appendix - blade element momentum theory

Appendix A-Blade Element Momentum Theory 176

APPENDIX A

Appendix A-Blade Element

Momentum Theory


A.1. Introduction

With the availability of large computing power and developments in grid generation and

numerical algorithms, it is very tempting to compute the turbulent flow around a rotating

blade using the equations of motion in their differential forms, (Reynolds-averaged Navier-

Stokes equations). These equations need a model of turbulence to make them a closed set.

A large number of models of turbulence have been proposed, for example, mixing length

models, one-equation models, two-equation models, Reynolds stress models, algebraic

stress models, etc. Tulapurkara (1996) gives a review of these models in computation of

flow past aerofoils and wings.

A rotating blade is experiencing a wide range of angle of attacks along its span, from

negative values to post-stall values. A high angle of attack causes a strong adverse-

pressure-gradient separating flow in which no turbulent model can be applied accurately.

Applying the equations of motion in their integral forms (linear and angular momentum)

around the wind turbine (rather than the blade) together with the experimental data of blade

aerodynamic characteristics is an alternative to solve the flow field and find the blade

loading. This method is known as Blade Element Momentum Theory, BEMT.

A.2. Induced, total and relative velocity fields

Extracting energy from wind slow downs the wind speed and rotation of blades induces

some circumferential and radial velocities to the wind. Wind speed retardation depends on

the amount of the extracted energy (wind turbine loading), while the induced

circumferential velocity depends on the wind turbine angular velocity. Induced radial

velocity due to centrifugal forces is much smaller than the two other components and is

usually neglected.

Flow in the plane of a wind turbine rotor, from now on referred as disk, can be considered

as a combination of upstream mean velocity field, wV and induced velocity field, iV .

iw VVV += (A.1)

Figure (A.1) shows the coordinate systems used to represent velocity fields. In this figure:


θ−− rx : Disk (rotor) system of coordinates

x : Rotor axis

θ−r : Rotor plane

r : Disk radial coordinate, 0<r≤ R; R: rotor radius

stn −− : Blade system of coordinates

n : Normal to the blade

st − : Blade plane

tn − : Aerofoil plane

s : Blade span-wise coordinates

γ : Yaw angle

δ : Conning angle

ψ : Azimuth angle

Ω : Rotor angular velocity

Here it is assumed that wind velocity has no vertical components ( 0=ZwV ).

Using the stn −− system of coordinates, wind velocity wV at a general point P can be

expressed as

( ) ( ) ( ) swtwnwPw eVeVeVV ˆsinsinˆcossinˆcos| ψδγψδγδγ +++++= (A.2)

Conning angle δ is usually very small and therefore δδ cossin << , but yaw angle γ can be

large enough such that γsin and γcos have the same order of magnitude. Neglecting terms

including δsin in comparison with the terms including δcos , Equation (A.2) can be

rewritten in a simpler form as


swtwnwPw eVeVeVV ˆsincossinˆcoscossinˆcoscos| ψδγψδγδγ ++= (A.3)

Figure A.1- Upstream wind velocity components in different coordinate systems

Neglecting radial induction component, induced velocity iV has only two components of

xiV and θ

iV in the disk plane. xiV is directed opposite to the axial component of the wind

velocity and θiV is due to induced angular velocity in the opposite direction of the rotor

angular velocity.

Induction factors are defined as

x

w

x

i VVa = (A.4)

Ω=′ rVa i

θ (A.5)

where a and a′ are axial and rotational or swirl induction factors respectively. One can

write the induced velocity vector at point P in θ−− rx system of coordinates in terms of

axial and rotational induction factors and then transform it to stn −− system as

t

x

wPi eariaVV θˆˆ| ′Ω−−= (A.6.a)

s-t

γ

x

wV

xiV

( )δγ +=− sinwts

w VV

θ−r

( )δγ += coswnw VV

δ

n

θiV

( ) ψδγ sinsin +wV z

P

s-t

s

ψ

)(θt

siV

( ) ψδγ cossin +wV

Ω

x


tnwPi eareaVV ˆˆcoscos| ′Ω−−= δγ (A.6.b)

Finally substituting wV from Equation (A.3) and iV from Equation (A.6.b) back into

