100 Years of Rotor Vortex Theory
Development of theories
of optimal rotors
V.L. Okulov, J.N. Sørensen, G.A.M. van Kuik
100 Years of Rotor Vortex Theory
N.E. Joukowsky has reported
“Vortex theory of the screw propeller”
1 October 1912
Before Rotor Vortex Theory
Momentum (or slipstream, or actuator disk) theory (1889)
W. Rankine
R.E. Froude
Strong discussion about a validation of the theory took place up to the
first formulation of the vortex theory but a distrust to the Froude’s
theory in English school kept long time after that.
Before Rotor Vortex Theory
Aerodynamics of aerofoil
Lord
Rayleigh
M. Kutta
N. Joukowsky
Strong discussion about aerodynamics of aerofoil results to
the formulation of Kutta-Joukowsky theorem (1902 - 1906).
Before Rotor Vortex Theory
Blade elements theory (1892)
S. Drzewiecki
The original theory of Drzewecki was incomplete
because it did not include induction velocity.
For this reason propellers designed in accordance with his theory in
beginning of XX century was inferior to ones after empirical selection.
Before Rotor Vortex Theory
Aerodynamics of wing
F. Lanchester
The first picture of the wing vortex by Lanchester (1907)
The first simplified vortices
of wing (Prandtl 1913)
L. Prandtl
Joukowsky used it to his
vortex theory of rotor
More accurate vortices
(Prandtl 1918)
Betz used it for rotor
100 Years of Rotor Vortex Theory
Rotor vortex theory
of Joukowsky
(October 1912)
was formulated in first article (1912)
of his famous cycle from 4 articles
“Vortex theory of screw propeller”
(1912-1918)
N.E. Joukowsky
+
=
Flamm’s visualization and the first wing vortex system result to his theory
100 Years of Rotor Vortex Theory
Model of helical tip vortex in his first article (1912)
Joukowsky was first who derived the vortex ring approximation for helical tip vortex
but he neglected a regular rest term which is sufficiently great without which
becomes impossible to find a correct solution for rotor with finite number of blades
Moor & Saffman (Phil. Trans. Roy. Soc., 1972)
re-suggested the vortex ring approximation 60 years later!
Ricca (JFM, 1994) estimated the regular term by numerical simulation
100 Years of Rotor Vortex Theory
Rotor with infinite number of blades in his first article (1912)
A complete solution with definition of induction
velocity and blade form was found by vortex
theory for rotor with infinite number of blades
(Joukowsky 1912)
In the first article (1912) Joukowsky proposed the rotor vortex theory
with finite number of blades but he could solve this infinite case only!
100 Years of Rotor Vortex Theory
2-d article of “Vortex theory of screw propeller” (1914)
V. Vetchinkin
(pupil of Joukowsky)
Vortex system for rotor with
arbitrary circulation along blade
(Vetchinkin, 1913)
Blade element approach
to the Vetchinkin’s rotor
(Joukowsky 1914)
In the second article (1914) Joukowsky only described the method
but he could not formulate an law for optimization
100 Years of Rotor Vortex Theory
3-d article of “Vortex theory of screw propeller” (1915)
In the third article (1914) Joukowsky for the first time created
theory of hydrodynamical cascades from the blade profiles
100 Years of Rotor Vortex Theory
4-th article of “Vortex theory of screw propeller” (1918)
The general momentum theory based on an understanding of the rotor
flow from the vortex theory of the screw propeller has been formulated
Propeller case
A partial case of the general theory for wind turbine rotor
with constant circulation includes a paradox of infinite
power for small tip speed ratio. This paradox has been
discussed by Sørensen & van Kuik in (WE, 2011)
100 Years of Rotor Vortex Theory
Rotor vortex theory of German school (1919)
L. Prandtl
A. Betz
(pupil of Prandtl)
The German vortex theory of rotor based on
Prandtl solution for wing with elliptical distribution of load.
100 Years of Rotor Vortex Theory
Prandtl’s correction of infinite number of blades (1919)
Idealized vortex system for
screw propeller by Prandtl
Associative plane flow with
the wake behind Betz rotor
In 1919 Prandtl’s school could only formulate the optimum for the rotor
vortex theory with finite number of blades but they have considered
case of infinite number of blades like the Russian consideration!
The First Results grounded by Rotor Vortex Theory
Betz-Joukowsky limit (1920)
The First Results grounded by Rotor Vortex Theory
Blade element momentum (BEM) theory (1912-1920)
+
=
The first combination was made in Russia (Sabinin & Yuriev 1912),
the next ones in Germany (Betz 1915) and English school adopted
it in book by Fage & Collins in 1919 only.
First optimization
of wind turbine by
BEM theory (1935)
H. Glauert
Betz-Joukowsky limit
Glauert’s correction
Development of Rotor Vortex Theory
Goldstein’s solution for Betz rotor (1929)
S. Goldstein
Unfortunately his solution was very complex to simulate and
Theodorsen used the electromagnetic analogy to design blades
for the screw propeller
Development of Rotor Vortex Theory
Theodorsen’s measurements for Betz rotor (1945)
Theodore Theodorsen prepares an electromagnetic equipment to give a
talk on the physics of a four-blade propeller in 1945.
Development of Rotor Vortex Theory
Progress of helical vortex theory in XX century
Goldstein’s circulation
Self-induced velocity of the helix
2
1.5

