In order to outline the generation of electricity by extracting kinetic energy from the wind field we consider the local balance equations for momentum (i.e., Newton’s 2^nd axiom), Equation (2.1), and total mass, Equation (2.2), for a macroscopic system given by (e.g., [13] -[16] ):

and

Here, is the air density, t is time, is the velocity of the flow, is the Stokes stress tensor given by

where is the bulk viscosity (near zero for most gases), is the identity tensor, is the gravity potential, and is the angular velocity of the Earth. Both and are symmetric second-rank tensors. Furthermore, the 1^st term of the left-hand side of Equation (2.1) describes the local temporal change of momentum, and the 2^nd term represents the exchange of momentum between the system under study and its surroundings, where exerts on the boundary of this system. The 1^st term on the right-hand side of this equation represents the gravity force, and the 2^nd one the Coriolis force. Equation (2.2) is the equation of continuity. In addition, local balance equations for various energy forms (i.e., internal energy, kinetic energy, potential energy, and total energy), various water phases (i.e., water vapor, liquid water, and ice), and gaseous and particulate atmospheric trace constituents exist. All these local balance equations can be derived from integral balance equations (e.g., [13] [14] ). Since and, Equation (2.1) is often written as (e.g., [17] - [19] )

where is the substantial derivative with respect to time. This equation form not only disguises its origin, namely the corresponding integral balance equation, but also is unfavorable if it has to be averaged, for instance, in the sense of Reynolds [20] to find a tractable equation for turbulent atmospheric layers. Nevertheless, in accord with Lamb’s transformation (e.g., [21] )

Equation (2.4) may be written as

The curl of Equation (2.6) leads to the prognostic equation for the vorticity

As the curl of the gradient of a scalar field is equal to zero, Equation (2.7) can be written as

This equation plays an important role in the description of rotational flows as occurred in the wake of the wind turbine. If the friction effect is negligible and the density is considered as spatially constant like in case of an incompressible fluid we will obtain

To deduce the local balance equation for the kinetic energy of the flow, Equation (2.1) has to be scalarly multiplied by the velocity vector. Using the identities

and

leads to

The colon expresses the double-scalar product (also called the double dot product) of the tensor algebra. Furthermore,. The 1^st term of the left-hand-side of Equation (2.12) describes the local temporal change of kinetic energy, and the 2^nd term is the energy exchange of the system with its surroundings which is performed by the surrounding air on the boundary of the system. The 1^st term of the right-hand-side represents the conversion of potential energy into kinetic energy and vice versa, the 2^nd term describes the reversible work rate of expansion, , or contraction, , and the 3^rd term represents the irreversible work rate owing to viscous friction. This term represents the dissipation of kinetic energy into the reservoir of heat. The term of our primary interest reads

It describes the transport of kinetic energy by the flow, and it may be called the kinetic energy stream density, but it is also denoted as wind power density. Inserting the definition of the total pressure,

into Equation (2.12) yields

Since the ABL is mainly governed by turbulent motion, the use of the macroscopic balance Equations (2.1), (2.2), and (2.12) is rather impracticable. Therefore, these balance equations are customarily averaged in the sense of Reynolds [20] . However, conventional Reynolds averaging will lead to various short-comings in the set of governing equations for turbulent atmospheric flow, even if these averaging techniques can be performed accurately [22] . If we ignore, for instance, density fluctuation terms, the possibility to describe physical processes as a whole will clearly be restricted (see [23] [24] ). The key questions that still remain are (a) how to average the governing macroscopic equations in the case of turbulent atmospheric flows and (b) what are the consequences of such an averaging, not only for momentum and total mass, but also for various energy forms like kinetic energy, potential energy, internal energy, and total energy, consisting of the sum of these three energy forms. In the terrestrial atmosphere, the total energy is conserved. As sketched in Figure 1 for a turbulent system (Hesselberg fluid), there are various ways of energy conversion.

