## Abstract

Hydroelasticity brings together hydrodynamics and elastic theories. It is concerned with deformations of elastic bodies responding to hydrodynamic excitations, which themselves depend on elastic deformation. This Theme Issue is intended to identify and to outline mathematical problems of modern hydroelasticity and to review recent developments in this area, including physically and mathematically elaborated models and the techniques used in their analysis.

## 1. Introduction

The problems of hydroelasticity are important in biology, medicine and offshore and polar engineering, as well as in many industrial applications. In particular, these days, industry is going into polar regions and deeper offshore, and is facing new challenging problems in modelling and design. The mathematical theory of hydroelasticity and the mathematical methods of solving associated complex problems are not well established and are still under development. The mathematical difficulties are viewed as the major obstacle to adequate and successful treatment of problems in hydroelasticity in both academic research and applications.

Hydroelasticity is concerned with the deformations of elastic bodies responding to hydrodynamic excitations and simultaneously the modification of these excitations owing to the body deformation. The problems of hydroelasticity are coupled, which implies that the elastic deformations of the body depend on the hydrodynamic forces and vice versa. The problems of coupled fluid motion and elastic deformations of a body in contact with the fluid are difficult to study both theoretically and numerically. In particular, the difficulties are due to the fact that the hydrodynamic forces acting on the surface of an elastic body depend strongly on the accelerations of the surface displacements. As a result, the hydrodynamic loads cannot be treated simply as ‘external’ pre-calculated loads. Instead, they must be determined together with the body deformations.

If the elastic deformations are relatively small and the body does not intersect the fluid free surface, then we can successfully use several techniques: the method of normal modes (which requires computation of the so-called added-mass matrix), or the method of boundary-integral equations with a non-local inertia term. The treatment of hydroelasticity in the presence of a liquid free surface or with relatively large body deformations is much more challenging. If an elastic body is floating on the liquid free surface (as in the problem of hydroelastic waves in the presence of ice cover), or if the body is in partial contact with the liquid (as in water entry of an elastic body and in violent sloshing), there are additional difficulties associated with the presence of the free-surface contact lines or contact points. The position of a contact line can be greatly modified by the elastic distortions of the body surface. Such problems are still difficult to analyse and compute.

The nearby fluid motion and the associated elastic deformations of a structure can be rather complicated. In such cases, numerical computations of the hydroelastic problem are necessary. However, it is not enough to run a well-established hydrodynamic code and a code for the structural analysis simultaneously, with some exchange of data between the codes, and just iterate with the aim of converging on some computational result. More sophisticated and adequate coupling between hydrodynamic and structural codes is needed. Such proper coupling can be designed on the basis of the mathematical theory of hydroelasticity.

The mathematical theory of hydroelasticity should provide a rigorous foundation for modern industrial and engineering projects. For example, hydroelastic waves in the presence of ice cover in cold regions [1]; springing and whipping analysis in the design of ultra-large ships, which can be as long as 400 m [2]; the dynamic behaviour of deep-water risers for the oil industry, with modern risers being as long as 2–3 km [3]; the sloshing in liquefied natural gas tanks with a flexible insulation system [4]; and the dynamic behaviour of very large floating structures such as floating airports [5].

The most mathematically developed area of hydroelasticity is that devoted to hydroelastic waves in the presence of ice cover. Hydroelastic waves are relevant in those cold regions where water is frozen in winter and where deep ice cover can be transformed into roads or aircraft runways, and where air-cushioned vehicles are used to break the ice [6]. The safe use of these transportation links is of obvious importance. One attractive aspect of mathematical modelling of waves beneath an ice sheet is that many experimental results are available [7,8]. The basic features of the ice response can be explained by modelling the ice sheet as an elastic plate on top of a fluid [8]. Even so, the theory and modelling of these waves are not as well developed as for the classical problem of water waves. Most of the theoretical work on the subject uses linear models, which describe waves of small amplitude, and there are fewer studies for the nonlinear models necessary for large-amplitude waves [9–11].

