Beyond Navier–Stokes: Burnett equations for flows in the continuum–transition regime

For computation of hypersonic flowfields about space vehicles in low earth orbits, where the local Knudsen numbers (Kri) lie in the continuum-transition regime, a set of extended three-dimensional hydrodynamic equations are required which are more accurate than the NavierStokes equations and computationally more efficient than the Direct Simulation Monte Carlo (DSMC) computations. One such set is the Burnett equations which are obtained from the Chapman-Enskog expansion of the Boltzmann equation (with Knudsen number (Kri) as a small parameter) to O(Kn). In this paper, the threedimensional augmented Burnett equations are derived from the Chapman-Enskog expansion of the Boltzmann equation to O(Kn) and adding the augmented terms (linear third-order super Burnett terms with coefficients determined from linearized stability analysis to ensure stability of the augmented Burnett equations to small wavelength disturbances). The equations are solved using an explicit time-stepping scheme with Steger-Warming flux-vector splitting algorithm for the convective flux terms and second-order central differencing for the stress and heat flux terms. Maxwell-Smoluchowski slip boundary conditions are employed at the wall. 3-D augmented Burnett equations solver is developed both in FORTRAN?? and object-oriented JAVA. Both the FORTRAN and JAVA codes are applied to compute the three-dimensional hypersonic blunt-body flows for various range of Knudsen numbers and Mach numbers producing identical results. The focus of this paper is on development of a JAVA code for a very complex set of equations and its application to 3-D complex configurations. x Graduate Research Assistant * Bloomfield Distinguished Professor and Executive Director, Fellow AIAA Copyright © by authors. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. NOMENCLATURE et total energy per unit mass E total flux vector in x direction Ej convective flux vector in x direction EV viscous flux vector in x direction F total flux vector in y direction F! convective flux vector in y direction Fv viscous flux vector in y direction G total flux vector in z direction GI convective flux vector in z direction GV viscous flux vector in z direction Kn Knudsen number L reference length M Mach number Pr Prandtl number P pressure qt heat flux terms Q unknown vector in physical domain rn nose radius Re Reynolds number R gas constant T temperature Tw wall temperature u velocity components in x direction v velocity components in y direction w velocity components in z direction a accommodation coefficient for wall surface a/, P/ coefficients of stress terms in augmented Burnett equations y/ coefficients of heat flux terms in augmented Burnett equations K thermal conductivity K mean free path of gas molecules ILI viscosity 0, coefficients of heat flux terms in augmented Burnett equations p density a,y stress tensor a reflection coefficient for wall surface Y specific heat ratio co, coefficients of stress terms in augmented Burnett equations w circular frequency of perturbation American Institute of Aeronautics and Astronautics

This paper presents a comparative study of the solutions of the augmented Burnett and BGK-Burnett equations with the Maxwell-Smoluchowski first- and second-order boundary conditions. The hypersonic blunt body problem has been solved to study the effects of these higher-order boundary conditions. Solutions are presented for free stream Knudsen numbers 0.01 and 0.1. The solutions of the augmented Burnett and BGK-Burnett equations with these higher-order boundary conditions are compared with the solutions of the Navier-Stokes equations with the first- and second-order slip boundary conditions.

Recently a new system of field equations for the accurate description of flows in rarefied gases, called regularized 13-moment equations, was obtained by means of a hybrid gas kinetic approach. The first part of this paper discusses the relationship of the new system to classical high-order theories like the Burnett and super-Burnett equations as well as to modified models like the augmented and regularized Burnett equations. In the second part, shock structure calculations with the new theory are presented and compared to direct-simulation Monte Carlo (DSMC) solutions and to solutions of the Burnett models. Owing to additional higher-order dissipation in the system, the profiles are smooth for any Mach number, in contrast to the results of Grad's 13-moment case. The results show reliable quantitative agreement with DSMC simulations for Mach numbers up to $M_{0}\approx 3.0$. The agreement is better for Maxwell molecules than for hard spheres. The results of the augmented Burnett equations are comparable, but these equations are shown to be spatially unstable. Additionally, a validiation procedure for the new equations is presented by investigating the positivity of Grad's distribution function.

