69 results found
Goss ZL, Coles DS, Kramer SC, et al., 2021, Efficient economic optimisation of large-scale tidal stream arrays, APPLIED ENERGY, Vol: 295, ISSN: 0306-2619
Duvernay T, Davies DR, Mathews CR, et al., 2021, Linking Intraplate Volcanism to Lithospheric Structure and Asthenospheric Flow, GEOCHEMISTRY GEOPHYSICS GEOSYSTEMS, Vol: 22
Kramer S, Davies R, Wilson C, 2021, Analytical solutions for mantle flow in cylindrical and spherical shells, Geoscientific Model Development, Vol: 14, Pages: 1899-1919, ISSN: 1991-959X
Computational models of mantle convection must accurately represent curved boundaries and the associated boundary conditions of a 3-D spherical shell, bounded by Earth's surface and the core–mantle boundary. This is also true for comparable models in a simplified 2-D cylindrical geometry. It is of fundamental importance that the codes underlying these models are carefully verified prior to their application in a geodynamical context, for which comparisons against analytical solutions are an indispensable tool. However, analytical solutions for the Stokes equations in these geometries, based upon simple source terms that adhere to physically realistic boundary conditions, are often complex and difficult to derive. In this paper, we present the analytical solutions for a smooth polynomial source and a delta-function forcing, in combination with free-slip and zero-slip boundary conditions, for both 2-D cylindrical- and 3-D spherical-shell domains. We study the convergence of the Taylor–Hood (P2–P1) discretisation with respect to these solutions, within the finite element computational modelling framework Fluidity, and discuss an issue of suboptimal convergence in the presence of discontinuities. To facilitate the verification of numerical codes across the wider community, we provide a Python package, Assess, that evaluates the analytical solutions at arbitrary points of the domain.
Pan W, Kramer SC, Piggott MD, 2021, A sigma-coordinate non-hydrostatic discontinuous finite element coastal ocean model, Ocean Modelling, Vol: 157, Pages: 1-21, ISSN: 1463-5003
A𝜎-coordinate non-hydrostatic coastal ocean model is developed using the discontinuous Galerkin fi-nite element method. With the selection of the low-order piecewise-constant PDG0and piecewise-linearPDG1discretisations in the vertical for the velocity and pressure fields, respectively, the proposed𝜎-coordinatemodel can naturally retain the wave dispersion characteristics of the widely-adopted multi-layer approach ofZijlema and Stelling (2005), which is demonstrated through both mathematical derivation and numerical tests.Under the finite element approach, higher-order vertical discretisation choices can also be readily made whichcan reduce the number of vertical layers required for the accurate representation of wave dispersion. Themodel is verified and validated through comparisons against a series of test cases with analytical solutions orexperimental measurements. All the comparisons demonstrate good agreement, indicating that the proposedmodel can accurately represent dispersive barotropic surface waves with as few as one vertical layer, and cansimulate baroclinic internal waves with reasonable accuracy using relatively coarse mesh resolution. It is alsodemonstrated that consistency in the coupling of barotropic and baroclinic flows can be properly ensured.
Kramer S, Wilson C, Davies R, et al., 2020, FluidityProject/fluidity: New test cases "Analytical solutions for mantle flow in cylindrical and spherical shells"
This release adds new test cases described in the GMD paper "Analytical solutions for mantle flow in cylindrical and spherical shells"
Angeloudis A, Kramer SC, Hawkins N, et al., 2020, On the potential of linked-basin tidal power plants: An operational and coastal modelling assessment, Renewable Energy, Vol: 155, Pages: 876-888, ISSN: 0960-1481
Single-basin tidal range power plants have the advantage of predictable energy outputs, but feature non-generation periods in every tidal cycle. Linked-basin tidal power systems can reduce this variability and consistently generate power. However, as a concept the latter are under-studied with limited information on their performance relative to single-basin designs. In addressing this, we outline the basic principles of linked-basin power plant operation and report results from their numerical simulation. Tidal range energy operational models are applied to gauge their capabilities relative to conventional, single-basin tidal power plants. A coastal ocean model (Thetis) is then refined with linked-basin modelling capabilities. Simulations demonstrate that linked-basin systems can reduce non-generation periods at the expense of the extractable energy output relative to conventional tidal lagoons and barrages. As an example, a hypothetical case is considered for a site in the Severn Estuary, UK. The linked-basin system is seen to generate energy 80–100% of the time over a spring-neap cycle, but harnesses at best 30% of the energy of an equivalent-area single-basin design.
