A Priori Assessment of Rotational Invariance in Multiscale CNN-Based Subgrid-Scale Model for Wall-Bounded Turbulent Flows

TL;DR

The paper tackles the challenge of achieving material objectivity in data-driven subgrid-scale closures for LES of wall-bounded turbulence. Starting from a multiscale CNN SGS model (MSC), it introduces two invariance-enhancing variants: MSC2, which excludes bias and batch normalization, and MSC3, which adds a spatial transformer network to learn orientation-consistent representations. Through a priori tests with rotated inputs, MSC2 and MSC3 demonstrate improved rotational invariance and accuracy in predicting $\tau_{ij}$ and key turbulence statistics, with MSC3 offering the best overall performance. The work highlights that enforcing invariance at both input and architectural levels is crucial for physically consistent data-driven SGS models, suggesting promising directions for applying such models in rotational flow configurations and turbomachinery applications. Quantitative results show substantial gains in rotational robustness, albeit with remaining challenges near walls and under certain rotated components, underscoring the need for further LES-scale validation and broader training data.

Abstract

This study proposes a rotationally invariant data-driven subgrid-scale (SGS) model for large-eddy simulation (LES) of wall-bounded turbulent flows. Building upon the multiscale convolutional neural network subgrid-scale model, which outputs SGS stress tensors ($τ_{ij}$) as the baseline, the deep neural network (DNN) architecture is modified to satisfy the principle of material objectivity by removing the bias terms and batch normalization layers while incorporating a spatial transformer network (STN) algorithm. The model was trained on a turbulent channel flow at $\mathrm{Re}_τ= 180$ and evaluated using both non-rotated and rotated inputs. The results show that the model performs well in predicting $τ_{ij}$ and key turbulence statistics, including dissipation, backscatter, and SGS transport. These quantities reflect the ability of the model to reproduce the energy transfer between the resolved scale and SGS. Moreover, it effectively generalizes to unseen rotated inputs, accurately predicting $τ_{ij}$ despite the input configurations not being encountered during the training. These findings highlight that modifying the DNN architecture and integrating the STN-based algorithm improves the ability to recognize and correctly respond to rotated inputs. The proposed data-driven SGS model addresses the key limitations of common data-driven SGS approaches, particularly their sensitivity to rotated input conditions. It also marks an important advancement in data-driven SGS modeling for LES, particularly in flow configurations where rotational effects are non-negligible.

Figures (14)

• Figure 1: Schematic of data-driven SGS model framework.
• Figure 2: Wall-normal distribution of turbulence statistics of (a) mean velocity and (b) root-mean-square velocity. Here, $\langle \cdot \rangle$ denotes spatial averaging over the $x$--$z$ plane and temporal averaging; these notations are used hereinafter in this paper.
• Figure 3: Schematic illustration of the DNN algorithm of the MSC model.
• Figure 4: Schematic of STN algorithm based on Ref.Jaderberg
• Figure 5: Schematic of the MSC3 model.