Simulations of the discrete Boltzmann Bhatnagar-Gross-Krook (BGK) equation are an important tool for understanding fluid dynamics in non-continuum regimes. Here, we introduce a discontinuous Galerkin finite element method (DG-FEM) for spatial discretization of the discrete Boltzmann equation for isothermal flows with Knudsen numbers (Kn~O(1)). In conjunction with a high-order Runge-Kutta time marching scheme, this method is capable of achieving high-order accuracy in both space and time, while maintaining a compact stencil. We validate the spatial order of accuracy of the scheme on a two-dimensional Couette flow with Kn = 1 and the D2Q16 velocity discretization. We then apply the scheme to lid-driven micro-cavity flow at Kn = 1, 2, and 8, and we compare the ability of Gauss-Hermite (GH) and Newton-Cotes (NC) velocity sets to capture the high non-linearity of the flow-field. While GH quadrature provides higher integration strength with fewer points, the NC quadrature has more uniformly distributed nodes with weights greater than machine-zero, helping to avoid the so-called ray-effect. Broadly speaking, we anticipate that the insights from this work will help facilitate the efficient implementation and application of high-order numerical methods for complex high Knudsen number flows.
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…
We analyze a large number of high-order discrete velocity models for solving the Boltzmann-BGK equation for finite Knudsen number flows. Using the Chapman-Enskog formalism, we prove for isothermal flows a relation identifying the resolved flow regimes for low Mach numbers. Although high-order lattice Boltzmann models recover flow regimes beyond the Navier-Stokes level we observe for several models significant deviations from reference results. We found this to be caused by their inability to recover the Maxwell boundary condition exactly. By using supplementary conditions for the gas-surface i…
A modified lattice Boltzmann model with a stochastic relaxation mechanism mimicking "virtual'' collisions between free-streaming particles and solid walls is introduced. This modified scheme permits to compute plane channel flows in satisfactory agreement with analytical results over a broad spectrum of Knudsen numbers, ranging from the hydrodynamic regime, all the way to quasi-free flow regimes up to Kn 30.
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…
A non-perturbative analysis of the Bhatnagar-Gross-Krook (BGK) model kinetic equation for finite values of the Knudsen number is presented. This analysis indicates why discrete kinetic versions of the BGK equation, and notably the Lattice Boltzmann method, can provide semi-quantitative results also in the non-hydrodynamic, finite-Knudsen regime, up to $Kn\sim {\cal O}(1)$. This may help the interpretation of recent Lattice Boltzmann simulations of microflows, which show satisfactory agreement with continuum kinetic theory in the moderate-Knudsen regime.
Porous structures of shales are reconstructed based on scanning electron microscopy (SEM) images of shale samples from Sichuan Basin, China. Characterization analyzes of the nanoscale reconstructed shales are performed, including porosity, pore size distribution, specific surface area and pore connectivity. The multiple-relaxation-time (MRT) lattice Boltzmann method (LBM) fluid flow model and single-relaxation-time (SRT) LBM diffusion model are adopted to simulate the fluid flow and Knudsen diffusion process within the reconstructed shales, respectively. Tortuosity, intrinsic permeability and …
In the current work we construct a multimolecule random process which leads to the Boltzmann equation in the appropriate limit, and which is different from the deterministic real gas dynamics process. We approximate the statistical difference between the two processes via a suitable diffusion process, which is obtained in the multiscale homogenization limit. The resulting Boltzmann equation acquires a new spatially diffusive term, which subsequently manifests in the corresponding fluid dynamics equations. We test the Navier-Stokes and Grad closures of the diffusive fluid dynamics equations in …
Size, morphology and distributions of pores in organic matters of shale matrix are discussed based on high resolution images from experiments in the literature. 150 nanoscale structures of the organic matters are then reconstructed by randomly placing pore spheres with different diameters and overlap tolerances. Effects of porosity, the mean diameter and the overlap tolerance on void space connectivity and pore size distribution are studied. Further, a pore-scale model based on the Lattice Boltzmann method is developed to predict the Knudsen diffusivity and permeability of the reconstructed or…
The state of a single-species monatomic gas from near-equilibrium to highly nonequilibrium conditions is investigated using analytical and numerical methods. Normal solutions of the Boltzmann equation for Fourier flow (uniform heat flux) and Couette flow (uniform shear stress) are found in terms of the heat-flux and shear-stress Knudsen numbers. Analytical solutions are found for inverse-power-law molecules from hard-sphere through Maxwell at small Knudsen numbers using Chapman-Enskog (CE) theory and for Maxwell molecules at finite Knudsen numbers using a moment-hierarchy (MH) method. Correspo…
Gas separation of a binary gaseous mixture is one of characteristic phenomena in the micro-scale flows that differ from the conventional size flows. In this work, the separation in a binary gas mixture flows through a microchannel is investigated by the lattice Boltzmann method with a diffuse-bounce-back (DBB) boundary condition. The separation degree and rate are measured in the He--Ar and Ne--Ar systems for different mole fractions, pressure ratios, and Knudsen numbers. The results show that the separation phenomenon in the He--Ar mixture is more obvious than that in the Ne--Ar mixture at th…
We present a new method in order to get variable slip coefficient in binary lattice Boltzmann models to simulate gaseous flows. Boundary layer theory is presented. We study both the single- and multi-fluid BGK-type models as well. The boundary slip and the Knudsen layer are analyzed in detail. Benchmark simulations are carried out in order to compare the analytical derivation with the numerical results. Excellent agreement is found between the two situations.
