Shakhov Model Research Articles

Lattice Boltzmann Simulation of Nonequilibrium Flows Using Spectral Multiple-Relaxation-Time Collision Model

Prediction of nonequilibrium flows is critical to space flight. The present work demonstrates that the recently developed spectral multiple-relaxation-time (SMRT) lattice Boltzmann (LB) model is theoretically equivalent to Grad’s eigensystem [“Principles of the Kinetic Theory of Gases,” Thermodynamik der Gase/Thermodynamics of Gases, Springer–Verlag, Berlin, 1958, pp. 205–294], where the eigenfunctions obtained by tensor decomposition of the Hermite polynomials are also those of the linearized Boltzmann equation. Numerical results of shock structure simulation using the Maxwell molecular model agree very well with those of a high-resolution fast spectral method code up to Mach 7, provided that the relaxation times of the irreducible tensor components match their theoretical values. If a reduced set of relaxation times is used, such as in the Shakhov model and lumped-sum relaxation of Hermite modes, non-negligible discrepancies start to occur as the Mach number is raised, indicating the necessity of the fine-grained relaxation model. Together with the proven advantages of LB, the LB-SMRT scheme offers a competitive alternative for nonequilibrium flow simulation.

Thermally induced oscillatory rarefied gas flow inside a rectangular cavity

Thermally induced oscillatory rarefied gas flow inside a two-dimensional rectangular cavity is investigated based on the hybrid macro-/mesoscopic scheme. The effects of the Knudsen (Kn) numbers and the oscillation frequency of lid temperature on the flow parameters are analyzed. The Shakhov model equation is solved numerically based on the mesoscopic approach in the near-wall region, and the macroscopic approach is adopted in the bulk flow region to reduce the computational cost. To close the numerical iteration procedure, the velocity distribution functions serving as the pseudo boundary between macroscopic and mesoscopic methods are reconstructed using the high-order Hermite polynomials. Numerical simulations demonstrate that the temperature profile at the central vertical of the cavity predicted by the hybrid method is in good agreement with results from the mesoscopic method, with a maximum error of 0.23%. In addition, the computational memory cost can be saved up to about 69.91%. The hybrid approach is able to capture the nonlinear phenomenon in the thermally induced oscillatory rarefied gas flow under high Kn numbers, where the horizontal velocity no longer obeys the law of periodic oscillating cosine function, and the rise time of the horizontal velocity is much longer than the fall time. The thickness of the viscous penetration layer and the disturbed region increases as the Kn number increases and decreases as the Strouhal number increases.

Spectral analysis and hydrodynamic manifolds for the linearized Shakhov model

We perform a complete spectral analysis of the linearized Shakhov model involving two relaxation times τfast and τslow. Our results are based on spectral functions derived from the theory of finite-rank perturbations, which allows us to derive an explicit expression for the critical wave number kcrit limiting the number of discrete eigenvalues above the essential spectrum together with the existence of a finite-dimensional slow manifold defining non-local hydrodynamics. We discuss the merging of hydrodynamic modes as well as the existence of second sound and the appearance of ghost modes beneath the essential spectrum in dependence of the Prandtl number.

Viscous and thermal velocity slip coefficients via the linearized Boltzmann equation with ab initio potential

The viscous and thermal velocity slip coefficients of various monatomic gases are computed via the linearized classical Boltzmann equation, with ab initio potential, subject to Maxwell and Cercignani–Lampis boundary conditions. Both classical and quantum interatomic interactions are considered. Comparisons with hard sphere and Lennard–Jones potentials, as well as the linearized Shakhov model are performed. The produced database is dense, covers the whole range of the accommodation coefficients and is of high accuracy. Using symbolic regression, very accurate closed form expressions of the slip coefficients, easily implemented in the future computational and experimental works, are deduced. The thermal slip coefficient depends, much more than the viscous one, on the intermolecular potential. For example, in the case of diffuse scattering, the relative differences in the viscous slip coefficient data between HS and AI potentials are less than 4%, whilst the corresponding ones in the thermal slip coefficient data are about 6% for He, reaching 15% for Xe. Quantum effects are considered for He, at temperatures 1–104 K to deduce that deviations from the classical behaviour are not important in the viscous slip coefficient, but they become important in the thermal slip coefficient, where the differences between the classical and quantum approaches reach 15% at 1 K. The computational effort of solving the linearized Boltzmann equation with ab initio and Lennard–Jones potentials is the same. Since ab initio potentials do not contain any adjustable parameters, it is recommended to use them at any temperature.