As argued by various authors [22] -[34] , the density-weighted averaging procedure suggested by Hesselberg [35] is very appropriate to formulate the balance equation for turbulent systems. It is given by

where is a field quantity like the wind vector, , the specific internal energy, e, and the specific enthalpy, h. Furthermore, the overbar () characterizes the conventional Reynolds mean. Whereas the hat () denotes the density-weighted average according to Hesselberg, and the double prime (") marks the departure from that. It is

obvious that. The Hesselberg mean of the wind vector, for instance, is given by.

Note that intensive quantities like the pressure, p, and the density, , of air are averaged in the sense of Reynolds. Arithmetic rules can be found, for instance, in [25] - [27] [29] [31] . As pointed out by Kramm and Meixner [22] and Lumley and Yaglom [36] , Hesselberg’s average is sometimes misnamed the Favre average.

In comparison with that of Reynolds, Hesselberg’s averaging calculus leads to several prominent advantages [22] [25] - [27] [29] - [31] [35] : (a) The equation of continuity,

keeps its form, and (b) the mean value of kinetic energy can exactly be split into the kinetic energy of the mean motion and mean value of the kinetic energy of the eddying motion, according to

This advantage is especially important in the theoretical description of the extraction of the kinetic energy from the wind field for generating electricity. The use of density-weighted averages is the common way to define averages in studies of highly compressible turbulent flows (see also [29] [32] ), probably the most natural way to define averages. The kinetic energy of the mean motion is usually abbreviated by MKE, and the kinetic energy of the eddying motion is usually called the turbulent kinetic energy abbreviated by TKE.

Hesselberg’s average procedure will be applied within the framework of this contribution. It can be related to that of Reynolds by (e.g., [22] [26] [30] [31] [37] )

Here, the prime () denotes the deviation from the Reynolds mean. Obviously, the different means, and

, are nearly equal if as used, for instance, in case of the Boussinesq approximation. In case

of a nearly incompressible fluid, the distinction between and is not necessary because the condition

is clearly fulfilled. However, to avoid any kind of confusion, we keep our notation.

In the averaged form, the local balance equation for momentum of the turbulent atmosphere reads (e.g., [22] [25] - [27] [29] [35] [38] )

where is the Reynolds stress tensor. It results from averaging the term in Equation (2.1) leading to. Similar local balance equations can be derived for various energy forms (i.e.,

internal energy, kinetic energy, potential energy, and total energy), various water phases (i.e., water vapor, liquid water, and ice), and gaseous and particulate atmospheric trace constituents [22] [25] - [31] [38] [39] .

Averaging Equation (2.12) provides the corresponding local balance equation for the kinetic energy

Obviously, the local derivative with respect to time not only contains the MKE, but also the TKE as outlined by Equation (2.18). Assuming, for instance, steady-state condition leads to

This means that the total kinetic energy is time-invariant, but MKE can be converted into TKE. In the inertial range, for instance, the TKE is transferred from lower to higher wave numbers until the far-dissipation range is reached, where kinetic energy is converted into heat energy by direct dissipation, , and turbulent dissipa-

tion,. Even though the fluctuations of the wind vector are usually small as compared to the mean wind vector, , the opposite is true for their gradients,. This phenomenon is connected with a

great intensity of rotation and is characteristic for all turbulent flows. Except for the immediate vicinity of rigid walls, turbulent dissipation exceeds direct dissipation by several orders of magnitude depending on the Reynolds number (e.g., [18] [22] ). Furthermore, the mean kinetic energy stream density reads

This equation describes the transfer of MKE and TKE by the mean wind field and the transfer of TKE by the eddying wind field. Ignoring the turbulent effects yields

i.e., is approximated by the MKE stream density. The magnitude of is given by

where. Apparently, this quantity expresses that the wind power density is proportional to the cube of the wind speed. The rotor of a wind turbine causes a divergence effect expressed by.