Linear models have been found to be useful in studying wave interactions with very large floating structures, such as airports or ultra-large ships [12,13]. Nonlinear models of fluid motions, combined with linear models of complex structures, are under development, with some important results having been obtained [14]. Unsteady nonlinear problems of wave impact onto elastic plates were studied by Korobkin & Khabakhpasheva [15]. Unsteady problems of hydroelasticity with a time-varying wetted area are important in studies of stresses induced in the insulation system of liquid natural gas (LNG) tanks owing to violent sloshing at low filling levels [16,17]. The dynamics of long marine risers has been studied numerically by the finite element method (FEM) and analytically with the help of asymptotic methods [18]. The theory of ship hydroelasticity was developed by Bishop & Price [19]. A ship is treated as a flexible structure that moves bodily and distorts when it encounters waves. This behaviour is potentially dangerous and it must be predicted as a necessary part of ship design. Nonlinear interaction between modern ships and surface waves is a very important topic, because some modern ships are so long that they must be treated as flexible [20].

The scattering of acoustic waves and the dynamics and stability analysis (using linear and nonlinear models of fluid-loaded elastic plates) have been intensively studied in recent decades [21–26]. Fluid–structure interactions, from the viewpoint of partial differential equations have also been considered recently, and progress has been made in developing mathematical theories on the existence and uniqueness of solutions, and on the well-posedness of models that couple the Navier–Stokes equations with the equations of elasticity [27–30].

This Theme Issue is aimed at bridging the gap between the mathematical theory of hydroelasticity and its modern industrial and engineering applications. The main topics covered in this Theme Issue include fluid–structure interactions and slamming problems, the behaviour of very large ships in the oceans, the sloshing impact in LNG tanks, ocean waves/sea ice interactions, and waves generated by moving loads on ice sheets or elastic plates. The scattering of linear hydroelastic waves by structures and the computation of nonlinear waves in two and three dimensions are supplemented by rigorous mathematical theories of nonlinear hydroelastic waves. The nonlinearity is shown to play an important role in certain cases. Applications of hydroelasticity in biological and other flows are also presented. Some future directions of research are also discussed.

## 2. Organization of the Theme Issue

The Theme Issue starts with a review by Squire [31] of the state of the art of hydroelasticity theory in the context of polar ice. It presents the challenges of hydroelasticity in polar and subpolar seas and focuses on advances made in sea-ice modelling. A classical model and its application to ocean waves’ interaction with continuous ice cover in various situations (different plate thicknesses, cracks in the ice cover, etc.), and solitary floes are presented. The shortcomings of the existing models are discussed and the modelling of waves interacting with fragmented ice in the marginal ice zone are reviewed. The study is complemented by a description of field and laboratory experiments with artificial ‘ice floes’. The very difficult problem of how to incorporate these developments into an oceanic general circulation model for forecasting purposes is discussed.

The next paper is concerned with scattering of waves by fixed structures such as a cylinder. Brocklehurst *et al.* [32] analyse the linear three-dimensional problem of hydroelastic wave diffraction by a bottom-mounted circular cylinder. The fluid is of finite depth and is covered by an ice sheet of infinite extent, which is clamped to the cylinder surface. The ice sheet is modelled as a linear elastic plate, and the fluid flow is assumed potential. The two-dimensional incident wave is regular and has small amplitude. An analytical solution of the coupled problem of hydroelasticity is found by using the Weber integral transform. The ice deflection, the vertical and horizontal forces acting on the cylinder and the strain in the ice sheet caused by the incident wave are of main interest in this study.

