By David Lannes
“Consider the action of the waves — the ebb and flow of the tides. What is the ocean? A prodigious force wasted.” Back in 1874, when French writer Victor Hugo published his novel Ninety-Three, mankind already knew that the ocean could provide energy. Yet the first industrial exploitation of ocean energy did not come to light until 1967—nearly a century later—at the Rance Tidal Power Station in France’s Brittany region. A revived interest in marine renewable energies arose in the 2000s, inspired by an oil shock and the need to increase renewable energy use due to the carbon emissions of fossil fuels. Technological and computational advances also made renewable energies more realistic.
Renewable marine energies include tidal energy, offshore wind energy, ocean currents, waves, thermal potential, salinity gradient power, and even biomass. The most widely spread of these energies—and the one with the most potential—is offshore wind energy, which has a technical potential of roughly 16,000 terawatt hours TW\(\cdot\)h per year by 2050 [2]. The technology is the same for onshore wind turbines, but additional engineering challenges arise — such as the construction of offshore foundations or floating support structures (in deeper water). To design the latter, one must understand the interactions between waves and floating structures.
Wave-structure interactions are also obviously important in the concept of wave energy. With an estimated technical potential of 5,600 (TW\(\cdot\)h) per year, wave energy is less powerful than offshore wind energy but still offers interesting perspectives. Researchers have proposed several devices to exploit wave energy; the Pelamis Wave Energy Converter in the U.K., which consists of several partially submerged connected cylinders whose relative motion generates electricity when a wave passes, was the first to actually generate electricity into the grid. Since then, scientists have developed dozens of other systems, including floating buoys, oscillating water columns, and flaps.
To design floaters, offshore industry has traditionally employed tank testing at small scales. However, approaches that are based on computational fluid dynamics (CFD) are beginning to convincingly replace model tests, especially for the study of turbulence-related issues. CFD computations are particularly relevant in the analysis of wave impact and vortex-induced motion that can endanger the mooring systems of floating structures, among other applications.
Although CFD is a common tool in aeronautics and the automotive industry, it remains at an early stage of development for floater design. A recent study [5] identified the following four unique challenges that offshore engineers face that differ from other industries:
(1) A highly separated flow with a Reynolds number of order \(10^7\) around the floater hull
(2) Large scale differences between the hull and the mooring and riser systems
(3) An open ocean environment that requires the computation of a large volume of fluid
(4) A non-Gaussian stochastic environment that necessitates multiple computations to provide reliable statistics for floater motion.
Due to these difficulties, the estimated cost of a CFD project for a numerical basin is equivalent to that of physical model tests.
An interesting alternative to CFD computations involves describing the flow via the so-called fully nonlinear potential flow (FNPF) approach [8], during which the flow is assumed to be irrotational and the velocity field derives from a scalar velocity potential; one can find the latter in the fluid domain by solving a Laplace equation with appropriate boundary conditions on the free surface and the object’s sides. Although this approach cannot account for turbulence-related issues, it can capture nonlinear effects in the wave-structure interactions.
Figure 1. Interior region \({\mathcal I}(t)\), exterior region \({\mathcal E}(t)\), and projection of the contact line \(\Gamma(t)\). Figure adapted from [6].
The performance of CFD and FNPF computations is steadily improving, but engineers use much simpler methods to construct “wave-to-wire” models that simulate the interactions of the various devices of arrays that count up to dozens of wave energy converters, as well as their connection to the network [4]. Such methods are typically based on the Cummins’ equations [3]: a set of coupled linear integro-differential equations, one for each degree of freedom. In the absence of an exciting force, the equations take the form
\[\sum_{j=1}^6\big[ (m_j \delta_{jk}+m_{jk})\ddot x_j+c_{jk} x_j+\int_{-\infty}^t K_{jk}(t-\tau)\dot x_j(\tau){\rm d}\tau \big]=0\qquad (1\leq k\leq 6).\tag1\]
Here, \(m_j\) denotes the mass/inertia in the \(j\)th degree of freedom, \(\delta_{jk}\) is the Kronecker symbol, and \(c_{jk}x_j\) is the hydrostatic force in the \(k\)th mode that results from a perturbation \(x_j\) in the \(j\)th mode. Finally, \((m_{jk})_{jk}\) and \((K_{jk})_{jk}\) are the matrices of added mass and radiation impulse response functions, respectively. In order to define these latter quantities and understand the limitations of the model, one can sketch the model’s derivation.
