Wind Turbine Aerodynamics and Vorticity-Based Methods: Fundamentals and Recent Applications

632 Pages·2017·16.52 MB·English

by  Emmanuel Branlard (auth.)

Research Topics in Wind Energy 7
Emmanuel Branlard
Wind Turbine
Aerodynamics and
Vorticity-Based
Methods
Fundamentals and Recent Applications
Research Topics in Wind Energy
Volume 7
Series editors
Joachim Peinke, University of Oldenburg, Oldenburg, Germany
e-mail: [email protected]
Gerard van Bussel, Delft University of Technology, Delft, The Netherlands
e-mail: [email protected]
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Emmanuel Branlard
Wind Turbine Aerodynamics
and Vorticity-Based Methods
Fundamentals and Recent Applications
123
EmmanuelBranlard
Department ofWind Energy, Aeroelastic
Design
Technical University of Denmark
Roskilde
Denmark
ISSN 2196-7806 ISSN 2196-7814 (electronic)
Research Topicsin Wind Energy
ISBN978-3-319-55163-0 ISBN978-3-319-55164-7 (eBook)
DOI 10.1007/978-3-319-55164-7
LibraryofCongressControlNumber:2017933865
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Preface
Thestandardapproachinthestudyofwindturbineaerodynamicsconsistsinusing
momentum analyses. The momentum theory of an actuator disk is an example of
momentum analysis. Blade element momentum (BEM) and the conventional
computation fluid dynamics (CFD) are two numerical methods also based on
momentum analyses. Velocity and pressure are the main variables used in
momentum analysis. The equations can also be formulated using vorticity as main
variable. This leads to an alternative approach referred to as vorticity-based
methods. The great potential of vorticity-based methods comes from the multitude
of formulations they offer, ranging from simple analytical models to advanced
numericalmethods.Theanalyticalmodel willbe referred toasvortextheories and
the numerical methods as vortex methods.
The term vorticity often intimidates the newcomer, but this fear vanishes when
one realizes that velocity and vorticity offer two different, but often equivalent,
points of view. For instance, the momentum theory of an actuator disk with con-
stant loadingcan be equivalently studied by considering the tubular vorticitysheet
thatispresent atthesurfaceofthestreamtube.Vorticityplaysanimportant rolein
wind turbine aerodynamics since strong vortices are present in the wakes in
particular.Vorticityandvorticity-basedmethodscannotbeomittedinabookonthe
topic. Most of the analytical models used in BEM methods are derived from ana-
lytical vortex models. Further, numerical vortex methods are now competing with
conventionalCFD methods interms ofaccuracy andcomputational time, andthey
are becoming a common tool for the study of wind turbine aerodynamics.
Theaimofthisbookistoshowtherelevanceofvorticity-basedmethodsforthe
study of wind turbine aerodynamics and to present historical and recent develop-
mentsinthefieldwithasufficientlevelofdetailsforthebooktobeself-contained.
This book is intended for students and researchers curious about rotor aerody-
namics and/or about vorticity-based methods. The book introduces the funda-
mentals of fluid mechanics, momentum theories, vortex theories, and vortex
methods necessary for the study of rotors and wind turbines in particular. Rotor
theoriesarepresentedinagreatlevelofdetailsatthebeginningofthebook.These
theories include the blade element theory, the Kutta–Joukowski theory, the
vii
viii Preface
momentumtheory,andtheBEMmethod.Differentmomentumtheoriesarederived
fromfirstprinciplesusingacriticalapproach.Theremainingofthebookfocuseson
vortex theory and vortex methods with application to wind turbine aerodynamics.
Examples of vortex theory applications that are discussed in this book are optimal
rotor design, tip-loss corrections, yaw models, and dynamic inflow models.
Historical derivations and recent extensions of the models are presented. The
cylindrical vortex model is another example of a simple analytical vortex model
usedinthisbook.Inthismodel,awindturbineanditswakearesimplifiedusinga
vortex system of cylindrical shape. Formulations equivalent to the ones used in a
BEMalgorithmareobtained.Themodelprovidesawake-rotationcorrectionwhich
greatly improves the accuracy of BEM algorithms. The cylindrical model is also
used to provide the analytical velocity field upstream of a turbine or a wind farm
(i.e., the induction zone) under aligned or yawed conditions. Such results are
obtainedinacoupleofsecondswithanimpressiveaccuracycomparedtonumerical
