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The Hydrodynamic Analysis of a Vertical Axis Tidal Current Turbine

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The Hydrodynamic Analysis of a Vertical Axis Tidal Current Turbine The hydrodynamic analysis of a vertical axis tidal current turbine Gareth I Gretton A thesis submitted for the degree of Doctor of Philosophy. The University of Edinburgh. 18th August 2009 Abstract Tidal currents can be used as a predictable source of sustainable energy, and have the potential to make a useful contribution to the energy needs of the UK and other countries with such a resource. One of the technologies which may be used to transform tidal power into mechanical power is a vertical axis turbine, the hydrodynamic analysis of which this thesis is concerned with. The aim of this analysis is to gain a better understanding of the power transformation process, from which position there is the possibility of improving the conversion efficiency. A second aim is to compare the results from different modelling approaches. Two types of mathematical modelling are used: a basic blade element momentum model and a more complex Reynolds-averaged Navier Stokes (RANS) model. The former model has been programmed in Matlab by the present author while the latter model uses a commercial computational fluid dynamics (CFD) code, ANSYS CFX. This RANS model uses the SST k-ω turbulence model. The CFD analysis of hydrofoils (equally airfoils), for both fixed and oscillating pitch conditions, is a significant proportion of the present work. Such analysis is used as part of the verification and validation of the CFD model of the turbine. It is also used as input to the blade element momentum model, thereby permitting a novel comparison between the blade element momentum model and the CFD model of the turbine. Both types of turbine model were used to explore the variation in turbine efficiency (and other factors) with tip speed ratio and with and without an angle of attack limiting variable pitch strategy. It is shown that the use of such a variable pitch strategy both increases the peak efficiency and broadens the peak. The comparison of the results from the two different turbine modelling approaches shows that when the present CFD hydrofoil results are used as input to the blade element model, and when dynamic effects are small and the turbine induction factor is low, there is generally good agreement between the two models. Declaration of originality I hereby declare that this thesis has been composed by myself and that except where stated, the work contained is my own. I also declare that the work contained in this thesis has not been submitted for any other degree or professional qualification except as specified. Gareth Gretton iii Acknowledgements I would like to begin by thanking Stephen Salter for proposing this thesis topic and for his unorthodox insight. For their formal and informal supervision of this project I would like to thank Tom Bruce and David Ingram; more specifically, I would like to thank Tom for being persistently encouraging and supportive, and David for his help with the theory and practice of CFD. Thanks are due to my office mates Jamie Taylor, Gregory´ Payne, Jorge Lucas and Remy´ Pascal (in order of appearance) for everything that good office mates should be thanked for. In a similar vein thanks also to the wider community of the Institute for Energy Systems and the School of Engineering. For doing what is often a thankless job I would like to thank the staff of the IT department within the School of Engineering. In particular, thanks to Bruce Duncan, Michael Gordon, and David Stewart for getting things working on the cluster and managing recalcitrant license servers. Others I would like to thank personally for their help with this project are Paul Cooper, Peter Fraenkel, Peter Johnson, Liz McRae and Joanna Whelan. For funding this work I would like to thank Edinburgh Designs and the Engineering and Physical Sciences Research Council (i.e. the UK taxpayer). Those staff of the ANSYS CFX support team who have given helpful and detailed replies are gratefully acknowledged. I would also like to say thanks to those people whom I have never met that provided helpful responses to emails. Such replies have often resulted in the very satisfying tying up of loose ends. Finally, and on a more personal level, I would like to thank my girlfriend Rebecca Day and my parents Ron and Kay for their support of, and perseverance with, me. iv Contents Declaration of originality . iii Acknowledgements . iv Contents . v List of figures . ix List of tables . xiv 1 Introduction 1 1.1 The need for sustainable energy . 