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

Larina, I. N.; Рыков, В. А.

doi:10.1134/s0015462813060148

Cited by 6 publications

(5 citation statements)

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“…The kinetic approach based on solving the Boltzmann equation is formally applicable for all degrees of rarefaction. However, in solving real applied problems by methods based on the kinetic approach, such as the Direct Simulation Monte Carlo (DSMC) method, 1 numerical solution of the Boltzmann equation, [2][3][4] and various model equations (BGK, ESM, Shakhov etc., [5][6][7], there are significant constraints associated with computational engineering capabilities. An alternative for computations of moderately rarefied flows is the use of continuum methods, which can simulate the behavior of significantly a) Electronic mail: timokhin@physics.msu.ru.…”

Section: Introductionmentioning

confidence: 99%

Study of the shock wave structure by regularized Grad’s set of equations

et al. 2015

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In this work, we continue to study the possibility of applying moment equations for strongly nonequilibrium flows by an example of the problem of the shock wave structure in a monatomic gas in a wide range of Mach numbers for various models of molecular interaction. The object of the study is the so-called regularized 13-moment Grad's system (R13). First time, both linear and nonlinear versions of this system of equations were considered for the problem at such wide range of parameters. The Godunov method with increased accuracy is used as a numerical tool for solving the R13 system. The numerical results for the R13 system are analyzed by using data obtained by the Direct Simulation Monte Carlo (DSMC) method, experimental data, and analytical results. As a whole, the R13 system provides an adequate description of the shock wave structure in a wide range of Mach numbers. For Mach numbers around 2, good agreement with experimental and DSMC results is observed for both linear and nonlinear versions of the system. For high Mach numbers, the result strongly depends on the molecular interaction model used: shock wave structure predictions of the nonlinear R13 system are better for Maxwell molecules and worse for hard spheres as compared to the linear version. Particular attention in this work is paid to studying nonmonotonicity of the total temperature profile (temperature overshoot) in the structure of a strong shock wave. It is shown that the moment equations correctly predict the existence of the temperature overshoot. At the same time, the solution of the moment equations overpredicts the temperature overshoot at least two-fold for Mach number M = 8, and the nonlinear version of the R13 system yields a better result for this parameter than the linear version. C 2015 AIP Publishing LLC. [http://dx.

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Section: Introductionmentioning

confidence: 99%

Study of the shock wave structure by regularized Grad’s set of equations

et al. 2015

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“…2000; Erofeev & Friedlander 2002; Torrilhon & Struchtrup 2004; Rykov, Titarev & Shakhov 2008; Xu & Huang 2010; Bobylev et al. 2011; Dodulad & Tcheremissine 2013; Larina & Rykov 2013). Along with the above-mentioned non-equilibrium, the reasons for this popularity include the importance of shock-wave phenomena in versatile real-life applications, simplicity of the mathematical formulation and the availability of experimental data (Cowan & Hornig 1950; Hansen & Hornig 1960; Robben & Talbot 1966; Schmidt 1969; Alsmeyer 1976; Pham-Van-Diep et al.…”

Section: Introductionmentioning

confidence: 99%

“…The shock-wave problem has become one of the key benchmarks for various molecular interaction models, mathematical approaches and numerical methods in the field of gas kinetic theory and rarefied gas dynamics (e.g. Grad 1952;Ruggeri 1993;Uribe et al 2000;Erofeev & Friedlander 2002;Torrilhon & Struchtrup 2004;Rykov, Titarev & Shakhov 2008;Xu & Huang 2010;Bobylev et al 2011;Dodulad & Tcheremissine 2013;Larina & Rykov 2013). Along with the above-mentioned non-equilibrium, the reasons for this popularity include the importance of shock-wave phenomena in versatile real-life applications, simplicity of the mathematical formulation and the availability of experimental data (Cowan & Hornig 1950;Hansen & Hornig 1960;Robben & Talbot 1966 ;Schmidt 1969;Alsmeyer 1976;Pham-Van-Diep et al 1989).…”

Section: Introductionmentioning

confidence: 99%

“…For the case of dilute simple gases, an accurate description of the shock structure is provided by the kinetic Boltzmann equation. Other models are less detailed and can be divided into kinetic models using simplified forms of the Boltzmann collision integral (Bhatnagar, Gross & Krook 1954;Shakhov 1968;Larina & Rykov 2013) and continuum models based on equations containing macroscopic parameters (moments of the velocity distribution function) (Grad 1949;Kogan 1969;Chapman & Cowling 1991;Struchtrup 2005). It is worth mentioning that, in addition to parameters that define intermolecular interaction, the shock-wave structure depends only on one dimensionless similarity criterion -the Mach number of the shock.…”

Section: Introductionmentioning

confidence: 99%

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Regular reflection of shock waves in steady flows: viscous and non-equilibrium effects

Bondar,

Shoev,

Timokhin

2024

J. Fluid Mech.

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Numerical analysis of a steady monatomic gas flow about the point of the regular reflection of a strong oblique shock wave from the symmetry plane is conducted with the Navier–Stokes–Fourier (NSF) equations, the regularized Grad 13-moment (R13) equations and the direct simulation Monte Carlo (DSMC) method. In contrast to the inviscid solution to this problem completely defined by the Rankine–Hugoniot (RH) relations, all three models predict a complicated flow structure with strong thermal non-equilibrium and a long wake with flow parameters not predicted by the RH relations. The temperature $T_y$ related to thermal motion of molecules in the direction normal to the symmetry plane has a maximum inside the reflection zone while in a planar shock wave the maximum is observed for the $T_x$ temperature. The R13 equations predict these features much better than the NSF equations and are in good agreement with the benchmark DSMC results. An analysis of the flow with the conservation equations was conducted in order to evaluate the effects of various processes on a fluid element moving along the symmetry plane. In contrast to the shock wave where effects of viscosity and heat conduction are one-dimensional with zeroth net contribution to the fluid-element energy across the shock, the flow across the zone of the shock reflection is dominated by two-dimensional effects with positive net contribution of viscosity and negative contribution of heat conduction to the fluid-element energy. These effects are believed to be the main source of the wake with parameters deviating from the RH values.

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Section: Introductionmentioning

confidence: 99%