Rayleigh–Brillouin scattering in molecular Oxygen by CT-DSMC simulations
Introduction
Light scattering in gases received renewed interest due to the possible applications of LIDAR (Light Detection and Ranging) to perform measurements of atmospheric properties remotely. We cite, in particular, measurements of wind speed distributions by the ADM-Aeolus mission of the European Space Agency [1], [2], [3]. The interpretation of light scattering experiments can be obtained within a well developed theoretical framework [4], [5]. In particular, experiments probe the target gas kinetic regime, where spontaneous fluctuations around a given equilibrium condition have length and time scales respectively comparable with the gas mean free path and mean free time [6]. Since the work of Van Leewen and Yip [7], showing that light scattering spectra from dilute gases can be described by the linearized Boltzmann equation, the theoretical analysis of experimental spectra has been based on kinetic model equations [8], [9], owing to the well known mathematical difficulties connected with the Boltzmann equation. In the case of a one-component gas, many studies have been based on the so called Tenti S6 model [8], consisting in a linear kinetic model for polyatomic gases, constructed according to the Gross–Jackson systematic approximation [10]. The model parameters are tuned to match the gas transport properties: shear and volume viscosity as well as thermal conductivity. In view of the following considerations, it should be noted that accurate experimental values of the shear viscosity and thermal conductivity are usually available. However, volume viscosity is a much more elusive quantity. Laboratory experiments have been conducted under controlled conditions in order to validate the use of the Tenti S6 model [11] and to measure bulk viscosity values to be used in the model [12], thus reducing the uncertainties related to poorly defined model parameters.
The present work is motivated by the consideration that kinetic models are based on simplified collision terms and their predictions should always be checked against those of the full Boltzmann equation, when possible. Nowadays, the Direct Simulation Monte Carlo (DSMC) method [13] provides numerically accurate solutions of the full Boltzmann equation with no additional assumptions beyond those intrinsic in the adopted molecular collision model. If the gas can be considered dilute, DSMC allows exploring the full range of Knudsen numbers, from the free molecular regime down to the hydrodynamic regime, although with increasing computational effort. In particular, DSMC reproduces the correct equilibrium fluctuations spectra in dilute monatomic gases [14] and provides a useful computational tool to investigate fluctuations dynamics out of equilibrium [15], [16]. Therefore, it is interesting to investigate whether DSMC can provide an interpretation of light scattering experiments based on basic molecular collision properties only, in the case of molecular gases.
In this work, spontaneous thermal fluctuations around a given equilibrium state of pure are studied by DSMC simulations. The computed scattering functions are compared with experimental light scattering data from Ref. [12]. Two different DSMC implementations are discussed. The first one is based on the phenomenological Variable Soft Sphere (VSS) model [17] for the total collision cross section. Translational–rotational coupling is described by the Borgnakke–Larsen model [18]. In principle, the phenomenological nature of the collision model requires some tuning. The VSS model coefficients can be tuned to match shear viscosity, thermal conductivity and self-diffusion coefficient, whereas the Borgnakke–Larsen model rotational relaxation number is defined by the volume viscosity. In this aspect, DSMC molecular collision models are similar to kinetic models, both requiring that their free parameters are adjusted to obtain the correct transport properties. However, it is not guaranteed that the two approaches will lead to the same results in the transition regime [19]. Hence, a comparison is in order. The second implementation attempts to go beyond phenomenological collision models by adopting the Classical Trajectories version of DSMC (CT-DSMC) [20], [21] which is based on the accurate description of the collision dynamics via an anisotropic Potential Energy Surface (PES) [22]. The adopted CT-DSMC implementation, contains no adjustable parameters and it has been shown to reproduce accurately experimental thermodynamic and transport properties of molecular Oxygen in the temperature range of interest [20]. The paper is organized as follows: Section 2 describes the basic concepts of light scattering experiments and the related theoretical tools; Section 3 describes how to obtain RBS spectra from DSMC simulations; in Section 4 the kinetic models used in the simulations are briefly described; Section 5 illustrates the simulations performed and compares the results with the experimental spectra. Concluding remarks are finally presented in Section 6.
Section snippets
Rayleigh–Brillouin scattering
Rayleigh–Brillouin scattering is the nonresonant scattering of light by neutral particles [5], [4]. Incident light with wavevector is scattered by fluctuations in the refraction index, in turn caused by spontaneous density fluctuations. Scattered light is measured at an angle . The wavevector of the density fluctuation probed by the experiment is related to that of the incident light via the Bragg condition [5]:
The spectrum of the scattered light , collected at distance
DSMC simulation of RBS spectra
Thermal fluctuations in gases, provided the density is low enough that only bimolecular collisions are effective, are described by the Boltzmann equation. The system is described in terms of the one-particle distribution function. By linearizing the equation around the equilibrium distribution a integro-differential equation for the space–time correlator of the fluctuations of the distribution function is obtained [23]. The density fluctuations are then readily obtained by integration over the
DSMC models
The details of the molecular processes occurring in the gas system are specified by assigning the appropriate set of collision cross sections.
We have used two types of models. One is based on standard DSMC implementations, with an analytical form for the kinetic cross sections (VSS model [17]) and a phenomenological model for the rotational relaxation (Larsen–Borgnakke model [18]). The other is a new implementation, based on a accurate Potential Energy Surface (PES) for the description of the
Results
The experimental spectra discussed in this work are described in Ref. [12]. The incoming laser has a wavelength ; the scattered light is collected at an angle . For molecular Oxygen, 6 different measurements have been performed for pressures between 1 and 3 bar. The precise experimental conditions are reported in Table 2.
The experimental error is estimated at 1% of the peak value [12].
The calculated spectra have been convoluted with the instrument function described in Ref. [12]
Conclusions
Rayleigh–Brillouin scattering experiments in molecular Oxygen have been simulated by DSMC solutions of the Boltzmann equation for a gas of rigid rotors. Different scattering models have been implemented and all reproduce the experimental spectra of Ref. [12] with accuracy comparable to the experimental uncertainty. It is pointed out that estimations of volume viscosity obtained from fitting of experimental spectra are model dependent. Simulations based on CT-DSMC technique and a prescribed
Acknowledgment
The authors gratefully acknowledge the support of NVIDIA Corporation within the framework of the “Hardware Donation Program”.
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