Topology of direct simulation Monte Carlo constitutive relations for neural network-accelerated finite-volume solutions of velocity-shear flows

A novel framework for modeling non-equilibrium gas flows was recently introduced by Garg et al. “Neural network-based finite volume method and direct simulation Monte Carlo solutions of non-equilibrium shock flow guided by nonlinear coupled constitutive relations,” Phys. Fluids 36, 106113 (2024) and “Fast neural network-based direct simulation Monte Carlo solutions of shock flow of diatomic gases with vibrational modes,” Phys. Fluids 37, 076105 (2025), in which a pre-trained deep neural network (DNN) machine learning (ML) model of constitutive relations, learned from direct simulation Monte Carlo (DSMC) data, was embedded into a finite volume method (FVM) solver, enabling the accurate reproduction of shock structures at a fraction of the computational cost. Both studies, however, focused solely on the compressive flow regime. The present study extends this framework to the velocity-shear flow regime for monatomic gases. Using DSMC data (up to Mach number 20, Knudsen number 50) for one-dimensional Couette flow, the topology of the constraints in the constitutive relations governing viscous stresses and heat fluxes is analyzed and compared with that of the nonlinear coupled constitutive relation (NCCR) theory. Building on this topological analysis, a comprehensive FVM-DSMC-ML framework is developed that integrates a DNN-based model of DSMC constitutive relations, specifically designed for velocity-shear dominated flows, within the FVM approach. The DNN model incorporates DSMC data to ensure generality and accuracy. The FVM-DSMC-ML framework also employs the modified Maxwell and Smoluchowski boundary conditions introduced by Myong “Theoretical description of the gaseous Knudsen layer in Couette flow based on the second-order constitutive and slip-jump models,” Phys. Fluids 28, 012002 (2016), whose coefficients are further refined using DSMC data. The resulting FVM-DSMC-ML solver demonstrates excellent agreement with DSMC predictions while reducing computational cost to approximately 1/15 that of conventional DSMC simulations. This study provides new physical insight into non-equilibrium velocity-shear flows and represents an important advance toward data-driven constitutive modeling in such regimes.