Pan W, Kramer S, Kärnä T, et al., 2020, Comparing non-hydrostatic extensions to a discontinuous finite element coastal ocean model, Ocean Modelling, Vol: 151, ISSN: 1463-5003
The unstructured mesh, discontinuous Galerkin finite element discretisation based coastal ocean model, Thetis, has been extended to include non-hydrostatic (buoyancy-driven and free surface) dynamics. Two alternative approaches to achieve this are described in this work. The first (a 3D based algorithm) makes use of prismatic element based meshes and uses a split-step pressure projection method for baroclinic and barotropic modes, while the second (a 2D based algorithm) adopts a novel multi-layer approach to convert a 3D problem into a combination of multiple 2D computations with only 2D triangle meshes required. Model development is carried out at high-level with the Firedrake library, using code generation techniques to automatically produce low-level code for the discretised model equations in an efficient and rapid manner. Through comparisons against several barotropic/baroclinic test cases where non hydrostatic effects are important, the implemented approaches are verified and validated, and the proposed algorithms compared. Depending on whether the problems are dominated by dispersive, baroclinic or barotropic features, recommendation are given over the use of full 3D or multi-layer 2D based approaches to achieve optimal computational accuracy and efficiency. It is demonstrated that while in general the 2D approach is well-suited for barotropic problems and dispersive free surface waves, the 3D approach is more advantageous for simulating baroclinic buoyancy-driven flows due in part to the high vertical resolution typically required to represent the active tracer fields. Keywords: Discontinuous Galerkin, Finite element, Unstructured mesh, Baroclinic flow, Non-hydrostatic, Dispersion, Free surface
Kramer S, Kärnä T, Hill J, et al., 2020, stephankramer/uptide: First release of uptide v1.0
python package for tidal calculations
Kramer S, 2020, stephankramer/assess: Version 1.0
First release of Assess, a python package that implements a number of analytical solutions to the Stokes equations in cylindrical and spherical shell domains. Documentation available from https://assess.readthedocs.io/
Wallwork JG, Barral N, Kramer SC, et al., 2020, Goal-oriented error estimation and mesh adaptation for shallow water modelling, SN Applied Sciences, Vol: 2, Pages: 1-11, ISSN: 2523-3971
This study presents a novel goal-oriented error estimate for the nonlinear shallow water equations solved using a mixed discontinuous/continuous Galerkin approach. This error estimator takes account of the discontinuities in the discrete solution and is used to drive two metric-based mesh adaptation algorithms: one which yields isotropic meshes and another which yields anisotropic meshes. An implementation of these goal-oriented mesh adaptation algorithms is described, including a method for approximating the adjoint error term which arises in the error estimate. Results are presented for simulations of two model tidal farm configurations computed using the Thetis coastal ocean model (Kärnä et al. in Geosci Model Dev 11(11):4359–4382, 2018). Convergence analysis indicates that meshes resulting from the goal-oriented adaptation strategies permit accurate QoI estimation using fewer computational resources than uniform refinement.
Kärnä T, Kramer S, Mitchell L, et al., 2020, thetisproject/thetis: Thetis coastal ocean model
This version is used in paper "A comparison of Bayesian inference and gradient-based approaches for friction parameter estimation."This release is specifically created to document the version of Thetis used in a particular set of experiments. Please do not cite this as a general source for Thetis. See https://thetisproject.org/publications.html for how to cite Thetis in your work.
