Boltzmann-BGK Equation Research Articles

Model order reduction for the 1D Boltzmann-BGK equation: identifying intrinsic variables using neural networks

Kinetic equations are crucial for modeling non-equilibrium phenomena, but their computational complexity is a challenge. This paper presents a data-driven approach using reduced order models (ROM) to efficiently model non-equilibrium flows in kinetic equations by comparing two ROM approaches: proper orthogonal decomposition (POD) and autoencoder neural networks (AE). While AE initially demonstrate higher accuracy, POD’s precision improves as more modes are considered. Notably, our work recognizes that the classical POD model order reduction approach, although capable of accurately representing the non-linear solution manifold of the kinetic equation, may not provide a parsimonious model of the data due to the inherently non-linear nature of the data manifold. We demonstrate how AEs are used in finding the intrinsic dimension of a system and to allow correlating the intrinsic quantities with macroscopic quantities that have a physical interpretation.

Local well-posedness for the compressible Navier–Stokes–BGK model in Sobolev spaces with exponential weight

Sprays are complex flows constituted of dispersed particles in an underlying gas. In this paper, we are interested in the equations for moderately thick sprays consisting of the compressible Navier–Stokes (NS) equations and Boltzmann BGK equation. Here the coupling of two equations is through a friction (or drag) force which depends on the density of compressible fluid and the relative velocity between particles and fluid. For the NS–BGK system, we establish the existence and uniqueness of solutions in Sobolev spaces with exponential weight.

Unlocking the Key to Accelerating Convergence in the Discrete Velocity Method for Flows in the Near Continuous/Continuous Flow Regimes.

How to improve the computational efficiency of flow field simulations around irregular objects in near-continuum and continuum flow regimes has always been a challenge in the aerospace re-entry process. The discrete velocity method (DVM) is a commonly used algorithm for the discretized solutions of the Boltzmann-BGK model equation. However, the discretization of both physical and molecular velocity spaces in DVM can result in significant computational costs. This paper focuses on unlocking the key to accelerate the convergence in DVM calculations, thereby reducing the computational burden. Three versions of DVM are investigated: the semi-implicit DVM (DVM-I), fully implicit DVM (DVM-II), and fully implicit DVM with an inner iteration of the macroscopic governing equation (DVM-III). In order to achieve full implicit discretization of the collision term in the Boltzmann-BGK equation, it is necessary to solve the corresponding macroscopic governing equation in DVM-II and DVM-III. In DVM-III, an inner iterative process of the macroscopic governing equation is employed between two adjacent DVM steps, enabling a more accurate prediction of the equilibrium state for the full implicit discretization of the collision term. Fortunately, the computational cost of solving the macroscopic governing equation is significantly lower than that of the Boltzmann-BGK equation. This is primarily due to the smaller number of conservative variables in the macroscopic governing equation compared to the discrete velocity distribution functions in the Boltzmann-BGK equation. Our findings demonstrate that the fully implicit discretization of the collision term in the Boltzmann-BGK equation can accelerate DVM calculations by one order of magnitude in continuum and near-continuum flow regimes. Furthermore, the introduction of the inner iteration of the macroscopic governing equation provides an additional 1-2 orders of magnitude acceleration. Such advancements hold promise in providing a computational approach for simulating flows around irregular objects in near-space environments.

The equivalence between BE-LSE and NS-LSEs under continuum assumption

The linear stability theory, which is based on the Navier–Stokes equations, has been extensively studied in the field of hydrodynamic stability. However, due to the continuum assumption, there exist various limitations when attempting to solve problems that involve rarefaction effects. In contrast, based on kinetic theory, the Boltzmann equation is well-suited for analyzing linear stability throughout the entire regime, ranging from continuous to rarefied flows. In this paper, a linear stability equation derived from the Boltzmann–Bhatnagar–Gross–Krook (Boltzmann-BGK) equation is introduced, and the relationship between small perturbations of the velocity distribution function and macroscopic physical variables is established. Under the continuum assumption, the linear stability equation based on the Boltzmann equation (BE-LSE) and the linear stability equations based on the Navier–Stokes equations (NS-LSEs) have the same numerical solutions, which indicates that the microscopic BE-LSE can recover to macroscopic NS-LSEs theoretically. Nevertheless, there is still a lack of mathematical proof for this theoretical outcome. To address this issue, the research on the equivalence of BE-LSE and NS-LSEs under continuum assumption is carried out in this paper, which can establish the theoretical relationship between BE-LSE and NS-LSEs. These efforts lay a solid theoretical foundation for stability research based on the BE-LSE.

