The actuator line (AL) was intended as a lifting line (LL) technique for computational fluid dynamics (CFD) applications. In this paper we prove – theoretically and practically – that smearing the forces of the actuator line in the flow domain forms a viscous core in the bound and shed vorticity of the line. By combining a near-wake representation of the trailed vorticity with a viscous vortex core model, the missing induction from the smeared velocity is recovered. This novel dynamic smearing correction is verified for basic wing test cases and rotor simulations of a multimegawatt turbine. The latter cover the entire operational wind speed range as well as yaw, strong turbulence and pitch step cases. The correction is validated with lifting line simulations with and without viscous core, which are representative of an actuator line without and with smearing correction, respectively. The dynamic smearing correction makes the actuator line effectively act as a lifting line, as it was originally intended.

The actuator line (AL) technique developed by

However, in contrast to LL vortex formulations, the blade forces are dispersed in the flow domain – most commonly in form of a Gaussian projection – to avoid numerical instabilities. A length scale – also referred to as smearing coefficient – controls this force redistribution, the lower limit of which is linked to the grid size by numerical stability requirements

Distribution of the tangential velocity component in a plane orthogonal to an infinite vortex line (along

The major contributions of this paper are as follows:

The development of a tuning-free, dynamic and numerically robust smearing correction, which is fully coupled to the AL model.

A theoretical proof of the force smearing – vortex core equivalence.

Proof of the vortex core inheritance in trailed vorticity.

The confirmation of the missing velocity assumption by comparing LL simulations with/without viscous core and AL results with/without correction.

The equivalence between the velocity field induced by an AL and a viscous vortex can be derived directly from the incompressible Navier–Stokes equations. This proof follows the approach by

Note that in the

The discretized incompressible Navier–Stokes equations are solved using DTU's CFD code EllipSys3D

Numerical box domain with a structured mesh and uniform spacing around the rotor at its centre. Only every eighth grid point is shown.

The numerical domain for the rotor simulations is discretized in a verified, standard manner

The sensitivity of the rotor thrust to the domain size, time step and grid size is explored in Fig.

Thrust sensitivity of the NREL 5-MW AL simulations at a 8

The in-house solver MIRAS has been employed to perform the free-wake lifting line simulations. MIRAS is a multi-fidelity computational vortex model, which is mainly used for predicting the aerodynamic behaviour of wind turbines and their wakes. It has been developed at DTU over the last decade and been extensively validated for small to large size wind turbine rotors by

The free-wake vortex method essentially models the wake of a wind turbine using a bundle of infinitely thin vortex filaments. To avoid numerical singularities, a viscous core must be introduced, which represents a more physical distribution of the velocities induced by each vortex filament, desingularizing the Biot–Savart law near the centre of the filament. The velocity induced by each one of the elements is obtained directly by evaluating the Biot–Savart law, and by summing the velocity induced by all filaments, the total wake induction is obtained as follows:

A viscous core model is applied to emulate the effect of viscosity by changing the vortex core radius as a function of time

For the sake of the present study, two different approaches to compute the angle of attack have been followed.

Inviscid (LL), where the non-regularized Biot–Savart law is used to compute the induction from the wake filaments at the quarter-chord location. This is the standard method used in a lifting line solvers.

Viscous (LL

Applying the velocity correction methodology introduced in Sect.

Trailed vorticity path. The blade rotates in the

The total missing induction at a blade section

Following the lifting line formulation of the NWM shown in Fig.

Finally the missing velocities computed in Eq. (

Interpolate the velocity vector

Compute the helix angles

Combine the CFD velocities with the respective correction from the previous time step

Compute the smearing factor

Determine the angle of attack and velocity magnitude from

Strictly, the influence of the shed vorticity on the velocity at the AL should be removed as remarked by

Compute the velocities from the newly released vortex element

At the first iteration of each time step, advance the previous elements in time.

Compute the velocity correction at the current time step

Update the velocity at the sections with some form of relaxation

Repeat steps 5–9 until convergence is reached.

