Computation of shock wave structure using a simpler set of generalized hydrodynamic equations based on nonlinear coupled constitutive relations
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Generalized hydrodynamic equations were originally proposed to describe the rarefied non-equilibrium flows beyond the Navier–Stokes–Fourier (NSF) equations by constructing a non-equilibrium canonical distribution function for the mesoscopic Boltzmann equation. Subsequently, nonlinear coupled constitutive relations (NCCR) were developed under the adiabatic assumption, the Eu’s closure, and Myong’s simplification. NCCR+ was also proposed to include the omitted terms in Myong’s simplification. The goal of this paper is to assess the improvements in the accuracy due to NCCR+ and the influence of the bulk viscosity in one-dimensional steady shock wave structure for monatomic and diatomic gases. In order to solve NCCR+ equations, a coupled solution process based on the time-independent method for non-conserved variables is employed, which is different from the previous uncoupled solution process used for the NCCR equations. Shock structures in argon and nitrogen are calculated up to Mach 50, where the shock profile, inverse shock thickness, asymmetry parameter, and temperature–density separation distance are validated by DSMC and available experimental measurements. The results show that NCCR+ could not provide much improvement in accuracy compared to NCCR but adds to the computational cost, suggesting that Myong’s simplification used in NCCR is satisfactory. Both NCCR and NCCR+ perform better than NSF in computing the one-dimensional shock wave structure at high Mach numbers. It is also shown that the bulk viscosity has significant influence on the accuracy of prediction of the shock wave structure in a diatomic gas using both NSF and NCCR. Stokes’ hypothesis in conventional NSF is valid only for flows in a monatomic gas and for very low Mach number flows in a diatomic gas. Additionally, it is also found that the viscosity exponent s = 0.81 in variable-hard-sphere model provides a good fit with the experimental data for shock wave in argon using NCCR.
KeywordsNonlinear constitutive relations Rarefied non-equilibrium flows Bulk viscosity Uncoupled and coupled solution methods
This research was funded by the National Natural Science Foundation of China (Grant Nos. 11502232, 51575487, 11572284, and 61627901) and the National Basic Research Program of China (Grant No. 2014CB340201). The first author of this paper (Zhongzheng Jiang) gratefully acknowledges the support of China Scholarship Council (Grant No. 201706320214). Furthermore, he would also like to thank Manuel Torrilhon and his group for their hospitality and Mathematics Division in the Center for Computational Engineering Science at RWTH Aachen University for providing the resources during his stay as joint-training Ph.D. student.
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