Reduced-order modeling of large-scale turbulence using Koopman $β$-variational autoencoders

TL;DR

The paper addresses the challenge of building accurate reduced-order models for scale-resolving turbulence by mitigating the influence of chaotic small-scale structures. It introduces a Koopman $β$-VAE that enforces approximately linear latent dynamics via a Koopman loss, yielding denoised latent representations that emphasize large-scale coherent flow features, and couples this with an LSTM ensemble for robust time-series forecasting of the latent variables. The approach preserves bulk flow characteristics and spectral content while reducing turbulent kinetic energy due to filtering of small scales, achieving stable long-horizon ROM predictions across multiple yaw angles of a Windsor-body LES. Practically, this framework offers a scalable, unsupervised method for real-time surrogate modeling in design optimization, with future potential extensions to unstructured meshes and fully 3D flows.

Abstract

Reduced-order models (ROMs) are very popular for surrogate modeling of full-order computational fluid dynamics (CFD) simulations, allowing for real-time approximation of complex flow phenomena. However, their application to CFD models including large eddy simulation (LES) and direct numerical simulaton (DNS) is limited due to the highly chaotic and multi-scale nature of resolved turbulent flow. Due to the large amounts of noise present in small-scale turbulent structures, error accumulation becomes a major issue, making long-term prediction of unsteady flow infeasible. While linear subspace methods like dynamic mode decomposition (DMD) can be used to pre-process turbulent flow data to remove small-scale structures, this often requires a very large number of modes and a non-trivial mode selection process. In this work, a ROM framework using Koopman $β$-variational autoencoders ($β$-VAEs) is introduced for reduced-order modeling of large-scale turbulence. The Koopman operator captures the variation of a non-linear dynamical system through a linear representation of state observables. By constraining the latent space of a $β$-VAE to grow linearly using a Koopman-inspired loss function, small-scale turbulent structures are filtered out in reconstructions of input data and latent variables are denoised in an unsupervised manner so that they can be sufficiently modeled over time. Combined with a long short-term memory (LSTM) ensemble for time series prediction of latent variables, the model is tested on LES flow past a Windsor body at multiple yaw angles, showing that the Koopman $β$-VAE can effectively denoise latent variables and remove small-scale structures from reconstructions while acting globally over multiple cases.

Figures (54)

• Figure 1: Schematic of the Koopman $\beta$-VAE, where small-scale structures are filtered out from turbulent input data through the use of the Koopman loss function.
• Figure 2: An example of bootstrapping, where random subsets of the original dataset are chosen through sampling with replacement.
• Figure 3: Geometry of the Windsor body (gray) and the plane in the wake on which flow data are interpolated onto (black).
• Figure 4: Instantaneous $Q$ criterion isocontours at $Q=100$ and $\delta=12.5$ portraying the streamwise velocity magnitude.
• Figure 5: Koopman $\beta$-VAE hyperparameters.