To verify the smearing hypothesis (Sect.

Definition of the wing test cases with either a rectangular or elliptic planform. Vortices are trailed in-between sections and the actuator line forces are computed and exerted at the sections' centres.

Input parameters common to both rectangular and elliptic wing simulations.

The theoretical predictions of the velocity field are achieved by representing the vortex system of Fig.

The vortex smearing hypothesis assumes the trailed vorticity inheriting the smeared velocity field from the bound vortex, which was confirmed theoretically by

Figure

Velocities induced perpendicular to a rectangular wing predicted by an actuator line (AL) without correction and by vortex segments with a viscous core (Vortex) with different smearing parameters. Velocities are shown along lines cutting the bound and trailed vortices at right angles and

As mentioned in Sect.

Analytical (An.) and corresponding model prediction of the velocity correction for varying force smearing (

The coupling between velocity correction and the flow domain is verified by comparing the corrected downwash at an elliptical wing to the theoretical expectation. The downwash should be constant along the wing and is given by

Figure

Downwash at an elliptical wing predicted by AL simulations with different smearing factors and smearing correction. The CFD components of the velocities are shown (dashed) as well as the total downwash incorporating the correction (solid). The theoretical value acts as reference.

The validity of the smearing hypothesis and its correction in rotor applications is demonstrated with simulations of the NREL 5-MW turbine

Input parameters for the NREL 5-MW simulations.

Figure

Normal and tangential forces on the NREL 5-MW blades at 8

Angle of attack with/without smearing correction on the NREL 5-MW blades at 8

The comparison of AL and LL is summarized in Fig.

Normal and tangential forces on the NREL 5-MW blades at 25

Local thrust and power coefficients along the NREL 5-MW blades at different wind speeds predicted by AL simulations with/without smearing correction and LL with/without core (

An overview of the simulation inputs and results for the NREL 5-MW in uniform inflow. Results are grouped by blade/grid resolution. For the actuator line (AL) and lifting line (LL) the simulation time step

As the smearing correction does not include yaw effects – the wake is assumed to advect normal to the rotor plane – we tested its influence at yaw angles

Normal

The pitch step is defined as

Here only the tangential force response after a

Inputs defining the pitch step.

Normalized tangential force response at two blade sections (middle and tip) of the NREL 5-MW following a pitch step of

Highly turbulent inflow should challenge the numerical stability of the new smearing correction by introducing strong and abrupt changes in the angle of attack. Comparing simulations with and without inflow turbulence should also reveal whether turbulence alters the nature of the correction. Figure

Time-averaged normal and tangential forces on the NREL 5-MW blades at an 8

Variation of normal and tangential forces on the NREL 5-MW blades at an 8

The actuator line was intended as a lifting line technique for CFD applications. In this paper we prove – theoretically and practically – that smearing the forces of the actuator line in the flow domain leads to smeared velocity fields. For the typical Gaussian force projection, the widely known Lamb–Oseen

The current implementation of the smearing correction relies on heavy bookkeeping. In future versions the latter will be removed without jeopardizing stability or accuracy, making it suitable for wind farm simulations in realistic atmospheric flows. Potentially, the correction might also enable accurate rotor simulations at lower discretization.

All data and parts of the code covering the smearing correction are available upon request. Commercial and research licenses for EllipSys3D can be purchased from DTU.

The equations governing the fixed-wake approach underlying the smearing correction (see Fig.

Direction vector from the vortex element to a control point along the blade:

ARMF was responsible for writing the paper, except for Sect.

The authors declare that they have no conflict of interest.

We would like to acknowledge DTU Wind Energy's internal project “Virtual Atmosphere” for partially funding this research. Furthermore many thanks to senior researcher Mac Gaunaa for his insights on vortex aerodynamics and senior researcher Niels Troldborg for his input on actuator line simulations/modelling (both from DTU Wind Energy). Thanks also to Ang Li for his help regarding the near-wake model.

This paper was edited by Alessandro Bianchini and reviewed by two anonymous referees.