Du Feu R, funke S, Kramer S, et al., 2019, The trade-off between tidal-turbine array yield and environmental impact: A habitat suitability modelling approach, Renewable Energy, Vol: 143, Pages: 390-403, ISSN: 1879-0682
In the drive towards a carbon-free society, tidal energy has the potential to become a valuable part of the UK energy supply. Developments are subject to intense scrutiny, and potential environmental impacts must be assessed. Unfortunately many of these impacts are still poorly understood, including the implications that come with altering the hydrodynamics. Here, methods are proposed to quantify ecological impact and to incorporate its minimisation into the array design process. Four tidal developments in the Pentland Firth are modelled with the array optimisation tool OpenTidalFarm, that designs arrays to generate the maximum possible profit. Maximum entropy modelling is used to create habitat suitability maps for species that respond to changes in bedshear stress. Changes in habitat suitability caused by an altered tidal regime are assessed. OpenTidalFarm is adapted to simultaneously optimise array design to maximise both this habitat suitability and to maximise the profit of the array. The problem is thus posed as a multi-objective optimisation problem, and a set of Pareto solutions found, allowing trade-offs between these two objectives to be identified. The methods proposed generate array designs that have reduced negative impact, or even positive impact, on the habitat suitability of specific species or habitats of interest.
Davies DR, Valentine AP, Kramer SC, et al., 2019, Earth's multi-scale topographic response to global mantle flow, NATURE GEOSCIENCE, Vol: 12, Pages: 845-+, ISSN: 1752-0894
Vouriot C, Angeloudis A, Kramer S, et al., 2019, Fate of large-scale vortices in idealized tidal lagoons, Environmental Fluid Mechanics, Vol: 19, Pages: 329-348, ISSN: 1567-7419
The generation and evolution of tidally-induced vortices in coastal and estuarine regions can influence water quality and sedimentary processes. These effects must be taken into consideration in the development of coastal reservoirs, barrages and lagoons, among other environmental flow applications. Results are presented here on the fate of large-scale vortices within confined tidally-forced domains. A computational approach is employed using the Thetis depth-averaged coastal ocean modeling framework. Initially, two test cases serve to demonstrate model capability in capturing the formation of dipoles downstream of oscillatory flow channels. Diagnostic quantities of vorticity and localized circulation are used to track the 2-D vortex evolution and dissipation. This approach is then applied to tidal lagoon geometries, where flows through the inlet induce a pair of counter rotating vortices (dipoles). Idealized model geometries and inlet conditions are used to determine the impact of three design parameters on large-scale vortical structures: (a) the lagoon geometry aspect ratio in the horizontal plane, (b) the inlet width and (c) the bathymetry profile as the coastline is approached. The dependence of vortex flushing behavior on the dimensionless ratio Wi/UT (where Wi is the width of the inlet channel, U is the maximum velocity and T is the tidal period) is reaffirmed, while the side walls and the sloping bathymetry are found to affect the vortex dissipation process.
Walker DW, Kramer SC, Biebl FRA, et al., 2019, Accelerating magnetic induction tomography-based imaging through heterogeneous parallel computing, Concurrency Computation, ISSN: 1532-0626
© 2019 John Wiley & Sons, Ltd. Magnetic Induction Tomography (MIT) is a non-invasive imaging technique, which has applications in both industrial and clinical settings. In essence, it is capable of reconstructing the electromagnetic parameters of an object from measurements made on its surface. With the exploitation of parallelism, it is possible to achieve high quality inexpensive MIT images for biomedical applications on clinically relevant time scales. In this paper we investigate the performance of different parallel implementations of the forward eddy current problem, which is the main computational component of the inverse problem through which measured voltages are converted into images. We show that a heterogeneous parallel method that exploits multiple CPUs and GPUs can provide a high level of parallel scaling, leading to considerably improved runtimes. We also show how multiple GPUs can be used in conjunction with deal.II, a widely-used open source finite element library.
