Actuator disc theory is the simplest rotor theory possible: the rotor is replaced by a permeable disc carrying an axisymmetric force field. It is more than a century old, with a first analytical result obtained by Froude in 1889. In 1918 Joukowsky published the first rotor performance prediction for a helicopter rotor in hover; in 1920 Betz and Joukowsky published the maximum efficiency of wind turbine rotors. In modern rotor design codes, this momentum theory still forms the basis, be it with many adaptations and engineering add-ons. This chapter treats the actuator disc theory in two versions. Best known is the classical theory relating to an actuator disc with thrust acting against the flow but without torque, so without wake swirl. This theory gives the Betz-Joukowsky limit. The results deviate when applied to a flow annulus instead of the entire stream tube, due to the role of the pressure exerted by one annulus to the other. The momentum theory for discs with thrust and torque is relevant for rotors operating with high torque at low rotational speed. For increasing rotational speed, the performance increases from zero to the Betz-Joukowsky limit. In all flow cases, with or without torque, the velocity vector in the meridional plane appears to be constant at the disc. For the performance per annulus and the performance with torque, the deviation from the classical momentum theory is explained by classifying force fields as conservative or non-conservative and investigating their impact on energy and momentum balances.

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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.".

Purchase print book: https://ebooks.iospress.nl/book/the-fluid-dynamic-basis-for-actuator-disc-and-rotor-theories-revised-second-edition

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Usually, the load on lifting bodies in incompressible, inviscid flow is determined by integration of the pressure on the body surface, once the flow is solved. Prandtl (Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch- Physikalische Klasse, vol. 1918, pp. 451-477) proposed an opposite method in which the body force field is the source term in the equation of motion to solve the flow. This force field method has not been used intensively but has regained importance in modern wind energy research. However, an analysis of which type of body force field generates vorticity and converts energy, and which body force field does not, is lacking. Prandtl's method is adopted here, but with the addition that the force field is allowed to be conservative or non-conservative. The relation between conservative/non-conservative body forces, vorticity generation, potential and kinetic energy and Helmholtz's vorticity theorems are derived. Similarly, the load on lifting bodies in two and three dimensions is classified as (non-)conservative, with some examples. To show that the force field method is consistent with the method where the load is output of an analysis, the expression for the Kutta-Joukowsky load and the relation between bound and trailing vorticity of a wing have been rederived using the force field method. The analysis confirms that this relation between bound and trailing vorticity is not governed by Helmholtz's theorems, as is often assumed, but by the non-conservative force field generating vorticity.

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The first version of the actuator disc momentum theory is more than 100 years old. The extension towards very low rotational speeds with high torque for discs with a constant circulation became available only recently. This theory gives the performance data like the power coefficient and average velocity at the disc. Potential flow calculations have added flow properties like the distribution of this velocity. The present paper addresses the comparison of actuator discs representing propellers and wind turbines, with emphasis on the velocity at the disc. At a low rotational speed, propeller discs have an expanding wake while still energy is put into the wake. The high angular momentum of the wake, due to the high torque, creates a pressure deficit which is supplemented by the pressure added by the disc thrust. This results in a positive energy balance while the wake axial velocity has lowered. In the propeller and wind turbine flow regime the velocity at the disc is 0 for a certain minimum but non-zero rotational speed. At the disc, the distribution of the axial velocity component is non-uniform in all actuator disc flows. However, the distribution of the velocity in the plane containing the axis, the meridian plane, is practically uniform (deviation < 0:2 %) for wind turbine disc flows with tip speed ratio > 5, almost uniform (deviation 2 %) for wind turbine disc flows with D 1 and propeller flows with advance ratio J D, and non-uniform (deviation 5 %) for the propeller disc flow with wake expansion at J D 2. These differences in uniformity are caused by the different strengths of the singularity in the wake boundary vorticity strength at its leading edge.

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The currently most used theory for rotor aerodynamics — Blade Element Momentum is based on the assumption of stationary wake conditions. However, an unsteady rotor loading results in an unsteady wake flow field. This work aims to study the impact of an unsteady actuator disc on the wake flow field using a free wake vortex ring model. The numerical results are compared to a wind tunnel measurement, where the wake flow of an actuator disc model undergoing transient load was obtained. The numerical results complement the experimental work while providing information such as the vorticity field and contributions from different vortex elements. The velocity at different locations is compared between the experimental and numerical results. The observed velocity peaks in the experimental results are also observed in the numerical results. A steeper ramp time results in a steeper velocity transient slope, and in turn in a larger amplitude of peak values. It is revealed that the rolling-up processes is the main cause for the velocity difference at various locations and in the three cases by decomposing velocity induced by different vortex element.

