The actuator line (AL) is a lifting line (LL) representation of aerodynamic surfaces in Eulerian computational fluid dynamics (CFD) applications. It allows simulating the interaction between the atmosphere and wind farms, as it captures all the important flow features of fully resolved rotors, at a fraction of the computational cost. However transferring a LL into the CFD domain requires dispersing the concentrated blade forces of the LL over a certain region – most commonly in the form of a Gaussian projection – to avoid causing numerical instabilities. This force smearing leads to the formation of a viscous core in the released vorticity, which subsequently reduces the induced velocity at the blade . Lower induction implies larger angles of attack and thus increased blade forces. Especially in regions presenting large load changes, as around the root and tip of the blade, does the AL thus overestimate the forces.

 – following the approach proposed by  – presented a correction to the AL that combines the fast and dynamic near-wake model by with a viscous core model to recover the missing induction. With the correction, the AL truly functions as a LL, which was verified over the entire operational wind speed range of modern turbines as well as in yaw and for dynamic pitch steps . The numerical stability of the correction was not challenged by any of those flow cases – not even by extreme inflow turbulence.

The only disadvantage of the new smearing correction is its computational cost. Though it is incorrect to apply conventional tip corrections to ALs – they correct actuator discs for missing discrete blades – their low cost makes them attractive. In this paper we present different methods that reduce the computational cost of the new correction to that of existing corrections without jeopardizing the stability or accuracy of the method.

Computing the missing induction requires re-evaluating the velocity contribution from each previously released vortex element at each time step. The velocity contribution from a single trailed vortex at some point along the blade is obtained by integrating along the vortex line 1u⋆=∫0∞fϵδũdl. Here δũ is the velocity induced by an infinitesimal element δl of a vortex line and fϵ represents the smearing factor, originating from the presence of a viscous core in the released vorticity. Integrating over the vortex length is equivalent to integrating over time, as at each time step an element is released. Originally, the near-wake model by provides directly the integrated velocities ũ.

Note that the integration only covers the near-wake region, from 0 to π/2.

This section compares the influence of the different speed-up methods presented in Sect.  with the original results of . All results are obtained with exactly the same computational set-up as presented in Sect. 3 of the same paper. The AL models the NREL 5 MW under uniform inflow. For the inflow wind speed specific turbine parameters refer to Table 2 in .

Figure  compares the force distributions for the NREL 5 MW at two wind speeds obtained with the speed-up methods presented in Sect.  to those obtained with the original model. The influence of reducing the wake length is only shown for a wind speed of 25 ms-1, but it is similar at lower wind speeds. With increasing wind speed, the peak force moves clearly from the tip to the root whilst the smearing correction ensures the smooth behaviour towards the blade ends. From pure visual inspection there is no change in the forces when applying any of the speed-up options. To highlight their impact, only the change in the force distributions with respect to the unmodified model is shown in Fig.  – here additionally the results for a wind speed of 14 ms-1 are presented. Reducing the wake length has a negligible effect on the forces as does reducing the inner loops, except close to the root. Fixing the smearing factor additionally to cutting the loops has the largest influence. However even at 25 ms-1 the deviation does not exceed 17 Nm-1. With respect to the local force the difference remains below 1 %.

A full result overview – the impact of the speed-up methods on thrust and power as well as their influence on the computational cost per blade – is given in Table . Results are shown for rotors discretized by 9 and 19 blade sections. Firstly, the greatest change in thrust or power across all methods occurs when fixing the smearing constant, yet never by more than 0.8 % and only at the highest wind speed. The positive influence of cutting the inner loops on performance grows with increasing resolution. However, the largest reduction in the computational cost comes from limiting the wake length and ultimately fixing the smearing factor. With the latter approach the longest of all smearing correction iterations lasted 8×10-4 s.

The smearing correction by recovered the lifting line behaviour of the actuator line, however at a larger computational cost than existing actuator disc tip corrections. This paper presents different methods for reducing the cost of the smearing correction to those levels. The number of wake elements manifests itself as the key cost driver. Reducing the wake length therefore significantly reduces the computational cost without negatively impacting the blade forces. The greatest speed-up comes from utilizing the near-wake model to avoid recomputing the contributions from each element at each time step, leading to a fall in the cost of at least 98 %. This is accompanied by changes in thrust and power of maximally 0.8 % and 0.7 %, respectively. Still, with respect to the great gain in performance this is acceptable and lies well within CFD simulation uncertainty. Furthermore, the new, faster method avoids any form of bookkeeping, greatly simplifying the implementation of the smearing correction. It also remains a dynamic correction that takes into account how the trailed vorticity changes over time and moves away from the blades. This faster and simpler version of the smearing correction is openly available .