Year: 2017
Type: article
Abstract: We investigate a model of thin layer turbulence that follows the evolution of the two-dimensional motions $\boldsymbol{u}_{2D}(x,y)$ along the horizontal directions $(x,y)$ coupled to a single Fourier mode along the vertical direction ( $z$ ) of the form $\boldsymbol{u}_{q}(x,y,z)=[v_{x}(x,y)\sin (qz),v_{y}(x,y)\sin (qz),v_{z}(x,y)\cos (qz)]$ , reducing thus the system to two coupled, two-dimensional equations. The model, despite its simplicity and ad hoc construction, displays a rich behaviour. Its reduced dimensionality allows a thorough investigation of the transition from a forward to an inverse cascade of energy as the thickness of the layer $H=\unicode[STIX]{x03C0}/q$ is varied. Starting from a thick layer and reducing its thickness it is shown that two critical heights are met: (i) one for which the forward unidirectional cascade (similar to three-dimensional turbulence) transitions to a bidirectional cascade transferring energy to both small and large scales and (ii) one for which the bidirectional cascade transitions to a unidirectional inverse cascade when the layer becomes very thin (similar to two-dimensional turbulence). The two critical heights are shown to have different properties close to criticality that we are able to analyse with numerical simulations for a wide range of Reynolds numbers and aspect ratios.
Source: Journal of Fluid Mechanics
DOI: 10.1017/jfm.2017.293
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2581360816