Abstract
Hydropower is available in inland waters and it can be recuperated using dams. Due to malfunction or an act of war, this equipment can collapse and destroy buildings, roads, etc. and kill people and animals, as well. This paper considers the dam-break problem in a horizontal smooth 1D channel. The fluid is water and the inertial effects are neglected versus the viscous ones in the momentum balance. Assuming the shallow water approximation, a non dimensional equation is built from the continuity and the Navier-Stokes equations in the limit of zero-inertia and solved analytically in two limits: short time and long time. These solutions are then combined into a single, universal model. Limitations of the model are examined by comparison to a converged finite difference numerical solution of the flow equation. Hydropower is also available in the ocean and it can be recuperated using Marine Current Turbines (MCT). The main existing MCT concepts are classified and described. At present stage, the ratio of their exploitation being difficult to improve via the concept itself, different tools susceptible to help in this aim are analyzed.
Keywords: Dam failure, Finite difference method, Flow regimes, Current Marine Turbines, Shallow water approximation, Similar solution.
1. Introduction
Due to gravity and Coriolis forces, earth waters are in large proportion in motion, so men naturally thought to control and use the extractable energy of free surface flows. Nowadays, Hydraulics which represents 90% of the sources of renewable energies used for producing electricity in the world is in progress mostly in the developing countries. In fact, hydropower is the only renewable energy which allows energy storage. Huge quantities of water are retained behind dams, constituent energy reservoirs available for electricity production. Bigger the dam is, greater the bottom pressure and risk of breakdown will be. Today, reliable techniques of dam design and building are available and do not arouse new research.
Meanwhile, due to the system malfunction or acts of war (e.g. Dnieproghes, Ukraine in 1941) among other causes, dams can collapse. The water released downstream can destroy fields, goods, infrastructure and be lethal. Since Ritter’s original work on dam-break flow [1], many studies have been performed focusing on experiments, theory and numerical methods [2]. Dam-break flow has become a classical hydraulic problem with such high level of complexity that a more realistic configuration raises new studies. In laboratory, the basic problem consists of a dam obstructing a 1D horizontal open channel thin and smooth, dry downstream and with a given volume of liquid upstream restrained between a fix plate and the dam. Unfortunately, no global study considering dam-break flow as a whole was available.
In that purpose, by image analysis methods, Nsom et al.[3] and Nsom [4] performed an experimental study with glucose-syrup fluids with adjustable viscosity and density. They identified three flow regimes: At initial time, the dam collapses and the fluid is released downstream (positive wave), while a negative wave propagates upstream. From damcollapse date to time where negative wave reaches the fix plate, an inertial regime governed by a Ritter’s type motion law [1] could be characterized. It was followed by a short time viscous regime and finally by an asymptotic (long time
viscous) solution. In the theoretical framework, the flow is described by the Navier Stokes and continuity equations, with the non slip condition and the shallow water approximation. Nsom et al. [5, 6] solved the resulting system of equations analytically and numerically. The solutions obtained were in good agreement with the experimental results.
To mitigate the climate change and increase its energy independence, European Union adopted an objective of 50% of renewable energy used in 2020. At national scale, the Environment Grenelle law brought that rate to 23%. These objectives aroused a huge research effort on renewable energies. West Brittany which is one the region in France with the largest coast affirmed her ambition to be the national leader in marine energies, with the help of specific instruments of research and development such as “Pôle Mer Bretagne“ and “France Energies Marines“.
The oceans can be the future of hydropower because it is constituted by more than 90% of the earth water reserved and energy can also be extracted by turbines from ocean waves, currents and tides. The existing marine turbines can be classified in two main groups. One group is based on the density difference between the water and the air. As the ratio of the densities of water vs. air is about 800, it allows to use the pressure exerted by a water stream on a surface exposed to the air in order to obtain an exploitable torque. The other group contains the turbines made of blades completely immerged in the water stream. In that case, the rotor motion is generated by the horizontal displacement of large quantity of water. The Betz limit (59%) gives the maximum ratio of the water recuperable kinetic energy by a marine turbine. To increase that ratio, the EU Commission now encourages not to invent new turbine concepts but to develop tools for improving the output of the available turbines.
In the second section of this work, we develop an analytical and a numerical short time and asymptotic viscous solution of the flow generated by a dam breach which takes place after the inertial solution, while in the third section, we present the main groups of existing marine turbines and we describe the main tools which can be developed for improving the marine turbines ratio.