We assume that the flow is linear and irrotational, and that the surface elevation’s variations are neglected for the velocity potential’s domain of definition. One can decompose the velocity potential \(\Phi\), which results from a small perturbation \(x_j(t)\) of the floating object in the \(j\)th mode, into
\[\Phi=\dot x_j \psi_j+\int_{-\infty}^t \varphi_j(t-\tau)\dot x_j(\tau){\rm d}\tau.\tag2\]
Here, \(\psi_j\) is the potential directly associated with the instantaneous impulsive velocity of the floating object, and \(\varphi_j\) is the potential associated with the radiating disturbance of the free surface (the time integral accounts for disturbances of the surface that are created by the object’s previous displacements). On the other hand, the pressure \(P\) at the wetted surface \(S_{\rm w}\) of the object (whose time variations are neglected) is given by the linear approximation of Bernoulli’s equation:
\[P=-\rho g z -\rho \partial_t\Phi \quad \mbox{ on } \;\; S\tag3\]
(\(\rho\) is the constant density of the fluid, \(g\) is the gravity, and \(z\) is the vertical variable). One can therefore write the resulting force that is exerted on the solid as
\[F=\int_{S_{\rm w}} \big(-\rho g z-(\partial_t\Phi) \big){\bf n}\tag4\]
(\(\bf n\) is the outward normal to the fluid surface under the object). A similar expression holds for the resulting torque. Using \((2)\)-\((3)\), Newton’s equations, and Archimedes’ principle, one readily obtains \((1)\) with
\[m_{jk}=\rho \int_{S_{\rm w}} \psi_j s_k \quad \mbox{ and } \quad K_{jk}(\tau)=\rho\int_{S_{\rm w}} \partial_t \varphi_j(\tau)s_k.\]
Here, \(s_k={\bf n}\cdot {\bf e}_k\) \((k=1,2,3)\) or \(s_k=({\bf r}\times {\bf n})\cdot {\bf e}_{k-3}\) \((k=4,5,6)\), where \({\bf e}_k\) is the unit vector in the \(k\)th direction and \({\bf r}\) is the position vector with respect to the object’s center of gravity. Engineers then compute the various components of the velocity potential with commercial software programs like WAMIT, which are based on a representation of the potential in terms of Green’s functions.
The assumption of linearity is ubiquitous in the derivation of Cummins’ equations \((1)\): the variations of the immersed part \(S_{\rm w}\) of the body are neglected, wave motion is linear, nonlinear terms are neglected in Bernoulli’s equation, and so forth. Models based on Cummins’ equations that have proven very useful for the study of floating structures are therefore not utilized when nonlinear effects are important, i.e., if one wants to assess maximum loads on the structure in extreme sea conditions. Such a situation is likely to occur quite often in shallow water, where the waves are of larger amplitude.
To mitigate this scenario, I have recently proposed another approach [6] that somehow fits between linear approaches that are based on Cummins’ equations and the very precise but computationally expensive CFD (and even FNPF) approaches. This method is essentially based on three steps:
(1) Use an asymptotic model to describe the waves
(2) Consider the pressure that is exerted on the object as a Lagrange multiplier
(3) Reduce the problem to a transmission problem.
Regarding the first step, scientists have made considerable progress in the last two decades on the derivation and justification of asymptotic models to the \(d+1\)-dimensional free surface Euler equations (also called water waves equations), where \(d\) is the horizontal dimension — especially in shallow water [7]. Such models typically couple the elevation \(\zeta\) of the free surface with respect to the rest state with the horizontal discharge \(Q\) (the vertical integral of the horizontal velocity). An obvious simplification involves casting the equations on the horizontal domain (with no dependence on the vertical variable \(z\)). Both quantities are connected by the exact relation
\[\partial_t \zeta+\mbox{div}_{\rm h} Q=0,\]
where \(\mbox{div}_{\rm h}\) is the divergence with respect to the horizontal variables. This equation is complemented by an approximate evolution equation for \(Q\), which takes the form
\[\partial_t Q + g(h_0+\zeta)\nabla_{\rm h} \zeta +\dots =-\frac{1}{\rho}(h_0+\zeta)\nabla_{\rm h} P_{\rm surf}.\tag5\]
Here, \(h_0\) is the water depth at rest and \(P_{\rm surf}\) is the pressure at the surface; the dots in the equation depend on the range of validity and precision of the asymptotic model.