Accelerated Simulation of Boltzmann-BGK Equations near the Diffusive Limit with Asymptotic-Preserving Multilevel Monte Carlo

Accelerated Simulation of Boltzmann-BGK Equations near the Diffusive Limit with Asymptotic-Preserving Multilevel Monte Carlo

Discrete unified gas-kinetic wave-particle method for flows in all flow regimes.

This work proposes a discrete unified gas-kinetic wave-particle (DUGKWP) method for simulation of flows in all flow regimes. Unlike the discrete velocity method (DVM) and the direct simulation Monte Carlo (DSMC) method which solve the governing equations by either the deterministic method or the stochastic method, the DUGKWP combines the advantages of these two methods. In the DUGKWP, the information of microscopic particles as well as macroscopic flow variables are both evolved. Specifically, the microscopic particles are updated by the free-transport and resampling processes, while the macroscopic flow properties are evolved via solving the macroscopic governing equations of conservation laws with the finite volume method. According to the discrete characteristic solution to the Boltzmann-BGK equation utilized in the DUGKWP, in the highly rarefied flow regime, the motion of microscopic particles greatly determines the fluxes for the macroscopic governing equations. Conversely, for the continuum flow, no microscopic particle exists in the computational domain and the DUGKWP is degraded to the Navier-Stokes solver. Numerical studies validate that the DUGKWP can accurately predict the flow properties in all flow regimes. Furthermore, compared with the deterministic method, the DUGKWP enjoys superior efficiency with less memory consumption for both high-speed rarefied flows and flows close to the continuum regime.

Adaptive partitioning-based discrete unified gas kinetic scheme for flows in all flow regimes

To improve the efficiency of the discrete unified gas kinetic scheme (DUGKS) in capturing cross-scale flow physics, an adaptive partitioning-based discrete unified gas kinetic scheme (ADUGKS) is developed in this work. The ADUGKS is designed from the discrete characteristic solution to the Boltzmann-BGK equation, which contains the initial distribution function and the local equilibrium state. The initial distribution function contributes to the calculation of free streaming fluxes and the local equilibrium state contributes to the calculation of equilibrium fluxes. When the contribution of the initial distribution function is negative, the local flow field can be regarded as the continuous flow and the Navier–Stokes (N-S) equations can be used to obtain the solution directly. Otherwise, the discrete distribution functions should be updated by the Boltzmann equation to capture the rarefaction effect. Given this, in the ADUGKS, the computational domain is divided into the DUGKS cell and the N-S cell based on the contribution of the initial distribution function to the calculation of free streaming fluxes. In the N-S cell, the local flow field is evolved by solving the N-S equations, while in the DUGKS cell, both the discrete velocity Boltzmann equation and the corresponding macroscopic governing equations are solved by a modified DUGKS. Since more and more cells turn into the N-S cell with the decrease of the Knudsen number, a significant acceleration can be achieved for the ADUGKS in the continuum flow regime as compared with the DUGKS.

A positivity-preserving and conservative high-order flux reconstruction method for the polyatomic Boltzmann–BGK equation

In this work, we present a positivity-preserving high-order flux reconstruction method for the polyatomic Boltzmann–BGK equation augmented with a discrete velocity model that ensures the scheme is discretely conservative. Through modeling the internal degrees of freedom, the approach is further extended to polyatomic molecules and can encompass arbitrary constitutive laws. The approach is validated on a series of large-scale complex numerical experiments, ranging from shock-dominated flows computed on unstructured grids to direct numerical simulation of three-dimensional compressible turbulent flows, the latter of which is the first instance of such a flow computed by directly solving the Boltzmann equation. The results show the ability of the scheme to directly resolve shock structures without any ad hoc numerical shock capturing method and correctly approximate turbulent flow phenomena in a consistent manner with the hydrodynamic equations.