Avital & Miloh [33] study the problem of sound scattering by free-surface piercing fluid-loaded cylindrical shells. A vertical cylindrical shell is clamped to a rigid base on the bed, in shallow water, and pierces the water surface. The hollow shell is composed of an isotropic and homogeneous material and is either empty or filled with a compressible fluid. This setting is of engineering interest across a variety of fields ranging from aeronautical to marine applications, such as modelling aircraft fuselages or offshore structures. The authors investigate the potential of using the shell’s flexibility to reduce sound scattering, often referred to as ‘making an object invisible to detection’. Linear acoustics and structural dynamics are used to model the sound scattering caused by an external incident sound wave. It is shown that zero sound scattering (acoustic invisibility) is theoretically attainable and can be achieved by a certain continuous distribution of oscillating pressure loads applied on the inner surface of the shell. Similarly, zero sound transmission into the shell’s contents can also be considered. The possibility of using a pre-determined discrete distribution of applied pressure loading is also discussed.

Favrie & Gavrilyuk [34] present mathematical and numerical models of nonlinear viscoplasticity. The hyperelasticity model is reviewed and then an extension is proposed to deal with plasticity. A macroscopic model, describing elastic–plastic solids, is derived in the special case when the internal specific energy has a separable form. It is assumed to be the sum of a ‘hydrodynamic’ part (which depends on the local density and entropy) and a shear part. In particular, the relaxation terms are constructed to be compatible with the von Mises yield criteria. It is shown that the deviatoric part of the stress tensor decays during plastic deformations, which is a Maxwell-type material behaviour. Numerical results, obtained with a high-order Godunov method, show the ability of this model to deal with real physical phenomena. A discussion of a diffusive interface model for the description of fluid–elastic solid interface concludes the paper.

Gudmestad [35] considers transient motions of oscillating systems, including terms proportional to the velocity of the structural motion. Damping limits the motions of an oscillator. The selection of formulations for damping is discussed. When the forcing of the dynamic system contains terms proportional to the velocity of motion of the oscillator, these effects add to the damping of oscillations. Should the total damping under certain conditions become apparently negative, the oscillations will grow until the damping again has become positive. Investigations into damping effects that apparently are negative and discussions on where apparent negative damping might appear in practical applications are of wide interest. Applications of this effect are presented for the motion of slender offshore structures, ice engineering and the ‘galloping’ of one degree of freedom systems in steady flow. Examples of the galloping phenomenon can be seen during the vibration of ice-coated power line cables or twin pipelines. This vibration changes the orientation of the structure, which oscillates with the fluid force. If this force tends to increase the vibration, then the structure will become unstable. It is this phenomenon which is called ‘galloping’.

Kapsenberg [36] reviews the state of the art in research on ship slamming. Here, the literature published on the problem of ship slamming in waves is reviewed from the point of view of someone working at a ship research institute. Such an institute is confronted with rather practical questions regarding the limitations of certain design parameters, such as the acceptable amount of bow flare angle for use at sea. The importance of these questions is illustrated by noting that actual slamming, or the presumed danger of slamming, is the main reason why ship operators reduce speed or change route. The review shows that such questions cannot yet be answered. Local problems of water impact are very complicated because of the importance of air inclusions, bubbles in the water, the compressibility of water and cavitation effects, according to the author. Only a computational method that properly includes all these effects will give an accurate answer. Model tests will not be capable of doing this, states Kapsenberg [36], because methods that extrapolate the results of model tests to full scale have not yet been developed. The problem of the global response of the ship to wave impacts is closer to being solved. A two-stage approach has been proposed, consisting of a computational fluid dynamics (CFD) method for individual impacts and an approximate method to be used for long-term simulations. However, to arrive at a realistic long-term distribution, one has to account for the seamanship of the captain; avoiding the worst conditions, adapting the ship’s speed and altering course all influence the actual extremes. Research on this topic has hardly begun.

On a related topic, Ten *et al.* [37] present a set of semi-analytical models of the hydroelastic sloshing impact in tanks of LNG vessels. The paper deals with the methods for evaluation of bending stresses in a complex structure of the tanks’ walls during the violent sloshing impacts inside the tanks of LNG carriers. The complexity of both the fluid flow and the structural behaviour of the containment system and the ship’s structure do not allow for a fully consistent direct approach, according to the present state of the art. Several simplifications are thus necessary in order to isolate the dominant physical aspects and to treat them properly. In this paper, the authors chose semi-analytical modelling for the hydrodynamic part and finite element modelling for the structural part. Depending on the impact type, different hydrodynamic models are proposed; the basic principles of hydroelastic coupling are clearly described and validated with respect to the accuracy and convergence of the numerical results.

