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.

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

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.

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.

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.

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.

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.