The rotor theories by Professor Joukowsky: Vortex theories

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The rotor theories by Professor Joukowsky: Vortex theories Valery L. Okulov a,b,n, Jens N. Sørensen a, David H. Wood c a b c

Wind Energy Department, Technical University of Denmark, DK-2800 Lyngby, Denmark Institute of Thermophysics, Siberian Branch of the Russian Academy of Sciences, 630090 Novosibirsk, Russia Schulich School of Engineering, University of Calgary, Canada

art ic l e i nf o

a b s t r a c t

Article history: Received 16 August 2014 Received in revised form 20 October 2014 Accepted 21 October 2014

This is the second of two articles with the main, and largely self-explanatory, title “Rotor theories by Professor Joukowsky”. This article considers rotors with ﬁnite number of blades and is subtitled “Vortex theories”. The ﬁrst article with subtitle “Momentum theories”, assessed the starring role of Joukowsky in aerodynamics in the historical context of rotor theory. The main focus in both articles is on wind turbine rotors, but much of the basic theory applies to propellers and helicopters as well. Thus this second article concentrates on the so-called blade element theory, the Kutta–Joukowsky theorem, and the development of the rotor vortex theory of Joukowsky. This article is to a large extent based on our own work, which constitutes the ﬁrst successful completion and further development of Joukowsky's work by deriving the ﬁrst analytical solution of his rotor. This rotor has a ﬁnite number of blades and will be compared with the rotor analysis of Betz and of others of the German school of aerodynamics. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Rotor aerodynamics Vortex theory Joukowsky History

Contents 1. 2.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Blade element momentum and vortex theories: retrospect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1. The history of the Kutta–Joukowsky (KJ) theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2. Origin of the blade element momentum (BEM) method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3. Development of the rotor vortex theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.4. Helical vortex theory as a basis for new solutions in rotor aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3. Modern BEM theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.1. A local blade element consideration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2. Tip correction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.3. Correction for heavily loaded rotors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.4. Yaw correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.5. Dynamic wake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.6. Airfoil data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.7. A cascade of blade elements with an overview of the KJ theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4. Modern stage of the rotor vortex theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.1. Lifting line theory of Joukowsky (NEJ) and Betz rotors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.2. Solution for the NEJ rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.3. Solution for the Betz rotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.4. Assessment of the different rotor theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.5. Comparison of NEJ and Betz rotors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.6. Experimental testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

n

Corresponding author.

http://dx.doi.org/10.1016/j.paerosci.2014.10.002 0376-0421/& 2014 Elsevier Ltd. All rights reserved.

Please cite this article as: Okulov VL, et al. The rotor theories by Professor Joukowsky: Vortex theories. Progress in Aerospace Sciences (2014), http://dx.doi.org/10.1016/j.paerosci.2014.10.002i

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1. Introduction This is the second of two articles with the main, and largely selfexplanatory, title “Rotor theories by Professor Joukowsky”. This article considers rotors with ﬁnite number of blades and is subtitled “Vortex theories”. The ﬁrst article with subtitle “Momentum theories” [1], assessed the starring role of Joukowsky in aerodynamics in the historical context of rotor theory. The main focus in both articles is on wind turbine rotors, but much of the basic theory applies to propellers and helicopters as well. Froude's momentum theory for an actuator disc was the ﬁrst elementary one-dimensional model of a rotor which was sufﬁciently accurate to describe correctly the averaged and simpliﬁed structure of the ﬂow, and establish the Betz–Joukowsky limit for the optimum performance of wind turbines. We have ascertained the role of Joukowsky in the derivation of this important limit on wind power conversion efﬁciency. Joukowsky's general momentum theory for actuator discs became the next stage in the development of rotor aerodynamics [1]. This theory followed from his understanding of the physical principles of the rotor operation based on vortex theory of screw propellers with constant circulation along the blade. In Ref. [1] we have analyzed the difﬁculties encountered when applying the general rotor momentum theory proposed by Joukowsky about a century ago on wind turbines. Undoubtedly, the general momentum theory of the rotor, as well as the simpler theory of Rankine–Froude, is still far from perfect due to simplifying assumptions and conditions. The simple momentum theories described in Ref. [1] were not sufﬁcient to design propeller blades, which was the most important rotor conﬁguration of the time. Therefore, in parallel, a speciﬁc design method was developed, which was based on dividing a blade into a number of span-wise sections and using vortex theory to determine the induced velocities. Thus this second article concentrates on the so-called blade element theory, the Kutta–Joukowsky theorem, and the development of the rotor vortex theory of Joukowsky. The article is to a large extent based on our own work, which constitutes the ﬁrst successful completion and further development of Joukowsky's work by deriving the ﬁrst analytical solution of the NEJ rotor. The acronym NEJ comes from the initial letters of Joukowsky's name (Nikolay Egorovich Joukowsky1). This rotor has a ﬁnite number of blades and will be compared with the rotor analysis of Betz and of others of the German school. Section 2 presents new historical facts regarding the Kutta– Joukowsky (KJ) theorem, as well as a review of the blade element momentum (BEM) method and the development of rotor vortex theories for estimating helical vortex structures, and Joukowsky's role in this development. Section 3 presents the most important steps in the development of the BEM theory, concluded with a derivation of the KJ theorem for a cascade of blade elements. The use of the KJ theorem to lifting line theory of rotors with ﬁnite number of blades is continued in Section 4, where the ﬁnal results are presented. Furthermore, the section provides a comparison and a critical review of different erroneous vortex theories of rotors.

Fig. 1. Originators of the Kutta condition and the KJ theorem: M. Kutta and N.E. Joukowsky.

For example, Anderson [2] (page 391) correctly credits Joukowsky [3] for deriving the KJ equation in 1906, but claims at the same time that Joukowsky was unaware of Kutta's work [4] (published in 1902: English translation in Ackroyd et al. [5]). This may not be strictly true, although there is no reference to Kutta in the 1906 paper. However, this may not be relevant, as Joukowsky without doubt was aware of the Kutta condition. Furthermore, Kutta did not derive the KJ equation. According to Panton [6] (page 427), the KJ equation was named “after the two people who discovered it independently”. This may be the result of a secondary referencing, as Lamb [7] (page 681) surprisingly also attributes the equation to both.2 Kutta [4] derived an equation for the lift for a thin, circular arc airfoil in an inviscid ﬂow obeying the Kutta condition, which states that the ﬂow leaves the airfoil smoothly at the trailing edge. He did not name the condition and he did not mention circulation and its relationship to the lift force. In 1910 Joukowsky [8] reviewed Kutta's work, re-derived his equation for lift, introduced the circulation and denoted the trailing edge conditions the Kutta condition, probably for the ﬁrst time.3 He then derived the Kutta– Joukowsky (KJ) theorem relating lift to the circulation around a body immersed in a two-dimensional ﬂow of an otherwise inviscid ﬂuid. This paper is one of the masterpieces of early aerodynamics research. In 1906, between these two papers, came Joukowsky [3], whose title can be translated as “On Bound Vortices”, although it appears in Ackroyd et al. [5] as “On Annexed Vortices”. He associated the body with the closed streamlines around a vortex of speciﬁed strength. He chose (in effect) a large circular control volume (CV) of radius R centered on the body and used the condition that the velocity potential of the ﬂow perturbed by a body decays as R 2 for sufﬁciently large R, so that there is no efﬂux of momentum from the CV. Thus the force due to pressure at R, determined using Bernoulli's equation, must be equal and opposite to that on the body. From this he derived the general form of what is now called the KJ equation, which is derived in a simpler manner below as Eq. (35). In 1910 Kutta [9] (English translation in Ackroyd et al. [5]) referred to Joukowsky [3] and extended the analysis of the circular arc airfoils. He also noted that his thesis [4] contained a much more detailed analysis including the KJ theorem. That thesis has

2. Blade element momentum and vortex theories: retrospect 2.1. The history of the Kutta–Joukowsky (KJ) theorem The history of Kutta and Joukowsky (Fig. 1) and their famous equation or theorem (KJ) is, even among experts, almost unknown. 1 Alternatively, his name is spelled Zhukovskii, Joukowski, Joukovskii or Žukovskij etc.

2 Ref. [7] is the last, 1945 edition. There is no reference to Kutta in the 1906 edition. The comments on the KJ equation cited here were also in the 1916 edition which introduced them. 3 In this connection it is interesting that the Kutta condition has been known as the Joukowsky–Chaplygin condition in Russia (Prof. S.A. Chaplygin was a colleague of Joukowsky). Consequently, in order to honor all initiators, this result should be re-named the “Kutta–Joukowsky–Chaplygin condition”. The well-established and convenient name Kutta condition should be considered as an easy abbreviation of this full name.

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been lost, according to Bloor [10] (page 297). On the available evidence, Kutta has been rightly credited with originating the Kutta condition, but his work was speciﬁc to thin circular arc sections. Joukowsky was solely responsible for the general form of the KJ equation. It is interesting to note that the Kutta condition explains how circulation is generated on a sharp-edge body such as an airfoil, but it is not necessary to assume the Kutta condition to derive the KJ equation, as will be demonstrated in Section 3.6. For the modern reader, the dates of these papers are tantalizing as they bracket Prandtl's introduction of the boundary layer concept [11]. Neither Kutta nor Joukowsky mention the role of viscosity in establishing the Kutta condition, nor is any consideration given to the effect of the viscous wake on the forces. Joukowsky [3,8] refers to the “pressure force” throughout. These omissions, which are perfectly easy to see in hindsight, are more interesting because Joukowsky also was versed in writing in German, which implies knowledge of the German literature [12]. Furthermore, Refs. [8,12] contain lift and drag results obtained from a wind tunnel that is now housed in the Joukowsky museum in Moscow, shown in Fig. 2, so we can assume that he was well versed in the practicalities of producing lift. Blasius' theorem [13] for the force acting on a body in an inviscid ﬂow is the other important contemporaneous development. As shown by Lamb [7], the KJ equation follows directly from Blasius theorem. His forcebased derivation of the KJ equation is a neat condensation of Joukowsky's original formulation, which he also derives from Blasius theorem.

2.2. Origin of the blade element momentum (BEM) method A representation of a rotor blade as the sum of its separate elements or blade sections was proposed in 1892 by Stepan

Fig. 2. Joukowsky's wind tunnel.

3

Karlovich Drzewiecki, a native of the Podolsk province of the Russian Empire (Fig. 3). According to his theory, the rotor is cut into cylindrical blade elements (Fig. 4a), with velocity triangles plotted for each element (Fig. 4b). The main importance of Drzewiecki's theory consists in the creation of an efﬁcient method for blade design. In his ﬁnal description of the theory in 1910, Drzewiecki [14] tried to determine the conditions for maximum performance of airscrews, consisting of a deﬁnition of the optimal angle of attack, α, methods for choosing the optimum pitch angle, φ (the angle between the chord and the plane of rotation), as well as the length of the blades, their cross-sectional shapes and chords, and a formula to determine a number of blades. Drzewiecki's theory was easy and convenient for engineering purposes. For that reason a promoted version of this method is still widely used when designing rotors. Nevertheless, his original theory even in its ﬁnal form [14] contains a signiﬁcant defect: it did not take into account the deviation from the initial upwind ﬂow at the rotor plane by the inﬂuence of the rotor itself, which changes the initial axial velocity V by v (the so called induced axial velocity). In his time, not all researchers accepted the results of the momentum actuator disc theory [1], because they did not understand the cause of additional velocity v due to the induction by the vortices in the wake downstream the rotor, explained by Joukowsky in 1912 [15]. Vetchinkin [16] wrote that “… an existence of the error remained, and we see that the screws of Drzewiecki (manufactured at the Ratmanov factory in Paris), which are absolutely correctly designed in accordance with his theory, are replaced in the market by purely empirical screws of Shovier, Regis, Levasseur, and others, who chose the form and size exclusively from tests, without any theoretical considerations”. The greatest failure of the original Drzewiecki's theory arose for hovering helicopter rotors, when there is no mean ﬂow through the rotor. This error was not solved, as noted by Vetchinkin [16], until 1910 when two students of the Imperial Technical School (G.H. Sabinin and B.N. Yuriev, who both were pupils of Joukowsky) used the Rankine–Froude elementary actuator disc theory for correction of the velocity ﬁeld in the rotor plane (Fig. 4c). After that, in 1910–1912, they started designing and studying new theoretical screws. Despite numerous sources describing this story, including a reference made by Joukowsky [17] and a description of their theory in the works of Vetchinkin [16,18], the authors were not able to publish their work in full at that time, because of their conscription and subsequent captivity in Germany. Later in 1923 Yuriev (see Fig. 3) explained their work in his article [19]. It should be noted that in Germany the effect of an additional axial velocity was introduced in Drzewiecki's theory in 1915 by Betz [20]. In England, it was taken into account from 1919 after the publication of Fage and Collins [21], while the correction to the axial velocity of the rotor was taken from experiments. For some reason, distrust in Froude's theory persisted in England until at least this publication in 1919.

Fig. 3. Founders of the blade element method for rotor calculation: S.K. Drzewiecki, B.N. Yuriev, G.A. Bothezat and G. Glauert (left to right).

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Fig. 4. Drzewiecki's blade element method: (a) sketch of the section; (b) the initial velocity triangle by Drzewiecki; and (c) new velocity triangle with correction factor by Yuriev and Sabinin.

Fig. 5. The original diagrams of screw vortex system with circulation distributions (J Γ): (a) by Joukowsky [15] and (b) by Vetchinkin [18].

