High-accuracy deterministic solution of the Boltzmann equation for the shock wave structure

Malkov, E. A.; Bondar, Ye. A.; Kokhanchik, A. A.; Poleshkin, S. O.; Иванов, М. С.

doi:10.1007/s00193-015-0563-6

Cited by 26 publications

(12 citation statements)

References 19 publications

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“…In validation studies of various kinetic and continuum gas flow models, the experiments are complemented with solutions of the Boltzmann equations, both direct solutions (Aristov & Cheremisin 1980) and those obtained by a statistical approach, namely, the direct simulation Monte Carlo (DSMC) method (Bird 1994). Both approaches provide essentially identical results on the shock structure including distribution functions (Ohwada 1993) and very fine details of the flow such as a tiny maximum of the temperature observed for high Mach numbers (Malkov et al 2015). The validity of these results is supported by excellent agreement with experiments both on macroparameter profiles (Belotserkovskii & Yanitskii 1975;Bird 1994;Timokhin et al 2015) and distribution functions (Pham- Van-Diep et al 1989).…”

Section: Introductionsupporting

confidence: 54%

“…The presence of an overshoot in the planar shock wave structure problem at large Mach numbers has been shown in many papers using both kinetic description (Elliott & Baganoff 1974; Erofeev & Friedlander 2002; Dodulad & Tcheremissine 2013; Malkov et al. 2015) and extended gas dynamics methods (Jin, Pareschi & Slemrod 2002; Torrilhon & Struchtrup 2004; Erofeev & Friedlander 2007; Ivanov et al. 2012; Timokhin et al.…”

Section: Resultsmentioning

confidence: 92%

“…Both approaches provide essentially identical results on the shock structure including distribution functions (Ohwada 1993) and very fine details of the flow such as a tiny maximum of the temperature observed for high Mach numbers (Malkov et al. 2015). The validity of these results is supported by excellent agreement with experiments both on macroparameter profiles (Belotserkovskii & Yanitskii 1975; Bird 1994; Timokhin et al.…”

Section: Introductionmentioning

confidence: 93%

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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: Introductionsupporting

confidence: 54%

Section: Resultsmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 93%

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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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“…Bird 1994); hence the kinetic approach -direct numerical solution of the Boltzmann equation or the direct simulation Monte Carlo (DSMC) method (Bird 1994) -should be employed for the shock wave structure problem (see e.g. Malkov et al 2015).…”

Section: Introductionmentioning

confidence: 99%

Viscous effects in Mach reflection of shock waves and passage to the inviscid limit

et al. 2023

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The influence of viscosity on the Mach reflection of shock waves in a steady flow of a monatomic gas is studied by solving the Navier–Stokes equations numerically. Based on the nested block grid refinement technique, the flow near the shock wave intersection is simulated, and its behaviour with increasing Reynolds number is studied. The computations are performed for the interaction of both strong (free-stream Mach number $M_\infty = 4$ ) and weak ( $M_\infty = 1.7$ ) shock waves. In the strong reflection of shock waves at all Reynolds numbers in the examined range, it is found that there exists a small-size zone behind the shock wave intersection where the flow parameters differ from those predicted by the Rankine–Hugoniot relations and hence deviate from the predictions of the inviscid three-shock theory. The structure of this zone is self-similar: in coordinates normalised to the mean free path of molecules in the free stream. The structure is identical at all Reynolds numbers considered in the study. As the Reynolds number increases, the size of this zone in physical coordinates decreases, but the maximum difference between the viscous and inviscid solutions in this zone remains constant, reaching approximately $10\,\%$ for pressure. In the weak reflection of shock waves, the flow structure behind the shock wave intersection is not self-similar, i.e. the flow fields at different Reynolds numbers do not coincide in the normalised coordinates, but converge, as the Reynolds number increases, to the parameters predicted by the inviscid three-shock theory.

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“…It has been employed by many scholars. Malkov et al [4] solved Boltzmann function of shock wave when Mach number is 4 and 8 based on hard sphere model. Bisi et al [5] solved the structure of the solution for binary mixed shock wave based on molecular dynamics.…”

Section: Introductionsmentioning

confidence: 99%

Solution to Propagation of Shock Wave Based on Continuum Hypothesis

Wang

Jing

2018

Shock and Vibration

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Based on the continuum hypothesis, problem of compressing wave was studied analytically. By exploring the temperature distribution, the propagation velocity, and the thickness of the transition region, the developing process of compressing wave turning into shock wave was revealed, and the following conclusions were reached: (1) the governing equations of compressing wave can be turned into two autonomous equations, and the phase diagram can be used as an effective tool for the analysis of the compressing wave; (2) the solution of compressing wave was composed of two parts: the inner solution and the outer solution; when Pr = 3/4, the analytical inner solution can be obtained; (3) as the disturbance velocity increases, the thickness of compressing wave will decrease, and the propagation velocity of the compressing wave will increase, and the compressing wave will become shock wave when the disturbance velocity is large enough; (4) velocity and temperature across compressing wave change monotonically and continuously, but the entropy generation changing tendency is closely related to Pr; therefore, the inner solution reveals the mechanism of irreversibility happening in compressing waves.

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High-accuracy deterministic solution of the Boltzmann equation for the shock wave structure

Cited by 26 publications

References 19 publications

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

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

Viscous effects in Mach reflection of shock waves and passage to the inviscid limit

Solution to Propagation of Shock Wave Based on Continuum Hypothesis

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