Neural network-based finite volume method and direct simulation Monte Carlo solutions of non-equilibrium shock flow guided by nonlinear coupled constitutive relations

For understanding many real-world problems involving rarefied hypersonic, micro-, and nanoscale gas flows, the primary method may be the direct simulation Monte Carlo (DSMC). However, its computational cost is prohibitive in comparison with the Navier–Stokes–Fourier (NSF) solvers, eclipsing the advantages it provides, especially for situations where flow is in the near continuum regime or three-dimensional applications. This study presents an alternate computational method that bypasses this issue by taking advantage of data-driven modeling and nonlinear coupled constitutive relations. Instead of using numerical solutions of higher-order constitutive relations in conventional partial differential equation-based methods, we build compact constitutive relations in advance by applying deep neural network algorithms to available DSMC solution data and later combine them with the conventional finite volume method for the physical laws of conservation. The computational accuracy and cost of the methodology thus developed were tested on the shock wave inner structure problem, where high thermal non-equilibrium occurs due to rapid compression, for a range of Mach numbers from 2 to 10. The simulation results obtained with the computing time comparable to that of the NSF solver showed almost perfect agreement between the neural network-based combined finite volume method and DSMC and original DSMC solutions. We also present a topology of DSMC constitutive relations that allows us to study how the DSMC topology deviates from the NSF topology. Finally, several challenging issues that must be overcome to become a robust method for solving practical problems were discussed.