Burnett Terms Research Articles

Status of the Navier–Stokes Equations in Gas Dynamics. A Review

The existing ideas on the status of the Navier–Stokes equations are changed in taking into account the following facts: generally speaking, the terms of these equations neglected in the boundary layer equations are of the order of certain Burnett terms in the conservation equations; the Navier–Stokes equations cannot be used to describe slow nonisothermal gas flows since in this case it is necessary to take the Burnett temperature stresses into account; and in the transport relations the Burnett terms determine certain effects (for example, the mechanocaloric effect).

Requirements to the accuracy of the Burnett transport coefficients

Some effects in gases are completely or partially determined by the Burnett terms of the stress tensor and the heat flux vector. Usually, approximate values of the Burnett transport coefficients for a monatomic gas are used to calculate the above-mentioned quantities. “Exact” expressions for these coefficients correct to values less than one percent are obtained for elastic-sphere molecules. The results of calculations of certain such effects are compared with one another using the approximate and exact Burnett transport coefficients for a monatomic gas consisting of elastic-sphere molecules.

Simulation of gas flows in micro/nano systems using the Burnett equations

The gaseous flow characteristics in micro/nano systems are studied using the Burnett equations. The Burnett equations are the second-order approximation of the Chapman–Enskog solution to the Boltzmann equation. This set of equation is appropriate to describe the rarefied gas flows. Velocity slip and temperature jump boundary conditions are applied on solid surface. Several techniques, including the usage of relaxation method on slip values and Burnett terms, are introduced to increase the stability of the Burnett equations. Hence, convergent results at Knudsen number up to 0.5 are achieved for the first time. The results of the Burnett equations are first verified using the corresponding experimental and the DSMC data. The Navier-Stokes and the Burnett equations give almost the same result when the flow is in slip regime (Kn<0.0l). But when the flow is in transition regime (0.1<Kn<0.5), the results of Navier-Stokes equations deviate from DSMC data while those of Burnett equations still agree very well. The Burnett equations are then used to study the gaseous flows in micro/nano systems. The pressure driven plane Poiseuille and the backward-facing step flows, the flows through micro filters are then studied. Different inlet to outlet pressure ratios, size effects, shape effects and boundary conditions are analyzed. The competitive relations of rarefied and compressible effects in micro gas flows are discussed.

Continuum Simulation of the Microscale Backward-Facing Step Flow in a Transition Regime

Microscale backward-facing step flows in transition regime are investigated in the present study using continuum-based Burnett equations. A relaxation method on the Burnett terms is proposed here, and convergent results at Knudsen number up to 0.5 are achieved. The results of Burnett equations agree very well with experimental data and direct simulation Monte Carlo (DSMC) results. The detailed flow characteristics of backward-facing step flow in transition regime are presented, and the difference from macro flow is analyzed. The effects of pressure ratio, Kn, and step ratio are discussed.

Viscous Compressible Slip Flows. Part 2 : Meshless Solver for Rotating Slip Flows

The Part 1 presented kinetic treatment of the slip flow and described kinetic flux vector splitting. The Part 2 of the paper presents the kinetic theory based approach for solving rotating slip flows. The paper describes numerical flow modeling of slip and rarefied flows. It also presents the meshless solver which makes use of least squares and Kinetic Flux Vector Splitting (KFVS) scheme and its Variance Reduction form (VRKFVS). The kinetic slip flow boundary condition is added with the Burnett terms which are quadratic in Knudsen number. The solver is able to capture slip flow features and typical features of the strongly rotating flows characterized by steep density gradient, supersonic flows, thin boundary layers towards the peripheral region with a rarefied central core.

