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The contribution of each blade element to the lift and drag force can be derived as follows. Consider an annular cross-section of a rotor blade as depicted in Fig. 3.11, and examine an element of length Ar of one blade.

ROTATING DIRECTION

Figure 3.11: Blade element velocities, and aerodynamic forces w.r.t the blade local coordinate frame with the chord line as reference.

AF Axial components of aerodynamic forces

AQ Tangential components of aerodynamic forces a Angle of attack of aerodynamic (or resultant) velocity

0 Pitch angle of rotor blade

6 Direction of aerodynamic velocity related to the rotor plane

Vp Local, undisturbed, perpendicular wind velocity

Vt Local, undisturbed, tangential wind velocity

W Local, undisturbed, aerodynamic wind velocity a Axial induction factor: represents the fractional decrease in wind velocity between the free stream and rotor plane a' Tangential induction factor: represents the swirl velocity of the air

The net effect on air flowing through this annular section of the rotor disk results from the forces and moments on all the blades. The instantaneous relative undisturbed wind velocity experienced by a blade element is

under an angle

It must be noted that the tangential induction factor a' in the above equation is as a rule an order smaller than the axial induction factor a.

Due to the special profile of a rotor blade, higher velocities will occur at the top of the blade rather than on the bottom side. According to the Bernoulli theorem, this leads to an underpressure at the first mentioned side of the blade and an overpressure at the latter. This air pressure difference is the driving force behind the rotation of the rotor. More precisely, the pressure distribution around an aerofoil can be represented by two forces, a lift L and a drag D force, and one torque, the pitching moment M. Both forces and the pitching moment are usually applied at a location 4 chord back from the leading edge (i.e. the so-called aerodynamic center) since, on most low speed aerofoils, the magnitude of the pitching moment is essentially constant up to maximum lift at that specific location. For symmetric aerofoils, the aerodynamic moment about the aerodynamic center is zero for all angles of attack. With camber, the moment is non-zero (normally negative for positive camber) and constant for thin aerofoils. Using the aerodynamic center as the location where the aerodynamic forces are applied simplifies the aerodynamic analysis. The effect of the pitching moment, however, is neglected in most design codes.

The angle of attack of the relative wind velocity (a) is determined by the difference between the angle of inflow (6) and the pitch angle (0):

Due to the resultant velocity W the blade cross-section exerts a quasi-steady aerodynamic lift force (AL ± W)

with c Local blade chord (varies along the blade: c = f (r))

C|2D] Blade element 2-D lift coefficient cdD] Blade element 2-D drag coefficient p Air density

### Ar Length of blade section

The dimensionless aerodynamic coefficients C\, Cd, and Cm are - among other things - functions of the angle of attack a, Reynolds number Re and Mach number Ma (compressibility of the airflow). These coefficients have to be either determined for each type of aerofoil separately by means of stationary windtunnel experiments and/or CFD computations or can be obtained from a database. An example of such a database is the Aerodynamische Tabel Generator (ATG) described by Timmer et al. [290].

Typical variation of these coefficients are shown in Fig. 3.12. The sudden change in the coefficients at 14° is due to flow separation from the suction side of the aerofoil; this is called stall.

The Reynolds number varies in practice between zero and 2 • 10 6 [119] depending on chord and undisturbed wind velocity of a blade-element. However, this dependency is often neglected, although the Reynolds number significantly affects the values for the lift and drag coefficients (see either Sharpe [259] or the aerofoil data option on DAWIDUM's Plot menu for the variation of aerofoil characteristics with the Reynolds number).

From the above discussion it follows that the quasi-steady aerodynamic lift and drag forces are proportional to the local blade chord c, are quadratic in resultant wind velocity W, and are approximately linear in the angle of attack a in the attached flow region.

In order to calculate the lift AL, and drag AD on a section of a rotor blade it suffices to determine the local, undisturbed, resultant wind velocity W, which consists of four components: the undisturbed wind velocity Vw (including yawed flow, wind shear, and tower shadow, see Section 2.3.4), the velocity of the blade element itself (including rotor shaft rotation, flap motion, lead-lag motion and the velocity of the tower top, resulting from the mechanical model which will be discussed in Section 3.4) and the induced velocities. Next, the determination of the induced wind velocities shall be described.

Figure 3.12: Typical variation of the 2-D lift, drag, and moment coefficient as function of the angle of attack a. ¡Solid! curve: lift coefficient Ci, dashed curve: drag coefficient Cd, and dashed-dotted curve: moment coefficient Cm about the aerodynamic center.

