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Vortex theory of the ideal wind turbine

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. G( r ). G. R. blade span. 0. blade span. R 0. Two definitions of the ideal rotor. Betz (1919). Joukowsky

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Vortex theory of the ideal wind turbine

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  1. Vortex theoryof the ideal wind turbine Jens N. Sørensen and Valery L. Okulov Department of Mechanical Engineering, Technical University of Denmark, DK-2800 Lyngby, Denmark

  2. G(r) G R blade span 0 blade span R0 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!

  3. Lifting-line theory for rotor with finite number of blades A (rotor plane): Kutta - Joukowsky Theorem B (wake approximation): From Helmholtz’s vortex theorem it results that: From symmetry considerations, neglecting expansion, it can be shown that: and

  4. Models of far wake for ideal rotors Goldstein circulation (Theodorsen, 1948): Characteristics of flow with helical vortices (Okulov, JFM, 2004)

  5. 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:

  6. 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 radiuse 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)

  7. Approximate attempts of simulating the wake motion Fragment of Goldstein’s 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.

  8. Final solutions for equilibrium motion of the wakes Definition of the vortex core size based on self-induced velocity by Okulov (JFM, 2004) Goldstein circulation functions forNb = 3 Points: Tibery & Wrench (1964) Lines: Okulov & Sørensen (2008) Elimination of singularity Vortex core radius e Averaged interference factor in far wake

  9. Comparison (1) of maximum power coefficients Solution of Betz rotor (Okulov&Sorensen,2008) Solution of Joukowsky rotor (present) Difference between the power coefficients Mass coefficient “Axial loss factor”

  10. Comparison (2) of maximum power coefficients Solution of Betz rotor (Okulov&Sorensen,2008) Approximation with Prandtl’s tip correction Difference between the power coefficients Mass coefficient Mass coefficient “Axial loss factor” “Axial loss factor”

  11. 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 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”

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