Bhatnagar Gross Krook Model Research Articles

Rheology of a dilute binary mixture of inertial suspension under simple shear flow

Abstract The rheology of a dilute binary mixture of inertial suspension under simple shear flow is analyzed in the context of the Boltzmann kinetic equation. The effect of the surrounding viscous gas on the solid particles is accounted for by means of a deterministic viscous drag force plus a stochastic Langevin-like term defined in terms of the environmental temperature Tenv. Grad’s moment method is employed to determine the temperature ratio and the pressure tensor in terms of the coefficients of restitution, concentration, the masses and diameters of the components of the mixture, and the environmental temperature. Analytical results are compared against event-driven Langevin simulations for mixtures of hard spheres with the same mass density m1/m2 = (σ(1)/σ(2))3, mi and σ(1) being the mass and diameter, respectively, of the species i. It is confirmed that the theoretical predictions agree with simulations of various size ratios σ(1)/σ(2) and for elastic and inelastic collisions in a wide range of parameter space. It is remarkable that the temperature ratio T1/T2 and the viscosity ratio η1/η2 (ηi being the partial contribution of the species i to the total shear viscosity η = η1 + η2) discontinuously change at a certain shear rate as the size ratio increases; this feature (which is expected to occur in the thermodynamic limit) cannot be completely captured by simulations due to the small system size. In addition, a Bhatnagar–Gross–Krook (BGK)-type kinetic model adapted to mixtures of inelastic hard spheres is exactly solved when Tenv is much smaller than the kinetic temperature T. A comparison between the velocity distribution functions obtained from Grad’s method, the BGK model, and simulations is carried out.

A phase-field-based multiple-relaxation-time lattice Boltzmann method for incompressible multiphase flows with density and viscosity contrasts

In this work, we present a unified framework of phase-field-based multiple-relaxation-time lattice Boltzmann (MRT-LB) method for incompressible multiphase flows with density and viscosity contrasts where a block-lower-triangular relaxation matrix is introduced. The present framework includes the classic MRT-LB model and central-moments-based LB model (CLBM, a particular version of the MRT-LB model). The governing equations of incompressible multiphase flows can be reproduced precisely through the direct Taylor expansion method at the second-order of expansion parameters. In this work, the pressure p is re-estimated, and we prove that the present pressure expression is suitable for classic MRT model, CLBM and various lattice structures with an accuracy of O(Δt2+ΔtMa2). In addition, the model construction and analysis are all carried out in the velocity space. As a result, the present model has a similar structure to the traditional Bhatnagar-Gross-Krook (BGK) model. Therefore the process of their analysis and implementation is also similar. We then test the present model by five classic physical problems. The results show that the present model is found to be accurate and has better numerical stability over the BGK model, and has the ability to simulate multiphase flow problems with large density ratio (1000).

Non-equilibrium flow of van der Waals fluids in nano-channels

The Enskog–Vlasov equation provides a consistent description of the microscopic molecular interactions for real fluids based on the kinetic and mean-field theories. The fluid flows in nano-channels are investigated by the Bhatnagar–Gross–Krook (BGK) type Enskog–Vlasov model, which simplifies the complicated Enskog–Vlasov collision operator and enables large-scale engineering design simulations. The density distributions of real fluids are found to exhibit inhomogeneities across the nano-channel, particularly at large densities, as a direct consequence of the inhomogeneous force distributions caused by the real fluid effects including the fluid molecules' volume exclusion and the long-range molecular attraction. In contrast to the Navier–Stokes equation with the slip boundary condition, which fails to describe nano-scale flows due to the coexistence of confinement, non-equilibrium, and real fluid effects, the Enskog–Vlasov–BGK model is found to capture these effects accurately as confirmed by the corresponding molecular dynamics simulations for low and moderate fluid densities.

Accuracy and Stability Analysis of the Semi-Lagrangian Method for Stiff Hyperbolic Relaxation Systems and Kinetic BGK Model

In this paper, we develop a family of third order asymptotic-preserving (AP) and asymptotically accurate (AA) diagonally implicit Runge–Kutta (DIRK) time discretization methods for the stiff hyperbolic relaxation systems and kinetic Bhatnagar–Gross–Krook (BGK) model in the semi-Lagrangian (SL) setting. The methods are constructed based on an accuracy analysis of the SL scheme for stiff hyperbolic relaxation systems and kinetic BGK model in the limiting fluid regime when the Knudsen number approaches 0. An extra order condition for the asymptotic third order accuracy in the limiting regime is derived. Linear von Neumann stability analysis of the proposed third order DIRK methods are performed to a simplified two-velocity linear kinetic model. Extensive numerical tests are presented to demonstrate the AA, AP, and stability properties of our proposed schemes.