Figure 3.12: Typical variation of the 2-D lift, drag, and moment coefficient as function of the angle of attack a. ¡Solid! curve: lift coefficient Ci, dashed curve: drag coefficient Cd, and dashed-dotted curve: moment coefficient Cm about the aerodynamic center.

### Induced velocities

In the equilibrium situation, the axial flow in the rotor plane of a wind turbine, depends on the wind velocity, and on the degree of loading (i.e. the size of axial force Dax = ^ AF) of the turbine. For instance, for a turbine with zero loading, the wind velocity at the rotor disk position (Vax) is equal to the undisturbed wind velocity (Vw), while an operating, and hence loaded turbine slows down the wind velocity to a lower value (see Fig. 3.7 on page 51). The difference between the axial component of the wind velocity and the axial flow velocity in the rotor plane is usually called the "axial induced" velocity, the velocity induced by the presence of the turbine. The tangential flow, on the other hand, is induced by the swirl velocity of the air flow around the blade.

Horizontal-axis wind turbine rotors are usually not aligned with the wind due to the continuously changing wind direction and the fact that no rotor is capable of following this variability. Furthermore, upwind rotors are sometimes tilted in order to increase the tower clearance, and hence to reduce tower shadow. In effect, the rotor is then yawed about a horizontal axis. For all these reasons it is thus necessary to include in the blade element momentum theory the effects of yaw. Here we will consider only the simple case of perpendicular flow.

The axial induced velocity can be determined by expressing the axial thrust AF on a blade element either as the rate of change of momentum in the annular ring swept out by this element

using Eq. (3.22) with A ~ n(r + 2Ar)2 — n(r — 2Ar)2 = 2nrAr the area of the annular ring, or as the force exerted by the wind on a blade element

Cn where Nb is the number of rotor blades. Assuming equality of Eqs. (3.32) and (3.33) gives

Cdax

The right-hand side term is defined as the dimensionless thrust coefficient Cdax (see Eq. (3.25)). Solving the above equation gives

as an expression for the axial induction factor in case of perpendicular flow. Alternatively, the axial induction factor can be calculated as arCN (3.36)

by substituting

Nb c 2nr in Eq. (3.34) and solving for a. The term ar (which is a function of the radius r) is called the local solidity or chord solidity.

The tangential induced velocity can, on the other hand, be determined by expressing the torque AQ on a blade element either as the rate of change of angular momentum

or as the torque exerted by the wind on a blade element

a r assuming Nb blades. Again assuming equality between the above expressions of the torque on a blade element we have

as an expression for the tangential induction factor. Alternatively, the tangential induction factor can be calculated as ar Ct with and

Nb c 2nr

Thus, Eqs. (3.35)/(3.36) and (3.39)/(3.40) are the set of non-linear relations that determine the dimensionless induced velocities a, and a' in case of perpendicular flow. Before these equations can be used, however, the local thrust coefficient must be modified to account for two effects: the departure of the local thrust coefficient from the momentum relation, and the non-uniformity of the induced velocities in the flow. These so-called tip losses will be considered after the treatment of the turbulent wake state.

Turbulent wake state

For heavily loaded wind turbines, which implies a high axial induction factor a as well as a high tangential induction factor a', the momentum and vortex theory are no longer applicable because of the predicted reversal of flow in the turbine wake. The vortex structure disintegrates and the wake becomes turbulent and, in doing so, entrains energetic air from outside the wake by a mixing process. Thereby thus altering the mass flow rate from that flowing through the actuator disk. The turbine is now operating in the so-called "turbulent wake state", which is an intermediate state between windmill, and propellor state (see Appendix B for an overview of the different flow states of a wind turbine rotor).

In the turbulent wake state the relationship between the axial induction factor and the thrust coefficient according to the momentum theory, Eq. (3.35), has to be replaced by an empirical relation. The explanation for this is that the momentum theory predicts a decreasing thrust coefficient with an increasing axial induction factor, while data obtained from wind turbines show an increasing thrust coefficient [279]. Thus, the momentum theory is considered to be invalid for axial induction factors larger than 0.5. This is consistent with the fact that when a = 0.5 the far wake velocity vanishes (i.e. a condition at which streamlines no longer exist), thereby violating the assumptions on which the momentum theory is based. The following approximations are implemented in DAWIDUM:

1. Anderson [2]. The empirical relation of Anderson is defined as:

or equivalently

Cdax =4(aT)2 +4(1 - 2aT) • a for Cdax > 4aT(1 - aT) (3.42)

where aT =1 — fV Cdaxl with Cdax1 = 1.816 as "best" fit [21]. Thus aT = 0.3262 and Cdax(aT) = 0.8792. The empirical relation of Anderson is a straight line in the Cdax - a diagram, and this line lies tangential to the momentum theory parabola at the transition point aT (marked * in Fig. 2.1);