In 1917 in Petersburg, Professor Georgy Bothezat (Fig. 3) proposed a more accurate version of the actuator disk theory in which the additional velocity at the blades and in the wake behind screw includes both axial and azimuthal components. The BEM theory was corrected by the deﬁnition of both induced velocity components [22]. On the basis of this completed version of the BEM theory, Henry Glauert (Fig. 3) corrected the Betz–Joukowsky limit [1] for each element of the rotor blade [23]. He proved that the limit of maximum power of a wind turbine should always be less than the Betz–Joukowsky limit, determined by momentum theory, and dependent on tip speed ratio:

λ0 ¼

Ω0 R U1

;

ð1Þ

where U 1 is wind speed and Ω0 is angular velocity of the rotor whose radius is R. It should be noted that the correction of Sabinin–Yuriev and its further clariﬁcation by Bothezat did not take into account the effect of a related interaction of the elements of different blades, i. e. this correction can only be applied for screws with a small number of thin blades. The exact calculation method by an unwinding of the cylindrical section of the rotor to a planar cascade of the blade proﬁles was introduced by Joukowsky [24,25]. Though Joukowsky referred to Chaplygin's analytical solution for the inﬂuence of the cascade of parallel plates on two-dimensional ﬂow [26], it was his achievement to apply Chaplygin's solution to determine the inﬂuence of number and spacing of the blade proﬁles to obtain a better approximation of

the rotor than the single blade or isolated airfoil in the original BEM theory. But none of the considerations included the ﬂuid interaction along the rotor blades. More convenient results are to be expected from the vortex theory of a rotor with a ﬁnite number of blades. 2.3. Development of the rotor vortex theories In the initial history of the aerodynamic theory two “schools” dominated the conceptual interpretation of the optimum rotor. In Russia, Joukowsky deﬁned the optimum rotor as one having constant circulation along the blades [15]. We call any rotor with constant circulation a NEJ-rotor in accordance with the initial letters of the full name – Nikolay Egorovich Joukowsky. The vortex wake for an Nb-bladed NEJ rotor consists of Nb helical tip vortices of strength Γ and an axial hub vortex of strength NbΓ . A simpliﬁed model of this vortex system is a rotating horseshoe vortex with a ﬁnite vortex core (see Fig. 5a). The other school, of Prandtl with his pupil Betz [27], assumed that optimum efﬁciency is obtained when the distribution of circulation along the blades generates a rigidly helicoidal wake that moves in the direction of its axis with a constant velocity. Betz used a vortex model of the rotating blades based on the lifting-line technique of Prandtl in which the vortex strength varies along the wingspan (Fig. 6a). We will call this a Betz rotor for the same reason. The creators of this new vortex theory were able to ﬁnd only partial solutions for a uniform azimuthal distribution of circulation

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Fig. 6. (a) An example of the vortex system with the helicoidal sheets corresponding to lifting line theory of Betz rotor [27] and (b) sketch of the Prandtl's “tip loss” correction from Ref. [28].

Fig. 7. (a) The vortex system for rotor with an inﬁnite number of blades [15,28] and (b) Betz sketch for different distributions of circulations along radial directions along the disk [29].

on the actuator disc replacing the rotor, or, as they called it, the limiting rotor with an inﬁnite number of blades (Fig. 7a). In 1912 Joukowsky wrote [15]: “The main idea of the bound vortices, which forms the basis for this article, could permit performing all calculations on the basis of the correct velocities of the relative liquid motion, but this analysis would be very complex”. It is interesting to note that Prandtl's 1923 review [28], after a brief description of the Betz vortex model, concluded: “In order to obtain a quantitative result, we shall for simplicity's sake next think of a screw having a large [inﬁnite – comment by present authors] number of blades”. In other words, both leading schools avoided searching for the solutions for rotors with a ﬁnite number of blades. As a result, Joukowsky created the general momentum

theory for wind turbines [17], and Prandtl determined an approximate correction for a transition from the disc theory to the rotor with a ﬁnite number of blades based on the solution to the simpler problem of the ﬂow around a cascade of semi-inﬁnite plates. Prandtl's “tip loss” correction (Fig. 6b) given in appendix to Betz [27], is still the main approach when designing rotor blades. In the general momentum theory Joukowsky employed the assumption that the circulation is constant over the surface of the rotor disc (top part of Fig. 7b). Alternatively, Betz assumed that the circulation could be distributed (middle part of Fig. 7b) and Prandtl added the “tip loss” correction (bottom part of Fig. 7b) to include the effect of ﬁnite Nb (Fig. 6) for the vortex rotor disk case (Fig. 7a). Betz [29] noted that the difference with a rotor with

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Fig. 8. S. Goldstein and a fragment of the page with his solution.

constant circulation is not as big as it may appear at a ﬁrst glance. Yuriev [19] notes that the papers with the theory of the Prandtl– Betz school arrived in Russia just after the First World War. On the other hand, before this time in 1913 Vetchinkin [18] suggested the screw showed Fig. 5b, which was quite different from the NEJ rotor (Fig. 5a) due to the arbitrary circulation along the blade. He proposed a variational method for calculating the circulation at each vortex element on the blades. Certainly, the piecewise approximation by Vetchinkin (Fig. 5b) does not directly lead to Betz's optimal screw condition [27], formulated in 1919 for continuous vortex sheets (Fig. 6a), but the conceptual idea is the same. Note that the study of Betz was a logical extension of Munk (another postgraduate student of Prandtl), who showed for the ﬁrst time that an elliptical distribution of load (or circulation) along the wing of a ﬁnite span minimizes the induced drag (see e.g. [30]). Extending their results to rotors, Betz established the requirements for minimization of the induced drag for rotating blades, which requires a regular wake structure (Fig. 6a) with a more complex load distribution than the elliptical one for the wing. Betz also gave the correct formulation of this theorem for the optimal vortex system of helicoidal sheets behind the rotor, though his arguments were not fully complete [31]. In 1937 Polyakhov [32] completed Betz's proof and suggested to use Ritz method for numerical optimization of the Betz rotor. Betz was not able to derive a new form of the load distributions for the rotor with ﬁnite number of blades, which instead was modeled using Prandtl's “tip loss” correction for the blade circulation (Fig. 7b in bottom). In 1929 Goldstein [33] derived an exact analytical solution for Betz rotors with 1, 2, 3 and 4 blades by complete determination of the equilibrium motion of inﬁnitely thin helicoidal vortex sheets (Fig. 6a). He found the optimum circulation distribution along blades using a complex form of trigonometric series with coefﬁcients expressed via modiﬁed cylindrical functions, which formed a series of the Kapteyn type. Later this circulation was named the Goldstein factor or Goldstein circulation function (Fig. 8). Though the analytical solution had been derived, it was very difﬁcult to use for designing an actual blade. In 1948 Theodorsen initiated the determination of the Goldstein factor using an electromagnetic analogy [34]. Fig. 9 illustrates how he prepared the measurement equipment to demonstrate the method and also how the electromagnetic model of the helical “wake” can be made from twisted metal foil. We should also note the approach differing from Ref. [33] for description of the non-uniform velocity ﬁeld caused by a ﬁnite number of blades. Instead of exact simulation of a vortex wake behind the rotor, the authors [35] (and references therein) extended the disk vortex theory by adding azimuthal harmonics depending on the number of blades. Each mode was analytically integrated along the semi-inﬁnite wake and the classical axisymmetric disk solution considered as the zeroth order term of the expansion. The authors of this approach claimed that a few low

Fig. 9. Theodorsen prepares his equipment to demonstrate an electromagnetic analogy to measure the Goldstein factor (left) and a fragment of the wake model from twisted metal foil (right).

Fig. 10. Elementary vortex structures: a straight line or a point vortex, vortex ring and a helical vortex (from left to right).

order harmonics were sufﬁcient to achieve an accurate result, though they did not prove and test it. They compared their results with the disk theory only, i.e. with the ﬁrst term of the same expansion and omitted any comparisons with other known solutions for the helicoidal vortex sheet [33] and a single helical vortex ﬁlament [36]. A complete analytical investigation of the original problem formulated by Joukowsky for the NEJ rotor [15] with a ﬁnite number of blades and ﬁnite size of the core of the tip vortices was recently performed by Okulov and Sørensen [37]. The results of this study are presented below in Section 4. In addition in Refs. [38,39] a new solution is provided for Betz rotors with Goldstein distributions of circulation using a new analytical representation of the velocity ﬁeld induced by helical vortex ﬁlaments [40]. This new approach gives an effective algorithm for the optimization of a wind turbine modeled as a Betz rotor. The long development time for these complete solutions of both rotors may be explained by the very complex analysis of the velocity induced by the helical trailing vortices. Indeed before these recent studies, there was neither a theory nor analytical solutions for the velocity ﬁeld due to helical vortices with ﬁnite cores, which, according to Joukowsky and Betz, are the basic elements of the vortex wake for the NEJ rotor. Furthermore, Goldstein's solution for the helical vortex

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sheets in the wake model of the Betz rotor remains very complex to analyze. In the next section we will describe the development of the helical vortex theory. 2.4. Helical vortex theory as a basis for new solutions in rotor aerodynamics Research on helical vortices has a long history, which dates back to the famous work of Kelvin in 1880 on the helical perturbations of the columnar vortex, i.e. helical vortices with a large pitch [41]. Helical vortices are of fundamental importance for ﬂuid mechanics because they describe one of the main states of swirling ﬂows. Examples of vortex structures with a helical form are described widely in the literature: tip vortices in the wakes of rotors (propellers, helicopter, turbines and wind turbines), concentrated vortices in rotating tanks and vortex devices; tornadoes; whirlpools in ﬂuids ﬂowing out from the tank; vortex structures caused by the vortex breakdown behind the delta wing and in pipes, etc. In particular, single and multiple helical vortices play a special role in the vortex theory of the rotor with a ﬁnite number of blades. An accurate estimation of velocities in the rotor plane requires knowing the velocity ﬁeld induced by the helical vortices of the wake. These velocities are signiﬁcant compared to the wind speed or velocity of the free stream at an optimal operating regime given by the Betz–Joukowsky limit [1]. In the retrospective review on the airscrew theory developed in Russian Central Aerohydrodynamic Institute (TsAGI), Maykopar [42] pointed out that at the increased speed of the ﬂight the mean velocity estimated by disk theories should be replaced by a correctly distributed velocity, representing a rotor with a ﬁnite number of blades, although this represents the most difﬁcult part of the screw calculation. In this context, the helical vortex theory is of special interest. Therefore, starting with the inspirational work of Joukowsky [15], the theory of helical vortices has been actively studied as a prerequisite to understand and analyze rotor aerodynamics. Doubly inﬁnite canonical helical vortices or vortex ﬁlaments are fundamental objects of vortex dynamics along with rectilinear (or point) vortices and vortex rings (Fig. 10). The simplest vortex is

7

an inﬁnitely long rectilinear vortex with no curvature or torsion. The vortex ring is more complex, since it has curvature but no torsion, whereas helical vortices have torsion in addition to curvature. These elementary structures also differ from the hydrodynamic point of view. Indeed, an isolated rectilinear vortex even with an inﬁnitely thin core does not distort in an inﬁnite undisturbed space because it only induces velocities in the plane normal to the vortex. In other words, there is no self-induced motion. For vortex rings, the self-induced motion is perpendicular to the ring plane with a velocity proportional to the circulation and radius with a logarithmic dependence on the vortex core thickness [43]. In addition to the forward motion of the vortex ring, the selfinduced motion of the helical vortex is accompanied by rotation as well, and its self-induced velocity additionally depends on the torsion. Unlike the theory of point vortices and vortex rings, helical vortex theory has not systematically been described in the literature and is not considered in textbooks and monographs on classical ﬂuid mechanics. For example, Saffman's book on vortex motion includes the large-pitch behavior of helical vortices only [44]. This was most likely due to a lack of a simple closed theory of helical vortices to facilitate the analysis of various hydrodynamic problems. An attempt to ﬁll this gap was made in the recent work of Okulov and colleagues [40,43,45–50]. The inﬁnitely thin helical vortex ﬁlament (Fig. 11a), is the fundamental geometry for vortices trailing from rotors in an inviscid incompressible ﬂuid, similar to a straight line ﬁlament for wing tip vortices and the inﬁnitely thin vortex ring as the generic vortex that moves under its self-induced velocity. The ﬁlament has radius R and pitch h ¼ 2π l (where appropriate, l will also be called the “pitch”) and the normal (n), tangential (t) and binormal (b) directions that are deﬁned in Fig. 11a. A doubly inﬁnite helix and the ring are the only vortex structures that can translate without deformation in their axial direction. In addition, the helix induces a ﬂuid rotation. In the rotor models described in the previous sections, the helical tip vortices or helicoidal wakes are entirely responsible for the induced ﬂow through the blades. Since a helicoidal wake can be viewed as a superposition of helical vortices, the latter is the fundamental vortex structure of any

Fig. 11. (a) The geometrical parameters of helical ﬁlament and (b) helical vortex of ﬁnite core.

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rotor wake. Unfortunately, in contrast to a straight line vortex, the Biot–Savart integral for the ﬂow induced by a helical ﬁlament does not have a closed form. It cannot be represented as a simple pole like that for a point vortex, or expressed as complete elliptic integrals, as in the case of an inﬁnitely thin vortex ring [43]. Therefore Maykopar [42] noted that the induced velocity of helical vortices must be calculated by special functions either through their integral representation, arising from the Biot–Savart integral, or by means of Kapteyn's type series of Bessel functions. Unfortunately the ﬁrst results of TsAGI attempts to conduct such direct calculations of the induced velocity are not very accurate. Nevertheless due to the very important need for propeller designs for the burgeoning aviation industry, these results were compiled in diagrams and tables [51], although this calculation technique was not sufﬁciently accurate. For example, they have calculated asymmetry velocity proﬁles along the circular sections of the rotor plane (Fig. 12a). The errors in these calculations remained even after the solution of the Biot–Savart integral in terms of Kapteyn series. This approach was used in one radial direction along the blade, Lerbs [52], who tried to describe the radial dependence of the induction factor by considering the blade contribution only. The

full trigonometric form of the Kapteyn's series was derived by Hardin [36] for the velocity ﬁeld induced by a single, singly inﬁnite helical vortex ﬁlament. In the notation of Ref. [40] the components of ﬂuid velocity induced by Nb helical vortices in cylindrical coordinates (r, θ, z) are given as ( ) I m mr=l K 0m mR=l N Γ 1 Γ R Nb 1 cos mχ n ; 2 ∑ ∑ m 0 uz r; χ ¼ b mR=l K mr=l I 2π l 0 m m πl n ¼ 1 m ¼ 1 ð2Þ N Γ uθ r; χ ¼ b 2π r

0 Γ R Nb 1 þ ∑ ∑ m π rl n ¼ 1 m ¼ 1 1

(

) I m mr=l K 0m mR=l cos mχ n ; I 0m mR=l K m mr=l

ð3Þ where I m ðxÞ and K m ðxÞ are modiﬁed Bessel functions and the derivatives (K0 and I0 ) are with respect to the order m. χ ¼ θ z=l and χ n ¼ χ þ2π ðn 1Þ=N b . In the notation “f:g” the upper symbols in brackets corresponds to radius r o R and the lower to r Z R. When χ ¼ 0 and χ n ¼ 0 the summations in (2) and (3) become classical Kapteyn series changing along the radial direction including the point on a helical ﬁlament when x ¼ r=R ¼ 1. The coefﬁcients of the Kapteyn series when x ¼ r=R-1 grow with an

Fig. 12. Different ways for calculations of azimuthal velocities induced by helical vortex ﬁlament along full azimuth with various radius: (a) asymmetry velocity proﬁles along the circular sections of the rotor plane [42] and (b) symmetric calculations in a cross-section the helical vortex by method of Ref. [49].

