H theorem, regularization, and boundary conditions for linearized 13 moment equations

Mesoscopic kinetic approach for studying nonequilibrium hydrodynamic and thermodynamic effects of shock wave, contact discontinuity, and rarefaction wave in the unsteady shock tube

This paper presents a detailed description of a molecular velocity distribution-based mesoscopic kinetic approach that enables a better understanding of various nonequilibrium hydrodynamic and thermodynamic effects in shock waves, contact discontinuities, and rarefaction waves. This builds on the mesoscopic kinetic approach in a previous investigation into regular reflection shocks by further addressing the mesoscopic physical meaning of kinetic moments from the view of kinetics and the implications of the magnitude and sign of nonequilibrium kinetic moments. To deepen understanding of nonequilibrium effects, this work focuses on the one-dimensional unsteady shock tube problem, which contains the typical and essential features of the discontinuous flows, and has no interference of two-dimensional flow direction. The approach uses a lattice Boltzmann method to solve the flow field, and describes nonequilibrium effects through the nonequilibrium kinetic moments of molecular velocity distribution functions. The mechanism of nonequilibrium effect in discontinuous flows is further probed. This work develops the mesoscopic kinetic approach and clarifies the mesoscopic physics of shock waves, contact discontinuities, and rarefaction waves.

Open Mathematical Aspects of Continuum Thermodynamics: Hyperbolicity, Boundaries and Nonlinearities

Thermodynamics is continuously spreading in the engineering practice, which is especially true for non-equilibrium models in continuum problems. Although there are concepts and approaches beyond the classical knowledge, which are known for decades, their mathematical properties, and consequences of the generalizations are less-known and are still of high interest in current researches. Therefore, we found it essential to collect the most important and still open mathematical questions that are related to different continuum thermodynamic approaches. First, we start with the example of Classical Irreversible Thermodynamics (CIT) in order to provide the basis for the more general and complex frameworks, such as the Non-Equilibrium Thermodynamics with Internal Variables (NET-IV) and Rational Extended Thermodynamics (RET). Here, we aim to present that each approach has its specific problems, such as how the initial and boundary conditions can be formulated, how the coefficients in the partial differential equations are connected to each other, and how it affects the appearance of nonlinearities. We present these properties and comparing the approach of NET-IV and RET to each other from these points of view. In the present work, we restrict ourselves on non-relativistic models.

Formulation of moment equations for rarefied gases within two frameworks of non-equilibrium thermodynamics: RET and GENERIC

In this work, we make a further step in bringing together different approaches to non-equilibrium thermodynamics. The structure of the moment hierarchy derived from the Boltzmann equation is at the heart of rational extended thermodynamics (RET, developed by Ingo Müller and Tommaso Ruggeri). Whereas the full moment hierarchy has the structure expressed in the general equation for the nonequilibrium reversible-irreversible coup- ling (GENERIC), the Poisson bracket structure of reversible dynamics postulated in that approach is a major obstacle for truncating moment hierarchies, which seems to work only in exceptional cases (most importantly, for the five moments associated with conservation laws). The practical importance of truncated moment hierarchies in rarefied gas dynamics and microfluidics motivates us to develop a new strategy for establishing the full GENERIC structure of truncated moment equations, based on non-entropy-producing irreversible processes associated with Casimir symmetry. Detailed results are given for the special case of 10 moments. This article is part of the theme issue 'Fundamental aspects of nonequilibrium thermodynamics'.

Modelling Thermally Induced Non-Equilibrium Gas Flows by Coupling Kinetic and Extended Thermodynamic Methods

Thermally induced non-equilibrium gas flows have been simulated in the present study by coupling kinetic and extended thermodynamic methods. Three different types of thermally induced gas flows, including temperature-discontinuity- and temperature-gradient-induced flows and radiometric flow, have been explored in the transition regime. The temperature-discontinuity-induced flow case has shown that as the Knudsen number increases, the regularised 26 (R26) moment equation system will gradually loss its accuracy and validation. A coupling macro- and microscopic approach is employed to overcome these problems. The R26 moment equations are used at the macroscopic level for the bulk flow region, while the kinetic equation associated with the discrete velocity method (DVM) is applied to describe the gas close to the wall at the microscopic level, which yields a hybrid DVM/R26 approach. The numerical results have shown that the hybrid DVM/R26 method can be faithfully used for the thermally induced non-equilibrium flows. The proposed scheme not only improves the accuracy of the results in comparison with the R26 equations, but also extends their capability with a wider range of Knudsen numbers. In addition, the hybrid scheme is able to reduce the computational memory and time cost compared to the DVM.

