The structure of hypersonic shock waves using Navier-Stokes equations modified to include mass diffusion

Howard Brenner has recently proposed modifications to the Navier-Stokes equations that relate to a diffusion of fluid volume that would be significant for flows with high density gradients. In a previous paper (Greenshields & Reese, 2007), we found these modifications gave good predictions of the viscous structure of shock waves in argon in the range Mach 1.0-12.0 (while conventional Navier-Stokes equations are known to fail above about Mach 2). However, some areas of concern with this model were a somewhat arbitrary choice of modelling coefficient, and potentially unphysical and unstable solutions. In this paper, we therefore present slightly different modifications to include molecule mass diffusion fully in the Navier-Stokes equations. These modifications are shown to be stable and produce physical solutions to the shock problem of a quality broadly similar to those from the family of extended hydrodynamic models that includes the Burnett equations. The modifications primarily add a diffusion term to the mass conservation equation, so are at least as simple to solve as the Navier-Stokes equations; there are none of the numerical implementation problems of conventional extended hydrodynamics models, particularly in respect of boundary conditions. We recommend further investigation and testing on a number of different benchmark non-equilibrium flow cases.Comment: written for the 2nd European Conference on AeroSpace Sciences (EUCASS), Belgium, 200

The structure of shock waves as a test of Brenner's modifications to the Navier-Stokes equations

Brenner has recently proposed modifications to the Navier-Stokes equations that are based on theoretical arguments but supported only by experiments having a fairly limited range. These modifications relate to a diffusion of fluid volume that would be significant for flows with high density gradients. So the viscous structure of shock waves in gases should provide an excellent test case for this new model. In this paper we detail the shock structure problem and propose exponents for the gas viscosity-temperature relation based on empirical viscosity data that is independent of shock experiments. We then simulate shocks in the range Mach 1.0-12.0 using the Navier-Stokes equations, both with and without Brenner's modifications. Initial simulations showed Brenner's modifications display unphysical behaviour when the coefficient of volume diffusion exceeds the kinematic viscosity. Our subsequent analyses attribute this behaviour to both an instability to temporal disturbances and a spurious phase velocity-frequency relationship. On equating the volume diffusivity to the kinematic viscosity, however, we find the results with Brenner's modifications are significantly better than those of the standard Navier-Stokes equations, and broadly similar to those from the family of extended hydrodynamic models that includes the Burnett equations. Brenner's modifications add only two terms to the Navier-Stokes equations, and the numerical implementation is much simpler than conventional extended hydrodynamic models, particularly in respect of boundary conditions. We recommend further investigation and testing on a number of different benchmark non-equilibrium flow cases

Mode identification in rapidly rotating stars

Context: Recent calculations of pulsation modes in rapidly rotating polytropic models and models based on the Self-Consistent Field method have shown that the frequency spectrum of low degree pulsation modes can be described by an empirical formula similar to Tassoul's asymptotic formula, provided that the underlying rotation profile is not too differential. Aims: Given the simplicity of this asymptotic formula, we investigate whether it can provide a means by which to identify pulsation modes in rapidly rotating stars. Methods: We develop a new mode identification scheme which consists in scanning a multidimensional parameter space for the formula coefficients which yield the best-fitting asymptotic spectra. This mode identification scheme is then tested on artificial spectra based on the asymptotic formula, on random frequencies and on spectra based on full numerical eigenmode calculations for which the mode identification is known beforehand. We also investigate the effects of adding random frequencies to mimic the effects of chaotic modes which are also expected to show up in such stars. Results: In the absence of chaotic modes, it is possible to accurately find a correct mode identification for most of the observed frequencies provided these frequencies are sufficiently close to their asymptotic values. The addition of random frequencies can very quickly become problematic and hinder correct mode identification. Modifying the mode identification scheme to reject the worst fitting modes can bring some improvement but the results still remain poorer than in the case without chaotic modes

A DSMC investigation of gas flows in micro-channels with bends

Pressure-driven, implicit boundary conditions are implemented in an open source direct simulation Monte Carlo (DSMC) solver, and benchmarked against simple micro-channel flow cases found in the literature. DSMC simulations are then carried out of gas flows for varying degrees of rarefaction along micro-channels with both one and two ninety-degree bends. The results are compared to those from the equivalent straight micro-channel geometry. Away from the immediate bend regions, the pressure and Mach number profiles do not differ greatly from those in straight channels, indicating that there are no significant losses introduced when a bend is added to a micro-channel geometry. It is found that the inclusion of a bend in a micro-channel can increase the amount of mass that a channel can carry, and that adding a second bend produces a greater mass flux enhancement. This increase happens within a small range of Knudsen number (0.02 Knin 0.08). Velocity slip and shear stress profiles at the channel walls are presented for the Knudsen showing the largest mass flux enhancement

Are the stars of a new class of variability detected in NGC~3766 fast rotating SPB stars?