Learning Invariance Preserving Moment Closure Model for Boltzmann–BGK Equation

As one of the main governing equations in kinetic theory, the Boltzmann equation is widely utilized in aerospace, microscopic flow, etc. Its high-resolution simulation is crucial in these related areas. However, due to the high dimensionality of the Boltzmann equation, high-resolution simulations are often difficult to achieve numerically. The moment method which was first proposed in Grad (Commun Pure Appl Math 2(4):331–407, 1949) is among the popular numerical methods to achieve efficient high-resolution simulations. We can derive the governing equations in the moment method by taking moments on both sides of the Boltzmann equation, which effectively reduces the dimensionality of the problem. However, one of the main challenges is that it leads to an unclosed moment system, and closure is needed to obtain a closed moment system. It is truly an art in designing closures for moment systems and has been a significant research field in kinetic theory. Other than the traditional human designs of closures, the machine learning-based approach has attracted much attention lately in Han et al. (Proc Natl Acad Sci USA 116(44):21983–21991, 2019) and Huang et al. (J Non-Equilib Thermodyn 46(4):355–370, 2021). In this work, we propose a machine learning-based method to derive a moment closure model for the Boltzmann–BGK equation. In particular, the closure relation is approximated by a carefully designed deep neural network that possesses desirable physical invariances, i.e., the Galilean invariance, reflecting invariance, and scaling invariance, inherited from the original Boltzmann–BGK equation and playing an important role in the correct simulation of the Boltzmann equation. Numerical simulations on the 1D–1D examples including the smooth and discontinuous initial condition problems, Sod shock tube problem, the shock structure problems, and the 1D–3D examples including the smooth and discontinuous problems demonstrate satisfactory numerical performances of the proposed invariance preserving neural closure method.

A discrete unified gas kinetic scheme on unstructured grids for viscid compressible flows and its parallel algorithm

<abstract><p>In this paper, we present a discrete unified gas kinetic scheme (DUGKS) on unstructured grids for high-speed viscid compressible flows on the basis of double distribution function (the density and the total energy distribution functions) Boltzmann-BGK equations. In the DUGKS, the discrete equilibrium distribution functions are constructed based on a D2Q17 circular function. In order to accelerate the simulation, we also illustrate a corresponding parallel algorithm. The DUGKS is validated by two benchmark problems, i.e., flows around the NACA0012 airfoil and flows past a circular cylinder with the Mach numbers range from 0.5 to 2.5. Good agreements with the referenced results are observed from the numerical results. The results of parallel test indicate that the DUGKS is highly parallel scalable, in which the parallel efficiency achieves $ 93.88\% $ on a supercomputer using up to $ 4800 $ processors. The proposed method can be utilized for high-resolution numerical simulation of complex and high Mach number flows.</p></abstract>

Entropy bounds for the space–time discontinuous Galerkin finite element moment method applied to the BGK–Boltzmann equation

This paper presents a numerical analysis for the time-implicit numerical approximation of the Boltzmann equation based on a moment system approximation in velocity dependence and a discontinuous Galerkin finite-element (DGFE) approximation in time and position dependence. The implicit nature of the DGFE moment method in position and time dependence provides a robust numerical algorithm for the approximation of solutions of the Boltzmann equation. The closure relation for the moment systems derives from minimization of a suitable φ-divergence. We present sufficient conditions such that this divergence-based closure yields a hierarchy of tractable symmetric hyperbolic moment systems that retain the fundamental structural properties of the Boltzmann equation. The resulting combined space-time DGFE moment method corresponds to a Galerkin approximation of the Boltzmann equation in renormalized form. We propose a renormalization map that facilitates the approximation of multidimensional problems in an implicit manner. Moreover, upper and lower entropy bounds are derived for the proposed DGFE moment scheme. Numerical results for benchmark problems governed by the BGK-Boltzmann equation are presented to illustrate the approximation properties of the new DGFE moment method, and it is shown that the proposed velocity-space-time DGFE moment method is entropy bounded.

Parallel unstructured finite volume lattice Boltzmann method for high-speed viscid compressible flows

Based on the double distribution function Boltzmann-BGK equations, a cell-centered finite volume lattice Boltzmann method on unstructured grids for high-speed viscid compressible flows is presented. In the equations, the particle distribution function is introduced on the basis of the D2Q17 circular function, and its corresponding total energy distribution function is adopted. In the proposed method, the advective term is evaluated by Roe’s flux-difference splitting scheme, and a limiter is used to prevent the generation of oscillations. The distribution functions on the interface are calculated by piecewise linear reconstruction, in which the gradient is computed by the least-squares approach. In order to do large-scale simulations, a parallel algorithm is illustrated. The present method is validated by a flow around the NACA0012 airfoil and a flow past a circular cylinder at high Mach numbers. The results agree well with the published results, which demonstrate that the present method is an efficient numerical method for high-speed viscid compressible flows. The parallel performance results show that the proposed parallel algorithm achieves 90% parallel efficiency on 4800 cores for a problem with [Formula: see text] unstructured triangle cells, which shows the potential to perform fast and high-fidelity simulations of large-scale high-speed viscid compressible flows in complicated computational domains.

