Comparing free surface and interface motion in electromagnetically driven thin-layer flows

Two-dimensional fluid dynamics is often approximated via laboratory experiments that drive a thin layer of fluid electromagnetically. That approximation would be most accurate if both the direction and magnitude of the flow were uniform over the depth of the layer. In practice, boundary conditions require the flow magnitude to drop to zero at the no-slip floor, but put no strong constraint on flow direction. We measure the velocity magnitude and direction simultaneously at the free surface and lower interface of a thin, two-layer vortex flow. We find that the flow direction is almost entirely independent of depth, though its slight misalignment grows as the Reynolds number (Re) increases. Similarly, we find that the ratio of speeds at the free surface and interface nearly matches an analytically derived profile based on idealized assumptions, even for complex flows, but deviates systematically as Re increases. We find that flows with thinner fluid layers are better aligned and more nearly match the predicted speed ratio than flows with thicker layers. Finally, we observe that in time-dependent flows, flow structures at the interface tend to follow flow structures at the free surface via complicated dynamics, moving along similar paths with a short time delay. Our results suggest that the depth-averaged equation of motion recently developed for thin-layer flows [Suri et al., Phys. Fluids 26, 053601 (2014)], which relies on flow alignment and idealized profiles and was previously tested for Kolmogorov flows with Re up to 30, is reasonably accurate for vortex flows with Re up to 470.

Figure 1

The experimental apparatus and its imaging capability. (a) A view from above shows the magnet grid, glass sheet, and experimental vessel, with one electrode visible at left. Magnet polarity is indicated by + or $-$. The direction of the current density $\mathit{J}$ is indicated with a black arrow. (b) A cross-section shows green and red tracer particles at the free surface and oil-acid interface, respectively. (c) The view through a filter blocking red shows only green particles, which are in the dish at bottom in the image. (d) The view through a filter blocking green shows only red particles, which are in the dish at top in the image.

Figure 2

Instantaneous velocity measurements at the free surface and interface. Each arrow indicates the position and velocity of one tracked particle; color indicates whether the particle followed fluid motion at the free surface or at the interface. Circles indicate vortex centroids at the top and bottom of the acid layer, located using an algorithm described below. Both snapshots show the same region, which covers 17% of the area visible to both cameras. The snapshots come from experiments using 6-mm layers at (a) $\mathrm{Re}=62$ and (b) $\mathrm{Re}=370$. By simultaneously tracking particles at the free surface and interface, we can quantify the variation of velocity with depth, even in time-dependent flows.

Figure 3

Idealized velocity profiles computed using the method of Suri et al. [34] with depths matching our experiments. Profiles for different layer thicknesses differ slightly, primarily because the magnetic field decays along $z$. The ratio of interface speed to free-surface speed is $s^{-1}=0.675$ for 6-mm layers and 0.648 for 3-mm layers.

Figure 4

Instantaneous alignment $cos\theta$ at (a) $\mathrm{Re}=62$ and (b) $\mathrm{Re}=370$, in the same region and at the same times shown in Fig. 2, from the same experiments with 6-mm layers. Arrows indicate velocity fields at the interface and free surface. Anti-alignment $\left(cos\theta \approx -1\right)$ occurs only near singularities where the surface speed is almost zero.

Figure 5

Statistics of alignment $cos\theta$, in experiments with 6-mm layers. (a) Distributions of $cos\theta$ from 19 experiments at different Reynolds numbers, plotted on a logarithmic scale, show that near-alignment $\left(cos\theta \approx 1\right)$ is most common. (b) The median value of $cos\theta$ is near 1 in all 19 experiments. (c) An enlargement of the same data shown in (b) demonstrates systematic variation with Re. Colors in (a) correspond to symbol colors in (b), (c) and indicate Re.

Figure 6

Instantaneous speed ratio $s^{-1}$ at (a) $\mathrm{Re}=62$ and (b) $\mathrm{Re}=370$, in the same region and at the same times shown in Fig. 2, from the same experiments with 6-mm layers. Blue regions have speed ratio lower than the theoretical prediction; red regions, higher. Arrows indicate velocity fields at the interface and free surface. The speed ratio is large near singularities where the surface speed is almost zero, but nearly matches the value for an idealized profile in most places.

