Formation of vortex structures by noncollinear waves on the water surface

The generation of a large-scale vortex flow by gravity waves on the water surface has been experimentally studied. It is shown that the mechanism of vortex system attenuation changes with an increase in the relative pump amplitude. The wave amplitudes have been experimentally determined at which the experiment is not described by a theoretical model. The experimental results agree well with the developed theoretical model.


Introduction
In nature, a significant part of gas and liquid flows occur in the form of vortex motion, have been studied for quite a long time.
A curious fact is that vortices can be generated as a result of the interaction of surface waves. This phenomenon was observed in the experiments [1,2,3], where the waves were excited in a vessel performing vertical oscillations as a result of the Faraday parametric instability. In [4], the vortex motion was formed by waves generated by wave generators. A theoretical model was also developed in this work, which describes the formation of vortices by perpendicular standing waves, taking into account the effect of a thin film that can be formed on the water surface and can change significantly the experimental results.
According to the developed theoretical model, the vorticity on the surface of a liquid Ω is described by the expression: where ε is the dimensionless parameter of a film, is the error function, = � ⁄ . This theoretical model is valid for both capillary [5] and gravity waves [4].

Experimental methods
The investigations were performed in an experimental setup, the scheme of which is presented in Fig.1 and a wall thickness of 1 cm which was made from glass. The bath was placed on a Standa vibration isolation table with an air suspension and was filled with distilled water with a total volume of about 50 l to a depth of 10 cm. The bath was covered on top with a plexiglass cover. Wave generators each consisting of an actuator and a plunger were mounted on a supporting frame; they excite waves on the water surface. in order to excite waves on the water surface, plungers were used which were stainless-steel rods with a diameter of 10 mm and a length of 300 mm. Pioneer TS-W254R subwoofers with a rated power of 250 W were used as the actuators. A sinusoidal signal generated by a dual-channel generator and amplified by a signal amplifier was supplied to the subwoofers.
To visualize the motion of a liquid, a white powder of PA-12 polyamide particles was deposited on the water surface. The Polyamide is was in a semi-immersed state, since the particle density is slightly less than the density of water. The particles on the surface were illuminated by the LEDs placed along the bath perimeter. An oscillating the water surface was recorded by a Canon EOS 70D camera at a rate of 24 fps which was located above the bath.
The PIVLab code [6] for MATLAB was used to process resulting video images. It allows calculating the field of displacements between the images by the method of cross-correlation processing of two images. Video Recording processing, as well as the processing algorithm, is presented in [7].
For each experiment, the dimensionless parameter of a film was determined by the method described in [4]. In the presented experiments, its value was in the range of 0.62 -0.65.

Experimental results and discussion
In the experiments, the waves were excited by two plungers, one of which was perpendicular to the wall of the experimental bath, and the second one was rotated by the angle α = 18° (see Figure 1 (b)).
The pump frequency was f = 2.34 Hz. Figure 2 (a) shows the modes that were excited on the water surface (red dots). The Wave vector difference was k1-k2 = 0.14 cm -1 . The similar difference was observed in the distribution of wave velocity amplitude [8] in the k-space 180 s after switching on pumping ( figure 2 (b)). The similar vector difference is also seen in the distribution of vorticity amplitude in the k-space at the same amplitude of the pump wave (Figure 3 (a)). In the distribution of a vorticity field in the space, depicted in Figure 3      With an increase in the pump amplitude, the character of attenuation changes. Figure 5 (b) shows the vorticity distribution in the k-space 96 s after switching off pumping with a wave amplitude of 1.1 mm. The figure demonstrates redistribution of vorticity from large wave vectors towards the small ones. After this, the entire vorticity passes completely into small wave vectors.
With an increase in the wave amplitude to 1.7 mm, the system attenuates by a different mechanism. The peaks presented in the vorticity distribution in the k-space135 s after the switching on pumping ( Figure 5 (c)) begin to shift towards small wave vectors which correspond to the vortex motion with a characteristic size comparable to the size of a pool.

Conclusions
It has been shown that the theoretical model describes well the experimental data when a vortex flow is generated by waves propagating at an acute angle to each other.
The wave amplitudes and the corresponding Reynolds numbers have been determined experimentally, at which the experiment ceases to be well described by the theoretical model. At the Reynolds number Re> 50, the nonlinear effects increase, and the theoretical model describes the obtained experimental data worse.
It has been found that at different pump amplitudes, the mechanism of vortex system attenuation is different: at a small wave amplitude (0.6 mm), the peaks characterizing the vorticity in the k-space shift towards large wave vectors; at a wave amplitude of 1.1 mm, the vorticity is redistributed from large wave vectors towards the small ones. When the amplitude of pumping wave reaches 1.7 mm, the attenuation mechanism is different from the previous ones: -the vorticity peaks in the k-space shift from large wave vectors towards the small ones.