Chapman-Enskog Approximation Research Articles

An efficient discrete unified gas-kinetic scheme for compressible thermal flows

In this paper, an efficient discrete unified gas-kinetic scheme (DUGKS) is developed for compressible thermal flows based on the total energy kinetic model for natural convection with a large relative temperature difference. A double distribution function model is designed with the second distribution representing the total energy. This efficient DUGKS enables the simulation of compressible thermal flows, governed by the compressible Navier–Stokes–Fourier system, using only a seventh-order, off-lattice Gauss–Hermite quadrature (GHQ) D3V27A7 combined with a fifth-order GHQ D3V13A5. The external force is included by truncated Hermite expansions. Based on the Chapman–Enskog approximation and Hermite projection, we propose a systematic approach to derive the discrete kinetic boundary conditions for the density and total energy distribution functions. The discrete kinetic boundary treatments are provided for the no-slip boundary condition, Dirichlet boundary condition and Neumann boundary condition. To validate our scheme, we perform simulations of steady natural convection (Ra=103−106) in two- and three-dimensional cavities with differentially heated sidewalls and a large temperature difference (ε=0.6), where the Oberbeck–Boussinesq approximation is invalid. The results demonstrate that the current efficient DUGKS is robust and accurate for thermal compressible flow simulations. With the D3V27A7 and D3V13A5 off-lattice discrete particle velocity model, the computational efficiency of the DUGKS is improved by a factor of 3.09 when compared to the previous partial energy kinetic model requiring the ninth-order Gauss–Hermite quadrature.

Consistent lifting relations for the initialization of total-energy double-distribution-function kinetic models.

A lifting relation connecting the distribution function explicitly with the hydrodynamic variables is necessary for the Boltzmann equation-based mesoscopic approaches in order to correctly initialize a nonuniform hydrodynamic flow. We derive two lifting relations for Guo etal.'s total-energy double-distribution-function (DDF) kinetic model [Z. L. Guo etal., Phys. Rev. E 75, 036704 (2007)1539-375510.1103/PhysRevE.75.036704], one from the Hermite expansion of the conserved and nonconserved moments, and the second from the O(τ) Chapman-Enskog (CE) approximation of the Maxwellian exponential equilibrium. While both forms are consistent to the compressible Navier-Stokes-Fourier system theoretically, we stress that the latter may introduce numerical oscillations under the recently optimized discrete velocity models [Y. M. Qi etal., Phys. Fluids 34, 116101 (2022)10.1063/5.0120490], namely a 27 discrete velocity model of the seventh-order Gauss-Hermite quadrature (GHQ) accuracy (D3V27A7) for the velocity field combined with a 13 discrete velocity model of the fifth-order GHQ accuracy (D3V13A5) for the total energy. It is shown that the Hermite-expansion-based lifting relation can be alternatively derived from the latter approach using the truncated Hermite-polynomial equilibrium. Additionally, a relationship between the order of CE expansions and the truncated order of Hermite equilibria is developed to determine the minimal order of a Hermite equilibria required to recover any multiple-timescale macroscopic system. Next, three-dimensional compressible Taylor-Green vortex flows with different initial conditions and Ma numbers are simulated to demonstrate the effectiveness and potential issues of these lifting relations. The Hermite-expansion-based lifting relation works well in all cases, while the Chapman-Enskog-expansion-based lifting relation may produce numerical oscillations and a theoretical model is developed to predict such oscillations. Furthermore, the corresponding lifting relations for Qi etal.'s total energy DDF model [Y. M. Qi etal., Phys. Fluids 34, 116101 (2022)10.1063/5.0120490] are derived, and additional simulations are performed to illustrate the generality of our approach.

Spatiotemporal evolution of electrical conductivity in current-limiting-fuse arc

To improve the interruption capacity of a current-limiting fuse, a detailed diagnosis of the spatial electrical conductivity distribution inside the fuse arc under the current limiting phase around current zero is required, because this distribution determines the distribution of transient recovery voltage inside the fuse. However, well-established methodologies applicable to fuse arcs are lacking, so the spatial distribution remains unknown. This study presents a borescope-integrated spectroscopic system that simultaneously obtains single-shot recordings of the axial distributions of the electron density and arc temperature in the fuse arc just before extinction. Combining the electron densities and arc temperatures, we can identify the fuse arc composition and hence calculate the axial electrical conductivity distribution under the first Chapman–Enskog approximation. The electrical conductivity provided by this systematic methodology includes no large uncertainties, thus demonstrating its superiority against previous estimation methods of the electrical conductivity.