Rathgeber F, Mitchell L, Luporini F, et al., 2019, OP2/PyOP2: Framework for performance-portable parallel computations on unstructured meshes
This release is specifically created to document the version of PyOP2 used in a particular set of experiments using Firedrake. Please do not cite this as a general source for Firedrake or any of its dependencies. Instead, refer to https://www.firedrakeproject.org/citing.html
Pan W, Kramer S, Piggott M, 2019, Multi-layer non-hydrostatic free surface modelling using the discontinuous Galerkin method, Ocean Modelling, Vol: 134, Pages: 68-83, ISSN: 1463-5003
A multi-layer non-hydrostatic version of the unstructured mesh, discontinuous Galerkinfinite element based coastal ocean model, Thetis, is developed. This is accomplishedusing the PDE solver framework, Firedrake, which is used to automatically produce thecode for the discretised model equations in a rapid and efficient manner. The motivationfor this work is a need to accurately simulate dispersive nearshore free surface processes.In order to resolve both frequency dispersion and non-linear effects accurately, additional non-hydrostatic terms are included in the layer-integrated hydrostatic equations,producing a form similar to the layered non-linear shallow water equations, but withextra vertical velocities at the layer interfaces. An implementation process is adoptedto easily handle the inter-layer connection, i.e. the governing equations are transformedinto a depth-integrated system through the introduction of depth-averaged variables.The model is verified and validated through comparisons against several idealisedand experimentally-based test cases. All the comparisons demonstrate good agreement,showing that the developed non-hydrostatic model has excellent capabilities in representing coastal wave phenomena including shoaling, refraction and diffraction of dispersive short waves, as well as propagation, run-up and inundation of non-linear tsunamiwaves.
Goss ZL, Piggott MD, Kramer SC, et al., 2019, Competition effects between nearby tidal turbine arrays—optimal design for alderney race, Pages: 255-262
Tidal renewable energy can be described as a fledgling industry, with the world’s pilot tidal stream turbine array only recently installed. Full-sized arrays will be developed if they prove their economic, engineering and environmental viability. Reliable numerical tools are needed to optimise power yields in arrays of potentially hundreds of turbines and assess viability of new sites and designs. To demonstrate our capability to optimise the number of turbines and their spatial distribution in a region, we focus on a test case based upon the Alderney Race. The site contains the majority of the Channel Islands resource with plans from both France and Alderney to develop adjacent arrays that could impact on each other. We present a shallow-water model of the English Channel using the Thetis ocean model. Together with the hydrodynamics modelling we employ adjoint technology to optimise the micrositing of turbines for a set of scenarios.
Angeloudis A, Hawkins N, Kramer SC, et al., 2019, Comparison of twin-basin lagoon systems against conventional tidal power plant designs, Pages: 159-168
Tidal power plant proposals present opportunities to deliver sustainable energy to the national grid in the UK and beyond. Multiple designs have been considered over the years that seek to competitively balance economic, environmental and technical challenges. Tidal range power plants have the advantage of reliable and predictable energy outputs with a certain degree of flexibility but typically feature non-generation periods in every tidal cycle. Twin-basin systems present an alternative that can significantly reduce this variability but very little information has been reported in terms of their performance to-date. In this work a coastal ocean modelling framework has been coupled with twin-basin and tidal power plant operation algorithms and is applied to compare tidal energy options. Results suggest that through use of a twin basin system the non-generation time can be substantially reduced at a notable expense to the overall energy output relative to conventional tidal lagoons and barrages.
