Formation and decay of eddy currents generated by crossed surface waves

The mass transport induced by crossed surface waves consists of Stokes and Euler contributions, which are very different in nature. The first contribution is a generalization of Stokes drift for a plane wave in ideal fluid and the second contribution arises due to the fluid viscosity and is excited by a force applied in the viscous sublayer near the fluid surface. We study the formation and decay of the induced mass transport theoretically and experimentally and demonstrate that the two contributions have different timescales for typical experimental conditions. The evolution of the Euler contribution is described by a diffusion equation, where the fluid kinematic viscosity plays the role of the diffusion coefficient, while the Stokes contribution instantly tracks the wave pattern and therefore it evolves faster due to the additional wave damping near the system boundaries. The difference becomes more pronounced if the fluid surface is contaminated. We model the effect of contamination by a thin insoluble liquid film present on the fluid surface, with the compression modulus being the only nonzero rheological parameter of the film. Then the Euler contribution to the mass transport becomes larger and the evolution of the Stokes contribution becomes faster than in the free surface case. We infer the value of the compression modulus of the film by fitting the results of transient measurements of eddy currents and demonstrate that the obtained value leads to the correct ratio of amplitudes of horizontal and vertical velocities of the wave motion and is in reasonable agreement with the measured dissipation rate of surface waves.

Figure 1

The scheme of the experimental setup: 1–plungers and driving mechanisms, 2–contrast pattern and bottom LED array used to highlight the pattern, 3–LED strips along the bath perimeter for illumination of tracers on the fluid surface, 4–photo camera that captures slow eddy currents and waves on the fluid surface.

Figure 2

(a) The amplitude of eddy currents $\left|\right|\mathrm{\Omega }\left|\right|=\left|\right|{\mathrm{\Omega }}_E\left|\right|+\left|\right|{\mathrm{\Omega }}_S\left|\right|$ depending on time for $38\%$ glycerine-water solution. The yellow curve corresponds to the experimental data and the blue curve shows numerical result based on Eq. (9). The red curve demonstrates the dependence of Stokes contribution $\left|\right|{\mathrm{\Omega }}_S\left|\right|$ on time based on the experimentally measured wave amplitudes $H_{1,2}\left(t\right)$ and $\psi \left(t\right)$. The horizontal lines show the time-asymptotic solutions, see expression (2), for the free surface case $\varepsilon =0$ and for the found parameter $\varepsilon =0.33$. (b) The decay of the wave amplitude after the pump is turned off.

Figure 3

Formation (a) and decay (b) of the eddy currents ($t_E\approx 125\text{s}$) on the fluid surface for the same parameters as in Fig. 2. The yellow curves correspond to the experimental data and the blue curves show numerical results based on Eq. (9). The red curves demonstrate the Stokes drift contribution based on the experimentally measured wave amplitudes $H_{1,2}\left(t\right)$ and the phase difference $\psi \left(t\right)$. The violet curves show analytical results based on expressions (14) and (17).

Figure 4

The time dependence ($t_E\approx 320\text{s}$) of the amplitude of eddy currents for $7.7\%$ glycerine-water solution for two levels of excitation with the waves steepnesses $k{H}_{1,2}\approx 4.7\times {10}^{-3}$ and $k{H}_{1,2}\approx 1.1\times {10}^{-2}$. The compression modulus of the film $\varepsilon =0.53\pm 0.02$. The yellow curves correspond to the experimental data and the blue curves show numerical results based on Eq. (9). The green curve shows the rms of large-scale velocity $V_L$ of slow motion with wave numbers $k_L\le 0.26{\text{cm}}^{-1}$ (vertical axis on the right) for the higher level of excitation.

Figure 5

(a) The comparison of the ratio of horizontal and vertical velocities on the fluid surface for the wave motion measured directly (vertical axis) and calculated theoretically (horizontal axis); see expression (4). Different colors correspond to the waves in $X$ and $Y$ directions. (b) The comparison of the wave decay time measured directly after the pumping is switched off (vertical axis) and calculated theoretically (horizontal axis) according to expression (5). The experimental values correspond to the largest decay time measured for $X$ and $Y$ waves. Solid lines show the perfect agreement between theory and experiment. Error bars are approximately the same for all data points and thus are shown only for one of them to make the figure more readable.