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Two new engineering models are presented for the aerodynamic induction of a wind turbine under dynamic thrust. The models are developed using the differential form of Duhamel integrals of indicial responses of actuator disc type vortex models. The time constants of the indicial functions are obtained by the indicial responses of a linear and a nonlinear actuator disc model. The new dynamic-inflow engineering models are verified against the results of a Computational Fluid Dynamics (CFD) model and compared against the dynamic-inflow engineering models of Pitt-Peters, Øye, and Energy Research Center of the Netherlands (ECN), for several load cases. Comparisons of all models show that two time constants are necessary to predict the dynamic induction. The amplitude and phase delay of the velocity distribution shows a strong radial dependency. Verifying the models against results from the CFD model shows that the model based on the linear actuator disc vortex model predicts a similar performance as the Øye model. The model based on the nonlinear actuator disc vortex model predicts the dynamic induction better than the other models concerning both phase delay and amplitude, especially at high load.

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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. This book 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.@en

The presence of conservative forces on rotor blades is neglected in the blade element theory and all the numerical methods derived from it (like e.g. the blade element momentum theory and the actuator line technique). This might seem a reasonable simplification of the real flow of rotor blades, since conservative loads, by definition, do not contribute to the power conversion. However, conservative loads originating from the chordwise bound vorticity might affect the tip vortex trajectory, as we discussed in a previous work. In that work we also hypothesized that this effect, in turn, could influence the wake induction and correspondingly the rotor performance. In the current work we extend a standard actuator line model in order to account for the conservative loads at the blade tip. This allows to isolate the influence of conservative forces from other effects. The comparison of numerical results with and without conservative loads enables to confirm qualitatively their relevance for the near wake and the rotor performance. However, an accurate quantitative assessment of the effect still remains out of reach due to the inherent uncertainty of the numerical model.

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The Blade Element Momentum model, which is based on the actuator disc theory, is still the model most used for the design of open rotors. Although derived from steady cases with a fully developed wake, this approach is also applied to unsteady cases, with additional engineering corrections. This work aims to study the impact of an unsteady loading on the wake of an actuator disc. The load and flow of an actuator disc are measured in the Open Jet Facility wind tunnel of Delft University of Technology, for steady and unsteady cases. The velocity and turbulence profiles are characterized in three regions: the inner wake region, the shear layer region and the region outside the wake. For unsteady load cases, the measured velocity field shows a hysteresis effect in relation to the loading, showing differences between the cases when loading is increased and loading is decreased. The flow field also shows a transient response to the step change in loading, with either an overshoot or undershoot of the velocity in relation to the steady-state velocity. In general, a smaller reduced ramp time results in a faster velocity transient, and in turn a larger amplitude of overshoot or undershoot. Time constants analysis shows that the flow reaches the new steady-state slower for load increase than for load decrease; the time constants outside the wake are generally larger than at other radial locations for a given downstream plane; the time constants of measured velocity in the wake show radial dependence.The data are relevant for the validation of numerical models for unsteady actuator discs and wind turbines, and are made available in an open source database (see Appendix).@en

Actuator disc theory is the basis for most rotor design methods, albeit with many extensions and engineering rules added to make it a well-established method. However, the off-design condition of a very low rotational speed Ω of the disc is still a topic for scientific discussions. Several authors have presented solutions of the associated momentum theory for actuator discs with a constant circulation, the so-called Joukowsky discs, showing the efficiency Cp → ∞ for the tip speed ratio λ → 0. The momentum theory is very sensitive to the choice of the radius δ of the core of the centreline vortex as the pressure and velocity gradients become infinite for δ → 0. Usually the vortex core area is not included in the momentum balance, as it vanishes for δ → 0. However, the pressure in the vortex core behaves as a Delta function and so contributes to the balance, thereby cancelling the singular behaviour. Applying this in the momentum balance results in Cp → 0 for λ → 0, instead of Cp → ∞. The Joukowsky actuator disc theory is confirmed by a very good match with numerically obtained results. At the disc the velocity in the meridian plane is shown to be constant. The Joukowsky calculations give higher Cp values than corresponding solutions for discs with a Goldstein-based wake circulation published in literature.@en

