Improved theory for shock waves using the OBurnett equations

Self Cite

In this work, we analyse wall-bounded flows in the continuum to transition regime with the help of higher-order transport equations (super-set of the Navier–Stokes equation). Towards this, we incorporate second-order in Knudsen number accurate terms in the single-particle distribution function, and with this complete representation, we first derive the second-order accurate extended-OBurnett (EOBurnett) and third-order accurate super-OBurnett (SOBurnett) equations. We then demonstrate that these newly derived equations exhibit unconditional linear stability. We finally validate the equations by solving for plane Poiseuille flow and derive closed-form analytical solutions for the pressure and velocity fields. The pressure and velocity results thus obtained have been compared with direct simulation Monte Carlo (DSMC) data in the transition regime. The results from both the EOBurnett and SOBurnett equations are found to yield better agreement with DSMC data than that obtained from the Navier–Stokes equations. This improved agreement is attributed to the presence of additional terms in the proposed equations, which effectively capture the effect of the Knudsen layer near the wall. The obtained higher-order transport equations and the closed-form solution presented in this work are novel. The ability of the equations to describe the flow in the transition regime should form the basis for conducting further realistic analytical studies of wall-bounded flows in the future.

Section: Analytical Solution Of Pressure-driven Poiseuille Flowmentioning

confidence: 99%

Section: Validation Of Analytical Solution Using Dsmc Datamentioning

confidence: 99%

Derivation of extended-OBurnett and super-OBurnett equations and their analytical solution to plane Poiseuille flow at non-zero Knudsen number

Yadav,

Jonnalagadda,

Agrawal

2024

Self Cite

“…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%

“…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, 2001 a , b ), the nonlinear coupled constitutive relations (Jiang et al. 2019) and extended thermodynamics (Ruggeri 1996; Taniguchi et al.…”

Section: Introductionmentioning

confidence: 99%

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

Zhu,

Wu,

Zhou

et al. 2024

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.

“…In the course of many years of research on normal shock wave internal structure, it has become a benchmark problem for testing new mathematical models of rarefied and non-equilibrium flows (Pham- Van-Diep, Erwin & Muntz 1991;Ohwada 1993;Torrilhon & Struchtrup 2004;Kudryavtsev, Shershnev & Ivanov 2008;Johnson 2013;Timokhin et al 2016;Velasco & Uribe 2019;Shoev, Timokhin & Bondar 2020;Jadhav, Gavasane & Agrawal 2021). It has been caused by the importance of shock-wave phenomena in real-life applications, simplicity of the mathematical formulation and availability of experimental data (Hansen & Hornig 1960;Schmidt 1969;Alsmeyer 1976;Pham-Van-Diep, Erwin & Muntz 1989;Timokhin et al 2020).…”

Section: Introductionmentioning

confidence: 99%

The Mott-Smith solution to the regular shock reflection problem

Timokhin

Kudryavtsev²,

Bondar³

2022

The classical Mott-Smith solution for one-dimensional normal shock wave structure is extended to the two-dimensional regular shock reflection problem. The solution for the non-equilibrium molecular velocity distribution function along the symmetry-plane streamline is obtained as a weighted sum of four Maxwellians. An analysis of applicability of the solution has been performed using the results of direct simulation Monte Carlo calculations for a range of incident shock wave intensities. Accuracy of the solution improves with increasing $Ma_n$ , the Mach number normal to the shock front, so that the solution becomes rather accurate for strong shocks with $Ma_n>8$ .