WAVE-GENERATED FLOWS ON THE WATER SURFACE

Predicting trajectories of fluid parcels on the water surface perturbed by waves is a difficult mathematical and theoretical problem. It is even harder to model flows generated on the water surface due to complex three-dimensional wave fields, which commonly result from the modulation instability of planar waves. We have recently shown that quasi-standing, or Faraday, waves are capable of generating horizontal fluid motions on the water surface whose statistical properties are very close to those in two-dimensional turbulence. This occurs due to the generation of horizontal vortices. Here we show that progressing waves generated by a localized source are also capable of creating horizontal vortices. The interaction between such vortices can be controlled and used to create stationary surface flows of desired topology. These results offer new methods of surface flow generation, which allow engineering inward and outward surface jets, large-scale vortices and other complex flows. The new principles can be also be used to manipulate floaters on the water surface and to form well-controlled Lagrangian coherent structures on the surface. The resulting flows are localized in a narrow layer near the surface, whose thickness is less than one wavelength.


Introduction
It has recently been demonstrated that Faraday waves, which are parametrically excited 3D nonlinear waves, create vortices on the fluid surface that interact and lead to the development of 2D turbulence [1][2][3] .The generation of horizontal vortices by quasi-standing nonlinear waves is an effect, which is impossible in planar irrotational waves.It was shown later that progressing nonlinear waves produced by a localized source are also capable of creating surface vortices 4 .The interaction between such vortices leads to the formation of large-scale surface flows, far away from a wave maker.
In this paper we show how waves generated by vertically oscillating plungers produce surface flows and how the topology of the flow can be controlled by changing the shape of the plunger and also the wave amplitude.

Wave Generation and Modulation Instability of Surface Waves
Waves in these experiments are generated by vertically oscillating plungers of different shapes.Spatially localized time-periodic perturbations of water surface generate waves propagating away from the plungers.The frequency of the plunger oscillations is varied in the range between 10 Hz, corresponding to gravity waves, and 200 Hz, corresponding to capillary waves.The dispersion relation is given by for gravity waves, and for capillary waves.Here g is the acceleration of gravity, α is the surface tension and ρ is the fluid density.
Both these branches of the surface waves are unstable with respect to the modulation instability, known as Benjamin-Feir instability for gravity waves 5 , which develops when the Lighthill criterion is satisfied: (1) The nonlinear frequency corrections have different signs for different branches: positive for gravity waves, and negative for capillary waves.Since the dispersion of the group velocity is negative for gravity waves and positive for capillary waves, the Lighthill criterion Eq. ( 1) is satisfied for both branches.
The transition from linear 2D regime to 3D wave fields, which is described in the main text (Figs.1a, b), has been studied before in Ref. 6 where it has been shown that the increase in the plunger acceleration (where A is a peak-to-peak displacement of a plunger and is the driving frequency) leads to the development of the instability and appearance of the cross-wave in the nonlinear stage 7 .The cross-wave instability modulates wave fronts in the transverse direction, destroying twodimensionality of the wave and breaking it into individual pulses, which then propagate away from the source.These propagating wave pulses oscillate vertically with half the driving frequency, , as expected for parametrically excited waves.The development of the cross-wave and the evolution of the wave spectra 6  which measures the gradient of the surface elevation.The wave fields are also visualized using the diffusive light imaging technique, Figs.1(g-h).Experiments are performed in a rectangular container 1.5×0.5 m 2 filled with water to the level of 80 mm.The wave makers are driven using a 4kN electrodynamic shaker (Bruel&Kjaer).The forcing is sinusoidal and monochromatic.The shaker frequency f 0 = 20 Hz corresponds to the wavelength of k = 12 mm.The experimental setup is shown in the photograph of Fig.  needs to be measured.The use of a and plasma treatment makes the particle wettability almost neutral.This prevents surface particle clustering and ensures homogeneous spreading on the water surface.Finite particle size effects are negligible under our experimental conditions as discussed in Ref. 9.
The diffusive light images and the horizontal particle motion are captured using a high-resolution fast camera (Andor Neo sCMOS).Three-dimensional Lagrangian trajectories are obtained using a combination of two-dimensional particle tracking velocimetry (PTV) technique and a subsequent evaluation of the local elevation along the trajectory.First, horizontal (x-y) coordinates of each point on a trajectory are tracked using a nearest neighbour algorithm 10 .Then, the particle elevation (z coordinate) is estimated as the mean of the wave elevation over a local window (500mm radius), which is centred on the x-y particle coordinates at a given time.The 3D trajectories of the particle and the wave elevation are visualized using the Houdini 3D animation tools (by Side Effects Software).Figure 3 shows examples of the reconstructed 3D trajectory and the wave elevation field.
The particle image velocimetry (PIV) technique is used to obtain the velocity field of the horizontal motion of the flow.In the experiment described in Fig. 3(c-f), the flow is recorded at 20fps with a spatial resolution of 1850×1850 pixels.The velocity fields are computed on a 60×60 grid with 1.2 mm resolution.In the experiments described in Fig. 4(d-f) of the main text, the flow is recorded at 120 fps with a spatial resolution of 512×512 pixels.The PIV velocity fields are computed on a 50×50 spatial grid with 1 mm resolution.