Numerical schemes for a multi-species BGK model with velocity-dependent collision frequency

We consider a kinetic description of multi-species gas mixture modeled with Bhatnagar-Gross-Krook (BGK) collision operators, in which the collision frequency varies not only in time and space but also with the microscopic velocity. In this model, the Maxwellians typically used in standard BGK operators are replaced by a generalization of such target functions, which are defined by a variational procedure [20]. In this paper we present a numerical method for simulating this model, which uses an Implicit-Explicit (IMEX) scheme to minimize a certain potential function, mimicking the Lagrange functional that appears in the theoretical derivation. We show that theoretical properties such as conservation of mass, total momentum and total energy as well as positivity of the distribution functions are preserved by the numerical method, and illustrate its usefulness and effectiveness with numerical examples.

A unified stochastic particle method based on the Bhatnagar-Gross-Krook model for polyatomic gases and its combination with DSMC

Simulating hypersonic flow around a space vehicle is challenging because of the multiscale and nonequilibrium nature inherent in these flows. To effectively deal with such flows, a hybrid scheme combining the stochastic particle Bhatnagar-Gross-Krook (BGK) method with direct simulation Monte Carlo (DSMC) was developed recently, but only for monatomic gases (Fei et al. (2021) [29]). In this paper, the particle-particle hybrid method is extended to polyatomic gas flows. In the near continuum regime, employing the Ellipsoidal–Statistical BGK model proposed by Dauvois et al. (2021) [22] with discrete levels of vibrational energy, the stochastic particle BGK method for polyatomic gases is first established following the idea of the unified stochastic particle BGK (USP-BGK) scheme. It is proven to be of second-order accuracy in the fluid limit. After that, the USP-BGK scheme with rotational and vibrational energies is combined with DSMC to construct a hybrid scheme. The present hybrid scheme for polyatomic gases is validated with numerical tests of homogeneous relaxation, 1D shock structure and 2D hypersonic flows past a wedge and a cylinder. Compared to the other stochastic particle methods, the proposed hybrid scheme can achieve higher accuracy at a much lower computational cost. Therefore, it is a more efficient tool to study multiscale hypersonic flows.

A meshfree arbitrary Lagrangian-Eulerian method for the BGK model of the Boltzmann equation with moving boundaries

In this paper we present a novel technique for the simulation of moving boundaries and moving rigid bodies immersed in a rarefied gas using an Eulerian-Lagrangian formulation based on least square method. The rarefied gas is simulated by solving the Bhatnagar-Gross-Krook (BGK) model for the Boltzmann equation of rarefied gas dynamics. The BGK model is solved by an Arbitrary Lagrangian-Eulerian (ALE) method, where grid-points/particles are moved with the mean velocity of the gas. The computational domain for the rarefied gas changes with time due to the motion of the boundaries. To allow a simpler handling of the interface motion we have used a meshfree method based on a least-square approximation for the reconstruction procedures required for the scheme. We have considered a one way, as well as a two-way coupling of boundaries/rigid bodies and gas flow. The numerical results are compared with analytical as well as with Direct Simulation Monte Carlo (DSMC) solutions of the Boltzmann equation. Convergence studies are performed for one-dimensional and two-dimensional test-cases. Several further test problems and applications illustrate the versatility of the approach.

Numerical Study of Grain Charging Kinetics on the Basis of BGK Kinetic Equation

An investigation of charging a spherical particle in partially ionized nonisothermal plasma is carried out on the basis of the numerical solution of the BGK (Bhatnagar–Gross–Krook) model kinetic equation. Stationary values of the particle charge and the electron and ion currents are calculated for various collisional regimes. It is verified that, for the strongly collisional regime, the effective potential has a Coulomb form at large distance from the particle surface. A new BGK-type model for binary gas mixtures is proposed. It is shown that this model satisfies all the basic properties of Boltzmann collision integrals including the correct exchange coefficients. A high-order implicit numerical method for solving the kinetic equations is developed. The method is conservative with respect to the collision integrals for arbitrary values of the Knudsen number.

