We propose an extension to recently developed Relativistic Lattice Boltzmann solvers (RLBM), which allows the simulation of flows close to the free streaming limit. Following previous works [Phys. Rev. C 98 (2018) 035201], we use product quadrature rules and select weights and nodes by separately discretising the radial and the angular components. This procedure facilitates the development of quadrature-based RLBM with increased isotropy levels, thus improving the accuracy of the method for the simulation of flows beyond the hydrodynamic regime. In order to quantify the improvement of this discretisation procedure over existing methods, we perform numerical tests of shock waves in one and two spatial dimensions in various kinetic regimes across the hydrodynamic and the free-streaming limits.
The system of hydrodynamic-type equations is derived from Alexeev's generalized Boltzmann kinetic equation by two-side distribution function for a stratified gas in gravity field. It is applied to a problem of ultrasound propagation and attenuation. The linearized version of the obtained system is studied and compared with the Navier-Stokes one at arbitrary Knudsen numbers. The problem of a generation by a moving plane in a rarefied gas is explored and used as a test while compared with experiment. It is good agreement between predicted propagation speed, attenuation factor and experimental re…
Rooted from the gas kinetics, the lattice Boltzmann method is a powerful tool in modeling hydrodynamics. In the past decade, it has been extended to simulate the rarefied gas flow beyond the Navier-Stokes level, either by using the high-order Gauss-Hermite quadrature, or by introducing the relaxation time that is a function of the gas-wall distance. While the former method, with a limited number of discrete velocities (i.e. D2Q36), is accurate up to the early transition flow regime, the latter method, with the same discrete velocities as that used in simulating hydrodynamics (i.e. D2Q9), is ac…
In this work, we have theoretically analyzed and numerically evaluated the accuracy of high-order lattice Boltzmann (LB) models for capturing non-equilibrium effects in rarefied gas flows. In the incompressible limit, the LB equation is proved to be equivalent to the linearized Bhatnagar-Gross-Krook (BGK) equation. Therefore, when the same Gauss-Hermite quadrature is used, LB method closely assembles the discrete velocity method (DVM). In addition, the order of Hermite expansion for the equilibrium distribution function is found not to be correlated with the approximation order in terms of the…
We present a systematic procedure for the construction of relativistic lattice Boltzmann models (R-SLB) appropriate for the simulation of flows of massless particles. Quadrature rules are used for the discretization of the momentum space in spherical coordinates. The models are optimized for one-dimensional flows. The applications considered in this paper are the Sod shock tube and the one-dimensional boost invariant expansion (Bjorken flow). Our models are tested against exact solutions in the inviscid and ballistic limits. At intermediate relaxation times (finite viscosity), we compare with …
A kinetic model is proposed for rarefied flows of molecular gas with rotational and temperature-dependent vibrational degrees of freedom. The model reduces to the Boltzmann equation for monatomic gas when the energy exchange between the translational and internal modes is absent, thus the influence of intermolecular potential can be captured. Moreover, not only the transport coefficients but also their fundamental relaxation processes are recovered. The accuracy of our kinetic model is validated by the direct simulation Monte Carlo method in several rarefied gas flows, including the shock wave…
In this paper, we employ the lattice Boltzmann method to solve the Boltzmann equation with the Shakhov model for the collision integral in the context of the 3D planar Couette flow. The half-range Gauss-Hermite quadrature is used to account for the wall-induced discontinuity in the distribution function. The lattice Boltzmann simulation results are compared with direct simulation Monte Carlo (DSMC) results for ${}^3{\rm He}$ and ${}^4{\rm He}$ atoms interacting via ab initio potentials, at various values of the rarefaction parameter $δ$, where the temperature of the plates varies from $1\ {\rm…
For multiscale gas flows, kinetic-continuum hybrid method is usually used to balance the computational accuracy and efficiency. However, the kinetic-continuum coupling is not straightforward since the coupled methods are based on different theoretical frameworks. In particular, it is not easy to recover the non-equilibrium information required by the kinetic method which is lost by the continuum model at the coupling interface. Therefore, we present a multiscale lattice Boltzmann (LB) method which deploys high-order LB models in highly rarefied flow regions and low-order ones in less rarefied …