We present a further theoretical extension to the kinetic theory based formulation of the lattice Boltzmann method of Shan et al (2006). In addition to the higher order projection of the equilibrium distribution function and a sufficiently accurate Gauss-Hermite quadrature in the original formulation, a new regularization procedure is introduced in this paper. This procedure ensures a consistent order of accuracy control over the non-equilibrium contributions in the Galerkin sense. Using this formulation, we construct a specific lattice Boltzmann model that accurately incorporates up to the th…
With the aim of better understanding the numerical properties of the lattice Boltzmann method (LBM), a general methodology is proposed to derive its hydrodynamic limits in the discrete setting. It relies on a Taylor expansion in the limit of low Knudsen numbers. With a single asymptotic analysis, two kinds of deviations with the Navier-Stokes (NS) equations are explicitly evidenced: consistency errors, inherited from the kinetic description of the LBM, and numerical errors attributed to its space and time discretization. The methodology is applied to the Bhatnagar-Gross-Krook (BGK), the regula…
We present a multi-scale lattice Boltzmann scheme, which adaptively refines particles' velocity space. Different velocity sets, i.e., higher- and lower-order lattices, are consistently and efficiently coupled, allowing us to use the higher-order lattice only when and where needed. This includes regions of either high Mach number or high Knudsen number. The coupling procedure of different lattices consists of either projection of the moments of the higher-order lattice onto the lower-order lattice or lifting of the lower-order lattice to the higher-order velocity space. Both lifting and project…
Is it possible to solve Boltzmann-type kinetic equations using only a small number of particles velocities? We introduce a novel techniques of solving kinetic equations with (arbitrarily) large number of particle velocities using only a lattice Boltzmann method on standard, low-symmetry lattices. The renormalized kinetic equation is validated with the exact solution of the planar Couette flow at moderate Knudsen numbers.
The effects of volume exclusion and long-range intermolecular attraction are investigated by the simplified kinetic model for surface-confined inhomogeneous fluids. Gas dynamics of the ideal gas, the hard-sphere fluid and the real gas are simulated by the Boltzmann equation, the Enskog equation and the simple kinetic equation, respectively. Only the Knudsen minimum appears for the ideal gas, while both the Knudsen minimum and the Knudsen maximum occur for the hard-sphere fluid and the real gas under certain confinements, beyond which the maximum and minimum may disappear. The Boltzmann equatio…
In Denicol et al., Phys. Rev. D 85, 114047 (2012), the equations of motion of relativistic dissipative fluid dynamics were derived from the relativistic Boltzmann equation. These equations contain a multitude of terms of second order in Knudsen number, in inverse Reynolds number, or their product. Terms of second order in Knudsen number give rise to non-hyperbolic (and thus acausal) behavior and must be neglected in (numerical) solutions of relativistic dissipative fluid dynamics. The coefficients of the terms which are of the order of the product of Knudsen and inverse Reynolds numbers have b…
The perfect fluid limit can be obtained from the Boltzmann equation in the limit of vanishing Knudsen number. By treating the collision term in an implicit manner, the implicit-explicit (IMEX) time stepping scheme allows this limit to be achieved at finite values of the time step. We consider the 9th order monotonicity-preserving (MP-9) scheme to implement the advection, which is treated explicitly in the IMEX approach. We reduce the computational costs using reduced distribution functions, which also permits the adiabatic index to be varied. We validate the capabilities of our model by consid…
For more than 150 years the Navier-Stokes equations for thermodynamically quasi-equilibrium flows have been the cornerstone of modern computational fluid dynamics that underpins new fluid technologies. However, the applicable regime of the Navier-Stokes model in terms of the level of thermodynamic non-equilibrium in the local flowfield is not clear especially for hypersonic and low-speed micro/nano flows. Here, we re-visit the Navier-Stokes model in the framework of Boltzmann statistics, and propose a new and more appropriate way of assessing non-equilibrium in the local flowfield, and the cor…
The one-dimensional nonlinear equations for the blood flow motion in distensible vessels are considered using the kinetic approach. It is shown that the Lattice Boltzmann (LB) model for non-ideal gas is asymptotically equivalent to the blood flow equations for compliant vessels at the limit of low Knudsen numbers. The equations of state for non-ideal gas are transformed to the pressure-luminal area response. This property allows to model arbitrary pressure-luminal area relations. Several test problems are considered: the propagation of a sole nonlinear wave in an elastic vessel, the propagatio…