Physics-informed neural networks for solving forward and inverse flow problems via the Boltzmann-BGK formulation

The Boltzmann equation with the Bhatnagar-Gross-Krook collision model (Boltzmann-BGK equation) has been widely employed to describe multiscale flows, i.e., from the hydrodynamic limit to free molecular flow. In this study, we employ physics-informed neural networks (PINNs) to solve forward and inverse problems via the Boltzmann-BGK formulation (PINN-BGK), enabling PINNs to model flows in both the continuum and rarefied regimes. In particular, the PINN-BGK is composed of three sub-networks, i.e., the first for approximating the equilibrium distribution function, the second for approximating the non-equilibrium distribution function, and the third one for encoding the Boltzmann-BGK equation as well as the corresponding boundary/initial conditions. By minimizing the residuals of the governing equations and the mismatch between the predicted and provided boundary/initial conditions, we can approximate the Boltzmann-BGK equation for both continuous and rarefied flows. For forward problems, the PINN-BGK is utilized to solve various benchmark flows given boundary/initial conditions, e.g., Kovasznay flow, Taylor-Green flow, cavity flow, and micro Couette flow for Knudsen number up to 5. For inverse problems, we focus on rarefied flows in which accurate boundary conditions are difficult to obtain. We employ the PINN-BGK to infer the flow field in the entire computational domain given a limited number of interior scattered measurements on the velocity without using the (unknown) boundary conditions. Results for the two-dimensional micro Couette and micro cavity flows with Knudsen numbers ranging from 0.1 to 10 indicate that the PINN-BGK can infer the velocity field in the entire domain with good accuracy. Finally, we also present some results on using transfer learning to accelerate the training process. Specifically, we can obtain a three-fold speedup compared to the standard training process (e.g., Adam plus L-BFGS-B) for the two-dimensional flow problems considered in our work.

A stochastic kinetic scheme for multi-scale flow transport with uncertainty quantification

Gaseous flows show a diverse set of behaviors on different characteristic scales. Given the coarse-grained modeling in theories of fluids, considerable uncertainties may exist between the flow-field solutions and the real physics. To study the emergence, propagation and evolution of uncertainties from molecular to hydrodynamic level poses great opportunities and challenges to develop both sound theories and reliable multi-scale numerical algorithms. In this paper, a new stochastic kinetic scheme will be developed that includes uncertainties via a hybridization of stochastic Galerkin and collocation methods. Based on the Boltzmann-BGK model equation, a scale-dependent evolving solution is employed in the scheme to construct governing equations in the discretized temporal-spatial domain. Therefore typical flow physics can be recovered with respect to different physical characteristic scales and numerical resolutions in a self-adaptive manner. We prove that the scheme is formally asymptotic-preserving in different flow regimes with the inclusion of random variables, so that it can be used for the study of multi-scale non-equilibrium gas dynamics under the effect of uncertainties.Several numerical experiments are shown to validate the scheme. We make new physical observations, such as the wave-propagation patterns of uncertainties from continuum to rarefied regimes. These phenomena will be presented and analyzed quantitatively. The current method provides a novel tool to quantify the uncertainties within multi-scale flow evolutions.

Pore-Scale Study of Rarefied Gas Flows Using Low-Variance Deviational Simulation Monte Carlo Method

Gaseous flow through ultra-tight porous media, e.g. shale and some high-performance insulation materials, is often rarefied, invalidating an analysis by the continuum flow theory. Such rarefied flows can be accurately described by the kinetic theory of gases which utilizes the Boltzmann equation and its simplified kinetic models. While discrete velocity methods have been successful in directly solving these equations, the immense potential of a particle-based solution of the variance-reduced Boltzmann-BGK (Bhatnagar–Gross–Krook) equation for rarefied flows in porous media has not been exploited yet. Here, a parallel solver based on the low variance deviational simulation Monte Carlo method is developed for 3D flows, which enables pore-scale simulations using digital images of porous media samples. The unique advantage of this particle-based formulation is in providing additional insights regarding the multi-scale nature of the flow and surface/gas interactions via two new parameters, i.e. pore and surface activity, respectively. Together, these two parameters can identify key flow properties of the porous media. The computational efficiency and accuracy of the current method has also been analysed, suggesting that this new solver is a powerful simulation tool to quantify flow properties of ultra-tight porous media.

