Deconvolutional artificial neural network (DANN) models are developed for subgrid-scale (SGS) stress in large eddy simulation (LES) of turbulence. The filtered velocities at different spatial points are used as input features of the DANN models to reconstruct the unfiltered velocity. The grid width of the DANN models is chosen to be smaller than the filter width in order to accurately model the effects of SGS dynamics. The DANN models can predict the SGS stress more accurately than the conventional approximate deconvolution method and velocity gradient model in the a priori study: the correlation coefficients can be made larger than 99% and the relative errors can be made less than 15% for the DANN model. In an a posteriori study, a comprehensive comparison of the DANN model, the implicit LES (ILES), the dynamic Smagorinsky model (DSM), and the dynamic mixed model (DMM) shows that the DANN model is superior to the ILES, DSM, and DMM models in the prediction of the velocity spectrum, various statistics of velocity, and the instantaneous coherent structures without increasing the considerable computational cost; the time for the DANN model to calculate the SGS stress is about 1.3 times that of the DMM model. In addition, the trained DANN models without any fine-tuning can predict the velocity statistics well for different filter widths. These results indicate that the DANN framework with the consideration of SGS spatial features is a promising approach to develop advanced SGS models in the LES of turbulence.

In this work, artificial neural network-based nonlinear algebraic models (ANN-NAMs) are developed for the subgrid-scale (SGS) stress in large eddy simulation (LES) of turbulence at the Taylor Reynolds number Re_λ ranging from 180 to 250. An ANN architecture is applied to construct the coefficients of the general NAM for the SGS anisotropy stress. It is shown that the ANN-NAMs can reconstruct the SGS stress accurately in the a priori test. Furthermore, the ANN-NAMs are analyzed by calculating the average, root mean square values, and probability density functions of dimensionless model coefficients. In an a posteriori analysis, we compared the performance of the dynamic Smagorinsky model (DSM), dynamic mixed model (DMM), and ANN-NAM. The ANN-NAM yields good agreement with a filtered direct numerical simulation dataset for the spectrum, structure functions, and other statistics of velocity. Besides, the ANN-NAM predicts the instantaneous spatial structures of SGS anisotropy stress much better than the DSM and DMM. The NAM based on the ANN is a promising approach to deepen our understanding of SGS modeling in LES of turbulence.

Long-term predictions of nonlinear dynamics of three-dimensional (3D) turbulence are very challenging for machine learning approaches. In this paper, we propose an implicit U-Net enhanced Fourier neural operator (IU-FNO) for stable and efficient predictions on the long-term large-scale dynamics of turbulence. The IU-FNO model employs implicit recurrent Fourier layers for deeper network extension and incorporates the U-net network for the accurate prediction on small-scale flow structures. The model is systematically tested in large-eddy simulations of three types of 3D turbulence, including forced homogeneous isotropic turbulence, temporally evolving turbulent mixing layer, and decaying homogeneous isotropic turbulence. The numerical simulations demonstrate that the IU-FNO model is more accurate than other FNO-based models, including vanilla FNO, implicit FNO (IFNO), and U-Net enhanced FNO (U-FNO), and dynamic Smagorinsky model (DSM) in predicting a variety of statistics, including the velocity spectrum, probability density functions of vorticity and velocity increments, and instantaneous spatial structures of flow field. Moreover, IU-FNO improves long-term stable predictions, which has not been achieved by the previous versions of FNO. Moreover, the proposed model is much faster than traditional large-eddy simulation with the DSM model and can be well generalized to the situations of higher Taylor–Reynolds numbers and unseen flow regime of decaying turbulence.

Deep neural network models have shown great potential in accelerating the simulation of fluid dynamic systems. Once trained, these models can make inferences within seconds, thus can be extremely efficient. However, it becomes more difficult for neural networks to make accurate predictions when the flow becomes more chaotic and turbulent at higher Reynolds numbers. One of the most important reasons is that existing models lack the mechanism to handle the unique characteristic of high-Reynolds-number turbulent flow; multi-scale flow structures are nonuniformly distributed and strongly nonequilibrium. In this work, we address this issue with the concept of visual attention: intuitively, we expect the attention module to capture the nonequilibrium of turbulence by automatically adjusting weights on different regions. We compare the model performance against a state-of-the-art neural network model as the baseline, the Fourier neural operator, on a two-dimensional turbulence prediction task. Numerical experiments show that the attention-enhanced neural network model outperforms existing state-of-the-art baselines, and can accurately reconstruct a variety of statistics and instantaneous spatial structures of turbulence at high Reynolds numbers. Furthermore, the attention mechanism provides 40% error reduction with 1% increase in parameters, at the same level of computational cost.

Fast and accurate predictions of turbulent flows are of great importance in the science and engineering field. In this paper, we investigate the implicit U-Net enhanced Fourier neural operator (IUFNO) in the stable prediction of long-time dynamics of three-dimensional (3D) turbulent channel flows. The trained IUFNO models are tested in the large-eddy simulations (LES) at coarse grids for three friction Reynolds numbers: ${\text{Re}}_{\tau }\approx 180$, 395, and 590. The adopted near-wall mesh grids are tangibly coarser than the general requirements for wall-resolved LES. Compared to the original Fourier neural operator (FNO), the implicit FNO (IFNO), and U-Net enhanced FNO (UFNO), the IUFNO model has a much better long-term predictive ability. The numerical experiments show that the IUFNO framework outperforms the traditional dynamic Smagorinsky model and the wall-adapted local eddy-viscosity model in the predictions of a variety of flow statistics and structures, including the mean and fluctuating velocities, the probability density functions (PDFs) and joint PDF of velocity fluctuations, the Reynolds stress profile, the kinetic energy spectrum, and the Q-criterion (vortex structures). Meanwhile, the trained IUFNO models are computationally much faster than the traditional LES models. Thus, the IUFNO model is a promising approach for the fast prediction of wall-bounded turbulent flow.

Modeling three-dimensional (3D) turbulence by neural networks is difficult because 3D turbulence is highly nonlinear with high degrees of freedom and the corresponding simulation is memory-intensive. Recently, the attention mechanism has been shown as a promising approach to boost the performance of neural networks on turbulence simulation. However, the standard self-attention mechanism uses O(n2) time and space with respect to input dimension n, and such quadratic complexity has become the main bottleneck for attention to be applied on 3D turbulence simulation. In this work, we resolve this issue with the concept of a linear attention network. The linear attention approximates the standard attention by adding two linear projections, reducing the overall self-attention complexity from O(n2) to O(n) in both time and space. The linear attention coupled Fourier neural operator (LAFNO) is developed for the simulation of 3D isotropic turbulence and free shear turbulence. Numerical simulations show that the linear attention mechanism provides 40% error reduction at the same level of computational cost, and LAFNO can accurately reconstruct a variety of statistics and instantaneous spatial structures of 3D turbulence. The linear attention method would be helpful for the improvement of neural network models of 3D nonlinear problems involving high-dimensional data in other scientific domains.