# Cylinder wakes in shallow oscillatory flow: the coastal island wake problem

## TL;DR

This study investigates the physical processes of island wakes in shallow, oscillatory tidal flows through laboratory experiments, identifying key parameters and wake classes, and developing scaling laws for real-world applications.

## Contribution

It introduces a new parameter S that accounts for bottom friction effects confined to a Stokes boundary layer, and classifies wake forms based on this parameter.

## Key findings

- Three classes of wake form identified: Steady Bubble, Unsteady Bubble, Vortex Shedding.
- Wake form transitions depend on the magnitude and evolution of the wake return flow.
- Scaling laws developed for upscaling laboratory results to natural island wakes.

## Abstract

Topographic complexity on continental shelves is the catalyst that transforms the barotropic tide into the secondary and residual circulations that dominate vertical and cross-shelf mixing processes. Island wakes are one such example that are observed to significantly influence the transport and distribution of biological and physical scalars. Despite the importance of island wakes, to date, no sufficient, mechanistic description of the physical processes governing their development exists for the general case of unsteady tidal forcing. Controlled laboratory experiments are necessary for the understanding of this complex flow phenomena. Three-dimensional velocity field measurements of cylinder wakes in shallow-water, oscillatory flow are conducted across a parameter space that is typical of tidal flow around shallow islands. Previous studies investigated the wake form dependance on $KC=U_0T/D$, where $KC$ is the Keulegan-Carpenter number, $D$ is the island diameter, $U_0$ the tidal velocity amplitude and $T$ the tidal period, and the stability parameter $S=c_fD/h$ where $h$ is the water depth and $c_f$ is the bottom boundary friction coefficient. In this study we demonstrate that when the influence of bottom friction is confined to a Stokes boundary layer the parameter $S=\delta^+/KC$ where $\delta^+=\delta/h$ and $\delta=2\pi\sqrt{2\nu/\omega}$ is the wavelength of the Stokes bottom boundary layer. Three classes of wake form are observed for decreasing wake stability: \emph{(1) Steady Bubble} for $S\gtrsim 0.1$; \emph{(2) Unsteady Bubble} for $0.06\lesssim S \lesssim 0.1$; and \emph{(3) Vortex Shedding} for $S\lesssim 0.06$. Transitions in wake form and wake stability are shown to depend on the magnitude and temporal evolution of the wake return flow. Scaling laws are developed to allow upscaling of the laboratory results to island wakes. Vertical and lateral transport depend on th...