Models based on blade element momentum (BEM) theory are commonly used in aeroelastic wind turbine codes. Blade element momentum theory describes how the forces acting on the wind slow down its velocity at the rotor disc by superimposing a so-called induced velocity on the free wind speed. The induced velocity depends on the loading of the wind turbine rotor. When the rotor loading changes, for example due to a change in pitch angle, the induced velocity does not immediately reach a new equilibrium value but instead slowly approaches it. Because BEM theory only predicts the steady-state induction, dynamic inflow submodels are added to account for the delays in the induction response. The time constants of these models can be based on analytic derivations, results of higher-fidelity models or measurements. compared analytically derived time constants for dynamic inflow with time constants derived from experimental data from the NASA Ames Phase VI measurements and numerical results from the free-wake vortex code AWSM . This comparison showed that the free-wake code could predict the dynamic variations in the turbine loading measured in the wind tunnel with high accuracy. The analytical vortex cylinder model, on the other hand, predicted a strong dependency of the time constants on the radial position on the blade. Neither the experiment nor the free-wake code results confirmed this predicted radial dependency. Investigating this disagreement was suggested as future research by .

also investigated the dynamic inflow runs from the Phase VI experiment. They performed computational fluid dynamics (CFD) simulations using the EllipSys code . Similar to Schepers, they found that the observed dynamic inflow effects could be predicted by high-fidelity codes, but the radial dependency of the time constants seemed much less pronounced than expected when deloading the rotor. Moreover the effect changed drastically when pitching from low to high loading, which is also in agreement with observations by Schepers. suggested using a dynamic inflow model with two time constants and scaling the time constants with the wake velocity. The time constants have been obtained from actuator disc simulations.

More recently, compared results from the AWSM code with dynamic inflow measurements from the New Mexico experiment . While the comparison is promising, the force overshoots are not as pronounced as in the Phase VI experiments due to the lower pitching rate relative to the rotor speed. A recent PhD thesis by contains both experimental and numerical investigations of dynamic inflow effects. The experimental investigations use discs with varying porosity , and the numerical model consists of unsteady vortex rings for the near wake and a vortex cylinder model for the far wake .

In the following section, the NASA Ames experiment and the previous analyses by Schepers and Sørensen and Madsen are presented in more detail. Then, a simple vortex cylinder model is introduced in Sect. . In Sect.  results from that model are used to explain the seemingly conflicting findings in previous work.

In the NREL/NASA Ames Phase VI Experiments by , a two-bladed wind turbine with 10 m diameter was placed in the NASA Ames open-loop wind tunnel. One of the turbine blades was instrumented at 30 %, 47 %, 63 %, 80 % and 95 % (corresponding to radii of 1.510, 2.343, 3.185, 4.023 and 4.780 m) with 22 pressure taps at each location. A large number of experiments in both parked and rotating conditions have been performed. The basis for the dynamic inflow investigations in are pitch step experiments at 5 m s-1 wind tunnel speed and at a rotor speed of 72 rpm. In the relevant run, denoted as Q0500000 in , 20 pitch steps between -5.9∘ pitch (heavily loaded rotor, induction factor a≈0.5) and 10.02∘ pitch (unloaded rotor, a≈0) have been performed. The maximum pitch rate was roughly 66∘ s-1. This pitch rate corresponds to 6.6∘ s-1 for a turbine with 100 m diameter because the dynamic inflow time constants are inversely proportional to the rotor radius . After each pitch step, a 15 s waiting time ensured that the flow conditions can reach an equilibrium. The previous analyses, as well as the present work, are based on the force response obtained by averaging the responses to the 20 pitch steps.