Shear viscosity coefficient of dilute gases via the ANC2s interaction potential

Molecular dynamics simulations and kinetic theory at the level of the Chapman-Enskog approximation were used to study repulsive and attractive contributions for a family of non-conformal potentials. The analysis was performed for the shear viscosity coefficient in the dilute-gas limit. To obtain a set of parameters to adequately model molecular interactions, collision integrals from kinetic theory were interpolated using experimental data on spherical and quasi-spherical molecular systems. Theoretical and simulation results show relative differences of 0.6%-2.3% and 0.4%-3.7% when comparing experimental data, respectively. The accuracy of the Chapman-Enskog first-order approximation was estimated using a Chapman-Cowling third-order correction. Results estimated to first order display differences of up to 0.8% when compared to higher-order corrections. Molecular interactions were modeled using the potential from the Approximate Non-Conformal (ANC) theory, which depends on length and energy scales, in addition to a softness parameter. A more refined version of the theory denoted ANC2s modulates repulsive and attractive contributions independently, for which the “2s” is mnemonic for the corresponding two softness parameters. In this work, we illustrate the robustness of the ANC2s representation by capturing the shear viscosity coefficient in the dilute-gas limit for a variety of molecular systems and by reproducing other interaction potentials, in particular, the Lennard-Jones 9-6, 12-6, and 20-6 forms.

Effects of the hidden errors in the bounce back scheme on the lattice Boltzmann simulation of the turbulent channel flow using the multiple-relaxation-time collision model

In our recent paper [Dong et al., Phys. Fluids 34, 093608 (2022)], it is shown that hidden errors can be introduced by a bounce back scheme at the boundary nodes, due to the fact that it may not be entirely consistent with the Chapman–Enskog approximation of the lattice Boltzmann equation applied to the interior nodes. In this paper, we investigate the effects of these hidden errors on the lattice Boltzmann simulation of the turbulent channel flow with a multiple-relaxation-time (MRT) collision model, extending our previous study using the Bhatnagar–Gross–Krook and two-relaxation-time collision models applied to laminar non-uniform viscous flows. A theoretical framework for identifying the hidden errors in the MRT model is developed, and the hidden errors in two bounce back schemes, namely, the off-wall and on-wall bounce back schemes, are derived in terms of the hydrodynamic variables and relaxation rates. The results reveal several important differences in the expression of hidden errors between the two bounce back schemes. The analysis also points to a correction for the on-wall bounce back scheme when the external force is present. A set of six simulations of the turbulent channel flow, using the two bounce back schemes and three grid resolutions, are, then, performed to demonstrate that the magnitude of the hidden errors can significantly affect the simulated turbulence statistics, the local consistency with the Navier–Stokes equations, and the numerical stability.

Charge transfer cross sections and transport coefficients of 39K+ − 39K and 41K+ − 39K at low-energy

The transport coefficients, diffusion D(T), and reduced mobility K0 are calculated for systems 39K+ − 39K and 41K+ − 39K in a thermal range between 1 and 3000 K within the Chapman–Enskog approximation. This is preceded by a theoretical study of the collision of alkaline ions in their gases for low and intermediate energies in the range 10−14 ≤ E ≤ 10−3 (a.u.), and within this field, we calculate the interaction energy potentials and elastic and charge-transfer cross sections and discuss the isotopic effects in the 39 K+ − 39K and 41K+ − 39K collisions.