Coupled constitutive relations: a second law based higher-order closure for hydrodynamics

In the classical framework, the Navier–Stokes–Fourier equations are obtained through the linear uncoupled thermodynamic force-flux relations which guarantee the non-negativity of the entropy production. However, the conventional thermodynamic descrip- tion is only valid when the Knudsen number is sufficiently small. Here, it is shown that the range of validity of the Navier–Stokes–Fourier equations can be extended by incorporating the nonlinear coupling among the thermodynamic forces and fluxes. The resulting system of conservation laws closed with the coupled constitutive relations is able to describe many interesting rarefaction effects, such as Knudsen paradox, transpiration flows, thermal stress, heat flux without temperature gradients, etc., which cannot be predicted by the classical Navier–Stokes–Fourier equations. For this system of equations, a set of phenomenological boundary conditions, which respect the second law of thermodynamics, is also derived. Some of the benchmark problems in fluid mechanics are studied to show the applicability of the derived equations and boundary conditions.

Evaporation Boundary Conditions for the Linear R13 Equations Based on the Onsager Theory

Due to the failure of the continuum hypothesis for higher Knudsen numbers, rarefied gases and microflows of gases are particularly difficult to model. Macroscopic transport equations compete with particle methods, such as the Direct Simulation Monte Carlo method (DSMC), to find accurate solutions in the rarefied gas regime. Due to growing interest in micro flow applications, such as micro fuel cells, it is important to model and understand evaporation in this flow regime. Here, evaporation boundary conditions for the R13 equations, which are macroscopic transport equations with applicability in the rarefied gas regime, are derived. The new equations utilize Onsager relations, linear relations between thermodynamic fluxes and forces, with constant coefficients, that need to be determined. For this, the boundary conditions are fitted to DSMC data and compared to other R13 boundary conditions from kinetic theory and Navier-Stokes-Fourier (NSF) solutions for two one-dimensional steady-state problems. Overall, the suggested fittings of the new phenomenological boundary conditions show better agreement with DSMC than the alternative kinetic theory evaporation boundary conditions for R13. Furthermore, the new evaporation boundary conditions for R13 are implemented in a code for the numerical solution of complex, two-dimensional geometries and compared to NSF solutions. Different flow patterns between R13 and NSF for higher Knudsen numbers are observed.

Discrete Boltzmann trans-scale modeling of high-speed compressible flows

We present a general framework for constructing trans-scale discrete Boltzmann models (DBMs) for high-speed compressible flows ranging from continuum to transition regime. This is achieved by designing a higher-order discrete equilibrium distribution function that satisfies additional nonhydrodynamic kinetic moments. To characterize the thermodynamic nonequilibrium (TNE) effects and estimate the condition under which the DBMs at various levels should be used, two measures are presented: (i) the relative TNE strength, describing the relative strength of the (N+1)th order TNE effects to the Nth order one; (ii) the TNE discrepancy between DBM simulation and relevant theoretical analysis. Whether or not the higher-order TNE effects should be taken into account in the modeling and which level of DBM should be adopted is best described by the relative TNE intensity and/or the discrepancy rather than by the value of the Knudsen number. As a model example, a two-dimensional DBM with 26 discrete velocities at Burnett level is formulated, verified, and validated.

Derivation of regularized Grad's moment system from kinetic equations: modes, ghosts and non-Markov fluxes

Derivation of the dynamic correction to Grad's moment system from kinetic equations (regularized Grad's 13 moment system, or R13) is revisited. The R13 distribution function is found as a superposition of eight modes. Three primary modes, known from the previous derivation (Karlin et al. 1998 Phys. Rev. E57, 1668-1672. (doi:10.1103/PhysRevE.57.1668)), are extended into the nonlinear parameter domain. Three essentially nonlinear modes are identified, and two ghost modes which do not contribute to the R13 fluxes are revealed. The eight-mode structure of the R13 distribution function implies partition of R13 fluxes into two types of contributions: dissipative fluxes (both linear and nonlinear) and nonlinear streamline convective fluxes. Physical interpretation of the latter non-dissipative and non-local in time effect is discussed. A non-perturbative R13-type solution is demonstrated for a simple Lorentz scattering kinetic model. The results of this study clarify the intrinsic structure of the R13 system.This article is part of the theme issue 'Hilbert's sixth problem'.

Derivation of stable Burnett equations for rarefied gas flows

A set of constitutive relations for the stress tensor and heat flux vector for the hydrodynamic description of rarefied gas flows is derived in this work. A phase density function consistent with Onsager's reciprocity principle and H theorem is utilized to capture nonequilibrium thermodynamics effects. The phase density function satisfies the linearized Boltzmann equation and the collision invariance property. Our formulation provides the correct value of the Prandtl number as it involves two different relaxation times for momentum and energy transport by diffusion. Generalized three-dimensional constitutive equations for different kinds of molecules are derived using the phase density function. The derived constitutive equations involve cross single derivatives of field variables such as temperature and velocity, with no higher-order derivative in higher-order terms. This is remarkable feature of the equations as the number of boundary conditions required is the same as needed for conventional Navier-Stokes equations. Linear stability analysis of the equations is performed, which shows that the derived equations are unconditionally stable. A comparison of the derived equations with existing Burnett-type equations is presented and salient features of our equations are outlined. The classic internal flow problem, force-driven compressible plane Poiseuille flow, is chosen to verify the stable Burnett equations and the results for equilibrium variables are presented.

