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

Abstract

Vortical impulse theory is used to investigate the relationship between turbine thrust and the near-wake velocity and vorticity fields. Three different hypotheses regarding the near-wake structure allow the derivation of novel expressions for the thrust on a steadily rotating wind turbine, and these are tested using stereoscopic particle-image velocimetry (PIV) data acquired just behind a rotor in a water channel. When one assumes that vortex lines and streamlines are aligned in a rotor-fixed frame of reference, one obtains a PIV-based thrust estimate that fails even to capture the trend of the directly measured thrust, and this failure is attributed to an implicit assumption that most of the generated thrust does useful work. When one neglects the axial gradients of radial velocity, the PIV-based thrust estimate captures the measured thrust trend, but underpredicts its magnitude by approximately $33\,\%$ . The third and most promising physical proposition treats the trailing vortices as purely ‘rolling’ structures that exhibit zero-strain rate in their cores, with the corresponding thrust estimates in close agreement with direct thrust measurements. This best-performing expression appears as a correction to the classical thrust expression from momentum theory, possessing additional squared-velocity terms that can account for the high-thrust regime of turbine operation that is typically addressed empirically.

A revision of blade element/momentum theory for wind turbines in their high-thrust region

“…What is not clear is whether the particular form of the wake vorticity term is sufficient, and ignoring the effects of the shed vorticity is justified. The thrust is consistently under-predicted (Figure 4), suggesting that additional shed vorticity terms are needed in the axial momentum equation, as argued by Limacher et al (2022). We also note that Ebert and Wood (2002) found the tip vortex had sufficient negative angular momentum at runaway to balance the kinetic energy deficit in the remainder of the wake.…”

“…In considering the high-thrust region, the main difference between the blade and cascade elements is that the wakes of the former can also contain shed vorticity from the radial variation in the blade loading. Thus, it is not possible to distinguish between the MT revision due to Limacher et al (2022) and the present revision on the basis of cascade analysis. One further comment is in the order: the present revision of MT makes no assumptions about the flow downwind of the rotor.…”

“…We now revise the right side of Eqs 1, 4 based on the cascade simulations of Golmirzaee and Wood (2023). The revision is different from that of Limacher et al (2022), who simplified the equations by assuming the shed vorticity was strain-free. They did not make a direct connection to blade element drag, and no consideration was given to the angular momentum equation.…”

“…Unfortunately, there have been very few subsequent experiments on the high-thrust region that provide sufficient detail for BEMT revision. One of the few that does was by Limacher et al (2022), who measured the thrust and all three velocities in the wake of a small model turbine by particle image velocimetry, for values of λ above and below λ R = 7.9. They proposed a revision to the conventional momentum equation that will be described later.…”

The effect of nacelle-to-rotor size on the wake of a miniature wind turbine

“…For utility-scale turbines, on the other hand, this ratio is typically less than 0.05 [2,[9][10][11]. Similarly, the ratio of nacelle length to rotor diameter is typically greater than 0.2 for miniature wind turbines [4][5][6]12], whereas it is less than 0.1 for utility-scale turbines [11].…”