Computation of shock wave structure using a simpler set of generalized hydrodynamic equations based on nonlinear coupled constitutive relations

Jiang, Zhongzheng; Zhao, Wenwen; Chen, W.; Agarwal, Ramesh K.

doi:10.1007/s00193-018-0876-3

Cited by 28 publications

(6 citation statements)

References 34 publications

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“…Since the number of moments is much less than the coefficients before Hermite polynomials, some special relations between these coefficients have to be assumed. It should also be noted that, although many numerical solutions of rarefied gas flows have been obtained by using the nonlinear coupled constitutive relations, the VDF has never been re-constructed from macroscopic quantities [35,46,47,82].…”

Section: Moment Methodsmentioning

confidence: 99%

See 1 more Smart Citation

On the accuracy of macroscopic equations for linearized rarefied gas flows

2020

Adv. Aerodyn.

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Many macroscopic equations are proposed to describe the rarefied gas dynamics beyond the Navier-Stokes level, either from the mesoscopic Boltzmann equation or some physical arguments, including (i) Burnett, Woods, super-Burnett, augmented Burnett equations derived from the Chapman-Enskog expansion of the Boltzmann equation, (ii) Grad 13, regularized 13/26 moment equations, rational extended thermodynamics equations, and generalized hydrodynamic equations, where the velocity distribution function is expressed in terms of low-order moments and Hermite polynomials, and (iii) bi-velocity equations and "thermo-mechanically consistent" Burnett equations based on the argument of "volume diffusion". This paper is dedicated to assess the accuracy of these macroscopic equations. We first consider the Rayleigh-Brillouin scattering, where light is scattered by the density fluctuation in gas. In this specific problem macroscopic equations can be linearized and solutions can always be obtained, no matter whether they are stable or not. Moreover, the accuracy assessment is not contaminated by the gas-wall boundary condition in this periodic problem. Rayleigh-Brillouin spectra of the scattered light are calculated by solving the linearized macroscopic equations and compared to those from the linearized Boltzmann equation. We find that (i) the accuracy of Chapman-Enskog expansion does not always increase with the order of expansion, (ii) for the moment method, the more moments are included, the more accurate the results are, and (iii) macroscopic equations based on "volume diffusion" do not work well even when the Knudsen number is very small. Therefore, among about a dozen tested equations, the regularized 26 moment equations are the most accurate. However, for moderate and highly rarefied gas flows, huge number of moments should be included, as the convergence to true solutions is rather slow. The same conclusion is drawn from the problem of sound propagation between the transducer and receiver. This slow convergence of moment equations is due to the incapability of Hermite polynomials in the capturing of large discontinuities and rapid variations of the velocity distribution function. This study sheds some light on how to choose/develop macroscopic equations for rarefied gas dynamics.

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Section: Moment Methodsmentioning

confidence: 99%

“…which, due to the decoupling between the shear stress and heat flux, are simpler than those in G13 equations (35). The RBS spectra are shown in Fig.…”

Section: Generalized Hydrodynamic Equationsmentioning

confidence: 96%

On the accuracy of macroscopic equations for linearized rarefied gas flows

2020

Adv. Aerodyn.

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“…The partial differential equations for the stress tensor and heat flux vector are transformed into nonlinear coupled algebraic equations using the adiabatic approximation (Myong 1999). The corresponding relations for the shock wave flow problem are obtained as (Jiang et al 2019;Liu, Yang & Zhong 2019), (6.12) where q(κ) is a nonlinear dissipation factor. Note that the equations are implicit, coupled and can be solved by iterative methods like the Newton method for given values of conserved variables and their derivatives.…”

Section: Comparison With Conventional Burnett Equationsmentioning

confidence: 99%

Improved theory for shock waves using the OBurnett equations

Jadhav

Gavasane²,

Agrawal

2021

J. Fluid Mech.

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The main goal of the present study is to thoroughly test the recently derived OBurnett equations for the normal shock wave flow problem for a wide range of Mach number ( $3 \leq Ma \leq 9$ ). A dilute gas system composed of hard-sphere molecules is considered and the numerical results of the OBurnett equations are validated against in-house results from the direct simulation Monte Carlo method. The primary focus is to study the orbital structures in the phase space (velocity–temperature plane) and the variation of hydrodynamic fields across the shock. From the orbital structures, we observe that the heteroclinic trajectory exists for the OBurnett equations for all the Mach numbers considered, unlike the conventional Burnett equations. The thermodynamic consistency of the equations is also established by showing positive entropy generation across the shock. Further, the equations give smooth shock structures at all Mach numbers and significantly improve upon the results of the Navier–Stokes equations. With no tweaking of the equations in any way, the present work makes two important contributions by putting forward an improved theory of shock waves and establishing the validity of the OBurnett equations for solving complex flow problems.

