3D structures and dispersion in shallow fluid layers

Abstract

Many experiments have been performed in electromagnetically driven shallow fluid layers to study two-dimensional (2D) turbulence. The shallowness of the fluid layer is commonly assumed to ensure 2D dynamics. However, contrary to the theory and numerical simulations on 2D turbulence, the experimental realisations are never purely 2D. For example, laboratory setups are bounded by a no-slip bottom and stress-free surface, which implies a vertical gradient. Surprisingly, deviations from two-dimensionality in such shallow fluid setups have hardly received any attention. The aim of this thesis was to investigate the influence of boundary and initial condition on the development of three-dimensional (3D) motion in-side shallow fluid layers. For this purpose, a dipolar vortex was considered as the canonical coherent structure in the shallow fluid layer. The dipolar vortex is one of the most elementary vortex structures in 2D turbulence. Such a vortex structure can be conveniently created by electromagnetic forcing. The first two configurations that have been investigated are the dipolar vortex in a shallow one- and two-fluid layer situation. The latter (stably stratified) two-layer setup was assumed to be an improvement with respect to the single layer configuration. Finally, the following shallow-fluid layer experiment has been considered: a periodically forced linear array of vortices near a lateral wall. All measurements have been performed with Stereoscopic Particle Image Velocimetry, providing the three-component velocity field on a horizontal plane inside the fluid layer. Furthermore, all these experiments were complemented by 3D numerical simulations of the Navier-Stokes equation. Based on the experimental and numerical results, the necessary condition for development of 3D motion in such shallow fluids was determined to be a vertical variation of the horizontal velocity field. Inside the two individual vortex cores an oscillating up- and downward motion was seen, as well as a spanwise vortex in front of the dipole. Free-surface deformations were proven to be of minor importance in generating 3D motions. Furthermore, friction exerted by the no-slip bottom and the flow initialisations were shown not to be primary actors in generating the observed complex and persistent 3D motions. Surprisingly, the 3D flow evolution of the dipole in the two-layer configuration evolved in a similar way as already seen in the single-layer setup. Contrary to statements in literature, the so-called frontal circulation was also observed in the two-layer configuration. The emergence of this structure has a different origin, however, it resulted from baroclinic vorticity production at the internal interface in stead of a propagating motion over the solid bottom of the single-layer dipole. Based on the comparison of the ratio of kinetic energy (contained in the vertical and horizontal flow components) between the single- and two-layer fluid, the two-layer fluid is not an improvement over the single-layer configuration. For the linear array of vortices, the influence of 3D motion and the presence of a lateral wall on the passive tracer transport was investigated. It was observed that particles released at the free surface form long filament-like structures related to the surface flow being convergent, in contrast with the purely 2D numerical simulations where the velocity field is by defini- tion divergence-free and a more homogeneous particle distribution remained throughout the time evolution. Particles released at mid-depth of the fluid illustrated the efficient vertical mixing: already after one forcing period the particles were almost dispersed homogeneously over the full depth of the fluid layer. With the presence of a lateral wall this rapid vertical dispersion is even further enhanced in the near-wall region. In summary, this thesis reveals the intrinsic three-dimensional flow behaviour of shallow fluid layers. Furthermore, experiments with a linear array of vortices illustrate the influence of the three-dimensional flow field and lateral walls on the dispersion of passive tracers. All the experimental and numerical results indicate that the interpretation of such experiments as two-dimensional realisations should be done with caution.