The state of the art engineering dynamic in ow models of Pitt-Peters, ¬ye and ECN have been used to correct Blade Element Momentum theory for unsteady load prediction of a wind turbine for two decades. However, their accuracy is unknown. This paper is to benchmark the performance of these engineering models by experimental and numerical methods. The experimental load and ow measurements of an unsteady actuator disc were performed in the Open Jet Facility at Delft University of Technology. The unsteady load was generated by a ramp-type variation of porosity of the disc. A Reynolds Averaged Navier-Stokes (RANS) model, a Free Wake Vortex Ring (FWVR) model and a Vortex Tube Model (VTM) simulate the same transient load changes. The velocity
eld obtained from the experimental and numerical methods are compared with the engineering dynamic in ow models. Velocity comparison aft the disc between the experimental and numerical methods shows the numerical models of RANS and FWVR model are capable to predict the velocity transient behaviour during transient disc loading. Velocity comparison at the disc between the engineering models and the numerical methods further shows that the engineering models predict much faster velocity decay, which implies the need for more advanced or better tuned dynamic in ow models.
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Wind turbine load alleviation has traditionally been addressed in the literature using either full-span pitch control, which has limited bandwidth, or trailing-edge flap control, which typically shows low control authority due to actuation constraints. This paper combines both methods and demonstrates the feasibility and advantages of such a combined control strategy on a scaled prototype in a series of wind tunnel tests. The pitchable blades of the test turbine are instrumented with free-floating flaps close to the tip, designed such that they aerodynamically magnify the low stroke of high-bandwidth actuators. The additional degree of freedom leads to aeroelastic coupling with the blade flexible modes. The inertia of the flaps was tuned such that instability occurs just beyond the operational envelope of the wind turbine; the system can however be stabilised using collocated closed-loop control. A feedforward controller is shown to be capable of significant reduction of the deterministic loads of the turbine. Iterative feedforward tuning, in combination with a stabilising feedback controller, is used to optimise the controller online in an automated manner, to maximise load reduction. Since the system is non-linear, the controller gains vary with wind speed; this paper also shows that iterative feedforward tuning is capable of generating the optimal gain schedule online.@en

The European Academy of Wind Energy (eawe), representing universities and institutes with a significant wind energy programme in 14 countries, has discussed the long-term research challenges in wind energy. In contrast to research agendas addressing short- to medium-term research activities, this eawe document takes a longer-term perspective, addressing the scientific knowledge base that is required to develop wind energy beyond the applications of today and tomorrow. In other words, this long-term research agenda is driven by problems and curiosity, addressing basic research and fundamental knowledge in 11 research areas, ranging from physics and design to environmental and societal aspects. Because of the very nature of this initiative, this document does not intend to be permanent or complete. It shows the vision of the experts of the eawe, but other views may be possible. We sincerely hope that it will spur an even more intensive discussion worldwide within the wind energy community.@en

In this paper the reductions of fatigue and extreme loads of wind turbine components are analysed. An individual flap controller was designed to reduce cyclic loads. The load reduction potential was computed for power production and start-up load cases with normal and extreme turbulence, extreme gust events, and direction changes according to the certification specifications. Additional to the highly investigated reduction of the blade root fatigue damage equivalent load, also significant reductions could be shown for both shaft and tower loads. When applying smart rotors, most components experience a fatigue load reduction of 5–15%, with the exception of the flapwise blade root moment, which is decreased by 23.8% and the blade root torsional moment which increases 14%. For the simulated ultimate loads, the flapwise root bending moment is reduced by 8%, while tip deflections get reduced by 6%. The most significant extreme load reduction can be found for loads in the tower that relate to asymmetry of the inflow, namely tower torsion and the fore-aft moment at the tower top. The blade root torsional moment is increased significantly. The changes in the ultimate load of all other components remain below 2%@en

Actuator disc theory is the basis for most rotor design methods, be it with many extensions and engineering rules added to make it a well-established method. However, the o_-design condition of a very low rotational speed of the disc is still a topic for scienti_c discussions. Several authors have presented solutions of the associated momentum theory for actuator discs with a constant circulation, the so-called Joukowsky discs, showing the e_ciency Cp ! 1 for ! 0. The momentum balance is very sensitive to the choice of the vortex core radius _ as the pressure and velocity gradients become in_nite for _ ! 0. Viscous vortex cores do not show this singular behaviour so an inviscid core model is sought which removes the momentum balance sensitivity to singular ow. A vortex core with a constant _ does so. Applying this results in Cp ! 0 for ! 0, instead of Cp ! 1. The Joukowsky actuator disc theory is con_rmed by a very good match with the numerically obtained results. It gives higher Cp values than corresponding solutions for discs with a Goldstein-based wake circulation published in literature. @en

Although the Blade Element Momentum method has been derived for the steady conditions, it is used for unsteady conditions by using corrections of engineering dynamic inflow models. Its applicability in these cases is not yet fully verified. In this paper, the validity of the assumptions of quasi-steady state and annuli independence of the blade element momentum theory for unsteady, radially varied, axi-symmetric load cases is investigated. Firstly, a free wake model that combines a vortex ring model with a semi-infinite cylindrical vortex tube was developed and applied to an actuator disc in three load cases: (i) steady uniform and radially varied, (ii) two types of unsteady uniform load and (iii) unsteady radially varied load. Results from the three cases were compared with Momentum Theory and also with two widely used engineering dynamic inflow models—the Pitt-Peters and the Øye for the unsteady load cases. For unsteady load, the free wake vortex ring model predicts different hysteresis loops of the velocity at the disc or local annuli, and different aerodynamic work from the engineering dynamic inflow models. Given that the free wake vortex ring model is more physically representative, the results indicate that the engineering dynamic inflow models should be improved for unsteady loaded rotor, especially for radially varied unsteady loads. @en