier-Stokes equations at low Mach numbers. The proposed direct approach of constructing thermal models overcomes the shortcomings of existing models and retains one of the most important advantages of the LB methods, the exact space discretization of the advection step, thus paving the way for direct numerical simulation of thermal flows. New thermal wall boundary condition capable of handling curved geometries immersed in a multispeed lattice is proposed by extending the Tamm-Mott-Smith boundary condition. Entropic realization of the current model ensures stability of the model also for subgrid simulations. Numerical validation and thermodynamic consistency is demonstrated with classical setups such as thermal Couette flow, Rayleigh-Bénard natural convection, acoustic waves, speed of sound measurements, and shock tube simulations.

For a one-dimensional benchmark shock tube problem, we implement the lattice Boltzmann method based on the H theorem [I. Karlin, A. Ferrante, and H. C. Ottinger, Europhys. Lett. 47, 182 (1999)]. Results of simulation demonstrate significant improvement of stability, as compared to realizations without explicit entropic estimations.

We develop a systematic method of constructing equilibria for kinetic models with discrete microscopic velocities. The approach is based on a suitable entropy maximum principle. The $H$ theorem is demonstrated in the continuous and discrete space-time realizations. In addition, we discuss an extension of the Lattice Boltzmann method to irregular grids.

Another question comes from the approximate character of the Grad equations, and is discussed in frames of the EIT: while the Grad equations are strictly hyperbolic at any level N ~i.e., predicting a finite speed of propagation!, will this feature will be preserved in the further corrections? These two questions are special cases of a more general one, namely, how does one derive a closed description with a given number of moments? Such a description is sometimes called mesoscopic @7# since it occupies an intermediate level between the hydrodynamic ~macroscopic! and the kinetic ~microscopic! levels of description. In this paper we aim at deriving the mesoscopic dynamics of thirteen moments in the simplest case when the kinetic description satisfies the linearized Boltzmann equation. Our approach will be based on the two assumptions: ~i! The mesoscopic dynamics of thirteen moments exists, and is invariant with respect to the microscopic dynamics. ~ii! The 13M Grad approximation is a suitable first approximation to this mesoscopic dynamics. The assumption ~i! is realized as the invariance equation for the ~unknown! mesoscopic distribution function. Following the assumption ~ii!, we solve the invariance equation iteratively, taking the 13M Grad approximation for the input approximation, and consider the first iteration~further we refer to this as to the dynamic correction, to distinguish from constructing another ansatz! .W e demonstrate that the correction results in the exact CE transport coefficients. We also demonstrate how the dynamic correction modifies the hyperbolicity of the Grad equations. A similar viewpoint on derivation of hydrodynamics was earlier developed in @8 #~ we will return to a comparison below!. First, we review the Boltzmann equation and the 13M Grad approximation. We denote as n0 , u050, and p0 the equilibrium values of the hydrodynamic parameters (n is the number density, u is the average velocity, and p5nkBT is the pressure!. The global Maxwell distribution function F is

Invariance of nonhydrodynamic variables is put forward as a working principle of extending hydrodynamics into a highly nonequilibrium domain. Following this principle, the leading modification of the viscosity due to the gradient of the average velocity is derived explicitly from nonlinear moment Grad equations [Commun. Pure Appl. Math. 2, 331 (1949)].

Derivation of macroscopic equations from the simplest dumbbell models is revisited. It is demonstrated that the onset of the macroscopic description is sensitive to the flows. For Peterlin's approximation [Makromol. Chem. 338, 44 (1961)] to Warner's finitely extensible nonlinear elastic spring-force model [Ind. Eng. Chem. Fundam. 11, 379 (1972)] (FENE-P), small deviations from the Gaussian solution undergo a slow relaxation before the macroscopic description sets on. Some consequences of these observations are discussed.

An isothermal model on the standard two-dimension nine-velocity lattice (D2Q9) is proposed and analyzed. It originates from the thermal model with energy conservation introduced by N. I. Prasianakis and I. V. Karlin [Phys. Rev. E 76, 016702 (2007)]. The isothermal and the thermal equivalent models are tested through the simulation of the decay of a shear wave and of a temperature wave. Both are shown to be Galilean invariant, reference temperature independent, and rotational isotropic through the measurement of the transport coefficients on a rotated moving frame of reference.

A lattice Boltzmann model is developed to simulate finite-rate catalytic surface chemistry. Diffusive wall boundary conditions are established to account for catalytic reactions in multicomponent mixtures. Implementation of wall boundary conditions with chemical reactions is based on a general second-order accurate interpolation scheme. Results of lattice Boltzmann simulations for a four-component mixture with a global catalytic methane oxidation reaction in a straight channel are in excellent agreement with a finite volume Navier-Stokes solver in terms of both the flow field and species concentrations.

Recently, analytical solutions for the nonlinear Couette flow demonstrated the relevance of the lattice Boltzmann (LB) models to hydrodynamics beyond the continuum limit [S. Ansumali, Phys. Rev. Lett. 98, 124502 (2007)]. In this paper, we present a systematic study of the simplest LB kinetic equation-the nine-bit model in two dimensions--in order to quantify it as a slip flow approximation. Details of the aforementioned analytical solution are presented, and results are extended to include a general shear- and force-driven unidirectional flow in confined geometry. Exact solutions for the velocity, as well as for pertinent higher-order moments of the distribution functions, are obtained in both Couette and Poiseuille steady-state flows for all values of rarefaction parameter (Knudsen number). Results are compared with the slip flow solution by Cercignani, and a good quantitative agreement is found for both flow situations. Thus, the standard nine-bit LB model is characterized as a valid and self-consistent slip flow model for simulations beyond the Navier-Stokes approximation.