Mitchell L, Ham D, Gibson T, et al., 2018, firedrakeproject/firedrake: an automated finite element system
This release is specifically created to document the version of firedrake used in a particular set of experiments using Firedrake. Please do not cite this as a general source for Firedrake or any of its dependencies. Instead, refer to https://www.firedrakeproject.org/citing.html
Kärnä T, Kramer SC, Mitchell L, et al., 2018, Thetis coastal ocean model: discontinuous Galerkin discretization for the three-dimensional hydrostatic equations, Geoscientific Model Development, Vol: 11, Pages: 4359-4382
<jats:p>Abstract. Unstructured grid ocean models are advantageous for simulating the coastal ocean and river–estuary–plume systems. However, unstructured grid models tend to be diffusive and/or computationally expensive, which limits their applicability to real-life problems. In this paper, we describe a novel discontinuous Galerkin (DG) finite element discretization for the hydrostatic equations. The formulation is fully conservative and second-order accurate in space and time. Monotonicity of the advection scheme is ensured by using a strong stability-preserving time integration method and slope limiters. Compared to previous DG models, advantages include a more accurate mode splitting method, revised viscosity formulation, and new second-order time integration scheme. We demonstrate that the model is capable of simulating baroclinic flows in the eddying regime with a suite of test cases. Numerical dissipation is well-controlled, being comparable or lower than in existing state-of-the-art structured grid models. </jats:p>
Kärnä T, Kramer SC, Mitchell L, et al., 2018, Thetis coastal ocean model: discontinuous Galerkin discretization for the three-dimensional hydrostatic equations, Geoscientific Model Development, Vol: 11, Pages: 4359-4382, ISSN: 1991-959X
Unstructured grid ocean models are advantageous for simulating the coastal ocean and river-estuary-plume systems. However, unstructured grid models tend to be diffusive and/or computationally expensive which limits their applicability to real life problems. In this paper, we describe a novel discontinuous Galerkin (DG) finite element discretization for the hydrostatic equations. The formulation is fully conservative and second-order accurate in space and time. Monotonicity of the advection scheme is ensured by using a strong stability preserving time integration method and slope limiters. Compared to previous DG models advantages include a more accurate mode splitting method, revised viscosity formulation, and new second-order time integration scheme. We demonstrate that the model is capable of simulating baroclinic flows in the eddying regime with a suite of test cases. Numerical dissipation is well-controlled, being comparable or lower than in existing state-of-the-art structured grid models.
Tidal range power plants represent an attractive approach for the large-scale generation of electricity from the marine environment. Even though the tides and by extension the available energy resource are predictable, they are also variable in time. This variability poses a challenge regarding the optimal transient control of power plants. We consider simulation methods which include the main modes of operation of tidal power plants, along with algorithms to regulate the timing of these. This paper proposes a framework where simplified power plant operation models are coupled with gradient-based optimisation techniques to determine the optimal control strategy over multiple tidal cycles. The optimisation results inform coastal ocean simulations that include tidal power plants to gauge whether the benefits of an adaptive operation are preserved once their hydrodynamic impacts are also taken into consideration. The combined operation of two prospective tidal lagoon projects within the Bristol Channel and the Severn Estuary is used as an example to demonstrate the potential benefits of an energy maximisation optimisation approach. For the case studies considered, the inclusion of pumping and an adaptive operation is shown to deliver an overall increase in energy output of 20–40% compared to a conventional two-way uniform operation. The findings also demonstrate that smaller schemes stand to gain more from operational optimisation compared to designs of a larger scale.
