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