100 years of rotor vortex theory development of theories ... · pdf filedevelopment of...

100 Years of Rotor Vortex Theory

V.L. Okulov, J.N. Sørensen, G.A.M. van Kuik

Development of theoriesof optimal rotors


N.E. Joukowsky has reported “Vortex theory of the screw propeller”

1 October 1912

Before Rotor Vortex Theory

R.E. FroudeW. Rankine

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.

Momentum (or slipstream, or actuator disk) theory (1889)


Lord Rayleigh M. Kutta

Strong discussion about aerodynamics of aerofoil results to the formulation of Kutta-Joukowsky theorem (1902 - 1906).

Aerodynamics of aerofoil

N. Joukowsky


S. Drzewiecki

Blade elements theory (1892)

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.


The first picture of the wing vortex by Lanchester (1907)

Aerodynamics of wing

F. Lanchester

L. Prandtl

The first simplified vortices of wing (Prandtl 1913)

Joukowsky used it to his vortex theory of rotor

More accurate vortices (Prandtl 1918)

Betz used it for rotor


N.E. Joukowsky

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)

+ =

Flamm’s visualization and the first wing vortex system result to his theory

100 Years of Rotor Vortex TheoryModel of helical tip vortex in his first article (1912)

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

Joukowsky was first who derived the vortex ring approximation for helical tip vortexbut 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

100 Years of Rotor Vortex TheoryRotor with infinite number of blades in his first article (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!

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 second article (1914) Joukowsky only described the method but he could not formulate an law for optimization

V. Vetchinkin(pupil of Joukowsky)

2-d article of “Vortex theory of screw propeller” (1914)

Vortex system for rotor with arbitrary circulation along blade

(Vetchinkin, 1913)

Blade element approach to the Vetchinkin’s rotor

(Joukowsky 1914)

100 Years of Rotor Vortex Theory3-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 Theory4-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

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)Propeller case

100 Years of Rotor Vortex TheoryRotor vortex theory of German school (1919)

The German vortex theory of rotor based on Prandtl solution for wing with elliptical distribution of load.

A. Betz(pupil of Prandtl)L. Prandtl

100 Years of Rotor Vortex TheoryPrandtl’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!

Betz-Joukowsky limit (1920)

The First Results grounded by Rotor Vortex Theory

Blade element momentum (BEM) theory (1912-1920)

The First Results grounded by Rotor Vortex Theory

Betz-Joukowsky limit

Glauert’s correction

+ =

H. Glauert

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)

Development of Rotor Vortex TheoryGoldstein’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 TheoryTheodorsen’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 TheoryProgress of helical vortex theory in XX century

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; Vozhdaevet all 1997; Boersma & Wood ,1999 etc.) were made in XX century to result analytical solutions for both rotor with finite number of blades.

Goldstein’s circulation

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

Self-induced velocity of the helix

2 4 6 8 101

0.5

0

0.5

1

1.5

2

pitch

Joukowsky’sapproximation

Black line is analytical solution (Okulov, 2004)

Development of Rotor Vortex TheoryMajor assumptions of the main rotor theories

Theories Number of blades

Definition of the pitch in the wake

Circulation along blade

Betz-Joukowsky limit (1920) actuator disk non specified non specified

Glauert’s optimization (1935) non specified non specified* non specified

Joukowsky consideration of the rotor I (1912) infinite without induction velocity constant

Betz consideration of the rotor II (1919) infinite without induction velocity Betz type

Prandtl tip correction(1919)

finite (N0) without induction velocity correction of Betz type

Goldstein’s solution of the rotor II (1929)

finite (N0=1-4) without induction velocity Goldstein’s type

Theodorsen’s considerationof the rotor II (1948) finite (N0)

with induction velocityof 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

Development of Rotor Vortex TheoryTest of the Betz-Goldstein’s and Theodorsen’s theories

0 5 10 15 200

0.2

0.4

0.6

0.8

1CP

N = 1 N = 2 N = 3 N = 5 N = 10N = 20N =

0 5 10 15 200

1

2

3

4

5

6CT

1 322PwC w I I

1 32 12 2Tw wC w I I

0 5 100

0.1

0.2

0.3

0.4

0.5

0.6

0 5 100

0.2

0.4

0.6

0.8CP CT

R/V R/V

N =2N = 3N = 4N = 5N = 6

2 1PC w w w 22TC w w

Lightly loaded propeller with finite number of blades(Betz, 1919; Goldstein,1929)The induction velocity did not influence on the pitch of the wake

Theodorsen’s consideration for propeller with finite number of blades(Theodorsen,1948)The pitch calculates via induction velocity in far wake

Points is Glauert’s optimization

Development of Rotor Vortex TheoryComparison of both rotors (Okulov & Sørensen, JFM 2010)

1 32 12 2Pw wC w I I

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.

Joukowsky vortex model of rotor Betz vortex model of rotor

Points is Glauert’s optimization


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.

Collaboration between Russian and German aerodynamic schools in the beginning of XX-th century was very successful


100 Years of Rotor Vortex TheoryConcluding remarks

The story of rotor vortex theorywill be continued!

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!

100 years of rotor vortex theory development of theories ... · pdf filedevelopment of...

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