Unfortunately, there is a notable inconsistency regarding the role of the turbulence intensity. According to de

Vries [40] , for instance, this quantity is, where is the standard deviation of the horizontal wind

speed and is the corresponding variance. If we assume that only a horizontal component of the mean wind field exists, for the purpose of convenience, in the direction of the x-axis of a Cartesian coordinate frame, Equation (2.23) would provide

or

i.e. we have still to consider the fluctuations of all components in this coordinate frame. On the other hand, de

Vries [40] argued that the instantaneous value is given by, and, hence, , i.e.,

Since, we have

The term is only equal to zero when the probability distribution of u is symmetrical. Nevertheless,

for estimating the effect owing to turbulence this term is ignored which leads to

Ignoring the similar term in Equation (2.27) yields

where and are the variances with respect to the y- and z-axis of a Cartesian coordinate frame. Thus, only in case of Equations (2.30) and (2.31) become identical, but such an equality does not generally exist. Figure 2 shows that the mean and the median of the turbulence intensity depending at the height of 90 m [41] . The observations were performed at the offshore measurement platform FINO1 which is located 45 km north of the island of Borkum in the German Bight. For wind speeds ranging from to the mean and the median of the turbulence intensity are smaller than 0.1. This means that according to Equation (2.30), the effects of the turbulence intensity are smaller than 3 percent. As reported by Türk and Emeis [41] , the same is true for this wind speed range at the 30 m height. The effect by turbulence may become more influential in case of aerodynamically rougher landscapes covered, for instance, with vegetation. In case of wind farms the effect by turbulence may considerably increase inside the array of wind turbines [42] [43] .

To obtain the local balance equation of MKE, Equation (2.20) has to be scalarly multiplied by leading to

or

with

The quantity H may be considered as the mean total pressure. Subtracting Equation (2.32) from Equation (2.21) yields

or

where

is a non-dimensional parameter characterizing the thermal stability of a turbulent flow. This stability parameter expresses the relative importance of the two TKE-terms. It may be interpreted as a generalized Richardson number. The difference between the well-known flux-Richardson number and the generalized Richardson number results from the parameterization of [22] [27] . Besides the vertical effects also horizontal effects have to be regarded under certain circumstances. In case of, mechanically produced TKE is mainly consumed by Archimedean effects. Consequently, there exists a critical -value given by. It characterizes that the mechanical gain of TKE is equal to the thermal loss of TKE, i.e., the term becomes equal to zero, and the net production rate of TKE vanishes. As the turbulent dissipation still acts as a sink of TKE, the turbulent flow will become more and more viscous (laminar). In case of, TKE is generated mechanically and thermally. If the mechanically generated TKE is much smaller than the thermal gain of TKE, and, hence, negligible, free convective conditions, , will occur. In the remaining range, forced convective conditions may prevail,. Thermally neutral stratification is characterized by. The 2^nd-order balance equation (2.35) is the only balance equation that additionally arises from averaging a macroscopic balance equation (e.g., [18] [23] ). In meteorological models, the balance equation of TKE (2.35) serves to derive the eddy diffusivities for momentum and―via the turbulent Prandtl number and the species-dependent turbulent Schmidt numbers―the eddy diffusivities for sensible heat, and water vapor. This method of parameterization is known as one-and-a-half-order closure (e.g. [22] [44] [45] ). In the mesoscale model of the National Centers for Environmental Prediction (NCEP) and the Weather Research and Forecasting (WRF) model, it is realized with respect to the level 2.5 of Mellor and Yamada [46] - [48] .

The local balance equation for the mean total energy can be deduced from Figure 1 leading to

This equation demonstrates that no production or destruction of mean total energy within any given fixed volume exists (e.g., [22] [26] [27] [29] ). Obviously, contributions of energy of different orders of magnitude are summed, where only a very small fraction of the total potential energy (or total internal energy), , is available for conversion into kinetic energy (e.g., [22] [49] - [52] ).