The impact of ion mobility coefficients on plasma discharge characteristics

In this paper, the high-accuracy ion mobility coefficient based on the Chapman–Enskog approximation to the solution of the Boltzmann equation for low pressure radio frequency plasma discharges is presented. We employ two-dimensional fluid simulations of the argon filled axisymmetric reactor, where the effect of new ion-kinetics-based fluid closure is compared to theoretical expressions and experimental data. The spatial profiles of plasma composition in the low pressure radio frequency capacitively coupled plasma are presented, which includes the metastable reactions in the simulation. Moreover, inelastic collision integrals terms, due to charge exchange inelastic collisions between ions and neutral species, have been also considered. A Monte Carlo simulation of kinetic ion energy distribution of impinging on the radio frequency powered electrode provides a measure of accuracy of the new transport model. From our simulation, the results that mirror the influence of ion mobility coefficient obtained by the Chapman–Enskog method on plasma physical quantities under different pressures, frequencies, and electrode gaps is in good agreement with experimental measurement results and theoretical expressions.

Fluctuation–dissipation relations in the absence of detailed balance: formalism and applications to active matter

We present a comprehensive study about the relationship between the way detailed balance is broken in non-equilibrium systems and the resulting violations of the fluctuation–dissipation theorem. Starting from stochastic dynamics with both odd and even variables under time-reversal, we derive an explicit expression for the time-reversal operator, i.e. the Markovian operator which generates the time-reversed trajectories. We then exploit the relation between entropy production and the breakdown of detailed balance to establish general constraints on the non-equilibrium steady-states (NESS), which relate the non-equilibrium character of the dynamics with symmetry properties of the NESS distribution. This provides a direct route to derive extended fluctuation–dissipation relations, expressing the linear response function in terms of NESS correlations. Such framework provides a unified way to understand the departure from equilibrium of active systems and its linear response. We then consider two paradigmatic models of interacting self-propelled particles, namely active Brownian particles and active Ornstein–Uhlenbeck particles. We analyze the non-equilibrium character of these systems (also within a Markov and a Chapman–Enskog approximation) and derive extended fluctuation–dissipation relations for them, clarifying which features of these active model systems are genuinely non-equilibrium.

An improved discrete unified gas kinetic scheme for simulating compressible natural convection flows

Discrete unified gas-kinetic scheme (DUGKS) has been developed recently as a general method for simulating flows at all Knudsen numbers. In this study, we extend DUGKS to simulate fully compressible thermal flows. We introduce a source term to the Boltzmann equation with the Bhatnagar-Gross-Krook (BGK) collision model [1] to adjust heat flux and thus the Prandtl number. The fully compressible Navier-Stokes equations can be recovered by the current model. As a mesoscopic CFD approach, it requires an accurate mesoscopic implementation of the boundary conditions. Using the Chapman-Enskog approximation, we derive the “bounce-back” expressions for both temperature and velocity distribution functions, which reveal the need to consider coupling terms between the velocity and thermal fields. To validate our scheme, we first reproduce the Boussinesq flow results by simulating natural convection in a square cavity with a small temperature difference (ϵ=0.01) and a low Mach number. Then we perform simulations of steady natural convection (Ra=1.0×106) in a square cavity with differentially heated side walls and a large temperature difference (ϵ=0.6), where the Boussinesq approximation becomes invalid. Temperature, velocity profiles, and Nusselt number distribution are obtained and compared with the benchmark results from the literature. Finally, the unsteady compressible natural convection with Ra=5.0×109,ϵ=0.6 is studied and the turbulent fluctuation statistics are computed and analyzed.

Finite scale theory: Predicting nature’s shocks

In this paper, we describe a continuum model that accurately reproduces the experimentally measured structure of physical shocks in a perfect gas. We begin by presenting a history of shock structure research, theoretical, experimental and numerical, to quantify the significant discrepancies between Navier–Stokes predictions and laboratory measurements. In our first main result, we discuss modifications that generalize the Chapman–Enskog approximation and lead to a new continuum model termed the Finite Scale Equations. In our second main result, we use this continuum model to calculate shock structure with results comparable in accuracy to numerical simulations based on the Boltzmann equation.

Scale matters.

