Existence of homoenergetic affine flows for the Boltzmann equation

An existence theorem is proved for homoenergetic affine flows described by the Boltzmann equation. The result complements the analysis of Truesdell and of Galkin on the moment equations for a gas of Maxwellian molecules. Existence of the distribution function is established here for a large class of molecular models (hard sphere and angular cut-off interactions). Some of the data lead to an implosion and infinite density in a finite time, in agreement with the physical picture of the associated flows; for the remaining set of data, global existence is shown to hold.     

Boundary value problems for a model Boltzmann equation with frequency proportional to the molecule velocity

Exact solutions of a model Boltzmann equation with a collision frequency that depends on the molecule velocity and with a BGK (Bhatnagar-Gross-Krook) collision operator are constructed for the problems of weak evaporation and temperature jump in a rarefied vapor above a plane surface. The numerical calculations and a comparison with previous results are given.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 140–153, May–June, 1996.     

Some one-dimensional unsteady adiabatic gas flows with plane symmetry

A method for the approximate analytic investigation of one-dimensional adiabatic gas flows in the form of arbitrary small perturbations of a simple wave is proposed. A class of exact solutions which, in particular, describes the flows arising from the short intense impact of a piston moving under gas pressure is obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 96–104, September–October, 1986.     

Nonlinear nonequilibrium kinetic model of the Boltzmann equation for a gas with power-law interaction

A model kinetic equation approximating the Boltzmann equation on a wide range of the intensities of nonequilibrium states of gases is derived to describe rarefied gas flows. The kinetic model is based on a distribution function dependent on the absolute velocity of gas particles. Themodel kinetic equation possesses a high computational efficiency and the problem of shock wave structure is solved on its basis. The calculated and experimental data for argon are compared.     

Benchmarking of the one-dimensional one-group interfacial area transport equation for reduced-gravity bubbly flows

In order to properly design and safely operate two-phase flow systems, especially those deployed on future space missions, it is necessary to have accurate predictive capabilities. The application of a novel predictive method, the interfacial area transport equation (IATE), to dynamically predict the change of interfacial area concentration for reduced-gravity two-phase flows is described in this paper. Fluid particle interaction mechanisms such as coalescence and breakup that are present in reduced-gravity two-phase flows have been studied experimentally as reported in a previous paper by the current authors [Vasavada et al., 2007]. These mechanisms represent the source and sink terms in the IATE and their mechanistic models are benchmarked using experimental data obtained in a 25 mm inner diameter ground-based test section wherein reduced-gravity conditions were simulated. The comparison of the predictions from the model against experimental data shows good agreement. It has been found that, in contrast to the hypothesis extended in the literature, the wake entrainment based coalescence mechanism is present in reduced-gravity two-phase flows and in some cases is more important than coalescence due to random collision. Physics based arguments are extended to support this conclusion.     

New results on the semidiscrete Boltzmann equation for a binary gas mixture

Summary A plane semidiscrete model of the Boltzmann equation for a binary gas mixture with molecular collisions ruled by the hard-spheres interaction potential is described. After establishing a model, a theorem demostrating the global existence of mild solutions of the initial-value problem is given and the propagation of unidimensional shock waves examined.

---

Sommario Si propone un modello semidiscreto piano dell'equazione di Boltzmann per una miscela binaria con collisioni molecolari soggette al potenziale di interazione delle sfere rigide. Costruito il modello, si dà un teorema di esistenza globale di soluzioni generalizzate per il problema di Cauchy, e si analizza la propagazione di onde d'urto unidimensionali.

     

Solution with a linear velocity field for a submodel of one-dimensional gas motions

A method for finding exact solutions of the equations of gas dynamics with a linear velocity field is proposed. This method was used to find exact solutions for one submodel of the evolutionary type which was fully integrated for the case of a polytropic gas. Examples of particle motion for the obtain exact solutions are given.     

Model Boltzmann equation for gas mixtures: Construction and numerical comparison

A new model for the nonlinear Boltzmann equation for gas mixtures is constructed by the method employed in the derivation of the McCormack model in the linearized kinetic theory [F.J. McCormack, Phys. Fluids 16 (1973) 2095]. Then it is compared numerically with other existing models proposed in [P. Andries, K. Aoki, B. Perthame, J. Stat. Phys. 106 (2002) 993] and in [L.H. Holway Jr., Phys. Fluids 9 (1966) 1658] (the so-called ES-BGK model) as well as with the original Boltzmann equation. The new model is not restricted to the Maxwell molecule, can fit to general molecular models, and reproduces well solutions of the Boltzmann equation at least in the case of weak nonequilibrium. The numerical comparison is performed in the case of a binary gas mixture consisted of the hard-sphere or pseudo Maxwell molecules, after parameters concerning the molecular interaction are adjusted appropriately.     