Fig. 13. (a) Different order m of the terms in the Kapteyn's series growing towards the helical ﬁlament at point r/R x ¼1. Cross-sections to axial direction of stream tubes around helical vortex ﬁlament, (b) inaccurate calculation using a ﬁnite number of terms of the Kapteyn's series, and (c) correct calculation by method of the singularity separation [49].

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increasing m (Fig. 13a), causing a singular behavior in the solution on the ﬁlament. The singularity was separated only in one radial direction by Lerbs [52] and Morgan and Wrench [53] who give the correct calculation of the induction factor of one rotor blade when χ ¼ 0 and χ n ¼ 0. It is clear that the regular behavior of the solution along other radii that do not pass through the ﬁlament is provided by alternating harmonics in the trigonometric series (2) and (3) of the Kapteyn type. To be precise: the series should contain inﬁnite harmonics since the calculation of the velocity ﬁeld by the ﬁnal term of the truncated series gives pseudo peaks on the central cylinder which contains the ﬁlament (Fig. 13b). An estimation of the velocity behavior in all directions on the disk is important for a transition from the ﬁlament to helical vortex with a ﬁnite core and for correct estimations of an induction factor of Nb -bladed rotor generated by the Nb helical vortices in the wake. The absence of a solution for the continuous ﬁeld induced by Nb vortices restricted

Fig. 14. Fragment of the Joukowsky's article [15] published in 1912 with the asymptotic representation of the singular behavior (12) of the velocity in the vicinity of the helical vortex ﬁlament.

9

applications of the radial approximation by Lerbs [52] and his followers and caused them to undertake direct simulations of the Goldstein function. In 1964, high-precision tables of this function were computed via direct calculation of the Goldstein solution in the form of Kapteyn series on an IBM 7090 by Tibery and Wrench [54]. These tables were reproduced by Wald [55]. The immediate application was to design efﬁcient and low noise marine propellers, so this solution was tabulated for relatively low tip speed ratios typical for shipbuilding, i.e., the solution was not complete, and was unsuitable for the whole range of operating parameters of the rotors used in other technical ﬁelds, particularly wind turbines. It is interesting to note that this problem of incorrect calculations of the induced velocities (Fig. 13b) is caused by speciﬁc behavior of the solution near the ﬁlament, found originally by Joukowsky in 1912 [15]. The asymptotic representation of the solution in the vicinity to a vortex ﬁlament via a sum of pole, logarithm and regular component is usually associated with Batchelor's analysis [56] (page 510). However Ricca [57] has found that this development was ﬁrst obtained by his countryman Da Rios in 1911; the fragment of the Russian article represented in Fig. 14 shows that it was derived independently by Joukowsky in 1912. Indeed, if we do not take into account these singularities then, as shown above on Fig. 13b, the calculation of the trigonometric series (2) and (3) of the Kapteyn type would present major difﬁculties. In 1997 Wood and Guang rediscovered this problem and posed it as problem no. 97-18 in the SIAM Review [45]. Its solution was found in the following year by separation of the singularities from the series in a form of the pole as a solution for a simple point vortex plus a logarithm from the distance [58]. In 1999, Boersma and Wood [46] used this separation to show that the regular remainder is signiﬁcant; therefore, they replaced it by an equivalent integral representation and calculated the integral numerically with an accuracy of eight signiﬁcant digits. In doing so, they conﬁrmed Kuibin and Okulov's ﬁnding [59] that the axial velocity of the helical ﬁlament (in the z direction in Fig. 11) has a leading

Fig. 15. Cross-sections in the axial direction of stream tubes around helical vortex ﬁlaments with monopole vortex core (top row) and with dipole vortex core (bottom row). Flows induced by helical ﬁlaments of different helical pitches: h/R¼ 8; h/R ¼2; h/R¼ 1 (from left to right) [48,49].

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10

term proportional to 1/l at small pitch l. As shown below, this limiting case is a fundamental result for wind turbine analysis. It should be pointed out, that a more efﬁcient solution to the total problem (for any value of l) is found in a separation of the singularity in a special form which takes into account the helical structure of the ﬂow as well as depends on the curvature and torsion. When the ﬁrst two dominant singularity terms are extracted, (2) and (3) are approximated by the formulas [40,49]: ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p 4 2 Nb Γ l þ R2 Nb ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ∑ Re uz r; χ ¼ þp 4 2 2π l 0 l þ r2 n ¼ 1 2 31 0 1 2 2 2 iχ n 2 7e l 3r 2l 3R 2l 6 B C ξ þ i χ n 7C þ @ 4 8 ξ 5A 3=2 þ 3=2 Aln 1 e e eiχ n 24 2 2 l þ r2 l þ R2 ð4Þ Γ uθ r; χ ¼ 2π r 2

(

0

)

Nb 0

p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4 2 l þ R2 N b ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ∑ Re p 4 2 l þ r2 n ¼ 1

1

31

iχ l B 3r 2 2l 3R2 2l C 6 7e n 7C þ @ þ 4 8 ξ Aln 1 eξ þ iχ n 5A 3=2 3=2 i χ e e n 24 2 2 2 2 l þr l þR 2

2

ð5Þ where qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2 2 r 1 þ 1 þ R2 =l exp 1 þ r 2 =l qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

eξ ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2 2 1 þR2 =l R 1 þ 1 þ r 2 =l exp (unfortunately, this expression was misprinted in Refs. [38,39,49] and velocities of (4) and (5) were misprinted in Ref. [39]). Here we use the notation “ 7 ” and “f:g” in which the upper sign or symbols in brackets corresponds to r o R and the lower to r Z R. Otherwise, it can be said that the pole and the logarithm were not written in the usual Cartesian system with the usual distance (r 2 2rR cos χ þ R2 ) as was done, for example, in Ref. [46], but in alternative distorted distance including torsion or helical pitch l: 2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ32 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 2 2 exp 1 þ r 2 =l exp 1 þR2 =l 6 exp 1 þr =l 7 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 5 2rR cos χ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

4r 2 2 2 1 þ 1 þ r 2 =l 1 þ 1 þ r 2 =l 1 þ 1 þ R2 =l

2

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ32 2 1 þ R2 =l 7 6 expqﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ5 : þ 4R 2 1 þ 1 þ R2 =l Note that the possibility of solving the problem by this method appeared in earlier works by Okulov [60,61]. A prototype for the solution can be independently found in Lerbs [52], but he could not describe the behavior for all locations around the helical ﬁlament and could only simulate the velocity in one radial direction. As a result of Okulov's approach, the approximation by ordinary functions of (4) and (5) included all information about the solution, while the regular remainder was so small that it could be ignored in most cases or corrected by an additional two terms of poli-logarithm types, as it was shown in Ref. [40]. A calculation of the velocity ﬁeld induced by the helical ﬁlament by (4) and (5) eliminates errors (Fig. 13c), obtained by direct calculation based on the Kapteyn series (2) and (3) (Fig. 13b). It is important to note that this approach is universal. The approximation in form of sum of the dipole, pole and the logarithm for the helical ﬁlament with a continuous distribution of dipoles along helical axis uses a procedure in the same distorted coordinates [48,49]. Both solutions are shown in Fig. 15. All the analysis to date (and all the discussion so far) assumes vortex ﬁlaments, that is, vortices whose core radius is too small to be dynamically important (Fig. 11a). This is only a ﬁrst approximation: the next important question is how the velocity ﬁeld induced by vortex with ﬁnite core (Fig. 11b) can be expressed by using the approximations (4) and (5). The helical vortex with ﬁnite core is important because it is used in the Joukowsky's vortex model of the NEJ rotor. An answer was found by a development of Dyson's vortex model for describing Saturn's rings [62]. Indeed, for both the case of a vortex ring and a helical vortex, the solution outside of the ﬁnite vortex core can be written in form of a superposition of singular solutions for vortex ﬁlaments with different distributions of multipole singularities [49]. Fig. 16 shows a comparison between the velocities induced by a helical vortex with ﬁnite core and an inﬁnitely thin helical ﬁlament. The difference is not very signiﬁcant even for the very large vortex core (0.3R) used in the plots. It should be noted that the difference disappears completely as the core size decreases. Thus, solutions for the ﬁlaments represented by formulas (4) and (5) are a good approximation for rotor optimization.

Fig. 16. Comparison of the axial (left) and azimuthal (right) velocity proﬁles induced by helical vortices: thin lines – inﬁnetely thin vortex ﬁlament, thick lines – the helical vortex with the core of radius ε ¼ 0:3R.

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A numerical calculation [63] provides additional veriﬁcation of the above conclusion. The use of just two terms in Dyson's development for helical vortices has been tested by a calculation where one turn of a helical vortex was replaced by 67 million vortex elements. The results of both calculations coincided to the numerical precision used in the calculations (Fig. 17). Both data of Figs. 16 and 17 conﬁrm that the model of helical ﬁlament (4) and (5) can be considered as a good approximation for the ﬂow outside a vortex core up to ε ¼ 0:3R. The ﬁnal element of the theory of helical vortex, which is required for obtaining solutions for the NEJ rotor, is a deﬁnition of the self-induced velocity of the helical vortex with a ﬁnite core. Self-induced motion of helical vortices is in practice manifested as a precession or rotation of the helix or the vortex core about the z-axis (see Fig. 11). Although each helical vortex ﬁlament is a singular object, those combining to form a ﬁnite vortex core with a dense distribution of the ﬁlaments are integrable. As mentioned previously, the helical vortex does not have a closed analytical expression for its self-induced velocity. Therefore for a long time, from the ﬁrst work by Kelvin in 1880 [41], various asymptotic methods were developed assuming either small or large values of the torsion or helical pitch [46,59,64,65], or by using approximate, but more general approaches, of the desingularization of the Biot– Savart integral [56,66–72]. Another technique is based on an explicit separation of the singularity. The simplest way is to separate a pole (as for example a point vortex) and a logarithm from the Cartesian distance (which occurs for any curved vortex) in the integrand of the Biot–Savart law before integrating, and

Fig. 17. Comparison of axial velocity uz induced by helical vortex of 2πl ¼ 0:8R with ﬁnite core along full azimuth radius inside and outside of the helix core of ε ¼ 0:3R (symbols) analytical solution based on the two terms of the Dyson's development [49]; (lines) high calculations based on the vortex method with using 67,000,000 vortex elements [63].

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determine the large regular remainder of the helical vortex numerically [46]. The next logical technique for separating out the singularity is to employ the so-called “osculating” vortex ring, whose self-induced motion is well known (see Fig. 18). This approach was proposed by Moore and Saffman in 1972 [70], but they were not aware that it was originally used by Joukowsky in 1912 [15]. Indeed, Joukowsky clearly demonstrated that a formal representation of the self-induced velocity of a helical vortex, after neglecting some terms in the Biot–Savart integral, may be transformed into a self-induced motion of an osculating vortex ring:

Γ R ε R ub ¼ ln : ð6Þ 4π R2 þ l2 2 R2 þ l2 We regret that western readers were (and probably still are) not familiar with this major contribution made by Joukowsky more than 60 years before being rediscovered by other scientists. At a ﬁrst sight, this approach is more accurate than the ﬁrst one, though the second regular integral term still remains sizeable and with very complex aspects to the integration. This term was neglected by Joukowsky when he tried to obtain an analytical form of the solution by simplifying the Biot–Savart integral. The neglected regular term is not small as it was found by direct numerical calculation [73]. An interesting property was also found in the calculations [73] for all calculated values of torsion or helical pitch of a helical vortex of a relatively small core with circular cross-section. The self-induced velocity differs from the average velocity induced by a vortex ﬁlament concentrated along the vortex axis by the same constant. Later it was theoretically proved that this constant is exactly 14 [46]. The same property is well known for vortex rings, where the difference is equal to 34 [59]. So, the determination of the self-induced velocity of helical vortex can be reduced to an already solved task, based on determining the velocity induced by an inﬁnitely thin vortex ﬁlament located at the core radius [40]. The ﬁrst numerical solutions of the problem of determining the induced equilibrium motion of multiple helical vortices in an unbounded domain were given in Refs. [47,74]. Both articles extended the determination of the velocity of a single constantdiameter helical vortex from Refs. [46,65] to include the effects of Nb 1 additional helical vortices. This arrangement approximates the tip vortices in the far wake of NEJ type of wind turbines, propellers etc. with Nb blades. The contributions from the additional vortices are determined from application of the Biot–Savart law that, apparently, cannot be evaluated in closed equation. An analytical solution of this task based on (4) and (5) was found in Ref. [40], but here we will use a more suitable notation of the

Fig. 18. Determination of the self-induced velocity of the helical vortex by vortex ring separation.

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12

Fig. 19. (a) Comparison of the angular velocity of the self-induced rotation of helical vortex: (solid line) – exact solution of Ref. [40] and (dotted line) – approximation by vortex ring [15]. (b) Comparison of maximum ampliﬁcation rate for unstable regimes of Nb helical vortices: (solid lines) – analytical solution [75] and (dashed lines) – numerical simulation of Ref. [74]

expressions introduced in Ref. [50]. The self-induced motion in the bi-normal direction of the multiple helices can be written by introducing the bi-normal velocity component ub as l ub ¼ uχ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ; 2 2 R þl

ð7Þ

where the velocity component in the χ -direction is given as uχ ¼ uθ ðr=lÞuz , which by Eq. (2.13) from Ref. [50] can be expressed for the motion of a helical vortex in the form pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ τ2 N b 4π R uχ ðσ Þ uχ ðσ Þ ¼ 2

Γ

τ

þ

1

τ 1þτ

2

1=2 ln

τ

3=2 σ N b 1 þ τ2

τ

þ

!