Burnett simulation of gas flow and heat transfer in micro Poiseuille flow

The Burnett equations with slip boundary conditions are used to simulate the compressible gas flow and heat transfer in micro Poiseuille flow in the slip and transition flow regime. A relaxation method on Burnett terms is proposed in the present study and the thermal creep effect is considered. Convergent results at Knudsen number up to 0.4 are achieved and the results agree very well with experimental data. It is found that with the increase of Knudsen number, the Poiseuille number decreases while Nusselt number increases. The local Poiseuille number decreases along the whole channel while the local Nusselt number decreases rapidly first and then increases slowly afterwards.

Damping and dispersion of oscillating modes of a multicomponent ionic mixture in a magnetic field

The collective-mode spectrum of a multicomponent magnetized ionic mixture for small wave number k is studied with the use of magnetohydrodynamics and formal kinetic theory. Apart from the usual thermal and diffusive modes, the spectrum contains a set of four oscillating modes. By evaluating the k 2 contributions to the eigenfrequencies, the damping and the dispersion of these oscillating modes are determined. The long-range nature of the Coulomb interactions is shown to imply that Burnett terms with higher-order gradients in the linear phenomenological laws have to be taken into account in order to obtain a full description of all damping and dispersion effects.

Constitutive relations and linear stability of a sheared granular flow

The constitutive relation for the granular flow of smooth, nearly elastic particles $((1 - e) \ll 1)$ is derived in the adiabatic limit, where the length scale for conduction, $\lambda/(1 - e)^{1/2},$ is small compared to the macroscopic scale. Here, $\lambda$ is the mean free path and e is the coefficient of restitution. In this case, energy is convected by the mean flow, and the rate of change of the energy in a Lagrangian reference frame is determined by a balance between the rate of production (due to the shear) and the rate of dissipation (due to inelastic collisions). A Gaussian approximation is used for the velocity distribution function, and the velocity variance is determined from a balance equation for the second moment of the velocity distribution using an asymptotic expansion in the small parameter $\epsilon = (1 - e)^{1/2}$. The stress tensor is then determined from the velocity variance. It is found that the leading-order, $O(\epsilon)$ and $O(\epsilon^2)$ contributions to the stress tensor are identical in form to those in the Euler, Navier–Stokes and Burnett approximations, and the numerical values of these coe.cients are close to those calculated using the Enskog procedure. The stress tensor is used to obtain analytical expressions for the growth rates of the hydrodynamic modes in a linear shear flow in the limit where the wavelength is long compared to the length scale of conduction. In the plane of flow, transverse momentum perturbations are found to be stable, while perturbations in the density and longitudinal momenta grow exponentially at short times and decay in the long-time limit. It is found that the Navier–Stokes approximation captures the leading behaviour of the growth rate in the small-wavenumber limit for perturbations in the plane of flow. In the vorticity direction, it is found that the Navier–Stokes approximation is not adequate to capture the leading behaviour in the small-wavenumber limit, and dominant terms in the growth rates of the transverse modes depend on the Burnett coefficients. Perturbations in this direction are unstable at low density, but become stable as the density is increased.When the wave length is small compared to the conduction length, the rate of conduction of energy is large compared to the rate of dissipation, and the hydrodynamic modes are identical to those for a gas of elastic particles at equilibrium. The hydrodynamic modes are all stable in this case. The transition between these two regimes is examined for a dilute granular flow, and a transition from unstable to stable modes is predicted as the wavenumber is increased. The analysis indicates that the minimal model which accurately captures the dynamics in both limits is one in which the constitutive relation for the stress incorporates the strain-rate-dependent Burnett terms in the stress equation (neglecting the temperature-dependent Burnett terms), and the constitutive relation for the heat flux is the Fourier law for heat conduction (neglecting all the Burnett terms).

In the Chapman-Enskog Expansion¶the Burnett Coefficients Satisfy¶the Universal Relation ω 3 +ω 4 + θ 3 = 0

The Chapman–Enskog expansion when applied to a gas of spherical molecules yields formal expressions for the stress deviator P and energy-flux vector q, P=μP(1)+μ2P(2)+…, q=μq(1)+μ2q(2)+…. The Burnett terms P(2), q(2) depend on 11 coefficients ωi, 1≦i≦6, θ&i, 1≦i≦ 5. This paper shows that ω3+ω4+θ3= 0.

