# Coexistence of Inverse and Direct Energy Cascades in Faraday Waves

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Setup

^{3}and uniformly disperse in the water volume. For that, 0.3 g of particles are wetted in a 10%-solids solution with a surfactant (1% Tween 80 solution, Polysorbate 80, non-ionic). No effects of buoyancy were observed; inertial effects are negligible and particles can be considered as perfect fluid tracers with Stokes numbers $St\approx {10}^{-4}$ [15]. Imaging was performed with a high-speed camera (2, Phantom VEO 640L, resolution: 2560 × 1600 px), which was triggered at ${f}_{\mathrm{rec}}=$ 400 Hz by a second signal from the function generator (standard 5V TTL signal). This allowed us to carefully synchronize the phase delay between the Faraday wave forcing and the imaging and thus to capture the flat surface of the waves two times (every eighth frame) per full wave period ${T}_{F}$ (corresponds to 16 frames). Furthermore, it helps to synchronize the imaging such that frame i and frames $i+n\xb78$, $\phantom{\rule{3.33333pt}{0ex}}n\in \mathbb{N}$ are taken at the same container height. For the horizontal plane measurements, the camera is positioned below the container (2a) and an optical mirror is used to deflect the camera’s line of sight in the vertical direction. The tracer particles are illuminated by a continuous wave argon-ion laser (8, wavelength of 488 nm, Ion Technologies), which is expanded to a planar laser sheet (2 mm thick, 80–100 mm wide) with a set of laser optics (9). From the forcing acceleration, the computed maximum container displacement of $(7.10\pm 0.71)\xb7{10}^{-3}$ mm was found to be sufficiently small in comparison to the laser sheet thickness such that it has a negligible effect onto the velocity fields and no correction for this external movement needs to be considered. PIV measurements were carried out at four horizontal planes at a distance h from the container bottom, $h=$ 30, 27, 21 and 4 mm. These heights were chosen due to previously published results [10], wherein three distinct flow regimes at different heights were observed. The measurement height $h=$ 21 mm was additionally selected because by observing the flow, we determined that this was the approximate height where vertical jets that rushed down from the surface would often disperse. The horizontal PIV measurements at the fluid surface were performed with LED light to prevent reflections of the laser sheet at the wave–air interface. For the vertical PIV measurements, the laser sheet was tilted by 90° in order to vertically cut through the flow domain to obtain temporally and spatially well-resolved velocity information in the vertical cross-section of the container.

#### 2.2. PIV Processing

#### 2.3. Spectral Analysis of Velocity Fields

## 3. Results and Discussion

^{2}, which is approximately one third of the entire field of view. An enlargement section of 25 × 25 mm