1 τ 3 ζ ð3 Þ 4 2 7=2 τ 3τ þ 4 8 N 2b 1 þ τ2

ð8Þ where τ ¼ l=R and σ ¼ ε=R are the non-dimensional pitch and radius of the vortex core, respectively, and ζ(3)¼ 1.20206… where ζ(.) is the Riemann zeta function. Fig. 19a shows a comparison between the analytical solution calculated by (6)–(8) and Joukowsky's vortex ring approximation [15]. On the plot we have reproduced a behavior of the regular part (without a singular term ln σ ) of the self-induced angular rotation Γ =4π R2 Ω of a single helix (N b ¼ 1), which was calculated by the formula [40]

Ω¼

2 τ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃub 1 þ τ2 1 þ τ2

The difference in the solution of (6) and (7), (8) in both limit cases of small and large pitch can be explained structurally. Indeed, for small pitch the helical vortex turns into an inﬁnite cylindrical vortex sheet, while Joukowsky's approximation remains a single ring of the same radius of the vortex cylinder (Fig. 18). The two different vortex structures should induce different speeds. At large pitch, both vortices tend to a rectilinear form without self-induction and both remainder terms in these approximations eventually become negligible. It should be mentioned that our comparison of the analytical solution deﬁned by (4)–(8), with the precise numerical simulation determined from a straightforward application of the Biot–Savart law [46,47], indicates exact agreement within 6–7 signiﬁcant digits [40]. Another example is the estimation of the vortex stability in helical multiplets [75] corresponding to the far wake of a Nb-bladed NEJ rotor. In this case, inﬁnitely long Nb-helical tip vortices of strength Γ with constant pitch and radius are combined with a root or hub vortex represented by an inﬁnitely long axial vortex of strength Γ 0 ¼ Nb Γ . Assuming a ﬁrst-order

Fig. 20. Cross sectional airfoil element.

perturbationof the position of the khelical vortex of the form, δrk ¼ δr~ k exp αt þ 2π iks=N b , where δr~ k is the amplitude vector of the perturbations, s is the sub-harmonic wave number that takes values within the range ½1; Nb 1, corresponding to N b 1 independent eigenfunctions, and α is the ampliﬁcation rate. An analysis of the stability [40,50,75] for the non-dimensional amplipﬃﬃﬃﬃﬃﬃ ﬁcation rate αð4π R2 =Γ Þ ¼ AB leads to the following analytical solutions: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

Nb s 1 þ τ 2 τ 4τ 2 3 A ¼ sðN b sÞ Eψ ð9Þ 5=2 Nb τ 4 1 þ τ2 s B ¼ 4N b γ þsðNb sÞ þ

1

τ 1 þ τ2

" 3=2

1 þ τ2

τ

3

3=2 2N þ2

Nb 2

τ2

1 þ 2τ 2 þ τ 1 þ τ2 1=2

1 s N Eþψ b 4 Nb s !!# 3=2 1 þ τ2

τ2

3 þ 2τ2 ln N b σ 4 τ

τ3 3 ζ ð3Þ 4 2 ; þ 9=2 2τ 6τ þ 4 Nb 2 1 þ τ2

ð10Þ

where γ ¼ Γ 0 =N b Γ is the circulation ratio, ζ(3) ¼1.20206… where ζ(.) is the Riemann zeta function, E ¼ 0:577215… is the Euler constant, and ψ ð U Þ is the psi function. The ﬁrst term of B describes the effect of the hub vortex. The vortex system is unstable if AB Z0 for any combination of s, τ, ε, and γ . As an illustration of (4), in Fig. 19b we plot the absolute maximum ampliﬁcation rate as function of helical pitch and number of tip vortices Nb in the far wake model without hub vortex (γ ¼ 0). In the plot, results from our analytical model are close to the numerical calculations of Ref.

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[74]. The comparison indicates that the analytical model (4)–(10) of single and multiple helical vortices [40,50] constitute new possibilities for designing novel solutions in rotor aerodynamics with ﬁnite number of blades. 3. Modern BEM theory 3.1. A local blade element consideration In the BEM theory the loading is computed using two independent methods, i.e. by a local blade element consideration using tabulated 2dimensional airfoil data and by use of the 1-dimensional momentum theorem. First, employing blade-element theory, axial load and torque are written respectively as follows: dT 1 ¼ N b F n ¼ ρcN b V 2rel U C n ; dr 2

ð11Þ

dM 1 ¼ Nb rF t ¼ ρcNb rV 2rel U C t dr 2

ð12Þ

where c is the blade chord, N b is the number of blades, V rel is the relative velocity, F n and F t denote the loading on each blade in axial and tangential direction, respectively, and C n and C t denote the corresponding 2-dimensional tabulated force coefﬁcients. From the velocity triangle at the blade-element (see Fig. 20), we deduce that sin ϕ ¼

U 1 ð1 aÞ ; V rel

cos ϕ ¼

Ωrð1 þa0 Þ

ð13Þ

V rel

where the induced velocity is deﬁned as W i ¼ ð aU 1 ; a ΩrÞ via both axial (a) and azimuthal (a0 ) induction factors. Using the above relations, we get 0

V 2rel ¼

U 21 ð1 aÞ2 sin ϕ 2

¼

U 1 ð1 aÞΩrð1 þ a0 Þ sin ϕ cos ϕ

0

ð16Þ

ð17Þ

where uR ¼ U 1 ð1 aÞ is the axial velocity in the rotor plane and uwake ¼ U 1 ð1 2aÞ is the axial velocity in the ultimate wake. Applying the moment of momentum theorem, we get dM ¼ ρruθ 2π ruR ¼ 4πρr 3 ΩU 1 a0 ð1 aÞ dr

ð18Þ

where uθ ¼ 2Ωra0 is the induced tangential velocity in the far wake. Combining (15) and (16) with (17) and (18), we get after some algebra a¼

1 4 sin ϕ=ðσ C n Þ þ 1

a0 ¼

2

;

1 : 4 sin ϕ cos ϕ=ðσ C t Þ 1

ð21Þ

dM ¼ 4πρr 3 ΩU 1 a0 Fð1 aÞ; dr

ð22Þ

where σ ¼ Nbc/2πr. An approximate formula of the Prandtl tip loss function (Fig. 6b) was introduced as follows:

2 N ðR rÞ ; ð23Þ F ¼ cos 1 exp b π 2r sin ϕ where ϕ ¼ ϕðrÞ is the angle between the local relative velocity and the rotor plane. The coefﬁcients (Cn, Ct) are related to the lift and drag coefﬁcients (Cl, Cd) by C n ¼ C l cos ϕ þ C d sin ϕ and C t ¼ C l sin ϕ C d cos ϕ, respectively. (Cl, Cd) depend on local airfoil shape and are obtained using tabulated 2D airfoil data corrected with 3D rotating effects. Equating (15) to (11), and (16) to (22), the ﬁnal expressions for the interference factors read a¼

1 ; 2 4F sin ϕ=ðσ C n Þ þ 1

ð24Þ

1 : 4F sin ϕ cos ϕ=ðσ C t Þ 1

ð25Þ

a0 ¼

By putting (21) into dimensionless form we get the following expression for the local thrust coefﬁcient: CT ¼

Next, applying axial momentum theory, the axial load is computed as dT ¼ rðU 1 uwake Þ2pruR ¼ 4prrU 21 að1 aÞ dr

dT ¼ 4π r ρU 21 aFð1 aÞ; dr

3.3. Correction for heavily loaded rotors

ð15Þ

dM ρN b cU 1 ð1 aÞΩr ð1 þ a Þ ¼ UC t dr 2 sin ϕ cos ϕ 2

inﬁnitely many blades. In order to correct for ﬁnite number of blades, Glauert [23] introduced Prandtl's tip loss factor (Fig. 6b) for wind turbines. In this method a correction factor F is introduced which corrects the loading. In a recent paper [76] the tip correction is discussed and various alternative formulations are compared. However, here we limit the correction to the original form given by Glauert [23]. In this model the induced velocities are corrected by the tip loss factor F modifying (7) and (8) as follows:

ð14Þ

Inserting these expressions into (11) and (12), we get dT ρN b cU 21 ð1 aÞ2 ¼ UC n dr 2 sin 2 ϕ

13

ð19Þ

ð20Þ

3.2. Tip correction Since the above equations are derived assuming azimuthally independent stream tubes, they are only valid for rotors with

dT 1=2 ρU 21 2π rdr

¼ 4aFð1 aÞ:

ð26Þ

For heavily loaded rotors, i.e. for a-values between 0.3 and 0.5, this expression ceases to be valid as the wake velocity tends to zero with an unrealistic large expansion as a result. It is therefore common to replace it by a simple empirical relation. Following Glauert [23], an appropriate correction is to replace the expression for a Z 1=3 with the following expression: a C T ¼ 4aF 1 ð5 3aÞ : ð27Þ 4 As discussed in e.g. Refs. [77,78] other expressions can also be used. 3.4. Yaw correction Yaw refers to the situation where the incoming ﬂow is not aligned with the rotor axis. In this case the wake ﬂow is not in line with the free wind direction and it is impossible to apply the usual control volume analysis. A way of solving the problem is to maintain the control volume and specify an azimuth-dependent induction. In practice, it works by computing a mean induction and prescribe a function that gives the azimuthal dependency of the induction. The following simple formula has been proposed in Ref. [79] χ r wi ¼ wi0 1 þ tan cos ðθblade θ0 Þ : ð28Þ R 2

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14

where wi0 is the annulus averaged induced velocity and χ is the wake skew angle, which is not identical to the yaw angle because the induced velocity in yaw alters the mean ﬂow direction in the wake ﬂow. In the notation used here θblade denotes the azimuthal position of the blade and θ0 is the azimuthal position where the blade is deepest in the wake. For more details, the reader is referred to the text book by Hansen [78]. 3.5. Dynamic wake Dynamic wake or dynamic inﬂow refers to unsteady ﬂow phenomena that affect the loading on the rotor. In a real ﬂow, the rotor is subject to unsteadiness from coherent wind gusts, yaw misalignment and control actions, such as pitching and yawing. When the ﬂow changes in time the wake is subject to a time delay when going from one equilibrium state to another. An initial change creates a change in the distribution of trailing vorticity which then is convected downstream and ﬁrst can be felt in the induced velocities after some time. However, the BEM method in its simple form is basically steady; hence unsteady effects have to be included as an additional “add-on”. In the European CEC Joule II project “Dynamic Inﬂow: Yawed Conditions and Partial Span Pitch” (see Ref. [80]) various dynamic inﬂow models were developed and tested. Essentially a dynamic inﬂow model predicts the time delay through an exponential decay with a time constant corresponding to the convective time of the ﬂow in the wake. As an example, the following simple model was suggested: Rf ðr=RÞ

dui ΔT þ 4ui ðU 1 ui Þ ¼ ; dt 2π r Δr

ð29Þ

where the function f ðr=RÞis a semi-empirical function associated with the induction. The equation can be seen to correspond to the axial momentum Eq. (26), except for the time term which is responsible for the time delay. 3.6. Airfoil data Prior to the BEM computations, 2-dimensional airfoil data had to be established from wind tunnel measurements. In order to construct a set of airfoil data to be used for a rotating blade, the airfoil data further need to be corrected for three-dimensional and rotational effects. A simple correction formula for rotational effects was proposed in Ref. [81] for incidences up to stall. For higher incidences (4401), 2-dimensional lift and drag coefﬁcients of a ﬂat plate can be used. These data, however, are too big because of aspect ratio effects and here the correction formulas of Ref. [82] are usually applied (see also Ref. [77]). Furthermore, since the angle of attack is constantly changing due to ﬂuctuations in the wind and control actions, it is necessary to include a dynamic stall model to compensate for the time delay associated with the dynamics of the boundary layer and wake of the airfoil. This effect can be simulated by a simple ﬁrst order dynamic model, as proposed in Ref. [83], or it can be considerably more advanced, taking into account also attached ﬂow, leading edge separation and compressibility effects, as in the model of Ref. [84]. 3.7. A cascade of blade elements with an overview of the KJ theorem Joukowsky [24] derived the circulation over a section of a constant diameter cylinder representing the streamsurface intersecting the blade element of a wind turbine or propeller. Eq. (13) in his paper considers only the mean velocities in a manner familiar to any student studying basic turbomachinery analysis. This derivation is repeated in Joukowsky [25] and followed by extensive analysis of the ﬂow over two-dimensional cascades of blades without providing a general derivation of the KJ equation.

This is sufﬁcient evidence to say that he originated the inﬁnite two-dimensional cascade model of the unfolded streamtube containing a ﬁnite number of wind turbine or propeller blade elements. It will be shown below that the cascade model allows a simple derivation of the KJ equation for blades of any solidity including zero for an (isolated) airfoil. The importance of the KJ equation comes mainly from its relation to Kelvin's requirement that circulation, once created in an inviscid ﬂuid, is conserved. Thus the vorticity shed from a wing or blade determines the circulation of the resulting tip and hub vortices in the wake. Joukowsky was clearly aware of this result, as reproduced Fig. 5a shows. The straight hub vortex lying along the axis of rotation has strength 2J (where J Γ ), whereas each bound and helical tip vortex has J. This “Joukowsky model” of the rotor wake – so named in Ref. [37] – and the KJ equation can be used for wind turbine blade design. A blade designer aims to achieve a ¼ 1=3 as this corresponds to the Betz–Joukowsky limit for optimum power output [1,85]. As demonstrated below, this occurs when the Goldstein parameter Nb Γλ0 =π ¼ 8=9, which ﬁxes Γ if N b and λ0 have already been chosen, as is normally the case. Blade element theory, as described above, then allows Γ to be related to the lift produced by, and hence the chord of, the airfoil making up the element. In turn, this determines the twist of the element. The chord and twist are main aerodynamic design parameters once the airfoil has been selected. There is much more to blade design than this simple description suggests, but the fundamental role of the KJ equation remains. Since the original derivation, the KJ equation has been rederived and extended in a number of ways. Taylor [86] considered the effects of the viscous wake of the body and argued that the contour around which the circulation is determined, should cut the wake at right angles. Lagally [87] analyzed the lift of two bodies using conformal mapping, which was further extended by Crowdy [88] to a “ﬁnite stack” of airfoils and Bai and Wu [89] for an arbitrary arrangement of bound and free vortices. With increasing interest in the aerodynamics of micro-aerial vehicles, there have been many extensions of the KJ to account for unsteady effects and low Reynolds numbers, e.g. Refs. [90–92]. There are three common modern derivations of the KJ equation: (1) a demonstration of its validity for a speciﬁc body, often a rotating circular cylinder, followed by an unproved statement of its generality, e.g. Ref. [93]; (2) a derivation using the Blasius theorem involving residue calculus and complex variables, e.g. Ref. [6], and (3) the moving and expanding control volume method of Batchelor [56], which requires a thorough knowledge of the unsteady Bernoulli equation and careful consideration of the decay of induced velocities at large distances. It is a nice irony that a very simple derivation of the KJ theorem follows from Joukowsky's representation of rotors as cascades. The derivation relies only on very basic ﬂuid mechanics concepts from an introductory course. Consider an inﬁnite cascade of identical bodies – usually airfoils – spaced distance s apart along the y-axis in Fig. 21a. Only four bodies are shown. The undisturbed velocity of the incompressible ﬂuid is U 1 along the x-axis. For simplicity, one body is located at the origin surrounded by a rectangular control volume (CV) with horizontal faces at y ¼ 7 s=2. The vertical faces are equi-distant from the y-axis: the actual distance is not important. The faces are labeled in clockwise order from the upstream one. Symmetry requires that for faces 2 and 4:

the pressures are equal at the same x; there is no net efﬂux of x- or y-direction momentum; and

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the contribution to Γ , the circulation around the contour, will cancel.

