Analysis of the Kelvin-Helmholtz instability in seagrass-oscillatory flow interaction (2019)
The aquatic vegetation is ubiquitous worldwide, from low temperature areas to tropical shallow coastal zones (Van Der Heide et al., 2007). These ecosystems provide a productive life cyle being the habitat for many marine animal species (Marion et al., 2014). At the coastal region, the aquatic vegetation (seagrass) attenuates the currents, dissipates the wave height and stabilises the coastlines (Maxwell et al., 2017; Pinsky et al., 2013) being considered as a soft system to reduce the risk of ooding and erosion under sea level rise and extreme wave events (Ondiviela et al., 2014). Based on the above described services, estimations of economical annual values provided by the aquatic vegetations is over $10 trillion (Nepf, 2013). Control _eld studies such as those by Schanz et al. (2002); Schanz and Asmus (2003) demonstrate the interdependence between the hydrodynamic and biological processes, since there is a cascading e_ect in which for a speci_c ow conditions, di_erent species can live and proliferate within the seagrass meadows or be washed away, depending on the energetic wave conditions and seagrass density. Regarding the interaction between the submerged seagrass and the surrounding ow, it is well known that the presence of the seagrass canopies attenuate the momentum by the work done on the ow by the stems (Finnigan, 2000). The e_ects in the velocity _eld can be di_erentiated and thus studied in terms of scales; 1) processes with spatial scales of the order of the stem diameter or spacing between stems; and 2) processes with scales of the order of the drag length scale. The turbulent structures at the scale of the stems are called wake scales and are produced by the shadow zone downstream the stems (Nepf, 2012; Zhang et al., 2018). Turbulent processes at the drag length scale are governed by the density of the canopy and the ow dynamics (Nepf, 2012). These processes modulate the water renewal between the water inside and above the canopy and the amount of suspended sediments along the water column (Luhar and Nepf, 2013). The turbulent processes at the drag length scale can be analysed as a plane mixing layer by two co-owing streams that present a shear layer at the top of the canopy (Raupach et al., 1996). This shear-layer-ow is characterized by an inexion point in the velocity pro_le (two water bodies moving at di_erent velocities), responsable for the vertical mass exchange at the top of the canopy (Ghisalberti and Nepf, 2009). The shear layer facilitates the generation of Kelvin-Helmholtz instability type vortex (Ghisalberti and Nepf, 2002). The Kelvin-Helmholtz type vortex has been widely studied in steady ows (Raupach et al., 1996; Finnigan, 2000; Ghisalberti and Nepf, 2002; Nepf, 2012; Mandel et al., 2017), characterizing its e_ect on the seagrass movement, the Reynolds stresses, the sediment distribution, the vertical mixing and the free surface. However, the formation and e_ect of Kelvin-Helmholtz type vortices in the wave oscillatory ow is still far to be completly understood. Indeed, it is still not clear which are the dominant terms in the Navier-Stokes equations for the oscillatory-seagrass-ow interaction. Ghisalberti and Schlosser (2013) reported some ǹecessary" conditions in the ow in order to produce Kelvin-Helmholtz instabilities; Abdolahpour et al. (2017) analysed a steady current released by the presence of the shear layer and its relation with the shear layer magnitude and Abdolahpour et al. (2018) used the seagrass-steady-ow interaction formulation of Ghisalberti and Nepf (2002) to estimate the Kelvin-Helmholtz frequency range in oscillatory ows. The evolution of vortices downstream submerged structures in oscillatory dominant ows is assumed to be dissipated by the viscosity and the e_ects on the wave breaking process have not been yet analysed. Indeed, a theoretical model to solve the Kelvin- Helmholtz instability modes as a function of the free surface and a general characterization of the turbulent spectra is an open question that will provide new insights in order to improve models and simulations of relevant hydrodynamic processes at coastal scale. The aim of this Thesis is to understand the relation between the free surface frequency and the Kelvin-Helmtholtz instability modes in seagrass-oscillatory-ow interaction. For this, I will _rst analyze the e_ects of a vortex by an isolated submerged stem interacting with a surface wave. Then, I develop an analytical model to determine the dominant terms in the momentum equation in seagrass-oscillatory-ow interaction. Finally I close the scienti_c question by solving the Kelvin-Helmholtz instability modes in seagrass-oscillatory ow interaction as a function of the free surface wave applying the Piecewise method to a simpli_ed velocity pro_le. This thesis is structured as follows. Chapter 2 analyses the e_ects of backwards wave breaking process induced by a strong transport of mass in a vortex produced by an isolated submerged stem. In chapter 3, a simpli_ed seagrass-oscillatory-ow model is developed by dimensional analysis of the Navier-Stokes equation. Here, some reference variables are de_ned according to the free surface wave parameters . The vali dation of the simpli_ed model is performed against experimental data from a ap type wavemaker system and a random seagrass distribution. Finally, chapter 4 presents a theoretical model for the Kelvin-Helmholtz instability modes as a function of the incoming free surface wave. The model applies the piecewise linear method to the Rayleigh's equation in an ideliazed vertical velocity pro_le. It is important to remark that this thesis is composed by three papers: Chapter 2 has been published in the Ocean Engineering Journal, Chapter 3 is in _nal revisión for the Experiments in Fluids Journal and Chapter 4 is under review for the Journal of Fluid Mechanics.