Three-dimensionality of shallow flows

Abstract

A large body of scientific effort has been devoted to study the dynamics of homogeneous, non-rotating shallow fluid layers in the laboratory [36]. These experiments are believed to be relevant in order to gain insight into some classes of geophysical flows like coastal and river flows, where the density stratification and rotation of the Earth does not play an important role. Experiments on shallow flows are also performed searching for similarities between real turbulent laboratory flows and idealized two-dimensional (2D) turbulence [81]. Finally, another type of research involves studies of vortex dynamics in shallow layers, where coherent structures like a dipolar vortex are created and measured [75]. These coherent vortices are of a special interest to geophysical flows since they are very frequently observed in such large-scale flow systems. A large ratio between the typical horizontal length scale (e.g. the size of the flow domain) to the typical vertical length scale (e.g. the fluid layer depth) is generally believed to ensure quasi-two-dimensional (Q2D) flow properties of shallow fluid layers. [36, 81, 75, 76, 9, 27]. If this ratio is large enough one usually refers to geometrical confinement of the flow. In this case Q2D dynamics practically means that geometrical confinement is believed to inhibit vertical velocities relative to horizontal motions and vertical profiles of the horizontal velocities become simply Poiseuille-like [25]. Following this reasoning the majority of the work done in the past focused on the planar properties of motion, usually at the free surface of a shallow fluid layer. Yet relatively little is known about the vertical flow structures inside a shallow fluid layer and their interactions with the horizontal motion of the fluid, including the free surface flows. As a response to that state-of-the-art, the research reported in this PhD thesis concerns the three-dimensional (3D) structures present in shallowflows. The main goal is to elucidate the importance or unimportance of 3D effects in a few shallow-flow configurations. Three typical flow structures are examined: a dipolar vortex colliding with a no-slip vertical wall (Chapter 3), a dipole moving over an inclined bottom (Chapter 4) and decaying turbulence (Chapter 5) in a shallow fluid layer. A dipolar vortex is chosen since it is a frequently occurring flow structure in real geophysical flows or in purely 2D turbulence [50]. Investigations on Q2D turbulence largely disregard the effects of lateral sidewalls and a simple and well-defined problem of dipolewall collision is believed to shed some light on this more complicated turbulence problem. A dipole approaching a no-slip inclined bottom corresponds with the realistic and common flow phenomenon of a vortex in coastal areas. Finally, the 3D flow structures in decaying turbulence provide new insights into this classic experiment with assumed quasi-two-dimensionality. The flows discussed in this thesis are driven by electromagnetic forcing. The flow phenomena are studied both by laboratory experiments and by 3D numerical simulations. Wherever the three components of the velocity vector in a single horizontal plane are measured in the laboratory the technique of Stereoscopic Particle Image Velocimetry (SPIV)was employed. In addition to these measurements, 3D numerical simulations have been performed with a realistic model for the Lorentz force. These experimental and numerical aspects of the three-dimensionality inside shallow fluid layers are summarized in the Chapter 6.