The flow characteristics and heat transfer in micro couette flow were studied numerically using three methods: Burnett equations, direct simulation Monte Carlo (DSMC) method and information preservation (IP) method. The results of these three methods are compared to each other. Convergent solutions to the conventional Burnett equations with second order velocity-slip and temperature-jump boundary conditions were obtained on arbitrary fine numerical grids for all Knudsen numbers (Kn) up to the limit of the equations' validity. The non-dimensional wall shear stress and heat flux shows good agreement with DSMC results and are better than first order slip boundary condition. The velocity slip and temperature jump calculated from Burnett equations fit DSMC results well when Kn < 0.3 but departure at higher Kn number. The IP method shows an identical velocity distribution with DSMC at low speed, but costs much less time. Perfect agreement was achieved in the density, velocity, temperature and pressure distributions between the results of Burnett equations and IP method at Kn = 0.02 and 0.1. Burnett equations and IP method are better than Navier-Stokes equation and DSMC method, respectively, in the flow and heat transfer in MEMS or NEMS systems

The structure of a normal shock waves in dilute gas mixtures of various compositions (Xenon-Helium and Argon-Helium) are studied using Molecular Dynamics (MD) and Direct Simulation Monte Carlo (DSMC). The MD solutions reproduce well the experimental data, with the exception of the parallel temperature profile in the 24.7% Ar-He mixture, despite the satisfactory agreement between the parallel velocity profiles. The Generalized Hard Sphere (GHS) DSMC solutions match almost perfectly the MD data, for all the cases considered. Using textbook parameters for like particles and their arithmetic averages for unlike particles, the Variable Hard Sphere (VHS) model fails to describe the shock wave structure in the Xe-He mixture. For the Ar-He cases, the VHS diffusion cross-section better approximates that obtained with the Lennard-Jones potential. Therefore, the VHS DSMC solutions exhibit less significant differences with those obtained with MD. If the VHS model is properly parametrized, the agreement between DSMC and MD solutions increases significantly. These cases exemplify situations where MD simulations based on a reliable interatomic potential can be used to inform simplified collision schemes used in DSMC.

This paper describes the computations of hypersonic flows in a diatomic gas in rotational nonequilibrium using a newly developed simplified set of Burnett equations designated as simplified conventional Burnett equations. Since the original formulation by Burnett, a number of variations to the original Burnett equations have been proposed, and the differences among these variants and their merits/shortcomings have been described in the literature. A new variant is created based on the conventional Burnett equations for hypersonic flows by neglecting higher-order terms that are inversely proportional to the Mach number. This set of simplified conventional Burnett equations is linearly stable for small disturbances in contrast to the conventional Burnett equations that suffer from Bobylev instability. To simulate the rotational nonequilibrium effect in a diatomic gas, both the Navier–Stokes and the simplified conventional Burnett equations are modified by including a rotational nonequilibrium relaxation model. The flow variables (density and translational and rotational temperature) for one-dimensional shock structure, flow past a two-dimensional cylinder, and an axisymmetric bicone in nitrogen are computed using both the simplified conventional Burnett and Navier–Stokes equations in the continuum-transition regime and are compared with the direct simulation Monte Carlo results. Simplified conventional Burnett calculations are in close agreement with the direct simulation Monte Carlo results at high Mach numbers.

The Direct Simulation Monte Carlo (DSMC) method was widely used to simulate low density gas flows with large Knudsen numbers. However, DSMC encounters limitations in the regime of lower Knudsen numbers (Kn<0.05). In such cases, approaches from classical computational fluid dynamics (CFD) relying on the continuum assumption are preferred, offering accurate solutions at acceptable computational costs. In experiments aimed at imaging aerosolized nanoparticles in vacuo a wide range of Knudsen numbers occur, which motivated the present study on the analysis of the advantages and drawbacks of DSMC and CFD simulations of rarefied flows in terms of accuracy and computational effort. Furthermore, the potential of hybrid methods is evaluated. For this purpose, DSMC and CFD simulations of the flow inside a convergent–divergent nozzle (internal expanding flow) and the flow around a conical body (external shock generating flow) were carried out. CFD simulations utilize the software OpenFOAM and the DSMC solution is obtained using the software SPARTA. The results of these simulation techniques are evaluated by comparing them with experimental data (1), evaluating the time-to-solution (2) and the energy consumption (3), and assessing the feasibility of hybrid CFD-DSMC approaches (4).

