Faraday waves occur on a fluid being subject to vertical shaking. Although it is well known that form and shape of the wave pattern depend on driving amplitude and frequency, only recent studies discovered the existence of a horizontal velocity field at the surface, called Faraday flow. This flow exhibits attributes of two-dimensional turbulence and is replicated in this study. Despite the increasing attention towards the inverse energy flux in the Faraday flow and other not strictly two-dimensional (2D) systems, little is known about the velocity fields developing beneath the fluid surface. In this study, planar velocity fields are measured by means of particle image velocimetry with high spatio-temporal resolution on the water surface and at different depths below it. A sudden drop in velocity and turbulent kinetic energy is observed at half a Faraday wavelength below the surface revealing that the surface flow is the main source of turbulent fluid motion. The flow structures below the surface comprise much larger spatial scales than those on the surface leading to very long-tailed temporal and spatial velocity (auto-) correlation functions. The three-dimensionality of the flow is estimated by the compressibility, which increases strongly with depth while the divergence changes its appearance from intermittent and single events to a large scale pattern resembling 2D cut-planes of convection rolls. Our findings demonstrate that the overall fluid flow beneath the surface is highly three-dimensional and that an inverse cascade and aspects of a confined 2D turbulence can coexist with a three-dimensional flow. Graphic abstract

In nature, turbulent flows exist that are neither simply 2D nor 3D but are forced towards one state or the other by boundary conditions such as varying stratification. Here, we report the first evidence of the co-existence of an inverse and a direct energy cascade in an experimental flow driven by Faraday waves in water. We find that an inverse energy cascade at the fluid surface and a direct energy cascade in the 3D bulk flow underneath co-exist. We base our analysis on temporally and spatially well-resolved velocity fields obtained by particle image velocimetry measurements at planes parallel and perpendicular to the water surface. The findings also provide strong evidence that the intense turbulent 2D surface flow drives the 3D bulk flow through sporadic vertical jets as a source of momentum to the bulk liquid.

Faraday waves form on the surface of a fluid which is subject to vertical forcing, and are researched in a large range of applications. Some examples are the formation of ordered wave patterns and the controlled walking or orbiting of droplets (Couder et al. (2005); Saylor and Kinard (2005)). Moreover, recent studies discovered the existence of a horizontal velocity field at  the fluid surface, called Faraday flow, which was shown to exhibit an inverse energy cascade and thus properties of two-dimensional turbulence (von Kameke et al., 2011, 2013; Francois et al., 2013). Additionally, three-dimensionality effects have been part of recent investigations in quasi-2D flows (both electromagnetically-driven (Kelley and Ouellette, 2011; Martell et al., 2019) or produced by parametrically-excited waves (Francois et al., 2014; Xia and Francois, 2017)). Furthermore, the occurrence of an inverse cascade in thick layers is also subject of current studies on the coexistence of 2D and 3D turbulence (Biferale et al., 2012; Kokot et al., 2017; Biferale et al., 2017). By performing 2D PIV measurements at horizontal planes beneath the Faraday waves, we recently showed that pronounced three dimensional flows occur in the bulk, with much larger spatial and temporal scales than those on the surface (Colombi et al., 2021), when the system is not shallow in comparison to typical length scales of the surface flow (fluid thickness exceeding half the Faraday wavelength λF). This in turn reveals that an inverse energy cascade and aspects of a confined 2D turbulence can coexist with a three dimensional bulk flow. In this work, 2D PIV measurements of the velocity fields are carried out at a vertical cross-section xz-plane and at four distinct horizontal xy-planes at different depths in Faraday waves. The results reveal that small and fast vertical jets penetrate from the surface into the bulk with fast accelerating bursts and strong momentum transport in the z−direction. Furthermore, the fraction of flow kinetic energy in the vertical direction is found to peak inside a layer of approximately 10 mm (one Faraday wavelength) below the fluid surface.

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