This paper presents the integration of a near-wake model for trailing
vorticity, which is based on a prescribed-wake lifting-line model proposed by

This work is based on a coupled aerodynamics model, where the
trailed vorticity effects in the near wake are computed based on a model
proposed by

The coupled model using the modified BEM approach for the far wake has been
proposed by

In the present paper, the iteration procedure of the NWM used by

The dynamic responses to pitch steps and prescribed blade vibrations are
validated by comparing them to results from the more complex free-wake code
GENUVP

This paper is structured as follows: in the next section a short description
of the NWM and a previous implementation of the coupling to a far-wake model
and shed vorticity model are presented. An overview of the current implementation is given in Sect. 3. In Sect.

The structure of the previous implementation

Using this far-wake induction, and the near-wake induction from the previous
time step, a new intermediate velocity triangle VT

The previous implementation of the coupled near- and far-wake model,
as described by

The NWM enables a fast computation of the induction due to the trailed
vorticity behind a rotor blade. The trailed wake can be discretized into
trailed vortex arcs from several positions on the blade, where each arc
consists of a number of vortex elements. The induction at a blade section due
to each vortex element can be computed using the Biot–Savart law, but this
computation is numerically expensive as the influence of each vortex element
on the induction at each blade section has to be determined.

The NWM, which only computes a fraction of the total rotor induction, is
complemented by a modified BEM model for the far wake. The total induced
velocity at a blade section is computed as

The far-wake component

To account for the far-wake dynamics, this work uses the dynamic inflow model
implemented in HAWC2. The model is applied in the same way for axial and
tangential induction, so in the following a scalar notation is used for
simplicity. Two parallel first-order low-pass filters are applied on the
quasi-steady induced velocities

Cambered airfoil in parallel inflow to the chord line. The shed wake corresponds to the time history of the bound circulation.

The sketch in Fig.

The structure of the current implementation of the coupled near- and far-wake
model is shown in Fig.

The weighting factors

The trailed vorticity is no longer based on the quasi-steady bound
circulation

The near-wake induction is computed in an iteration loop, which is
detailed in Sect.

The coupling factor is no longer needed as input but instead continually
updated during the computation, as described by

The trailed vorticity is assumed to follow helix arcs to account for the
downwind convection of the trailed vorticity. To achieve this,

The computation of

Overview of one time step in the coupled near- and far-wake model
used in this work. Relevant equation numbers are included. [P] refers to

The dynamic inflow model described in Sect.

This requires a modification of the constants

The influence of shed vorticity on the bound circulation buildup has to be
considered when determining the strength of the trailed vortices of the NWM.
Joukowski's relation between quasi-steady lift

In this paper, the step response of the circulation is approximated by the
three-term indicial function used by

Any change in bound circulation

Effect of including camber in the unsteady aerodynamics model on effective angle of attack during a step in relative velocity.

Comparison of viscous drag and induced drag during oscillations of a
cambered airfoil parallel to the inflow at 1

The impact of this modification is shown for basic cases of relative velocity
changes in Figs.

In the unsteady circulation computation described in the previous section,
the camber is accounted for through the quasi-steady circulation

The NWM can become numerically unstable depending on the time step, operating
point of the turbine, blade geometry and radial calculation point
distribution

Coarse and fine blade geometry for the NREL 5 MW reference turbine. The coarse definition is a typical geometry definition for BEM-based computations. The finer geometry is smoothed for use in computational fluid dynamics and free-wake codes.

Maximum stable time step depending on the number of points for the
coarse and fine blade geometries of the NREL 5 MW reference turbine. The
points are distributed using a full cosine distribution

The numerical instability which occurs at larger time steps can be explained as follows: the axial induction due to trailed vortices typically reduces the angle of attack at a blade section, which in attached flow leads to a reduced lift. In the original implementation of the NWM the constant circulation trailed during a time step is only depending on the flow conditions at the blade at the beginning of a time step. Therefore, a longer time step will lead to a larger induction and thus a further reduction in lift in the next time step. If the time step is too large, the induction can become large enough to create a negative lift in the next time step that is larger in absolute value than the previous positive lift. This in turn leads to stronger trailed vortices of opposite sign, which will cause even larger induced velocities in the opposite direction, which again leads to stronger vortices.