The measurement at 5 m s-1 exhibits prominent aerodynamic force overshoots that are characteristic for dynamic inflow. The dynamic response of the axial force after the upward and downward pitch steps is shown in Fig. a and b, respectively. Note that upward pitching refers to the deloading of the rotor and downward pitching to the loading of the rotor. This is based on the convention that the pitch angle is positive towards lower angles of attack. The force is scaled such that the overshoot after the pitch step is 1 and that the steady-state force some time after the pitch step is 0: Fscaled=(F-F2)/(Fm-F2), where F is the axial force, F2 the equilibrium value of the force some time after the pitch step, and Fm the maximum or minimum force overshoot reached during the pitch step, depending on the pitching direction.

Some dynamics in the downward pitching step at the 80 % radial station and to a much smaller extent at the 63 % radial station appeared before the pitching was finished at roughly 0.35 s. The maximum value at the 80 % section has been increased to 1.2; see Fig. b. This was not necessary for the 63 % station because the maximum force overshoot at that station coincided with the remaining stations on the blade. The force response has been previously analyzed by and . This previous work will be introduced in the following.

The engineering model that predicts the strong radial dependency of the time constants in is based on an analytical cylindrical vortex wake model. In this model, which has been presented by , a semi-infinite cylindrical vortex is trailed from the tip radius at the rotor disc. Thus it is assumed that the rotor has an infinite number of blades with a radially constant bound circulation, and the root vortex is neglected. The induction from the trailed vortex cylinder is azimuth independent in the rotor plane but can vary radially. A sketch of the cylindrical wake is shown in Fig. .

The axial induced velocity ui at a radius r on the disc due to such a cylindrical wake is ui=R4π∫x=0xend∫ϕ=02πγt(x)R-rcos⁡ϕx2+R2+r2-2rRcos⁡ϕ3/2dϕdx, where x is the stream-wise coordinate, with xend denoting the length of the wake behind the rotor plane, and ϕ is the azimuthal coordinate. The rotor radius is denoted as R and γt is the tangential component of the wake vorticity, which induces an axial velocity in the rotor plane.

Previously this model was used to analytically derive the time constant for a step change in circulation at the rotor disc, which is transported downstream with the free-stream velocity. In the present work, the position of the step change is varied and the wake velocity vw (see Eq. ), is chosen as the downstream convection velocity. Equation () is integrated numerically.

Figure  shows the induction as function of the dimensionless length of the cylindrical vortex wake behind the rotor. As written above, a constant downwind convection velocity is assumed in the computations. In the unloaded case (a=0), this velocity is vw=5 m s-1 =1R/s. Therefore the dimensionless length of the wake in the unloaded case corresponds directly to the time, since the wake length after 1 s is 1 radius (x/R=1). In the loaded case (a=0.5) the wake velocity is 4 times smaller (see Eq. ); thus, the nondimensional wake length has to be multiplied by a factor of 4 to obtain the time.

The induced velocity buildup clearly shows a large radial dependency at the beginning, but for a larger distance x/R>1.5 of the vorticity step change from the rotor disc, the radial dependency disappears. Then the dynamic induction buildup is similar on the whole disc.

A simple vortex wake model has been used to investigate the question why neither the unsteady NASA Ames Phase VI experiments nor the corresponding high-fidelity simulations showed the expected radial dependency of the dynamic inflow time constants. Results from the model show that the dynamic inflow response initially exhibits a strong radial dependency, but a short time after the step change in loading, the time constants for the different stations become similar, suggesting that two time constants are necessary for accurate modeling of dynamic inflow effects. When using the force response as a basis for time constant investigations, time constants can only be analyzed shortly after the pitch step when the forces start decaying from the maximum overshoot. This delayed analysis obscures some of the fast induction development and can lead to an apparent decreased or even reversed radial dependency.

This indicates that it is difficult to obtain time constants for dynamic inflow models based only on the force time series from measurements, CFD or free-wake vortex code simulations of a wind turbine. These force measurements and computations, though, have a large value in validating the aerodynamic force predictions from engineering models. Computing unsteady induced velocities from CFD simulations is an active field of research, and there is still some uncertainty, especially towards the blade root and tip. Free-wake vortex codes, on the other hand, could be used more easily to investigate dynamic inflow effects and to tune engineering models because the induced velocity is directly available from the computations.