A wind turbine experiences an overshoot in loading after, for example, a collective step change in pitch angle. This overshoot occurs because the wind turbine wake does not immediately reach its new equilibrium, an effect usually referred to as dynamic inflow. Vortex cylinder models and actuator disc simulations predict that the time constants of this dynamic inflow effect should decrease significantly towards the blade tip. As part of the NASA Ames Phase VI experiment, pitch steps have been performed on a turbine in controlled conditions in the wind tunnel. The measured aerodynamic forces from these experiments seemed to show much less radial dependency of the dynamic inflow time constants than expected when pitching towards low loading. Moreover the dynamic inflow effect seemed fundamentally different when pitching from low to high loading, and the reason for this behavior remained unclear in previous analyses of the experiment. High-fidelity computational fluid dynamics and free-wake vortex code computations yielded the same behavior as the experiments. In the present work these observations from the experiments and high-fidelity computations are explained based on a simple vortex cylinder wake model.

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.

More recently,

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 the NREL/NASA Ames Phase VI Experiments by

The measurement at 5 m s

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.

Scaled axial force measurements from the Phase VI experiment at
5 m s

In the work by

For both the upward and downward pitching case, the time constants based on measurements and on free-wake code computations vary much less with radial position than the analytically derived time constants predicted by the cylindrical wake model.

While the analytically derived time constants predict that the induction develops the fastest at the 95 % section, this cannot be seen in the measurements or AWSM computations. In the downward pitching case, the tip time constants are even found to be largest at the blade tip.

The analytic time constants, which are based on the undisturbed inflow speed, are independent of the
pitching direction, but there is considerable influence of the pitching direction on the dynamics seen in
the measurements (Fig.

Time constants (in seconds) from measurements, AWSM calculations and
a cylindrical wake model from

The NASA Ames Phase VI dynamic inflow measurements were also analyzed by

The CFD computations agree well with the measurements showing a small radial variation in the time constants in general. In the CFD computations of the downward pitch step, loading the rotor, it was found that the tip forces seem to develop the slowest.

In a range of additional pitch step cases with a much smaller pitch angle variation computed by the CFD code, the radial variation in the dynamic inflow time constants also appeared to be much less pronounced than in the actuator disc simulations.

Sketch of cylindrical wake with radius

The engineering model that predicts the strong radial dependency of the time
constants in

The axial
induced velocity

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

The aim of the vortex cylinder computations presented here is to demonstrate
that some of the fundamental questions from the NASA Ames Phase VI experiment
and regarding dynamic inflow in general can be answered based on a very
simple model. Therefore the following assumptions are made, neglecting some
aerodynamic effects:

The vortex strength of the cylindrical vortex wake changes from zero to a constant value at

Two-dimensional unsteady airfoil aerodynamics and dynamic stall are neglected. To justify this, the force response to a step change in angle of attack has been computed using Jones' equations for a flat plate and the relative speeds at the disc. After 0.35 s (corresponding roughly to the pitch step duration), the force response reaches between 90 % (most-inboard section) and 99 % (tip section) of its quasi-steady value, showing that this effect is much faster than the dynamic inflow effect.

Pitching speed is neglected.

The downwind convection velocity of the trailed vortices is constant and equal to the wake
velocity before the pitch step as given in Eq. (

Wake expansion is neglected, which introduces an error in the heavily loaded case.

Figure

Normalized induced velocity at different radial stations depending
on the length of the vortex wake behind the rotor plane. The dashed lines
correspond to a time instant

The induced velocity buildup clearly shows a large radial dependency at the
beginning, but for a larger distance

Based on Eqs. (

Applying Eq. (

Estimated dynamic inflow time constants at different radial stations depending on the length of the vortex wake behind the rotor plane.

Scaled induced velocities predicted by the cylindrical wake model
for the deloading

In

Due to the assumption of a constant wake velocity and trailed vortex
strength, the different pitching directions only differ in how the wake
length translates to the time after the pitch step, as explained at the
beginning of Sect.

This conclusion holds also if the force time series are obtained from CFD
simulations. Another option is to use dynamic inductions from the CFD
solution to estimate the time constants. However, a recent comparison has
shown that there is still considerable uncertainty in induced velocities from
CFD towards the root and tip of the blade

While the approach for loads with respect to the rotor-plane coordinate system such as distributed driving force or thrust is straightforward, a comparison of aerodynamic characteristics is more challenging.

They present a promising methodology for extracting the unsteady induced velocities that can capture the increasing induction towards the blade tip.The induced velocities at the blades or at an actuator disc are directly
available in vortex codes.

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.

Data from the NASA Ames Phase VI measurements are available
from the National Renewable Energy Laboratory (NREL) upon request. A
repository

GP wrote the numerical code. GP and HM interpreted the results. GP wrote the manuscript with support from HM. GP and HM collaborated on the revisions.

The authors declare that they have no conflict of interest.

The work has been carried out within the project “Dansk deltagelse i IEA Wind Task 29 Mexnext III” granted by the Danish funding agency EUDP, grant journal no. 64014-0543.Edited by: Raúl Bayoán Cal Reviewed by: two anonymous referees