Burton T. (et. al.) Wind energy Handbook

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

2

 sin łp( , ł) d dł ¼

1



6iD



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1  

d

2

sin

ł dł (3:178)

which gives

¼

C

(3:179)

To establish a relationship between the yawing moment coefﬁcient and the axial

velocity induced by the pressure distribution of Equation (3.176) the velocity distri-

bution has to be obtained by integrating Equations (3.145). Unfortunately, no

analytical solution has been determined for the anti-symmetric case, as Mangler

and Squire have done for the symmetric case. Numerical values of induced velo-

cities need to be calculated from Equations (3.145) using the pressure distribution

deﬁned by Equations (3.174) and (3.175).

Pitt and Peters (1981) have determined the axial velocity distribution for values of

the yaw angle from 08 to 908: the yaw angle ﬁxes the far upstream conditions where

the integration commences. The velocity distribution found corresp onds to that of

Equation (3.167) for the axi-symmetric case. Pitt and Peters again impose the form

of Equation (3.170) and determine the average value of the axial induced velocity a

and the value of a

, using the same method of Equation (3.171); in both cases, of

course, numerical integration is necessary.

The values of a

, are not zero, as might have been expected from the anti-

symmetric pressure distribution, but are equal and opposite to the values of a

found for the corrected axi-symmetric pressure distribution of Equation (3.166). The

variation of the two coefﬁcients a

and a

with yaw angle ª is determined

numerically but, using the Mangler and Squire analytical forms for guidance,

analytical variations can be inferred. Pitt and Peters found that the linearized axial

induced velocity distribution is

¼

128

 tan

(3:180)

and

¼ 1  tan



(3:181)

Pitt and Peters also include a cosine term in the linearized axial induced ﬂow factor

representation of Equation (3.170) which will only arise if C

6¼ 0

a ¼ a

þ a

 sin ł þ a

 cos ł (3:182)

In which case there will be an additional pressure distribution given by

p( , ł) ¼ P

( )Q

(i0)C

cos ł ¼ 6iC



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1  

cos ł (3:183)

THE METHOD OF ACCELERATION POTENTIAL 135

The tilting moment coefﬁcient is given by



2

 cos łp( , ł) d dł ¼

1



6iC



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1  

d

2

cos

ł dł (3:184)

Therefore

¼

C

(3:185)

The axial induced velocity distribution resulting from the pressure ﬁeld of Equation

(3.183) is calculated by integration of Equations (3.145) and is then matched with

the linear velocity distribution of Equation (3.182) using the same method as for

Equation (3.171).

2

 cos ł(a

þ a

 cos ł)2 d dł

2

 cos łC

(, ª)

k¼1

( , ª)cos kł

 d d ł (3:186)

The functions A

( , ª) being determined numerically. Again, using the Mangler

and Squire results as guidance, an expression for a

is found.

¼sec

(3:187)

3.11.5 The Pitt and Peters model

Pitt and Peters (1981) have developed the linear theory that relates the axia l induced

ﬂow factors to the thrust and moment coefﬁcients given in Equations (3.169),

(3.172), (3.180), (3.181) and (3.187) which collect together in matrix form

0 

128

 tan

0 sec

128

 tan

0  1  tan



(3:188a)

(a) ¼ [L](C)(3:188b)

The procedure for using Equation (3.188) is to assume initial values for (a) from

which the values of (C) can be calculated from the blade element theory. New

values of (a) are then found from Equation (3.188) and an iteration proceeds.

136 AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

For the wind turbine the value of a

may not be small compared with 1 and so the

above procedure will converge on values of a

which are too small compared with

what the momentum theory would deliver.

To produce more realistic results that is, results in line with Glauert’s momentum

theory where

¼ 4a

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1  a(2 cos ª  a)

¼ 4aA

(a)(3:105)

or the Coleman theory, where

¼ 4a cos ª þ tan



sin ª  a sec





¼ 4aA

(a)(3:111)

Also, the wake skew angle should be used in matrix [L] instead of the yaw angle.

The matrix [L] should then be modiﬁed to become

[L] ¼

4A(a

)

0 

128

 tan



0 sec



128A(a

)

 tan



0  1  tan





(3:189)

where A(a

) is chosen according to which momentum theory is to be used.

The Pitt and Peters method does not include any determination of induced

velocities in the plane of the rotor disc and as a consequence it is not possible to

account for wake rotation. However, it is possible that the Kinner solutions Q

()

that were excluded from the analysis bec ause they give inﬁnite pressure at 

¼ 1,

which lies along the axis of rotation, may give velocity distributions which provide

for wake rotation; the momentum theory of Section 3.3 also predicts an inﬁnite

pressure at the axis of rotation because of wake rotation. In practice, of course, the

rotor disc would not extend to the axis of rotation and the singularity would not

occur.