Laser-induced fluorescence modulation techniques for velocity measurements in gas flows

Broadband fluorescence of iodine, excited at 514.5 nm by a single-mode argon-ion laser tuned to the quasi-linear part of an absorption line, was used to detect the Doppler shift and hence the velocity of iodine molecules seeded in a nitrogen jet flow. The slope of the absorption line profile was measured directly using a frequency shift introduced by acoustooptic modulators (AOMs). A velocity of 36 m/s was measured in a jet of N₂ at 60 Torr in 2 ms with an accuracy of 11%. To reduce experimental noise, the laser beams were switched at 125 KHz and signal-tuned amplification was used.     

One-dimensional unsteady motions of a viscous heat-conducting gas with a linear velocity distribution with respect to the coordinate

The one-dimensional motions of a perfect gas are considered in cases of spherical, cylindrical and plane symmetry, when the velocity is proportional to the distance to the center of symmetry. The solutions obtained are an extension of the known solutions of Sedov [1, 2] to the case of a viscous heat-conducting gas with a power-law temperature dependence of the coefficient of viscosity and thermal conductivity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 130–133, March–April, 1987.The author is grateful to L. I. Sedov for his interest in the work and to A. G. Kulikovskii for useful discussions.     

Variance‐reduced Monte Carlo solutions of the Boltzmann equation for low‐speed gas flows: A discontinuous Galerkin formulation

We present and discuss an efficient, high‐order numerical solution method for solving the Boltzmann equation for low‐speed dilute gas flows. The method's major ingredient is a new Monte Carlo technique for evaluating the weak form of the collision integral necessary for the discontinuous Galerkin formulation used here. The Monte Carlo technique extends the variance reduction ideas first presented in Baker and Hadjiconstantinou (Phys. Fluids 2005; 17 , art. no. 051703) and makes evaluation of the weak form of the collision integral not only tractable but also very efficient. The variance reduction, achieved by evaluating only the deviation from equilibrium, results in very low statistical uncertainty and the ability to capture arbitrarily small deviations from equilibrium (e.g. low‐flow speed) at a computational cost that is independent of the magnitude of this deviation. As a result, for low‐signal flows the proposed method holds a significant computational advantage compared with traditional particle methods such as direct simulation Monte Carlo (DSMC). Copyright © 2008 John Wiley & Sons, Ltd.     

Formation of one-dimensional flows of viscous heat-conducting gas for small parameter perturbations

The formation of unsteady one-dimensional flows is studied, using the solution of the problem of reflection of a normally incident plane shock wave from a heat-conducting wall as an example. The process is considered for low intensities of the incident wave, behind which the gas temperature hardly differs from the initial wall temperature. The flows with complicated internal structures that arise are investigated on the basis of Navier-Stokes equations linearized near the initial state. An analytic solution of the problem describing the discontinuous structure of the reflected flow is constructed, which can serve as a test in the numerical solution of the original nonlinearized Navier-Stokes equations. The influence of the Prandtl numbers, the specific heat ratio, and dissipative and other factors is considered. The features of the effects of viscosity, thermal conductivity, and accommodation on the formation of flows and ideal (inviscid, nonheat-conducting) and dissipative zones are traced. It is shown that the solution of the linearized system agrees with the solution for asymptotic flow regimes.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 105–111, September–October, 1986.     

Analysis of the transition of one-dimensional flows of a viscous heat conducting gas to asymptotic regimes

Asymptotic solutions are presented which describe the formation of one-dimensional flows of a viscous heat conducting gas at short and long times. Comparison of them with numerical calculations made in the study for the system of Navier-Stokes equations in conjunction with an estimate of the ratio of each term in the asymptotic expansion to the preceding term leads to dimensionless-number dependences that determine the limits of applicability of the obtained solutions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 110–118, September–October, 1981.     

Shock-wave structure for a binary gas mixture: finite-difference analysis of the Boltzmann equation for hard-sphere molecules

The structure of a normal shock wave for a binary mixture of hard-sphere gases is analyzed numerically on the basis of the Boltzmann equation by a finite-difference method. In the analysis, the complicated collision integrals are computed efficiently as well as accurately by means of the numerical kernel method, which is the generalization to the case of a binary mixture of the method devised by Ohwada in 1993 in the shock-structure analysis for a single-component gas. The transition from the upstream to the downstream uniform state is clarified not only for the macroscopic quantities but also for the velocity distribution functions.