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Convolutional-network models to predict wall-bounded turbulence from wall quantities
L. Guastoni, A. Güemes, A. Ianiro, S. Discetti, P. Schlatter, H. Azizpour, R. Vinuesa
TL;DR
The paper addresses non-intrusive prediction of turbulent flow fields from wall measurements, motivated by the difficulty of monitoring instantaneous flow states for closed-loop control. It compares two convolutional neural-network models across wall-normal locations and Reynolds numbers, including transfer learning. Both nonlinear models outperform EPOD in most tested cases, with FCN strongest near the wall, FCN-POD farther away, and transfer learning retaining reference-like performance with reduced training data.
Problem
Monitoring instantaneous turbulent flow states for closed-loop control is difficult, especially at high Reynolds numbers, motivating prediction from time-resolved wall measurements.
Method
Two convolutional models predict velocity fluctuations at wall-normal locations from wall quantities, using direct FCN prediction or POD-based reconstruction, and are compared across DNS datasets and transfer-learning settings.
Results
Both models outperform EPOD in almost all tested locations and features; FCN is more accurate nearer the wall, while FCN-POD is more accurate farther away.
Takeaways & Limitations
Transfer learning allows FCN training with 50% and 25% of the original dataset while achieving performance similar to the reference model up to y+ = 50.
Takeaways & Limitations
The models assume local wall information is sufficient to predict flow farther from the wall, and their architectures and training remain improvable.
Abstract
from arXiv · showhide
Two models based on convolutional neural networks are trained to predict the two-dimensional velocity-fluctuation fields at different wall-normal locations in a turbulent open channel flow, using the wall-shear-stress components and the wall pressure as inputs. The first model is a fully-convolutional neural network (FCN) which directly predicts the fluctuations, while the second one reconstructs the flow fields using a linear combination of orthonormal basis functions, obtained through proper orthogonal decomposition (POD), hence named FCN-POD. Both models are trained using data from two direct numerical simulations (DNS) at friction Reynolds numbers $Re_τ = 180$ and $550$. Thanks to their ability to predict the nonlinear interactions in the flow, both models show a better prediction performance than the extended proper orthogonal decomposition (EPOD), which establishes a linear relation between input and output fields. The performance of the various models is compared based on predictions of the instantaneous fluctuation fields, turbulence statistics and power-spectral densities. The FCN exhibits the best predictions closer to the wall, whereas the FCN-POD model provides better predictions at larger wall-normal distances. We also assessed the feasibility of performing transfer learning for the FCN model, using the weights from $Re_τ=180$ to initialize those of the $Re_τ=550$ case. Our results indicate that it is possible to obtain a performance similar to that of the reference model up to $y^{+}=50$, with $50\%$ and $25\%$ of the original training data. These non-intrusive sensing models will play an important role in applications related to closed-loop control of wall-bounded turbulence.
1. Introduction
Deep neural networks are being applied to fluid-mechanics and turbulence problems, including nonlinear prediction of flow fields from wall measurements. This work compares convolutional approaches with linear methods and examines their use for non-intrusive sensing and transfer across Reynolds numbers.
- Machine learning in fluid mechanics: DNNs have been applied to turbulence modelling, flow-pattern extraction, inflow generation, and temporal prediction of turbulent dynamical systems.Reported applications include RANS and LES improvement, low-order near-wall turbulence prediction, and dimensionality reduction.
- Motivation for non-intrusive sensing: Closed-loop turbulence control requires monitoring the instantaneous flow state, but direct measurements are especially challenging at high Reynolds numbers.Near-wall structures become progressively smaller as Reynolds number increases.
- Motivation for non-intrusive sensing: Non-intrusive sensing instead uses time-resolved wall quantities, such as wall-shear stress or pressure, to correlate wall information with the flow away from the wall.
- Linear baselines: Earlier flow-field prediction relied mainly on linear methods, including LSE and POD-based extensions, with LSE reconstructing only large wall-attached eddies in the outer logarithmic region.Spectral and POD-based extensions were developed to improve linear predictions, including performance with noisy measurements.
- Study objectives: The study compares FCN and FCN-POD nonlinear predictors with EPOD and evaluates whether FCN knowledge transfers between different friction Reynolds numbers.Both models characterize the relation between wall features and flow fields on wall-parallel planes.
2. Methodology
The study uses DNS databases of turbulent open-channel flow to train convolutional models that infer velocity fluctuations from wall quantities. It compares direct FCN prediction with POD-based reconstruction, using spatially localized convolutions and reduced-order representations.
- Datasets: DNS databases represent turbulent open-channel flow at Re_τ = 180 and 550 with periodic streamwise/spanwise boundaries and a no-slip lower wall.A symmetry condition is imposed at the upper boundary, excluding interaction of large scales with both walls.
- Datasets: The datasets contain 19,920 fields with a 4:1 training-to-validation split, while independent test datasets contain more than 3,000 fields for each Reynolds number.Test simulations use a different random seed to ensure spatial and temporal decorrelation from training data.
- Model inputs and FCN: The models use two-dimensional streamwise and spanwise wall-shear-stress components plus wall pressure to predict velocity fluctuations at different wall-normal locations.The inputs contain coherent features, motivating convolutional processing of the wall-parallel fields.
- Model inputs and FCN: The FCN is trained with mean-squared error between predicted and DNS velocity fluctuations after consistent scaling and is evaluated after restoring the outputs' original magnitude.Convolutional kernels reduce trainable parameters relative to fully connected networks because their dimensions are smaller than the input fields.