Equation (A.1), velocity field in the plane of blade can be written as

swtwnw eVearVeaVV ˆsincossinˆcoscossinˆ)1(coscos ψδγψδγδγ +′Ω−+−=

(A.7)

For a moving blade the relative velocity of the flow at a point P located on the blade is

PbladePflowPrel VVV ||| −= (A.8)

where

tPblade erV ˆ| Ω= (A.9)

Combining Equations (A.7),(A.8) and (A.9) leads to

nwrel eaVV ˆ)1(coscos −= δγ

tw earV ˆ)1(coscossin ′+Ω−+ ψδγ

sw eV ˆsincossin ψδγ+ (A.10)

Figure (A.2) shows the relative velocity in the plane of blade aerofoil. Inflow angle and

normalised in-plane relative velocity can be derived from Figure (A.2) as

ψδγλδγ

ϕcoscossin)1(

)1(coscostan

−′+−

=a

a

r

(A.11)

and

ϕδγ

sin

)1(coscos| a

V

V

w

tnrel −=− (A.12)

where local velocity ratio, rλ is defined as


w

rV

rΩ=λ (A.13.a)

or in terms of blade span coordinate s (δcos

rs = ):

w

rV

s Ω=

δλ

cos (A.13.b)

Figure A.2- Relative velocity in aerofoil plane

A.2.1. Special case – Zero yaw

In the case of zero yaw angle, 0=γ , Equations (A.4), (A.10), (A.11) and (A.12) can be re-

written as

w

x

i VVa = (A.14)

tnwrel eareaVV ˆ)1(ˆ)1(cos ′+Ω−−= δ (A.15)

)1(

)1(costan

a

a

r′+

−=

λδ

ϕ (A.16)

ϕδsin

)1(cos| a

V

V

w

tnrel −=− (A.17)

and the velocity diagram changes to Figure (A.3).

n

t

tnrelV −| ϕ

)1(coscos aVw −δγ

ψδγ coscossin)1( wVar −′+Ω


Now, flow through the rotor disk can easily be found by superposing the x-components of

the upstream wind and the induced velocities.

)1( aVVVV w

x

i

x

wd −=−= (A.18)

Figure A.3- Relative velocity in aerofoil plane; zero yaw

A.3. Momentum Theory

Momentum theory applied to the wind turbine aerodynamic is based on three basic

assumptions:

1. Axi-symmetric flow

2. Steady flow

3. Frictionless flow

Figure (A.4) shows an axi-symmetric flow through a wind turbine and typical qualitative

variations of pressure, velocity and rotation between upstream of a wind turbine and far

wake behind it.

n

t ϕ

)1(cos aVw −δ

)1( ar ′+Ω

tnrelrel VV −= |


Figure A.4- Pressure, velocity and rotation distributions

A.3.1. Thrust and torque coefficients

Applying the x-component of linear momentum equation to the annulus control volume

shown in Figure (A.5), gives thrust force as )( FWrotor VVdQdT −= ∞ρ ,where the volume

flow rate diskd dAVdQ = , wVV =∞ and )1( aVV wd −= and therefore

diskFWwwrotor dAVVaVdT ))(1( −−= ρ (A.19)

Applying the energy equation for the same control volume gives the turbine power.

)(5.0 22

FWwdiskd VVdAVdP −= ρ (A.20)

∞ Disk Far Wake

Pressure, P

Velocity, V

Rotation, ω


Figure A.5- Annulus control volume; Linear momentum balance

Turbine power can also be obtained by multiplying the thrust force and flow velocity at the

disk.

ddTVdP = (A.21)

Combining Equations (A.19), (A.20) and (A.21) concludes

wFW VaV )21( −= (A.22)

By substituting FWV back in Equation (A.19), rotordT will be determined in terms of the

wind velocity at the upstream and the axial induction factor.

diskwrotor dAVaadT 2)1(2 ρ−= (A.23)

Thrust coefficient by definition is

diskw

TdAV

dTC

2

21 ρ

= (A.24)

And therefore as a result of Momentum Theory it becomes:

)1(4 aaCT −= (A.25)

CV

FWdQVρ dT ∞dQVρ


To determine torque coefficient by the Momentum Theory one can start from applying the

angular momentum equation about the x-axis for the control volume shown in Figure (A.6)

to find a relation between the rotation in far wake and circumferential velocity at disk as

( ) ( ) ( ) θωωω diskdiskdiskFW rVrrrr 22/222 === .