In their paper, Plotnikov & Toland [38] derive rigorously a mathematical model for two-dimensional nonlinear hydroelastic waves. They use the special Cosserat theory of hyperelastic shells satisfying Kirchoff’s hypothesis and the irrotational flow theory to model the interaction between an elastic sheet and an infinite ocean beneath it. The mathematical model in Eulerian coordinates for two-dimensional nonlinear hydroelastic travelling waves with two wave-speed parameters, one for the sheet and another for the fluid, is derived from a general discussion of three-dimensional motions, involving a Eulerian description of the flow and a Lagrangian description of the elastic sheet. The liquid under the elastic sheet is considered inviscid and incompressible. The nonlinear equations derived describe the propagation of waves of finite amplitude on the surface of an ocean covered by an ice sheet, regarding the ice sheet as an elastic shell. Some particular features of the derived system are discussed, such as singular waves and jumps.

Vanden-Broeck & Părău [39] present computations of two-dimensional generalized solitary waves and of periodic waves travelling under an ice sheet. The flow is assumed to be potential. Weakly nonlinear solutions are derived and fully nonlinear solutions are calculated numerically. The authors use a model described in Forbes [40] and Părău & Dias [9], in which the ice sheet is modelled by a Kirchhoff–Love plate and a nonlinear term involving the plate curvature is used. Comparisons with the classical gravity–capillary waves problem are made and similarities and differences are discussed. Their results complement the weakly nonlinear results obtained by Părău & Dias [9], who derived a forced nonlinear Schrödinger equation for the ice sheet deflection, and more recent numerical results by Bonnefoy *et al.* [41], which are based on higher order spectral methods.

Părău & Vanden-Broeck [42] report in the next paper computations of three-dimensional waves under an ice sheet generated by a moving load. The ice sheet is modelled as a thin elastic plate and the linear elastic plate equations are used. This approximation is often appropriate [8]. The fluid underneath the plate is of infinite depth and is assumed to be incompressible and inviscid, while the flow is irrotational. Fully nonlinear boundary conditions are used for the interface between the ice sheet and the fluid. Solutions are computed using a desingularized boundary integral equation method. Different types of solutions are found depending on the velocity of the moving disturbance generating the flow. An artificial Rayleigh viscosity [43] is introduced in the dynamic boundary condition when the waves are non-symmetric. Davys *et al.* [44] considered waves generated by a concentrated point source and obtained wave crest patterns. More recently, Milinazzo *et al.* [45] have computed the linear deflection generated by a rectangular load. The present analysis complements the findings in these linear studies, by including the effect of the nonlinearity of the fluid flow.

Whittaker *et al.* [46] describe the energetics of flow through a rapidly oscillating tube with slowly varying amplitude. This study is motivated by the problem of self-excited oscillations in fluid-filled collapsible tubes. There is much interest in the study of flow-induced instabilities in flows through elastic-walled tubes, as such systems occur in a wide range of industrial and biological applications. Examples are pipe flutter and wheezing during forced expiration from the lungs. In this paper, the authors examine the flow structure and energy budget of flow through an elastic-walled tube by considering the case in which a background axial flow is perturbed by prescribed small-amplitude, high-frequency, long-wavelength oscillations of the tube wall, with a slowly growing or decaying amplitude. A multiple-scale analysis is used to show that at leading-order previous constant amplitude equations derived are recovered [47], with the effects of growth or decay entering only at first order. The effects on the flow structure and energy budget are also quantified. The paper ends with a discussion of how these new results should be understood and how they can be used to predict an instability that can lead to self-excited oscillations in collapsible tube systems.

