Note on the Application of Transient Wave Packets for Wave–Ice Interaction Experiments
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water Article Note on the Application of Transient Wave Packets for Wave–Ice Interaction Experiments Marco Klein 1,* , Moritz Hartmann 2 and Franz von Bock und Polach 2,3 1 Offshore Dynamics Group, Hamburg University of Technology, 21073 Hamburg, Germany 2 Ship Structural Design and Analysis, Hamburg University of Technology, 21073 Hamburg, Germany; [email protected] (M.H.); [email protected] (F.v.B.u.P.) 3 Institut für Meereskunde, Universität Hamburg, 20146 Hamburg, Germany * Correspondence: [email protected] Abstract: This paper presents the transient wave packet (TWP) technique as an efficient method for wave–ice interaction experiments. TWPs are deterministic wave groups, where both the amplitude spectrum and the associated phases are tailor-made and manipulated, being well established for efficient wave–structure interaction experiments. One major benefit of TWPs is the possibility to determine the response amplitude operator (RAO) of a structure in a single test run compared to the classical approach by investigating regular waves of different wave lengths. Thus, applying TWPs for wave–ice interaction offers the determination of the RAO of the ice at specific locations. In this context, the determination of RAO means that the ice characteristics in terms of wave damping over a wide frequency range are obtained. Besides this, the wave dispersion of the underlying wave components of the TWP can be additionally investigated between the specific locations with the same single test run. For the purpose of this study, experiments in an ice tank, capable of generating tailored waves, were performed with a solid ice sheet. Besides the generation of one TWP, regular waves of different wave lengths were generated as a reference to validate the TWP results for specific Citation: Klein, M.; Hartmann, M.; wave periods. It is shown that the TWP technique is not only applicable for wave–ice interaction von Bock und Polach, F. Note on the investigations, but is also an efficient alternative to investigations with regular waves. Application of Transient Wave Packets for Wave–Ice Interaction Keywords: wave–ice interaction; wave–ice experiments; transient wave packets; wave damping; Experiments. Water 2021, 13, 1699. wave dispersion https://doi.org/10.3390/w13121699 Academic Editor: Luca Martinelli 1. Introduction Received: 14 May 2021 The interaction of ice and waves is a very complex process involving ice mechanics Accepted: 16 June 2021 Published: 19 June 2021 and wave theory. The influencing parameters are the ice type, size, thickness, distribution and ice covered area as well as mechanical properties of the ice. The ice field affects Publisher’s Note: MDPI stays neutral the incoming waves and the waves have a strong influence on the ice field. The waves with regard to jurisdictional claims in displace and may break the ice while the ice causes a change in wave properties due to published maps and institutional affil- its flexural deformation as well as attenuation of waves due to scattering, damping and iations. frictional effects. The effect of ice on wave propagation is related to the dispersion of the waves and can be divided into a direct effect on the wave number and the attenuation of waves due to scattering and dissipation. The presence of the ice, and therewith a changed boundary condition, alters the wave dispersion immediately compared to the open water solution, Copyright: © 2020 by the authors. Licensee MDPI, Basel, Switzerland. affecting wave properties in terms of wave length, group velocity and amplitude. In This article is an open access article addition, the attenuation of the wave energy during the propagation under ice causes a distributed under the terms and decrease in the wave amplitude. conditions of the Creative Commons Most experimental studies related to wave–ice interaction have investigated the atten- Attribution (CC BY) license (https:// uation of waves, as field measurements are easier to obtain (e.g., [1–4]). The attenuation creativecommons.org/licenses/by/ can be described as an exponential decay of the wave energy when passing the ice field 4.0/). due to wave scattering and dissipation [5,6]. The attenuation depends on the ice state and Water 2021, 13, 1699. https://doi.org/10.3390/w13121699 https://www.mdpi.com/journal/water Water 2021, 13, 1699 2 of 21 mechanical properties of the ice as well as the wave length. Short waves within a wave spectrum are scattered and damped quickly, whereby the longer waves penetrate deep into the ice field [7,8], where they can travel long distances, dissipating energy only slowly (e.g., [9,10]). However, major uncertainties exist regarding the attenuation coefficient and its dependency on ice state and temperature [5,6]. The effect on the dispersion of waves under ice is complex and also strongly depends on the ice state and temperature. Field observations have shown that the presence of ice can lead to shorter or longer waves compared to waves in open water (e.g., [9,11–13]). The