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“…On the computational side, numerical techniques are becoming ubiquitous in solving the problems of fluid dynamics. With the development of computational science, the shock structure problem has benefited a lot, especially from the numerical methods of molecular dynamics (Valentini &Schwartzentruber 2009), direct simulation Monte Carlo (DSMC) (Bird 1970(Bird , 1994, direct simulation of the Boltzmann equation (Ohwada 1993;Kosuge, Aoki & Takata 2001), simplified models of the Boltzmann equation such as the Bhatnagar-Gross-Krook (BGK) model (Liepmann, Narasimha & Chahine 1962;Xu & Tang 2004), the discrete velocity model/method (DVM) (Broadwell 1964;Gatignol 1975;Inamuro & Sturtevant 1990;Malkov et al 2015), the classical and modified Navier-Stokes equations (Holian et al 1993;Greenshields & Reese 2007;Uribe & Velasco 2018), the higher-order hydrodynamic equations represented by the Burnett equations (Reese et al 1995;Agarwal, Yun & Balakrishnan 2001;García-Colín, Velasco & Uribe 2008;Bobylev et al 2011;Zhao et al 2014;Jadhav, Gavasane & Agrawal 2021), Grad's moment equations and variants (Torrilhon & Struchtrup 2004;Torrilhon 2016;Cai & Wang 2020;Cai 2021), generalised hydrodynamics (Al-Ghoul & Eu 1997, 2001a, the nonlinear coupled constitutive relations (Jiang et al 2019) and extended thermodynamics (Ruggeri 1996;Taniguchi et al 2014), as well as their hybrid approaches, such as Boltzmann-MC (numerical calculation of the Boltzmann equation with collision integral evaluated by the Monte Carlo method) (Hicks, Yen & Reilly 1972), DVMC (DVM with Monte Carlo evaluations of the collision integral) (Kowalczyk et al 2008;…”

Section: Introductionmentioning

confidence: 99%

A comprehensive study on the roles of viscosity and heat conduction in shock waves

Zhu,

Wu,

Zhou

et al. 2024

J. Fluid Mech.

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Shock waves are of great interest in many fields of science and engineering, but the mechanisms of their formation, maintenance and dissipation are still not well understood. While all transport processes existing in a shock wave contribute to its compression and irreversibility, they are not of equal importance. To figure out the roles of viscosity and heat conduction in shock transition, the existence of smooth shock solutions and the counter-intuitive entropy overshoot phenomenon (the specific entropy is not monotonically increasing and exhibits a peak inside the shock front) are theoretically and numerically investigated, with emphasis on the effects of viscosity and heat conduction. Instead of higher-order hydrodynamics, the Navier–Stokes formalism is employed for its stability and simplicity. Supplemented with nonlinear thermodynamically consistent constitutive relations, the Navier–Stokes equations are adequate to demonstrate the general nature of shock profiles. It is found that heat conduction cannot sustain strong shocks without the presence of viscosity, while viscosity can maintain smooth shock transition at all strengths, regardless of heat conduction. Hence, the critical role in shock compression is played by viscosity rather than heat conduction. Nevertheless, the dispensability of heat conduction would not compromise its essential role in the emergence of an entropy peak. It is the entropy flux resulting from heat conduction that neutralises the positive entropy production and thus prevents the decreasing entropy from violating the second law of thermodynamics. This mechanism of entropy overshoot has not been addressed previously in the literature and is revealed using the entropy balance equation.

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Computation of shock wave structure using a simpler set of generalized hydrodynamic equations based on nonlinear coupled constitutive relations

Cited by 28 publications

References 34 publications

On the accuracy of macroscopic equations for linearized rarefied gas flows

On the accuracy of macroscopic equations for linearized rarefied gas flows

Improved theory for shock waves using the OBurnett equations

A comprehensive study on the roles of viscosity and heat conduction in shock waves

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