Solution Of Boltzmann Equation Research Articles

Exact solution of Boltzmann equation in a longitudinal expanding system

Exact solution of Boltzmann equation in a longitudinal expanding system

On the investigation of shock wave/boundary layer interaction with a high-order scheme based on lattice Boltzmann flux solver

Shock wave/boundary layer interaction (SWBLI) continues to pose a significant challenge in the field of aerospace engineering. This paper aims to address this issue by proposing a novel approach for predicting aerodynamic coefficients and heat transfer in viscous supersonic and hypersonic flows using a high-order flux reconstruction technique. Currently, finite volume methods are extensively employed for the computation of skin aerodynamic coefficients and heat transfer. Nevertheless, these numerical methods exhibit considerable susceptibility to a range of factors, including the inviscid flux function and the computational mesh. The application of high-order flux reconstruction techniques offers promising potential in alleviating these challenges. In contrast to other high-order methods, the flux reconstruction is combined with the lattice Boltzmann flux solver in this study. The current method evaluates the common inviscid flux at the cell interface by locally reconstructing the lattice Boltzmann equation solution from macroscopic flow variables at solution points. Consequently, this framework performs a positivity-preserving, entropy-based adaptive filtering method for shock capturing. The present approach is validated by simulating the double Mach reflection, and then simulating some typical viscous problems. The results demonstrate that the current method accurately predicts aerodynamic coefficients and heat transfer, providing valuable insights into the application of high-order methods for shock wave/boundary layer interaction.

High‐order gas kinetic flux solver for viscous compressible flow simulations

AbstractAlthough the gas kinetic schemes (GKS) have emerged as one of the powerful tools for simulating compressible flows, they exhibit several shortcomings. Since the local solution of continuous Boltzmann equation with the Maxwellian distribution function is used to calculate the numerical fluxes at the cell interface, the flux expression in GKS is usually more complicated. In this paper, a high‐order simplified gas kinetic flux solver (GKFS) is presented for simulating two‐dimensional compressible flows. Circular function‐based GKFS (C‐GKFS), which simplifies the Maxwellian distribution function into the circular function, combined with an improved weighted essentially non‐oscillatory (WENO‐Z) scheme is applied to capture more details of the flow fields with fewer grids. As a result, a simple high‐order accurate C‐GKFS is obtained, which improves the computing efficiency and reduce its complexity to facilitate the practical application of engineering. A series of benchmark‐test problems are simulated and good agreement can be obtained compared with the references, which demonstrate that the high‐order C‐GKFS can achieve the desired accuracy.

Second order hydrodynamics based on effective kinetic theory and electromagnetic signals from QGP

We study the thermal dilepton and photon production from relativistic heavy ion collisions in presence of viscosities by employing the recently developed second order dissipative hydrodynamic formulation estimated within a quasiparticle description of thermal quantum chromo-dynamics (QCD) medium. The sensitivity of shear and bulk viscous pressures to the temperature dependence of relaxation time is analyzed within one- dimensional boost invariant expansion of quark gluon plasma (QGP). The dissipative corrections to the phase-space distribution functions up to first order in gradients are obtained from the Chapman–Enskog like iterative solution of effective Boltzmann equation in the relaxation time approximation. Thermal dilepton and photon production rates for QGP are calculated by employing this viscous modified distribution function. Yields of these particles are quantified for the longitudinal expansion of QGP with different temperature dependent relaxation times. Our analysis employing this second order hydrodynamic model indicates that the spectra of dileptons and photons get enhanced by both bulk and shear viscosities and are well behaved. Also, these particle yields are found to be sensitive to relaxation time. Further, we do a comparison of these particle spectra with a standard hydrodynamic formulation.

Boundary Layer Solution of the Boltzmann Equation for Specular Boundary Condition

In the paper, we establish the existence of steady boundary layer solution of Boltzmann equation with specular boundary condition in L 2x,v ∩L ∞x,v in half-space. The uniqueness, continuity and exponential decay of the solution are obtained, and such estimates are important to prove the Hilbert expansion of Boltzmann equation for half-space problem with specular boundary condition.