results indicate that ice type, ice thickness and the incoming wave period affect the dispersion in different ways. Different dispersion models have been developed with increasing complexity to account for the relevant physical mechanisms—from added mass and elastic plate models to viscous-layer models. A detailed study on different dispersion models and their effects on wave length as well as the area of validity of the different models can be found in Collins III et al. [14]. Measurements of wave dispersion in ice—field measurements as well as model tests in a more controllable environment—have indicated that a distinct conclusion on which dispersion model fits best is not possible. Moreover, due to the variability of the ice state, temperature and incoming wave field in terms of wave frequencies, all dispersion models appear to be suitable for certain conditions (e.g., [15–19]). Summing up, the attenuation and dispersion of waves under ice are two important interacting physical mechanisms and integral parts of the breaking up of the ice-covered sea [2,8]. Thus, exact knowledge of the physics of wave–ice interaction is indispensable for an accurate description of the dynamic processes within the marginal ice zone (MIZ). Therefore, modelling of the processes in both the long and short term is required such for questions related to climate change investigations or weather forecasts, including maintaining and predicting ship routes. Besides full-scale measurements, which give real-world insights, wave–ice investigations in the controlled environment of a test facility are essential for fundamental research. In this context, experiments exclusively focused on wave–ice interactions are a quite recent research field compared to the originally intended structure–ice investigations featuring the classical application area of ice tanks. Thus, the established model ice has been developed to physically test its interaction with structures on a small scale. Following Froude and Cauchy scaling laws, the model ice has particular mechanical properties, which are adjusted to scale the downward breaking of a ship in ice correctly (e.g., [20,21]). The adjustment is done by crystal size and added chemicals. The main objective here is the correct scaling of the global forces (resistance) on the ship, while accepting that some of the resistance components might not be scaled correctly. Generally, model ice has been developed over recent decades to overcome or reduce certain problems such as plasticity or ductility (e.g., [21–23]). Although model ice might represent the force level correctly, it has a significant amount of plasticity. The latter has been shown numerically for a different model ice type (fine-grained) by von Bock und Polach [24] and on the basis of measurement data for most existing model ice types [25]. Due to the plasticity of the model ice, the ice undergoes larger deformation to reach the critical failure stress than actual ice (sea ice, lake ice) would [26,27]. Consequently, the higher proportion of plasticity in model ice is likely to affect the energy dissipation of waves underneath the ice sheet which will introduce a scale effect in terms of higher damping. However, several wave–ice interaction experiments in the controlled environments of test facilities have been conducted in recent years, focusing either on regular waves or irregular sea states. The research questions are manifold, from fundamental research on theoretical wave–ice propagation models to ships and structures in waves. Most of the studies deal with fundamental research questions such as damping and attenuation, wave characteristics and dispersion as well as the consequences for the ice under various different ice conditions. These studies are normally conducted with regular waves, as they can be used to answer specific research questions in a clearly defined manner [16–18,28–36]. Those experiments comprise the generation of a set of regular waves to cover a wide Water 2021, 13, 1699 3 of 21 frequency range, so that a long test campaign may affect the ice characteristics. On the one hand, the cooling control system regulates the temperature with the target to keep ice properties constant during the experiments. However, ice properties are affected by various parameters, which vary over the time of the experiments, such as water temperature, ice thickness or entrapped chemicals and, consequently, a certain dynamic or variation of the ice properties is inevitable. On the other hand, the persistent bending of the ice due to the presence of the waves may affect the micro-structure and the mechanical behaviour of the ice by various causes, such as the growth of micro-cracks and increased plasticity up to the limit of fatigue strength. This strongly implies that the ice characteristics in terms of wave damping can also change during the course of the test campaign. In consequence, the scalability of experiments is affected by both inherent ice scale effects, such as excessive plasticity, and dynamic variations of ice properties, which can also cause unintended breakup and macro-crack formation due to fatigue.