Exact solution to the hierarchy of nonlinear lattice Boltzmann (LB) kinetic equations in the stationary planar Couette flow is found at non-vanishing Knudsen numbers. A new method of solving LB kinetic equations which combines the method of moments with boundary conditions for populations enables to derive closed-form solutions for all higher-order moments. Convergence of results suggests that the LB hierarchy with larger velocity sets is the novel way to approximate kinetic theory.
In the CLB method, a Bosanquet-type effective viscosity is employed to account for the rarefaction effect on the gas viscosity. To gain accurate simulations and to match the Bosanquet-type effective viscosity, the combined bounce-back/specular-reflection boundary scheme with a modified second-order slip boundary condition is adopted in the CLB method. The present method is applied to study gas flow in a microchannel with periodic boundary condition and gas flow in a long microchannel (the ratio of the length to the height L/H=100) with pressure boundary condition over a wide range of Knudsen n…
The Chapman-Enskog expansion was used in the lattice Boltzmann method (LBM) to derive a Navier-Stokes-like equation and a formula was obtained to correlate the LBM model parameters to the kinematic viscosity implicitly implemented in LBM simulations. The obtained correlation formula usually works as long as the model parameters are carefully selected to make the Mach number and Knudsen number small although the validity of Chapman-Enskog expansion that has a formal definition of time derivative without tangible mathematical sense is not recognized by many mathematicians.
Many modern production and measurement facilities incorporate multiphase systems at low pressures. In this region of flows at small, non-zero Knudsen- and low Mach numbers the classical mesoscopic Monte Carlo methods become increasingly numerically costly. To increase the numerical efficiency of simulations hybrid models are promising. In this contribution, we propose a novel efficient simulation approach for the simulation of two phase flows with a large concentration imbalance in a low pressure environment in the low intermediate Knudsen regime. Our hybrid model comprises a lattice-Boltzmann…
This paper is a continuation of our earlier work [Z.L. Guo {\it et al.}, Phys. Rev. E {\bf 88}, 033305 (2013)] where a multiscale numerical scheme based on kinetic model was developed for low speed isothermal flows with arbitrary Knudsen numbers. In this work, a discrete unified gas-kinetic scheme (DUGKS) for compressible flows with the consideration of heat transfer and shock discontinuity is developed based on the Shakhov model with an adjustable Prandtl number. The method is an explicit finite-volume scheme where the transport and collision processes are coupled in the evaluation of the flu…
It is challenging to solve the Boltzmann equation accurately due to the extremely high dimensionality and nonlinearity. This paper addresses the idea and implementation of the first flux reconstruction method for high-order Boltzmann solutions. Based on the Lagrange interpolation and reconstruction, the kinetic upwind flux functions are solved simultaneously within physical and particle velocity space. The fast spectral method is incorporated to solve the full Boltzmann collision integral with a general collision kernel. The explicit singly diagonally implicit Runge-Kutta (ESDIRK) method is em…
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…
Finite-difference Lattice Boltzmann (LB) models are proposed for simulating gas flows in devices with microscale geometries. The models employ the roots of half-range Gauss-Hermite polynomials as discrete velocities. Unlike the standard LB velocity-space discretizations based on the roots of full-range Hermite polynomials, using the nodes of a quadrature defined in the half-space permits a consistent treatment of kinetic boundary conditions. The possibilities of the proposed LB models are illustrated by studying the one-dimensional Couette flow and the two-dimensional driven cavity flow. Numer…
In a recent paper by Lasseux, Valdés-Parada and Porter (J.~Fluid~Mech. \textbf{805} (2016) 118-146), it is found that the apparent gas permeability of the porous media is a nonlinear function of the Knudsen number. However, this result is highly questionable, because the adopted Navier-Stokes equations and the first-order velocity-slip boundary condition are first-order (in terms of the Knudsen number) approximations of the Boltzmann equation and the kinetic boundary condition for rarefied gas flows. Our numerical simulations based on the Bhatnagar-Gross-Krook kinetic equation and regularized …