A kinetic model called the $ν$-model is proposed to replace the complicated Boltzmann collision operator in the simulation of rarefied flows of monatomic gas. The model follows the relaxation-time approximation, but the collision frequency (i.e, inverse relaxation time) is a function of the molecular velocity to reflect part of the collision details of the Boltzmann equation, and the target velocity distribution function (VDF) to which the VDF relaxes is close to that used in the Shakhov model. Based on the numerical simulation of strong non-equilibrium shock waves, a half-theoretical and half…
We present the formulation of a kinetic mapping scheme between the Direct Simulation Monte Carlo (DSMC) and the Lattice Boltzmann Method (LBM) which is at the basis of the hybrid model used to couple the two methods in view of efficiently and accurately simulate isothermal flows characterized by variable rarefaction effects. Owing to the kinetic nature of the LBM, the procedure we propose ensures to accurately couple DSMC and LBM at a larger Kn number than usually done in traditional hybrid DSMC-Navier-Stokes equation models. We show the main steps of the mapping algorithm and illustrate detai…
We study a two phase flow with interactions of liquid and rarefied gas inside the gas phase. The gas phase is modeled by the BGK model of the Boltzmann equation. The liquid phase is modeled by the incompressible Navier-Stokes equations. In this paper we have excluded the heat transfer between two phases. The interface boundary conditions for the liquid and gas phases is presented. The BGK model is solved by a semi-Lagrangian scheme with a meshfree reconstruction since the droplet moves inside the gas phase and its interface deforms with respect to time and space. Therefore, the similar meshfre…
A system of fluid-dynamic-type equations and their boundary conditions derived from a system of the Boltzmann equation is of great importance in kinetic theory when we are concerned with the motion of a slightly rarefied gas. It offers an efficient alternative to solving the Boltzmann equation directly and, more importantly, provides a clear picture of the flow structure in the near-continuum regime. However, the applicability of the existing slip-flow theory is limited to the case where both the boundary shape and the kinetic boundary condition are smooth functions of the boundary coordinates…
An efficient third-order discrete unified gas kinetic scheme (DUGKS) with efficiency is presented in this work for simulating continuum and rarefied flows. By employing two-stage time-stepping scheme and the high-order DUGKS flux reconstruction strategy, third-order of accuracy in both time and space can be achieved in the present method. It is also analytically proven that the second-order DUGKS is a special case of the present method. Compared with the high-order lattice Boltzmann equation {LBE} based methods, the present method is capable to deal with the rarefied flows by adopting the Newt…
Efficient modeling of rarefied flow has drawn widespread interest for practical engineering applications. In the present work, we proposed the Grad's distribution function for 13 moments-based moment gas kinetic solver (G13-MGKS) and the macroscopic governing equations are derived based on the moment integral of discrete Boltzmann equation in the finite volume framework. Numerical fluxes at the cell interface related to the macroscopic variables, stress and heat flux can be reconstructed from the Boltzmann integration equation at surrounding points of the cell interface directly, so the compli…
A high-performance gas kinetic solver using multi-level parallelization is developed to enable pore-scale simulations of rarefied flows in porous media. The Boltzmann model equation is solved by the discrete velocity method with an iterative scheme. The multi-level MPI/OpenMP parallelization is implemented with the aim to efficiently utilise the computational resources to allow direct simulation of rarefied gas flows in porous media based on digital rock images for the first time. The multi-level parallel approach is analyzed in details confirming its better performance than the commonly-used …
The transverse force exerted on a rotating sphere immersed in an otherwise uniform flow of a rarefied gas is investigated based on the Bhatnagar-Gross-Krook (BGK) model of the Boltzmann equation assuming the Maxwell boundary condition on the sphere. In several existing studies, it has been shown that the transverse force acting on the sphere, also known as the Magnus force, has opposite signs in the free molecular and continuum flows. The present study intends to clarify the force's transition in terms of the Knudsen number (i.e., the reciprocal ratio of the sphere radius to the molecular mean…
A rigorous asymptotic analysis of the Boltzmann equation for small Knudsen numbers leads, in the general case, to more complicated sets of differential equations than widely used to describe the behavior of gas in terms of classical fluid dynamics. The present paper deals with such one that is valid for significant temperature variations and finite Reynolds numbers at the same time (slow nonisothermal flow equations). A finite-volume solver developed on the open-source CFD platform OpenFOAM is proposed for computer simulation of a slightly rarefied gas in an arbitrary geometry. Typical tempera…