Γ is positive in the clockwise direction. Only the ﬂow through faces 1 and 3 contributes to the momentum balance. The x-velocity at any point in the ﬂow is U 1 þ u where the latter is due to the bodies, as is the vertical velocity, v. Applying the Reynolds transport theorem to the CV gives for the vertical force on the body, Fy: Z s=2 Z s=2 F y ¼ ρ ðU 1 þ u3 Þv3 dy ρ ðU 1 þ u1 Þv1 dy; ð30Þ s=2

s=2

where the subscripts on u and v denote the face, or Z s=2 ðu1 v1 u3 v3 Þdy: F y ¼ ρU 1 Γ þ ρ

ð31Þ

s=2

Note carefully, that (31) only uses the deﬁnition of Γ ; it has not been assumed that the bodies can be represented by vortices. Similarly, the x-direction force, Fx, is found from Z s=2 Z s=2 Z s=2 Fx þ ðP 1 P 3 Þdy ¼ ρ ðU 1 þ u3 Þ2 dy ρ ðU 1 þ u1 Þ2 dy; s=2

s=2

s=2

ð32Þ

y

U∞

U∞ s

y

2

4

s/2

where P is the gauge pressure which is removed using Bernoulli's equation as was done by Joukowsky [3]. Thus o 1 1 n 1 1 P ¼ ρU 21 ρ ðU 1 þ uÞ2 þ v2 ¼ ρ U 1 u þ u2 þ v2 ; ð33Þ 2 2 2 2 Eq. (32) can then be rewritten as Z s=2

2 1 u u23 þ2U 1 ðu1 u3 Þ þ v23 v21 dy: Fx ¼ ρ 2 s=2 1

ð34Þ

The ﬁrst term in (31) makes it necessary to represent a lifting body by a vortex of strength Γ . This representation is sufﬁcient as (31) and (32) are then fully satisﬁed. If all the bodies in the cascade are replaced by vortices of strength Γ , u is an even function of y and v is an odd function. Thus uv is odd and the integral in (31) is identically zero. Equispacing of the CV faces 1 and 3 about the y-axis requires u1(y) ¼u3(y) and v1(y)¼ v3(y) so the integrand in (32) is zero for any y. Eqs. (31) and (32) reduce to F x ¼ 0;

F y ¼ ρU 0 Γ ;

ð35Þ

which is the usual form of the KJ equation. Note that the result is independent of the spacing s and that the Kutta condition was not used in the derivation. From the cascade in Fig. 21a, bodies can now be removed to ultimately produce Fig. 21b for an isolated body. Any vortex in Fig. 21a induces a velocity on faces 1 and 3 that equals the velocity induced by the vortex at the origin on the equivalent of faces 1 and 3 for that vortex. In other words, the integrals in (30)–(32) and (34) between y¼ 7 s/2 for an inﬁnite cascade are equal to the same integrals with limits changed to 7 1 for a single body. Alternatively, it is noted that (35) is independent of s and remains valid as s-1. This establishes the equivalence of KJ equation for a cascade and an isolated body. It turns out that a single lifting body and an inﬁnite cascade are the simplest arrangements in which to establish the KJ equation. For a ﬁnite “stack” of lifting bodies, the analysis becomes considerably more complex, e.g. Ref. [88], and momentum balances quickly lose their attraction.

4. Modern stage of the rotor vortex theory

s/2 1

15

x

x

3

Based on the concepts outlined by Joukowsky [15] and Betz [27] nearly a century ago, analytical aerodynamic optimization models are developed for rotors of a ﬁnite number of blades with constant circulation distribution [37] and with Goldstein's circulation [38]. In this section we also examine the basic rotor theories with different deﬁnition of the helical pitch in the vortex wake. 4.1. Lifting line theory of Joukowsky (NEJ) and Betz rotors

Fig. 21. (a) Control volume for a cascade of identical, equi-spaced bodies and (b) control volume for an isolated body.

In the vortex theory each of the blades is replaced by a lifting line on which the radial distribution of bound vorticity is

Fig. 22. Sketch of the vortex system corresponding to lifting line theory of NEJ (a) and Betz (b) [37] rotors of radius R.

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Fig. 23. Velocity and power triangles for a blade element moving upwards of Joukowsky rotor (a) and Betz rotor (b).

represented by the circulation Γ ¼ Γ ðrÞ where r is a radial distance along the rotor blade. This results in a free vortex system consisting of helical trailing vortices, as sketched in Fig. 22a and b. Using the KJ equation, the bound vorticity serves to produce the local lift on the blades while the trailing vortices induce the velocity ﬁeld in the rotor plane and in the wake. As illustrated in Fig. 22 the velocity vector in the rotor plane is made up by the rotor angular velocity Ω0 , the undisturbed wind speed U 1 , the axial and circumferential velocity components uz0 and uθ0 , respectively, induced at a blade element in the rotor plane by the tip vortices, and vθ0 , the circumferential velocity induced by the hub vortex. The fundamental expressions for the forces acting on a blade (Fig. 23) are most conveniently expressed by the KJ theorem, Eq. (35), which in elemental vector form reads ! ! ! d L ¼ ρ V 0 Γ dr;

ð36Þ

where dL is the lift on a blade element of radial dimension dr, V 0 is the resultant relative velocity and ρ is the density of the air. From Eq. (36), we can write the local torque dQ of a rotor blade as follows: ð37Þ dQ ¼ ρΓ U 1 uz0 rdr: Integrating Eq. (37) along the blades and summing up, we get the following expression for the power output, P ¼ Ω0 Q : Z R P ¼ ρN b Ω 0 Γ U 1 uz0 rdr: ð38Þ 0

To determine the theoretical maximum efﬁciency of a rotor the power coefﬁcient is introduced as follows:

1 ð39Þ ρπ R2 U 31 : CP ¼ P 2 The maximum power that can be extracted from a stream of air contained in an area equivalent to that swept out by the rotor corresponds to the maximum value of the wind power coefﬁcient. To determine the velocity ﬁeld vθ0 , uz0 and uθ0 induced at a blade element the free half-inﬁnite helical vortex system behind the rotor is replaced by an associated vortex system that extends to inﬁnity in both directions. Neglecting deformations or changes in the wake the vortex system is uniquely described by the far wake properties in the so-called Trefftz plane, which is the plane normal to the relative wind far behind the rotor. For an ideal ﬂuid the ﬂow through the Trefftz plane is independent of the axial position. Thus, in accordance with Kelvin's theorem, the bound circulation Γ about a blade element is uniquely related to the circulation of a corresponding vortex in the Trefftz plane. By symmetry, it is readily seen that the induced velocities at a point in the rotor

plane (Fig. 23) equals half the induced velocity at a corresponding point in the Trefftz plane [15,27]: vθ0 ¼

1 v ; 2 θ

uθ 0 ¼

1 u 2 θ

uz 0 ¼

and

1 uz : 2

ð40Þ

4.2. Solution for the NEJ rotor In the vortex theory of the Joukowsky rotor [15] each of the blades is replaced by a lifting line about which the circulation associated with the bound vorticity is constant, resulting in a free vortex system consisting of helical vortices trailing from the tips of the blades and a rectilinear hub vortex. The vortex system may be interpreted as consisting of rotating horseshoe vortices with cores of ﬁnite size, as sketched in Fig. 22a (or Fig. 5a which is reproduced from the original drawing of Joukowsky). The associated vortex system consists of N b helical tip vortices of ﬁnite vortex cores (ε 5R) with constant pitch h 2π l and circulation Γ . The free vortex lines comprise vortex cores of ﬁnite size in order to avoid singular behavior of the Biot–Savart law. The vortex cores are collinear to the axes of the helical lines and their vorticity is assumed to be uniformly distributed across the core cross-section. The multiplet moves downwind (in the case of a propeller) or upwind (in the case of a wind turbine) with a constant velocity U 1 ð1 7 υÞ in the axial direction where υ denotes the difference between the wind speed and axial translational velocity of the vortices. Denoting the angle between the axis of the tip vortex and the Trefftz plane as Φ (see Fig. 23a), the helical pitch is given as h ¼ 2π R tan Φ;

or

l=R ¼ h=2π R ¼ tan Φ τ:

ð41Þ

Introducing the azimuthally averaged induced axial velocity as R 2π huz iθ ¼ 21π 0 uz dθ, from (2) we get huz iθ ¼ 0

for

r 4R

and

huz iθ ¼

Nb Γ const 2π l

for

r 4R

ð42Þ

It should be mentioned that the dimensionless averaged induced axial velocity in the wake (r o R), which is identical to the total axial wake interference factor a, takes the same constant value N Γ aU 1 huz iθ 0 o r o R ¼ b : 2π l

ð43Þ

The vortex system also includes a rectilinear hub vortex of strength N b Γ , resulting in a simple formula for the additional induced velocity that only consists of the circumferential component: vθ ¼

Nb Γ : 2π r

ð44Þ

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Deﬁning the azimuthally azimuthal velocity induced by averaged R 2π the helical vortices as uθ θ ¼ 21π 0 uθ dθ, and combining Eqs. (3) and (44), we get vθ r ¼ R ¼ uθ θ r ¼ R : ð45Þ For an unexpanded wake originating from a rotor with inﬁnitely many blades, the self-induced velocity of the tip vortices equals half the averaged induced axial velocity in the wake (Note that the axial velocity of the tip vortices is not inﬂuenced by the straight hub vortex). This is sometimes referred to as the “rollerbearing analogy” which holds for a continuous vortex sheet [56,94] and in the limiting case of small pitch [46,59]. Although this approximation cannot be rigorously justiﬁed for a vortex system consisting of an arbitrary ﬁnite number of vortices (see Section 4.4), we employ the same analogy by assuming that the helical vortices are transported with a relative axial speed, υ, that corresponds to half the averaged induced velocity: 1 ðR þ εÞ ; 2 R

υ¼ a

ð46Þ

where a correction of a small expansion of the cross-section of the wake is made in order to include the radius, ε, of the vortex cores. Thus, the associated vortex system is assumed to translate in equilibrium along the bi-normal direction to the helical axis of the tip vortices with the velocity, ub , (Fig. 22a): ub υcosΦ ¼

aðR þ εÞ aðR þ εÞ R pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ; cosΦ 2 2R 2R R2 þ l

ð47Þ

in which motion of the far wake was determined in accordance with the “roller-bearing analogy” of (46). This assumption we can use for a deﬁnition of the vortex core of multiple helical tip vortices. Substituting (6) and (8) in (47) ﬁnally the dimensionless radius σ of the tip vortex core for this wake motion must satisfy the equation pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ τ2 τ τ4 3τ2 þ 38 ζ ð3Þ N b σ þ 2 7=2 τ τ N2b 1 þ τ2 ! 3=2 σ N b 1 þ τ2 1 1 þ ¼0 ð48Þ 1=2 ln 4 τ τ 1 þ τ2 for any given value of l or τ (from (41)) and Nb . Fig. 24 shows the “total” core size, which is equal to the vortex core radius computed by solving (48) and multiplied by the number of blades, as function of the tip vortex pitch for different numbers of blades. It may be noted that the limit of small pitch has the same asymptote for all number of blades and the “total” core tends to zero faster than the pitch with a ﬁxed ratio h=N b ε ¼ 6. This implies that the distance between the tip vortices, independent of the value of the pitch, always is greater than 4 core radii, as depicted in Fig. 24. This also implies that it is impossible to reach a dense cylindrical vortex surface as used in actuator disk theory. In the other limit, when the pitch tends to inﬁnity, the vortex core disappears which shows that an equilibrium vortex multiplet motion is achieved when the vortices become rectilinear. The axial velocity ﬁeld induced by the tip vortices can subsequently be determined in all points of the Trefftz plane because we have deﬁned the ﬁnite radius of the tip vortex core by the “rollerbearing analogy”. The velocities outside the vortices are determined by (4) and inside the vortex cores by taking the average value of the velocity on the boundary of the vortex cores. The axial velocity, made dimensionless with the azimuthally averaged induced axial velocity huz iθ of (44) for the n-blade (θ ¼ 2π n=N b , z ¼0, 1r nr Nb), takes the form:

2π n aU 1 u~ z r; Nb

Fig. 24. The vortex core radius for equilibrium motion of tip vortices as function of pitch for different numbers of blades (the number on the solid curves refer to the number of blades). Dotted line indicates asymptotic behavior of the core radius for small pitch and the sketch shows the helical tip vortices representing the limit case of the wake.

