The force term F in Eq. () is obtained using F=G⋅(ftip⋅(FLeL+FDeD)), with the lift and drag forces shown in Fig. , and defined as FL=12clUmag2cls,FD=12cdUmag2cls, with cl and cd being the lift and drag coefficient, Umag the sampled velocity magnitude in the blade reference frame, c the chord width, ls the length of the actuator segment, G the Gaussian kernel and ftip the tip correction.

The same unmodified airfoil coefficients were used as shown in . These data were obtained from 2-D experiments without rotation in a wind tunnel, and therefore they do not include the stall delay due to boundary layer stabilizing effects such as Coriolis and centrifugal forcing, which enhance the lift of the airfoil near the root of the blade. As shown in using unmodified airfoil data results in problems predicting the blade forces in the root region for high wind speed flows. They seem to handle the moderate wind speeds which are in the focus of this work fairly well. Hence, despite their shortcomings, the unmodified airfoil data are used throughout this work. Then, the forces in Eqs. () and () are projected onto the blade reference frame using the unit vectors eL and eD in Eq. ().

While the Glauert tip correction ftip was originally intended to represent the otherwise absent tip vortices in the actuator disk model, it still proves advantageous for the ALM at lower resolutions as shown in . Due to the relatively low resolution, the shed tip vortices from an ALM are much larger than the ones observed experimentally. Hence, the induction caused by the simulated vortices is weaker than in reality, and the Glauert tip correction permits us to compensate for it .

Finally, in order to avoid spurious oscillations around the point of the inserted force, the punctual force is distributed using a kernel function G⋅(⋅). As done in previous works such as or this work adheres to a normal distribution with the distribution width ϵ=σΔx, with σ being the distribution width of the normal function and Δx the cell width.

This work is realized within the open-source framework OpenFOAM (version 2.2.2)

OPENFOAM (Open source Field Operation And Manipulation) is a registered trade mark of OpenCFD Limited, producer and distributor of the OpenFOAM software via https://www.openfoam.com/ (last access: 1 March 2018).

The implementation was validated against EllipSys3D and verified against the Model Experiment in Controlled Conditions (MEXICO) and New Model Experiment in Controlled Conditions (NEW MEXICO) experiment in . The MEXICO rotor is a three-bladed rotor with a radius R=2.25 m. It rotates at a constant rpm of 424.5 min-1 for the three different flow cases with U∞={10,15,24} m s-1; hence, their respective tip speed ratios are 10.0, 6.7 and 4.2. As shown in , the chord-based Reynolds number Re=U∞c/ν varies from 0.4×106 for the low-velocity case towards the hub up to 0.6×106 for the high-velocity case in the tip region. The power coefficients for the observed cases are CP={0.28,0.36,0.19}. A comparison of the axial profiles of the velocity components can be seen (Fig. ) showing that both codes reproduce very similar results in the near wake further away from the rotor. Apart for the high-velocity case where 3-D effects become important, an excellent agreement can be observed for the other two cases.

For the high-velocity case (U∞=24 m s-1), the vortex sheets shed from the blades become visible by the oscillations in the axial velocity component Ux.

When refining the grid using the actuator line method, the distribution parameter ϵ has to be adjusted to obtain a global torque 〈T〉 close to the reference value Tref. In the following, only the case for U∞=15 m s-1 will be examined as the other cases in served as extreme cases for determining how the model behaves at its limits.

Instead of relying on a constant ϵ/Δx for different grid resolutions, this work adapts ϵ/Δx depending on the grid resolution or number of cells across the rotor diameter N=2R/Δx. The results are shown in Eq. (). A confidence interval of ±1 % was established around the reference torque value Tref. Through iterations, an optimal distribution parameter is found to fall in this range.

The lower bound for the distribution parameter here is ϵ=1.7Δx for the sake of the numerical stability of the applied method. Other frameworks applying a different numerical discretization can go even lower, e.g. in . By doing so, it can be seen that the best solution in terms of global torque for a resolution of N=D/Δx=32 is off by around 4 % in Fig. .

As a general trend, it can be seen that ϵ/Δx has to be increased with increasing resolution. This stems from the fact that by refining the mesh with a constant ϵ/Δx, the punctual induction caused by the blade would be too high and eventually the torque would be below the reference value, e.g. for ϵ=2Δx for N≥64. By contrast, when having a very low resolution, a constant ϵ distributes the force too widely, causing a lower induction around the rotor resulting in an overestimation of the torque, e.g. for ϵ=2Δx for N≤48.