2. Garrad Hassan [29]. The empirical relation of Garrad Hassan is defined as:

or equivalently

a = 7ZE + 7ZE V-15239 + 31600 • Cdax for Cdax > 0.96 (3.44) 158 158

3. Glauert [55, 82]. The empirical relation of Glauert is defined as:

a = 0.143 +^0.6427 • Cdax - 0.55106 for Cdax > 0.96 (3.45)

4. Johnson [118]. The empirical relation of Johnson is obtained from interpolation of the expressions for the wind turbine and propellor state by a third order polynomial:

1.491

Cdax or equivalently

1.491

1.991 a

5. Wilson [280, 308]. The empirical relation of Wilson is defined as:

or equivalently a = 1 , Cdax -4aC for Cdax > 4ac(1 - ac) (3.49)

This is a linear extrapolation of the from the momentum theory parabola at the transition point aC (marked o in Fig. 2.1);

Observe that the empirical relation of Wilson is identical to that of Anderson, only the location of the transition point is different. The five mentioned approximations were already compared in Fig. 2.1 on page 25 for perpendicular flow. From the observed differences it has been concluded that the listed empirical approximations must be regarded as being only approximate at best. The prediction is, nevertheless, more realistic than the one from the momentum theory as illustrated in Fig. B.1 on page 228. It must be noted that, in general, the value of the axial induction factor rarely exceeds 0.6 and for a well-designed blade it will be in the vicinity of 0.33 for much of its operating range [37].

Thus for values of Cdax greater than the (empirical model dependent) transition point, the right-hand side term of Eq. (3.34) needs to be substituted in one of the above mentioned empirical relations (i.e. either Eq. (3.41), (3.44), (3.45), (3.46), or (3.49)) to compute a.

### Blade tip and root effects

The blade momentum theory, as previously developed, does not account for the effect of a finite number of rotor blades. Therefore a correction has to be applied for the interaction of the shed vorticity with the blade's bound vorticity. This effect is usually greatest near the blade tip, and it significantly affects the rotor torque and thrust.

Either an approximate solution by Prandtl or a more exact solution by Goldstein can be used to account for the non-uniformity of the induced axial velocity [55]. Both approximations give similar results. The expression obtained by Prandtl is however commonly used, since this has a simple closed form, whereas the Goldstein solution is represented by an infinite series of modified Bessel functions.

This expression is in literature denoted with the misleading term tip-loss factor. Misleading because it corrects for the fact that induction is not uniform over the annulus under consideration owing to the finite number of blades, and not for the finiteness of the blades. Prandtl's tip-loss factor is defined as

n a c with

6i Angle between relative wind vector and the plane of rotation at blade section i

Note that at the blade tips, where T = R, the factor equals zero, as can be reasoned by the fact that the circulation at the blade tips is reduced to zero by the wake vorticity.

A similar loss takes place at the blade root where, as at the blade tip, the bound circulation must fall to zero, and therefore a vortex must be trailed into the wake. The blade root-loss factor is defined as

n with to the radial position of start root loss (typically 10% to 30% of the blade radius [118]).

The effective total loss factor at any blade section is, according to Eggleston and Stoddard [55], then the product of the two:

Figure 3.13 shows Prandtl's combined blade tip and blade root loss factor FL as a function of the normalized radius T = t/R.

The incorporation of the combined blade tip and root loss factor FL into the expressions for the induction factors depends upon whether the azimuthal averaged values of the induction factors, or the maximum values (local to a blade element) are to be determined. If the former alternative is chosen then, in the momentum terms the induction factors remain unmodified, but in the blade element terms the induction factors must appear as the average value divided by FL. The latter choice, however, allows the simplest modification of Eqs. (3.35)/(3.36) and (3.39)/(3.40). In this case, the induction factors in the momentum terms are to be multiplied by FL while the values of 6 and CN that are obtained from the blade element calculations are not multiplied by FL. Equation (3.35) then becomes a = 1 - ^l- Cdax (3.53)

for a wind turbine operating in windmill state or, when the turbine is operating in turbulent wake state, by dividing the right-hand side of either Eq. (3.41), (3.44), (3.45), (3.46), or (3.49) by FL. In addition, Eq. (3.39) becomes

root

Figure 3.13: Prandtl's combined blade tip and blade root loss factor Fl as function of the normalized radius r. It is assumed that the inflow angle fa is constant over the radius of the blade, and ro is taken as 0.10 • r.

tip root

+1 0

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### Responses

• nikola
Why we use axial induction factor in blade element monentum theory?
9 years ago