^{2}additionally shows the corresponding instantaneous velocity fields. For comparison, the Faraday wavelength ${\lambda}_{F}=(9.5\pm 1.0)$ mm (approximately corresponding to twice the energy injection length scale) is depicted. The observations and conclusions apply for both accelerations ${a}_{f}$ as the same characteristic structures appear in the velocity field for both cases. Three distinct flow regimes at different heights can be identified from the horizontal planar measurements. On the fluid surface, the main features of the significantly turbulent Faraday flow can be identified by the presence of multiple vortices, with a typical size varying between 1/2 and 2 Faraday wavelengths, and rapid bursts of horizontal jet-like flow accelerated between them. The Reynolds number (based on half the Faraday wavelength and average absolute velocity of one component $\langle |u|\rangle $ varies in this regime between 21 for the lower forcing and 41 for ${a}_{f}=0.70$ g. This highly turbulent flow, with short time scales, is exclusively confined in a thin layer of the fluid surface (approximately 2 mm thick), with vorticity vectors pointing perpendicular to the fluid surface and velocity components in the horizontal plane. A second “transition" regime was identified for h between 20 and 28 mm. In this regime, the velocity magnitude exponentially decreases before reaching the plateau presented in Figure 4. The Reynolds numbers, based on characteristic length determined from spatial correlations as shown in [10], range here between 3 and 17. Here, the flow structures become slower and less turbulent, but still carry the imprint of vortical structures of the surface flow, as well as the effects of the vertical jets. This can be seen particularly well in Figure 3c and Supplementary Figure S4c, which show the presence of positive divergence, identified by velocity vectors pointing outwards from the source point, which indicates the presence of a jet as it dissolves by impinging on the slower flow at lower depths. The third regime starts at $h\approx 20$ mm right below the depth at which vertical jets typically dissolve, and extends down to the container bottom. Here, no more vortical structures are observed in the horizontal velocity fields, but instead, only slowly varying large scale motions with very long temporal scales and Reynolds numbers varying in this domain between 3 and 9. The vorticity and divergence of the horizontal velocity fields at these heights were analyzed in [10]. These large-scale motions are, however, different from the previously reported streaming effects caused by standing waves as theoretically described [23] and experimentally observed for longitudinal Faraday waves [24]. The measured structures are several Faraday wavelengths in size and excessively large to be explained by classical streaming patterns that are of the order of half the wavelength of the standing wave [23]. It is thus likely that they are also largely driven by the downward jets. In addition to the description of flow structures and kinetic energy profiles, the spectral analysis of energy transport and net energy and enstrophy fluxes through the scales was performed at the aforementioned planes. The results are depicted in Figure 6 for $h=30$, 27 and 21 mm and both forcing accelerations (note the different scales among the panels). The results for $h=4$ mm are not shown.

^{−1}(computed with half the Faraday wavelength as the dominant forcing scale for energy injection). At wavenumbers $k<{k}_{\mathrm{inj}}$, the slope of ${k}^{-5/3}$ is captured fairly well, and the negative net energy flux ${\Pi}_{E}(k)$ validates the presence of an inverse energy cascade, although the transition from negative to positive is not localized at one wavenumber as in theory or simulated flows [1,25] but occurs more gradually. Our data also resolve the direct enstrophy cascade with a positive net enstrophy flux ${\Pi}_{Z}(k)$ for $k>{k}_{\mathrm{inj}}$. The slope of $E(k)$ is, however, slightly steeper than the ${k}^{-3}$ scaling predicted by Kraichnan [26]. This phenomenon is not uncommon in experimental 2D turbulence, where friction and damping effects can cause deviations at larger wavenumbers in contrast to theory or simulated data [9,25]. The situation is substantially different in the plane immediately below the fluid surface—at $h=27$ mm—shown in Figure 6b. The slope of the energy spectra is more homogeneous throughout the entire wavenumber range and no distinct bend can be seen at the energy injection wavenumber. Here, in sharp contrast to the surface flow, the net energy flux remains positive for all wavenumbers, proving that the flow exhibits a direct energy cascade and that there is a transition from the 2D turbulent Faraday flow on the surface to a direct energy cascade right below the surface. This observation is validated by the zero net enstrophy flux, indicating the vanishing influence of the direct enstrophy cascade. The results for the energy fluxes and spectra are similar at further submerged planes below the surface (e.g., $h=21$ mm, Figure 6c and $h=4$ mm, not shown), where both trends of positive net energy flux and zero net enstrophy cascade are confirmed, although with much smaller magnitudes than at $h=27$ mm. This is also observed in the velocity fields, as the flow becomes less turbulent, and the velocity structures become larger, less chaotic and with longer temporal scales [10]. In Figure 6c, at $h=21\phantom{\rule{0.166667em}{0ex}}$ mm, there is a valley in the energy flux at intermediate scales ($k\approx 7\phantom{\rule{0.166667em}{0ex}}$ mm

^{−1}corresponding to 9 mm $\approx {\lambda}_{F}$) that we cannot definitely explain. A possible interpretation is that the positive divergence (sources are of sizes ${\lambda}_{F}$, see Figure 3c) caused by the impinging vertical jets caused some upscale energy transport by stretching out on that plane, diminishing the positive mean value of ${\Pi}_{E}(k)$. This would also explain why the effect is not seen at $h=27\phantom{\rule{0.166667em}{0ex}}$ mm.

