Abstract
Abstract. Using the concept of impulse in control volume analysis, we derive general expressions for wind turbine thrust in a constant, spatially uniform wind. The absence of pressure in the impulse equations allows for their application in the near wake, where the flow is assumed to be steady in the frame of reference rotating with the blades. The assumption of circumferential uniformity in the near wake – as applies when the number of blades or the tip speed ratio tends to infinity – is needed to reduce these general expressions to the Kutta–Joukowsky (KJ) equation for blade-element thrust. The present derivation improves upon the classical derivation based on the Bernoulli equation by allowing the flow to be rotational in the near wake. The present derivation also yields intermediate expressions for thrust that are valid for a finite number of blades and trailing vortex sheets of finite thickness. For the circumferentially uniform case, our analysis suggests that the magnitudes of the radial velocity and the axial induction factor must be equal somewhere on the plane containing the rotor, and we cite previous studies that show this to occur near the rotor tip across a wide range of thrust coefficients. The derivation reveals one further complication; when deriving the KJ equations using annular control volumes, the existence of vorticity on the lateral control surfaces may cause the local blade loading to differ from the KJ equation, but the magnitude of these deviations is not explored. This complication is not visible to the classical derivation due to its neglect of vorticity.
Highlights
Blade-element theory (BET) for wind turbines uses the fundamental assumption that the forces acting on the elements comprising the rotor blades are given by the Kutta– Joukowsky (KJ) theorem
This paper describes the application of vortical impulse theory to determine the thrust on a steadily rotating wind turbine in a steady, spatially uniform wind
We assume that the vorticity field is steady when viewed by an observer rotating with the blades, so the vortex lines and streamlines in the near wake are aligned in this frame
Summary
Blade-element theory (BET) for wind turbines uses the fundamental assumption that the forces acting on the elements comprising the rotor blades are given by the Kutta– Joukowsky (KJ) theorem. The KJ equations are generally introduced as assumptions in BEM theory, they can be derived using the unsteady Bernoulli equation. This was shown as early as Glauert (1935) and can be found in Sørensen (2016) and van Kuik (2018). Assuming the trailing wake to be infinitely thin to permit the use of the Bernoulli equation, and representing the rotor’s power extraction by a sudden drop in pressure across the rotor plane, sectional thrust can be expressed as follows: 2π. We offer an improved derivation of the KJ equation for blade-element thrust that avoids this contradiction. 2. Section 3 manipulates the general CV equations furnished by impulse theory to yield simplified integral expressions of wind turbine thrust.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have