In this paper a kinetic-theory-based upwind algorithm for the Bhatnagar ‐ Gross‐ Krook (BGK)‐ Burnett equations is presented. The Boltzmann equation, with the BGK approximation for the collision integral, describes the spatial and temporal variations of the second-order distribution function that forms the basis of this formulation. The second-order distribution function is derived by considering the e rst three terms in the Chapman ‐ Enskog expansion and using the Navier ‐ Stokes equations to express the material derivatives, present in the second-order terms, in terms of the spatial derivatives. The BGK ‐ Burnett equations are derived by taking moments of the BGK ‐ Boltzmann equation with the collision invariant vector. A kinetic wave/particle split scheme for the BGK ‐ Burnett equations is derived by taking moments of the upwind discretized BGK ‐ Boltzmann equation. This algorithm is applied to a hypersonic shock structure problem. This is the e rst time that a kinetic-theory-based method has been developed for solving the BGK ‐ Burnett equations.

Large statistical scatter and effective pressure boundary conditions are two critical problems in the computation of microchannel flows with the direct simulation Monte Carlo (DSMC) method. To address these issues, an extension of the DSMC-IP (information preservation) coupled method is developed from the one-dimensional case to the two-dimensional case for microchannel flow. Simulation results in a microchannel flow from DSMC, IP, and numerical and analytical solutions to the Navier-Stokes equations are compared. The DSMC-IP coupled method successfully reduces the large statistical scatter usually obtained with DSMC in such low-speed flow systems. It also provides a suitable implementation of pressure boundary conditions

In this study, direct simulation Monte Carlo (DSMC) method, the Navier-Stokes equations, and the Burnett equations are applied to micro-Couette flow to investigate flow and heat transfer characteristics in the slip and transition flow regimes. The unique solutions of the Burnett equations are obtained through determination of additional boundary condition for pressure using the results of DSMC. The effect of rarefaction on velocity, temperature, and pressure distributions is analyzed. The nondimensional parameter PrE characterizing convective heat transfer in Coutte flow increases as flow becomes more rarefied. The prediction of shear stress and heat flux at wall shows that the solution of the Burnett equations is superior than that of the Navier-Stokes equations at relatively high Kn number in the slip flow regime, but the Burnett equations with the slip boundary condition, which is proportional to Kn number, cannot be extended to the transition flow regime.

Recent success in applying the Burnett equations to the one-dimensional shock-structure problem has raised the issue of whether the full Burnett equations can be used replace the Navier-Stokes equations for solving boundary-value problems in rarefied gasdynamics. As is familiar from the classical rarefied gasdynamics literature, the Burnett equations, if not solved as a successive approximation to the Navier-Stokes equations for a small Knudsen number, would require more boundary conditions than those in the Navier-Stokes system, owing to the presence of the higher-order derivatives. In this paper, this issue is examined with concrete solution examples for the steady Couette flows, addressing specifically whether solutions to the full Burnett equations can be uniquely determined without adding more boundary conditions than those in the Navier-Stokes system. The analysis, supported by detailed numerical solutions, confirms that additional boundary conditions are needed as long as the Knudsen number is not identically zero, lest the solution to the Burnett equations is not unique.

We study the rotational relaxation process in nitrogen using all-atom molecular dynamics (MD) simulations and direct simulation Monte Carlo (DSMC). The intermolecular model used in the MD simulations is shown to (i) reproduce very well the shear viscosity of nitrogen over a wide range of temperatures, (ii) predict the near-equilibrium rotational collision number in good agreement with published trajectory calculations done on ab initio potential energy surfaces, and (iii) produce shock wave profiles in excellent accordance with the experimental measurements. We find that the rotational relaxation process is dependent not only on the near-equilibrium temperature (i.e., when systems relax to equilibrium after a small perturbation), but more importantly on both the magnitude and direction of the initial deviation from the equilibrium state. The comparison between MD and DSMC, based on the Borgnakke-Larsen model, for shock waves (both at low and high temperatures) and one-dimensional expansions shows that a judicious choice of a constant Zrot can produce DSMC results which are in relatively good agreement with MD. However, the selection of the rotational collision number is case-specific, depending not only on the temperature range, but more importantly on the type of flow (compression or expansion), with significant limitations for more complex simulations characterized both by expansion and compression zones. Parker's model, parametrized for nitrogen, overpredicts Zrot for temperatures above about 300 K. It is also unable to describe the dependence of the relaxation process on the direction to equilibrium. Finally, we present a demonstrative cell-based formulation of a rotational relaxation model to illustrate how, by including the key physics obtained from the MD data (dependence of the relaxation process on both the rotational and the translational state of the gas), the agreement between MD and DSMC solutions is drastically improved.

