The relationship between surface features and sub-surface turbulence: simulations and experiments

TL;DR

This work investigates the coupling between free-surface deformations and sub-surface turbulence by combining DNS of deformable free-surface flows and new laboratory experiments with simultaneous surface profilometry and velocity measurements. The authors detect surface features as dimples and scars using a wavelet-based method and show that the relative surface-area covered, $S(t)$, correlates strongly with the mean-square horizontal divergence, $\overline{\beta^2}$, both at the surface and at depths extending beyond the blockage layer when scaled by the integral scale $L_\infty$. DNS and experimental results align under integral-scale normalization, with correlations remaining significant (up to $\sim0.7$) at depths around $z\approx -1.7\lambda$ and down to about $2L_\infty$, driven by large intermittent upwelling events. The findings support using surface imprints as proxies for sub-surface turbulence and cross-surface transport, enabling remote sensing approaches for gas and heat flux estimation in natural flows.

Abstract

Turbulent flows close beneath a free water surface are of central importance in the Earth system, yet remain relatively little studied with many open questions. We study the degree of correlation between surface motion and sub-surface velocities, comparing data from direct numerical simulation to a new experimental setup where the free-surface motion and sub-surface velocity field are measured simultaneously in a jet-stirred tank. A previous study showed a close correlation between the time series of mean-square surface divergence and the surface features (Babiker \& al., 2023 \emph{J.~Fluid Mech.} {\bf 964}, R2); we find that the normalised cross-correlation between mean-square horizontal divergence at vertical position $z$ ($β(z)=\partial_xu+\partial_yv$, with $z$ the vertical coordinate and $u,v$ the velocity in horizontal directions $x,y$, respectively) and relative surface area covered by persistent surface features (`dimples' and `scars') decreases slowly with depth but remains significant even two integral scales beneath the surface. Although turbulent Reynolds numbers differ by an order of magnitude between simulation and experiment, the same scaling and behaviour is found for all quantities we consider when scaled by the integral scale. Our results demonstrate that although surface-to-bulk correlations may appear weak when covariance between surface and velocity quantities is made point-to-point for the full surface, conditional quantities guided by visible surface features can yield significant information about the largest, most energetic flow events even beyond the blockage layer.

Figures (11)

• Figure 1: (a) Schematic of the experiment, showing the projector, profilometry camera above the tank, projected fringe pattern on the surface, PIV camera, and horizontal laser sheet. (b) Picture of the turbulence tank with the 32 submersible bilge pumps.
• Figure 2: Illustration of the computational set-up for the isotropic turbulence interacting with a deformable free surface, including details on the structure of the free region. Regions and surface deformations are not to scale. Original figure in aarnes2025. Reprinted with permission.
• Figure 3: Root-mean-square of turbulent (solid lines) and mean (dashed line) velocity components in the vertical (blue) and horizontal (red) directions, as a function of depth for the cases A (a) and B (b). Mean velocities are calculated along the vertical centreline of the field of view.
• Figure 4: Energy spectrum for experimental cases A and B calculated along a line of the vertical PIV fields at $z=z_\text{ref}$. The Kolmogorov scaling $E_{xx}\sim k^{-5/3}$ is shown for reference.
• Figure 5: Example of surface elevation from the profilometry measurements (a,c) and the identified structures on the surface (b,d) of two time steps from Case A.