Since the circumferential velocity θdiskV is only due to induction, one can substitute θθ

iVV =

from Equation (A.5) in the above equation to find ( )FWr ω2 .

( ) Ω′= arr FW

22 2ω (A.26)

Figure A.6- Annulus control volume; Angular momentum balance between disk and Far

Wake

Applying angular momentum equation about x–axis for the control volume shown in

Figure (A.7), the applied torque on the rotor will be determined.

( ) ( ) ∞−= ωωρ 22 rrdQdM FWx (A.27)

Combining Equations (A.26) and (A.27) gives the rotor torque as

diskwx dAraaVdM 2)1(2 ′−Ω= ρ (A.28)

Torque coefficient is defined as

CV

FWrdQ )( 2ωρ diskrdQ )( 2ωρ


diskw

x

M

rdAV

dMC

2

21 ρ

= , (A.29)

and finally as a result of the Momentum Theory it becomes

)1(4 aaC rM −′= λ (A.30)

Figure A.7- Annulus control volume; Angular momentum balance

A.3.2. Tip and Hub Losses

In momentum theory, the axi-symmetric flow is the basic assumption, which holds if the

turbine rotor has an infinite number of blades with zero chord length. In the case of a real

turbine with a finite number of blades, the induced velocity on the blades is different from

the mean induced velocity in the flow annulus and therefore circumferential symmetry

does not hold. The non-uniformity of the induced flow field makes the actual local TC and

MC to be smaller than the expected values by the optimum actuator disk theory. The

departure of the induced velocity, TC and MC from their momentum theory values is more

significant near the tip and root of the blade. These deviations from the uniform induced

velocity flow field are called tip and hub losses. The overall loss factor, F is defined as

hubtipFFF = (A.31)

CV

FWrdQ )( 2ωρ

dM

∞)( 2ωρ rdQ


In which tipF is unity at inboard parts of the blade and takes smaller values near the tip of

the blade and hubF is unity at outboard parts of the blade and takes smaller values near the

hub of the blade. Overall loss factor F can be applied on the induced velocity or disk

velocity.

A.3.3. Loss Models

Depending on which parameter or parameters are affected by F , different models will be

introduced. Most of commercial codes use two models that are known as

1. Classical or Wilson

2. Standard or Advanced or Wilson-Walker.

a. Classical Model

In Classical model, loss factor F is applied to the axial induced velocity:

w

x

i aFVV = (A.32)

and therefore

)1( aFVV wd −= (A.33)

)21( aFVV wFW −= (A.34)

)1(4 aFFaCT −= (A.35)

)1(4 aFaC rM −′= λ (A.36)

According to Figure (A.8), which shows the relative velocity diagram for this model, the

normalised relative velocity is given by Equation (A.37).

ϕδ

sin

cos)1(| aF

V

V

w

tnrel −=− (A.37)

b. Wilson-Walker Model


In Wilson-Walker model, loss factor F is directly applied to the disk

velocity, )1( aFVV wd −= and the difference between the free stream velocity and far wake

velocity is defined as wFWw aVVV 2=− . With the above assumptions thrust and torque

coefficients can be calculated as:

)1(4 aFaCT −= (A.38)

)1(4 aFaC rM −′= λ (A.39)

For this model the relative velocity diagram becomes as shown in Figure (A.9) and the

normalised relative velocity becomes as given by Equation (A.40).

ϕδ

sin

cos)1(| Fa

V

V

w

tnrel −=− (A.40)

Figure A.8- Relative velocity in aerofoil plane; zero yaw; Classical model

Figure A.9- Relative velocity in aerofoil plane; zero yaw; Wilson-Walker model

n

t ϕ )1( ar ′+Ω

tnrelrel VV −= |

)1(cos aFVw −δ

n

t ϕ )1( ar ′+Ω

tnrelrel VV −= |

)1(cos aFVw −δ


A.3.4. Prandtl Tip and Hub loss factors

Among some theories for estimating the tip and hub loss factors, Prandtl theory is simple

and efficient and also gives acceptable results. In Prandtl theory tip and hub loss factors are

defined as

)exp(cos2 1

tiptip fF −= −

π if 7≤tipf (A.41.a)