Surface Flows Produced by Cylindrical Wave Makers
If a wave maker oscillates at low amplitude as in the photo of Fig. 2(a), the wave maker produces nearly planar propagating wave fronts.To visualize the fluid motion, buoyant tracer particles are uniformly dispersed over the fluid surface.The particles are pushed in the direction of the wave propagation, forming an outward jet 4 .As a consequence, a compensating return flow converges towards the sides of the wave maker.The flow changes dramatically as the wave amplitude is increased above the threshold for the onset of the modulation instability.As the modulation grows and the cross-wave instability breaks the wave front into trains of propagating pulses, the wave field becomes three-dimensional, as seen in Fig. 3(b).Simultaneously, the direction of the central jet reverses.It now pushes floaters towards the wave maker and against the wave propagation, as in Fig. 4. The flow is strong enough to move floating objects far from the plunger on the water surface.The motion of the floater can thus be reversed by simply changing the amplitude of the wave maker oscillations.
The reversal of the central jet in the nonlinear regime of the wave generation using a cylindrical plunger is a robust effect observed in a broad range of the plunger accelerations with cylinders of various lengths.The stability of the pattern can be optimized by adjusting the excitation frequency, which affects the mode number of the cross wave.The reversal of the jet direction appears to be independent of different driving frequencies.Fig. 4 shows inward flow patterns produced by the cylindrical plunger (white rectangle in the centres of the plots) at two frequencies, in the gravity and in the capillary wave range.The motion of floaters on the surface perturbed by capillary-gravity waves propagating away from a wave maker seems inconsistent with the Stokes drift model.This should not be surprising since the original model was developed for planar waves of very small amplitude.However the floaters uniformly move in the direction of the wave 1660179-6 Int.J. Mod.Phys.Conf.Ser.2016.42.Downloaded from www.worldscientific.comby AUSTRALIAN NATIONAL UNIVERSITY on For personal use only.
propagation in the initial stage of the flow development, before a return flow starts to develop.Surface particle streaks (moving averaged over 5 wave periods) are illustrated in Fig. 6(a), where they are filmed shortly after the plunger is activated.During this time most floaters in front of a plunger move in the direction of the wave propagation.However even then their velocities disagree with the Stokes drift expectation, , where a is the wave amplitude, k and ω are the wave number and frequency.Since the wave amplitude decays away from the plunger, particle velocities should be the highest near the plunger and should decay with the distance.This is not the case, as seen in Fig. 6(c).At a later stage, after a stationary flow develops, this discrepancy becomes more pronounced.We conclude that in all wave-driven flows described here, the velocity field is not directly determined by the Stokes drift of the underlying wave field.The velocity maxima in Figs. 5 and 6 can be considered as separating near-field from far-field flow pattern.In particular, Fig. 6 shows that initially the jet velocity is closer to the wave maker (blue diamonds), while in the steady state (red squares), the velocity profile is broader with the maximum velocity being further away from the plunger.The green triangles in Fig. 6(c) and the solid line show that the squared wave intensity a 2 strongly decays as a function of the distance from the plunger.This suggests that the flow velocity is not determined by the local wave field, but results from a global flow pattern.

Surface Flows Produced by Various Wave Makers
As discussed in the main text, stable flow patterns exhibiting inward and outward jets as well as stationary vortices can be generated by appropriately shaping wave makers.Some examples of such flows are shown in Fig. 7.
Similarly to the cylindrical wave maker, pyramidal plungers produce outwards jets normal to the sides of a triangle or a square, while the return flows are directed towards vertices.Above the threshold of modulation instability pyramidal wave makers produce inward central jets and reverse the flow direction.An example of the tractor beam driven by the triangular pyramid is shown in Fig. 8.

Fig. 1 .
Fig. 1.Schematics of the plunger, waterline and the underwater view of the surface perturbation at different vertical accelerations: (a) a c = 3g, (b) a c = 5g and (c) a c = 8g.The corresponding frequency spectra of the surface gradient are shown in (d-f).The driving frequency of the plunger is f 0 = 20 Hz.The wave fields produced by a conical plunger are visualized using the diffusive light imaging technique at (g) low and (h) higher plunger accelerations.These wave fields correspond to the schematics (a) and (c).

Fig. 2 . 3 .
Fig. 2. (a) Photo and (b) schematics of the experimental A vertically movable plunger is attached to the table of the electromagnetic shaker via the plunger frame.Linear LED arrays on the sides of the transparent water tank illuminate surface particles whose motion is filmed from above using Andor Neo sCMOS camera.

Fig. 5 .
Fig. 5. Structure of the flow produced by the cylindrical plunger (130 mm long) as described in the main text (Figs.1b,d) for the cases of (a) outward, and (b) inward jets.(c) Measured (normalized) jet velocity versus the distance away from the wave maker.The excitation frequency f 0 = 20 Hz.

Fig. 6 .
Fig. 6.Surface particle streaks (a) shortly after the plunger is turned on, and (b) in the steady state, after the establishment of a stable quadrupole vortex flow.(c) Central jet velocity as a function of the distance from the wave maker during the start-up phase (diamonds), and in the steady state (squares).Triangles and the solid line show squared wave amplitude a 2 in front of the cylindrical wave maker.

Fig. 7 .
Fig. 7. Surface flow patterns produced by the wave makers of different shapes in the linear regime: (a) elliptical, (c) triangular pyramid, and (d) square pyramid.Waterlines are shown as solid white lines on the plungers.The patterns correspond to the weakly nonlinear waves, below the modulation instability threshold.The excitation frequency in these examples is f 0 = 60 Hz.1660179-8 Int.J. Mod.Phys.Conf.Ser.2016.42.Downloaded from www.worldscientific.comby AUSTRALIAN NATIONAL UNIVERSITY on 06/26/16.For personal use only.

Fig. 8 .
Fig. 8. Visualization of the surface flow produced by a triangular pyramid above the threshold of modulation instability.The wave maker frequency f = 60 Hz.The central jet is directed inward (tractor beam mode).A corresponding flow in the linear regime is shown in Fig. 7(b) (outward jet).