Avdis A, Candy AS, Hill J, et al., 2017, Efficient unstructured mesh generation for marine renewable energy applications, Renewable Energy, Vol: 116, Pages: 842-856, ISSN: 0960-1481
Renewable energy is the cornerstone of preventing dangerous climate change whilst maintaining a robust energy supply. Tidal energy will arguably play a critical role in the renewable energy portfolio as it is both predictable and reliable, and can be put in place across the globe. However, installation may impact the local and regional ecology via changes in tidal dynamics, sediment transport pathways or bathymetric changes. In order to mitigate these effects, tidal energy devices need to be modelled in order to predict hydrodynamic changes. Robust mesh generation is a fundamental component required for developing simulations with high accuracy. However, mesh generation for coastal domains can be an elaborate procedure. Here, we describe an approach combining mesh generators with Geographical Information Systems. We demonstrate robustness and efficiency by constructing a mesh with which to examine the potential environmental impact of a tidal turbine farm installation in the Orkney Islands. The mesh is then used with two well-validated ocean models, to compare their flow predictions with and without a turbine array. The results demonstrate that it is possible to create an easy-to-use tool to generate high-quality meshes for combined coastal engineering, here tidal turbines, and coastal ocean simulations.
du Feu RJ, Funke SW, Kramer SC, et al., 2017, The trade-off between tidal-turbine array yield and impact on flow: A multi-objective optimisation problem, Renewable Energy, Vol: 114, Pages: 1247-1257, ISSN: 1879-0682
This paper introduces a new approach for investigating trade-offs between different societal objectives in the design of tidal-turbine arrays. This method is demonstrated through the trade-off between the yield of an array, and the extent to which that array alters the flow. This is posed as a multi-objective optimisation problem, and the problem is investigated using the array layout optimisation tool OpenTidalFarm. Motivated by environmental concerns, OpenTidalFarm is adapted to not only maximise array yield but also to minimise the effect of the array upon the hydrodynamics of the region, specifically the flow velocity. A linear scalarisation of the multi-objective optimisation problem is solved for a series of different weightings of the two conflicting objectives. Two idealised test scenarios are evaluated and in each case a set of Pareto solutions is found. These arrays are assessed for the power they generate and the severity of change they cause in the flow velocity. These analyses allow for the identification of trade-offs between these two objectives, while the methods proposed can similarly be applied to the two key societal objectives of energy production and conservation, thus providing information that could be valuable to stakeholders and policymakers when making decisions on array design.
Culley DM, Funke SW, Kramer SC, et al., 2017, A surrogate-model assisted approach for optimising the size of tidal turbine arrays, International Journal of Marine Energy, Vol: 19, Pages: 357-373, ISSN: 2214-1669
The new and costly nature of tidal stream energy extraction technologies can lead to narrow margins of success for a project. The design process is thus a delicate balancing act – to maximise the energy energy extracted, while minimising cost and risk. Scenario specific factors, such as site characteristics, technological constraints and practical engineering considerations greatly impact upon both the appropriate number of turbines to include within a tidal current turbine array (array size), and the individual locations of those turbines (turbine micro-siting). Both have been shown to significantly impact upon the energy yield and profitability of an array.The micro-siting arrangement for a given number of turbines can significantly influence the power extraction of a tidal farm. Until the layout has been optimised (a process which may incorporate turbine parameters, local bathymetry and a host of other practical, physical, legal, financial or environmental constraints) an accurate forecast of the yield of that array cannot be determined. This process can be thought of as ‘tuning’ an array to the proposed site to maximise desirable outcomes and mitigate undesirable effects.The influence of micro-siting on the farm performance means that determining the optimal array size needs to be coupled to the micro-siting process. In particular, the micro-siting needs to be repeated for any new trial array size in order to be able to compare the performance of the different farm sizes. Considering the large number of design variables in the micro-siting problem (which includes at least the positions of each turbine) it becomes clear that algorithmic optimisation is a key tool to rigorously determine the optimal array size and layout.This paper proposes a nested optimisation approach for solving the array size and layout problem. The core of this approach consists of two nested optimisation procedures. The ‘outer’ optimisation determines the array
