Deep structured neural networks for turbulence closure modeling

Abstract

Despite well-known limitations of Reynolds-averaged Navier–Stokes (RANS) simulations, this methodology remains the most widely used tool for predicting many turbulent flows due to computational efficiency. Machine learning is a promising approach to improve the accuracy of RANS simulations. One major area of improvement is using machine learning models to represent the complex relationship between the mean flow field gradients and the Reynolds stress tensor. In the present work, modifications to improve the stability of previous optimal eddy viscosity approaches for RANS simulations are presented and evaluated. The optimal eddy viscosity is reformulated with a non-negativity constraint, which promotes numerical stability. We demonstrate that the new formulation of the optimal eddy viscosity improves conditioning of RANS equations for a periodic hills test case. To demonstrate the suitability of this proportional/orthogonal tensor decomposition for use in a physics-informed data-driven turbulence closure, we use two neural networks (structured on this specific tensor decomposition, which is incorporated as an inductive bias into the network design) to predict the newly reformulated linear and non-linear parts of the Reynolds stress tensor. Injecting these network model predictions for the Reynolds stresses into RANS simulation improves predictions of the velocity field, even when compared to a sophisticated (state of the art) physics-based turbulence closure model. Finally, we apply shapley additive explanations values to obtain insights from the learned representation for inner workings of the neural network used to predict the optimal eddy viscosity from the input feature data.

Current and emerging deep-learning methods for the simulation of fluid dynamics

“…An additional consideration in turbulence modelling is that the inferred Reynolds-stress tensor must not threaten the numerical stability of the RANS equations. In this regard, McConkey et al [194] developed a stable model by decomposing the anisotropy tensor into a linear and a nonlinear part, each modelled by an independent MLP. The first MLP is trained to return the optimal turbulent viscosity, ν t , assuming a linear relationship between A and S (i.e.…”

“…In this regard, McConkey et al. [194] developed a stable model by decomposing the anisotropy tensor into a linear and a nonlinear part, each modelled by an independent MLP. The first MLP is trained to return the optimal turbulent viscosity, νt, assuming a linear relationship between bold-italicA and S (i.e.…”

Application of Machine Learning Algorithms to Predict Turbulent Viscosity

“…𝑇 (1) = 𝑠, 𝑇 (6) = 𝑟 2 𝑠 + 𝑠𝑟 2 − 2 3 𝐼 ⋅ 𝑇𝑟(𝑠𝑟 2 ), 𝑇 (2) = 𝑠𝑟 − 𝑟𝑠, 𝑇 (7) = 𝑟𝑠𝑟 2 − 𝑟 2 𝑠𝑟, 𝑇 (3) = 𝑠 2 − 1 3…”

Development and Validation of a Machine Learned Turbulence Model

“…McConkey et al [45] used deep neural network with 14 hidden layers and 30 neurons in each layer to train separate models for turbulent eddy viscosity and non-linear stresses, which were then combined to obtain the Reynolds stresses.…”

“…For research area (d) involving Reynolds stresses for RANS, a few studies have directly inferred the Reynolds stresses [25,[44][45], whereas the largest volume of work has been devoted to augmenting the existing RANS model. [25] used a tensor basis random forest model to train for Reynolds stresses using DNS datasets for the periodic hill and converging-diverging nozzle.…”