When a wind turbine is yawed, the shape of the wake changes and a curled wake profile is generated. The curled wake has drawn a lot of interest because of its aerodynamic complexity and applicability to wind farm controls. The main mechanism for the creation of the curled wake has been identified in the literature as a collection of vortices that are shed from the rotor plane when the turbine is yawed. This work extends that idea by using aerodynamic concepts to develop a control-oriented model for the curled wake based on approximations to the Navier–Stokes equations. The model is tested and compared to time-averaged results from large-eddy simulations using actuator disk and line models. The model is able to capture the curling mechanism for a turbine under uniform inflow and in the case of a neutral atmospheric boundary layer. The model is then incorporated to the FLOw Redirection and Induction in Steady State (FLORIS) framework and provides good agreement with power predictions for cases with two and three turbines in a row.

This work was authored by the National Renewable Energy Laboratory, operated by Alliance for Sustainable Energy, LLC, for the U.S. Department of Energy (DOE) under Contract No. DE-AC36-08GO28308. Funding provided by the U.S. Department of Energy Office of Energy Efficiency and Renewable Energy Wind Energy Technologies Office. The views expressed in the article do not necessarily represent the views of the DOE or the U.S. Government. The U.S. Government retains and the publisher, by accepting the article for publication, acknowledges that the U.S. Government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to do so, for U.S. Government purposes.

A curled wake is a phenomenon observed in the wake of a wind turbine when the
turbine is yawed relative to the free-stream velocity. When a wind turbine is
yawed, the wake is not only deflected in a direction opposite to the yaw
angle, but its shape changes. The mechanism behind this effect has drawn
attention not only from fluid dynamicists because of the interesting physics
phenomena happening in the wake but also from the wind turbine controls
community who intends to use it to control wind farm flows

It has been shown that wake steering (redirection of the wake through yaw
misalignment) can lead to an increase in power production of wind turbine
arrays

FLOw Redirection and Induction in Steady State (FLORIS) is a software
framework used for wind plant performance optimization

In this work, we describe the aerodynamics of the curled wake, and propose a new, simple, and computationally efficient model for wake deficit, based on a linearized version of the Navier–Stokes equations with approximations. The model is tested and compared to LESs using actuator disk and line models.

Here, we develop a simplified model of the wake deficit considering the
aerodynamics of the curled wake. We start by writing the Reynolds-averaged
Navier–Stokes streamwise momentum equation for an incompressible flow:

the streamwise velocity profile,

the rotational velocity from the shed vortices caused by yawing, and

the rotational velocity due to the blade rotation.

Linearizing the Euler momentum equation for the streamwise component leads
to

The curled wake effect is added to the model by adding a distribution of
counter-rotating vortices to the base flow solution. Figure

Diagram showing a collection of vortices shed from the rotor plane with the corresponding downstream distribution of spanwise velocities due to the superposition of the vortices.

Each vortex is described as a Lamb–Oseen vortex with a tangential velocity
distribution given by

It is important to include wake rotation in the model, because the rotation
will move the wake in a preferred direction. Wake rotation is taken into
account by adding a tangential velocity distribution that is caused by the
rotation inside the rotor area. The tangential induction factor is defined
as

The atmospheric boundary layer can be specified as part of the background
flow. A profile including streamwise and spanwise velocity components can be
specified. The streamwise profile is described by using a power law:

The turbulent viscosity in Eq. (

The presence of the ground will have an effect on the shed vortices. The
ground effect is incorporated by applying a symmetry boundary condition at
the ground

Superposing all the effects mentioned earlier leads to a base flow that
includes all the features presented. The linearized equation allows us to add
features by superposing them onto the velocity components. Notice that, in
this implementation of the model, the base solutions are a function of only
the spanwise directions,

The initial condition for the perturbation velocity,

It is now possible to discretize Eq. (

The proposed numerical method uses a forward-time, centered-space method

This stability criterion is based on a two-dimensional equation and the
equation we are solving is three-dimensional. However, after testing various
conditions, this criterion served as a good guideline for the
three-dimensional version of the equation. After some algebraic manipulation,
it is possible to show that the maximum grid spacing in the streamwise
direction,

In this section, we compare the proposed model to LESs with an actuator disk/line model. Different simulations are used to test the proposed model: (1) a simulation of an isolated turbine using the ADM under uniform inflow, (2) a simulation of an isolated turbine using the ALM under uniform inflow, and (3) a simulation of a turbine using the ALM inside the atmospheric boundary layer under neutral stability conditions.