Gaseous microflow modeling using the Fokker-Planck equation

We present a comparative study of gaseous microflow systems using the recently introduced Fokker-Planck approach and other methods such as: direct simulation Monte Carlo, lattice Boltzmann, and variational solution of Boltzmann-BGK. We show that this Fokker-Plank approach performs efficiently at intermediate values of Knudsen number, a region where direct simulation Monte Carlo becomes expensive and lattice Boltzmann becomes inaccurate. We also investigate the effectiveness of a recently proposed Fokker-Planck model in simulations of heat transfer, as a function of relevant parameters such as the Prandtl, Knudsen numbers. Furthermore, we present simulation of shock wave as a function of Mach number in transonic regime. Our results suggest that the performance of the Fokker-Planck approach is superior to that of the other methods in transition regime for rarefied gas flow and transonic regime for shock wave.

Onsager's-principle-consistent 13-moment transport equations

A new set of generalized transport equations is derived for higher-order moments which are generated in evolution equation for stress tensor and heat flux vector in 13-moment equations. The closure we employ satisfies Onsager's symmetry principle. In the derivation, we do not employ a phase density function based on Hermite polynomial series in terms of higher-order moments, unlike Grad's approach. The distribution function is rather chosen to satisfy collision invariance, and H-theorem and capture relatively strong deviations from equilibrium. The phase density function satisfies the linearized Boltzmann equation and provides the correct value of the Prandtl number for monatomic gas. The derived equations are compared with Grad's 13-moments equations for gas modeled as Maxwellian molecule. The merits of the proposed equations against Grad's and R13 equations are discussed. In particular, it is noted that the proposed equations contain higher-order terms compared to these equations but require a fewer number of boundary conditions as compared to the R13 equations. The Knudsen number envelope which can be covered to describe flows with these equations is therefore expected to be larger as compared to the earlier equations.

High-order thermal lattice Boltzmann models derived by means of Gauss quadrature in the spherical coordinate system

We use the spherical coordinate system in the momentum space and an appropriate discretization procedure to derive a hierarchy of lattice Boltzmann (LB) models with variable temperature. The separation of the integrals in the momentum space into angular and radial parts allows us to compute the moments of the equilibrium distribution function by means of Gauss-Legendre and Gauss-Laguerre quadratures, as well as to find the elements of the discrete momentum set for each LB model in the hierarchy. The capability of the high-order models in this hierarchy to capture specific effects in microfluidics is investigated through a computer simulation of Couette flow by using the Shakhov collision term to get the right value of the Prandtl number.

Higher-order Galilean-invariant lattice Boltzmann model for microflows: single-component gas

We introduce a scheme which gives rise to additional degree of freedom for the same number of discrete velocities in the context of the lattice Boltzmann model. We show that an off-lattice D3Q27 model exists with correct equilibrium to recover Galilean-invariant form of Navier-Stokes equation (without any cubic error). In the first part of this work, we show that the present model can capture two important features of the microflow in a single component gas: Knudsen boundary layer and Knudsen Paradox. Finally, we present numerical results corresponding to Couette flow for two representative Knudsen numbers. We show that the off-lattice D3Q27 model exhibits better accuracy as compared to more widely used on-lattice D3Q19 or D3Q27 model. Finally, our construction of discrete velocity model shows that there is no contradiction between entropic construction and quadrature-based procedure for the construction of the lattice Boltzmann model.

Variational principles for thermal transport in nanosystems with heat slip flow

Some years ago, Prigogine minimum entropy production principle was generalized by Lebon and Dauby to account for the nonlocal heat transport equation derived by Guyer and Krumhansl. Here, this criterion is extended to incorporate the effects of heat slip flow along the walls. This formulation is shown to be useful for the description of steady heat flow in nanosystems, where the heat slip flow plays a decisive role.

Thermodynamically admissible 13 moment equations from the Boltzmann equation

We use guiding principles from nonequilibrium thermodynamics to develop an admissible set of 13 moment equations for rarefied gas flows. The main benefits of our thermodynamic approach are an explicit entropy expression fulfilling an H theorem and a sound Hamiltonian formulation of the reversible free flight transport. To calculate the entropy and to find explicit closure approximations, we propose a simple set of approximate 13 parameter solutions to Boltzmann's kinetic equation. We discuss how standard hydrodynamics is recovered as a limiting case.