With or without wake rotation a ﬂow angle  can be determined from which a

torque can be found. If the normal force on an eleme nt of the rotor disc is equal to

L cos  then the tangential force will be L sin .

3.11.6 The general acceleration potential method

Peters with a number of associates has developed the theory further and a reading

of these references (Pitt and Peters, 1981, Goanker and Peters, 1988, HaQuang and

Peters, 1988) is recommended. The accel eration potential method has been devel-

oped speciﬁcally for wind turbines by van Bussell (1995) where a much more

comprehensive account of the theory is given.

THE METHOD OF ACCELERATION POTENTIAL 137

3.11.7 Comparison of methods

A project to compare existing methods of predicting yaw behaviour, among other

aspects of the aerodynamic behaviour of wind turbines, was reported upon (Snel

and Schepers, 1995). Figure 3.73 shows results obtained by various methods for

predicting the yawing moment of the 2 MW, three-blade turbine at Tjæreborg in

Denmark at a yaw angle of 328 and a wind speed of 8:5m=s.

Most of the theoretical predictions in Figure 3.73 have the correct phasing and

about the correct mean yawing moment but the amplitude of the yawing moment

varies. In this example the second method bears the closest comparison with the

measured data. Generally, the amplitude of the yawing moment variation is under-

estimated by the theoretical predic tions, whereas the mean yawing moment is quite

well predicted.

ECN.i.w.

ECN.d.e.

Unist

NTUA

TUD

meas.

-50

-100

-150

-200

0 90 180 270 360

Azimuth angle in degree

Vortex cylinder method

Method using eqn. 3.173 to determine F(µ)

Pitt and Peters eqn. 3.189

CFD method

Method using eqn. 3.123

CFD method

Measured data

Yawing moment (kNm)

Figure 3.73 Yawing Moment on the Tjæreborg Turbine at 328 Yaw and 8.5 m/s, (Srel and

Schepers 1995)

138

AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

3.12 Stall Delay

A phenomenon ﬁrst noticed on propellers by Himmelskamp (1945) is that of lift

coefﬁcients being attained at the inboard section of a rotating blade which are

signiﬁcantly in excess of the maximum value possible in two-dimensional static

tests. In other words the angle of attack at which stall occurs is greater for a rotating

blade than for the same blade tested statically. The power output of a rotor is

measurably increased by the stall-delay phenomenon and, if included, improves

the comparison of theoretical prediction with measured output. It is noticed that

the effect is greater near the blade root and decreases with radius.

The reason for the stall delay has been the cause of much discussion but a

convincing physical proce ss has not yet been established. What is agreed is that, for

whatever reason, the adverse pressure gradient experienced by the ﬂow passing

over the downwind surface of the blade is reduced by the blade’s rotation. The

adverse pressure gradient slows down the ﬂow as it approaches the trailing edge of

the blade after the velocity peak reached close the the leading edge. In the boundary

layer viscosity also slows down the ﬂow and the combination of the two effects, if

sufﬁciently large, can bring the boundary layer ﬂow to a standstill (relative to the

blade surface) or even cause a reversal of ﬂow direction. When ﬂow reversal takes

place the ﬂow separates from the blade surface and stall occurs giving rise to loss of

lift and a dramatic increase in pressure drag.

Aerodynamic analyses (Wood (1991) and Snel et al. (1993)) of rotating blades

using computational ﬂuid dynamic techniques, which include the effects of viscos-

ity, do sho w a decrease in the adverse pressure gradie nt but it is not obvious from

these numerical calculations as to what exactly is occurring physically.

It is also agreed that the parameter that inﬂuences stall delay predominantly is

the local blade solidity c(r)=r. The evidence which does exist shows that for

attached ﬂow conditions, below what would otherwise be the static (non-rotating)

stall angle of attack, there is little difference between two-dimensional ﬂow condi-

tions and rotating conditions. When stall does occur, however, the air in the

separated region, which is moving very slowly with respect to the blade surface, is

rotating with the blade and so is subject to centrifugal force causing it to ﬂow

radially outwards. Prior to stalling taking place, centrifugal forces on the ﬂuid in

the boundary layer, again causing radial ﬂow, may reduce the displacement

thickness and so increase the resistance to separation.