To stabilize the NWM the balance between trailed vortex strength based on the
sectional circulation and the induced velocities are iterated to equilibrium
in each time step, which removes the need for small time steps to stabilize
the aerodynamics model. The iteration is structured as follows:

The quasi-steady circulation is computed according to Joukowski's law using the velocity triangle at the airfoil section based on the induction from the last iteration.

The unsteady circulation is computed including shed vorticity effects
(see Sect.

This unsteady circulation defines the constant vortex strengths trailed during a time step.

These constant vortex strengths lead to an induction at all airfoil sections.

The new induction is combined from the inductions from step 1 and 4
by applying a relaxation factor:

The BEM model for the far-wake is excluded from this iteration procedure. The AOA and relative velocity used to compute the far-wake induction are the values from the converged iteration in the previous time step. This is accelerating the computation and is feasible because the near-wake effects are on a much faster timescale than the dynamic inflow effects in the BEM model.

In the following, an estimation of the relaxation factor for a blade section
is described. A conservative estimation is based on the least stable case,
which is characterized by the following properties:

One single blade section with one vortex trailing from each side.
Adjacent sections would tend to have similar circulations and therefore
reduce the vortex strengths and the corresponding induction at the blade
section. The trailed vortices on both sides of the section depend only on the
bound circulation

The lift coefficient is linearly dependent on the angle of attack,

No prior trailed vorticity is present. It would stabilize the model, because the induction would be determined not only by the momentary circulation at the section but also by the decaying influence of the wake trailed before. If the model converges in the very first time step, with a given induction at the section from the previous iteration, then the iterations will also converge with prior trailed vorticity.

The helix angle at which the vortices are trailed is assumed to be small. Thus, all the induction due to trailed vorticity is assumed to be axial induction.

With these assumptions, the downwash after a time step

Equation (

Downwash after an iteration as a function of the downwash from the preceding iteration in the case of a single section with trailed vortices.

The intersections of the curves with the blue curve (

Instead of reducing time step and point density until a simulation is stable,
which can lead to time steps orders of magnitude smaller than commonly used
in aeroelastic codes and low spatial resolution, a relaxation factor

In the initial phase of the simulation, the maximum relaxation factor for all
blade sections can be quickly determined by setting

In this section, an approach to accelerate the model is presented. The number
of exponential terms used to approximate the decreasing induction with
increasing distance from the blade in Eq. (

The reduced approximation function is defined as

A comparison of the buildup of induction in time, corresponding to the
integral of the exponential functions, is shown in Fig.

Comparison of induction buildup between full NWM and reduced NWM, depending on the length of a trailed vortex filament with constant circulation.

GENUVP is a potential flow solver combining a panel representation of the
solid boundaries (blades) with a vortex particle representation of the wake.
In the present work, the blades are considered as thin lifting surfaces
carrying piecewise constant dipole distribution (equivalent to horseshoe-type
vortex filaments). Blades shed vorticity in the wake along their trailing
edges and their tips (vorticity emission line). In the model a hybrid wake
approach is followed, which refers to the mixed formulation used in the
representation of the wake. In this formulation, the dipole representation is
retained for the near part (equivalent to horseshoe filaments), while the far
part is modeled by free vortex particles. The near-wake part, consisting of
the newly shed vorticity trailed within the current time step, is modeled as
a vortex sheet also carrying piecewise constant dipole distribution. Within
every time step, a strip of wake panels is released that are in contact with
the emission line. By applying the no-penetration boundary condition at the
center of each solid panel and the Kutta condition along the emission line,
the unknown dipole intensities are determined. Then, at the end of each time
step, the newly shed vorticity is transformed into vortex particles and
all vortex particles are convected downstream with the local flow velocity
(free-wake representation) into their new positions. The layout of the
modeling is shown in Fig.

Layout of the free-wake modeling of a blade: black lines define the blade surface panels, red lines define the wake generated within a time step, and symbols represent freely moving particles.

Since GENUVP is defined as a potential flow solver, the loads need correction
in order to account for viscous effects. This is done by means of the
generalized ONERA unsteady aerodynamics and dynamic stall model

In the case of a flexible blade, flow equations are solved for the deformed blade geometry, while deformation velocities are accounted for in formulating the non-penetration boundary condition.