- POD-based predictions: The POD formulation is applied to smaller subdomains because the larger domain spreads energy across more wave numbers and modes, whereas subdomain decomposition concentrates energy in leading modes.The method is intended to filter noise associated with small, uncorrelated scales through POD's energy-optimal representation.
- POD-based predictions: A reduced-order representation is also used to address ill-conditioning of the wall-quantity correlation matrix caused by correlations between successive snapshots.The text introduces decompositions of wall and flow quantities to construct this reduced representation.
3. Results
The nonlinear FCN and FCN-POD models predict instantaneous fields and turbulence statistics more effectively than EPOD, with FCN strongest near the wall and FCN-POD increasingly competitive farther away. Spectral results explain this distance-dependent behavior through the models’ treatment of energetic scales and receptive-field limitations.
- Instantaneous predictions: At y+ = 15, all methods predict the instantaneous fields accurately, while EPOD increasingly attenuates fluctuations farther from the wall.The reduced footprint of large wall-attached scales away from the wall limits EPOD’s linear reconstruction.
- Instantaneous predictions: At y+ = 50, FCN-POD produces smoother fields and FCN slightly noisier fields than the DNS reference.Both convolutional models begin to show deviations from the reference at this wall-normal location.
- Instantaneous predictions: At Reτ = 550, FCN maintains performance similar to Reτ = 180 across all wall-normal locations despite using the same architecture and parameter count.The FCN-POD architecture was modified at the higher Reynolds number to reconstruct approximately the same amount of energy.
- Power-spectral-density predictions: At y+ = 100, POD-based methods outperform FCN because they predict temporal dynamics without separately learning the spatial energy-scale distribution.The FCN receptive field is sufficient closer to the wall but too small to reproduce the larger scales present farther away.
- Power-spectral-density predictions: At Reτ = 550, FCN outperforms POD-based approaches for y+ = 15–50, while FCN-POD matches FCN farther from the wall.FCN-based methods lose a wider range of small scales with distance, whereas EPOD has greater difficulty reproducing large scales.
4. Transfer learning
Transfer learning initializes the Reτ = 550 FCN with weights from Reτ = 180 to reduce data and training demands. It improves performance with equal data and remains comparable to the reference model up to y+ = 50 using 25% or 50% of the original dataset, while y+ = 100 requires more samples.
- Method: Weights from the Reτ = 180 FCN initialize the Reτ = 550 model, with a lower learning rate used to prevent rapid divergence from the transferred weights.The approach targets the data and computational costs of training neural networks, which increase with Reτ.
- Full-dataset initialization: The initialized model outperforms random initialization with the full dataset, especially near the wall, while both models are approximately equivalent at y+ = 100 after 150,000 updates.This comparison uses validation and test loss.
- Full-dataset initialization: Transfer learning is feasible and advantageous when models trained at different Reynolds numbers use the same amount of data.The authors report this conclusion after comparing initialized and randomly initialized models.
- Reduced-data training: Up to y+ = 50, initialized networks achieve performance very similar to the reference model with 25% or 50% of the original training data.The reduced-data models provide comparable performance with the same number of updates and substantial data savings.
- Reduced-data training: At y+ = 100, 50% of the training data stops improving after 100,000 updates, whereas 25% causes overfitting, making sufficient samples necessary for convergence.The validation loss can overestimate test error with 25% of the data and underestimate it with 50%.
- Statistical predictions: Initialized models achieve comparable RMS-statistics accuracy, although small error variations reflect the stochastic optimization process.The comparison is reported for models evaluated after 250,000 updates at Reτ = 550.
5. Conclusions
The FCN and FCN-POD predict wall-normal velocity fluctuations accurately across Reynolds numbers, with FCN stronger near the wall and FCN-POD farther away. Transfer learning enables similar FCN performance up to y+ = 50 using reduced training data, while architectural and training limitations remain.
- The FCN and FCN-POD predict velocity fluctuations in good agreement with DNS up to y+ = 50 and outperform EPOD in most tested features.Evaluation covers instantaneous predictions, RMS turbulence statistics, and spectral energy content; the advantage is attributed to nonlinear scale-interaction prediction.
- Reτ = 180 and 550 can be handled with minimal architectural modifications while providing satisfactory results on both datasets.
- FCN is more accurate near the wall, while FCN-POD becomes more accurate farther away because POD modes encode additional spatial information.FCN performs better up to y+ = 30 at Reτ = 180 and y+ = 50 at Reτ = 550.
- Both models remain improvable through network-architecture and training changes, including FCN filtering and FCN-POD hyperparameter exploration.The FCN-POD also has scaling and smoothness issues identified for potential improvement.
- Transfer learning achieves performance similar to the reference FCN up to y+ = 50 with 50% or 25% of the original training dataset.Weights trained at the lower Reynolds number initialize training at the higher Reynolds number, reducing data and training-time requirements.
- Once trained, the networks are computationally cheap to evaluate and may be pruned for deployment on low-powered hardware or real-time operation.
Appendix A.
Appendix A provides wall-normal and spanwise fluctuation fields corresponding to fields shown for Reτ = 180 and 550.
- The appendix contains wall-normal and spanwise fluctuations corresponding to fields in Figures 5 and 7 for Reτ = 180 and 550.