1=tipF if )5.0(5.0 85.0 += tip

new

tip FF (A.41.b)

Where

ϕsin2

)(

r

rRBf tip

−= (A.41.c)

and

)exp(cos2 1

hubhub fF −= −

π if 7≤hubf (A.42.a)

1=hubF if 7>hubf (A.42.b)

Where

ϕsin2

)(

hub

hub

hubR

RrBf

−= (A.42.c)

In the above equations, B is the number of blades, R and hubR are rotor and hub radii and

ϕ is the inflow angle from Equation (A.16).

A.3.4.1. Xu modification

An improvement to Prandtl’s tip loss factor has presented by Xu (2001) as follows

a. Pre-stalled condition ( sαα < ):

)5.0(5.0 85.0 += tip

new

tip FF if 1/7.0 ≤≤ Rr (A.43.a)


R

FrF

Rrtipnew

tip7.0

)|1(1

7.0/ =−−= if 7.0/ <Rr (A.43.b)

b. Post-stalled condition ( sαα ≥ ):

8.0=new

tipF if 1/8.0 ≤≤ Rr (A.43.c)

1=new

tipF if 8.0/ <Rr (A.43.d)

A.3.5. Heavy loading (High axial induction factor)

Momentum theory predicts a parabolic variation for thrust coefficient with a maximum

value of 1 at 5.0=a , while the experimental data show that TC keeps increasing for

5.0>a . For small axial induction factors, 4.00 ≅<< caa , known as light loading state,

predicted thrust coefficient by the momentum theory is in a good agreement with the

experimental data. In the case of heavy loading state, where caa > , predicted TC departs

dramatically from its actual value. In the extreme loading situation, 1=a , wind turbine acts

as a drag driven device with a thrust coefficient of 2)( max, == DragTT CC rather than

0=TC as predicted by Equation (A.25). Extrapolating Equation (A.25), with a maximum

value of 2=TC at 1=a , predicts reasonable values for TC . Separating light and heavy

loading state, Equations (A.35) and (A.38) can be re-written as follows

a. Classical model

)1(4 aFFaCT −= if caa ≤ (A.44.a)

21

2

0 AaAaACT ++= if caa > (A.44.b)

where

2

2

0)1(

)2(442

c

cc

a

aaFFA

−

−+−= (A.44.c)

2

22

1)1(

8)1(44

c

ccc

a

aFaFaA

−

−++−= (A.44.d)


2

2222

2)1(

442

c

ccc

a

aFFaaA

−

+−= (A.44.e)

b. Wilson-Walker model

)1(4 aFaCT −= if caa ≤ (A.45.a)

21

2

0 BaBaBCT ++= if caa > (A.45.b)

where

Fa

Bc

4)1(

220 −

−= (A.45.c)

Fa

aB

c

c 4)1(

421 +

−

−= (A.45.d)

22)1(

242

c

c

a

aB

−

−+= (A.45.e)

A.4. Blade Element Force Analysis

Figure (A.10) shows a blade segment (element) subjected to the aerodynamic forces in the

same system of coordinates as introduced in Figure (A.1). Assuming 2-dimentional flow

on the aerofoil and neglecting radial forces on the blade ( 0=sdF ), thrust force on the

element can be obtained as δcosndFdT = or

( ) δϕϕ cossincos dDdLdT += (A.46)

Lift and drag coefficients are defined as

( ) etn

rel

L

dAV

dLC

−

=2

21 ρ

(A.47)

( ) etn

rel

D

dAV

dDC

−

=2

21 ρ

(A.48)


Figure A.10- Blade element force analysis

where ( )tnrelV − is the relative velocity in the tn − plane (see Figures (A.2) and (A.3)) and

δcoscdrcdsdAe == is the element area. Combining Equations (A.46), (A.47) and (A.48)

gives thrust force on a blade element as

( ) ( )drCCVcdT DLtnrel ϕϕρ sincos2

1 2 += − (A.49)

and for a turbine with B blades it becomes

( ) ( )drCCVBcdT DLtnrel ϕϕρ sincos2

1 2 += − (A.50.a)

or in terms of span coordinate

( ) ( ) δϕϕρ cossincos2

1 2dsCCVBcdT DLtnrel += − (A.50.b)

