Model Collision Integral Research Articles

Arbitrary Lagrangian-Eulerian discrete velocity method with application to laser-induced plume expansion

We propose a direct arbitrary Lagrangian-Eulerian (ALE) variant of the previously developed discrete velocity method to solve the kinetic equation with the Bhatnagar–Gross–Krook (BGK) model collision integral. The effectiveness and robustness of the new scheme are demonstrated by computing the axisymmetric plume expansion into a low pressure gas due to evaporation from the solid surface caused by a nanosecond laser pulse. The numerical studies show that the new numerical scheme provides a very significant reduction of the required computing time as compared to the discrete velocity method on a fixed mesh. Cross-comparisons with the direct simulation Monte Carlo (DSMC) approach further confirm the accuracy of the new scheme and model kinetic equations as applied to plume expansion studies. Finally, the good parallel scalability of the new ALE scheme as implemented into the in-house code Nesvetay is demonstrated. The use of two very different approaches to numerical prediction of the transient flow pattern allows us to obtain a reliable numerical solution, which can be regarded as a new transient benchmark test.

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.

Solitary waves in a two-dimensional graphene-based superlattice

The article is about the features of the propagation of solitary electromagnetic waves in the two-dimensional graphene superlattice both in the collisionless mode and in the collision mode. The quasiclassical approach has been used, in which the law of the dispersion of charge carriers is determined by the approximation of the quantum mechanical calculations. The magnitude of the electric current has been calculated by using the classic kinetic Boltzman equation with the model collision integral in the constant relaxation frequency approximation. The effect of the high-frequency electric field and the nonadditivity of the energy spectrum on the propagation of a solitary electromagnetic pulse in the arbitrary directions inside a sample has been determined. The numerical simulation of the evolution of solitary electromagnetic pulses is performed by using the method of difference schemes.

Influence of Runaway Electrons on the Formation Time of Nanosecond Discharge

In this paper, we present the theoretical results of a self-consistent kinetic simulation, clarifying the influence of runaway electron flows on a delay time of the switching stage of a nanosecond discharge. The simulation is based on an accurate numerical solution of the Boltzmann kinetic equation, with model collision integrals that take into account elastic and ionization collisions of the electrons with neutral atoms. As an example, the breakdown of a cylindrical gap is investigated with respect to the variations of the voltage rise time value. The current and voltage time profiles, power spectrum, and current of high-energy electrons, as well as the electron distribution function, are calculated accordingly. Our numerical solution agrees with the existent experimental data.

Stable Runge-Kutta discontinuous Galerkin solver for hypersonic rarefied gaseous flow based on 2D Boltzmann kinetic model equations

A stable high-order Runge-Kutta discontinuous Galerkin (RKDG) scheme that strictly preserves positivity of the solution is designed to solve the Boltzmann kinetic equation with model collision integrals. Stability is kept by accuracy of velocity discretization, conservative calculation of the discrete collision relaxation term, and a limiter. By keeping the time step smaller than the local mean collision time and forcing positivity values of velocity distribution functions on certain points, the limiter can preserve positivity of solutions to the cell average velocity distribution functions. Verification is performed with a normal shock wave at a Mach number 2.05, a hypersonic flow about a two-dimensional (2D) cylinder at Mach numbers 6.0 and 12.0, and an unsteady shock tube flow. The results show that, the scheme is stable and accurate to capture shock structures in steady and unsteady hypersonic rarefied gaseous flows. Compared with two widely used limiters, the current limiter has the advantage of easy implementation and ability of minimizing the influence of accuracy of the original RKDG method.

Derivation of model kinetic equation

A new form of the model collision operator is derived. One-component and many-component systems are considered. The collision operator proposed takes properly into account the relaxation of the first 13 hydrodynamic moments. A technique for construction of the model collision integral based on a known expression for the linearized model operator is proposed. It is shown that, within our model, the model collision operator does not contain the complicated exponential, common for ellipsoidal statistical models. Boltzmann's H-theorem is proved for our model.

Dissipative transport in superlattices within the Wigner function formalism

We employ the Wigner function formalism to simulate partially coherent, dissipative electron transport in biased semiconductor superlattices. We introduce a model collision integral with terms that describe energy dissipation, momentum relaxation, and the decay of spatial coherences (localization). Based on a particle-based solution to the Wigner transport equation with the model collision integral, we simulate quantum electronic transport at 10 K in a GaAs/AlGaAs superlattice and accurately reproduce its current density vs field characteristics obtained in experiment.

