Deep learning closure of the Navier-Stokes equations and slip boundary conditions for transition-continuum flows

Despite having good predictive accuracy for transition-continuum flows, the Direct Simulation Monte Carlo (DSMC) method remains computationally more expensive than continuum CFD methods that solve the Navier-Stokes equations (NSE), for which the transport models and boundary conditions are often inaccurate at these conditions.

We propose neural network closure models to augment both the NSE and slip boundary conditions. The modelling approaches include modifications of the viscosity, thermal conductivity, mean free path, stress tensor, and heat flux. Physical constraints, such as the second law of thermodynamics, are considered. The neural networks are designed to ensure the positivity of the transport properties, and a projection method ensures that the modified stress tensor and heat flux satisfy the Clausius-Duhem inequality. A framework based on the discrete -adjoint equations is utilized to optimize the closure models.

Our example case is a two-dimensional hypersonic blunt cone at Mach numbers between 2 and 10, and Knudsen numbers between 0.01 and 0.1. The database is divided into a training set and a test set. Preliminary results show a significant reduction in loss for the training set. An evaluation of the predictive capabilities of the models is conducted for out-of-sample (i.e., unseen in training) conditions.