^{−1}(Figure 6a,b). There, energy might be lost due to the clashing of the vertical surface jets which then cause the sporadic downward jets, thereby fueling the direct cascade found in the bulk flow below. The interpretation that energy is transferred from the surface flow to the bulk and not vice versa is also supported by the estimates of the mean energy injection rates ${\u03f5}_{in}$ that are calculated as ${\u03f5}_{in}=|{\langle {\Pi}_{E}\rangle}_{in}|=|{\int}_{{k}_{1}}^{{k}_{2}}{\Pi}_{E}(k)/{k}_{2}-{k}_{1}|$, where $[{k}_{1},{k}_{2}]$ denotes the interval of a potential inertial range in the corresponding cascade, even though the energy flux is not constant (see Supplementary Figures S5 and S6). The approximate values for the energy injection rates ${\u03f5}_{in}$ of the direct cascades at $h=27$ mm are only approximately $14\%$ of the energy injection rates at the fluid surface $h=30$ mm, for both forcing accelerations.

## 4. Conclusions

^{−1}and an inverse energy cascade with negative net energy flux towards larger scales. At further depths, the transition to a direct cascade becomes evident due to the exclusively positive net energy flux and zero enstrophy flux throughout the entire wavenumber range. Additionally, measurements of velocities $(u,w)$ in the container’s vertical cross-section unveiled the presence of strong and confined vertical jets. These originate from the surface and dissolve at approximately one Faraday wavelength ($\approx 9$ mm) below it. The jets, together with the simultaneous formation of vortices by shear effects, appeared to be the main fuel for the three-dimensional bulk flow beneath Faraday waves. Our results further reveal that the vertical component of velocity decreases at a smaller rate than the horizontal components in the aforementioned transition layer, directly beneath the fluid surface. Conversely, we found that the average ratio of the flow kinetic energy in the z-direction to the total kinetic energy increases in this layer, indicating a shift from the 2D Faraday flow confined to the fluid surface towards a 3D bulk flow.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

2D3C | Two-dimensional three-component |

PIV | Particle image velocimetry |

EMD | Electromagnetically driven |

PTV | Particle Tracking Velocimetry |

## Appendix A. PIV Processing Parameters

h | ${\mathit{a}}_{\mathit{f}}$ | $\mathbf{\Delta}\mathit{x}$, $\mathbf{\Delta}\mathit{y}$ | ${\mathit{L}}_{\mathit{x}}\times {\mathit{L}}_{\mathit{y}}$ | $\mathbf{\Delta}\mathit{t}$ | $\mathit{\tau}$ | Conv. | N |
---|---|---|---|---|---|---|---|

mm | m s^{−2} | px|mm | mm | ms | ms | px/mm | |

30 | 0.70 g | 14 1.13 | 169 × 127 | 40 | 7680 | 12.42 | 6 |

30 | 0.47 g | 14 1.13 | 169 × 127 | 40 | 7680 | 12.42 | 4 |

27 | 0.70 g | 14 1.21 | 175 × 99 | 80 | 12,800 | 11.53 | 4 |

27 | 0.47 g | 14 1.21 | 175 × 99 | 120 | 12,800 | 11.53 | 4 |

21 | 0.70 g | 14 1.20 | 173 × 98 | 160 | 12,800 | 11.65 | 4 |

21 | 0.47 g | 14 1.20 | 173 × 98 | 240 | 12,800 | 11.65 | 4 |

4 | 0.70 g | 14 1.18 | 170 × 96 | 480 | 12,800 | 11.90 | 4 |

4 | 0.47 g | 14 1.18 | 170 × 96 | 480 | 12,800 | 11.90 | 4 |

Vertical | ${\mathit{a}}_{\mathit{f}}$ | $\mathbf{\Delta}\mathit{x}$, $\mathbf{\Delta}\mathit{z}$ | ${\mathit{L}}_{\mathit{x}}$ | $\mathbf{\Delta}\mathit{t}$ | $\mathit{\tau}$ | Conv. | N |
---|---|---|---|---|---|---|---|