Baryon asymmetry and lower bound on right handed neutrino mass in fast expanding Universe: an analytical approach

The expansion rate of the Universe deviates from its standard value when the total energy density includes contribution from a new scalar field apart from the radiation energy density. The non-trivial modifications incurred in the Boltzmann equations render the well known analytical solutions unsuitable in non standard scenario. In the present study we derive analytical expressions for the efficiency factor (which is nothing but solution of set of Boltzmann equations) using certain legible approximations. A fair degree of accuracy of these formulas have been observed by juxtaposing the analytical results with that obtained through numerical solution of Boltzmann equations. Faster expansion of the Universe results in decrement of the effective decay parameter which brings down the amount of washout of asymmetry due to inverse decay. Thus in non-standard cosmology scenario, a larger fraction of the asymmetry (generated at early epoch) is expected to survive till present epoch. Alteration of the cosmology does not affect the underlying particle physics model responsible for the generation of the CP asymmetry. Therefore starting from an identical particle physics model we will end up with a larger final baryon asymmetry in the non-standard scenario. It hints towards the possible relaxation of the lower bound of the lightest right handed neutrino mass required to produce adequate asymmetry which is in agreement with current experimental data.

Anisotropic expansion, dissipative hydrodynamics from kinetic theory

In this paper, we consider Kasner space-time describing anisotropic three-dimensional expansion of the fluid and obtain the dissipative evolution equations for shear stress tensor and energy density from kinetic theory. For this, we use the iterative solution of relativistic Boltzmann equation with relaxation time approximation. We show that our results for second- and third-order evolution equations reduce to those of one-dimensional expansion case under suitable conditions for the anisotropic parameters in Kasner space-time.

A combined flux reconstruction and lattice Boltzmann flux solver for inviscid compressible flow simulation

AbstractA numerical method, which combines the high order flux reconstruction (FR) and the lattice Boltzmann flux solver (LBFS), has been developed to solve inviscid compressible flows in this article. Extensive research has shown that LBFS has absorbed the advantages of both finite volume method (FVM) and lattice Boltzmann method (LBM). On one hand, compared with FVM, FR can achieve arbitrary high order accuracy and is compact for parallel computing by avoiding wide stencils on meshes, which shows FR is a better choice than FVM. On the other hand, different from the discrete velocity Boltzmann equation based methods, the present method, that is, FR‐LBFS, solves the Euler equations essentially, and then the calculation efficiency can be improved. In FR‐LBFS, the common inviscid flux at the cell interface is evaluated by the local reconstruction of the lattice Boltzmann equation (LBE) solution from macroscopic flow variables at solution points. Unlike the traditional flux solution at the interface where the nonlinear convective terms have to be considered in the computation, the feature of linear convection term in LBE can make it easier to calculate flux in the current method. Furthermore, by considering real physical effects in LBM, the positive density and pressure can be achieved in the case of strong shocks and discontinuities. Some typical inviscid problems, including the advection of density perturbation, sod shock tube, Riemann problem for two‐dimensional gas dynamics, supersonic flow in a convergent nozzle with a ramp, subsonic flow around a cylinder, and transonic flow over a NACA0012 airfoil, are simulated to demonstrate the accuracies and robustness of the proposed FR‐LBFS.

Propagation of Gevrey regularity for solution of non-cutoff Boltzmann equation

In this paper, we focus on the Gevrey regularity of the solution to non-cutoff Boltzmann equation. We prove the propagation of Gevrey regularity for γ<0, this extends the result of Desvillettes et al. (2009) on the case of Maxwellian molecules. Our proof relies on the elementary L2 weighted estimate.

Grad's distribution functions-based gas kinetic scheme for simulation of flows beyond Navier–Stokes level

Conventional gas kinetic scheme (GKS) has been successfully utilized to obtain the accurate solution of Navier–Stokes (NS) equations. However, when it comes to flows beyond NS level, the conventional GKS is not reliable because its initial gas distribution function is approximated by the first-order Chapman–Enskog expansion, which merely recovers the NS equations. In order to make an extension for flows beyond NS level, we propose the Grad's distribution functions-based GKS in this paper. This scheme retains the advantage of conventional GKS and constructs the numerical fluxes through a time-dependent gas distribution function, which is derived from the integral solution of Boltzmann equation. In the present scheme, the initial gas distribution function in the local solution of Boltzmann equation is approximated by Grad's 13 and 26 distribution functions. Furthermore, the high-order moments in the initial Grad's distribution function are calculated by moment relationship directly, and thus, the solution of complicated partial differential equations for these high-order moments is avoided. Four benchmark numerical examples are tested to validate the performance of the present scheme, and the results demonstrate that the present scheme can not only recover NS solutions in the continuum region but also predict reasonable results for flows in the slip and transition regimes.

