VCNN-e: A vector-cloud neural network with equivariance for emulating Reynolds stress transport equations

Abstract

Developing robust constitutive models is fundamental and a longstanding problem for accelerating the simulation of complicated physics. Machine learning provides promising tools to construct constitutive models based on various calibration data. In this work, we propose a new approach to emulate constitutive tensor transport equations for tensorial quantities through a vector-cloud neural network with equivariance (VCNN-e). The VCNN-e respects all the invariance properties desired by constitutive models and faithfully reflects the region of influence in physics. By design, the model guarantees that the predicted tensor is invariant to the frame translation and ordering (permutation) of the neighboring points. Furthermore, it is equivariant to the frame rotation, i.e., the output tensor co-rotates with the coordinate frame. We demonstrate its performance on Reynolds stress transport equations, showing that the VCNN-e can effectively emulate the Reynolds stress transport model for Reynolds-averaged Navier-Stokes (RANS) equations. Such a priori evaluations of the proposed network pave the way for developing and calibrating robust and nonlocal, non-equilibrium closure models for the RANS equations.

Ensemble Kalman method for learning turbulence models from indirect observation data

“…First, as with their traditional counterparts, these data-driven models addressed only the Boussinesq assumption of the linear models as their strain-stress relations are still local, and thus they cannot address the weak equilibrium assumption described above. This is in contrast to the data-driven non-local Reynolds stress models (Han et al 2022;Zhou et al 2022), which emulate the Reynolds stress transport equations and fully non-equilibrium models. Second, the training of such models often requires full-field Reynolds stresses (referred to as direct data hereafter), which are rarely available except from high-fidelity simulations such as direct numerical simulations (DNS) and wall-resolved large eddy simulations (LES) (Yang & Griffin 2021).…”

A PDE-free, neural network-based eddy viscosity model coupled with RANS equations

“…This is crucial in eventually using VCNN to emulate Reynolds stress transport equations. Such adaptation for equivariant tensor outputs of the network has already been addressed in a follow-up work (Han et al, 2022). Another line of development stems from the limitation that the performance of the network was only evaluated in predicting the transport of passive scalar in a laminar flow.…”

A PDE-free, neural network-based eddy viscosity model coupled with RANS equations

“…This is crucial in eventually using VCNN to emulate Reynolds stress transport equations. Such adaptation for equivariant tensor outputs of the network has already been addressed in a follow-up work [26]. Another line of devel-opment stems from the limitation that the performance of the network was only evaluated in predicting the transport of passive scalar in a laminar flow.…”

“…Rather, this work serves as a proof of concept regarding the applicability of the VCNN to turbulence modelling problems. By combining present work with the VCNN architecture with equivariance [26], the framework can be further extended to predict Reynolds stress tensor anisotropy.…”