Turbulence-induced anti-Stokes flow: experiments and theory

Abstract

We report experimental evidence of an Eulerian-mean flow, $\overline{u}(z)$, created by the interaction of surface waves and tailored ambient sub-surface turbulence, which partly cancels the Stokes drift, $u_s(z)$, and present supporting theory. Water-side turbulent velocity fields and Eulerian-mean flows were measured with particle image velocimetry before vs after the passage of a wave group, and with vs without the presence of regular waves. We compare different wavelengths, steepnesses and turbulent intensities. In all cases, a significant change in the Eulerian-mean current is observed, strongly focused near the surface, where it opposes the Stokes drift. The observations support the picture that when waves encounter ambient sub-surface turbulence, the flow undergoes a transition during which Eulerian-mean momentum is redistributed vertically (without changing the depth-integrated mass transport) until a new equilibrium state is reached, wherein the near-surface ratio between $|\mathrm{d}\overline{u}/\mathrm{d}z|$ and $|\mathrm{d}u_s/\mathrm{d} z|$ approximately equals the ratio between the streamwise and vertical Reynolds normal stresses. This accords with a simple statistical theory derived here and holds regardless of the absolute turbulence level, whereas stronger turbulence means faster growth of the Eulerian-mean current. We present a model based on Rapid Distortion Theory which describes the generation of the Eulerian-mean flow as a consequence of the action of the Stokes drift on the background turbulence. Predictions are in qualitative, and reasonable quantitative, agreement with experiments on wave groups, where equilibrium has not yet been reached. Our results could have substantial consequences for predicting the transport of water-borne material in the oceans.

Figures (12)

• Figure 1: Experimental set-up: (a) side view of water channel with flow from left to right and field of view (FOV) indicated with a green rectangle, (b) top view of measurement region for the stereo particle image velocimetry (PIV) set-up in experiment 1, including positions of wave probes (WPs) and laser-induced fluoresence (LIF) camera for surface detection; (c) longitudinal view of the planar PIV set-up in experiments 2 and 3. Experiment 2 employed three stacked cameras as shown, whereas in experiment 3, a single PIV camera was used.
• Figure 2: (a) Example surface elevation of a single wave group from Experiment 1 as a function of time, measured by a wave probe at the measurement location. (b) Example from experiment 1 of an ensemble-average group surface elevation amplitude envelope as a function of time normalised by the measured group temporal width $\tau$ for case 1.D (see \ref{['eq:wamp']}). The time intervals for SPIV measurement (1-3) are shown with vertical dashed lines. (c) Surface elevation measurements of one ensemble from experiment 3 (cases 3.A and 3.B), which shows the onset of a regular wave train. The red box indicates the interval used for analysis.
• Figure 3: Change in Eulerian-mean current due to the passage of a wave group. The waves travelled in the positive $x$-direction, against the current. (a) An example of mean streamwise velocity depth profile before the arrival of a wave group, $U_1(z)$, and after the group has passed, $U_3(z)$, here for case 1.D; (b) mean streamwise velocity difference $\Delta U = U_3-U_1$ as a function of depth for the flow cases of Experiment 1. Error bars are omitted for visibility -- see analysis in Appendix \ref{['app:errors']}; (c) the slope of $\Delta U(z)$ relative to the Stokes drift gradient (a prime denotes derivation with respect to $z$). Light smoothing (moving average with window size $8$ mm) was applied to the curves in panel c for better visibility.
• Figure 4: The wave-induced current $\Delta U$ under regular waves as a function of depth $k_0z$ for cases 2.A.1, 2.A.2, 2.B.1, 2.B.2, 3.A and 3.B in panels (a)--(f), respectively. For each case, different wave steepness values $ak_0$ are shown as indicated in the legend. The dashed lines are the theoretical Stokes drift profiles at the same location for each case, shown as $-u_s(z)$, that is, with opposite sign to the Stokes drift. The filled circles at $k_0z=-4$ and $0$ indicate the theoretical value of the Eulerian return flow, $u_\mathrm{rf}$.
• Figure 5: The wave-induced current under regular waves at the 'reference' depth $k_0z=-0.27$ as a function of wave steepness for cases 2.A--3.B as indicated in the legend. The dashed line is proportional to $(ak_0)^2$.