A recent photometric survey in the NGC~3766 cluster led to the detection of stars presenting an unexpected variability. They lie in a region of the Hertzsprung-Russell (HR) diagram where no pulsation are theoretically expected, in between the $\delta$ Scuti and slowly pulsating B (SPB) star instability domains. Their variability periods, between $\sim$0.1--0.7~d, are outside the expected domains of these well-known pulsators. The NCG~3766 cluster is known to host fast rotating stars. Rotation can significantly affect the pulsation properties of stars and alter their apparent luminosity through gravity darkening. Therefore we inspect if the new variable stars could correspond to fast rotating SPB stars. We carry out instability and visibility analysis of SPB pulsation modes within the frame of the traditional approximation. The effects of gravity darkening on typical SPB models are next studied. We find that at the red border of the SPB instability strip, prograde sectoral (PS) modes are preferentially excited, with periods shifted in the 0.2--0.5~d range due to the Coriolis effect. These modes are best seen when the star is seen equator-on. For such inclinations, low-mass SPB models can appear fainter due to gravity darkening and as if they were located between the $\delta$~Scuti and SPB instability strips.Comment: 6 pages, 2 figures, to appear in the proceedings of the IAU Symposium 307, New windows on massive stars: asteroseismology, interferometry, and spectropolarimetr

Gravity modes in rapidly rotating stars. Limits of perturbative methods

CoRoT and Kepler missions are now providing high-quality asteroseismic data for a large number of stars. Among intermediate-mass and massive stars, fast rotators are common objects. Taking the rotation effects into account is needed to correctly understand, identify, and interpret the observed oscillation frequencies of these stars. A classical approach is to consider the rotation as a perturbation. In this paper, we focus on gravity modes, such as those occurring in gamma Doradus, slowly pulsating B (SPB), or Be stars. We aim to define the suitability of perturbative methods. With the two-dimensional oscillation program (TOP), we performed complete computations of gravity modes -including the Coriolis force, the centrifugal distortion, and compressible effects- in 2-D distorted polytropic models of stars. We started with the modes l=1, n=1-14, and l=2-3, n=1-5,16-20 of a nonrotating star, and followed these modes by increasing the rotation rate up to 70% of the break-up rotation rate. We then derived perturbative coefficients and determined the domains of validity of the perturbative methods. Second-order perturbative methods are suited to computing low-order, low-degree mode frequencies up to rotation speeds ~100 km/s for typical gamma Dor stars or ~150 km/s for B stars. The domains of validity can be extended by a few tens of km/s thanks to the third-order terms. For higher order modes, the domains of validity are noticeably reduced. Moreover, perturbative methods are inefficient for modes with frequencies lower than the Coriolis frequency 2Omega. We interpret this failure as a consequence of a modification in the shape of the resonant cavity that is not taken into account in the perturbative approach.Comment: 8 pages, 6 figures, Astronomy & Astrophysics (in press

Determining the metallicity of the solar envelope using seismic inversion techniques

The solar metallicity issue is a long-lasting problem of astrophysics, impacting multi- ple fields and still subject to debate and uncertainties. While spectroscopy has mostly been used to determine the solar heavy elements abundance, helioseismologists at- tempted providing a seismic determination of the metallicity in the solar convective enveloppe. However, the puzzle remains since two independent groups prodived two radically different values for this crucial astrophysical parameter. We aim at provid- ing an independent seismic measurement of the solar metallicity in the convective enveloppe. Our main goal is to help provide new information to break the current stalemate amongst seismic determinations of the solar heavy element abundance. We start by presenting the kernels, the inversion technique and the target function of the inversion we have developed. We then test our approach in multiple hare-and-hounds exercises to assess its reliability and accuracy. We then apply our technique to solar data using calibrated solar models and determine an interval of seismic measurements for the solar metallicity. We show that our inversion can indeed be used to estimate the solar metallicity thanks to our hare-and-hounds exercises. However, we also show that further dependencies in the physical ingredients of solar models lead to a low accuracy. Nevertheless, using various physical ingredients for our solar models, we determine metallicity values between 0.008 and 0.014.Comment: Accepted for publication in MNRA

A Sunyaev-Zel'dovich Effect Survey for High Redshift Clusters

Interferometric observations of the Sunyaev-Zel'dovich Effect (SZE) toward clusters of galaxies provide sensitive cosmological probes. We present results from 1 cm observations (at BIMA and OVRO) of a large, intermediate redshift cluster sample. In addition, we describe a proposed, higher sensitivity array which will enable us to survey large portions of the sky. Simulated observations indicate that we will be able to survey one square degree of sky per month to sufficient depth that we will detect all galaxy clusters more massive than 2x10^{14} h^{-1}_{50}M_\odot, regardless of their redshift. We describe the cluster yield and resulting cosmological constraints from such a survey.Comment: 7 pages, 6 figures, latex, contribution to VLT Opening Symposiu