We present a general framework for constructing trans-scale \emph{discrete Boltzmann models} (DBMs) for high-speed compressible flows ranging from continuum to transition regime. This is achieved by designing a higher-order discrete equilibrium distribution function which satisfies additional nonhydrodynamic kinetic moments. In order to characterize the \emph{% thermodynamic non-equilibrium} (TNE) effects and estimate the condition under which the DBMs at various levels should be used, two novel measures are presented: (i) the relative TNE strength, describing the relative strength of the ($N+…
Mathematically, the typical difference of Discrete Boltzmann Model (DBM) from the traditional hydrodynamic one is that the Navier-Stokes (NS) equations are replaced by a discrete Boltzmann equation. But physically, this replacement has a significant gain: a DBM is roughly equivalent to a hydrodynamic model supplemented by a coarse-grained model of the Thermodynamic Non-Equilibrium (TNE) effects, where the hydrodynamic model can be and can also beyond the NS. Via the DBM, it is convenient to perform simulations on systems with flexible Knudsen number. The observations on TNE are being obtaining…
A double-distribution-function based lattice Boltzmann method (DDF-LBM) is proposed for the simulation of polyatomic gases in the supersonic regime. The model relies on an extended equilibrium state that is constructed to reproduce the first 13 moments of the Maxwell-Boltzmann distribution exactly. This extends the validity of the standard 5-constraint (mass, momentum and energy) approach and leads to the correct simulation of thermal, compressible flows with only 39 discrete velocities in 3D. The stability of this BGK-LBM is reinforced by relying on Knudsen-number-dependent relaxation times t…
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…
Lattice Boltzmann method (LBM) has been applied to predict flow properties of porous media including intrinsic permeability, where it is implicitly assumed that the LBM is equivalent to the incompressible (or near incompressible) Navier-Stokes equation. However, in LBM simulations, high-order moments, which are completely neglected in the Navier-Stokes equation, are still available through particle distribution functions. To ensure that the LBM simulation is correctly working at the Navier-Stokes hydrodynamic level, the high-order moments have to be negligible. This requires that the Knudsen n…
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…
Conventional lattice Boltzmann models only satisfy moment isotropy up to fourth order. In order to accurately describe improtant physical effects beyond the isothermal Navier-Stokes fluid regime, higher order isotropy is required. In this paper, we present some basic results on moment isotropy and its relationship to the rotational symmetry of a generating discrete vector set. The anslysis provides a geometric understanding for popular lattice Boltzmann models, while offering a systematic procedure to construct higher order models.
Various experiments have found a boundary slip in hydrophobic microchannel flows, but a consistent understanding of the results is still lacking. While Molecular Dynamics (MD) simulations cannot reach the low shear rates and large system sizes of the experiments, it is often impossible to resolve the needed details with macroscopic approaches. We model the interaction between hydrophobic channel walls and a fluid by means of a multi-phase lattice Boltzmann model. Our mesoscopic approach overcomes the limitations of MD simulations and can reach the small flow velocities of known experiments. We…
Gas slippage occurs when the mean free path of the gas molecules is in the order of the characteristic pore size of a porous medium. This phenomenon leads to the Klinkenberg's effect where the measured permeability of a gas (apparent permeability) is higher than that of the liquid (intrinsic permeability). A generalized lattice Boltzmann model is proposed for flow through porous media that includes Klinkenberg's effect, which is based on the model of Guo et al. (Z.L. Guo et al., Phys.Rev.E 65, 046308 (2002)). The second-order Beskok and Karniadakis-Civan's correlation (A. Beskok and G. Karniad…
In this work, closure of the Boltzmann--BGK moment hierarchy is accomplished via projection of the distribution function $f$ onto a space $\mathbb{H}^{N}$ spanned by $N$-order Hermite polynomials. While successive order approximations retain an increasing number of leading-order moments of $f$, the presented procedure produces a hierarchy of (single) $N$-order partial-differential equations providing exact analytical description of the hydrodynamics rendered by ($N$-order) lattice Boltzmann--BGK (LBGK) simulation. Numerical analysis is performed with LBGK models and direct simulation Monte Car…