Figure 7

Statistics of speed ratio $s^{-1}$ in experiments with 6-mm layers. (a) Distributions of $s^{-1}$ from the same 19 experiments characterized in Fig. 5. (b) Median and most likely values of $s^{-1}$, varying with Re. Colors in (a) correspond to symbol colors in (b) and indicate Re. In all cases, the most likely value of $s^{-1}$ falls within 11% of the value $s^{-1}=0.675$ expected for an idealized profile with 6-mm layers. However, typical values of $s^{-1}$ drop systematically as Re increases, showing that the steady case becomes a poorer approximation as flows become more turbulent.

Figure 8

Statistics of speed ratio $s^{-1}$, conditioned on $cos\theta$, in experiments with 6-mm layers. Median values appear as thin lines, mean values appear as dotted lines, and the value for an idealized profile appears as a dashed line. Colors indicate varying Reynolds numbers, as in Fig. 5. Panel (a) shows the entire range $-1\le cos\theta \le 1$; panel (b) shows the range $0.8\le cos\theta \le 1$. The speed ratio $s^{-1}$ decreases as Re increases. When $cos\theta \approx 1$, mean and median values of $s^{-1}$ nearly match the value for an idealized profile.

Figure 9

Statistics of alignment $cos\theta$, in experiments with 3-mm layers. (a) Distributions of $cos\theta$ from six experiments at different Reynolds numbers, plotted on a logarithmic scale, show that near-alignment $\left(cos\theta \approx 1\right)$ is most common. (b) The median value of $cos\theta$ is near 1 in all experiments. (c) An enlargement of the same data shown in (b) demonstrates systematic variation with Re. Colors in (a) correspond to symbol colors in (b, c) and indicate Re.

Figure 10

Statistics of speed ratio $s^{-1}$ in experiments with 3-mm layers. (a) Distributions of $s^{-1}$ from the same six experiments characterized in Fig. 9. (b) Median and most likely values of $s^{-1}$, varying with Re. Colors in (a) correspond to symbol colors in (b) and indicate Re. In all cases, the mostly likely value of $s^{-1}$ falls within 2% of the value $s=0.648$ expected for an idealized profile with 3-mm layers.

Figure 11

Statistics of speed ratio $s^{-1}$, conditioned on $cos\theta$, in experiments with 3-mm layers. Median values appear as thin lines, mean values appear as dotted lines, and the value for an idealized profile appears as a dashed line. Colors indicate varying Reynolds numbers, as in Fig. 9. Panel (a) shows the entire range $-1\le cos\theta \le 1$; panel (b) shows the range $0.8\le cos\theta \le 1$. The speed ratio $s^{-1}$ decreases as Re increases.

Figure 12

Comparison of the speed ratio relative error $\epsilon$ for different layer thicknesses. (a) Speed ratio relative error for median $s^{-1}$ varying with Re for both 6-mm and 3-mm layers. (b) Speed ratio relative error for most likely $s^{-1}$ varying with Re for both 6-mm and 3-mm layers. For thinner layers, the error in the most likely value of $s^{-1}$ is lower, although the error in the median value is higher.

Figure 13

Snapshots of the Okubo-Weiss criterion $Q$ and vortex centroids. (a), (b) Snapshots at the free surface and interface, respectively, in the same experiment, at the same time, and in the same region as Fig. 2. (c), (d) Snapshot at the free surface and interface, respectively, in the same experiment, at the same time, and in the same region as Fig. 2. Corresponding velocity fields are indicated with arrows. $Q>0$ in regions where rotation dominates shear, and the weighted centroids of regions identified as vortices are plotted as circles.

Figure 14

Paths of coupled vortices at the free surface and interface, all from the $\mathrm{Re}=370$ experiment with 6-mm layers. Each curve shows the path of the weighted centroid of a region identified as a vortex, with thinner curves indicating vortices on the free surface and thicker curves indicating vortices at the interface. Time is indicated in color. In these and many other examples, vortices appear to be coupled, with a vortex at the interface moving along a similar path to a nearby vortex at the free surface, with some lag.

Figure 15

Statistics of alignment $cos\theta$, conditioned on Okubo-Weiss parameter $Q$ normalized by its standard deviation ${\sigma }_Q$, at (a) the top of the acid layer, and (b) the bottom of the acid layer, in experiments with 6-mm layers. Colors indicate varying Reynolds numbers, as in Fig. 5. Alignment is best near $Q=0$, where vorticity and strain are nearly balanced.