RelaxNet: A structure-preserving neural network to approximate the Boltzmann collision operator

The extremely high dimensionality and nonlinearity in the Boltzmann equation bring tremendous difficulties to the study of rarefied gas dynamics. This paper addresses a neural network-based surrogate model that provides a structure-preserving approximation for the fivefold collision integral. The notion originates from the similarity in structure between the BGK-type relaxation model and residual neural network (ResNet) when a particle distribution function is treated as the input to the neural network function. Therefore, we extend the ResNet architecture and construct what we call the relaxation neural network (RelaxNet). Specifically, two feed-forward neural networks with physics-informed connections and activations are introduced as building blocks in RelaxNet, which provide bounded and physically realizable approximations of the equilibrium distribution and velocity-dependent relaxation time respectively. The evaluation of the collision term is thus significantly accelerated due to the fact that the convolution in the fivefold integral is replaced by tensor multiplication in the neural network. We fuse the mechanical advection operator and the RelaxNet-based collision operator into a unified model named the universal Boltzmann equation (UBE). We prove that UBE preserves the key structural properties in a many-particle system, i.e., positivity, conservation, invariance, H-theorem, and correct fluid dynamic limit. These properties promise that RelaxNet is superior to strategies that naively approximate the right-hand side of the Boltzmann equation using a machine learning model. A novel sampling strategy based on closure hierarchies of the moment system of the Boltzmann equation is developed to generate reliable and unbiased sampling for the supervised learning. The construction of the RelaxNet-based UBE and its solution algorithm are demonstrated in detail. Several numerical experiments, where the ground-truth datasets are produced by the Shakhov model, velocity-dependent ν-BGK model, and the full Boltzmann equation, are investigated. The capability of the current approach for simulating non-equilibrium flow physics is validated through satisfactory in- and out-of-distribution performance. The open-source codes to reproduce the numerical results are available under the MIT license [1].

The Shakhov model near a global Maxwellian

Shakhov model is a relaxation approximation of the Boltzmann equation proposed to overcome the deficiency of the original BGK model, namely, the incorrect production of the Prandtl number. In this paper, we address the existence and the asymptotic stability of the Shakhov model when the initial data is a small perturbation of global equilibrium. We derive a dichotomy in the coercive estimate of the linearized relaxation operator between zero and non-zero Prandtl number, and observe that the linearized relaxation operator is more degenerate in the former case. To remove such degeneracy and recover the full coercivity, we consider a micro–macro system that involves an additional non-conservative quantity related to the heat flux.

Discrete Boltzmann modeling of detonation: Based on the Shakhov model

A Discrete Boltzmann Model(DBM) based on the Shakhov model for detonation is proposed. Compared with the DBM based on the Bhatnagar–Gross–Krook (BGK) model, the current model has a flexible Prandtl numbers and consequently can be applied to a much wider range of detonation phenomena. Besides the Hydrodynamic Non-Equilibrium (HNE) behaviors usually investigated by the Navier–Stokes model, the most relevant Thermodynamic Non-Equilibrium (TNE) effects can be probed by the current model. The model is validated by some well-known benchmarks, and some steady and unsteady detonation processes are investigated. As for the von Neumann peak relative to the wave front, it is found that (i) (within the range of numerical experiments) the peak heights of pressure, density, and flow velocity increase exponentially with the Prandtl number, the maximum stress increases parabolically with the Prandtl number, and the maximum heat flux decreases exponentially with the Prandtl number and (ii) the peak heights of pressure, density, temperature and flow velocity, and the maximum stress within the peak are parabolically increase with the Mach number, the maximum heat flux decreases exponentially with the Mach number.

Capturing the influence of intermolecular potential in rarefied gas flows by a kinetic model with velocity-dependent collision frequency

A kinetic model called the$\nu$-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-empirical collision frequency is designed for different intermolecular potentials: the$\nu$-model shows significantly improved accuracy, and the underlying mechanism is analysed. The$\nu$-model also performs well in canonical rarefied microflows, especially in thermal transpiration, where kinetic models with velocity-independent collision frequency lack the capability to distinguish the influence of intermolecular potentials.