Hydrodynamic limits and numerical errors of isothermal lattice Boltzmann schemes

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 regularized and the multiple relaxation time (MRT) collision models in the isothermal framework. Deviation terms are systematically confronted to linear analyses in order to validate their expressions, interpret them and provide explanations for their numerical properties. The low dissipation of the BGK model is then related to a particular pattern of its error terms in the Taylor expansion. Similarly, dissipation properties of the regularized and MRT models are explained by a phenomenon referred to as hyperviscous degeneracy. The latter consists in an unexpected resurgence of high-order Knudsen effects induced by a large numerical pre-factor. It is at the origin of over-dissipation and severe instabilities in the low-viscosity regime.

A lattice Boltzmann model for the Navier-Stokes equation

The lattice Boltzmann model is a vital numerical calculation tool, which has become a useful solution to fluid problems because of the limitations of the Navier-Stokes equation and other macroscopic models in solving such problems. Here, the effectiveness of this model in the nearly incompressible Navier-Stokes equation is investigated as follows. First, the principle and excellent characteristics of the lattice Boltzmann model in numerical calculations are described and analyzed. Second, the approximate incompressible Navier-Stokes equations of the BGK (Bhatnagar-Gross-Krook) model in lattice Boltzmann method are analyzed. Finally, the two-dimensional five discrete velocity (D2N5) model in BGK model is taken as an example, and the vector rebound boundary scheme and the single point second-order boundary scheme based on the Dirichlet boundary condition are derived. Numerical results based on the Poiseuille flow and Taylar-Green vortex show that the BGK-based vector rebound boundary scheme has second-order accuracy only when the proportional distance is 0.5, and in other cases it cannot reach the accuracy. For the single-point second-order boundary scheme, the second-order accuracy can be achieved both in the Poiseuille flow based on the straight boundary region and in the Taylar-Green vortex based on the curved boundary. Through Maxwell iteration, the vector BGK model based on discretization is stable. When the boundary position is in the middle of two lattice points, the vector rebound boundary scheme has second order accuracy, and the single point second order boundary scheme has better performance in second order accuracy. This provides an idea for dealing with boundary conditions in incompressible Navier-Stokes equations.

Research on Electromagnetic Wave Propagation Characteristics of Fully Ionized Inhomogeneous Dusty Plasma in a Magnetized BGK Model

Starting from Boltzmann's approximate equation, the Bhatnagar-Gross-Krook (BGK) collision model of inhomogeneous dusty plasma is derived. In the case of an external magnetic field, the expression of the dielectric coefficient of the dusty plasma under the collision condition of the fully ionized BGK is obtained by combining the collision effect and charging effect of the dusty plasma. By using the transfer matrix method, the reflection coefficient and transmission coefficient of electromagnetic (EM) waves under different external magnetic fields and dusty plasma parameters are calculated, and the propagation characteristics of dusty plasma under the nonuniform fully ionized BGK collision model are analyzed. Results show that the inelastic collision absorption effect between electrons and dust particles is the strongest when the incident frequency of EM waves is close to or equal to the charging frequency of dusty plasma. Meanwhile, the size and density of dust particles can affect the effective collision frequency of dusty plasma and strongly influence the EM wave propagation characteristics. By analyzing the propagation characteristics of different external magnetic fields to circularly polarized waves, changing the size and direction of the external magnetic field can artificially control the low coefficient band range and size of the EM wave transmission coefficient in dusty plasma. These results provide a theoretical basis for the study of EM wave propagation in weakly impacted fully ionized dusty plasma and communication problems in near space.