Momentum and energy relaxation in femtosecond-scale energy transport in metals

In the current report we investigate the processes of heat transport and energy relaxation in metals using Boltzmann transport equation for the electron distribution function perturbation, simultaneously accounting both energy and momentum relaxation. Mutual influence of electron-electron and electron-phonon energy relaxation on the electron distribution function is considered using linearized integro-differential collision integrals. These energy exchange effects provide main contribution to the difference between results obtained using Boltzmann equation solution and widely considered two-temperature model.

S-, P- and R-striations as attractors for electron phase trajectories in spatially periodic resonance fields

A new point of view on the appearance of S-, P- and R-striations in a positive column of inert gases is proposed, based on a dynamic analysis of the resonance properties of electron phase trajectories in spatially periodic fields. The positive column may be considered as a resonator containing a set of resonant modes. Like a tuning fork, being disturbed, it responds with one of the modes, in particular with of S-, P-, or R-modes or striations, depending on the discharge conditions. The dynamic approach eliminates the difficulties of the kinetic theory associated with the long length of the solution of Boltzmann equation, which is much greater than the length of the positive column.

Non-equilibrium evaporation: 1D benchmark problem for single gas

Evaporation is one of the most effective processes for heat transfer enhancement in the confined spaces. The development of efficient tools for modeling of evaporation processes requires a clear understanding of the applicability of existing approaches. The planar evaporation of a monoatomic gas at low and arbitrary evaporation speed (Mach number lower than 0.5) is simulated using the numerical solution of the S-model kinetic equation. These results are compared with the Molecular Dynamics simulations available from literature and our simulations performed using the LAMMPS tools. The S-model kinetic equation is shown to be an excellent time-effective model to simulate the structure inside and beyond from the Knudsen layer. It is also shown that non-linear Moment Method of the solution of Boltzmann equation gives the results in good agreement with the S-model and Molecular Dynamics data and it can be used for engineering applications when the heat flux in the gas phase above the Knudsen layer is negligible. The useful tool for evaporation rate calculation based on analytical Moment Method formulas is provided as well. It is shown that the well-known Schrage expression considerably overestimate the evaporation rates with the evaporation coefficients calculated using Molecular Dynamics approach. The provided results could be used as the benchmark data to test evaporation models.

Synergism analysis in dielectric strength of CO2+N2+O2 ternary mixtures

The synergy effect of O2 content in dielectric strength of CO2+N2+O2 ternary mixtures with constant N2 ratio is investigated from microscopic properties of the component gases. AC Breakdown voltage measurements in the ternary mixtures subjected to uniform electric fields verify the two term Boltzmann equation solutions carried out with the Bolsig + program in the ternary mixtures. The dielectric strength of the ternary mixture increases as the O2 content is increased and the effect is pronounced at certain fixed N2 component ratios in the mixtures. The experimental results are valid in the range of 1 cm-bar where the Paschen’s Law apply. For the uniform field gap, lightning impulse (LI) voltage breakdown voltages are also measured in the ternary mixtures, however the synergy effect is not significant under the present experimental parameters.

A multi-temperature universe can allow a sub-MeV dark photon dark matter

An analysis of sub-MeV dark photon as dark matter is given which is achieved with two hidden sectors, one of which interacts directly with the visible sector while the second has only indirect coupling with the visible sector. The formalism for the evolution of three bath temperatures for the visible sector and the two hidden sectors is developed and utilized in solution of Boltzmann equations coupling the three sectors. We present exclusion plots where the sub-MeV dark photon can be dark matter. The analysis can be extended to a multi-temperature universe with multiple hidden sectors and multiple heat baths.

Phonon vortex dynamics in graphene ribbon by solving Boltzmann transport equation with ab initio scattering rates

In this work, we study thermal phonon vortex in graphene ribbon by a discrete-ordinate solution of phonon Boltzmann equation under Callaway's dual relaxation model. The phonon scattering rates of normal and resistive processes are acquired from ab initio calculation without need of any empirical input parameters. The temperature, size and isotope effects on transition from phonon vortex transport to conventional Fourier's heat conduction in both simple and complex geometries are investigated. The physical mechanism for the evolution of phonon vortex is declared by wide phonon mean free path distribution of resistive processes. A hierarchical vortex series is obtained with primary, secondary and terniary vortexes in a complicated geometry. The present work provides an accurate and efficient multi-scale numerical framework for modeling hydrodynamic phonon transport in high-thermal-conductivity materials and also sheds light on the heat dissipation applications.

