100 Years of Rotor Vortex Theory

8/22/2019 100 Years of Rotor Vortex Theory

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100 Years of Rotor Vortex Theory

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

Development of theories

of optimal rotors


2/25


N.E. J oukowskyhas reportedVortex theory of the screw propeller

1 October 1912


3/25

Before Rotor Vortex Theory

R.E. FroudeW. Rankine

Strong discussion about a validation of the theory took place up to thefirst formulation of the vortex theory but a distrust to the Froudes

theory in English school kept long time after that.

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


4/25


LordRayleigh M. Kutta

Strong discussion about aerodynamics of aerofoil results to

the formulation of Kutta-J oukowsky theorem (1902 - 1906).

Aerodynamics of aerofoil

N. J oukowsky


5/25


S. Drzewiecki

Blade elements theory (1892)

The original theory of Drzewecki was incompletebecause 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.


6/25


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

Aerodynamics of wing

F. Lanchester

L. Prandtl

The first simplified vorticesof wing (Prandtl 1913)

J oukowsky used it to hisvortex theory of rotor

More accurate vortices(Prandtl 1918)

Betz used it for rotor


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N.E. J oukowsky

Rotor vortex theory

of Joukowsky

(October 1912)

was formulated in f irst article (1912)

of his famous cycle from 4 articles

Vortex theory of screw propeller

(1912-1918)

+ =

Flamms visualization and the first wing vortex system result to his theory


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Model 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 (J FM, 1994) estimated the regular term by numerical simulation

J oukowsky was first who derived the vortex ring approximation for helical tip vortex

but he neglected a regular rest term which is sufficiently greatwithout which

becomes impossible to find a correct solution for rotor with finite number of blades


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Rotor with infinite number of blades in his first article (1912)

In the first article (1912) Joukowsky proposed the rotor vortex theory

with f inite number of blades but he could solve this inf inite 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

(J oukowsky1912)


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In the second article (1914) Joukowsky only described the method

but he could not formulate an law for optimization

V. Vetchinkin

(pupil of J oukowsky)

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

Vortex system for rotor witharbitrary circulation along blade

(Vetchinkin, 1913)

Blade element approach

to the Vetchinkins rotor(J oukowsky1914)


11/25


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


12/25


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

A partial case of the general theory for wind turbine rotorwith constant circulation includes a paradox of infinitepower for small tip speed ratio. This paradox has been

discussed by Srensen & van Kuik in (WE, 2011)Propeller case


13/25


Rotor 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


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Prandtls 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 Prandtls 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!


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Betz-Joukowsky limit (1920)

The First Results grounded by Rotor Vortex Theory


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

The First Results grounded by Rotor Vortex Theory

Betz-Joukowsky limit

Glauerts co rrection

+ =

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)


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Development of Rotor Vortex Theory

Goldsteins 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


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Theodorsens 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.


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Progress 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.

Goldsteins circulation

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

Self-induced velocity of the helix

2 4 6 8 101

0.5

0

0.5

1

1.5

2

pitch

Joukowskysapproximation

Black line is analytical solution (Okulov, 2004)


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Major assumptions of the main rotor theories

TheoriesNumber of

blades

Definition of

the pitch in the wake

Circulation

along blade

Betz-Joukowsky limit

(1920) actuator disk non specified non specified

Glauerts optimization

(1935)non specified non specified* non specified

Joukowsky consideration

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

Betz considerationof the rotor II (1919)

infinite without induction velocity Betz type

Prandtl tip correction

(1919)finite (N0) without induction velocity correction of Betz type

Goldsteins solut ion

of the rotor II (1929)finite (N0=1-4) without induction velocity Goldsteins type

Theodorsens considerationof the rotor II (1948)

finite (N0)with induction velocity

of far wakeGoldsteins type

New OS solut ion of rotor II (2008) finite (N0)with induction velocity

in rotor planeGoldsteins type

New OS solution of rotor I (2010) finite (N0)with induction velocity

in rotor planeconstant


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Test of the Betz-Goldsteins and Theodorsens theories

0 5 10 15 20

0

0.2

0.4

0.6

0.8

1

CP

N= 1N= 2

N= 3

N= 5

N= 10

N= 20

N=

0 5 10 15 20

0

1

2

3

4

5

6

CT

1 322

P

wC w I I

1 32 1 2 2T

w 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 2

2T

C w w

Lightly loaded propeller withfinite number of blades

(Betz, 1919; Goldstein,1929)The induction velocity did notinfluence on the pitch of thewake

Theodorsens considerationfor propeller with finitenumber of blades

(Theodorsen,1948)The pitch calculates viainduction velocity in far wake

Points is Glauerts optimization


22/25


Comparison of both rotors (Okulov & Srensen, JFM 2010)

1 32 1

2 2

P

w wC w I I

For the first time the comparison between the famous rotor conceptionswas made by analytical solutions but in both cases the wake expansion

has been neglected.

J oukowsky vortex model of rotor Betz vortex model of rotor

Points is Glauerts optimization


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The centennial is also celebrated in

Russia.

The Russian institute TsAGI, founded by

Joukowsky, makes a special issue onthis topic

Development

of theories of an optimal rotor

by V. Okulov, J. Srensen G. van Kuik.


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



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Concluding remarks

The story of rotor vortex theory

will be continued!

Both Russian and German schools created their vortex theories

independently because World War I and Russian revolutionseparated their collaboration.

100 years of the theory should celebrate from 2012 to 2019!

100 Years of Rotor Vortex Theory

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