Numerical Simulation of Three-Dimensional Augmented Burnett Equations for Hypersonic Flow

For computation of hypersonic e owe elds about space vehicles in low Earth orbits, where the local Knudsen numbers Kn lie in the continuum-transition regime, a set of extended three-dimensional hydrodynamic equations is required that is more accurate than the Navier ‐Stokes equations and computationally more efe cient than the direct simulation Monte Carlo (DSMC) computations. The three-dimensional augmented Burnett equations are derived from the Chapman ‐Enskog expansion of the Boltzmann equation to O(Kn 2) and adding the augmented terms (linear third-order super Burnett terms with coefe cients determined from linearized stability analysis to ensure stability of the augmented Burnett equations to small wavelength disturbances ). The three-dimensional augmentedBurnettequationsareappliedtocomputethehypersonicblunt-bodye owsforvariousrangeofKnudsen numbers (0:0884 < ‐ Kn < ‐ 0:227) and Mach numbers (10 < ‐ M < ‐ 25:3). The computational results are compared with the Navier ‐Stokes solutions, the existing augmented Burnett solutions, and the available DSMC results. The comparisons show that the difference between the Navier ‐Stokes and the augmented Burnett solutions is very small (less than 3% for the shock layer thickness ) at Knudsen numbers less than 0.01; the difference becomes signie cant as the Knudsen number increases. The comparisons also show that the augmented Burnett solutions are signie cantly closer to the DSMC results for the temperature distributions in the continuum-transition regime than the Navier ‐Stokes calculations.

Frame-indifferent kinetic theory

The second-order transport terms in a monatomic gas, originally derived by Burnett from Boltzmann's equation via the Chapman–Enskog iteration, have the curious property of being dependent on the observer's reference frame. However, frame-indifference has long been accepted as an essential property of constitutive equations in continuum mechanics. Various attempts have been made to find errors in the kinetic theory, but these have been countered by physical accounts of the framedependent terms that are independent of the Chapman–Enskog theory.It is shown in this paper that both the Chapman–Enskog theory and the physical models are based on an inappropriate definition of the peculiar velocity. The Chapman–Enskog theory is easily corrected, and the result is a kinetic theory in harmony with the principle of material frame-indifference. A brief survey of the debate on this topic is presented, and corrected expressions are given for the Burnett terms.

Statistical mechanics of nonlinear hydrodynamic fluctuations

The paper studies nonlinear hydrodynamic fluctuations by the methods of nonequilibrium statistical mechanics. The generalized Fokker-Planck equation for the distribution function of coarse-grained densities of conserved quantities is derived from the Liouville equation and then is investigated by using the gradient expansions in the flux correlation matrix. We have obtained the functional-differential Fokker-Planck equation describing the nonlinear hydrodynamic fluctuations in spatially nonuniform systems to second order in gradients of coarse-grained fluctuating fields. An outline of the derivation of Fokker-Planck equations containing the Burnett terms is also given. The explicit coordinate representation for the hydrodynamic Fokker-Planck equation is discussed in the case of one-component simple fluid. The general scheme of a change of coarse-grained functional variables is developed for hydrodynamic Fokker-Planck equations. The corresponding transformation rules are found for “drift” terms, “diffusion coefficients” and thermodynamic forces. The dynamical equations and stationary conditions for averages of functions (functionals) of hydrodynamic fields are discussed by using the Fokker-Planck operators acting on such functions. The explicit form of these operators are found for various sets of fluctuating fields. As an application of the formalism the calculation of the stationary correlation functions is presented for a simple nonequilibrium steady state.