Blade surface pressures have been measured by Ronsten (1991) on a blade while

static and while rotating. Figure 3.74 shows the comparison of surface pressure

coefﬁcients for similar angles of attack in the static and rotatin g conditions (tip

speed ratio of 4.32) for three span-wise locations. At the 30% span location the

estimated angle of attack at 30:418 is well above the static stall level which is

demonstrated by the static pressure coefﬁcient distribution. The rotating pressure

coefﬁcient distribution at 30% span shows a high leading edge sucti on pressure

peak with a uniform pressure recovery slope over the rear section of the upper

surface of the chord. The gradual slope of the pressure recovery indicates a reduced

adverse pressure gradient with the effect on the boundary layer that it is less likely

to separate. The level of the leading edge suction peak, however, is very much less

STALL DELAY 139

than it would be if, in the non-rotating situation, it were possible for ﬂow still to be

attached at 30:418.

The situation at the 55% span-wise location is similar to that at 30%; the static

pressures indicate that the section has stalled but the rotating pressures show

leading edge suc tion peak which is small but signiﬁcant. At the 75% span location

there is almost no difference between static and rotatin g blade pressure coefﬁcient

distributions at an angle of attack of 12.948, which is below the static stall level; the

leading edge suction pressure peak is a little higher than that at 30% span, much

higher than that at 55% but the pressure recovery slope is much steeper. The ﬂow

appears to be attached at the 30% and 55% span locations on the rotating blade, but

the suction pressures are too low for that actually to be the case, so stall appears to

be greatly delayed and the low adverse pressure gradient shown by the gentle slope

of the pressure recovery appears to indicate the reason for the delay. At 30% span

the ratio c=r ¼ 0:374, c=r ¼ 0:161 at 55% span and at the 75% location c=r ¼ 0:093.

The increased lift also occurs in the post stall region and is attributed to the radial

ﬂow in the separated ﬂow regions.

Snel, et al. 1993, have proposed a simple, empirical modiﬁcation to the usually

available two dimensional, static aerofoil lift coefﬁcient data which ﬁts the meas-

ured lift coefﬁcients by Ronsten (1991) and computed results using a three-dimen-

sional computational ﬂuid dynamics code.

Table 3.2 Summary of Ronsten’s measurements of lift coefﬁcient and lift coefﬁcients

corrected to rotating conditions using Equation (3.190)

r=R



100 30% 55% 75%

c=R 0.374 0.161 0.093

Angle of attack Æ 30.418 18.128 12.948

static (measured) 0.8 0.74 1.3

rotating (measured) 1.83 0.93 1.3

rotating (Snel, 1995) 1.87 0.84 1.3

Rotating Static

30.41

18.12

12.94

r/R=30% 55% 75%

-4

-3

-2

-1

Figure 3.74 Pressure Measurements on the Surface of a Wind Turbine Blade while Rotating

and while Static by Ronsten (1991)

140

AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

If the linear part of the static, two-dimensional, C

 Æ curve is extended beyond

the stall then let ˜C

be the difference between the two curves. Then the correction

to the two-dimensional curve to account for the rotational, three-dimensional,

effects is 3(c=r)

˜C

3-D

¼ C

2-D

þ 3



˜C

(3:190)

Table 3.2 compares the measured static (C

2-D

) and rotating (C

3-D

) lift coefﬁcients

with the calculated values for the rotating values using Snel’s correction of Equation

(3.190). The correction is quite good and is very simple to apply. An example of the

correction is given by Snel et al. (1993) and is shown in Figure 3.75.

3.13 Unsteady ﬂow – Dynamic inﬂow

3.13.1 Introduction

Natural winds are almost never steady in either strength or direction and so it is

seldom that the conditions for the momentum theory apply. It takes a ﬁnite time for

the wind to travel from far upwind of a rotor to far downwind and in that time

wind conditions will change so an equilibrium state is never achieved. Even if the

‘average’ wind speed changes only slowly small-scale turbulence will cause a

continuous unsteadiness in the velocities impinging on a rotor blade.

Several approximate solutions offer themselves for the determination of the

Wind speed

(

m/s

)

Aerodynamic power (kW)

0 10 20 30

400

300

200

100

Measurement

2D coefficients

3D coefficients

Figure 3.75 A Comparison of Measured and Snel’s Predicted Power Curves for a NOR-

TANK 300 kW Turbine

UNSTEADY FLOW – DYNAMIC INFLOW 141

dynamic ﬂow conditions at the rotor disc. It could be assumed that the induced

velocity remains ﬁxed at the level determined by the average wind speed blowing

over a set period of time, that may be quite short. The wake remain s frozen while

the unsteady component of the wind passes through the rotor disc unattenuated by

the presence of the rotor. The unsteady forces, that would impose zero net force on

the rotor, would be determined by the blade element theory. Alternatively, the

induced velocity through the rotor disc could be determined from the instantaneous

wind velocity as if that velocity was steady. The induced velocity will change as the

wind speed changes but it must be assumed that the entire wake changes instanta-

neously to remain in step. Equilibrium in the wake is maintained at all times. The

truth lies somewhere between the two scenarios given above, both of which rely on

simple assumptions about the state of the wake.