The lattice Boltzmann method is a highly promising approach to the simulation of complex flows. Here, we realize recently proposed multispeed lattice Boltzmann models [S. Chikatamarla et al., Phys. Rev. Lett. 97 190601 (2006)] by exploiting the flexibility offered by off-lattice Boltzmann methods. The approach is based on the general characteristic-based algorithm for off-lattice Boltzmann simulations that preserves all appealing properties of the standard lattice Boltzmann method while extending the method to unstructured grids. We show that the use of multispeed models indeed gives rise to major improvements in accuracy. The suggested approach thus renders truly large-scale off-lattice Boltzmann computations practical.

A lattice Boltzmann (LB) model for the simulation of realistic multicomponent mixtures is constructed. In the hydrodynamic limit, the LB model recovers the equations of continuum mechanics within the mixture-averaged diffusion approximation. The present implementation can be used to simulate realistic mixtures with arbitrary Schmidt numbers and molecular masses of the species. The model is applied to the mixing of two opposed jets of different concentrations and the results are in excellent agreement with a continuum model. An application to the simulation of mixtures in microflows is also presented. Results compare well with existing kinetic theory predictions of the slip coefficient for mixtures in a Couette flow.

An alternative approach, the kinetically reduced local Navier-Stokes (KRLNS) equations for the grand potential and the momentum, is proposed for the simulation of low Mach number flows. The Taylor-Green vortex flow is considered in the KRLNS framework, and compared to the results of the direct numerical simulation of the incompressible Navier-Stokes equations. The excellent agreement between the KRLNS equations and the incompressible nonlocal Navier-Stokes equations for this nontrivial time-dependent flow indicates that the former is a viable alternative for computational fluid dynamics at low Mach numbers.

We demonstrate that the requirement of galilean invariance determines the choice of H function for a wide class of entropic lattice Boltzmann models for the incompressible Navier-Stokes equations. The required H function has the form of the Burg entropy for D=2, and of a Tsallis entropy with q=1-2/D for D>2, where D is the number of spatial dimensions. We use this observation to construct a fully explicit, unconditionally stable, galilean invariant, lattice-Boltzmann model for the incompressible Navier-Stokes equations, for which attainable Reynolds number is limited only by grid resolution., 4 pages

We revisit recent derivations of kinetic equations based on Tsallis' entropy concept. The method of kinetic functions is introduced as a standard tool for extensions of classical kinetic equations in the framework of Tsallis' statistical mechanics. Our analysis of the Boltzmann equation demonstrates a remarkable relation between thermodynamics and kinetics caused by the deformation of macroscopic observables.

A general method of constructing dissipative equations is developed, following Ehrenfest's idea of coarse graining. The approach resolves the major issue of discrete time coarse graining versus continuous time macroscopic equations. Proof of the H theorem for macroscopic equations is given, several examples supporting the construction are presented, and generalizations are suggested.

In Ref. 1 , we derived a correction to the Navier-Stokes expression for the stress , in the one-dimensional case, for large values of the average velocity u . This correction has the form R(g) xu , where g xu is the longitudinal rate. The viscosity factor R(g) is a solution of a differential equation, subject to a certain initial condition Eq. 1 of Ref. 2 . This equation was not studied completely in Ref. 1 . In their Comment, Uribe and Pina indicated some interesting features of this equation. In particular, they asked what happens to the relevant solution at negative values of g? Let us denote the points of the (g ,R) plane as P (g ,R). The relevant solution R(g) emerges from point P0 (0, 4 3 ), and, as we demonstrate below, it can be unequally continued to arbitrary values of g , positive and negative. This solution can be constructed, for example, with the Taylor expansion used in Ref. 2 , and which is identical to the relevant subseries of the Chapman-Enskog expansion cf. Ref. 1 , Eq. 8 . However, the difficulty in constructing this solution numerically for g 0 originates from the fact that the same point P0 is the point of essential singularity of other irrelevant solutions to Eq. 1 . Indeed, for g 1, let us consider R(g) R(g) , where R(g) 4 3 8 9 ( 2)g is the relevant solution for small g , and (g) is a deviation. Neglecting in Eq. 1 all regular terms of the order g), and also neglecting g in comparison to , we derive the following equation: (1 )g(d /dg) 3 2 . The solution is (g) (g0)exp a(g 1 g0 ) , where a (3/2)(1 ) . The essential singularity at g 0 is apparent from this solution, unless (g0) 0 that is, no singularity exists except for the relevant solution R R). Let (g0) 0. If g 0, then →0, together with all its derivatives, as g→0. If g 0, the solution expands, as g→0. The complete picture for 1 is as follows: The lines g 0 and P (g ,g ) define the boundaries of the domain of attraction A A A , where A P g 0,R g 1 , and A P g 0,R g 1 . The graph G „g ,R(g)... of the relevant solution belongs to the closure of A , and goes through the points P0 (0, 4 3 ), P ( ,0), and P ( ,0). These points at the boundaries of A are the points of essential singularity of any other irrelevant solution with the initial conditions P A , and P A G . That is, if P A , P A G , the solution expands at P0, and is attracted to P . If P A , and P A G , the solution expands at P , and is attracted to P0. It is this latter case that was found numerically by Uribe and Pina. The above consideration is supported by our independent numerical study of Eq. 1 see Fig. 1, corresponding to the case of hard spheres, 1 2 ). The difficulty of numerical integration from P0 to negative values of g is quite clear: the integration then goes in a direction opposite to the direction of attraction of the irrelevant solutions. However, the same feature becomes an advantage for the integration to the positive values of g: because of the attraction of all irrelevant solutions to the relevant one, roundoff errors will be suppressed. This explains why no difficulty was encountered in this part of integration in Ref. 2 . One can utilize the attraction in the negative domain in