A physics-assisted deep neural network-based closure framework for velocity gradient dynamics in compressible flows with vibrational non-equilibrium

In this study, we propose a dynamical model for the evolution of velocity gradients in compressible turbulent flows with vibrational non-equilibrium effects, using physics-assisted deep neural networks. Such models provide a powerful framework for understanding the nonlinear physics associated with small-scale structures. In compressible flows, the influence of thermodynamic fields on velocity-gradient dynamics is represented through the thermodynamic gradient field (TGF) tensor. The TGF tensor is one of the primary unclosed terms in the velocity-gradient evolution equations. The TGF tensor comprises contributions from the pressure-Hessian tensor, rho bold italic upper H ρ H $\rho \boldsymbol{H}$ , and the baroclinic tensor, bold italic upper B B $\boldsymbol{B}$ . Accordingly, the proposed framework incorporates closures for both bold italic upper H H $\boldsymbol{H}$ and bold italic upper B B $\boldsymbol{B}$ tensor dynamics. Building upon existing phenomenological closures for the bold italic upper H H $\boldsymbol{H}$ tensor governing mechanisms, we develop a neural-network-based closure for the inviscid mechanism responsible for generating the bold italic upper B B $\boldsymbol{B}$ tensor. Unlike the other recently used tensor bases, the presented work employs a novel tensor basis allowing for the inclusion of non-symmetric features in the model. The framework also incorporates a data-driven closure for vibrational non-equilibrium effects. The resulting framework combines phenomenological and data-driven representations of various bold italic upper H H $\boldsymbol{H}$ and bold italic upper B B $\boldsymbol{B}$ tensors governing mechanisms, termed as the hybrid enhanced homogenised Euler equation (H-EHEE) model. Model predictions are evaluated across a range of turbulent Mach numbers and compared against direct numerica