Analysis of Rarefied Gas Flow Full Solutions for Burnett Equations

This paper concerns the numerical analysis of rarefied gas flowing through an orifice in a rotating cylinder. The full solutions for the Burnett equations in the steady state are successfully obtained and compared with those for the Navier-Stokes equations. Then, the pressure in the downstream side is chosen as a parameter to give an appropriate Knudsen number and a Much number. The results show construction of the rarefied gas flow, where the Burnett terms, or the higher order of approximation, reduces the contribution of the viscosity term in the equation of motion and of the conduction term in the equation of energy. The criteria of invalidity for the Navier-Stokes equations and the Burnett equations are discussed, too.

Analysis of rarefied gas flow. Full solutions for burnett equations.

This paper concerns the numerical analysis of rarefied gas flowing through an orifice in a rotating cylinder. The full solutions for the Burnett equations in the steady state are successfully obtained and compared with those for the Navier-Stokes equations. Then, the pressure in the downstream side is chosen as a parameter to give an appropriate Knudsen number and a Much number. The results show construction of the rarefied gas flow, where the Burnett terms, or the higher order of approximation, reduces the contribution of the viscosity term in the equation of motion and of the conduction term in the equation of energy. The criteria of invalidity for the Navier-Stokes equations and the Burnett equations are discussed, too.

Radial diffusion in the poiseuille flow of a gas mixture

In a number of experiments (see [1], in which experimental papers are listed), diffusion has been observed in the radial direction in the process of flow of a mixture along tubes at low pressures. The heavier molecules accumulate near the tube axis. The attempt made in [1] to explain this phenomenon by the influence of the Burnett contribution to the diffusion did not lead to success, and the Burnett terms in the radial diffusion velocity indicate a motion of heavy molecules away from the tube axis. In the present paper, a complete analysis is given of this phenomenon. We consider the problem of the flow of a mixture along a cylindrical tube of finite length for given pressure difference Δp between its ends. On the basis of the hydrodynamic equations of the Burnett and super-Burnett approximations, a consistent asymptotic (with respect to the small parameter ɛ) solution is given; ɛ = (Δp/p)R/L is the relative change in the pressure along the tube at a distance of order R (R and L are the radii and length of the tube). Radial diffusion occurs in the quadratic approximation in ɛ. It is shown that the radial diffusion velocity contains new terms not present in [1]; these are due to the inhomogeneity of the temperature and the pressure over the tube section, the expansion of the gas, and the super-Burnett correction to the diffusion velocity. The most important is the thermodiffusion term, which is determined by the hydrodynamic equations of the Navier-Stokes approximation. The remaining terms have order relative to it of Kn2 (Kn = 〈 1 〉/R is the Knudsen number, and 〈 1 〉 is the mean free path of the molecules). The expression obtained for the diffusion velocity agrees in sign with the experiment.

Reaction surface (flame-sheet) model in the theory of combustion of unmixed fuels. Molecular-kinetic solution in the reaction region

In the first part [1] of this investigation, a study was made of the main characteristic features of the flow resulting from the mixture of two flows with components that enter into an irreversible chemical reaction. The main errors in the solution of the corresponding gas-dynamic problem resulting from the use of various assumptions were also analyzed. It was found that the influence of the usually ignored Burnett and super-Burnett terms in the transport properties (i.e., in the stress tensor and in the heat flux and diffusion vectors) can be much stronger in this problem than in ordinary mixing problems. In the present, second part, a molecular-kinetic study is made of the flow in the thin reaction region. For the finding of the “external” solution of the problem in the mixing layer or when the classical flame-sheet model is used, this region is treated as a surface in the flow field on which all the chemical transformations of the components of the gas mixture take place. Gas-dynamic equations are obtained that describe the main and some of the subsequent approximations in the reaction region; these equations confirm the arguments put forward in the first part concerning the part played by the non-Navier-Stokes terms in the transport properties.

On the bounds of the thickness of a steady shock wave

Following the reasoning of von Mises, the author attempts to answer the question, “Does the use of higher order Burnett terms from the solution of the Boltzmann equation in the von Mises approach improve the bounds of the thickness of steady shock?” The answer is a negative one.