Analysis of the slip coefficient and defect velocity in the Knudsen layer of a rarefied gas using the linearized moment equations

The linearized R13 and R26 moment equations are used to study Kramers' problem. Analytical solutions for the defect velocity and slip coefficient are derived and compared with numerical results from the kinetic theory. It is found that the linearized R26 equations can capture the Knudsen layer fairly accurately in terms of the defect velocity and slip coefficient, while the linearized R13 equations underpredict the kinetic data. At the wall, however, the kinetic models predict a slightly higher value for the defect velocity than the linearized R26 equations. In general, the linearized R26 equations perform well for both specular and diffusive walls.

Effects of rarefaction in microflows between coaxial cylinders

Microscale gas flows between two rotating coaxial circular cylinders of infinite length with different temperatures are investigated. Navier-Stokes-Fourier (NSF) and regularized 13-moment (R13) equations in their linear form are used to independently analyze velocity and temperature fields in shear-driven rotary flows, i.e., cylindrical Couette flows. Knudsen boundary layers, which present non-Newtonian stress and non-Fourier heat flow, are predicted as the dominant rarefaction effects in the linear theory. We show that the R13 system yields more accurate results for this boundary value problem by predicting the Knudsen boundary layers, which are not accessible for NSF equations. Furthermore, a set of second-order boundary conditions for velocity slip and temperature jump are derived for the NSF system. It is shown that the proposed boundary conditions effectively improve the classical hydrodynamics. The accuracy of NSF and R13 equations is discussed based on their comparison with available direct simulation Monte Carlo data.

Boltzmann equation and hydrodynamic fluctuations

We apply the method of invariant manifolds to derive equations of generalized hydrodynamics from the linearized Boltzmann equation and determine exact transport coefficients, obeying Green-Kubo formulas. Numerical calculations are performed in the special case of Maxwell molecules. We investigate, through the comparison with experimental data and former approaches, the spectrum of density fluctuations and address the regime of finite Knudsen numbers and finite frequencies hydrodynamics.

Switching criteria for hybrid rarefied gas flow solvers

Switching criteria for hybrid hydrodynamic/molecular gas flow solvers are developed, and are demonstrated to be more appropriate than conventional criteria for identifying thermodynamic non-equilibrium. For switching from a molecular/kinetic solver to a hydrodynamic (continuum-fluid) solver, the criterion is based on the difference between the hydrodynamic near-equilibrium fluxes (i.e. the Navier–Stokes stress and Fourier heat flux) and the actual values of stress and heat flux as computed from the molecular solver. For switching from hydrodynamics to molecular/kinetic, a similar criterion is used but the values of stress and heat flux are approximated through higher order constitutive relations; in this case, we use the R13 equations. The efficacy of our proposed switching criteria is tested within an illustrative hybrid kinetic/Navier–Stokes solver. For the test cases investigated, the results from the hybrid procedure compare very well with the full kinetic solution, and are obtained at a fraction of the computational cost.

Higher-order effects in rarefied channel flows

The regularized 13 moment (R13) equations and their boundary conditions are considered for plane channel flows. Chapman-Enskog scaling based on the Knudsen number is used to reduce the equations. The reduced equations yield second-order slip conditions, and allow us to describe the characteristic dip in the temperature profile observed in force driven flow. Due to the scaling, the R13 equations' ability to describe Knudsen layers is lost. Solutions with Knudsen layers are discussed as well, and it is shown that these give a better match to direct solutions of the Boltzmann equations than the reduced equations without Knudsen layers. For a radiatively heated gas the R13 equations predict a dependence of the average gas temperature on the Knudsen number with a distinct minimum around Kn = 0.2 , similar to the well-known Knudsen minimum for Poiseuille flow.

Velocity profile in the Knudsen layer according to the Boltzmann equation

Flow of a dilute gas near a solid surface exhibits non-continuum effects that are manifested in the Knudsen layer. The non-Newtonian nature of the flow in this region has been the subject of a number of recent studies suggesting that the so-called ‘effective viscosity’ at a solid surface is half that of the standard dynamic viscosity. Using the Boltzmann equation with a diffusely reflecting surface and hard sphere molecules, Lilley & Sader discovered that the flow exhibits a striking power-law dependence on distance from the solid surface where the velocity gradient is singular. Importantly, these findings (i) contradict these recent claims and (ii) are not predicted by existing high-order hydrodynamic flow models. Here, we examine the applicability of these findings to surfaces with arbitrary thermal accommodation and molecules that are more realistic than hard spheres. This study demonstrates that the velocity gradient singularity and power-law dependence arise naturally from the Boltzmann equation, regardless of the degree of thermal accommodation. These results are expected to be of particular value in the development of hydrodynamic models beyond the Boltzmann equation and in the design and characterization of nanoscale flows.