Model Kinetic Description For Many-Component Plasma

A consistent derivation of the model linearized collision operator for a multicomponent system is presented. In these results an ambiguity in the choice of coefficients is eliminated, in contrast to the BGK type models. A technique for reconstruction of the model collision integral form based on a known expression for the model linearized operator is proposed. It is shown that the model collision integral in the local (not complete) equilibrium approximation does not contain a complicated exponential, that is common for the BGK type integrals. Boltzmann's H-theorem is proved for our model.

Construction and comparison of parallel implicit kinetic solvers in three spatial dimensions

The paper is devoted to the further development and systematic performance evaluation of a recent deterministic framework Nesvetay-3D for modelling three-dimensional rarefied gas flows. Firstly, a review of the existing discretization and parallelization strategies for solving numerically the Boltzmann kinetic equation with various model collision integrals is carried out. Secondly, a new parallelization strategy for the implicit time evolution method is implemented which improves scaling on large CPU clusters. Accuracy and scalability of the methods are demonstrated on a pressure-driven rarefied gas flow through a finite-length circular pipe as well as an external supersonic flow over a three-dimensional re-entry geometry of complicated aerodynamic shape.

Rarefied gas flow in a circular pipe of finite length

The paper is devoted to the study of a rarefied gas flow through a circular pipe caused by small pressure differences between the reservoirs attached to the ends of the pipe. The analysis is based on of the direct numerical solution of the Boltzmann kinetic equation with the linearized model collision integral of Shakhov. The solution of the problem is computed for a wide range of the rarefaction parameter values and is compared with the kinetic and continuum solutions for an infinitely long circular pipe as well as with the flow into vacuum.

A comparative analysis of approaches for investigating hypersonic flow over blunt bodies in a transitional regime

Hypersonic flows of a viscous perfect rarefied gas over blunt bodies in a transitional flow regime from continuum to free molecular, characteristic when spacecraft re-enter Earth's atmosphere at altitudes above 90-100km, are considered. The two-dimensional problem of hypersonic flow is investigated over a wide range of free stream Knudsen numbers using both continuum and kinetic approaches: by numerical and analytical solutions of the continuum equations, by numerical solution of the Boltzmann kinetic equation with a model collision integral in the form of the S-model, and also by the direct simulation Monte Carlo method. The continuum approach is based on the use of asymptotically correct models of a thin viscous shock layer and a viscous shock layer. A refinement of the condition for a temperature jump on the body surface is proposed for the viscous shock layer model. The continuum and kinetic solutions, and also the solutions obtained by the Monte Carlo method are compared. The effectiveness, range of application, advantages and disadvantages of the different approaches are estimated.

Analytical solution of Stokes’ second problem on the behavior of rarefied gas over an oscillating surface

Stokes’ second problem on the behavior of rarefied gas occupying a half-space is analytically solved. The plane bounding the half-space executes harmonic oscillations. The kinetic equation with the model collision integral in the form of a τ-model is used and the case of diffuse reflection of gas molecules from the wall is considered. The distribution function of gas molecules is constructed and the mass velocity of the gas in the half-space, together with its value directly at the wall, is determined. The drag force acting from the gas on the boundary executing oscillatory motion in its plane is obtained. Moreover, the energy dissipation rate per unit area of the oscillating plate bounding the gas is determined.

Efficient Deterministic Modelling of Three-Dimensional Rarefied Gas Flows

AbstractThe paper is devoted to the development of an efficient deterministic framework for modelling of three-dimensional rarefied gas flows on the basis of the numerical solution of the Boltzmann kinetic equation with the model collision integrals. The framework consists of a high-order accurate implicit advection scheme on arbitrary unstructured meshes, the conservative procedure for the calculation of the model collision integral and efficient implementation on parallel machines. The main application area of the suggested methods is micro-scale flows. Performance of the proposed approach is demonstrated on a rarefied gas flow through the finite-length circular pipe. The results show good accuracy of the proposed algorithm across all flow regimes and its high efficiency and excellent parallel scalability for up to 512 cores.

On the accuracy of the BGK model for ion drift in own gas

The model of collisions of ions with gas atoms, considering resonant charge exchange of ions, polarization and elastic (gas-kinetic) interactions is constructed. Ion drift characteristics in the dc electric field are calculated. The results are compared to calculations based on the Bhatnagar-Gross-Krook model collision integral (BGK integral). It is shown that the use of the BGK collision integral leads to significant errors due to the specificity of ion-atom collisions.

