The analysis of different variants of R13 equations applied to the shock-wave structure

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 $Ma_n>8$ .

Section: Resultsmentioning

confidence: 79%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The Mott-Smith solution to the regular shock reflection problem

Timokhin

Kudryavtsev²,

Bondar³

2022

“…The exceptions include the overestimated 1-D temperature overshoot in the IS and temperature distributions in the region of formation of the leading front of shock waves. Both of these effects are well known (see the results of Timokhin et al (2015) and Timokhin et al (2017)). Another small difference is slight overestimation of the temperature maximum on the symmetry plane by the R13 equations.…”

Section: General Features Of Flow Fieldsmentioning

confidence: 77%

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

Bondar,

Shoev,

Timokhin

2024

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.

“…However, an extension of the R13 equations to hard-sphere molecules by changing the viscosity index did not yield satisfactory results. The authors remarked that the validity of the R13 moment equations is questionable for Ma > 4; however, recent works by Timokhin et al (2015Timokhin et al ( , 2016Timokhin et al ( , 2017 show that the applicability of the R13 equations can be extended up to a Mach number equal to 8 for different interaction potentials. When shock profiles are evaluated by the R13 equations for hard-sphere molecules proposed by Cai & Wang (2020), the results agreed well with the DSMC results.…”

Section: Comparison With Moment Equationsmentioning

confidence: 99%

Improved theory for shock waves using the OBurnett equations

Jadhav

Gavasane²,

Agrawal

2021

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.