From the perspective of the generation of electricity by extracting kinetic energy from the wind field, Equations (2.17), (2.20), and (2.32) play the dominant role. To obtain a tractable set of equations, effects caused by

molecular and turbulent friction, , , and, are usually ignored. In addition, incompressibility () and steady state () conditions are presupposed. In doing so, the set of

approximated equations reads

and

Because of the condition of incompressibility, , Equation (2.41) may also be written as

Based on this condition, Bernoulli’s equation, which plays an important role in describing the conversion of wind energy, can simply be derived by considering this condition along a streamline. In accord with the natural coordinate frame for streamlines, the Nabla operator reads

Here, we consider a natural coordinate frame with the unit vectors, , and that form a right- handed rectangular coordinate system at any given point of a curve in space (moving trihedron) like a trajectory or a streamline (see Figure 3), where the subscript s characterizes the streamline-related quantities. The velocity vector at a given point along the streamline is given by, where V is its magnitude, is the unit tangent of the streamline, and is the unit tangent of the corresponding trajectory. The unit vectors and are the principal normal and the binormal, respectively (e.g., [53] [54] ). The different meaning of trajectories and streamlines is explained in the Appendix A.

With respect to Equation (2.43), the condition (2.42) results in

This means that for any value of, the condition

is fulfilled along a streamline. Equation (2.45) is Bernoulli’s equation (e.g., [14] [15] [17] [55] [56] ). Even though air density is considered as spatially constant, Bernoulli’s equation can often be applied to atmospheric flows. If the streamlines are mainly horizontally oriented and the variation of the gravity potential with height is small like in case of the swept area of a wind turbine, the variation of the gravity effect may be considered as negligible

so that Equation (2.45) results in (e.g., [21] [56] )

This approximation of Bernoulli’s equation customarily serves as the foundation of, and is used to derive the Rankine-Froude theorem.

The integration of Equations (2.39) to (2.41) over a time-independent control volume, encompassing the rotor of the wind turbine, yields [6] [56]

and

In accord with Gauss’ integral theorem, Equation (2.47) and the left-hand side of Equation (2.48) can be written as

and

where is the thrust. Since is a second-rank tensor, it is advantageous to scalarly multiply Equation (2.48) by the unit vector from the left to get the more tractable equation

Here, is the axial force acting on the rotor (e.g., [6] ). If we assume that the axial direction coincides with any horizontal direction, the term will be nearly equal to zero. Since the Coriolis acceleration is given by, where is the latitude, and, , and are the components of the mean wind vector in west-east direction (characterized by the unit vector

), south-north direction (characterized by the unit vector), and the vertical direction (characterized by the

unit vector), respectively; the term is very small

for any wind speed smaller than the cut-out wind speed because. Thus, this term is negligible, and Equation (2.52) may be approximated by

The second term of the left-hand side of this equation is usually ignored in the blade element momentum (BEM) theory. However, this term is not zero [6] [10] [57] .

The velocity vector may be expressed by, where, , and are the cylin-

drical polar coordinates, respectively; and, , and are the corresponding unit vectors pointing in axial, radial, and azimuthal direction. The azimuthal velocity component acting on the rotor at a certain radius r causes a torque given by

In accord with Gauss’ integral theorem, the left-hand side of Equation (2.49)reads

This term represents the power extracted by the rotor of the wind turbine. In case of a quasi-horizontal flow, the right-hand side of Equation (2.49) can be neglected because is quasi-perpendicular to. The effect of the gravity potential was already considered as negligible in Bernoulli’s Equation (2.45). The integral relation (2.55) underlines the importance of Bernoulli’s equation in wind power studies.

Rearranging the left-hand side of Equation (2.49) yields

Because of, the divergence term can be expressed by leading to

Obviously, the first term on the right-hand side of this equation is missing in that of Gorban’ et al. [1] , repeated here by Equation (1.4). This means that the filtration Equation (1.2) that leads to Equation (1.4) is meritless in determining the maximum efficiency of propeller-type wind turbines. Thus, the argument of van Kuik et al. [9] seems to be justified by Equation (2.57).