The effects of surface roughness on accommodation coefficients are studied using the direct simulation Monte-Carlo (DSMC) approach. DSMC simulations are carried out by varying the roughness parameters. An empirical relationship between surface roughness and accommodation coefficients is established.

In rarefied gas dynamics scattering kernels deserve special attention since they contain all the essential information about the effects of physical and chemical properties of the gas–solid surface interface on the gas scattering process. However, to study the impact of the gas–surface interactions on the large-scale behavior of fluid flows, these scattering kernels need to be integrated in larger-scale models like Direct Simulation Monte Carlo (DSMC). In this work, the Gaussian mixture (GM) model, an unsupervised machine learning approach, is utilized to establish a scattering kernel for monoatomic (Ar) and diatomic (H2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {H}_{2}$$\\end{document}) gases directly from Molecular Dynamics (MD) simulations data. The GM scattering kernel is coupled to a pure DSMC solver to study isothermal and non-isothermal rarefied gas flows in a system with two parallel walls. To fully examine the coupling mechanism between the GM scattering kernel and the DSMC approach, a one-to-one correspondence between MD and DSMC particles is considered here. Benchmarked by MD results, the performance of the GM-DSMC is assessed against the Cercignani–Lampis–Lord (CLL) kernel incorporated into DSMC simulation (CLL-DSMC). The comparison of various physical and stochastic parameters shows the better performance of the GM-DSMC approach. Especially for the diatomic system, the GM-DSMC outperforms the CLL-DSMC approach. The fundamental superiority of the GM-DSMC approach confirms its potential as a multi-scale simulation approach for accurately measuring flow field properties in systems with highly nonequilibrium conditions.

Recently, as aerodynamics was applied to flying vehicles with very high speed and flying at high altitude, the numerical simulation based on the Navier–Stokes (NS) equations was found that cannot correctly predict certain aero-thermo-dynamic properties in a certain range of velocity and altitude while the Knudsen number indicates that the flow is still in the continuum regime. As first noted by Zhou and Zhang (Science in China, 2015), the invalidity of NS equations for such flows might be attributed to an non-equilibrium effect originating from the combined effects of gas rarefaction and strong shear in the boundary-layer flows. In this paper, we present the scope, physical concept, mathematical model of this shear non-equilibrium effect in hypersonic flows, as well as the way of considering this effect in conventional computational fluid mechanics (CFD) for engineering applications. Several hypersonic flows over sharp bodies and blunt bodies are analyzed by the proposed new continuum model, named direct simulation Monte Carlo (DSMC) data-improved Navier–Stokes (DiNS) model. Recently, as aerodynamics was applied to flying vehicles with very high speed and flying at high altitude, the numerical simulation based on the Navier–Stokes (NS) equations was found that cannot correctly predict certain aero-thermo-dynamic properties in a certain range of velocity and altitude while the Knudsen number indicates that the flow is still in the continuum regime. As first noted by Zhou and Zhang (Science in China, 2015), the invalidity of NS equations for such flows might be attributed to an non-equilibrium effect originating from the combined effects of gas rarefaction and strong shear in the boundary-layer flows. In this paper, we present the scope, physical concept, mathematical model of this shear non-equilibrium effect in hypersonic flows, as well as the way of considering this effect in conventional computational fluid mechanics (CFD) for engineering applications. Several hypersonic flows over sharp bodies and blunt bodies are analyzed by the proposed new continuum model, named direct simulation Monte Carlo (DSMC) data-improved Navier–Stokes (DiNS) model..