Abstract
When investigating limits of rotor aerodynamic models, the Betz limit serves as a solid marker of an upper limit which no model should be able to exceed. A century ago Joukowsky (1912) proposed a rotor aerodynamic model utilizing a rotating actuator disc with a constant circulation. This model has since then been the subject of much controversy as it predicts a power performance that for all tip speed ratios exceeds the Betz limit and which goes to infinity when the tip speed ratio goes to zero. Recently, it was demonstrated that the Joukowsky model is fully consistent with the inviscid Euler equations and that the apparent inconsistency partly can be explained by the lack of viscous effects (Sprensen and van Kuik [4]). However, even including a term to account for the effect of viscosity at small tip speed ratios, the model still predicts a power yield that exceeds the Betz limit. In the present work we study in detail, using a CFD actuator line model, the flow behavior for rotors at small tip speed ratios. It is shown that the excessive swirl appearing towards the rotor center at small tip speed ratios generates vortex breakdown, causing a recirculating zone in the wake that limits the power yield of the rotor. The appearance of vortex breakdown has a similar effect on the flow behavior as the vortex ring state that usually appears at higher tip speed ratios. Limits to where vortex breakdown might occur with tip speed ratio and rotor loading as parameter are investigated and presented in the paper. The limits found correspond to well-known criterion for vortex breakdown onset for swirling flows in general. By applying a criterion for vortex breakdown in combination with the general momentum theory, the power performance always stays below the Betz limit.
Highlights
Independent of the various theories, the upper limit for the aerodynamic power performance of wind turbine rotors tends to converge towards the Betz limit through the normal range of operating tip speed ratios
A century ago Joukowsky (1912) proposed a rotor aerodynamic model utilizing a rotating actuator disc with a constant circulation. This model has since been the subject of much controversy as it predicts a power performance that for all tip speed ratios exceeds the Betz limit and which goes to infinity when the tip speed ratio goes to zero
In the present work we study in detail, using a CFD actuator line model, the flow behavior for rotors at small tip speed ratios
Summary
Independent of the various theories, the upper limit for the aerodynamic power performance of wind turbine rotors tends to converge towards the Betz limit through the normal range of operating tip speed ratios. Published under licence by IOP Publishing Ltd study (see Sørensen and Mikkelsen [2]) an axi-symmetric CFD actuator disc model formulated in streamfunction-vorticity variables, was used to investigate model limits associated with the onset of vortex breakdown, which appear in the wake of an actuator disc operating at low tip speed ratio. The general momentum theory combined with known criteria for limits in swirling flows was compared to the axi-symmetric CFD prediction of rotor-induced vortex breakdown in the wake. The result showed that vortex breakdown can be predicted numerically as part of the wake of low tip speed ratio operating rotors. It was further shown that a simple criterion for vortex breakdown used in combination with the general momentum theory, by limit, the power so that it always stays below the Betz limit. The wake is represented by a system of distinct helical tip and root vorticies instead of an axi-symmetric vortex sheet
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