results from CFD methods which would require days of computation. Different
applications of numerical vortex methods are presented in this book. Numerical
methods are used for instance to investigate the influence of a wind turbine on the
incomingturbulence.Shearedinflowsarealsoinvestigated.Itisshowninparticular
that most vortex methods omit a term resulting in excessive upward displacement
ofthewind turbine wake.Many analytical flowsare derivedindetailinthisbook:
vortexrings,Hill’svortex,vortexblobs,etc.Theyareusedthroughoutthebookto
devisesimplerotormodelsortovalidatetheimplementationofnumericalmethods.
Several MATLAB programs are provided to ease some of the most complex
implementations: BEM codes, vortex cylinder velocity functions, Goldstein’s cir-
culation, lifting-line codes, Karman–Trefftz conformal map, projection functions
for vortex particle methods, etc.
Part I introduces the fluid mechanics foundations relevant to this book. Part II
introduces rotor aerodynamics, including momentum analyses, vortex models, and
the BEM method. Part III focuses on classical vortex theory results which origi-
nated from the study of rotors with optimal circulation. Part IV presents the recent
developments in rotor aerodynamics based on analytical vortex flows. Part V
presentsrecentapplicationsofvortexmethods.PartVIprovidesdetailedanalytical
solutionsthatarerelevantforrotoraerodynamics,eitherforthederivationofvortex
models or for the implementation and validation of vortex methods. Part VII is
dedicatedtovortexmethods.PartVIIIprovidesmathematicalcomplementstosome
chapters of the book.
Roskilde, Denmark Emmanuel Branlard
January 2017
Acknowledgements
Thecurrentworkwouldnothavebeenpossiblewithoutthesupportandhelpofmy
PhD supervisor Mac Gaunaa and the contributions from Spyros Voutsinas, Ewan
Machefaux, Philippe Mercier, Gregoire Winckelmans, Niels Troldborg, Giorgios
Papadakis,andHenrikBrandenborgSørensen.Iwouldliketothankmycolleagues
for their inspiration and fruitful discussions: Jakob Mann, Niels Sørensen, Curran
Crawford, Philippe Chatelain, Torben Larsen, Anders Hansen, Georg Pirrung,
Frederik Zahle, Mads Hejlesen, Juan Pablo Murcia, Alexander Forsting, Christian
Pavese, Michael McWilliams, Lucas Pascal, and Jacobus De Vaal.
Iamgratefultothepersonswho acceptedtoreviewsomechapters ofthis book
despitealimitedtime:DamienCastaignet,MichaelMcWilliams,MacGaunaa,Jens
Gengenbach, Gil-Arnaud Coche, Julien B., and Björn Schmidt.
Aboveall,IamgladforthemomentsoflifeandloveIexperiencedthankstomy
family and friends. I wish to share more of those with all of you: Ewan, François,
Aghiad,Mika,Dim,Heidi,Mike,K,Ozi,Bertille,Julie,Kiki,Loïc,Milou,Romain,
Sofie, Lucas P., Lucas M., Philipp, Jeanne, Alessandro, Julien, Sophie, Dad, and
Mom.
ix
Contents
1 Introduction .... .... .... ..... .... .... .... .... .... ..... .. 1
References .. .... .... .... ..... .... .... .... .... .... ..... .. 6
Part I Fluid Mechanics Foundations
2 Theoretical Foundations for Flows Involving Vorticity ... ..... .. 11
2.1 Fluid Mechanics Equations in Inertial and Non-inertial
Frames... .... .... ..... .... .... .... .... .... ..... .. 11
2.1.1 Physical Quantities ... .... .... .... .... ..... .. 11
2.1.2 Conservation Laws ... .... .... .... .... ..... .. 12
2.1.3 Fluid-Mechanic Equations
in a Non-inertial Frame.... .... .... .... ..... .. 17
2.1.4 Fluid Mechanics Assumptions... .... .... ..... .. 26
2.1.5 Usual Cases - Equations of Euler and Bernoulli.. .. 29
2.2 Flow Kinematics and Vorticity.. .... .... .... .... ..... .. 32
2.2.1 Flow Kinematics. .... .... .... .... .... ..... .. 32
2.2.2 Vorticity and Related Definitions .... .... ..... .. 33
2.2.3 Helmholtz (First) Law. .... .... .... .... ..... .. 36
2.2.4 Helmholtz-(Hodge) Decomposition... .... ..... .. 36
2.2.5 Bounded and Unbounded Domain - Surface
Map - Generalized Helmholtz Decomposition.... .. 37
2.3 Main Dynamics Equations Involving Vorticity.. .... ..... .. 38
2.3.1 Circulation Equation.. .... .... .... .... ..... .. 38
2.3.2 Vorticity Equation.... .... .... .... .... ..... .. 40
2.3.3 Stretching and Dilatation of Vorticity. .... ..... .. 40
2.3.4 Alternative Forms of the Vorticity Equation..... .. 42
2.3.5 Vorticity Equation in Particular Cases. .... ..... .. 43
2.3.6 Pressure... ..... .... .... .... .... .... ..... .. 44
2.3.7 Vortex Force, Image/Generalized/Bound Vorticity,
Kutta–Joukowski Relation.. .... .... .... ..... .. 45
xi