The GENUVP free-wake code has been thoroughly validated over the past years
against measured data both on wind turbines and helicopter rotors in the
framework of numerous EU-funded projects. Blade loads and wake velocities
comparisons against measurements have been performed on the MEXICO rotor in
the context of Innwind.eu project

In the following section, the effectiveness of the iteration procedure and
the estimation of the relaxation factor are demonstrated for a horseshoe
vortex. Then, in Sect.

To illustrate the efficiency of the iterative implementation, induction
buildups for a simplified case are shown in Fig.

The relaxation factors estimated as proposed in Sect.

Buildup of the downwash for a horseshoe vortex depending on the time step. The NWM tends to be unstable (left) but can be stabilized by iterating to convergence of the downwash (right).

Estimated relaxation factor compared with the lowest stable
relaxation factor from trial and error depending on the number of aerodynamic
sections. The time step is 0.02

In the following Sect.

This paragraph contains an overview of the inherent modeling differences between the different models. It is also detailed how these differences are investigated in the following comparisons of aerodynamic response to pitch steps and prescribed blade vibrations.

The free-wake code uses a lifting-surface approach, while the near-wake model
uses a lifting-line approach. The free-wake code models the airfoils as the
camber line of the airfoil sections (thin airfoil approach) using potential
theory. Thereby, the different sections of the blade have a lift gradient of

The time constants for the shed vorticity model in the BEM-based codes do not contain a correction for airfoil thickness but are instead the approximations for a flat plate originally obtained by Jones. The flat plate approximation agrees with the thin lifting surface in the free-wake simulations. The comparisons of the aerodynamic response to prescribed blade vibrations contain also results from a BEM model with deactivated shed vorticity model. These results are included to evaluate the isolated influence of the shed vorticity modeling and distinguish the dynamic effects of shed and trailed vorticity. In the free-wake code, the shed vorticity is inherently modeled and can not be turned off. In contrast to the fast dynamic effects due to trailed and shed vorticity close to the blade, there are also slow dynamic inflow effects. This term is used here to describe effects that would also be visible in actuator disk simulations where the individual blades are not modeled at all. The slow dynamic effects are modeled directly in the free-wake code and by means of a dynamic inflow model in the other codes. The influence of these effects is compared for pitch steps where the free-wake code results directly model the influence of wake expansion. In the case of the blade vibrations, on the other hand, the main effects occur in the direct wake close to the blade. The influence of dynamic inflow in these cases is very small, which is reflected in the large time constants in the modeling. Wake expansion is also expected to be of minor importance in these cases.

Scaled axial force at different radial positions during and after a
pitch step by 5

Scaled axial force at different radial positions during and after a
pitch step by 5

Another difference between the free-wake code and the BEM-based models is that the dynamic interaction between a blade and the wake of the other blades is only modeled in the free-wake code. This introduces dynamic variations in the pitch step cases that are missing in the BEM-based codes. If this blade–wake interaction were to play an important role in the vibration cases, the agreement between the codes should be better in the high wind speed cases with a larger helix angle than in the low wind cases, where the wake of previous blades is closer to the rotor plane when a blade is passing.

In order to avoid additional uncertainties due to dynamic stall modeling, all
cases have been chosen such that stall is mostly avoided. To obtain this, the
pitch steps are conducted from 5

Pitch steps with stiff blades have been performed where the NREL 5 MW
reference turbine is operating at a wind speed of 8

Left plot: force distribution before the partial pitch step and
50

Figure

The results of the slower pitch step in Fig.

Axial force distributions for a partial pitch comparison at
8

The aerodynamic response to blade vibrations is investigated for normal
operation at 8 and 25

Mode shapes used in the work computations, which are simplified to be purely in-plane or out-of-plane deflections.

Modal parameters and time steps prescribed in the work comparison. The time steps for the first flap were chosen as 180 steps per revolution and thus depend on the rotor speed.

In the BEM and coupled model, the blade section velocities due to the vibrations are added to the relative wind speed. The deflection of the blade and the resulting change in the section positions have been neglected because the amplitudes are small compared to the blade radius. In the free-wake code, not only deflection velocities but also deformation of the blade shape is considered.

Aerodynamic work per oscillation of first flap motion at
8

Aerodynamic work per oscillation of first edge motion at
8

To distinguish the effects of shed and trailed vorticity, the following
comparisons include a BEM computation with a disabled shed vorticity model.
The aerodynamic work during out-of-plane motion according to the first flap
mode shape is shown in Fig.