The applicability of Navier-Stokes equations is limited to near-equilibrium flows in which the gradients of density, velocity and energy are small. Here I propose an extension of the Chapman-Enskog approximation in which the velocity probability distribution function (PDF) is averaged in the coordinate phase space as well as the velocity phase space. I derive a PDF that depends on the gradients and represents a first-order generalization of local thermodynamic equilibrium. I then integrate this PDF to derive a hydrodynamic model. I discuss the properties of that model and its relation to the discrete equations of computational fluid dynamics.This article is part of the theme issue 'Hilbert's sixth problem'.

Effect of vibrational degrees of freedom on the heat transfer in polyatomic gases confined between parallel plates

Conductive stationary heat transfer through rarefied nonpolar polyatomic gases, confined between parallel plates maintained at different temperatures, is investigated. It is assumed that gas molecules possess both rotational and vibrational degrees of freedom, described by the classical rigid rotator and quantum harmonic oscillator models, respectively. The flow structure is computed by the Holway kinetic model and the Direct Simulation Monte Carlo method. In both approaches the total collision frequency is computed according to the Inverse Power Law intermolecular potential. Inelastic collisions in DSMC simulations are based on the quantum version of the Borgnakke–Larsen collision model. Results are presented for N2, O2, CO2, CH4 and SF6 representing diatomic as well as linear and nonlinear polyatomic molecules with 1 up to 15 vibrational modes. The translational, rotational, vibrational and total temperatures and heat fluxes are computed in a wide range of the rarefaction parameter and for various ratios of the hot over the cold plate temperatures. Very good agreement, between the Holway and DSMC results is observed as well as with experiments. The effect of the vibrational degrees of freedom is demonstrated. In diatomic gases the vibrational heat flux varies from 5% up to 25% of the total one. Corresponding results in polyatomic gases with a higher number of vibrational modes show that even at low reference temperatures the contribution of the vibrational heat flux may be considerably higher. For example in the case of SF6 at 300K and 500K the vibrational heat flux is about 67% and 76% respectively of the total heat flux. Furthermore, it is numerically proved that the computed solutions are in agreement with the Chapman–Enskog approximation in a central strip of the computational domain even at moderately large values of the rarefaction parameter, as found in previous investigations. This property has been used to compute the gas thermal conductivity predicted by the adopted models.

Thermodynamic properties and transport coefficients of air thermal plasmas mixed with ablated vapors of Cu and polytetrafluoroethylene

Because the fault arc in aircraft electrical system often causes a fire, it is particularly important to analyze its energy and transfer for aircraft safety. The calculation of arc energy requires the basic parameters of the arc. This paper is mainly devoted to the calculations of equilibrium composition, thermodynamic properties (density, molar weight, enthalpy, and specific heat at constant pressure) and transport coefficients (thermal conductivity, electrical conductivity, and viscosity) of plasmas produced by a mixture of air, Cu, and polytetrafluoroethylene under the condition of local thermodynamic equilibrium. The equilibrium composition is determined by solving a system of equations around the number densities of each species. The thermodynamic properties are obtained according to the standard thermodynamic relationships. The transport coefficients are calculated using the Chapman-Enskog approximations. Results are presented in the temperature range from 3000 to 30 000 K for pressures of 0.08 and 0.1 MPa, respectively. The results are more accurate and are reliable reference data for theoretical analysis and computational simulation of the behavior of fault arc.

Transport coefficients of Quark-Gluon Plasma in a Kinetic Theory approach

One of the main results of heavy ions collision at relativistic energy experiments is the very small shear viscosity to entropy density ratio of the Quark-Gluon Plasma, close to the conjectured lower bound η/s = 1/4π for systems in the infinite coupling limit. Transport coefficients like shear viscosity are responsible of non-equilibrium properties of a system: Green- Kubo relations give us an exact expression to compute these coefficients. We computed shear viscosity numerically using Green-Kubo relation in the framework of Kinetic Theory solving the relativistic transport Boltzmann equation in a finite box with periodic boundary conditions. We investigated different cases of particles, for one component system (gluon matter), interacting via isotropic or anisotropic cross-section in the range of temperature of interest for HIC. Green-Kubo results are in agreement with Chapman-Enskog approximation while Relaxation Time approximation can underestimates the viscosity of a factor 2. Another transport coefficient of interest is the electric conductivity σel which determines the response of QGP to the electromagnetic fields present in the early stage of the collision. We study the σel dependence on microscopic details of interaction and we find also in this case that Relaxation Time Approximation is a good approximation only for isotropic cross-section.

