Adjoint-Trained Deep-Learning Closures of the Navier–Stokes Equations for 2D Nonequilibrium Flows

Rarefied and nonequilibrium flows may be accurately simulated by solving the Boltzmann equations, though this can be computationally expensive in regimes relevant to hypersonic flight. The Navier–Stokes equations, while computationally tractable, are unreliable in these regimes due to the failure of the continuum assumption. To address this, a recent study introduced a deep learning framework to augment the Navier–Stokes equations for one-dimensional shocks in the transition-continuum regime [1]. The framework trains closure models consistently with the Navier–Stokes equations and the second law of thermodynamics using the adjoint method, which calculates the loss sensitivities with respect to the dependent variables to enable online optimization over the system of partial differential equations. This study extends this framework to two-dimensional, steady, hypersonic boundary-layer flows. Target data is obtained through direct simulation Monte Carlo (DSMC) solutions of the Boltzmann equation. Quasi-out-ofsample results are presented for idealized boundary conditions, obtained from DSMC, and preliminary out-of-sample results are provided for slip-wall boundary conditions.