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

Abstract

Wall-bounded turbulent flows are chaotic and multiscale, rendering real-time prediction at high Reynolds numbers computationally prohibitive in applications such as wind farms. Classical data assimilation methods are based on repeated solution of the governing equations and thus inherit this cost. Generative models instead learn the probability distribution of flow states, enabling scalable probabilistic reconstruction. Using plane Couette flow as a canonical configuration, we develop a generative framework that couples a $β$-variational autoencoder with a transformer-based diffusion model 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 at $Re_h=1300$, where the corresponding DNS resolution in this region requires $O(10^6)$ spatial degrees of freedom. The diffusion model reproduces two-point correlations, energy spectra, and single-point statistics up to fourth order using $O(10)$ latent spatial degrees of freedom, yielding a compression ratio of $O(10^5)$ - one to two orders of magnitude above prior reports. Two assimilation scenarios demonstrate that conditional diffusion models with the proposed sampling strategy can enforce complex statistical constraints. However, enforcing these constraints while preserving physical fidelity and sample diversity introduces an inherent trade-off. Excessive conditioning can distort the learned diffusion prior, paralleling limitations of classical ensemble-based data assimilation. These results highlight both the promise of diffusion models as probabilistic surrogates for turbulent wall-bounded flows and the challenges of conditioning such models, establishing a foundation for future real-time reconstruction from operational data.

Figures (19)

• Figure 1: Schematic of the proposed three-step data assimilation framework using stochastic generative models. $(a)$ Our overarching goal is to enable real-world data assimilation of full wind farm flow fields under operating conditions. First, offline high-fidelity simulations of representative operating states are performed to construct a data distribution. Second, a stochastic generative model is trained to learn this distribution, producing statistically consistent flow-field samples from a compact sample space. Third, during deployment and data assimilation, field observations e.g., from unmanned aerial vehicles (UAVs) and light detection and ranging (LiDAR) systems, are collected under conditions not seen during training, and assimilated through conditional generation. These observations guide the generative model toward the true operating state, enabling predictions beyond the original training distribution. $(b)$ In this work, we establish the foundation for this vision by indirectly enforcing turbulent statistics. We first construct a data distribution from DNS of four-dimensional turbulent plane Couette flow. A latent diffusion transformer model is then trained to reproduce the associated turbulent statistics. Finally, the model performs data assimilation for the same flow configuration using diffusion posterior sampling with two observation types: scattered observations and a localized rectangular data block.
• Figure 2: Schematic of the latent conditional diffusion framework. $(a,f)$ The input and output of the framework is a four-dimensional spatiotemporal flow field in physical space. ($b$) The encoder of a $\beta$-VAE maps a temporal sequence of DNS snapshots $\hbox{\boldmath $\phi$}$ of velocities $u, v, w$ and pressure $p$ into a low-dimensional latent space $\mathsf{z}$$(c)$. ($d$) A diffusion transformer (DiT) learns the latent dynamics and generates new latent trajectories, where $\mathsf{z}_0$ denotes the clean latent sample and $\mathsf{z}_T$ the fully diffused one. ($e$) The decoder then reconstructs four-dimensional spatiotemporal flow fields $\hbox{\boldmath $\tilde{\phi}$}$ from the generated latents. Training of the $\beta$-VAE and the DiT is performed in two separate stages.
• Figure 3: Schematic of a VAE architecture. The encoder takes a DNS flow field sample and predicts the mean $\hbox{\boldmath $\mu$}$ and variance $\hbox{\boldmath $\varSigma$}$ of the variational distribution. In the bottleneck, a sample is drawn from a normal distribution and the latent representation is constructed as a sample from the variational distribution. The decoder reconstructs the input data. The training is conducted by minimizing the loss function in equation \ref{['eq:vae_loss']}. The detailed $\beta$-VAE architecture is given in appendix \ref{['subsec:app_VAE']}.
• Figure 4: Schematic of the sampling strategy used to impose statistical quantities. The time series is divided into $M$ segments $\varPsi_1, \varPsi_2, \cdots, \varPsi_M$. Each generated sample sees one of the segments $M$ segments.
• Figure 5: DNS of turbulent plane Couette flow is used to construct the spatiotemporal dataset used for training. $(a)$ First, The computational domain of the DNS is divided at the channel centreline. We use the top half with 16 subdomains (grey) and generate one subdomain (one snapshot) with 1/32 size of the computational domain of the DNS (blue). Depicted is the plane Couette flow domain with channel half-height $h$; both walls move in streamwise direction $x$ with wall velocity $U_\text{w}$. $(b)$ Second, for the training of the Diffusion Transformer (DiT) each trajectory is segmented into clips. Illustrated is the segmentation of the 2000-snapshot trajectory of the DNS for one subdomain ${\hbox{\boldmath $\phi$}}(t_i)$ with $t_i = i \times 10\Updelta t$. Each snapshot is a three-dimensional image (grey box). Every ${\hbox{\boldmath $\varPhi$}}_i$ is a clip including 10 snapshots with timestep $\Updelta t_{\mathrm{gen}}$ starting at $t_i$. A stride of $30\Updelta t$ produces $N_\text{c}=637$ clips for one subdomain and $10{,}192$ clips in total for all 16 subdomains in the top half of the simulation domain.