Turbulence generation and data assimilation in wall-bounded flows with a latent diffusion model

Abstract

Wall-bounded turbulent flows are chaotic and multiscale, rendering fast prediction at high Reynolds numbers computationally prohibitive in applications such as wind farms. Classical data assimilation is based on repeated solutions of the governing equations and thus inherits this cost. Generative models learn the probability distribution of flow states, enabling scalable probabilistic reconstruction. Our generative framework couples a variational autoencoder with a diffusion transformer to generate four-dimensional spatiotemporal samples. Bayesian conditioning enables data assimilation without retraining and allows statistical constraints to be imposed through sampling. The framework is applied to a subdomain of turbulent plane Couette flow, where the corresponding DNS in this generation region requires $O(10^6)$ spatial degrees of freedom. Using $O(10)$ latent spatial degrees of freedom, the model achieves a compression ratio of $O(10^5)$, which is one to two orders of magnitude above prior reports. It reproduces single-point statistics up to fourth order and the energy spectra, as well as the intermittency and phase-sensitive structure captured by velocity-increment and $Q$--$R$ statistics. Two assimilation scenarios demonstrate that, when observations are statistically consistent with the prior, conditional diffusion models with the proposed sampling strategy preserve complex turbulent statistics in the posterior. However, enforcing these constraints while preserving physical fidelity and sample diversity introduces an inherent trade-off. Excessive conditioning can distort the learned prior, paralleling limitations of classical ensemble-based data assimilation, where this can likewise degrade the prior covariance. These results highlight both the promise of diffusion models as probabilistic surrogates and the challenges of conditioning them.