In this paper, computations of transient, incompressible, turbulent, plane jets using the discrete lattice BGK Boltzmann equation are reported. A priori derivation of the discrete lattice BGK Boltzmann equation with a spatially and temporally dependent relaxation time parameter, which is used to represent the averaged flow field, from its corresponding continuous form is given. The averaged behavior of the turbulence field is represented by the standard {\it k-$ε$} turbulence model and computed using a finite-volume scheme on non-uniform grids. Computed results are compared with analytical sol…
Thermodynamic nonequilibrium effects in three-dimensional high-speed compressible flows: Multiscale modeling and simulation via the discrete Boltzmann method
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…
The discrete Boltzmann equation for both the ideal and a non-ideal fluid is extended by adding Langevin noise terms in order to incorporate the effects of thermal fluctuations. After casting the fluctuating discrete Boltzmann equation in a form appropriate to the Onsager-Machlup theory of linear fluctuations, the statistical properties of the noise are determined by invoking a fluctuation-dissipation theorem at the kinetic level. By integrating the fluctuating discrete Boltzmann equation, the fluctuating lattice Boltzmann equation is obtained, which provides an efficient way to solve the equat…
High-resolution porous structures of catalyst layer (CL) with multicomponent in proton exchange membrane fuel cells are reconstructed using a reconstruction method called quartet structure generation set. Characterization analyses of nanoscale structures are implemented including pore size distribution, specific area and phase connectivity. Pore-scale simulation methods based on the lattice Boltzmann method are developed and used to predict the macroscopic transport properties including effective diffusivity and proton conductivity. Nonuniform distributions of ionomer in CL generates more tort…
Magnetron sputtering used for physical vapor deposition processes often requires gas pressures well below 1 Pa. Under these conditions the gas flow in the reactor is usually determined by a Knudsen number of about one, i.e., a transition regime between the hydrodynamic and the rarefied gas regime. In the first, the gas flow is well described by the Navier-Stokes equations, while in the second a kinetic approach via the Boltzmann equation is necessary. In this paper the neutral gas flow of argon and molecular nitrogen gas inside an industrial scale plasma reactor was simulated using both a flui…
A type of discrete Boltzmann model for simulating shallow water flows is derived by using the Hermite expansion approach. Through analytical analysis, we study the impact of truncating distribution function and discretizing particle velocity space. It is found that the convergence behavior of expansion is nontrivial while the conservation laws are naturally satisfied. Moreover, the balance of source terms and flux terms for steady solutions is not sacrificed. Further numerical validations show that the capability of simulating supercritical flows is enhanced by employing higher order expansion…
In the present work, the Tensor-Train decomposition algorithm is applied to reduce the memory footprint of a stochastic discrete velocity solver for rarefied gas dynamics simulation. An energy-conserving modification to the algorithm is proposed, along with an interleaved collision/convection routine which allows for easy application of higher-order convection schemes. The performance of the developed algorithm is analyzed for several 0- and 1-dimensional model problems in terms of solution error and reduction in memory use requirements.
The three-dimensional Couette flow between parallel plates is addressed using mixed lattice Boltzmann models which implement the half-range and the full-range Gauss-Hermite quadratures on the Cartesian axes perpendicular and parallel to the walls, respectively. The ability of our models to simulate rarefied flows are validated through comparison against previously reported results obtained using the linearized Boltzmann-BGK equation for values of the Knudsen number (Kn) up to $100$. We find that recovering the non-linear part of the velocity profile (i.e., its deviation from a linear function)…
A one-dimensional discrete Boltzmann model for detonation simulation is presented. Instead of numerical solving Navier-Stokes equations, this model obtains the information of flow field through numerical solving specially discretized Boltzmann equation. Several classical benchmarks including Sod shock wave tube, Colella explosion problem, and one-dimensional self-sustainable stable detonation are simulated to validate the new model. Based on the new model, the influence of negative temperature coefficient of reaction rate on detonation is further investigated. It is found that an abnormal deto…
Learning a generalized Navier-Stokes model beyond the continuum regime
CAPT, LMAM, and School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Beijing Computational Science Research Center（北京计算科学研究中心）；School of Mathematical Sciences, Peking University（北京大学数学科学学院）
