Learning Temporally Consistent Turbulence Between Sparse Snapshots via Diffusion Models

TL;DR

The paper tackles the challenge of generating temporally plausible turbulence between sparse, decorrelated states to enable richer statistics, inflow generation, and data augmentation. It proposes a conditional video diffusion (DDPM) framework that conditions on a subset of frames to infill the trajectory between decorrelated snapshots, trained on DNS data for both a statistically stationary Kolmogorov flow and a non-stationary Kelvin-Helmholtz Instability. Key findings show that the method reproduces PDFs, POD modes, turbulent kinetic energy spectra, and, for KHI, evolving statistics across billow collapse, with some artefacts in rapidly changing regimes. This approach offers a practical data-driven surrogate that can reduce DNS storage needs and enhance analysis of time-evolving turbulence from sparse observations.

Abstract

We investigate the statistical accuracy of temporally interpolated spatiotemporal flow sequences between sparse, decorrelated snapshots of turbulent flow fields using conditional Denoising Diffusion Probabilistic Models (DDPMs). The developed method is presented as a proof-of-concept generative surrogate for reconstructing coherent turbulent dynamics between sparse snapshots, demonstrated on a 2D Kolmogorov Flow, and a 3D Kelvin-Helmholtz Instability (KHI). We analyse the generated flow sequences through the lens of statistical turbulence, examining the time-averaged turbulent kinetic energy spectra over generated sequences, and temporal decay of turbulent structures. For the non-stationary Kelvin-Helmholtz Instability, we assess the ability of the proposed method to capture evolving flow statistics across the most strongly time-varying flow regime. We additionally examine instantaneous fields and physically motivated metrics at key stages of the KHI flow evolution.

Figures (10)

• Figure 1: Time evolution of a Kelvin-Helmholtz instability, generated via DNS. Spanwise component of velocity omitted. $\phi^*$ represents a dimensionless scalar, (i.e. $\theta^*)$ in smythLengthScalesTurbulence2000. $U^*, V^*$ are the $x, y$ dimensionless velocity components, respectively. $t^*$ is dimensionless time, given by $t^* = t\frac{U_0}{L_x}$.
• Figure 2: Simulation domain and intial flow conditions. Gravity vector depicted as $g$. Initial velocity profile and temperature profile given as a function of the velocity and length scales in the shear layer. $L^*_x = 6.8$, $L^*_y=13.65$, $L^*_z =1.6$, are the dimensionless lengths of the domain bounding box as a ratio to $h_0$. The top and bottom of the domain, $(y_{max}, y_{min})$, have zero gradient boundary conditions for $U, V, \phi$. The left, right, front, and back boundaries at $(x_{min}, x_{max}),\ (z_{min}, z_{max})$ are periodic.
• Figure 3: A comparison of two choices of $\{\lambda_{min}, \lambda_{max}\}$ in terms of their impact on a sample from the Kolmogorov flow dataset.
• Figure 4: A comparison of the PDFs of vorticity for: the reference DNS sequences, and DDPM-generated outputs with different numbers of conditioning frames. Upper and lower limits are indicated for each PDF in dashed lines.
• Figure 5: A comparison of POD mode-based reconstructions of a sample DNS sequence from the test set (top), the corresponding DDPM-generated sequence (middle), and the error field between them (bottom) shown for $u_x$ of the mid-sequence snapshot. From left to right, the POD modes used for reconstruction are: $1, 5, 10, 20, 50$, with the final plot showing the actual field.