Jones TD, Davies DR, Campbell IH, et al., 2017, The concurrent emergence and causes of double volcanic hotspot tracks on the Pacific plate, NATURE, Vol: 545, Pages: 472-+, ISSN: 0028-0836
Perrin A, Goes S, Prytulak J, et al., 2016, Reconciling mantle wedge thermal structure with arc lava thermobarometric determinations in oceanic subduction zones, Geochemistry, Geophysics, Geosystems, Vol: 17, Pages: 4105-4127, ISSN: 1525-2027
Subduction zone mantle wedge temperatures impact plate interaction, melt generation, and chemical recycling. However, it has been challenging to reconcile geophysical and geochemical constraints on wedge thermal structure. Here we chemically determine the equilibration pressures and temperatures of primitive arc lavas from worldwide intraoceanic subduction zones and compare them to kinematically driven thermal wedge models. We find that equilibration pressures are typically located in the lithosphere, starting just below the Moho, and spanning a wide depth range of ∼25 km. Equilibration temperatures are high for these depths, averaging ∼1300°C. We test for correlations with subduction parameters and find that equilibration pressures correlate with upper plate age, indicating overriding lithosphere thickness plays a role in magma equilibration. We suggest that most, if not all, thermobarometric pressure and temperature conditions reflect magmatic reequilibration at a mechanical boundary, rather than reflecting the conditions of major melt generation. The magma reequilibration conditions are difficult to reconcile, to a first order, with any of the conditions predicted by our dynamic models, with the exception of subduction zones with very young, thin upper plates. For most zones, a mechanism for substantially thinning the overriding plate is required. Most likely thinning is localized below the arc, as kinematic thinning above the wedge corner would lead to a hot fore arc, incompatible with fore-arc surface heat flow and seismic properties. Localized subarc thermal erosion is consistent with seismic imaging and exhumed arc structures. Furthermore, such thermal erosion can serve as a weakness zone and affect subsequent plate evolution.
Funke SW, Kramer SC, Piggott MD, 2016, Design optimisation and resource assessment for tidal-stream renewable energy farms using a new continuous turbine approach, Renewable Energy, ISSN: 1879-0682
This paper presents a new approach for optimising the design of tidal stream turbine farms. In this approach, the turbine farm is represented by a turbine density function that specifies the number of turbines per unit area and an associated continuous locally-enhanced bottom friction field. The farm design question is formulated as a mathematical optimisation problem constrained by the shallow water equations and solved with efficient, gradient-based optimisation methods. The resulting method is accurate, computationally efficient, allows complex installation constraints, and supports different goal quantities such as to maximise power or profit. The outputs of the optimisation are the optimal number of turbines, their location within the farm, the overall farm profit, the farm's power extraction, and the installation cost.We demonstrate the capabilities of the method on a validated numerical model of the Pentland Firth, Scotland. We optimise the design of four tidal farms simultaneously, as well as individually, and study how farms in close proximity may impact upon one another.
Kramer SC, Piggott MD, 2016, A correction to the enhanced bottom drag parameterisation of tidal turbines, Renewable Energy, Vol: 92, Pages: 385-396, ISSN: 0960-1481
Hydrodynamic modelling is an important tool for the development of tidal stream energy projects. Many hydrodynamic models incorporate the effect of tidal turbines through an enhanced bottom drag. In this paper we show that although for coarse grid resolutions (kilometre scale) the resulting force exerted on the flow agrees well with the theoretical value, the force starts decreasing with decreasing grid sizes when these become smaller than the length scale of the wake recovery. This is because the assumption that the upstream velocity can be approximated by the local model velocity, is no longer valid. Using linear momentum actuator disc theory however, we derive a relationship between these two velocities and formulate a correction to the enhanced bottom drag formulation that consistently applies a force that remains close to the theoretical value, for all grid sizes down to the turbine scale. In addition, a better understanding of the relation between the model, upstream, and actual turbine velocity, as predicted by actuator disc theory, leads to an improved estimate of the usefully extractable energy. We show how the corrections can be applied (demonstrated here for the models MIKE 21 and Fluidity) by a simple modification of the drag coefficient.
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