On Deep-Learning-Based Closures for Algebraic Surrogate Models of Turbulent Flows

TL;DR

This work tackles energy loss in POD-based reduced-order models for turbulent flows by introducing a deep-learning closure that learns the spatial PDF of the truncation error, $p(\mathcal{E}|\mathcal{X_P})$, using a transformer with easy-attention. A common POD basis is built by clustering coherent modes across training yaw angles via PSD analysis and Hotelling’s $T^2$, enabling robust closure for unseen flow conditions in the Windsor wake. The closure significantly improves energy accuracy and statistical fidelity, reducing the mean energy error from $\sim$37% to $\sim$12% and lowering the velocity PDF divergence $D_{KL}$ from about $0.2$ to below $0.026$, with larger attention sizes providing better coverage but risking slight overfitting. Collectively, the method provides a practical, geometry- and condition-generalizable approach to enhance POD-based surrogate models for complex turbulent flows, with direct implications for aerospace and automotive aero-surrogates.

Abstract

A deep-learning-based closure model to address energy loss in low-dimensional surrogate models based on proper-orthogonal-decomposition (POD) modes is introduced. Using a transformer-encoder block with easy-attention mechanism, the model predicts the spatial probability density function of fluctuations not captured by the truncated POD modes. The methodology is demonstrated on the wake of the Windsor body at yaw angles of [2.5,5,7.5,10,12.5], with 7.5 as a test case. Key coherent modes are identified by clustering them based on dominant frequency dynamics using Hotelling T2 on the spectral properties of temporal coefficients. These coherent modes account for nearly 60% of the total energy while comprising less than 10% of all modes. A common POD basis is created by concatenating coherent modes from training angles and orthonormalizing the set, reducing the basis vectors from 142 to 90 without losing information. Transformers with different size on the attention layer, (64, 128 and 256), are trained to model the missing fluctuations. Larger attention sizes always improve predictions for the training set, but the transformer with an attention layer of size 256 overshoots the fluctuations predictions in the test set because they have lower intensity than in the training cases. Adding the predicted fluctuations closes the energy gap between the reconstruction and the original flow field, improving predictions for energy, root-mean-square velocity fluctuations, and instantaneous flow fields. The deepest architecture reduces mean energy error from 37% to 12% and decreases the Kullback--Leibler divergence of velocity distributions from KL=0.2 to below KL=0.026.

Figures (20)

• Figure 1: (Red) geometry of the Windsor body and (blue) working plane where the data is interpolated to develop the model. The plane is perpendicular to the vertical axis and is located at $z/L=0.186$. The arrow indicates the flow direction.
• Figure 2: Streamline comparison between $\delta=2.5^{\circ}$ (left) and $\delta=12.5^{\circ}$ (right) at the plane $z/L=0.186$. The green, red and blue dots represent the core of the leeward side vortex, the core of the windward side vortex and the saddle point, respectively.
• Figure 3: Mean streamwise velocity at $z/L=0.186$ for $\delta=2.5^{\circ}$ (left) and $\delta=12.5^{\circ}$ (right).
• Figure 4: Root mean square of the streamwise velocity fluctuations at $z/L=0.186$ for $\delta=2.5^{\circ}$ (left) and $\delta=12.5^{\circ}$ (right).
• Figure 5: Mean streamwise velocity for all simulated angles at $y/L=0$ (a) and root-mean square value of the velocity fluctuations at $x/L=1.3$ (b).