We introduce a numerical solver for the spatially inhomogeneous Boltzmann equation using the Burnett spectral method. The modelling and discretization of the collision operator are based on the previous work [Z. Cai, Y. Fan, and Y. Wang, Burnett spectral method for the spatially homogeneous Boltzmann equation, arXiv:1810.07804], which is the hybridization of the BGK operator for higher moments and the quadratic collision operator for lower moments. To ensure the preservation of the equilibrium state, we introduce an additional term to the discrete collision operator, which equals zero when the…
An implicit high-order discontinuous Galerkin (DG) method is developed to find steady-state solution of rarefied gas flow described by the Boltzmann equation with full collision operator. In the physical space, velocity distribution function is approximated by the piecewise polynomials of degree up to 4, while in the velocity space the fast spectral method is incorporated into the DG to evaluate the collision operator. A specific polynomial approximation for the collision operator is proposed to reduce the computational complexity of the fast spectral method by $K$ times, where for two-dimensi…
The non-equilibrium gas dynamics is described by the Boltzmann equation, which can be solved numerically through the deterministic and stochastic methods. Due to the complicated collision term of the Boltzmann equation, many kinetic relaxation models have been proposed and used in the past seventy years for the study of rarefied flow. In order to develop a multiscale method for the rarefied and continuum flow simulation, by adopting the integral solution of the kinetic model equation a DVM-type unified gas-kinetic scheme (UGKS) has been constructed. The UGKS models the gas dynamics on the cell…
The radiometric force on a sphere due to its thermal polarization in a rarefied gas flow being in equilibrium is investigated on the basis of a kinetic model to the linearized Boltzmann equation. The scattering kernel proposed by Cercignani and Lampis to model the gas-surface interaction using two accommodation coefficients, namely the tangential momentum accommodation coefficient and the normal energy accommodation coefficient, is employed as the boundary condition. The radiometric force on the sphere, as well as the flow field of the gas around it, are calculated in a wide range of the gas r…
One of the central problems in the study of rarefied gas dynamics is to find the steady-state solution of the Boltzmann equation quickly. When the Knudsen number is large, i.e. the system is highly rarefied, the conventional iteration scheme can lead to convergence within a few iterations. However, when the Knudsen number is small, i.e. the flow falls in the near-continuum regime, hundreds of thousands iterations are needed, and yet the "converged" solutions are prone to be contaminated by accumulated error and large numerical dissipation. Recently, based on the gas kinetic models, the implici…
In this study, a phase-field lattice Boltzmann model based on the Allen-Cahn equation with a filtered collision operator and high-order corrections in the equilibrium distribution functions is presented. Here we show that in addition to producing numerical results consistent with prior numerical methods, analytic solutions, and experiments with the density ratio of 1000, previous numerical deficiencies are resolved. Specifically, the new model is characterized by robustness at low viscosity, accurate prediction of shear stress at interfaces, and removal of artificial dense bubbles and rarefied…
In this paper, the original discrete unified gas kinetic scheme (DUGKS) is extended to arbitrary Lagrangian-Eulerian (ALE) framework for simulating the low-speed continuum and rarefied flows with moving boundaries. For ALE method, the mesh moving velocity is introduced into the Boltzmann-BGK equation. The remapping-free scheme is adopted to develop the present ALE-type DUGKS, which avoids the complex rezoning and remapping process in traditional ALE method. As in some application areas, the large discretization errors will be introduced into the simulation if the geometric conservation is not …
The rarefied effect of gas flow in microchannel is significant and cannot be well described by traditional hydrodynamic models. It has been know that discrete Boltzmann model (DBM) has the potential to investigate flows in a relatively wider range of Knudsen number because of its intrinsic kinetic nature inherited from Boltzmann equation. It is crucial to have a proper kinetic boundary condition for DBM to capture the velocity slip and the flow characteristics in the Knudsen layer. In this paper, we present a DBM combined with Maxwell-type boundary condition model for slip flow. The tangential…
In the frame of the Boltzmann equation, wall-bounded flows of rarefied gases require the implementation of boundary conditions at the kinetic level. Such boundary conditions induce a discontinuity in the distribution function with respect to the component of the momentum which is normal to the boundary. Expanding the distribution function with respect to half-range polynomials allows this discontinuity to be captured. The implementation of this concept has been reported in the literature only for force-free flows. In the case of general forces which can have non-zero components in the directio…