8 > if r o R ε and r 4 R þ ε < uz r; 2Nπbn ¼ Rþεr > : 2ε uz R ε; 2Nπbn R 2εε ruz R þ ε; 2Nπbn if

Rεor oRþε

ð49Þ From Eq. (43) we get the following relation between the bound circulation and the interference factor: Nb Γ ¼ 2π laU 1

ð50Þ

From simple geometric considerations in the rotor plane (Fig. 23a), using Eqs. (45) and (46), the angular pitch is given as U 1 uz0 r ¼ R tan Φr ¼ R ¼ Ω0 R þ uθ0 r ¼ R vθ0 r ¼ R U 1 ð1 υÞ U 1 1 12 að1 þ σ Þ l ð51Þ ¼ ¼τ ¼ R Ω0 R Ω0 R Eq. (51) can be also written as 1 2

Ω0 l ¼ U 1 aU 1 ð1 þ σ Þ

ð52Þ

Inserting (40), (49), (50) and (52) into (38), the power can be determined from the following integral: ! Z 1 a 2 3 ~ P ¼ ρ π R U 1 a 1 ð1 þ σ Þ 1 a ð53Þ u z ðx; 0Þxdx 2 0 Performing the integration and introducing the dimensionless wind power coefﬁcient (39), we get ð54Þ C P ¼ 2a 1 12 aJ 1 1 12 aJ 3 R1 where J 1 ¼ 1 þ σ and J 3 ¼ 2 0 u~ z ðx; 0Þxdx. For a given helicoidal wake structure, the power coefﬁcient is seen to be uniquely determined, except for the parameter a. Differentiation of C P with respect to a yields the maximum value, C P;max , resulting in qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2 J 1 þJ 3 J 21 J 1 J 3 þ J 23 : ð55Þ aðC P ¼ C P; max Þ ¼ 3J 1 J 3 4.3. Solution for the Betz rotor To compare the efﬁciency of the Joukowsky rotor with the Betz rotor, here we outline the main points of the model of Okulov and Sørensen [38]. In this model the vortex strength of the lifting line varies along the blade span, following the so-called Goldstein circulation. This results in a vortex sheet that is continuously shed

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from the trailing edge (Figs. 5 and 22b). Betz [27] showed that the ideal efﬁciency is obtained when the distribution of circulation along the blade produces a rigidly moving helicoidal vortex sheet with constant pitch, h, that moves downwind (in the case of a propeller) or upwind (in the case of a wind turbine) in the axial direction of its axis with a constant velocity U 1 ð1 7 wÞ. The associated vortex system to the wake consists of a regular helicoidal sheet extended to inﬁnity in both directions. Denoting the angle between the vortex sheet and the rotor plane as Φ (see Fig. 23b), the pitch is given as h ¼ 2π r tan Φ

or

l=r ¼ h=2π r ¼ tan Φ τ;

ð56Þ

where r is the radial distance along the sheet. Since the sheet is translated with constant relative axial speed, wU 1 , the induced velocity comprises only the component wU 1 cosΦ that is “pushed” normal to the screw surface (Fig. 23b). The axial and circumferential velocity components uz and uθ induced by the inﬁnite sheet at the sheet itself are therefore given as uθ ¼ wU 1 cosΦsinΦ

and

uz ¼ wU 1 cos2 Φ

ð57Þ

From simple geometric considerations these equations are rewritten as uθ ¼ wU 1

xl 2

l þ x2

and

uz ¼ wU 1

x2 2

l þ x2

;

ð58Þ

where x ¼ r=R is the dimensionless radius. Goldstein (Fig. 8) ﬁrst found an analytical solution to the potential ﬂow problem of the moving associated vortex system consisting of an inﬁnite helical vortex sheet [34]. Using inﬁnite series of Bessel functions, he succeeded in obtaining an analytical solution to the problem, but for Nb ¼ 2 and 4 only. In his model a dimensionless distribution Gðx; lÞ of circulation was introduced as follows: N b Γ ðx; lÞ ¼ 2π lwU 1 Gðx; lÞ:

(5) for a single helical ﬁlament has been successfully applied by Okulov and Sørensen [38,39] to determine the Goldstein circulation Gðx; lÞ. The vortex sheet was discretized by 100 uniformly spaced helical ﬁlaments, provided the condition of its motion together with ﬂuid. The results coincide with the tabular data given in Refs. [54,95] for the Goldstein solution (Fig. 25). Thus a new efﬁcient algorithm has been developed for solving the Betz rotor for any tip speed ratio. In addition, the discretization technique made it possible to calculate curved blades often used for marine propellers. Comparison of Goldstein functions for the curved blades and straight-line blades (Fig. 26) shows that the curvature is effective only for the wakes with large pitch e.g. for ship propellers with a low rotation speed. The optimization of the Betz rotor for curved blades can be now precisely counted by the vortex method. Semi-empirical models and approximate approaches, such as, for example, presented in work [96], have been completely superseded. To compute the power coefﬁcient we employ the same procedure as in the previous Section, that is we integrate (38) using (40) and the Goldstein's distribution in (59). In addition to this, from geometric considerations in the rotor plane (Fig. 23b), using (56) and (57), the angular pitch is given as tanΦ ¼

U 1 12 uz

Ω0 r þ 12 uθ

¼

U 1 1 12 w l ¼ : r Ω0 r

ð60Þ

The relation given in (60) can be deduced as follows:

tan Φ ¼

U 1 12 uz U 1 ð1 12 w cos 2 ΦÞ sin Φ ¼ ¼ 1 cos Φ Ω0 r þ 2 uθ Ω0 r þ 12 U 1 w cos Φ sin Φ

) Ω0 r sin Φ þ 12 U 1 w cos Φ sin Φ ¼ U 1 cos Φð1 12 w cos 2 ΦÞ 2

) Ω0 r sin Φ ¼ U 1 cos Φð1 12 wð cos 2 Φ þ sin 2 ΦÞÞ

ð59Þ

It should be noted that for any given value of the wake pitch l and number of rotor blades N b the solution in the form of (4) and

) tan Φ ¼

sin Φ U 1 ð1 12 wÞ ¼ cos Φ Ω0 r

ð61Þ

Fig. 25. Comparisons of the Goldstein circulations [34] for different pitches (left: h ¼ 1; right: h ¼ 1/12) and number of vortex sheets (indicated by curve number); lines: calculation using 100 vortex ﬁlaments [38]; symbols: direct calculations by the Kapteyn series [54].

Fig. 26. See Fig. 25: lines indicate the Goldstein circulations but for rotors with curved blades.

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from (61) it follows that U 1 12 uz

U 1 ð1 12 wÞ : ¼ 1 Ω0 r Ω0 r þ 2 uθ Eq. (60) can be also written as Ω0 l ¼ U 1 1 12 w :

ð62Þ

Inserting (58), (59) and (62) into (38), the power can be determined from the following integral:

Z w 1 w x2 P ¼ ρπ R2 U 31 w 1 xdx: ð63Þ 2Gðx; lÞ 1 2 0 2 x2 þ l2 Performing the integration and introducing the dimensionless wind power coefﬁcient (39), we get C P ¼ 2w 1 12 w I 1 12 wI 3 ; ð64Þ where Z I1 ¼ 2

1

Z Gðx; lÞxdx

and

1

I3 ¼ 2

0

Gðx; lÞ

0

x3 dx x2 þ l

2

:

ð65Þ

The coefﬁcients I 1 and I 3 are usually referred to as the mass coefﬁcient and the axial energy factor, respectively. For a given helicoidal wake structure, the power and thrust coefﬁcients are seen to be uniquely determined, except for the parameter w. Differentiating C P from Eq. (64) with respect to w yields the maximum value, C P;max , resulting in qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2 wðC P ¼ C P; max Þ ¼ I 1 þ I 3 I 21 I 1 I 3 þ I 23 : ð66Þ 3I 3 4.4. Assessment of the different rotor theories It should be noted that the vortex approach for the rotors with a ﬁnite number of blades, proposed by Joukowsky and Betz, is not sufﬁcient for a complete formulation of the problem

19

of determining the optimum rotor [97–99]. Some additional assumptions are needed in order to close the formulation of the problems. First of all, a correct choice of the helical pitch in the vortex wake is essential for both rotor conﬁgurations. Table 1 presents existing rotor theories based on different assumptions. These assumptions are sketched in Table 2, where it is readily seen that the differences are related to different assumptions regarding the convective velocity of the vortices. The early model (i), which simply assumes the convective velocity to be equal to the wind speed, is simplest. This was considered a good approximation for weakly loaded rotors, but was only applied as a simpliﬁcation only, in order for the helical pitch to become independent of the induced velocities uz and uθ. Otherwise, an iterative procedure was required as the velocities depend on the pitch themselves, which complicates the solution of the problem. Table 2 demonstrates that the ﬁrst pitch deﬁnition (i) avoids this iterative process [27,33]. According to the second model (ii), the pitch was determined by the velocities in the far wake, because previous investigators believed that the steady vortex structure formed in the far wake had a constant pitch everywhere, including in the rotor plane [34,95]. The third and ﬁnal model (iii), described by (54) and (64), relates the circulation to the pitch in the rotor plane, which subsequently uses half of the value of the wake velocity for computing the helical pitch [37–39]. For all the basic models listed above it was possible to introduce a single-valued parameter for the rotor optimization. For the NEJ rotor it is an axial induction factor а based on the azimuthally averaged ﬂow (43). In the case of the Betz rotor it is w – the axial self-induced velocity of the vortex sheet. Both parameters are constant and independent of the radial position in the wake. All rotor models could be optimized by considering these parameters for each ﬁxed pitch of the vortex wake as an extension of the basic analysis that leads to the Betz–Joukowsky limit through optimizing the axial induction factor. Examples of optimization of Betz rotor for the three models of wake were

Table 1 Main assumptions underlying various models of the rotor wake. Theory

Number of blades Determination of the vortex system pitch

Distribution of the circulation along the blade

Betz–Joukowsky limit BEM method by Glauert Joukowsky rotor Betz rotor Prandtl correction Goldstein solution Theodorsen solution Current solution for NEJ rotor (Section 4.3) Current solution for Betz rotor (Section 4.4)

Actuator disc Not speciﬁed Inﬁnite Inﬁnite Finite Finite Finite Finite Finite

Not speciﬁed Not speciﬁed Constant Betz distribution Betz distribution with Prandtl correction Goldstein's function Goldstein's function Constant Goldstein's function

a

Not speciﬁed Not speciﬁeda (i) Without correction for axial induction (i) Without correction for axial induction (i) Without correction for axial induction (i) Without correction for axial induction (ii) With correction for axial induction in far wake (iii) With correction for axial induction in rotor plane (iii) With correction for axial induction in rotor plane

Axial and azimuthal induction factors are introduced in the rotor plane.

Table 2 Pitch deﬁnitions and rotor solutions for different wake models. Wake

Pitch

(i)

U1 or lB ¼ Ω 0

(ii)

lT ¼ U 1 ðΩ10 wÞ or

(iii)

Power and thrust coefﬁcients

lO ¼

Ω0 lB U1

1 U1 1 2 w Ω0

C T B ¼ w 2I 1 lB þ wI2 lB C P B ¼ w 2I 1 lB wI 3 lB w I 1 lB ð2þ wÞ wI3 lB

¼1

Ω0 lT U1

¼ 1 w

or

Ω0 lO U1

¼ 1 12 w

Optimal value of factor w wðlS Þ ¼ II13

C T T ¼ w 2ð1 wÞI 1 lT þ wI 2 lT C P T ¼ wð1 wÞ 2I1 lT wI 3 lT w I 1 lT ð2þ wÞ wI3 lT

wðlr Þ ¼

C P O ¼ 2wð1 12 wÞðI 1 ðlO Þ 12 wI 3 ðlO ÞÞ C T O ¼ 2wðI 1 ðlO Þ 12 wI 3 ðlO ÞÞ

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

wðlO Þ ¼ 3I23 I 1 þ I 3 I21 I 1 I 3 þ I 23

2I1 þ I3

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 2 2 4I 1 2I 1 I 3 þ I 3 3I 3

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considered in Refs. [97–99]. In Table 2 we reproduce the three variants of the rotor solutions for different wake models in which the pitch index indicates by the ﬁrst letter of the authors who proposed it: (B), Betz and Goldstein [27,33]; (T), Theodorsen [34]; and (O), Okulov and Sørensen [37,38]. The third column of Table 2 contains the integrals of I1 and I3 deﬁned by (65) and I1 I2 ¼I3; the last row corresponds to the solution for the optimal values of w (see Refs. [97–99]). It is very important that power and thrust coefﬁcients for all wake models for a ﬁxed value of the helical pitch are functions of one variable. This is the induction factor w of the moving wakes, which is the same for all points of the sheets and which represents a convenient and unique parameter for the optimization. Taking into consideration that the thrust coefﬁcient is not critical for wind turbines, we ﬁnd a value of the induction factor w for a maximum power coefﬁcient. For each rotor model, differentiating of CP with respect to w yields the maximum value, resulting in the last column of Table 2. For a correct comparison of the different rotor theories of Table 2 it is necessary to replace an abstract pitch by a real value of the operating regimes (1) for wind turbines which for each theory according to the pitch deﬁnition of the second column of the table transform to the form: 1

λ0 ¼ ; lB

λ0 ¼

1 wðlT Þ ; lT

or

λ0 ¼

1 12 wðlO Þ lO

ð67Þ

where w is deﬁned in the last column of Table 2. Results of the optimization of wind turbines for these three models of the wake behind the Betz rotor are presented in Figs. 27 (Betz model), 28 (Theodorsen model) and 29 (Okulov and Sørensen model).

For the limiting case of N b ¼ 1, the value of the Goldstein circulation takes a simple form G1(x,l)¼ x2/(x2 þl2). Then I1 and I3 from (65) can be presented in a simple analytical form [38]: 2

I1 1 ¼ 1 l ln

2

1þl l

2

and

I1 3 ¼ 1þ

l

2

1þl

2

2

2l ln

1þl

2

2

ð68Þ

l

i.e., the analytical forms of the solutions for all rotor theories at Nb ¼ 1 are written by a combination of conventional functions. This limiting case is of great importance for a comparison of the rotor theories because it corresponds to the maximum possible value of the power coefﬁcient for an ideal rotor. The results of the optimization for all cases in the limit case when N b ¼ 1 are shown in Fig. 30. On the ﬁgure is also shown the well-known Betz–Joukowsky limit derived from actuator disk theory [1]. An additional wellexamined result is Glauert's calculations [23] by the blade element momentum method for the effects of ﬁnite tip speed ratio. The comparison of the results of the power coefﬁcient obtained using the rotor theories based on (i) and (ii) rotor show their inadmissibility (Figs. 27, 28 and 30). The (i) model gives an absurd prediction of 100% utilization of wind energy, and the (ii) model underestimates the limiting value. Only the (iii) model is well correlated with both the Betz–Joukowsky limit and Glauert's calculation (Figs. 29 and 30). This means that the rotor theory with wake model (iii) [38,39] is correct and that the choice of this wake model seems completely to have closed the classical problem of the Betz rotor with ﬁnite number of blade with a non-expanding wake expansion. We suggest that the reason why the erroneous models of (i) and (ii) persisted for so long is related to the initial effort to describe only propeller performance. In this case the ratio C T =C P should be maximized instead of C P , which is the parameter to be

Fig. 27. Optimization according to the wake model (i) (weakly loaded rotor) introduced by Betz and Goldstein [27,33]. (the horizontal dotted line is the Betz–Joukovsky limit [1]; and circles – Glauert calculations [23]; CP is power coefﬁcient and CT is thrust coefﬁcient (Table 2) as function of tip speed ratio (67) for different number of blades Nb).