The in-plane vibrations at 8

With the airfoil polars of the NREL 5 MW reference turbine the agreement
between the codes is not as good in the 8

Aerodynamic work per oscillation of first edge motion at
8

However, the coupled model produces results much closer to the free-wake code
close to the blade tip than the BEM model. At 25

Figure

Aerodynamic work per oscillation of second flap (top) and edge
(bottom) motion at 0.25 m amplitude at 8

The BEM model results compare similarly well with the GENUVP results as for
the first modes. Because the frequencies of the second modes are higher the
shed vorticity model is more important in these cases. The importance of the
trailed vorticity model at higher frequencies does not increase by the same
amount, because the higher reduced frequencies affect the buildup of the
unsteady bound circulation (see
Eq.

For easier evaluation of the force response differences, the aerodynamic work
can be expressed in terms of a damping ratio of a respective blade mode.
Because the computations have been based on prescribed purely in-plane and
out-of-plane structural mode shapes, these dampings do not correspond to any
aeroelastic blade modes. For a system with a single degree of freedom with the
modal mass

Aerodynamic damping estimations for the first in-plane vibrations at
8

In the out-of-plane prescribed vibration cases investigated, the trailed
vorticity reduces the aerodynamic work. Further, a previous study by

In this paper, several modifications of a coupled model consisting of a trailed vorticity model for the near-wake and a BEM-based model for the far wake have been presented and validated. Results from the coupled model are compared to free-wake panel code and a BEM model to evaluate the benefits and limitations of the added trailed vorticity modeling.

Estimated logarithmic decrements [%] corresponding to the
aerodynamic work of first in-plane vibrations at 8

It has been shown that the acceleration of the model by reducing the number of exponential functions in the trailing wake approximation from two to one is possible with negligible effect on the results. The approach presented here does not change the steady results predicted by the NWM.

An iteration scheme to stabilize the model has been presented. It applies a relaxation factor that is computed dynamically based on the blade discretization and the operating point of the turbine. To evaluate the computed relaxation factors, minimum necessary relaxation factors have been determined by trial and error and the estimated factors are found to be conservative. The iterative process enables stable computations without the need for very small time steps and reduces oscillations of the near-wake induction.

The 2-D shed vorticity modeling, based on thin airfoil theory, has been extended by including the unsteady effects on the bound circulation. Further, it has been found that it is necessary to include airfoil camber in the modeling of the influence of varying inflow velocity on the dynamic angle of attack to obtain good results if the direction of vibration is close to parallel to the inflow direction.

A comparison of pitch step responses of the NREL 5 MW reference turbine using the coupled near- and far-wake model, a BEM model based on the aerodynamics model in HAWC2 and the free-wake panel code GENUVP has been presented. The trailed vorticity modeling in the coupled model gives results closer to the free-wake code than the BEM model during the pitching motion, and for a slow pitching rate a clear improvement is seen in the computation of the overshoot. Fast pitch rates resulted in oscillations due to the motion of the wake in the free-wake code, which could not be achieved in the coupled model due to the prescribed wake assumption. The response to a partial pitch of the outer half of the blade demonstrated the cross-sectional aerodynamic coupling, which will have an influence on the load distribution in the presence of trailing edge flaps.

The coupled model agreed better than the BEM model with the free-wake code in all prescribed vibration cases investigated. The main improvement due to the trailed vorticity is found close to the tip of the blade, even in the case of the higher modes investigated. The work response to the edgewise vibrations has been found to be difficult to model if the direction of vibration is close to parallel to the inflow direction. The results in this case compare much better if no drag forces are computed. If drag is included, the coupled model still compares well with the free-wake code close to the blade tip, but there are larger deviations in the results of all models further inboard. In general, the simulations agreed better for out-of-plane vibrations than in-plane vibrations.

The implementation of the coupled near- and far-wake model presented here delivers promising results and will be further investigated and validated against computational fluid dynamics results and measurements in future work. In particular, the more accurate prediction of aerodynamic work for edgewise vibrations is considered to be important for stability analyses and load predictions due to the low aeroelastic damping typically associated with these vibrations.

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The authors declare that they have no conflict of interest.