The result of the Betz-Jowkowsky limit is based on simplified description of the flow field. Even though the flow field exhibits a pure axial behavior in front of the rotor, the exertion of a torque on the rotor disk by the air passing through it causes an equal, but opposite torque to be imposed on the air. Because of this reaction torque, the air starts to rotate in a direction opposite to that of the rotor; the air gains angular momentum and so in the wake of the rotor disk the air particles have a velocity component in a direction which is tangential to the rotation as well as having an axial velocity component [65] . Since the stream tube is opening behind the propeller, there is also a velocity component in the radial direction. Thus, by interacting with the rotor also velocity components in radial and azimuthal directions occur. The velocity vector at the rotor may be expressed by cylindrical polar co-ordinates; the velocity vector in the wake behind the rotor may be expressed in a similar manner.

To consider these rotational effects, Glauert [10] developed a simple model for the optimum rotor. In his approach, the rotor is a rotating axisymmetric actuator disk, corresponding to a rotor with an infinite number of blades [6] [11] [12] .

As outlined in Appendix B, the general equations of the General Momentum Theory lead to (see Equation (B.24))

where is, again, the undisturbed wind speed far upstream of the wind turbine, is the angular velocity of the rotor, is the axial velocity through the propeller disk, is the angular velocity imparted to the slipstream, the axial velocity in the final wake, and the corresponding angular velocity at radial distance

from the axis of the slipstream. Equation (3.25) already derived by Glauert [10] for an engine-driven propeller and by Wilson and Lissaman [62] for propeller-type wind turbines suffice to determine the relationship between the thrust and torque of the propeller and the flow in the slipstream. Owing to the complexity of the equations, however, it is customary to adopt certain approximations based on the fact that the rotational velocity in the slipstream is generally very small.

In case of finite-bladed rotor Equations (3.42) and (3.43) are imprecise. Based on the vortex theory, each of the rotor blades has to be replaced by a lifting line on which the radial distribution of bound vorticity is represented

by the circulation depending on the radial distance along the blade [11] [12] . This results in a free vortex system consisting of helical trailing vortices, as sketched in Figure 11. With respect to the vortex theory, 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 the wake [11] [12] . The velocity vector in the rotor plane is made up by the rotor angular velocity, , the undisturbed wind speed, , the axial and circumferential velocity components, , and, respectively. These velocity components induced at a blade element in the rotor plane by the tip vortices (see Figure 12). Another circumferential velocity, is induced by the hub vortex (see Figure 12). To determine the velocity field given by, , and induced at a blade element in the rotor plane, the free half-infinite helical vortex system behind the rotor is replaced by ‘an associated vortex system’ that extends to infinity in both directions [12] . Neglecting deformations or changes in the wake, the vortex system is uniquely described by the far wake properties in the Trefftz plane [66] . It is defined as the plane normal to the relative wind far downstream of the rotor. In accordance with Helmholtz’ vortex theorem, the bound circulation around a blade element is uniquely related to the circulation of a corresponding vortex in the Trefftz plane [12] . By symmetry, the induced velocities at a point in the rotor plane equals half the induced velocity at a corresponding point in the Trefftz plane [67] -[70] , i.e., , , and (see also [12] ).

Okulov and Sørensen [12] distinguished between two different concepts that dominated the conceptual interpretation of the optimum rotor: (a) Joukowsky [67] defined the optimum rotor as one having constant circulation along the blades, such that the vortex system for an -bladed rotor consists of helical tip vortices of strength and an axial hub vortex of strength. A simplified model of this vortex system can be obtained by representing it as a rotating horseshoe vortex (Figure 11(a)). Betz and Prandtl [71] argued 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 wing-span (Figure 11(b)). This distribution, usually referred to as the Goldstein circulation function, is rather complex and difficult to determine accurately [12] [72] .