The acceleration potential method avoids reference to the wake and allows the

ﬂow conditions at the rotor disc to be determined by the upwind ﬂow ﬁeld, which

is much simpler to determine than that of the wake.

In steady-ﬂow conditions the velocity at a ﬁxed point in the upwind ﬂow ﬁeld is

constant; acceleration of the ﬂow from point to point tak es place (e.g., U

(@u=@x)in

the x-direction, assuming u, the induced velocity, is much smaller than U

) but no

rate of change of velocity with time (e.g., @u=@ t) occurs at a single point. In

unsteady ﬂow, conditions at a ﬁxed point do change with time and the total

acceleration in the x-direction is then (@u=@t) þ U

(@u=@x). The additional accele ra-

tion requires an additional inertia force the reaction to which will change the force

on the rotor disc. The additional force is often termed the added mass force because

if the unsteadiness in the relative ﬂow past a blade can be attributed not to ﬂow

turbulence but to an unsteady motion of the blade itself some of the air will be

forced to move (accelerate) with the blade, effectively adding to the mass of the

blade.

3.13.2 Adaptation of the acceleration potential method to unsteady

ﬂow

If the unsteady acceleration terms are added to Equations (3.145 ), which are

simpliﬁed to account for the induced velocities being very much smaller than the

wind velocity, then those equations become

@ t

þ U



¼

@ p

þ U



¼

@ p

@ y

@ t

þ U



¼

@ p

(3:191)

As before, differentiating each equation with respect to its particular directi on and

adding together the results gives

142 AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES

@ t

@ y



þ U

@ y



¼

@ y

but, for continuity of the ﬂow,

@ y

¼ 0

Therefore the condition

@ y

¼ 0

applies together with the Kinner pressure distributions of Section 3.11.2.

The accelerations

@ t



can be determined directly from Equations (3.191 ) without integration being

necessary but it is only the component @u=@ t that is required because it is normal to

the rotor disc and so will give rise to a normal force.

The solutions for

¼

@ p

have already been obtained in Sections 3.11.2–3.11.4 and so it remains to determine

the solutions for

¼

@ p

(3:192)

Equation (3.192) cannot be solved for the velocity u because that is the solution of

the complete equation that is the ﬁrst of Equations (3.191). What can be det ermined

from Equation (3.192) is the acceleration @u=@ t for which it is necessary to differ-

entiate the chosen pressure distribution. The Kinner pressure distributions, that are

solutions of Equations (3.149), are given as functions of the ellipsoidal co-ordin ates

(, , ł) so to obtain the derivative with respect to x a co-ordinate transformation is

required. The relationships between the ellipsoidal co-ordinates and the Cartesian

co-ordinates (x, y, z) are given in Equations (3.157) from which can be obtained the

derivatives @x=@v, @x=@, @x=@ł, etc., but what are really needed are the inverses

of these derivatives.

We can ﬁnd by appropriate differentiations of Equations (3.147)

@x 0

@ y9

UNSTEADY FLOW – DYNAMIC INFLOW 143

for example, and so the complete Jacobian can be determined:

@

@ł

@x 0

@ y9

@x 0

@

@ y9

@

@x 0

@ł

@ y9

@ł

@x 0

@ y9

(3:193)

the inverse of which is what is required:

@x 0

@ y9

@x 0

@

@x 0

@ł

@x0

@ y9

@

@ y9

@ł

@ y9

@

@ł

@

@ł

(3:194)

The Jacobian matrix of Equation (3.193) can be determined algebraically from

Equations (3.147) and this can then be inverted algebraically to give the inverse

Jacobian of Equation (3.194). From the inverse Jacobian it is found that

@x 0

(1  v

)

R(

þ v

)

v(1 þ 

)

R(

þ v

)

@

(3:195)

However, only the acceleration at the ro tor disc itself is required and there the value

of  is zero so

@x 0

@

(3:196)

If the corrected axi-symmetric pressure drop distribution of Equation (3.166) is

chosen, to conform with the steady ﬂow case then, for the whole ﬂow ﬁeld,

p(v, ) ¼

7 tan

1



þ 4(1  v

)





þ 15v



 tan

1



 1





þ 9þ ( v

 

)tan

1





(3:197)

in wh ich the pressure is normalized by (1 = 2)rU

. The term C

is the contri bution

to the total thrust coefﬁcient of the dynamic acceleration @u=@ t. Note that, as

explained at the end of Section 3.11.1, the pressure level just upwind of the rotor

disc, as given by Equation (3.197), is half the magnitude of the pressure drop across

the disc given by Equation (3.166).

144 AERODYNAMICS OF HORIZONTAL-AXIS WIND TURBINES