Fig. 28. Optimization according to the wake model (ii), introduced by Theodorsen [34] (for horizontal dotted and circles see caption of Fig. 27).

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Fig. 29. Optimization according to Okulov and Sørensen's wake model (iii) from Sections 4.2 and 4.3 [38,39] (for horizontal dotted and circles see caption of Fig. 27).

Fig. 30. The highest CP and the corresponding CT for different rotor models: (i) dashed line: Betz and Goldstein [27,33]; (ii) dash-dotted line: Theodorsen [34] and (iii) solid line: Okulov and Sørensen [38,39].

optimized in the case of a wind turbine. If we rule out an anomalous behavior of the thrust coefﬁcient C T for the (i) model at small λ0 (Figs. 27 and 30), the ratio C T =C P takes nearly the same value for the three wake models. For a propeller, this means that the choice of the pitch is not important, as it is for a wind turbine. This fact has for a long time prevented the mistake to be revealed in Goldstein and Theodorsen rotor theories. Though the choice of the wake model (iii) completely closes the problem for the Betz rotor, for the NEJ rotor, the selection of wake pitch is not sufﬁcient for a complete statement of the problem. Returning to the NEJ rotor model, we note again that Joukowsky considers tip vortices with a ﬁnite core size by using an ideal ﬂuid model. In this regard, the following questions remain open: how is the tip vortex formed on the blade, where exactly is it formed, and what is the convective velocity of the tip vortex as it moves downstream from rotor into the wake, as compared the free with wind speed and the average velocity in the wake. Those questions cannot be answered for an ideal ﬂuid model. For this reason it is necessary to make some additional assumptions for the NEJ rotor. For the system associated with the wake of Fig. 22a, which essentially consists of doubly-inﬁnite helical vortices (vortex multiplet), the questions can be reformulated as follows: (1) How should the radius of the vortex multiplet coincide to the rotor radius? (2) How should the velocity of the vortex multiplet be chosen? The ﬁrst question can be reduced to analyzing two basic approximations of the non-expanding wake. These cover two possible basic positions of tip vortices on the blade: (A) The vortex core lies wholly on the blade. (B) Only a part of the vortex core coincides with the blade surface.

In fact, the proposed options differ from each other by a small quantity equal to the radius of the tip vortex whose size is signiﬁcantly smaller than the multiplet size (see Fig. 24), but both cases should be studied at least. The second question is relevant because the doubly-inﬁnite multiplet is only a model of the semi-inﬁnite rotor wake. The multiplet motion includes a relationship between several velocity components: an axial interaction between wind speed and the velocity in wake resulting in the “roller-bearing analogy” in (46) for the tip vortex motion [56,94]; an azimuthal component induced by the hub vortex (Fig. 22a); and the rotation of the multiplet constrained by the self-induced motion of the helical vortices (6). The following reasonable assumptions were tested with respect to the multiplet motion: (C) The azimuthal component of the self-induced velocity of the tip vortices in the multiplet is completely compensated by the azimuthal velocity in the other direction due to the hub vortex, see condition (45). (D) The relative axial motion of the multiplet obeys the “bearing” analogy (46). Hypothesis (C) can be implemented by using the formulas (6)–(8) to choose the helical vortex core whose self-induced motion equals the swirl velocity in the opposite direction caused by hub vortex. The idea to deﬁne the vortex core by self-induced velocity of the multiplet again goes back to Joukowsky [15]. In the early 20th century this idea could not be properly considered due to the lack of experimental information and the rough approximation of the helical vortex motion by the formula for vortex ring (Fig. 19a). In spite of that, Joukowsky in 1912 proposed a model of the “frozen-in” wake based on the photos of the wake behind screw propellers made by Flamm [100]. That suggests a

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rotation of the wake together with the rotor to maintain the steady-state helical shape. The assumptions (C) and (D) stated above support an opposite hypothesis to Joukowsky's proposal about the equal rotation of the free tip vortices in the wake together with rotor blades. Both current assumptions allow the free vorticity leaving from tip to move axially with the ﬂow velocity without rotation like bound vorticity of the blades. According to (C) these free vortices move downstream with a constant undisturbed velocity (wind speed) minus half the ﬂow deceleration in the wake. This motion of free vortices results in the tip vortices with a ﬁxed helical form and looks similar to the “frozen-in” wake model which both coincide with the wake visualizations on the instantaneous photos. Though the wake approximation of the free vortex system by (C) and (D) did not include a wake expansion recorded in the experiments. Just below, in the description of our recent experiments, we will show that the conditions (C) and (B) are much closer to reality than the “frozenin” model by Joukowsky. As the ﬁrst step to test our hypotheses we will examine the effect of various deviations from the proposed assumptions on the power characteristics of the rotor solutions. As for the Betz rotor we should transit from the abstract pitch of the wake to the operating parameter – tip-speed ratio (1) which for the NEJ rotor can be done in accordance with (51): R R a ε : ð69Þ ¼ ð1 υÞ 1 1 þ U1 l l 2 R Testing of the different assumptions (A)–(D) for the NEJ rotor model with the ﬁxed pitch by (iii) wake model was arranged by comparison with the Betz–Joukowsky limit. As a result of the searching through various choices, it was found that any deviations from the assumptions (C) and (D) for the multiplet motion

λ0

Ω0 R

and option (B) of its radial size of the tip vortex conﬂicts with the Betz–Joukowsky limit and give higher values of the power coefﬁcient (Fig. 31). Thus, to agree with the Betz–Joukowsky limit the additional assumptions for the NEJ rotor of the Section 4.2 should be selected in the form: (iii) The pitch of the wake vortices behind rotor depends on the velocities induced in the rotor plane: uz0 ; uθ0 or formulated mathematically, tan Φ ¼

U 1 uz0

Ω0 R þ uθ0

l ¼τ ; R

(B) The multiplet radius coincides with the rotor radius; (C) The azimuthal component of the self-induced velocity of the tip vortices completely cancels the azimuthal velocity induced by the hub vortex, see the condition (45); (D) The relative convective motion of the multiplet in the axial direction takes place in accordance with the “bearing” analogy under a half of the axial induction factor (46). The last question that should be considered for the analytical models of the rotor is the estimation of the inﬂuence of wake expansion on the power coefﬁcient. We can analyze this question using numerical vortex model [101]. The result presented in Fig. 32 indicates the very slight inﬂuence the wake expansion has on the power coefﬁcient of the wind turbine, with a difference of about 1–2%. Thus, the current analytical rotor models for rotors with a ﬁnite number of blades and without wake expansion are quite correct for performance calculations. The calculations give some justiﬁcation for the additional assumption to different rotor theories, though the ﬁnal criterion here should be based on experimental results.

Fig. 31. Examples of the test for the additional assumptions of the NEJ rotor: (top row) – calculation with the option (A) and a deviation from assumptions (C) and (D); (lower row) – calculation in accordance with the assumptions (C) and (D) and option (B).

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4.5. Comparison of NEJ and Betz rotors In the following we present some representative results from the rotor models described above. To compare the efﬁciency of the two rotor concepts – the NEJ rotor and the Betz rotor – it is necessary to use some unambiguous parameters. As usual in rotor aerodynamics, we employ the axial interference factor, a, and the tip-speed ratio, λ0 . However, since a appears only for the NEJ rotor in (43) and (54), but are not explicitly included in (64) and (67) for power coefﬁcient and λ0 of the Betz rotor, it is needed to derive some additional relations. For both rotors, λ0 is connected to the

23

helical pitch l by direct speciﬁcation of a through (55) and w through (66), we use directly (69) for the NEJ and the last formula from (67) for the Betz rotor. Fig. 33 presents the maximum power coefﬁcient of both rotors for different number of blades as function of tip speed ratio. From the plots it is evident that the optimum wind power coefﬁcient of the Joukowsky rotor for all number of blades is larger than it for the Betz rotor. The difference, however, vanishes for λ0 -1 or for N b -1, where both models tend towards the Betz limit. An expression for the axial interference factor of the Betz rotor R1 can be obtained by combining Eqs. (2) and (59), a ¼ w 0 Gðx; lÞdx.

Fig. 32. Inﬂuence of the wake extension for different number of blades: 1 ( ), 3 ( þ) and 7 ( ). (Left) The wake extends in the axial direction z. (Right) The difference between values of the maximal power coefﬁcient of numerical calculation with the wake extension [100] and the exact solution without the wake expansion in accodance with Section 4.3.

Fig. 33. Maximal power coefﬁcients as function of tip speed ratio and number of blades (here and hereby on the next two ﬁgures – left: NEJ rotor; right: Betz rotor).

Fig. 34. Axial interference factor as function of tip speed ratio and number of blades.

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In Fig. 34 we display the axial interference factor of the two rotors as function of tip-speed ratio and number of blades. An approximation for the optimum circulation of the NEJ rotor can easily be determined by the induction factor (Fig. 34) and the pitch of the wake using (50). Because l ¼ Rð1 0:5að1 þ σ ÞÞ=λ0 in accidence with (51), we can estimate the dimensionless circulation on the blade as Nb Γ a a ¼ 1 ð1 þ σ Þ ; 2 2π U 1 R λ0

ð70Þ

where the size of the vortex core σ was found from (48) or Fig. 24 and the induction factor by (55) or Fig. 34. It should be mentioned that the procedure used in Section 3.6 to approximate the optimum circulation for inﬁnitely small vortex cores, is valid only for the NEJ rotor because it takes a constant value along the blade. For the Betz rotor, the procedure involves a more complex algorithm with an estimation of the Goldstein circulation along the blade (Figs. 25 and 26). Further comparison of the two plots shows that the Betz rotor for the same tip speed ratio decelerates the ﬂow less than the NEJ rotor. As a consequence, if we employ the axial interference factor as independent variable, an optimum Betz rotor can produce more power than a NEJ rotor under the same deceleration of the wind (see Fig. 35). Thus in Sections 4.2 and 4.3 new analytical models have been developed for rotors with a ﬁnite number of blades, including a rotor with constant circulation (NEJ rotor) and a rotor subject to Goldstein's (Betz rotor) circulation under new conditions (iii) for the pitch of the rotor wakes. Both methods are based on an analytical solution of the problems of equilibrium motion of multiple helical vortices and vortex sheets (by a new approach via a simpliﬁcation of Goldstein's solution), modeling the associated vortex systems for rotor wakes of both rotors. The vortex multiplet is a set of helical vortices with ﬁnite core, introduced to eliminate the singularity of the induced velocity ﬁeld in the vicinity of each ﬁlament. The ﬁnite core radius is determined in the framework of an ideal ﬂuid by assuming that the relative wake motion is governed by a constant axial speed equal to half the averaged induced velocity in the wake and wind speed. The main achievement of the model is that it eliminates the singularity of the solution at all operating conditions. In contrast to earlier models, both new models enable for the ﬁrst time the determination of the theoretical maximum efﬁciency of NEJ and Betz rotors with an arbitrary number of blades. Optimum conditions for ﬁnite number of blades as function of tip speed ratio were compared for two models: (a) NEJ rotor with constant circulation along the blade, and (b) Betz rotor with circulation given by Goldstein's function. For all tip speed ratios the Joukowsky rotor achieves a higher efﬁciency than the Betz

rotor, but the efﬁciency of the Betz rotor is larger if we compare it for the same deceleration of wind speed.

4.6. Experimental testing It should be noted that the experimental aerodynamics of the rotor developed more slowly than the theories described above and sometimes hindered theoretical developments. As already mentioned in Ref. [1], the wrong perception and interpretation of experimental results by Parsons delayed acceptance of Froude's theory for about 30 years. The ﬂow downstream the impeller is very complex under strong swirling [102,103] and requires special experimental techniques and methods for correct investigation [104,105]. Nevertheless, initial experiments were performed using simple experimental techniques, such as visualization of the vortex structure in the water behind a screw by small air bubbles, and a determination of changes in the ﬂow structure by paper strips or hairsprings around rotors in wind tunnels. Flamm's bubble visualization behind the screw and Riabouchinsky's hairspring visualization of the ﬂow directions behind a propeller (described in Ref. [15]) gave Joukowsky the basic arguments to develop his vortex theory of screw propeller for the NEJ rotor [15]. Then came a long period of studying the average characteristics of the swirling ﬂow in the rotor wake. First this was done applying various intrusive methods, such as Pitot tubes and hot wire anemometers, and then using non-intrusive velocimeters (such as laser-Doppler anemometry and particle image velocimetry) [106]. This stage of rotor ﬂow testing produced some useful results [107], though the averaging of velocity resulted in loss of important information about the vortex structure of the ﬂow downstream the rotor that was observed by visualization, and used in development of a classical vortex theory of the rotor. Knowledge of the average ﬂow regularities has stimulated the development of some semi-empirical and engineering methods of calculating the wake, but it certainly has been limited and has not met expectations, both in terms of the development of theories, and for the veriﬁcation of numerical computations. Perhaps that is why the recent European project «MEXICO» [108] was a full-scale investigation of the ﬂow downstream the rotor in a wind tunnel. Despite the processing of the ﬂow diagnostic results being incomplete, the ﬁrst comparisons of these measurements are in poor agreement with the calculated results [109,110]. Another recent and comprehensive study of the ﬂow structure downstream of a ship propeller was based on the visualization combined with LDA measurements of the ﬂow ﬁeld using phase averaging in several cross sections of the wake [111]. There is a need for detailed testing of the theories described here, especially since modern measuring technologies allow such

Fig. 35. Maximal power coefﬁcients (54) and (64) as function of axial interference factor and number of blades.

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veriﬁcation. At present, the ﬁrst two authors have initiated such comprehensive experimental study of a laboratory model of a three-bladed rotor [112–114] and we note other recent results [115–121]. In our experiment, a stereo PIV system was used to determine the velocity ﬁeld behind the wind turbine model in a water channel. A three-bladed rotor was designed to operate at λ ¼5. Three-dimensional distributions of the instantaneous velocity ﬁelds were obtained. These are presented in Fig. 36 showing the cylindrical polar velocity components u, w, v. The data clearly captures the complex vortex system in the wake, speciﬁcally the three tip vortices near the blades at x/R¼ 0. From the distributions of the velocity components u and v in the plane of a laser light sheet, the sections of vortex cores are well deﬁned from the blade tip downstream in the axial direction. Further, a vortex sheet drifting down from a sharp blade edge along its entire length is well captured. This becomes apparent due to the periodic variation of velocities with approximately equal pitch in axial direction behind the blade, as shown by pronounced radially extended tongues. There is a concentration zone of ﬂow rotation along the rotor axis that corresponds to a concentrated hub vortex. The poor deﬁnition of the vortex cores and sometimes their total absence from the third component w is due to the disappearance of the peripheral azimuthal velocity at the wake boundary behind the rotor in the domain of the tip vortices (middle pictures in Fig. 36).