Joukowsky’s
approximation
1
0.5
0
0.5
1
Points: Tibery & Wrench (1964)
Lines: Okulov & Sørensen (WE, 2008)
pitch
2
4
6
8
10
Black line is analytical solution (Okulov, 2004)
Many famous contributions in helical vortex theory (e.g. Forsdyke,1928; Goldstein,
1929; Rosenhead, 1930; Morgan & Wrench, 1965; Crow, 1970; Widnall et al 1971 &
1972; Moore & Saffman, 1972; Batchelor, 1973; Callegari & Ting, 1978; Fukumoto &
Miyazaki, 1991; Okulov 1993&1995; Ricca, 1994; Kuibin & Okulov 1998; Vozhdaev
et all 1997; Boersma & Wood ,1999 etc.) were made in XX century to result
analytical solutions for both rotor with finite number of blades.
Development of Rotor Vortex Theory
Major assumptions of the main rotor theories
Number of
blades
Definition of
the pitch in the wake
Circulation
along blade
actuator disk
non specified
non specified
non specified
non specified*
non specified
infinite
without induction velocity
constant
infinite
without induction velocity
Betz type
finite (N0)
without induction velocity
correction of Betz type
finite (N0=1-4)
without induction velocity
Goldstein’s type
Theodorsen’s consideration
of the rotor II (1948)
finite (N0)
with induction velocity
of far wake
Goldstein’s type
New OS solution of rotor II (2008)
finite (N0)
with induction velocity
in rotor plane
Goldstein’s type
New OS solution of rotor I (2010)
finite (N0)
with induction velocity
in rotor plane
constant
Theories
Betz-Joukowsky limit
(1920)
Glauert’s optimization
(1935)
Joukowsky consideration
of the rotor I (1912)
Betz consideration
of the rotor II (1919)
Prandtl tip correction
(1919)
Goldstein’s solution
of the rotor II (1929)
Development of Rotor Vortex Theory
Test of the Betz-Goldstein’s and Theodorsen’s theories
Lightly loaded propeller with
finite number of blades
(Betz, 1919; Goldstein,1929)
The induction velocity did not
influence on the pitch of the
wake
CT
CP
5
0.8
w
w

C T  2 w  1   I1  I 3
2
2

4
0.6
3
w 

C P  2 w  I1  I 3 
2 

0.4
0.2
0
Theodorsen’s consideration
for propeller with finite
number of blades
(Theodorsen,1948)
The pitch calculates via
induction velocity in far wake
6
1

0
5
10
15
N
N
N
N
N
N
N
=
=
=
=
=
=
=
1
2
3
5
10
20
2
1

0
20

0
5
10
15
20
0.6
CP
CT
0.8



C T  2 w   w     
2


0.5
C P  2 w 1  w   w 
0.4
0.6
0.3
0.4
0.2
0.1
0
R/V
0
5
10
N=2
N=3
N=4
N=5
N=6
0.2
0
R/V
0
Points is Glauert’s optimization
5
10
Development of Rotor Vortex Theory
Comparison of both rotors (Okulov & Sørensen, JFM 2010)
Joukowsky vortex model of rotor
Betz vortex model of rotor
w 
 w 
CP  2w1  I1  I3 
2 
 2 
Points is Glauert’s optimization
For the first time the comparison between the famous rotor conceptions
was made by analytical solutions but in both cases the wake expansion
has been neglected.
100 Years of Rotor Vortex Theory
The centennial is also celebrated in
Russia.
The Russian institute TsAGI, founded by
Joukowsky, makes a special issue on
this topic
“Development
of theories of an optimal rotor”
by V. Okulov, J. Sørensen G. van Kuik.
100 Years of Rotor Vortex Theory
Collaboration between Russian and German aerodynamic schools
in the beginning of XX-th century was very successful
100 Years of Rotor Vortex Theory
Concluding remarks
Both Russian and German schools created their vortex theories
independently because World War I and Russian revolution
separated their collaboration.
100 years of the theory should celebrate from 2012 to 2019!
The story of rotor vortex theory
will be continued!