Mechanism of the anisotropic distribution of molecules freely moving in a channel with rough walls

A mechanism of the anisotropic distribution of molecules with respect to their directions of motion in a channel with rough walls is proposed. Based on a solution of the kinetic equation with a model collision integral and with allowance for the multiple scattering of gas molecules from the rough channel walls, it is shown that a flow driven by pressure gradient is governed by the Fick’s law with an effective diffusion coefficient determined by the geometry of surface inhomogeneities.

On the model kinetic description of plasma and a Boltzmann gas of hard spheres

A new form of the collision operator for a Boltzmann gas of hard spheres and Coulombplasma is proposed. One-component and many-component systems are considered. Theproposed collision operator properly takes into account the relaxation of the first 13hydrodynamic moments. Besides this, it accounts for the non-diagonal componentcontribution in the quadratic approximation in the expansion of the linearized collisionoperator with respect to the complete system of Hermite polynomials. It is shown thatfor a system of charged particles with the Coulomb interaction potential, thesecontributions are essential and lead to Spitzer corrections to the transport coefficients. Anexpression for the intensity of the Langevin source in the kinetic equation is obtainedin the same approximation. A new form of the model collision operator for aBoltzmann gas of hard spheres is proposed. For a many-component system we havereconstructed a non-linear model collision integral by using the linearized collisionintegral found. Unlike previous ones, it does not contain complicated exponentialdependence and avoids coefficient ambiguity in the many-component collision integral.

Conservative numerical methods for model kinetic equations

A new conservative discrete ordinate method for nonlinear model kinetic equations is proposed. The conservation property with respect to the collision integral is achieved by satisfying at the discrete level approximation conditions used in deriving the model collision integrals. Additionally to the conservation property, the method ensures the correct approximation of the heat fluxes. Numerical examples of flows with large gradients are provided for the Shakhov and Rykov model kinetic equations.

Kinetic theory of electrostatic ion cyclotron waves (QPESIC) in multicomponent plasmas with negative ions

The instabilities of the quasi-perpendicular electrostatic (?B = 0) ioncyclotron waves (QPESIC) are investigated. The kinetic theory with BGK model collision integrals is used to estimate the critical electron drift velocity in the presence of positively or negatively charged resonant ions in multi-component plasma. Analytical evaluation for the ion-cyclotron modes and instabilities in the long-wave range in a weakly-ionized Maxwellian plasma with two positive ion species, one negative ion species and with electrons, drifting along magnetic lines of force is demonstrated. The spectrum in these situations is also given. It is shown that the critical drift decreases as the state of plasma approaches the isothermic state.

Kinetic approach for the ion drag force in a collisional plasma

The linear kinetic approach to calculate the ion drag force in a collisional plasma is generalized. The model collision integral (for ion-neutral collisions) is discussed and employed to calculate the plasma response for arbitrary velocity of the plasma flow and arbitrary frequency of the collisions. The derived plasma response is used to calculate the self-consistent force on the test charged particle. The obtained results are compared to those of the traditional pair collision approach, and the importance of the self-consistent kinetic consideration is highlighted. In conclusion, the applicability of the proposed approach is discussed.

Modeling of Coulomb collisions in a kinetic description of the electron cyclotron resonance plasma heating

The electron distribution function is modeled numerically with allowance for Coulomb collisions and quasilinear effects under cyclotron resonance conditions by solving a two-dimensional kinetic equation containing the quasilinear diffusion operator and the Coulomb collision operator in the Landau form. Two simplified model collision integrals that make it possible to describe electron heating by microwave radiation are considered. The first model collision operator is obtained by introducing the parametric time dependence of the temperature of the background Maxwellian electrons into the linear collision integral. It is shown that the heating of the bulk electrons can be described in a noncontradictory way if the temperature dynamics of the background electrons is calculated from the equation of energy balance, which is governed by the amount of the microwave power absorbed by the resonant electrons with the distribution function modified due to quasilinear effects. This conclusion is confirmed in a more rigorous fashion by comparing the solutions obtained using the first model Coulomb collision integral with those obtained using the second model integral, namely, the nonlinear operator derived by averaging the distribution function of the scattering electrons over pitch angles. The time-dependent linear collision integral is used to obtain analytic solutions describing quasi-steady electron heating with allowance for the quasilinear degradation of microwave power absorption.