Using Equations (A.24) and (A.17), thrust coefficient can be written as

( )ϕ

ϕϕδσ2

22

sin

sincos)1(cos DLrT

CCaC

+−= (A.51)

where rσ , local solidity ratio, is defined as

δππσ

cos22 s

Bc

r

Bcr == (A.52)

Rotor Plane

Element

Blade Span

ϕ dD

tdF dL

ndF n

sdF

δ

dT

s

t

ndF δ

x

r


Aerodynamic forces on the blade element also produce a torque about the rotor axis equal

to tx rdFdM = (Figure (A.10)). Recalling Equations (A.47) and (A.48), for a turbine

with B blades the generated torque about the rotor axis can be expressed as

( ) ( ) δϕϕρ coscossin2

1 2rdrCCVBcdM DLtnrelx −= − (A.53.a)

or in terms of span coordinate

( ) ( ) δϕϕρ coscossin2

1 2sdsCCVBcdM DLtnrelx −= − (A.53.b)

Inserting the above result into the definition of the torque coefficient MC , Equation (A.29),

yields to

( )ϕ

ϕϕδσ2

23

sin

cossin)1(cos DLrM

CCaC

−−= (A.54)

A.4.1. Blade Aerodynamic Characteristics

Lift and drag coefficients are functions of the angle of attack and Reynolds number. Angle

of attack is in turn a function of the velocity field and the blade geometry and can be

expressed as

αββϕα ∆+−−+= pitche 0 (A.55)

In the above equationϕ is the inflow angle, (Equation (A.16)), eβ and 0β stand for the

blade elastic twist and pre-twist, pitch is the blade pitch angle and α∆ accounts for the

angle of attack corrections as given by the following equation.

mc ααα ∆+∆=∆ (A.56)

In Equation (A.56) cα∆ refers to cascade correction and mα∆ refers to the other corrections.

A.4.1.1. Cascade Correction, cα∆

Cascade correction to the angle of attack has two components

21 ααα ∆+∆=∆ c (A.57)


where 1α∆ accounts for the effect of finite aerofoil thickness and 2α∆ accounts for the

effect of finite aerofoil width

rc

AB a

πϕ

α2

cos 0

1 =∆ (A.58)

−

−′+

−=∆ −−

R

ra

Ra

ra )1(tan

)21(

)1(tan

4

1 11

2α (A.59)

0ϕ is the inflow angle prior to rotational induction, ( 0=′a in Figure (A.3)) and aA is the

aerofoil cross section area, normally taken as max68.0 ctAa ≈ , where maxt is the maximum

thickness of the aerofoil.

A.4.1.2. Effect of Finite Aspect Ratio, Pre-Stall Condition

The effect of finite aspect ratio at small angle of attacks can be considered as either

Estimating the finite aerofoil data by applying Lanchester-Pradtl theory to 2-D data

LL CC ′= (A.60)

AR

CCC L

DD π

2

+′= (A.61)

AR

CL

παα +′= (A.62)

where LC ′ , DC ′ and α ′ are infinite aerofoil data and AR stands for the aspect ratio, or using

2-D data and applying a tip loss factor to the HAWT aerodynamic model.

A.4.1.3. Effect of Finite Aspect Ratio, Post-Stall Condition

For larger angle of attacks, two models can be used to estimate lift and drag coefficients of

an aerofoil.

a. Viterna-Corrigan model

Viterna-Corrigan model (Viterna and Corrigan, 1981) uses only three values from the 2-

Dim data to estimate lift and drag coefficients in post-stall condition, 90<<αα s .