For the second step, one can divide the horizontal plane into three parts: the interior region (the projection of the object’s time-dependent wetted surface \(S_{\rm w}(t)\)), the projection of the contact line, and the exterior region (see Figure 1). The surface in the exterior region is free but the pressure is constrained; \(P_{\rm surf}\) is typically equal to a constant atmospheric pressure so that the right side vanishes in \((5)\). The reverse occurs in the interior region: the pressure is free but the fluid’s surface is constrained and must coincide with the wetted part of the object. One can interpret the right side in \((5)\), which does not vanish, as the Lagrange multiplier that is associated with this constraint.
The third step involves finding the pressure for the incompressible Euler equations in the interior domain by solving a rather simple elliptic equation. When combined with some coupling conditions at the contact line, this solution allows one to reduce the whole problem to a transmission problem in the exterior region for the asymptotic model, with nonstandard transmission conditions.
This three-step approach is quite simple and very efficient. It accounts for nonlinear effects and benefits from the recent progress with numerical simulations of shallow water flows. The method also yields several nonlinear generalizations of Cummins’ equations and shows that researchers should generally not expect to describe a floater’s motion via simple integro-differential equations; they must instead consider transmission problems. The resulting models are mathematically challenging. Most importantly, one should carefully investigate this approach’s range of validity by comparing it with CFD computations or numerical approaches that are fully based on nonlinear flows or experiments.
Ultimately, wave structure comprises only a small part of the problems that are associated with the development of marine renewable energies. For example, seawater is corrosive and warrants the creation of specific materials; precise weather and wave forecasts are needed; and researchers must evaluate wave farms’ effect on the wave field, in addition to the impact of sedimentation, erosion, and water quality on biodiversity, local fisheries, and so on. Seemingly more remote issues—such as legal aspects, funding policies, economic viability, and social acceptance of altered landscapes—are also relevant.
Although marine renewable energies may be “greener” than fossil energies, they are certainly not “green.” As scientists, we can help to fairly evaluate the merits and disadvantages that will allow citizens to make informed choices about the society in which they want to live. Some communities have already created scientific structures to assist local governments that are deciding on strategies to address global warming [1]. Answering such concerns in turn raises new scientific issues; this is a new and fascinating form of interdisciplinarity.
This article is based on David Lannes’ invited presentation at the 2020 SIAM Annual Meeting, which took place virtually last year.
References
[1] AcclimaTerra, & Le Treut, H. (Ed). (2020). Anticipating climate change in Nouvelle-Aquitaine. To guide policy at local level (Executive Report). Éditions AcclimaTerra.
[2] Borthwick, A.G. (2016). Marine renewable energy seascape. Engineer., 2(1), 69-78.
[3] Cummins, W. (1962). The impulse response function and ship motions. (Hydromechanics Laboratory Research and Development Report). U.S. Department of the Navy.
[4] Forehand, D.I., Kiprakis, A.E., Nambiar, A.J., & Wallace, A.R. (2015). A fully coupled wave-to-wire model of an array of wave energy converters. IEEE Trans. Sustain. Ener., 7(1), 118-128.
[5] Kim, J.W., Jang, H., Baquet, A., O’Sullivan, J., Lee, S., Kim, B., …, Jasak, H. (2016). Technical and economic readiness review of CFD-based numerical wave basin for offshore floater design. In Offshore Technology Conference. Houston, TX.
[6] Lannes, D. (2017). On the dynamics of floating structures. Annals of PDE, 3(1), 11.
[7] Lannes, D. (2020). Modeling shallow water waves. Nonlinearity, 33, R1.
[8] Penalba, M., Giorgi, G., & Ringwood, J.V. (2017). Mathematical modelling of wave energy converters: A review of nonlinear approaches. Renew. Sustain. Ener. Rev., 78, 1188-1207.
David Lannes is a CNRS research director at the Institut de Mathématiques de Bordeaux. He works on the modelling and mathematical analysis of waves in coastal regions, and is interested in applications like the evaluation of submersion risks or renewable energies in marine environments. Lannes is also involved in various interdisciplinary structures that aim to coordinate the scientific community’s answers to ongoing environmental issues.