A coupled gas-kinetic BGK scheme for the finite volume lattice Boltzmann method for nearly incompressible thermal flows

This work presents a coupled gas-kinetic Bhatnagar-Gross-Krook (BGK) scheme based finite volume lattice Boltzmann method (FVLBM) for nearly incompressible thermal flows. In the present formulation, a thermal lattice BGK model with double distribution functions coupled through the Boussinesq approximation is adopted. Compared with the conventional thermal finite volume/finite difference lattice Boltzmann schemes, the present method has two striking advantages. One is that the fluxes on cell interfaces are determined by the formal analytical solution of the lattice BGK equation rather than interpolation or solving the Riemann problem, which reduces numerical dissipation considerably. Another is that, an implicit collision of the lattice Boltzmann equations (LBE) is adopted to remove the time-step constrain from the relaxation time, which improves the efficiency for high Rayleigh number thermal flows. Additionally, the accuracy of the proposed scheme are validated by the numerical simulations of four test cases in two dimensions: (a) the thermal Couette flow with wall injection; (b) the natural convection in a square cavity with differently heated vertical walls; (c) the Rayleigh-Bénard convection in a rectangle heated from bottom; and (d) the turbulent Rayleigh-Bénard convection in a square cavity heated from bottom. The numerical results show that the present scheme has second-order accuracy and can be a reliable tool for incompressible thermal flows.

Fokker–Planck model for binary mixtures

Abstract

A new BGK model to compute the scattering characteristics of electromagnetic waves by weakly ionized dusty plasma shroud

The dynamics theory of dust charging in the Bhatnagar–Gross–Krook (BGK) model is changed, and the updated expression of the complex permittivity for weakly ionized dusty plasma is derived. Then, by using the physical optics method, the radar-cross section (RCS) of electrically large blunt cones covered by dusty plasma with weak ionization is calculated under the new complex permittivity and compared with the traditional results. The numerical results indicate that the new values of the RCS are significantly smaller than their traditional values. In addition, the influence of dust plasma parameters on the scattering characteristics of EM waves is discussed under the new BGK model. The results enriched the physical mechanisms of the scattering characteristics of weakly ionized dusty plasma shroud and provide a theoretical basis for future studies.

Gaussian Lattice Boltzmann method and its applications to rarefied flows

A novel discretization approach for the Bhatnagar-Gross-Krook (BGK) kinetic equation is proposed. A hierarchy for the Lattice Boltzmann models starting from the one-dimensional three-velocity D1Q3 model is derived. The equilibrium states for the models in the hierarchy converge to the Maxwell distribution. The method inherits the properties of the Lattice Boltzmann method such as a linear streaming step and conservation of moments. Similar to the finite-difference methods for the BGK model, the presented approach describes high-order moments of the distribution function with a controllable error. The Sod shock tube problem, the Poiseuille flow between parallel plates, the plane Couette flow, and the lid-driven cavity are considered. Good stability and accuracy are observed for the test problems.

Discrete effect on single-node boundary schemes of lattice Bhatnagar–Gross–Krook model for convection-diffusion equations

The discrete effect on the boundary condition has been a fundamental topic for the lattice Boltzmann method (LBM) in simulating heat and mass transfer problems. In previous works based on the anti-bounce-back (ABB) boundary condition for convection-diffusion equations (CDEs), it is indicated that the discrete effect cannot be commonly removed in the Bhatnagar–Gross–Krook (BGK) model except for a special value of relaxation time. Targeting this point in this paper, we still proceed within the framework of BGK model for two-dimensional CDEs, and analyze the discrete effect on a non-halfway single-node boundary condition which incorporates the effect of the distance ratio. By analyzing an unidirectional diffusion problem with a parabolic distribution, the theoretical derivations with three different discrete velocity models show that the numerical slip is a combined function of the relaxation time and the distance ratio. Different from previous works, we definitely find that the relaxation time can be freely adjusted by the distance ratio in a proper range to eliminate the numerical slip. Some numerical simulations are carried out to validate the theoretical derivations, and the numerical results for the cases of straight and curved boundaries confirm our theoretical analysis. Finally, it should be noted that the present analysis can be extended from the BGK model to other lattice Boltzmann (LB) collision models for CDEs, which can broaden the parameter range of the relaxation time to approach 0.5.

Uniformly accurate machine learning-based hydrodynamic models for kinetic equations

A framework is introduced for constructing interpretable and truly reliable reduced models for multiscale problems in situations without scale separation. Hydrodynamic approximation to the kinetic equation is used as an example to illustrate the main steps and issues involved. To this end, a set of generalized moments are constructed first to optimally represent the underlying velocity distribution. The well-known closure problem is then solved with the aim of best capturing the associated dynamics of the kinetic equation. The issue of physical constraints such as Galilean invariance is addressed and an active-learning procedure is introduced to help ensure that the dataset used is representative enough. The reduced system takes the form of a conventional moment system and works regardless of the numerical discretization used. Numerical results are presented for the BGK (Bhatnagar-Gross-Krook) model and binary collision of Maxwell molecules. We demonstrate that the reduced model achieves a uniform accuracy in a wide range of Knudsen numbers spanning from the hydrodynamic limit to free molecular flow.

