In hypersonic flows about space vehicles in low earth orbits or flows in microchannels of microelectromechanical devices, the local Knudsen number lies in the continuum–transition regime. Navier–Stokes equations are not adequate to model these flows since they are based on small deviation from local thermodynamic equilibrium. To model these flows, a number of extended hydrodynamics or generalized hydrodynamics models have been proposed over the past fifty years, along with the direct simulation Monte Carlo (DSMC) approach. One of these models is the Burnett equations which are obtained from the Chapman–Enskog expansion of the Boltzmann equation [with Knudsen number (Kn) as a small parameter] to O(Kn2). With the currently available computing power, it has been possible in recent years to numerically solve the Burnett equations. However, attempts at solving the Burnett equations have uncovered many physical and numerical difficulties with the Burnett model. As a result, several improvements to the conventional Burnett equations have been proposed in recent years to address both the physical and numerical issues; two of the most well known are the “augmented Burnett equations” and the “BGK–Burnett equations.” This paper traces the history of the Burnett model and describes some of the recent developments. The relationship between the Burnett equations and the Grad’s 13 moment equations is elucidated by employing the Maxwell–Truesdell–Green iteration. Numerical solutions are provided to assess the accuracy and applicability of Burnett equations for modeling flows in the continuum–transition regime. The important issue of surface boundary conditions is addressed. Computations are compared with the available experimental data, Navier–Stokes calculations, Burnett solutions of other investigators, and DSMC solutions as much as possible.

Open Archive TOULOUSE Archive Ouverte (OATAO) OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. Abstract Great prominence is put on the design of aeronautical gas turbines due to increasingly stringent regulations and the need to tackle rising fuel prices. This drive towards innovation has resulted sometimes in new concepts being prone to combustion instabilities. In the particular field of annular combustion chambers, these instabilities often take the form of azimuthal modes. To predict these modes, one must compute the full combustion chamber, which remained out of reach until very recently and the development of massively parallel computers. Since one of the most limiting factors in performing Large Eddy Simulation (LES) of real combustors is estimating the adequate grid, the effects of mesh resolution are investigated by computing full annular LES of a realistic helicopter combustion chamber on three grids, respectively made of 38, 93 and 336 million elements. Results are compared in terms of mean and fluctuating fields. LES captures self-established azimuthal modes. The presence and structure of the modes is discussed. This study therefore highlights the potential of LES for studying combustion instabilities in annular gas turbine combustors.

As the flow becomes rarefied it has been seen that predictions of continuum formulations, such as the Navier–Stokes equations, become inaccurate. These inaccuracies stem from the linear approximations to the stress and heat flux in the viscous flux terms in the Navier–Stokes equations. Hence, it has long been conjectured that the inclusion of higher-order terms in the constitutive relations for the stress and heat flux may improve the predictive capabilities of such continuum formulations. Following this approach, second-order systems of hydrodynamic equations, such as the Burnett and Woods equations, were applied to the shock structure problem. While it was observed that these equations afforded a better description of the shock structure on coarse grids, they were prone to small wavelength instabilities when the grids were refined. The cause of this instability was subsequently traced to the fact that these equations can potentially violate the second law of thermodynamics when the local Knudsen number exceeds a critical limit. This leads to the fundamental question: is entropy consistency achievable in a system of second-order hydrodynamic equations? To answer this question, a novel set of equations, known as the BGK-Burnett equations, is constructed by taking moments of the Boltzmann equation for the second-order distribution function.The formulation of second-order hydrodynamic equations by moment methods is beset by three hurdles: (i) the highly nonlinear collision integral in the Boltzmann equation needs to be evaluated, (ii) the second-order distribution function does not satisfy the moment closure criterion and (iii) identification of the approximations to the material derivatives in the second-order distribution function that will correctly account for the difference in time scales between the first- and second-order fluxes. The first three terms of the Chapman–Enskog expansion, that defines the second-order distribution function, and the Bhatnagar–Gross–Krook model of the collision integral, form the basis of the BGK-Burnett equations. The entropy-consistent behaviour of the equations depend on the moment closure coefficients and the approximations to the material derivatives in the second-order fluxes. The requirement of moment closure alone, however, results in non-unique closure coefficients and a family of BGK-Burnett equations, from which an entropy-consistent set must be identified. From this family, two sets of BGK-Burnett equations have been considered, and this paper presents the details of the formulation of these two sets of equations, the identification of entropy-consistent approximations to the material derivatives by a novel entropy consistent relaxation technique, and shock structure computations in a monatomic hard sphere gas for a range of Mach numbers.

Abstract. The sensitivities of idealized large-eddy simulations (LESs) to variations of model configuration and forcing parameters on quantities of interest to wind power applications are examined. Simulated wind speed, turbulent fluxes, spectra and cospectra are assessed in relation to variations in two physical factors, geostrophic wind speed and surface roughness length, and several model configuration choices, including mesh size and grid aspect ratio, turbulence model, and numerical discretization schemes, in three different code bases. Two case studies representing nearly steady neutral and convective atmospheric boundary layer (ABL) flow conditions over nearly flat and homogeneous terrain were used to force and assess idealized LESs, using periodic lateral boundary conditions. Comparison with fast-response velocity measurements at 10 heights within the lowest 100 m indicates that most model configurations performed similarly overall, with differences between observed and predicted wind speed generally smaller than measurement variability. Simulations of convective conditions produced turbulence quantities and spectra that matched the observations well, while those of neutral simulations produced good predictions of stress, but smaller than observed magnitudes of turbulence kinetic energy, likely due to tower wakes influencing the measurements. While sensitivities to model configuration choices and variability in forcing can be considerable, idealized LESs are shown to reliably reproduce quantities of interest to wind energy applications within the lower ABL during quasi-ideal, nearly steady neutral and convective conditions over nearly flat and homogeneous terrain.

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