Session PJ: Waves II

Evolution of depressed and elevated tsunami waves

Tsunamis differ greatly between positive (elevated) and negative (depressed) waves. These differences are explained using a hydraulic model based on the conservation of impulse. Laboratory experiments of depression waves, conducted using a novel wave-maker, are compared with model predictions. Over a sloping beach, they have a nearly constant V-shaped depression trailed by a growing lambda- shaped positive wave. The shoreline recedes (draw-down) over a significant distance, caused by shoreward water being drawn toward the V-shaped depression. When the lambda- shaped positive wave breaks (spills), an energetic hydraulic bore develops and moves up the beach. The model leads to general formulae for wave slopes, draw-down and run-up. The run up of negative waves can be larger or smaller than positive waves, depending on the wave amplitude and beach parameters. The predictions are consistent with videos and photographs taken during the 2004 Sumatra tsunami. Using the amplitude data in the new tsunami warning systems, the properties of tsunamis on beaches could be estimated in real time using the present work, thus improving emergency response strategies.   

Harmonic Generation of Internal Waves Reflected from a Slope

The reflection of oceanic internal waves from bottom topography can generate harmonics and cause mixing that could be important in sustaining the meridional overturning ciruclation. Internal wave reflection is often treated as a linear or a weakly nonlinear inviscid problem\footnote[1]{T. Dauxois and W.R. Young, J. Fluid Mech. {\bf390}, 271-295 (1999)}. Under these assumptions, and for a linearly stratified fluid, Thorpe\footnote[2]{S. A. Thorpe, J. Fluid Mech., {\bf178}, 279-302 (1987)} and Tabaei et al.\footnote[3]{A. Tabaei, T. R. Akylas and K. G. Lamb, J. Fluid Mech. {\bf526}, 217-243 (2005)} derived predictions for the boundary angle where second harmonic generation should be most intense. We have conducted laboratory experiments and two-dimensional numerical simulations of the Navier-Stokes equation in the Boussinesq approximation to test these predictions. The results from experiments and simulations are in agreement but differ from both theories, except for very low intensity incoming waves. However, we obtain an empirical geometric relationship between the wave beam and boundary angles that predicts a condition for maximal second harmonic generation, and that agrees with the results from our experiment and simulation.   

Deep Ocean Wave Cancellation Using a Cycloidal Turbine

We investigate the use of a cycloidal turbine for deep ocean wave termination for the purpose of converting wave energy to shaft power. Cycloidal turbines consist of one or more hydrofoils that rotate around a central shaft and can be pitched during rotation. In the present investigation, the shaft is parallel to the wave crests, and the turbine operates in sync with the wave frequency by means of feedback control. The approach differs from traditional approaches in that it is a lift based system and therefore has the potential to be more efficient than existing drag based converters. It also allows for feathering of the blades in order to survive storms. We present two-dimensional inviscid results of potential flow simulations modeling the turbine blades as single point vortices of constant circulation rotating under a linearized free water surface. With suitable parameter choices for the turbine radius, blade number, submersion depth and airfoil circulation up to 97{\%} of the incoming deep ocean Airy wave energy can be converted to shaft power. For a typical North Atlantic deep ocean wave this corresponds to 100 kW of power per meter of wave crest. The remaining energy is lost to harmonic waves travelling both in the up- and down wave directions.   

Synchronous Sloshing in a Free Container

A standing wave in a container partially filled with liquid and free to move along one horizontal axis is analyzed. Interaction of the sloshing liquid with the container drives the acceleration of the vessel, which oscillates back-and-forth, out-of-phase with the liquid oscillations. Linearized shallow water theory is employed to obtain lowest-mode frequencies for rectangular and cylindrical containers. Validity of the results require $h/\lambda <<1$ where $h$ is the fluid depth and $\lambda $ is the sloshing wavelength, and $\eta _o /h<<1$ where $\eta _o $ is the wave amplitude. Experiments using containers with water, supported on a low-friction cart constrained to move in one dimension, reveal excellent agreement with theory up to a certain liquid depth corresponding to the shallow-water limit. Beyond that critical depth, the observed frequencies of oscillation are lower than linear predictions and thus full potential theory is required.   

Numerical and Experimental Study of the Dynamics of Imploding Hydraulic Jumps

The dynamics of imploding hydraulic jumps is investigated in this study. Experimental and numerical studies are performed to identify the critical conditions at which circular shallow water waves can be produced and amplify as they propagate toward the center without wave breaking. These conditions enable water waves to behave analogously to gaseous shock waves through the hydraulic analogy. The stability of the imploding jumps is also analyzed by introducing obstacles in the path of the implosion. Experimentally, a gate-type water table is constructed and the creation of a circular converging hydraulic jump is achieved by retracting the gate which separate two volumes of water by mean of three pneumatic pistons. A CCD camera is used to visualize the dynamics of the implosion. The acquired images are processed on Matlab using an image processing toolbox based algorithm which detects the shape and trajectory of the imploding wave. To compare the characteristics of the imploding jump and the mechanism of wave breaking, numerical simulations using Volume of Fluid (VOF) and Smoothed Particles Hydrodynamics (SPH) methods are performed. The experimental and numerical results are compared with the Chester-Chisnell-Whitham (CCW) approximate solution of the shallow water wave equations.   

Similarity solution for strong exploding shock waves in water

In single bubble sonoluminescence, strong bubble oscillations often lead to the generation of spherical shock waves which expand outward in the surrounding media. These shock waves offer valuable information to help diagnose the cavitation events at the center. In this work, the similarity behavior of these waves is investigated using analyses and simulations. A fitted analytical equation of state (EOS) is first extracted from a tabular EOS for water (LANL), and then incorporated into the Euler equations. A similarity solution for the flow variables behind a shock wave is derived following the theory by Taylor (1950). Numerical simulations are performed using the original tabular EOS, to compare with and validate the analytical solution. The result indicates that the shock wave propagation can be divided into three regimes. 1. A strong-shock regime where the wave-front location scales to the 2/5 power of time, and the similarity solution well-describes the flow dynamics. 2. A sonic regime where the shock wave weakens and propagates with the local speed of sound. 3. A transition regime connecting the two above. Based on these results, a time-of-flight theory is developed, from which the energy of the initial shock wave can be estimated given the arrival time of the shock wave-front at a defined detector location.   

Do Waveless Ships Exist?

Consider two-dimensional ideal and low-speed flow past a ship modeled as a semi-infinite body with constant draft. In the 1970s, on the basis of numerical evidence, it was conjectured that ships with a single front face will always generate a wake. Later in the 1980s, seemingly waveless ships with bulbous profiles were discovered. And finally, conflicting evidence in the 1990s suggested that the waves were in fact present, but simply too small to be recognized. In this talk, we will show how recent techniques in exponential asymptotics can be used in order to study the ship-wave problem. In particular, we will show how the formation of waves near a ship are a necessary consequence of singularities in ship's geometry, such as those corresponding to sharp corners or stagnation points. Finally, we will show how the theory can be applied in order to prove that certain ship profiles will or will not produce a wake in the low-speed limit.