Thermophoretic force on a sphere of arbitrary thermal conductivity in a rarefied gas

The thermophoretic force on a sphere with arbitrary thermal conductivity immersed in a rarefied gas subject to a uniform temperature gradient is calculated in the framework of kinetic theory of gases in a range of the Knudsen number which covers the free molecular, transition and continuum regimes of the gas flow around the sphere. The condition of continuity of the radial heat flux at the gas–solid interface is used to calculate the thermophoretic force as a superposition of the thermophoretic force on a sphere with an uniform temperature and the radiometric force on a sphere due to its thermal polarization. Both solutions were previously obtained from the Shakhov model to the linearized Boltzmann equation and the Cercignani–Lampis model of gas–surface interaction. To analyze the influence of the gas and particle thermal conductivities on the thermophoretic force, spheres of PolyStyrene Latex (PSL), glass and nickel in helium and argon gases are considered. The force is calculated for several values of tangential momentum accommodation coefficient and normal energy accommodation coefficient as introduced in the Cercignani–Lampis scattering kernel. A comparison with experimental data available in the literature allowed to extract the values of the accommodation coefficients for helium interaction with the particle surface.

Study of Shock Structures Using the Unified Gas-Kinetic Wave-Particle Method with Various BGK Models

Recently, the unified gas-kinetic wave-particle (UGKWP) method was proposed to describe rarefied flows efficiently in all Knudsen numbers. However, the initial UGKWP method with the Bhatnagar–Gross–Krook (BGK) model has its own limitations due to the unit Prandtl number. To further improve its performance in non-equilibrium flows, the UGKWP method is extended for the ellipsoidal statistical model (ESBGK) and Shakhov model (SBGK) in this paper. The numerical results of shock structures of the above BGK-type models show that the SBGK model fits the reference data best except the asymmetry parameter, and the ESBGK model comes the second. Meanwhile, in order to overcome the deficiency of negative probability density function (PDF) in SBGK, the UGKWP method with positive-preserving Shakhov PDF is also constructed. It weakens the early rising of temperature phenomenon that existed in the SBGK model, and improves the performance in predicting the stress and heat flux evidently at high Mach numbers.

Experimentally-Benchmarked kinetic simulations of heat transfer through rarefied gas with constant heat flux at the boundary

In many applications, the heat flux at the surface is known instead of the surface temperature. In addition, in some applications, like vacuum drying or high altitude flights, the pressure is below atmospheric pressure, so the rarefaction effects become important, and therefore, the Navier-Stokes-Fourier equations fail to predict gas thermal behavior. In this paper, a constant heat flux boundary condition is developed and implemented in the frame of the Shakhov model kinetic equation, with the possibility to simulate the diffuse-specular reflexion of the molecules from the surface. The developed technique is implemented for the simulation of gas heat transfer in a two concentric cylinders configuration, similar to vacuum drying of used nuclear fuel canisters. The numerical results obtained using developed approach are compared with experimental data of heat transfer through rarefied gas between two concentric cylinders.

Numerical solution of the Boltzmann equation with S-model collision integral using tensor decompositions

The paper presents a new solver for the numerical solution of the Boltzmann kinetic equation with the Shakhov model collision integral (S-model) for arbitrary spatial domains. The numerical method utilizes the Tucker decomposition, which reduces the required computer memory for up to 100 times, even on a moderate velocity grid. This improvement is achieved by representing the distribution function values on a structured velocity grid as a 3D tensor in the Tucker format. The resulting numerical method makes it possible to solve complex 3D problems on modern desktop computers. Our implementation may serve as a prototype code for researchers concerned with the numerical solution of kinetic equations in 3D domains using a discrete velocity method. Program summaryProgram Title: Boltzmann-TCPC Library link to program files:https://doi.org/10.17632/29wv8nmbgn.1Developer’s repository link:https://github.com/chikitkin/Boltzmann-TuckerLicensing provisions: MITProgramming language: Python 3External libraries: Solver is based on the customized version of the tucker3d library [1]Nature of problem: Numerical solution of the Boltzmann kinetic equation with the S-model collision integral in an arbitrary 3D spatial domainSolution method: Discrete velocity method utilizing tensor decomposition for memory reductionAdditional comments including restrictions and unusual features: At present, 1st order advection scheme is used, solver supports unstructured hexagonal meshes written in StarCD ASCII formatReference tucker3d (https://github.com/rakhuba/tucker3d) library contains Python implementations of several important procedures for working with tensors in the Tucker format.