The impulsive swirl of a gas

Abstract

A kinetic generator for classical field theories with conservation laws

In this study, we demonstrate that, by suitably modifying the equilibrium distribution, the Bhatnagar-Gross-Krook form of the Boltzmann equation can serve as a generator of a broad class of nonlinear partial differential equations via a suitable generalization of the Chapman-Enskog method. We achieve this by properly choosing the relaxation time and moments of the equilibrium state in such a way as to suppress the dynamics of the zeroth moment, density, at increasingly faster time scales. As concrete examples, explicit analytical formulations of the Burgers', Korteweg-de Vries and Kuramoto-Sivashinsky equations are presented, along with a discussion of the associated conservation laws of the corresponding moments. The concept is numerically demonstrated by solving the BGK Boltzmann equation with the finite-element method. Because the Boltzmann-BGK equation can be equipped with a H-theorem, the framework developed in this work indicates a path towards the development of new algorithms for the numerical solution of the aforementioned equations with enhanced stability.

Tensor methods for the Boltzmann-BGK equation

We present a tensor-decomposition method to solve the Boltzmann transport equation (BTE) in the Bhatnagar-Gross-Krook approximation. The method represents the six-dimensional BTE as a set of six one-dimensional problems, which are solved with the alternating least-squares algorithm and the discrete Fourier transform at N collocation points. We use this method to predict the equilibrium distribution (steady-state simulation) and a non-equilibrium distribution returning to the equilibrium (transient simulation). Our numerical experiments demonstrate Nlog⁡N scaling. Unlike many BTE-specific numerical techniques, the numerical tensor-decomposition method we propose is a general technique that can be applied to other high-dimensional systems.

A strong-coupled method combined finite element method and lattice Boltzmann method via an implicit immersed boundary scheme for fluid structure interaction

Fluid structure interaction (FSI) problems are often encountered in ocean science and technology. In this present, an implicit immersed boundary (IB) method combined with Lattice Boltzmann Method (LBM) and finite element method (FEM) is presented for the strong-coupled simulations of fluid structure interaction. Velocity field is solved rapidly by lattice Boltzmann-BGK equations without external force, and the deformation of a flexible structure was computed by FEM. Fluid structure coupling was processed by immersed boundary implicit forced method. The resulting combination gives an easy-to-use, inexpensive and accurate technique to solve FSI problems. Simultaneously, the body force on the immersed boundary can be calculated directly and acts as the load for structure solver, so the displacement of structure can be obtained at the same time, which is avoid the time lag between the fluid solver and solid solver. Several examples including flow over a static and deformable plate, and the flow problems of an oscillating and cantilever flexible plate were simulated and analyzed. The results agree well with the previous experimental and numerical investigations, which indicate the IB-LB-FEM proposed in this paper is an efficient and reliable fluid-structure interaction numerical simulation method.

Simplified unified wave-particle method with quantified model-competition mechanism for numerical calculation of multiscale flows.

A quantified model-competition (QMC) mechanism for multiscale flows is extracted from the integral (analytical) solution of the Boltzmann-BGK model equation. In the QMC mechanism, the weight of the rarefied model and the weight of the continuum (aerodynamic and hydrodynamic) model are quantified. Then, a simplified unified wave-particle method (SUWP) is constructed based on the QMC mechanism. In the SUWP, the stochastic particle method and the continuum Navier-Stokes method are combined together. Their weights are determined by the QMC mechanism quantitatively in every discrete cell of the computational domain. The validity and accuracy of the present numerical method are examined using a series of test cases including the high nonequilibrium shock wave structure case, the unsteady Sod shock-tube case with a wide range of Kn number, the hypersonic flow around the circular cylinder from the free-molecular regime to the near continuum regime, and the viscous boundary layer case. In the construction process of the present method, an antidissipation effect in the continuum mechanism is also discussed.