An impulse-based derivation of the Kutta–Joukowsky equation for wind turbine thrust

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.

On the relationship between turbine thrust and near-wake velocity and vorticity

“…By applying vortical impulse theory to the CV in figure 1, Limacher & Wood (2021) derived the following expression for the thrust on a steadily rotating turbine in an unbounded, incompressible and steady free-stream flow (neglecting viscous terms on ): where the two components of vorticity are defined as The only additional assumption inherent to the derivation of (2.4) is that the flow field within appears steady in a frame of reference rotating with the blades. It is worth emphasizing that the underlying impulse theory is an exact expression of the conservation of linear momentum, and thus the stated assumptions are rather minor restrictions on the generality of (2.4).…”

“…Noca (1997) and Wu, Ma & Zhou (2015). Limacher & Wood (2021) employed impulse theory to derive general expressions for the thrust on a steadily rotating turbine, with no assumptions about the far wake whatsoever. One of their key results serves as the starting point for the present analysis.…”

Wake Expansion and the Finite Blade Functions for Horizontal-Axis Wind Turbines

“…Neither the angular momentum equation nor the conventional axial momentum equation for the flow through the rotor contain v. It does appear in the impulse-based derivation of the axial momentum equation by Limacher and Wood [3] but is removed by the assumptions of continuity through the rotor and the near-zero value of the expansion integral. Wood and Limacher [16] show that the part of the expansion integral covering the rotor is equal to the contribution to the axial thrust on the rotor due to the pressure acting on the expanding upwind flow, ∆C T .…”

“…Their axial and angular momentum depend on the average axial (u) and azimuthal (w) velocities of each streamtube whereas the lift and drag depend on the velocities at the blade, u b and w b . In addition, the radial velocity (v) associated with flow expansion, is of the same magnitude as u and thus needs to be considered in determining the accuracy of BEMT, Limacher and Wood [3].…”

“…The expectation of large effects of expansion arise from its association with a significant v at the rotor. Limacher and Wood [3] showed that the flow immediately upwind of the rotor is constrained by the "expansion integral":…”