Medium effects on the thermal conductivity of a hot pion gas

We investigate the effect of the medium on the thermal conductivity of a pion gas out of chemical equilibrium by solving the relativistic transport equation in the Chapman-Enskog and relaxation time approximations. Using an effective model for the $\ensuremath{\pi}\ensuremath{\pi}$ cross section involving $\ensuremath{\rho}$ and $\ensuremath{\sigma}$ meson exchange, medium effects are incorporated through thermal one-loop self-energies. The temperature dependence of the thermal conductivity is observed to be significantly affected.

On the right side of Einstein's equation

Every story has two sides and so does any good equation. Recent developments in observational cosmology have led to attempts to make modifications on both sides of the Einstein equation to explain some of the puzzling new findings. The work described here is concerned with an examination of the source of gravity that we usually find on the right-hand side of Einstein's equation. My aim is to describe a modified version of the stress-energy tensor that is the source of the gravitational field. The derivation is based on the kinetic theory of a gas of identical particles with no internal structure. The presentation here is in two parts. In Part I, I describe the stress tensor that Xinzhong Chen and I have proposed for the matter tensor for a nonrelativistic gas with input from Hongling Rao and Jean-Luc Thiffeault. Our derivation of the equations of fluid dynamics is based on kinetic theory and without recourse to the standard Chapman–Enskog approximation. The nonrelativistic treatment reveals the underlying physics clearly and it facilitates comparison with experimental results on acoustic propagation. Further, it provides a setting for a Newtonian cosmology, which remains a topic of some interest. I regret though that there is room only for the relativistic version of cosmology here. In Part II, I derive the stress-energy tensor in the relativistic case in a similar manner and exhibit its application to the usual isotropic cosmological model. The surprising result of that discussion is that, in addition to the Friedmann solution, we find a second solution that arises from terms that the usual Chapman–Enskog approximation discards. The new solution is a temporal analogue of a spatial shock wave. Just as the usual shock waves make transitions in properties within a mean free path, the new solution can change its properties in a mean flight time. Whereas the Friedmann solution is not dissipative, the new solution produces entropy at a rate that may be of cosmological interest. For the calculation of cosmic entropy production I work in the ultrarelativistic limit in which particle masses are negligible. A stability calculation to decide whether and when the new solution may be realized has yet to be done since the microphysics seems so uncertain.

Medium effects on the viscosities of a pion gas

The bulk and shear viscosities of a pion gas is obtained by solving the relativistic transport equation in the Chapman-Enskog approximation. In-medium effects are introduced in the $\pi\pi$ cross-section through one-loop self-energies in the propagator of the exchanged $\rho$ and $\sigma$ mesons. The effect of early chemical freeze-out in heavy ion collisions is implemented through a temperature dependent pion chemical potential. These are found to affect the temperature dependence of the bulk and shear viscosities significantly.

Shear viscosity and chemical equilibration of the QGP

We have investigated, in the frame work of the transport approach, different aspects of the QGP created in Heavy Ion Collisions at RHIC and LHC energies. The shear viscosity η has been calculated by using the Green-Kubo relation at the cascade level. We have compared the numerical results for η obtained from the Green-Kubo correlator with the analytical formula in both the Relaxation Time Approximation (RTA) and the Chapman-Enskog approximation (CE). From this comparison we show that in the range of temperature explored in a Heavy Ion collision the RTA underestimates the viscosity by about a factor of 2, while a good agreement is found between the CE approximation and Gree-Kubo relation already at first order of approximation. The agreement with the CE approximation supplies an analytical formula that allows to develop kinetic transport theory at fixed shear viscosity to entropy density ratio, η/s. We show some results for the build up of anisotropic flows in a transport approach at fixed shear viscosity to entropy density ratio, η/s. We study the impact of a T-dependent η/s(T) on the generation of the elliptic flows at both RHIC and LHC. We show that the transport approach provides, in a unified way, a tool able to naturally describe the v2(pT) in a wide range of pt, including also the description of the rise and fall and saturation of the v2(pT) observed at LHC. Finally, we have studied the evolution of the quark-gluon composition employing a Boltzmann-Vlasov transport approach that include: the mean fields dynamics, associated to the quasi-particle model, and the elastic and inelastic collisions for massive quarks and gluons. Following the chemical evolution from an initial gluon dominated plasma we predict a quark dominance close to Tc paving the way to an hadronization via quark coalescence.

