Laboratory study of the wave-induced mean flow and set-down in unidirectional surface gravity wave packets on finite water depth

The net movement of Lagrangian particles under water waves comprises a Stokes drift in the direction of wave propagation and an Eulerian return flow in the opposing direction. Accurate prediction of the Eulerian return flow in the ocean is of importance in modeling the transport of plastic pollution, oil, wreckage, and sediment. Herein, we derive a multiple-scales solution for the Eulerian mean flow under wave packets that is valid for all water depths, relative to both the length of the wave and the length of the wave packet. To validate this solution, we carry out particle tracking velocimetry experiments in a long flume to extract the mean motion from Lagrangian seeding particles under wave packets, finding good agreement. The extraction technique is able to deal with small background motion and subharmonic error waves associated with wave generation by the paddle, the latter being relatively large in finite-depth flume experiments. In finite depth, the return flow is forced by both the divergence of the Stokes transport on the wave-packet scale and the formation of a non-negligible mean set-down underneath the packet, which acts like a bounding streamtube in the form of a convergent-divergent duct. The magnitude of the horizontal return flow is thus enhanced, with particular relevance to transport in the finite-depth coastal environment.

Figure 1

Theoretical normalized horizontal (left panel) and vertical (right panel) displacements as a function of the relative depth $k_{0}d$ at $z=0$ [from (23) and (25)]. The displacements by the Eulerian mean flow displacement are shown as black lines, the displacements by the Stokes drift are shown by red lines, and the Lagrangian displacements are shown by blue lines. The solid lines show the displacements when the set-down is fully accounted for, whereas the dashed lines ignore the set-down.

Figure 2

Experimental setup used to obtain PTV measurements of fluid trajectories beneath focused wave packets generated by a double-element piston-type wavemaker at the COAST laboratory, University of Plymouth, UK.

Figure 3

The experimental linear free surface elevation measured at the focus location (black lines). The linear part has been extracted with a band-pass filter between $0.7f_0$ and $1.3f_0$. The envelope (red dashes lines) has been calculated using a Hilbert transform of the linear free surface.

Figure 4

Time series of the subharmonic components of the free surface elevation representing a “set-down,” from experiments (solid black lines), extracted from the free surface elevation by a low-pass filter at $0.5f_0$, and theory (dashed red lines) for the different experiments.

Figure 5

Time series of the subharmonic components of the free surface at each wave gauge (extracted from the free surface elevation using a low-pass filter at $0.5f_0$). The $y$-axis is offset by the product of the distance of each gauge from the focus location $x_g$ and $C=8.0\times {10}^{-4}$. Also shown are the group speed of the wave packet $c_{g,0}$ with which the set-down moves (continuous black lines), the speed of the error wave $\sqrt{gd}$ (dashed black lines), and the speed of its reflection $-\sqrt{gd}$ (dash-dotted black lines).

Figure 6

Quantification of the measurement error of the set-down. Panel (a) shows the subharmonic set-down for experiment 5 ($f_0=0.88$ Hz, $\epsilon =0.12$) as a function of time: three repeats are plotted using black dotted lines, and the mean is overlaid in solid black. The gray region shows a confidence band of $\pm 2$ standard deviations around the mean. The theoretical set-down is plotted using a dashed red line. Panel (b) is a comparison of the magnitude of the set-down between the theoretical (labeled ${\eta }_{\mathrm{SD},\mathrm{T}}^{\left(2\right)}$ on the horizontal axis) and experimentally measured (labeled ${\eta }_{\mathrm{SD},\mathrm{M}}^{\left(2\right)}$ on the vertical axis) magnitude of the set-down for each experiment. The error bars show $\pm 2$ standard deviations of the absolute calibration error.

Figure 7

Experimental particle trajectories under wave packets with $f_0=1.00\text{Hz}$ and $\epsilon =0.18$ at three different depths and before removal of the background motion. The left panel is closest to the surface, the central panel is close to the vertical position where there is no mean drift, and the right panel is the deepest trajectory, showing the large negative mean flow. The blue and red crosses denote the position before the wave packet has arrived and after it has passed, respectively.

Figure 8

Profiles of horizontal Lagrangian displacement between $t_1=-0.1\sigma /c_{g,0}$ and $t_2=0.1\sigma /c_{g,0}$ (with $t=0$ the center of the packet) as a function of depth: postprocessed experimental data (black crosses), theoretical prediction from (22) (continuous red lines), and theoretical prediction ignoring the set-down from (22) (dotted black lines). The theoretical prediction ignoring the set-down is obtained by replacing the term $[1-c_{g,0}^2/\left(gd\right)]$ in the denominator of the second term in the square brackets in (22) by 1.

Figure 9

Scaled, nondimensional profiles of horizontal Lagrangian displacement between $t_1=-0.1\sigma /c_{g,0}$ and $t_2=0.1\sigma /c_{g,0}$ (with $t=0$ the center of the packet) as a function of depth: postprocessed experimental data (black crosses) and theoretical predictions from (22) (continuous red lines).

Figure 10

Profiles of vertical Lagrangian displacement between $t_1=-3\sigma /c_{g,0}$ and $t_2=0$ (corresponding to the center of the packet $t=0$) as a function of depth: postprocessed experimental data (black crosses) and theoretical predictions from (24) (continuous red lines).

Figure 11

Scaled, nondimensional profiles of vertical Lagrangian displacement between $t_1=-3\sigma /c_{g,0}$ and $t_2=0$ (corresponding to the center of the packet $t=0$) as a function of depth: postprocessed experimental data (black crosses) and theoretical predictions from (24) (continuous red lines).