Computational methods for pore-scale simulation of rarefied gas flow

Direct simulation at the pore-scale is crucial to unravel rarefaction effect on gas transport in tight porous media. To satisfy the dual demands on modeling accuracy and computation effort, an appropriate method must be chosen. This work, therefore, evaluates four numerical methods for pore-scale rarefied gas flows in a two-dimensional (2D) model porous media over a wide range of rarefaction. These methods are the incompressible Navier-Stokes equation with the first-order velocity-slip boundary condition, and three gas-kinetic solvers i.e. a finite-difference (FD) iterative solver for the linearized Bhatnagar, Gross and Crook (BGK) model kinetic equation, a finite-volume (FV) solver for the non-linear Shakhov model, and an open-source direct simulation Monte Carlos (DSMC) solver. The benchmark cases cover the characteristic Knudsen number ranging from 0.0231 to 4.62 while the characteristic Reynolds numbers are kept to be less than 1.0. All the solvers are developed in OpenFOAM, except the FD solver, allowing us to investigate the effect of local grid refinements using the automatic Cartesian grid generator in OpenFOAM. The flow fields and the apparent permeabilities predicted by all the solvers have been compared in detail. Besides, the computational time of these solvers is measured and analyzed to demonstrate the relative cost of the three kinetic solvers. It is found that the FD solver is the most efficient one and gives accurate results over the whole range of Knudsen number. Finally, this study also evaluates the feasibility of the recently developed discrete unified gas-kinetic scheme (DUGKS), the algorithm in the aforementioned non-linear Shakhov model equation solver, for pore-scale rarefied gas flow. It is found the results predicted by this solver agree well with the other two kinetic solvers.

Pressure and temperature driven fully-developed rarefied gas flow in a channel with uniform injection/suction through its permeable walls

The pressure and temperature driven fully-developed rarefied gas flow between two parallel permeable plates, with uniform gas injection and suction from the bottom and top plates respectively, is investigated, based on the linearized Shakhov (S) model and the linearized Boltzmann equation (BE). Both flow configurations are characterized by the gas rarefaction parameter and the injection/suction velocity magnitude. Computational results for the so-called kinetic coefficients, as well as for the macroscopic quantities are provided in a wide range of the two parameters. The Onsager-Casimir reciprocity relation between the mechanocaloric and thermal creep coefficients has been proven to hold for arbitrary values of the injection velocity and has been used to validate the accuracy of the obtained results. It is shown that, the kinetic coefficients and macroscopic quantities are significantly affected as the injection velocity is increased. In addition, the kinetic results properly recover the corresponding analytical solutions in the free molecular and slip regimes. The examined prototype rarefied injection/suction flows may be useful to the investigation of more complex and realistic flow configurations in channels of various cross-sections with permeable walls where the injection velocity is not constant in the whole flow domain.

Comparison of the Shakhov and ellipsoidal models for the Boltzmann equation and DSMC for ab initio-based particle interactions

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 initiopotentials. 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 simulations are performed in the non-linear regime for the 3He, 4He, and Ne gases. The reference temperature ranges between 1 K and 3000 K for 3He and 4He and between 20 K and 5000 K for Ne. While good agreement is seen up to the transition regime for the direct phenomena (shear stress, heat flux driven by temperature gradient), the relative errors in the cross phenomena (heat flux perpendicular to the temperature gradient) exceed 10% even in the slip-flow regime. The kinetic model equations are solved using the finite difference lattice Boltzmann algorithm based on half-range Gauss-Hermite quadratures with the third order upwind method used for the implementation of the advection.

On the total enthalpy behavior inside a shock wave

The total enthalpy behavior inside a shock wave in a dilute monatomic gas has been numerically studied for various values of Mach and Prandtl numbers with the continuum (the Navier–Stokes–Fourier equations) and kinetic (the Shakhov model and the direct simulation Monte Carlo method) approaches. A significant difference between the results by the continuum and kinetic approaches has been observed for the internal shock wave structure. In a wide range of the free-stream Mach numbers, the continuum approach predicts qualitatively similar behavior of total enthalpy distributions that can be of a concave, constant, or convex shape depending on the Prandtl number. The more sophisticated kinetic approach predicts a more complicated form of total enthalpy profiles: e.g., an inflection point for Mach numbers around two and Prandtl numbers close to unity. The evolution of the total enthalpy in the shock is determined by the balance of heat conduction and mechanical work of normal viscous stress—processes that are predicted inaccurately by using the Navier–Stokes–Fourier equations at high Mach numbers.

