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Ebook: The Fluid Dynamic Basis for Actuator Disc and Rotor Theories
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The first rotor performance predictions were published by Joukowsky exactly 100 years ago. Although a century of research has expanded the knowledge of rotor aerodynamics enormously, and modern computer power and measurement techniques now enable detailed analyses that were previously out of reach, the concepts proposed by Froude, Betz, Joukowsky and Glauert for modelling a rotor in performance calculations are still in use today, albeit with modifications and expansions.
This book is the result of the author's curiosity as to whether a return to these models with a combination of mathematics, dedicated computations and wind tunnel experiments could yield more physical insight and answer some of the old questions still waiting to be resolved. Although most of the work included here has been published previously, the book connects the various topics, linking them in a coherent storyline
“The Fluid Dynamic Basis for Actuator Disc and Rotor Theories” was first published in 2018. This Revised Second Edition (2022) will be of interest to those working in all branches of rotor aerodynamics – wind turbines, propellers, ship screws and helicopter rotors. It has been written for proficient students and researchers, and reading it will demand a good knowledge of inviscid (fluid) mechanics.
Jens Nørkær Sørensen, DTU, Technical University of Denmark: “(…) a great piece of work, which in a consistent way highlights many of the items that the author has worked on through the years. All in all, an impressive contribution to the classical work on propellers/wind turbines.”
Peter Schaffarczyk, Kiel University of Applied Sciences, Germany: “(…) a really impressive piece of work!”
Carlos Simão Ferreira, Technical University Delft: “This is a timely book for a new generation of rotor aerodynamicists from wind turbines to drones and personal air-vehicles. In a time where fast numerical solutions for aerodynamic design are increasingly available, a clear theoretical and fundamental formulation of the rotor-wake problem will help professionals to evaluate the validity of their design problem. ‘The Fluid Dynamic Basis for Actuator Disc and Rotor Theories’ is a pleasure to read, while the structure, text and figures are just as elegant as the theory presented.”
The cover shows ‘The Red Mill’, by Piet Mondriaan, 1911, collection Gemeentemuseum Den Haag. Cover image: © 2022 Mondrian/Holtzman Trust.
To the first edition, May 2018:
Exactly 100 years ago prof. Joukowsky, one of the founders of modern aerodynamics, published the first rotor performance prediction. The tools of Joukowsky and his contemporaries were mathematics and wind tunnel experiments, nowadays we use mathematics, numerics, full scale experiments and, still, wind tunnel experiments. A century of research has expanded the knowledge of rotor aerodynamics enormously, with modern computer power and measurement techniques enabling detailed analyses of flows which were out of reach 100 years ago. However, the concepts for modelling a rotor in performance calculations as proposed by Froude, Betz, Joukowsky and Glauert are still used, be it with modifications and expansions. Especially the fast development of wind energy has revitalised the use of these concepts as they enable fast and accurate rotor designs.
Although the early models for a rotor have a proven track record, there is room for improvement in knowledge. The author returned to these models expecting that the combination of mathematics, dedicated computations and wind tunnel experiments would bring more physical insight. Furthermore, to the author’s opinion several old questions were still waiting to be resolved. The result of this curiosity driven work is this book. Although most work has been published in papers, the book adds a storyline and connects topics. When the research started, there was not a storyline at all, only questions. None of the hours spent by the author has been part of any official R&D project as it would have been impossible to define objectives, deliverables or deadlines.
Although numerical methods and experiments have contributed to the book, most of the content is a mathematical treatment of the fluid dynamic aspects of rotor modelling as, quoting Maxwell (1831-1879): ‘there is nothing more practical than a good theory’.(see http://listverse.com/2009/02/26/another-10-most-influential-scientists/, last visited March 21, 2022. The quote is sometimes attributed to social psychologist Lewin (1890–1947) but he lived after Maxwell.) A good knowledge of (inviscid) fluid mechanics is required to read the book, which is written for proficient students and researchers. Finally: on average the name of Joukowsky appears once per page, as his ideal rotor concept is examined in detail. The book is published one century after his first power prediction for a propeller in still air, a ship screw in still water or a hovering helicopter rotor, so it is dedicated to Nikolay Yegorovich Joukowsky.
To the second edition, May 2022:
In the first edition the question why the absolute velocity at the disc is practically constant, was not yet answered satisfactorily. Additional research gave this answer (van Kuik, 2020), which is included in the new chapter 7. This chapter now contains all information about the velocity distribution, which were part of chapters 5 and 6 in the previous edition. In this second edition these chapters treat only the average velocity. Other changes in the book concern textual corrections and rearrangements, an update of the reference list, and corrections in equations. Luckily these corrections turned out to be without consequences for the derivations and (numerical) results. In appendix D.1 of the first edition, the complete elliptic integrals were presented as programmed in the computer code, instead of how these integrals are described in literature like Branlard (2017). In this edition appendix D.1 has been adapted accordingly. Inspection and calibration by Peter Schaffarczyk confirmed previous calculations.
Gijs van Kuik, May 2018, May 2022.
Reading guide
The book intends to be interesting for all branches in rotor aerodynamics: wind turbines, propellers, ship screws, helicopter rotors. However, with the background of the author in wind energy, many references relate to the wind turbine field of science. Chapter 1 treats the historical context, gives the motivation of the book and formulates several research questions. Chapters 2 and 3 are best read together, as they treat the advantage of the dynamic method (force fields as input in the equations of motion) over the kinematic method (force fields determined once the flow is solved), with emphasis on the two phenomena governing the flow: energy, and vorticity. Chapter 4 shows why the actuator disc concept still is the correct basis for rotor design and analysis. The performance prediction by actuator disc theory is the topic of chapters 5 and 6: in 5 the Froude disc concept (no torque, no angular momentum in the wake) and in 6 the Joukowsky disc (with torque and angular momentum). Both analyses are complemented by results of potential flow calculations. In chapter 7 the velocity distribution at the disc is discussed, for both types of disc. As a side step chapter 8 treats two special topics: the role for conservative forces acting on a disc with thickness, and on a rotor blade with non-zero chord and thickness. Chapter 9 is a chapter with an open end. Herein some results of previous chapters are combined to explore improvements in design and analysis methods. Finally chapter 10 looks back.
The book is not written as a summation of independent chapters but follows a certain storyline. There is no forward referencing but quite some backward referencing. Chapters may be read apart from the others when these referenced equations and results are accepted, but it is advised to begin at the beginning and end at the end. An extract of the book has been published as chapter in the Handbook of Wind Energy Aerodynamics, (van Kuik, 2021)