2. Dam-break flow revisited
2.1. Equation of motion Let 0 h denote the height of fluid at negative time in a smooth horizontal rectangular channel, g the gravity, ρ and μ the fluid density and dynamic viscosity, respectively. Using a cartesian coordinates system with the origin at the dam site, x-axis lying on the channel-length and the z-axis in the increasing vertical direction (Fig. 1).
The fluid is assumed to flow mainly in the direction of x-axis with height h at the given control section of the abscissa x, at time t.
3. Tools for improving the output of marine turbines
3.1. The existing marine turbines
The existing marine turbines are generally designed as a transposition of wind turbines in the water so, they are gifted with a horizontal or a vertical axis. One type of horizontal axis turbine consists in a single or a twinned rotor, each rotor comprising one or three blades [11]-[14]. The rotors are fixed on a tubular tower set in a foundation in the seabed and emerging above the sea surface, where the system maintenance can be performed. Another type of horizontal axis turbine consists in surrounding the blades by a duct capturing a large area of the tidal stream and accelerating the flow through a narrowing channel into the turbine, the whole system being entirely anchored to the seabed [15]-[16]. An alternative concept is a buoyant system requiring no support structure. It consists in two contra-rotating turbines fixed on a crossbeam. The whole system is tethered to the seabed by a series of mooring chains [17]. The vertical axis turbines that operate in marine currents are based on the the Darrieus turbine principles. The Darrieus turbine is a cross flow machine whose rotation axis meets the working flow at right angles. In marine current applications, cross flow turbines allow the use of a vertically oriented rotor which can transmit the torque directly to the water surface without the need of complex transmission systems [18]. Different versions of vertical turbines were developed such as the Gorlov Helical Turbine [19], among others [20-21]. Other systems exist as for example the oscillating profiles [22]-[23] or the floating system of marine energy [24], but they must be considered as demonstrators in the sense of technical performances. Different tools should be developed for improving the marine energy turbines.
3.2. Better knowledge of marine current
The oceanic ecosystem needs to be understood thoroughly to provide efficient tools for the development of marine renewable energy convertors. The ocean being a complex system characterized by a hierarchy of levels interacting among each other in a non-linear manner, such a goal is intrinsically multi-scale. As such, it requires methods to transfer information between scales and account for cross-scale interactions. The complexity of the oceanic system requires the use of numerical modelling to cover the numerous aspects at once. Until the 21st century, numerical models
were able to cover the different factors of variability and the different scales only through a nesting approach ([25], [26]) of several structured grid configurations, and the use of physical parameterizations for small-scale processes ([27]-[29]). Following the increase in computing power and the development of unstructured grid capabilities, a new generation of models has emerged which allows for enhancement of the spatial resolution only where is needed ([30]-[31]). This new flexibility in adapting the resolution to the scale of processes of interest leads to multi-scale numerical simulations, especially relevant for the challenges encountered by the development of marine turbines. Different tools can be derived from such computational investigations for improving the output of marine turbines. Such tool is for example a better use of the current. Indeed, an accurate determination of the main current axis allows adjusting a fixed marine turbine following the strongest current axis. Only a difference of a few degrees between the turbine and the strongest current axis would make the turbine output collapse.
3.3. Array of Marine Current Turbines (MCT)
An array of MCT’s consists of a given number of single turbine placed one behind the other. Different types of turbines have been developed with either horizontal or vertical axis, but it is now commonly accepted that whatever the turbine considered, the maximum energy can be obtained with an array, i.e. a given number of single turbine placed one behind the other, much more closely together than wind turbines, with advantages in terms of installation costs and cable costs [32]. Indeed, some a priori considerations have been postulated recently, but there is a crucial need of real condition studies.
The basic case to address is the interaction between two turbines. Myers & Bahaj [33] used the tidal current data from the Race of Alderney situated in the Channel Islands to run simulations combined with an analytical method that details the layout energy capture potential throughout a large-scale array. Several array configurations, sizes and the impact of spacing on the velocity distribution were considered in the analysis. For this specific set of data, the optimum configuration predicted an annual energy output at an assigned rated array capacity. A generalization of such result is not easy to postulate.
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fonte: Materials and processes for energy (A. Méndez-Vilas, Ed.)