Stability Analysis of Quadrature-Based Moment Methods for Kinetic Equations

In this paper, we give a systematic stability analysis of the quadrature-based moment method (QBMM) for the one-dimensional Boltzmann equation with BGK or Shakhov models. As reported in recent literature, the method has revealed its potential for modeling nonequilibrium flows, while a thorough theoretical analysis is largely missing but desirable. We show that the method can yield nonhyperbolic moment systems if the distribution function is approximated by a linear combination of $\delta$-functions. On the other hand, if the $\delta$-functions are replaced by their Gaussian approximations with a common variance, we prove that the moment systems are strictly hyperbolic and preserve the dissipation property (or $H$-theorem) of the kinetic equation. In the proof, we also determine the equilibrium manifold that lies on the boundary of the state space. The proofs are quite technical and involve detailed analyses of the characteristic polynomials of the coefficient matrices.

Kinetic Methods for Solving Unsteady Problems with Jet Flows

The study of nonstationary rarefied gas flows is currently paid much attention. Such interest to these problems is caused by the creation of pulsed jets used for the deposition of thin films and special coatings on solid surfaces. However the problems of nonstationary rarefied gas flows have not been studied sufficiently fully because of their large computational complexity. In this paper the computational aspects of investigating the nonstationary flows of a reflected gas from a wall and flowing through a suddenly formed gap is considering. The objective of this study is to analyze the possible numerical kinetic approaches for solving such nonstationary problems and to identify the difficulties encountered in solving.When studying the gas flows in strong rarefaction regimes one should consider the Boltzmann kinetic equation, but its numerical implementation is rather laborious. In order to use more simple approaches based for example on approximation kinetic equations (Ellipsoidal-Statistical model, Shakhov model), it is important to estimate the difference of the solutions of the model equations and the Boltzmann equation. For this purpose two auxiliary problems are considered: reflection of the gas flow from the wall and outflow of the free jet into the rarefied background gas. Numerical solution of these problems shows a weak dependence of the solution on the type of the collision operator in the rarefied region, but a strong dependence on the velocity grid step . The detailed velocity grid is necessary to avoid non-monotonous behavior of macroparameters caused by the “ray effect”. To reduce numerical costs on detailed grid a hybrid method based on the synthesis of model equation and the Boltzmann equation is proposed. Such approach can be promising since it reduces the domain in which the Boltzmann collision integral should be used.The results presented in this paper were obtained using two different software packages Unified Flow Solver (UFS) [13] and Nesvetay 3D [14-15]. Note that UFS uses the discrete ordinate method for velocity space on a uniform grid and a hierarchical adaptive mesh refinement in physical space. The possibility of calculating both the Boltzmann equation and model equations is realized. The Nesvetay 3D complex was created to solve the Shakhov model equation, (S-model) and makes it possible to calculate on non-structured non uniform grids in velocity and physical spaces.Translated from Russian. Original text: Mathematics and Mathematical Modeling. 2018. no. 4. Pp. 27-44.

The kinetic Shakhov–Enskog model for non-equilibrium flow of dense gases

When the average intermolecular distance is comparable to the size of gas molecules, the Boltzmann equation, based on the dilute gas assumption, becomes invalid. The Enskog equation was developed to account for this finite size effect that makes the collision non-local and increases the collision frequency. However, it is time-consuming to solve the Enskog equation due to its complicated structure of collision operator and high dimensionality. In this work, on the basis of the Shakhov model, a gas kinetic model is proposed to simplify the Enskog equation for non-ideal monatomic dense gases. The accuracy of the proposed Shakhov–Enskog model is assessed by comparing its solutions of the normal shock wave structures with the results of the Enskog equation obtained by the fast spectral method. It is shown that the Shakhov–Enskog model is able to describe non-equilibrium flow of dense gases, when the maximum local mean free path of gas molecules is still greater than the size of a molecular diameter. The accuracy and efficiency of the present model enable simulations of non-equilibrium flow of dense gases for practical applications.

Comparison of the Shakhov kinetic equation and DSMC method as applied to space vehicle aerothermodynamics

The paper is devoted to the comparative study of Shakhov model kinetic equation and Direct Simulation Monte Carlo (DSMC) solutions as applied to high-speed flow of a monatomic gas over generic three-dimensional space vehicle under angle of attack. The corresponding calculations are carried out using Nesvetay and SMILE software packages, developed by the authors. Satisfactory agreement is shown for surface distribution of pressure, friction and heat transfer coefficients despite very different modeling approaches and computational meshes employed. It is thus shown that the use of the model kinetic equations with approximate collision integrals provides accurate results for high-speed external monatomic gas flows over complex three-dimensional geometries at acceptable computational cost.