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Additionally, Fig. 37 shows proﬁles of the same velocities in the cross section at a distance of 0.4R in front of the rotor (the squares), and at a distance of 2.5R behind the rotor (circles). The ﬁlled circles show the proﬁles that correspond to half the sum of velocities in front of the blade and behind it. The middle plot of the Figure supports above observation about absent azimuthal velocity in the wake boundary r/R 41.2. It should be emphasized that the results conﬁrm many points of the existing rotor theories. First of all, this applies to the simplest Froude theory which for the optimal regimes involves double decelerating of the axial velocity in the far wake, as compared to its decelerating in the rotor plane [1]. Indeed, for the tested rotor at the optimal tip speed ratio equal to 5, Fig. 37, u/V at the rotor (dark circles) is close to 1/3, and in far wake it is close to 2/3, as predicted by Froude theory. The next element of the classical theories is the prediction of the wake expansion behind the wind turbines by approximately 1.41 times at the optimal modes [78]. Indeed, in Fig. 36, as the distance from the turbine increases, the radius of the tip vortices increases up to the lower horizontal limit of the observation domain at about 1.28R. In Fig. 32 the wake expansion was also determined for rotor with ﬁnite number of blades according to the calculation of Ref. [101]. An important issue for ﬁnite Nb, the assumption of constant pitch of the vortices, despite the reduction in velocity producing different average velocities in the near and the far wake. This

Fig. 36. Distribution of instantaneous velocity components: axial – u, azimuthal – w, and radial – v (top to bottom) in the vortex wake behind the rotor model for λ ¼ 5.

1

w/V

u/V

v/V

0.4

0

0.2

-0.2

0.667

0.333 -0.4

0 0

r/R 0

0.5

1

r/R

r/R 0

0.5

1

0

0.5

1

Fig. 37. Proﬁles of the average axial, azimuthal and radial velocity components in front of the rotor (squares), during motion of the rotor (ﬁlled circles), and in the far wake (open circles) for λ¼ 5.

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caused misunderstanding and dismissal of that assumption of the classical rotor theories. The constancy of the pitch follows from two pitch deﬁnitions: in the rotor plane, Ref. [37], and in the far wake, Ref. [122]. In the ﬁrst article the wake pitch was assumed to be the pitch angle of the blade (Fig. 23), that is, l=R ¼ ð1 0:5aÞ=λ0 when a is the induction factor in far wake if we neglect the very small radius of the vortex core. The same formula in Ref. [122] was found from a simple circulation calculation around a rectangular contour (of axial length h) containing a tip vortex in far wake, exploiting that the velocity of the vortex is the average of the wake velocity and wind speed. It means that the tip vortex formed on the rotor blade moves into the free far wake with the same pitch value. Experimental data from the «MEXICO» project [110] and the results in Fig. 36 show the nearly constant axial distance between the cores. Another important assumption of the analytical model considered in this paper, was the total absence of the tangential rotation of the tip vortices, as opposed to the concept of “frozen-in” wake, rotating with the angular velocity of the rotor, according to Joukowsky [15]. This assumption is supported by the azimuthal velocity of the tip vortices being close to zero (see the instantaneous plot with very small w in the tip vortex positions in Fig. 36 and a consistency of this fact for average velocity w/V in Fig. 37), but in Joukowsky's “frozen-in” wake the azimuthal velocity should be equal to the blade tip rotation. Thus, this experimental study not only allowed us to diagnose the instantaneous structure of the three-dimensional velocity ﬁeld in the longitudinal section of the ﬂow downstream of a three bladed rotor at different ﬂow regimes, but it also made it possible to verify and validate some of the assumptions and hypotheses underlying the classical rotor theories.

5. Conclusions This is the second of two articles dedicated to the 100th Anniversary of a remarkable publishing event: the ﬁrst article of the famous series “Vortex theory of screw propeller” by N.E. Joukowsky. These articles together give a retrospective view on the key historical stages in the development of consistent theories for determining optimal wind turbine rotors with ideal load distributions along the blades, and describe the current status of these theories to illustrate the importance of Joukowsky's contribution. The analysis of primary sources resulted in the conclusion, that at the time of the ﬁrst development of a vortex theory for the rotor, the leading role in the evolution of rotor aerodynamics belonged to the scientiﬁc school headed by Professor Joukowsky. The results of Joukowsky and his followers were several years ahead of those of foreign scientiﬁc schools. In particular, the development of a rotor theory based on the vortex concept was proposed by Joukowsky seven years before the German aerodynamic school of Professor Ludwig Prandtl. The main result of the retrospective analysis presented in the second section of the ﬁrst article was devoted to acknowledging the principal result in wind power engineering due to Joukowsky: the limiting value of power which can be produced from the kinetic energy of wind. Henceforth this result should be called the Betz–Joukowsky limit, since it was published by Joukowsky (simultaneously with A. Betz, Prandtl's pupil) in 1920. The article turned out to be the last in the scholar's life and long remained unknown to the world scientiﬁc community. It seems paradoxical that the last of his efforts for the development of wind power engineering was not immediately followed up because of the lack of interest in wind energy due to the availability of unlimited natural resources. It would be comforting to think that he had foreseen its unprecedented growth

occurring in the world today, knowing that his results would again be relevant and in demand after a century. In this paper, we document another instance in which history has short-changed Joukowsky: it was he alone who introduced the famous Kutta– Joukowsky (KJ) equation relating airfoil lift and circulation. Joukowsky also introduced the cascade model for blade element analysis so it is pleasing that this model is shown in Section 3.6 to allow a very simple derivation of the KJ equation. In addition, the ﬁrst article examines a paradoxical result of the general momentum theory of a rotor proposed by Joukowsky [17]. This general actuator disc model was formulated by Joukowsky to obtain analytical solutions, considering the disc as a limiting case of a rotor with an inﬁnite number of blades. However, in the early 20th century, a solution for a rotor with ﬁnite number of blades had failed due to its complexity, and Joukowsky [15] at the conclusion gave up his search, noting that “… this analysis would prove to be very complicated”. However, even in this extreme case, an adequate solution for the general momentum theory was not easy to deduce, because for the operation of wind turbines at low tip speed ratios this would lead to an unnatural inﬁnite growth of power generated by the ﬂow, or to an unlimited increase of the power coefﬁcient. This paradox of the ideal model for many years has stimulated numerous attempts to refute the exact value of the Betz–Joukowsky limit, obtained from a simple momentum actuator disc theory, without specifying the nature of the energy extraction. The general theory was introduced for an inﬁnite number of blades. Finite blade numbers induce additional losses, which, naturally, makes the power coefﬁcient lower than the Betz– Joukowsky limit. This is exactly what happens when blade sections are introduced (using the KJ theory) by alternative theories, such as the momentum rotor theory: with increasing tip speed ratio, the power coefﬁcient increases monotonically from zero to the Betz–Joukowsky limit, but always remains below it. The authors' analysis of the causes of abnormal power prediction for the general momentum theory has shown that it is determined by the limitations of the ideal ﬂuid model. The adaptation of a solution by introducing the effect of pressure and friction on the boundary of the control stream tube allowed the authors to eliminate the inﬁnity power yield. As revised, the behavior of the maximum power coefﬁcient was to ﬁt in with the solutions for other rotor models and determined its value as being less than the Betz–Joukowsky limit, as it actually should be when specifying the ﬂow energy sink in the form of rotating blades as this was done in the general momentum theory. Derivation of an analytical solution for ﬁnite blade numbers and ﬁnite size of the core of tip vortices within the vortex concept of NEJ rotor, proposed a century ago, though never solved by Joukowsky, was the concluding result of the current article. The sticking point here was the lack of solution of the problem of helical tip vortices with a ﬁnite radius of the vortex core. It was the one that Joukowsky had in mind, referring to the complexity of the analysis. It should be noted that he justiﬁed the approximation of helical vortex with an osculating vortex ring in his ﬁrst article “Vortex theory of screw propeller” 60 years earlier than this approximation was rediscovered in 1970. However, its accuracy was still not sufﬁcient to solve the problem. The analytical solution of the helical vortex problem with ﬁnite core was obtained only in the beginning of the 21st century that allowed the ﬁrst two authors of this work to be the ﬁrst who have brought the solution of Joukowsky's rotor (NEJ) to its conclusion. Moreover, the approach suggested by the authors to determine the optimal load along the blade proved to be suitable for the study and for comparison of other rotor models. For the basic concepts of optimal rotor proposed by Betz, Goldstein and Theodorsen, the solutions were obtained and the results were analyzed and

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compared. In particular it was found out that the model of NEJ rotor was more effective than other models for wind turbines. Thus, this article justiﬁes the priority of the scientiﬁc school headed by Professor Joukowsky in the development of the vortex theory of a rotor and acknowledges his equal priority in ﬁnding the crucial result on the maximum value of power produced from the kinetic energy of wind. We then eliminated the problems that arose in Joukowsky's general momentum actuator disc theory in the operational modes of a low-speed wind turbine. The analytical solution for the ideal NEJ rotor with a ﬁnite number of blades and ﬁnite core of tip vortices was found for the ﬁrst time and became a signiﬁcant contribution to the further development of the vortex theory of a rotor by Professor Joukowsky.

Acknowledgments This work has been carried out with a support of the Danish Council for Strategic Research for the project COMWIND – Center for Computational Wind Turbine Aerodynamics and Atmospheric Turbulence: Grant 2104_09_067216/DSF (http://www.comwind. org) and the Russian Science Foundation (Grant no. 14-1900487). DHW acknowledges the support of the Canadian Natural Science and Engineering Research Council through its Industrial Research Chair program in conjunction with the ENMAX Corporation (File No: 417872 - 10) References [1] van Kuik GAM, Sørensen JN, Okulov VL. The rotor theories by professor Joukowsky: momentum theories. Prog Aerosp Sci 2014 this issue. http://dx. doi.org/10.1016/j.paerosci.2014.10.001. [2] Anderson Jr JD. Fundamentals of aerodynamics. 2nd ed.. McGraw-Hill, New York; 1991. [3] Joukovsky NE. On annexed [bounded] vortices. Trudy Otdeleniya Fizicheskikh Nauk Obshchestva Lubitelei Estestvoznaniya 1906;13(2):12–25 [in Russian[. [4] Kutta WM. Auftriebskräfte in strömenden Flüssigkeiten Ill. Aeronaut Mitt 1902;6:133–5. [5] J.A.D. Ackroyd, B.P. Axcell and A.I. Ruban, Early developments of modern aerodynamics, 2001, Reston, VA : AIAA ; Oxford : Butterworth-Heinemann. [6] Panton RL. Incompressible ﬂow. 3rd ed.John Wiley & Sons, New York; 2005. [7] Lamb H. Hydrodynamics. Dover, New York; 1945. [8] Joukowsky NE. Geometrische Untersuchungen über die Kutta'sche Strömung. Trudy Otdeleniya Fizicheskikh Nauk Obshchestva Lubitelei Estestvoznaniya 1910;15(1):10–22; Joukowsky NE. Geometrische Untersuchungen über die Kutta'sche Strömung. Trudy Otdeleniya Fizicheskikh Nauk Obshchestva Lubitelei Estestvoznaniya 1911;15(2):36–47. [9] Kutta WM. Über eine mit den Grundlagen des Flugsproblems in Beziehung stehende zweidimensionale Strömung. Sitzungsberichte Bayer Akad Wiss 1910;40:1–58. [10] Bloor D. The enigma of the aerofoil. Chicago University Press, Chicago; 2011. [11] L. Prandtl Über Flüssigkeitsbewegung bei sehr kleiner Reibung. Verhandlungen des III. Internationalen Mathematiker-Kongresses, Heidelberg, 1904 (Teubner, Leipzig, 1905), 484–491; Reprinted in: L. Prandtl, A. Betz (Eds.), Vier Abhandlungen zur Hydrodynamik und Aerodynamik, 1927, Institut für Strömungsforschung, Göttingen; Reprinted in: W. Tollmien, H. Schlichting, H. Görtler (Eds,), Gesammelte Abhandlungen zur angewandten Mechanik, Hydro- und Aerodynamik, vol. 2, 1961, Springer, Berlin; English translation in: On the motion of ﬂuids of very small viscosity, 1928, NACA Tech. Memo. 452, Washington, DC; English translation (Prandtl L. Ergebnisse und Ziele der Göttinger Modellversuchsanstalt. Zeitschrift für Flugtechnik und MotorluftschiffartZeitschrift für Flugtechnik und Motorluftschiffart 1913) in: J.A.K. Ackroyd, B.P. Axcell, A.I. Ruban (Eds.), 2001, Butterworth-Heinemann, Oxford, UK. [12] Joukowsky NE. Über die Konturen der Tragﬂächen der Drachenﬂieger. Z Flugtech Motorluftschiffahrt 1910;22:281–4. [13] Blasius H. Funktionentheoretische Methoden in der Hydrodynamik. Z Math Phys 1910;58:90–110. [14] Drzewiecki SK. Theory of air propellers and the method of their calculation; foreword by N.B. Delone. Publ R.K. Lubkovsky, Kiev. 1910. [15] Joukowsky NE. Vortex theory of screw propeller. Trudy Otdeleniya Fizicheskikh Nauk Obshchestva Lubitelei Estestvoznaniya 1912;16(1):1–31 [in Russian]. [16] Vetchinkin VP. Calculation of screw propeller, Part II. Trudy Avia RaschetnoIspytatelnogo Byuro 1918;4:1–129 [in Russian].

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Please cite this article as: Okulov VL, et al. The rotor theories by Professor Joukowsky: Vortex theories. Progress in Aerospace Sciences (2014), http://dx.doi.org/10.1016/j.paerosci.2014.10.002i

The rotor theories by Professor Joukowsky: Vortex theories

The rotor theories by Professor Joukowsky: Vortex theories

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