Smith & Wilson [48] study the fluid–body interactions and in particular clashing, skimming and bouncing. The motivation of their paper comes from industrial, biological and sports applications. Solid–solid and solid–fluid impacts and bouncing are investigated in detail. For fluid–body interactions a theoretical study is presented in which the motions of the body and the fluid influence each other nonlinearly. The authors describe clashing as the solid–solid impact arising from fluid–body interaction in a channel, while skimming refers to another situation in which a body impacts obliquely upon a fluid surface without deep penetration into the liquid. Bouncing usually then follows in both situations. The main new contributions of the paper are the influences of thickness and camber. These lead to a different and more general form of clashing and hence bouncing. Numerical and analytical aspects are presented.

## 3. Future research directions

There is a strong demand nowadays for adequate models of hydroelasticity that include the relevant physical phenomena. Researchers are responding to this demand in a way which depends on their experience, research interests and skills. Problems of hydroelasticity arising in theoretical, computational and experimental studies are complex. It would be too naive to expect that CFD properly combined with FEM code can solve all problems. It is impossible, on the other hand, to deal with complex configurations by using only analytical or semi-analytical models. To make significant progress in hydroelasticity, we need *stronger interaction and better collaboration between computational, experimental and theoretical researchers and scientists*.

Several conferences on hydroelasticity have been held recently, such as the *5th International Conference on Hydroelasticity in Marine Technology* (Southampton, September 2009), the *International Workshop in Fluid and Elasticity* (Marseilles, June 2009) and the *Mathematical Challenges and Modelling of Hydroelasticity* (Edinburgh, June 2010, ICMS). These conferences and many others where papers on hydroelasticity were presented, as well as this Theme Issue, helped us to identify topics which have to be addressed in research to meet the current challenges in hydroelasticity.

We found that the theoretical part of research on hydroelasticity at present is less developed than the numerical and experimental studies. This inadequacy in the body of theory is the main obstacle to progress in this discipline, including progress in computations and experiments. We feel that *proper coupling between CFD and structural FEM codes* cannot be achieved without rigorous mathematical justification of this procedure. *CFD approaches should be extended with further relevant effects being identified and incorporated in models*. The important effects to model are *cavitation, ventilation, complex rheology, evaporation, melting/freezing of ice, crushing of ice and composite structures, large deformations of the structure or its elements, heterogeneity of the structure and poroelasticity*. The influence of these effects on fluid–structure interaction should be studied both theoretically and numerically, and supported by *experiments of increasing complexity*.

A rigorous mathematical theory of hydroelasticity is needed for both *linear and nonlinear processes*. There are already some important developments in nonlinear waves in infinite elastic plates lying over a fluid, but much less is known about *the behaviour of finite elastic plates in nonlinear waves*.

Several important problems of hydroelasticity are nonlinear in nature. The nonlinearity of fluid–structure interaction can be due to the nonlinearity of the fluid flow (as in the problem of nonlinear water waves interacting with a floating platform) or due to large deformations of the structure, or due to the nonlinear interaction of the linear flow with the linear behaviour of the structure, as in the problem of slamming, whipping, violent sloshing and others where *the contact region between the fluid and structure changes quickly* and *the loads are transient*.

Nowadays, the important questions in hydroelasticity to answer, and to investigate deeper, are concerned with motions of elastic shells and membranes in compressible fluids, and elastic microcapsules in collapsing vessels; with high-speed water transport and landing/take off of aircrafts on water, slamming and whipping of modern large ships and sloshing LNG tanks; with interaction of ice and offshore structures; and with scattering of water waves by random ice floes. The stability of hydroelastic interactions is still an important issue for analysis and application.

Many directions of research on hydroelasticity mentioned above require very sophisticated computations. However, simple mathematical models (extended, if necessary, to *more complex and realistic geometries*) are always needed to improve our understanding of complex physical systems, and to identify the most important physical effects.

## Acknowledgements

The authors thank ICMS for the help in organizing the workshop *Mathematical Challenges and Modelling of Hydroelasticity* in June 2010. Some of the discussions during this workshop helped us in shaping this Theme Issue.

## Footnotes

One contribution of 13 to a Theme Issue ‘The mathematical challenges and modelling of hydroelasticity’.

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