Vortex theory of the ideal wind turbine

Jens N. Sørensen and Valery L. Okulov

Department of Mechanical Engineering, Technical University of Denmark,

DK-2800 Lyngby, Denmark

Two definitions of the ideal rotor

Joukowsky(1912)

Betz(1919)

blade span

r

R 0

V w const constV w

In both cases only conceptual ideas were outlined for rotors with finite number of blades,

whereas later theoretical works mainly were devoted to rotors with infinite blades!

blade span

R0

Lifting-line theory for rotor with finite number of blades

0 0bdL N U dr

cosdT dL

sindP r dL

A (rotor plane): Kutta - Joukowsky Theorem

0 00

bdT N r u dr

00 0

b zdP N V u rdr

B (wake approximation):

From Helmholtz’s vortex theorem it results that:

From symmetry considerations, neglecting expansion, it can be shown that:

and

0

10 2

zz

u u 10 2

u u

Models of far wake for ideal rotors

2

bN rG r

l w

Goldstein circulation (Theodorsen, 1948):

2b

z z

Nru u u a u

l l

Characteristics of flow with helical vortices(Okulov, JFM, 2004)

2bN r l wG r 2bN la

120

2 zdP a l V u r dr 120

2 zdP w l G r V u r dr

Velocity triangles determining geometry of the wakes

12

00

tgV w

r

2 210 2 1l V a l R 1

0 2l V w

22

2 2sinz

wru w

r l

2

11222

2 1 zl

dP a V a V u r drR

2121

2 2 22wr

dP w V w V G r r drr l

The model assumption: 1 12 2zw u a The model assumption: 0tg tg

l

r

tg zV u

r u

1

0 2

tgV w l

R v R

2bN al

vR R

θ

z

l lw u u u

R R

From definition of u

(Okulov, JFM, 2004)

From definition of w(Goldstein, 1929)

Equilibrium motion for both far-wake models

20

0 20

R rG r u r r dr

l

Definition of Goldstein circulation G(r):Uniform motion of the helical sheets in

axial direction with velocity

2 4 4 Ind

b bN Na lw u

l R R

Definition of the vortex core size:

Uniform axial motion of all helical vortices in vortex core with unknown vortex core

radius and constant velocity

2bN r l wG r

2

2Ind

Ru

l

0 0 02u r r r u r r give us an equation for definition

of the vortex core size:give us an integral equation for

definition of G(r)

by using dimensionless variables

00 0

R

b

lw N r u r r dr

r

zu u r u l

z

l lu u w u

R r

From definition velocity

for flows with helical symmetry we can write

Approximate attempts of simulating the wake motionFragment of Goldstein’s solution (1929)

Measurements of Theodorsen (1945)

The “ring” term was introduced by Joukowski in 1912.

Approach by Moore & Saffman (1972)

HELIX SELF-INDUCED MOTION

Asymptotic for large and small pitch:

Kelvin (1880); Levy & Forsdyke (1928);

Widnall (1972) etc …

Approximations (cat-off method, …):

Thomson, 1883; Rosenhead, 1930;

Crow, 1970; Batchelor, 1973;

Widnall et al, 1971; etc …

Final solutions for equilibrium motion of the wakes

Averaged interference factor in far wake

Goldstein circulation functions for Nb = 3

Points: Tibery &Wrench (1964) Lines:Okulov &Sørensen(2008)

2

2

3 22

1 22

4 2 27 2 22

4 1

11 1 1ln ln

41

333

81

bInd b

b

b

NRu N

N

l

N R

Vortex core radius Elimination of

singularity

Definition of the vortex core size based on self-induced velocity by Okulov (JFM, 2004)

Comparison (1) of maximum power coefficients

1

1

0

2 I G x xdx

1 3

3 2 20

2

x dxI G x

x l

w

wV

Mass

coefficient

“Axial loss

factor”

1 32 12 2P

w wC w I I

1 2 32 12 2P

a aC a J J J

Solution of Betz rotor

(Okulov&Sorensen,2008)Solution of Joukowsky rotor

(present)

Difference between

the power coefficients2 2 2

1 2 3 1 2 1 2 3 3

1 33

J J J J J J J J Ja

J J

2 21 3 1 1 3 3

3

2

3

I I I I I Iw

I

zua

V

1

3

0

2 zu xJ l xdx

a

2

1 21 2

lJ

R

2

2 2

1 112 6

JR R

Comparison (2) of maximum power coefficients

31

1 2 20

2F x x

I dxl x

51

3 22 20

2

F x xI dx

x l

w

wV

Mass

coefficient

“Axial loss

factor”

1 32 12 2

P

w wC w I I1 32 1

2 2P

w wC w I I

1

1

0

2 I G x xdx

1 3

3 2 20

2

x dxI G x

x l

Mass

coefficient

“Axial loss

factor”

Approximation with

Prandtl’s tip correction

Solution of Betz rotor

(Okulov&Sorensen,2008)

21 12arccos

2

bx lN

Fl

2 21 3 1 1 3 3

3

2

3

I I I I I Iw

I

Difference between

the power coefficients

w

wV

• An analytical optimization model has been developed for a rotor with finite number of blades and constant circulation (“Joukowsky rotor”)

• Optimum conditions for finite number of blades as function of tip speed ratio has been compared for two models:

(a) “Joukowsky rotor” with constant circulation along blade (b) “Betz rotor” with circulation given by Goldstein’s function (Okulov & Sorensen, WE, 2008)

• The optimum power coefficients evaluated by approximation with Prandtl‘s tip correction correlates well with the original analytical solution using Goldstein’s circulation for “Betz rotor”

• For all tip speed ratios the “Joukowsky rotor” achieves a higher efficiency that the “Betz rotor”

Summary