In rarefied gas flows, the spatial grid size could vary by several orders of magnitude in a single flow configuration (e.g., inside the Knudsen layer it is at the order of mean free path of gas molecules, while in the bulk region it is at a much larger hydrodynamic scale). Therefore, efficient implicit numerical method is urgently needed for time-dependent problems. However, the integro-differential nature of gas kinetic equations poses a grand challenge, as the gain part of the collision operator is non-invertible. Hence an iterative solver is required in each time step, which usually takes a…
Here we introduce new coarse-grained variables for a turbulent flow in the form of moments of its Reynolds-averaged Boltzmann equation. With the exception of the collision moments, the transport equations for the new variables are identical to the usual moment equations, and thus naturally lend themselves to the variety of already existing closure methods. Under the anelastic turbulence approximation, we derive equations for the Reynolds-averaged turbulent fluctuations around the coarse-grained state. We show that the global relative entropy of the coarse-grained state is bounded from above by…
The general synthetic iteration scheme (GSIS) is extended to find the steady-state solution of nonlinear gas kinetic equation, removing the long-standing problems of slow convergence and requirement of ultra-fine grids in near-continuum flows. The key ingredients of GSIS are that the gas kinetic equation and macroscopic synthetic equations are tightly coupled, and the constitutive relations in macroscopic synthetic equations explicitly contain Newton's law of shear stress and Fourier's law of heat conduction. The higher-order constitutive relations describing rarefaction effects are calculated…
Generalizing the collision term in the relativistic Boltzmann equation to include nonlocal effects, and using Grad's 14-moment approximation for the single-particle distribution function, we derive evolution equations for the relativistic dissipative fluid dynamics and compare them with the corresponding equations obtained in the standard Israel-Stewart and related approaches. Significance of this generalization on hydrodynamic evolution is demonstrated in the framework of one-dimensional scaling expansion.
In this paper, we consider the capabilities of the Boltzmann equation with the Shakhov and ellipsoidal models for the collision term to capture the characteristics of rarefied gas flows. The benchmark is performed by comparing the results obtained using these kinetic model equations with direct simulation Monte Carlo (DSMC) results for particles interacting via ab initio potentials. The analysis is restricted to channel flows between parallel plates and we consider three flow problems, namely: the heat transfer between stationary plates, the Couette flow and the heat transfer under shear. The …
A quadrature-based finite-difference lattice Boltzmann model is developed that is suitable for simulating relativistic flows of massless particles. We briefly review the relativistc Boltzmann equation and present our model. The quadrature is constructed such that the stress-energy tensor is obtained as a second order moment of the distribution function. The results obtained with our model are presented for a particular instance of the Riemann problem (the Sod shock tube). We show that the model is able to accurately capture the behavior across the whole domain of relaxation times, from the hyd…
The relativistic Boltzmann equation for a single particle species generally implies a fixed, unchangeable equation of state that corresponds to that of an ideal gas. Real-world systems typically have more complicated equation of state which cannot be described by the Boltzmann equation. The present work derives a 'Boltzmann-like' equation that gives rise to a conserved energy-momentum tensor with an arbitrary (but thermodynamically consistent) equation of state. Using this, a Lattice Boltzmann scheme for diagonal metric tensors and arbitrary equations of state is constructed. The scheme is ver…
We present new analytic solutions to the relativistic Boltzmann equation within the relaxation time approximation. We first obtain spherically expanding solutions which are the kinetic counterparts of the exact solutions of the Israel-Stewart equation in the literature. This allows us to compare the solutions of the kinetic and hydrodynamic equations at an analytical level. We then derive a novel boost-invariant solution of the Boltzmann equation which has an unconventional dependence on the proper time. The existence of such a solution is also suggested in second order hydrodynamics and fluid…
Using a Lattice Boltzmann hydrodynamic computational modeler to simulate relativistic fluid systems we explore turbulence in two-dimensional relativistic flows. We first a give a pedagogical description of the phenomenon of turbulence and its characteristics in a two-dimensional system. The classical Lattice Boltzmann Method and its extension to relativistic fluid systems is then described. The model is tested against a system incorporating a random stirring force in k-space and then applied to a realistic sample of graphene. Part I: We investigate the relativistic adaptation of the Lattice Bo…