αα

ααsin

coscossin

2

max, LDL KCC += (A.63)

αα cossin 2

max, DDD KCC += (A.64)

where

( )s

s

ssDstallLL CCKαα

αα2max,,

cos

sincossin−= (A.65)

s

sDstallD

D

CCK

α

α

cos

sin 2

max,, −= (A.66)

ARCD 018.11.1max, += if 50≤AR (A.67.a)

01.2max, =DC if 50>AR (A.67.b)

For other angle of attacks in the range of [-180, 180] reflected values for DC and reflected-

reduced values for LC can be used, (see Figure (A.11)).

b. Flat-Plate Theory

Flat-plate theory estimates post-stall aerodynamic characteristics of an aerofoil as

αcosNL CC = (A.68)

αsinND CC = (A.69)

where

+= 98.1,

sin

238.0222.0

1min

α

NC if 1≥F (A.70.a)

−−=

F

FCN

2

)1(22.12tanh81.098.1 if 15.0 << F (A.70.b)


17.1=NC if 5.0≤F (A.70.c)

-1

-0.5

0

0.5

1

1.5

-180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180

Angle of attack

AD C B C D

Drag

Lift

A: Available data

B: Viterna model

C: Viterna model with reflection and reduction

D: Interpolation between limits

Figure A.11- Viterna post-stall model

A.5. Blade Element Momentum Theory, BEMT

Equating thrust and torque coefficients obtained from the blade element force analysis

(with the assumption of zero drag force) and those obtained from momentum theory is the

base of the BEMT. Neglecting drag force in Equations (A.51) and (A.54), thrust and torque

coefficients will become

ϕϕδσ

2

22

sin

cos)1(cos0

LrT

CaC

−= (A.71)

ϕϕδσ

2

22

sin

sin)1(cos0

LrM

CaC

−= (A.72)

Depending on which brake state model in the momentum theory is used, formulation for

BEMT will be different.

a. Classical Model

Combining Equations (A.71) and (A.72) with equations (A.44) and (A.36) gives


2

110T

Ca

−−= if 96.04.0

0≤≡≤ TCa (A.73.a)

0

20

2

11

2

)(40

A

CAAAAa

T−−+−= if 96.04.0

0>≡> TCa (A.73.b)

δλϕ

cos

tan

r

aFa =′ (A.74)

b. Wilson-Walker Model

Combining Equations (A.71) and (A.72) with Equations (A.45) and (A.39) gives:

2

11 0

F

C

a

T−−= if FCa T 96.04.0

0≤≡≤ (A.75.a)

0

20

2

11

2

)(40

B

CBBBBa

T−−+−= if FCa T 96.04.0

0>≡> (A.75.b)

δλϕ

cos

tan

r

aa =′ (A.76)

6 Equations (A.16), (A.31), (A.55), (A.71), (A.73)/(A.75), (A.74)/(A.76) and two set of

tabulated data for LC and DC can be solved to find a , a ′ , F ,ϕ ,α , LC , DC and 0T

C .

Knowing a ,ϕ , LC and DC one can use Equations (A.50) and (A.51) to calculate T and

TC and Equations (A.53) and (A.54) to find M and MC . Having rotor torqueM , turbine

mechanical power, P can be easily calculated by

MdMP Ω=Ω= ∫ (A.77)

and the power coefficient PC can be determined from the following equation.

233

21

21 RV

P

AV

PC

wrotorw

P

πρρ== (A.78)


A.6. Flap bending

Referring to Figure (A.12), tangential and normal force increments on the blade element at

locationη , tdF and ndF , are given by Equation (A.79).

( )ηηη

ϕϕϕϕ

ηρϕϕϕϕ

.

2

..sincos

cossin

21

sincos

cossin

atD

L

tnrel

atatn

t

C

CdVc

dD

dL

dF

dF

−=

−=

−

(A.79)

Figure A.12- Blade element force analysis for flap bending calculation

Flap-bending moment produced by these forces at location s can be calculated as:

( ) ( ) ( )( )∫−=

=

−−+=s

R

ntsatFB sR

dFdFM δη

η ηη ηδ

ξξcos

0.cos

sincos (A.80)

In the above equation,

−− sR

ηδcos

is the moment arm and ξ is the angle between the

normal axis and the chord line at s - location which can be expresses in terms of the inflow

and attack angles as:

sat.)( αϕξ −= (A.81)

r

Blade

element

δ

s

η

x

R

ϕ

dD

dL

n

t

Chord line at

s -location

ξ

ndF

tdF


A.7. References

Tulapurkara, E.G., 1996. Turbulence Models for the Computation of Flow past Airplanes.

Progress in Aerospace Sciences, 33, pp 71-165.

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appendix - blade element momentum theory

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