1 1.2 The advantages of tidal current energy . 1 1.3 A brief history of tidal energy . 2 1.4 Current developments . 6 1.4.1 Horizontal axis turbines . 7 1.4.2 Vertical axis turbines . 8 1.4.3 Other concepts . 10 1.5 Vertical axis turbines – principle of operation . 11 1.6 Horizontal axis versus vertical axis and wind versus tide . 13 1.7 The Edinburgh Designs turbine . 17 1.8 Aims and objectives . 18 2 Literature review 23 2.1 Introduction . 23 2.2 Airfoil section data and modelling . 23 2.2.1 Basics of airfoil behaviour and design . 24 2.2.2 Experimental measurements . 28 2.2.3 Numerical modelling . 36 2.3 Variable pitch . 49 2.4 Mathematical models of vertical axis turbines . 50 2.4.1 Blade element momentum models . 51 2.4.2 Extensions to blade element models . 53 2.4.3 Vortex models . 54 2.4.4 Potential flow solutions . 55 2.4.5 CFD models . 56 2.4.6 Coupled CFD and blade element momentum models . 59 2.4.7 Free surface effects for tidal current turbines . 59 2.5 Physical tests of vertical axis turbines . 60 2.5.1 Experimental models . 60 2.5.2 Large-scale devices . 61 2.5.3 Porous bodies . 61 2.6 The tidal resource . 63 2.6.1 Origin and characteristics of the tides . 63 2.6.2 Resource assessment . 64 2.6.3 Feedback effects from power extraction . 67 2.6.4 Turbulence in the marine boundary layer . 70 v Contents 3 Theory 75 3.1 Basic fluid dynamics theory . 75 3.1.1 Conservation of mass . 75 3.1.2 Conservation of momentum . 75 3.1.3 Conservation of energy . 77 3.1.4 Secondary thermodynamic properties and equations of state . 78 3.2 Turbulence . 79 3.2.1 Mathematical description . 80 3.2.2 The Reynolds-averaged Navier-Stokes equations . 81 3.2.3 The Boussinesq approximation . 83 3.2.4 Mixing length models . 84 3.2.5 Governing equation for the turbulent kinetic energy . 85 3.2.6 The k-ε turbulence model . 86 3.2.7 The Wilcox k-ω turbulence model . 87 3.2.8 The Baseline (BSL) k-ω turbulence model . 88 3.2.9 The shear stress transport (SST) k-ω turbulence model . 89 3.2.10 Modelling flow near the wall . 91 3.2.11 The decay of inlet turbulence . 93 3.3 CFX discretization and solution theory . 94 3.3.1 The finite volume method . 94 3.3.2 Shape functions . 97 3.3.3 The Rhie-Chow interpolation method . 98 3.3.4 Transient term . 101 3.3.5 Convection term . 101 3.3.6 Compressibility . 102 3.3.7 The coupled solution . 102 3.3.8 The iterative solution method . 103 3.4 Boundary conditions . 106 3.4.1 Inlet . 106 3.4.2 Outlet . 106 3.4.3 Opening . 106 3.4.4 Symmetry plane . 106 3.5 Dimensional analysis and scaling . 107 3.5.1 Reynolds number . 107 3.5.2 Mach number . 107 3.5.3 Strouhal number or reduced frequency . 107 3.5.4 Force and moment coefficients for airfoil sections . 108 3.6 Errors and uncertainty in CFD modelling . 109 3.6.1 Categorization of errors and uncertainties . 110 3.6.2 Verification and validation . 112 3.6.3 Estimating the spatial discretization error . 116 3.6.4 Estimating the far-field boundary error . 121 3.7 Bibliographic note . 123 4 Blade element momentum models 125 4.1 Introduction . 125 4.2 Turbine parameters . 125 vi Contents 4.2.1 Tip speed ratio and solidity . 125 4.2.2 Coefficients of power, torque and thrust . 126 4.2.3 Blade force and moment coefficients . 126 4.3 A description of the model . 127 4.3.1 Preliminaries . 127 4.3.2 Momentum theory . 128 4.3.3 The Betz limit . 129 4.3.4 Blade element theory . 130 4.3.5 Force normal to the streamline . 132 4.3.6 Integrated forces and torques . 133 4.3.7 Streamtube expansion . 134 4.3.8 θ-angles of.
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