Viscous Coefficients of a Hot Pion Gas

The steps essentially involved in the evaluation of transport coefficients in linear response theory using Kubo formulas are to relate the defining <i >retarded</i> correlation function to the corresponding <i >time-ordered</i> one and to evaluate the latter in the conventional perturbation expansion. Here we evaluate the viscosities of a pion gas carrying out both the steps in the <i >real-time</i> formulation. We also obtain the viscous coefficients by solving the relativistic transport equation in the Chapman-Enskog approximation to leading order. An in-medium <svg style="vertical-align:-0.1638pt;width:20.2125px;" id="M1" height="8.1499996" version="1.1" viewBox="0 0 20.2125 8.1499996" width="20.2125" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,7.888)"><path id="x1D70B" d="M574 449q-23 -58 -38 -79q-16 -4 -79 -4q-27 -100 -46 -219q-14 -87 7 -87t74 36l13 -27q-74 -81 -139 -81q-48 0 -48 62q0 22 8 59l60 258l-154 4q-21 -103 -54.5 -209t-64.5 -151q-38 -23 -83 -23l-7 15q46 36 94 152.5t64 216.5q-81 0 -138 -54l-18 23q22 27 39.5 43&#xA;t61.5 33.5t100 17.5q43 0 131.5 -2.5t129.5 -2.5q23 0 33.5 5t24.5 25z" /></g><g transform="matrix(.017,-0,0,-.017,10.109,7.888)"><use xlink:href="#x1D70B"/></g> </svg> cross-section is used in which spectral modifications are introduced in the propagator of the exchanged <svg style="vertical-align:-3.56265pt;width:9.2875004px;" id="M2" height="12.175" version="1.1" viewBox="0 0 9.2875004 12.175" width="9.2875004" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path id="x1D70C" d="M516 277q0 -126 -140 -231q-36 -27 -75.5 -42.5t-63.5 -15.5q-38 0 -73 43q-10 -44 -18 -106q-18 -134 -32 -151q-12 -16 -40 -25t-46 -10l-5 14q22 38 84 406q16 92 44.5 146.5t88.5 95.5q68 47 136 47q63 0 101.5 -47.5t38.5 -123.5zM432 259q0 61 -27 101t-79 40&#xA;t-82 -49t-49 -148l-24 -127q42 -36 88 -36q79 0 126 61.5t47 157.5z" /></g> </svg>.

Shear viscosity of a strongly interacting system: Green-Kubo correlator versus Chapman-Enskog and relaxation-time approximations

The shear viscosity $\ensuremath{\eta}$ has been calculated by using the Green-Kubo relation in the framework of a partonic transport approach solved at cascade level. We compare the numerical results for $\ensuremath{\eta}$ obtained from the Green-Kubo correlator with the analytical formulas in both the relaxation time approximation (RTA) and the Chapman-Enskog (CE) approximation. We investigate and emphasize the differences between the isotropic and anisotropic cross sections and between the massless and massive particles. We show that in the range of temperature explored in a heavy ion collision and for perturbative-QCD-like cross section, the RTA significantly underestimates the viscosity by about a factor of 2--3, while a good agreement is found between the CE approximation and Gree-Kubo relation already at first-order of approximation. The agreement with the CE approximation supplies an analytical formula that allows us to develop a kinetic transport theory at fixed shear-viscosity to entropy-density ratio, $\ensuremath{\eta}/s$. This opens the possibility of exploring dissipative nonequilibrium evolution of the distribution functions versus $T$-dependent $\ensuremath{\eta}/s$ and particle momenta in the dynamics of the quark-gluon plasma created in ultrarelativistic heavy-ion collisions.