Here, we compare the results from the model to LESs of a wind turbine under
uniform inflow of a turbine using an actuator disk/line model under uniform
inflow from

Comparison of streamwise velocity contours between a large-eddy
simulation (LES) using an actuator disk model without rotation under uniform
inflow from

Comparison of streamwise velocity contours between a LES using an actuator line model under uniform inflow from

Comparison of axial velocity along a horizontal line between a
large-eddy simulation using an actuator disk model (ADM)

Comparison of axial velocity along a vertical line passing through
the center of the rotor between a large-eddy simulation using an actuator
disk model

Figure

Figure

Figure

Atmospheric boundary layer inflow

Comparison of streamwise velocity contours for the proposed
model

The framework presented can easily be extended by adding more features. As an
example, we present a comparison of the model with an LES of a wind turbine
inside a neutral atmospheric boundary layer with a yaw angle,

To add the effects of the atmosphere to the curled wake model, a vertical
profile of velocity in the streamwise direction is added to the base
solution. Also, a linear spanwise velocity component is added to the base
solution to take veer into account, although this had little effect on the
results presented. The veer profile was chosen as a linear profile that
matched the inflow from the LES results. Figure

Plots across horizontal and vertical lines passing through the
center of the rotor for a LES using a neutral ABL simulation of a turbine in
20

Figure

Figure

In the LES and proposed model, the curled wake shape is produced in the near
wake. As the wake evolves, the turbulence diffuses the curled wake shape, and
eventually it becomes more similar
to a Gaussian wake, as observed by

Wake displacement

FLORIS simulation of two aligned wind turbines yawing the first
turbine 25

FLORIS simulation of three aligned wind turbines yawing the first
turbine 25

It is difficult to track a wake centerline in the curled wake model and the
LES. The curled wake is characterized by a complex three-dimensional
structure and a wake center is not really descriptive of this mechanism,
especially in the near wake. Figure

We now test the proposed model inside the FLORIS framework

First, we run a case of two turbines aligned with the flow and the upstream
turbine is yawed by 25

Now, we present results for three turbines aligned with the upstream turbine
yawed 25

Table

Power percentage improvements for the cases with and without steering for the Gaussian model, curled wake model, and SOWFA.

The curled wake model provides improvements in predicting power gains for more than two turbines in a row. This outcome is because the vortices from the first turbine are propagated downstream. However, because the vortices do not decay in time, the power may be overpredicted.

The key differences between the model and simulations can be summarized as
follows:

The vortices caused by the curl effect in the model do not change their position and do not decay. In reality, these vortices induce motion on each other and are advected by the free-stream flow, which may have a lateral component.

The turbulence model does not take into account the wind turbine wake. It can only take into account the turbulence from the atmospheric boundary layer background flow. This is why the wake decays faster in the large-eddy simulations compared to the model.

The vortices in the model do not decay with downstream distance. In reality, vortices decay because of the radial diffusion of tangential momentum.

The model does not take into account all the nonlinear interactions present in the simulation. For this reason, the model is only able to capture the behavior of the larger scales, and hence, not all the details of the flow (such as the deformation of the vortices) can be captured.

A new model has been proposed to study the aerodynamics of the curled wake.
The model solves a linearized version of the Navier–Stokes momentum equation
with the curl effect added as a collection of vortices with an elliptic
distribution shed from the rotor plane. The main difference between the model
presented and the Gaussian models from

The model has the ability to include several features of the wake including effects due to yaw (“curl”), wake rotation, a boundary layer profile, and turbulence modeling. The model has been implemented and tested to reproduce curled wake profiles. Good agreement is observed when comparing the model to large-eddy simulations of flow past a yawed turbine using an actuator disk/line model. The model was implemented and tested using the FLORIS framework. Good agreement was observed in predicting power extraction by yawing the first turbine in a row of two and three turbines. We observe that the effects of the vortices shed by a yawed turbine propagate for downstream distances longer than the separation between two turbines. This means that a yawed turbine can be used to redirect not only its own wake but the wake of other downstream turbines as well. Also, we note that the shed vortices allow for spanwise velocity components, which are vital when considering wake redirection and wind farm controls. The vortices generated are not limited to only yawing, as they can also be used for tilt and combinations of tilt and yaw. This work sets a foundation for a simplified wake steering model to be used in a more general wind farm control-oriented framework. Future work consists of improving the curled wake model with emphasis on implementing a robust decay model for the vortices and comparing the model to experimental data.

This code is currently under development and not publicly available yet. Information can be obtained in the meantime from the corresponding author.

The authors declare that they have no conflict of interest.

The authors would like to acknowledge Charles Meneveau and Patrick Hawbecker for suggestions on turbulent modeling in the atmospheric boundary layer and Patrick Moriarty for providing wind turbine aerodynamics insights. Simulations for the NREL code SOWFA were performed using the National Renewable Energy Laboratory's Peregrine high-performance computing system. Numerical implementation and plots were done using Python with NumPy, Matplotlib, and SciPy libraries. Edited by: Jens Nørkær Sørensen Reviewed by: three anonymous referees