Mapping surface height dynamics to subsurface flow physics in free-surface turbulent flow using a shallow recurrent decoder

TL;DR

This work addresses the challenge of inferring near-surface turbulence beneath a free surface from sparse surface measurements. It introduces SHRED, a SHallow REcurrent Decoder that uses a two-part architecture—a temporal LSTM encoder and a shallow spatial decoder—to map delay-embedded surface height data to full subsurface velocity fields, trained in a compressed, low-rank space. The approach is demonstrated on both DNS and PIV-based experimental data, achieving faithful reconstructions of large- to intermediate-scale structures up to about $2L_ ext{\infty}$ depth using only three surface sensors, with quantified performance across multiple metrics. SHRED offers a promising path toward nonintrusive remote sensing of subsurface turbulence and related gas/heat exchange processes, with potential extensions to generalization, derived quantities, and integration with other data-driven dynamics tools.

Abstract

Near-surface turbulent flows beneath a free surface are reconstructed from sparse measurements of the surface height variation, by a novel neural network algorithm known as the SHallow REcurrent Decoder (SHRED). The reconstruction of turbulent flow fields from limited, partial, or indirect measurements remains a grand challenge in science and engineering. The central goal in such applications is to leverage easy-to-measure proxy variables in order to estimate quantities which have not been, and perhaps cannot in practice be, measured. Specifically, in the application considered here, the aim is to use a sparse number of surface height point measurements of a flow field, or drone video footage of surface features, in order to infer the turbulent flow field beneath the surface. SHRED is a deep learning architecture that learns a delay-coordinate embedding from a few surface height (point) sensors and maps it, via a shallow decoder trained in a compressed basis, to full subsurface fields, enabling fast, robust training from minimal data. We demonstrate the SHRED sensing architecture on both fully resolved DNS data and PIV laboratory data from a turbulent water tank. SHRED is capable of robustly mapping surface height fluctuations to full-state flow fields up to about two integral length scales deep, with as few as three surface measurements.

Figures (13)

• Figure 1: Illustration of the SHRED architecture, training and deployment phase. Left: Time series from three sensors combined with the time dynamics of the DNS or PIV data ($V$-matrix from SVD of slice planes) to form matrix $W$. Middle: Matrix $W$ processed by long-short time memory and shallow decoder network. Right: The model is deployed by taking new measurements of the surface. After processing we reconstruct the full surface and velocity slice planes.
• Figure 2: (a): Schematics for the simulation domain, with undulating grid following the free surface, depth-dependent forcing function $f(z)$ ($=0$ in the free region), and periodic boundaries along the horizontal directions. (b): Experimental setup, where turbulence is generated by randomly actuated jets and data is captured by profilometry (projector and camera 1) and PIV (sheet-making optics and camera 2; laser and mirrors not in the frame).
• Figure 3: Top 2 rows: Eight representative spatial SVD modes ($u_i$) and temporal modes ($v_i$) of the decomposed horizontal velocity component $u_y$ at the surface for the S2 case, chosen for illustrative purposes, for a snapshot at an arbitrary point in time. Middle two rows: Evolution of the same modes in time (represented by frame number). Lower left: singular values of the SVD modes. Lower right: Normalized turbulent power-density spectrum (PSD) for velocity fields compressed by retaining only modes from $1$ to rank $r$.
• Figure 4: Same as Fig. \ref{['fig:SVD_DNS']}, but for the high-turbulence experimental E2 case. SVD modes are found from the turbulent streamwise velocity component 1 cm ($0.14 L_{\infty}$)below the free surface. Note that the total number of SVD modes is 900 in the experimental case.
• Figure 5: Example of typical MSE loss profile for the validation dataset from case S2 (left) and E2 (right).