INVESTIGATION OF ELECTRONS DIFFUSION COEFFICIENT DEPENDENCE ON TEMPERATURE IN PLASMA GAS CO2, N2, HE AND MIXTURES

The evolution of the μNth value at different temperatures was achieved through the drift velocity of electron. The results were show when the temperature was increased, the number of the electrons will be decreased because using the momentum transfer cross section for CO2 molecules through collisions. The calculation of the diffusion coefficient was used to deduce the μNth values of CO2 electrons at temperature between 288 to 573 k by utilization numerically the Boltzmann equation solution. The results were appearing the agreement with the theoretical and experimental data

Formulações espectrais para solução de problemas de transporte de partículas

In this work, a deterministic approach to the solution of the integro differential linear Boltzmann equation, in one and twodimensional media, is presented. Explicit solutions, in terms of the spatial variables, for the discrete ordinates approximation of the original model are obtained from a spectral formulation. A relevant feature of the methodology is the reduced order of the eigenvalue problems, which is given as half of the number of the discrete directions. Such aspect, along with the use of arbitrary quadrature schemes to represent the integral term of the equation, are fundamental to provide fast, concise and accurate solutions to several problems of interest. In particular, the solutions here are derived for models associated with two different applications: rarefied gas dynamics and neutron transport. Mathematical aspects present in the solution of the two models are emphasized and extensions of the formulation to other applications are commented and referenced.

Grad's Second Problem and Its Solution Within the Framework of Burnett Hydrodynamics

Abstract In his seminal work, Grad not only derived 13 moment equations but also suggested two problems to check his derived equations. These problems are highly instructive as they bring out the character of the equations by examining their solutions to these problems. In this work, we propose Grad's second problem as the potential benchmark problem for checking the accuracy of different sets of higher-order transport equations. The problem definition can be stated as: examination of steady-state solution for a gas at rest in infinite domain upon application of a one-dimensional heat flux. With gas at rest (no bulk velocity), the interest lies in obtaining the solution for pressure and temperature. The problem is particularly interesting with respect to the solution for pressure when Maxwell and hard-sphere molecules are considered. For Maxwell molecules, it is well known that the exact normal solution of Boltzmann equation gives uniform pressure with no stresses in the flow domain. In the case of hard-sphere molecules, direct simulation Monte Carlo (DSMC) results predict nonuniform pressure field giving rise to stresses in the flow domain. The simplistic nature of the problem and interesting results for pressure for different interaction potentials makes it an ideal test problem for examining the accuracy of higher-order transport equations. The proposed problem is solved within the framework of Burnett hydrodynamics for hard-sphere and Maxwell molecules. For hard-sphere molecules, it is observed that the Burnett order stresses do not become zero; they rather give rise to a pressure gradient in a direction opposite to that of temperature gradient, consistent with the DSMC results. For Maxwell molecules, the numerical solution of Burnett equations predicts uniform pressure field and one-dimensional temperature field, consistent with the exact normal solution of the Boltzmann equation.

An improved hybrid lattice Boltzmann flux solver for 3D integrated hypersonic fluid-thermal-structural analysis

In this paper, a hybrid lattice Boltzmann flux solver (LBFS) is proposed for simulation of 3D integrated hypersonic fluid-thermal-structural problems. In the solver, the macroscopic 3D Navier–Stokes equations and structural heat transfer equation are discretized by the finite volume method and the numerical fluxes at the cell interface are reconstructed by the local solution of Boltzmann equation. To compute the numerical fluxes, two lattice velocity models are introduced. One is the D1Q4 discrete velocity model for calculating the inviscid flux across the cell interface of N–S equations, and the other one is the D3Q6 model for evaluating the flux of structural energy equation. Furthermore, a new dual-thermal-resistance model is proposed to calculate the thermal properties on the fluid-structure interface. To validate the accuracy and stability of the present solver, applications for hypersonic fluid-thermal-structural analysis are demonstrated on aerodynamically heated blunt cone body at [Formula: see text]. Numerical results showed that the present solver can predict accurately the thermal properties of hypersonic fluid-thermal-structural problems and offer the potential for significant improvements in predicting fluid-structural-thermal problems of long-endurance high speed vehicles.