Vortex theory of the ideal wind turbine

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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

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Two definitions of the ideal rotor
Betz (1919)
Joukowsky (1912)
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!
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Lifting-line theory for rotor with finite number
of blades
A (rotor plane) Kutta - Joukowsky Theorem
B (wake approximation) From Helmholtzs vortex
theorem it results that From symmetry
considerations, neglecting expansion, it can be
shown that
and
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Models of far wake for ideal rotors
Goldstein circulation (Theodorsen, 1948)
Characteristics of flow with helical
vortices (Okulov, JFM, 2004)
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Velocity triangles determining geometry of the
wakes
From definition of u? (Okulov, JFM, 2004)
From definition of w (Goldstein, 1929)
The model assumption
The model assumption
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Equilibrium motion for both far-wake models
From definition velocity for flows with
helical symmetry we can write
Definition of Goldstein circulation G(r) Uniform
motion of the helical sheets in axial direction
with velocity
Definition of the vortex core size Uniform
axial motion of all helical vortices in vortex
core with unknown vortex core radius e and
constant velocity
by using dimensionless variables
give us an equation for definition of the vortex
core size
give us an integral equation for definition of
G(r)
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Approximate attempts of simulating the wake motion
Fragment of Goldsteins solution (1929)
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
Measurements of Theodorsen (1945)
Approach by Moore Saffman (1972)
The ring term was introduced by Joukowski in
1912.
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Final solutions for equilibrium motion of the
wakes
Goldstein circulation functions for Nb 3
Definition of the vortex core size based on
self-induced velocity by Okulov (JFM, 2004)
Points Tibery Wrench (1964) Lines Okulov
Sørensen (2008)
Elimination of singularity
Vortex core radius e
Averaged interference factor in far wake
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Comparison (1) of maximum power coefficients
Solution of Betz rotor (OkulovSorensen,2008)
Solution of Joukowsky rotor (present)
Difference between the power coefficients
Mass coefficient
Axial loss factor
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Comparison (2) of maximum power coefficients
Approximation with Prandtls tip correction
Solution of Betz rotor (OkulovSorensen,2008)
Difference between the power coefficients
Mass coefficient
Mass coefficient
Axial loss factor
Axial loss factor
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Summary

• 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 Goldsteins
  function (Okulov Sorensen, WE, 2008)
• The optimum power coefficients evaluated by
  approximation with Prandtls tip correction
  correlates well with the original analytical
  solution using Goldsteins circulation for Betz
  rotor
• For all tip speed ratios the Joukowsky rotor
  achieves a higher efficiency that the Betz
  rotor