Using the Kutta-Joukowsky-theorem

where is the lift force on a blade element of radial dimension, is the resultant relative velocity and is the bound circulation, Okulov and Sørensen [11] deduced the local thrust and the local torque of a rotor blade given by

and

Here, we only discuss the torque. Since the related power is given by, the integration over the total blade span provides for blades yields

Using the analytical solution to the induction of helical vortex filaments developed by Okulov [73] , Okulov & Sørensen [11] extended Goldstein’s [72] original formulation by a simple modification to handle heavily loaded rotors in accord with the general momentum theory. Assuming that the induction in the rotor plane equals half the induction in the Trefftz plane in the far wake, as described before, they found for the power efficiency

Here, is the dimensionless translational velocity of the vortex sheet (see Figure 11(b))

and

where is the Goldstein [72] circulation function, , , is the pitch of the vortex sheet, and is the angle between the vortex sheet and the rotor plane. Thus, l may be expressed by. The first derivative test yields

The result of the second derivative test shows that Equation (3.63) characterizes the maximum of. As pointed out by Okulov and Sørensen [11] , for a rotor with infinitely many blades, both functions, and, tend to unity when the pitch tends to zero. In this case, Equation (3.60) degenerates to the expression. This result is completely consistent with the axial momentum theory because leads to Equation (3.22). Furthermore, Equation (3.63) provides and, hence,. This result is in

agreement with that designates the maximum of in Figure 5.

In the vortex theory of the Joukowsky rotor [67] -[70] , 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 [12] . As sketched in Figure 11(a), the vortex system may be interpreted as consisting of rotating horseshoe vortices with cores of finite size, where the radius of the core is. The “associated vortex system” consists of a multiplet of helical tip vortices of finite vortex cores () with constant pitch h and circulation. The multiplet moves downwind (in case of a propeller) or upwind (in case of a wind turbine) with a constant velocity in the axial direction, where is the difference between the wind speed and axial translational velocity of the vortices [12] . Using the analytical solution to the induction of helical vortex filaments developed by Okulov [73] again, Okulov & Sørensen [12] derived for the power efficiency

where and

Here, is the non-dimensional radius of the vortex core, and is a non-dimensional axial velocity. For a given helicoidal wake structure, the power coefficient is seen to be uniquely determined, except for the parameter a. The first derivative test yields

The result of the second derivative test shows that for this value of a characterizes the maximum of the power efficiency.

Figure 13 illustrates that the power coefficient computed by Okulov and Sørensen [11] [12] for various number of blades depends on the tip-speed ratio given by Equation (3.35). Also shown is the result provided by Glauert’s [10] optimum actuator disk. Obviously, the optimum power coefficient of the Joukowsky rotor for all number of blades is larger than that for the Betz rotor, but the efficiency of the Betz rotor is larger if we compare it for the same deceleration of the wind speed [12] . The difference, however, vanishes for or for, where in both models tend towards the Betz-Joukowsky limit [12] .

Since neither the axial interference factor a nor explicitly occurs in Equations (3.60) and (3.63), and a have to be connected to the helical pitch l and the generic parameter w by [12]

and

In case of the Joukowsky rotor the tip-speed ratio can be expressed by [12]

Figure 14 shows the power curves of seven wind turbines of different rated power listed in Table 1. The power curves were determined by considering the listed values (only the Enercon machines) or by taking discrete values from the power curves illustrated in the actual brochures found at the manufacturers’ websites. Based on these discrete values, the parameters A, K, Q, B, M, and u of the generalized logistic function (e.g., [56] )

were numerically determined for each of the seven wind turbines (see Table 2). The function represents the power generated by the corresponding wind turbine at the wind speed v.

Figure 15 illustrates the wind power density, , for (a) a flow far upstream to the wind turbine given Equation (2.25) with, (b) an “ideal” wind turbine generating wind power by obeying the Betz-

Joukowsky limit of, and (c) the power curves of the seven wind turbines shown in Figure 14, but normalized by the corresponding swept areas. As shown in Figure 15, the wind power densities are lower than the Betz-Joukowsky limit, but follow it up to a wind speed at hub height of or so. Beyond this wind speed, they approach plateau values at the rated wind speeds; these plateau values are ranging from (General Electric 1.6-82.5) to (Enercon E-82 E4). The two other wind power densities continuously increase because of the -law. Thus, the power efficiencies of all wind turbines considered here are notably higher than the 30-percent limit of Gorban’ et al. [1] for wind speeds between and (see Figure 16). In case of wind speeds higher than, the power efficiencies eventually fall below this 30-percent limit and approach very low values in the vicinity of the cut-out wind speed.