A unified stochastic particle Bhatnagar-Gross-Krook method for multiscale gas flows

The stochastic particle method based on Bhatnagar-Gross-Krook (BGK) or ellipsoidal statistical BGK (ESBGK) model approximates the pairwise collisions in the Boltzmann equation using a relaxation process. Therefore, it is more efficient to simulate gas flows at small Knudsen numbers than the counterparts based on the original Boltzmann equation, such as the Direct Simulation Monte Carlo (DSMC) method. However, the traditional stochastic particle BGK method decouples the molecular motions and collisions in analogy to the DSMC method, and hence its transport properties deviate from physical values as the time step size increases. This defect significantly affects its computational accuracy and efficiency for the simulation of multiscale flows, especially when the transport processes in the continuum regime is important. In the present paper, we propose a unified stochastic particle ESBGK (USP-ESBGK) method by combining the molecular convection and collision effects. In the continuum regime, the proposed method can be applied using large temporal-spatial discretization and approaches to the Navier-Stokes solutions with second-order accuracy. Furthermore, it is capable to simulate both the small scale non-equilibrium flows and large scale continuum flows within a unified framework efficiently and accurately. The applications of USP-ESBGK method to a variety of benchmark problems, including Couette flow, thermal Couette flow, Poiseuille flow, Sod tube flow, cavity flow, Taylor-Green vortex, and flow through a slit, demonstrated that it is a promising tool to simulate multiscale gas flows ranging from rarefied to continuum regime.

A numerical method for coupling the BGK model and Euler equations through the linearized Knudsen layer

The Bhatnagar-Gross-Krook (BGK) model, a simplification of the Boltzmann equation, in the absence of boundary effect, converges to the Euler equations when the Knudsen number is small. In practice, however, Knudsen layers emerge at the physical boundary, or at the interfaces between the two regimes. We model the Knudsen layer using a half-space kinetic equation, and apply a half-space numerical solver [19,20] to quantify the transition between the kinetic to the fluid regime. A full domain numerical solver is developed with a domain-decomposition approach, where we apply the Euler solver and kinetic solver on the appropriate subdomains and connect them via the half-space solver. In the nonlinear case, linearization is performed upon local Maxwellian. Despite the lack of analytical support, the numerical evidence nevertheless demonstrate that the linearization approach is promising.

Quantification of thermally-driven flows in microsystems using Boltzmann equation in deterministic and stochastic contexts

When the flow is sufficiently rarefied, a temperature gradient, for example, between two walls separated by a few mean free paths, induces a gas flow—an observation attributed to the thermostress convection effects at the microscale. The dynamics of the overall thermostress convection process is governed by the Boltzmann equation—an integrodifferential equation describing the evolution of the molecular distribution function in six-dimensional phase space—which models dilute gas behavior at the molecular level to accurately describe a wide range of flow phenomena. Approaches for solving the full Boltzmann equation with general intermolecular interactions rely on two perspectives: one stochastic in nature often delegated to the direct simulation Monte Carlo (DSMC) method and the others deterministic by virtue. Among the deterministic approaches, the discontinuous Galerkin fast spectral (DGFS) method has been recently introduced for solving the full Boltzmann equation with general collision kernels, including the variable hard/soft sphere models—necessary for simulating flows involving diffusive transport. In this work, the deterministic DGFS method, Bhatnagar-Gross-Krook (BGK), Ellipsoidal statistical BGK (ESBGK), and Shakhov kinetic models, and the widely used stochastic DSMC method, are utilized to assess the thermostress convection process in micro in-plane Knudsen radiometric actuator—a microscale compact low-power pressure sensor utilizing the Knudsen forces. The BGK model underpredicts the heat-flux, shear-stress, and flow speed; the S-model overpredicts; whereas, ESBGK comes close to the DSMC results. On the other hand, both the statistical/DSMC and deterministic/DGFS methods, segregated in perspectives, yet, yield inextricable results, bespeaking the ingenuity of Graeme Bird who laid down the foundation of practical rarefied gas dynamics for microsystems.