Abstract
We explore the initial perturbations that form on a liquid free surface as a result of the submersion of a circular cylinder beneath the surface, a scenario that arises in a number of diverse applications. The behaviour of the free surface is determined by transforming the equations of motion of the system via the Wehausen scheme, to variables for the free surface. A small-time series expansion is utilized to construct a recursive scheme that can be implemented numerically, and the time frame over which this approximation is valid is analysed. The resulting numerical model allows one to extend the results in the literature to study arbitrary cylinder sizes, including those where the cylinder is close to the free surface, and arbitrary cylinder motions. Of particular interest in this study was identifying the conditions under which strong jets would appear, and those were the free surface exhibited gravity waves. The formation of a central jet is found to be related to the growth of secondary, nonlinear waves, which rapidly merge as the obstacle is submerged. Classification maps are presented as a function of obstacle size and submersion speed, to identify the conditions which lead to jetting. Furthermore, the acceleration profile of the cylinder is shown to significantly affect the conditions under which jets form, which we argue is due to the rate at which energy is injected into the system.
Highlights
Surface disturbances can be beneficial, for example in wave energy devices and offshore structures (Siddorn & Eatock Taylor 2008; McCauley et al 2018; Orszaghova et al 2019), or harmful, such as for imploding liquid liners as encountered in magnetized target fusion (Huneault et al 2019) or for waves effects on ship motion (Tuck 1965)
Prior studies of initial perturbations on the free surface due to the motion of submerged obstacles have been studied via the use of small-time series (Tyvand & Miloh 1995) and reducing the equations of motion to the free surface (Kostikov & Makarenko 2018)
These analytical results are limited to constant velocity and acceleration submersion profiles, and to low-order approximations in time
Summary
Surface disturbances can be beneficial, for example in wave energy devices and offshore structures (Siddorn & Eatock Taylor 2008; McCauley et al 2018; Orszaghova et al 2019), or harmful, such as for imploding liquid liners as encountered in magnetized target fusion (Huneault et al 2019) or for waves effects on ship motion (Tuck 1965). 574), who constructed a recursive scheme based on the alternating application of the MilneThomson operator and the Kochin operator† on the free stream potential F = U z to obtain a sequence that converges for large depths of cylinder submersion and satisfies the linearised free-surface condition These successive approaches to solve the linear problem are, in general, less important than the contribution of surface nonlinearities, which were not considered in the analysis of Havelock (1927, 1936) and Wehausen & Laitone (1960), but addressed by Tuck (1965). If jetting is expected to occur, studying nonlinear features observed on the surface provides a physical foundation to the process of jet formation, and serves as initial conditions for more complete analytical and numerical models found in the literature (see for example the works of Eggers & Dupont (1994), Eggers & Villermaux (2008) and Howell (2015))
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