Section | m s^{−2} | px|mm | mm | ms | ms | px/mm | |

Upper | 0.70 g | 16 1.06 | 102 | 40 | 15,360 | 15.16 | 4 |

Middle | 0.70 g | 16 1.06 | 102 | 160 | 15,360 | 15.16 | 4 |

Bottom | 0.70 g | 16 1.06 | 102 | 320 | 15,360 | 15.16 | 4 |

Upper | 0.47 g | 16 1.06 | 102 | 40 | 15,360 | 15.16 | 4 |

Middle | 0.47 g | 16 1.06 | 102 | 200 | 15,360 | 15.16 | 4 |

Bottom | 0.47 g | 16 1.06 | 102 | 400 | 15,360 | 15.16 | 4 |

## Appendix B. PIV Algorithms

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**Figure 1.**(

**a**) Schematics of the experimental setup for the horizontal- and vertical-plane PIV measurements, respectively. The zoom-in region shows the location of the horizontal planes measured as distance from container bottom, height h. (

**b**) Pattern of Faraday waves on the fluid at forcing frequency ${f}_{f}=50$ Hz. The green and blue planes qualitatively indicate the position of the laser sheet for the horizontal and vertical PIV measurements, respectively.

**Figure 2.**Instantaneous velocity field in the vertical plane on top of the corresponding particle images averaged over 16 successive images (640 ms) for ${a}_{f}=0.47$ g (

**a**) and ${a}_{f}=0.70$ g (

**b**). Every second arrow shown. Air–water surface at $h=30$ mm. Color online.

**Figure 3.**Visualization of tracer particles at different horizontal planes at a forcing acceleration of ${a}_{f}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.47$ g. The particle streaks are obtained by averaging subsequent raw particle images over time span $\Delta t$. The enlargement panels further show the overlaid corresponding instantaneous velocity fields from the central frame. Note the presence of positive divergence in panel (

**c**) as velocity vectors point outwards from the source point, indicating the presence of a downward pushing jet. Corresponding presentation for ${a}_{f}=0.70$ g is in Supplementary Figure S4. Color online.

**Figure 4.**Profiles of absolute values of velocity components $\langle |u|\rangle $, $\langle |v|\rangle $, and $\langle |w|\rangle $ against the distance from the container bottom h for vertical and horizontal PIV measurements (empty and filled markers, respectively). Forcing accelerations ${a}_{f}=0.47$ g (

**a**) and ${a}_{f}=0.70$ g (

**b**). Values are averaged across available time-steps, measurement runs (4 or 6, see Appendix A) and PIV grid points. Color online.

**Figure 5.**Average ratio of flow kinetic energy in the vertical direction to the total flow kinetic energy along the height from the container bottom h for forcing accelerations ${a}_{f}=0.47$ g and $0.70$ g (blue squares and red circles). The horizontal dashed black line shows the 3D isotropic turbulence case at a value of 1/3. Color online.

**Figure 6.**Wavenumber spectrum of flow kinetic energy, net energy fluxes and net enstrophy fluxes at different horizontal planes for forcing accelerations ${a}_{f}=0.47$ g and $0.70$ g (blue and red). Results are averaged for all available time steps and measurement runs (4 or 6, Appendix A). Color online.

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**MDPI and ACS Style**

Colombi, R.; Rohde, N.; Schlüter, M.; von Kameke, A.
Coexistence of Inverse and Direct Energy Cascades in Faraday Waves. *Fluids* **2022**, *7*, 148.
https://doi.org/10.3390/fluids7050148

**AMA Style**

Colombi R, Rohde N, Schlüter M, von Kameke A.
Coexistence of Inverse and Direct Energy Cascades in Faraday Waves. *Fluids*. 2022; 7(5):148.
https://doi.org/10.3390/fluids7050148

**Chicago/Turabian Style**

Colombi, Raffaele, Niclas Rohde, Michael Schlüter, and Alexandra von Kameke.
2022. "Coexistence of Inverse and Direct Energy Cascades in Faraday Waves" *Fluids* 7, no. 5: 148.
https://doi.org/10.3390/fluids7050148