The optimal distribution parameter ϵ/Δx found in Fig.  is now shown in dependence on the grid resolution N=D/Δx in Fig. . It seems as if this value would eventually reach an asymptotic limit for higher resolutions. Looking at a more theoretical approach in suggests that the optimal distribution width ϵ lies between 0.14 and 0.25 of the chord c whereas in this case for D/Δx=128 the ϵ/c lies between 0.5–8.9 depending on the span-wise location. The observation made by is backed by , where ϵ/c falls in the same range. But it should be kept in mind that use a much higher grid resolution, allowing ϵ/Δx≥4 and in the case of even ϵ/Δx≥5.

The curvature in Fig.  also confirms the findings of that keeping the relation ϵ/Δx=const while increasing the resolution is not a very good solution. While suggests choosing the smallest possible distribution width ϵ in order to minimize interactions with the vortical structures (ϵ/Δx=1), the observations made here fall more in line with work such as , suggesting ϵ/Δx to be adapted to the physical model in order to distribute the force over a meaningful length scale.

For an excerpt of the resolutions presented in Fig. , the radial profiles of the velocity components can be found in Fig. . It can be seen that the method seems to converge towards a solution when refining the mesh. As shown in Fig.  the lowest resolution at N=32 over-predicts the torque by distributing the force to widely, which is also reflected in the low axial induction downstream at x/R=0.13. Despite following the trend of the experimental values well, the method seems to converge towards radial profiles which are especially off in the tip and hub region where the strongest vortices are shed. These are limitations intrinsic to the ALM, which is less apparent when using high-fidelity approaches such as full rotor simulations . In order to ameliorate the results at the tip, a non-isotropic kernel could be investigated .

In Fig.  the shed vortical structures can be seen in dependence on the grid resolution. While the root vortex is rather diffuse, a clear tip vortex can be noticed. It is interesting to note the vortices shed around mid-span due to the suboptimal choice of the airfoils of the blade causing a sudden change in circulation.

In order to estimate the resolution necessary to obtain tip vortex radii as seen in the MEXICO experiment, the vortex radii are shown in Fig. . The vortex radius rcore is defined as the limit containing 99 % of the circulation. The Gaussian distribution is used as an approximation for the vorticity distribution within the vortex. This assumption is normally applied for low Reynolds number flows, while this case exhibits a Reynolds number of ReΓ=Γ/ν=220×103. As these findings are related to the vortex dynamics of the flow ReΓ is used instead of Re defined earlier. Nevertheless, this approximation is used in order to be able to draw an analogy between the experimental and the numerical results. It holds fairly well when comparing the Gaussian distribution and the vorticity for N=128 as shown in . Hence, by assuming a Gaussian distribution for the vortices in the MEXICO experiment, a corresponding distribution parameter ϵ can be deduced as shown in Fig. .

This would necessitate a resolution of N≥4096 for the presented case, which would result in a computational grid beyond any justifiable computational scope. Full rotor calculations as conducted by allowed us to obtain tip vortices of rcore/R≈0.012 for N≈900 in the tip region, which corresponds very well to results in Fig. . Another result for the vortex radius can be found in , where for ϵ/Δx=1 and N≈244 in the tip region a vortex core radius of rcore/R≈0.055 was found. Despite the radii in this work and the references being calculated based on three different methods, the results fall within the same range.

In Fig.  the longitudinal evolution of the turbulence intensities can be seen. There is a stronger decay for higher turbulence intensities, which was also found in . In that work EllipSys3D was compared to a solution based on OpenFOAM, and it was found that over the same longitudinal distance of 10R an absolute difference in the turbulence intensity of 48 % and 44 % occurred for each framework respectively. This stands in a stark contrast to the 4 % in this case for the high turbulence-intensity case. This huge decay, which is even more significant for EllipSys3D, necessitates approaching the introduction of the turbulence close to the turbine for high turbulence-intensity cases .

An important aspect when imposing a synthetic turbulence as boundary conditions of a Computational Fluid Dynamics (CFD) simulation is respecting the Nyquist–Shannon sampling theorem as also mentioned by . Hence, a study considering different ratios between the grid resolution of the synthetic turbulence and the simulation was undertaken. It is found that the higher the computational resolution is compared to the one of the synthetic turbulence, the less the turbulence intensity decays in longitudinal direction. While the criterion of Nyquist–Shannon states that the resolution of the computational domain should be dx/Δx>2, this work uses the ratio of dx/Δx=2.5, with dx being the cell width of the synthetic field and Δx the cell width of the computational mesh.

When taking the case for TIsyn=5 %, it is interesting to note that while the resolved TI (green dashed line) is around 4.2 % at the rotor position x/R=0, a huge part of the difference in relation to the imposed turbulence falls in the SGS model with 〈TIres〉+〈TIsgs〉=4.8 % and finally just a relatively small amount of the turbulent intensity or turbulent kinetic energy is “lost” by numerical dissipation.

Despite the fact that the computational grid respects the Nyquist–Shannon criterion for signal sampling in respect to the synthetic grid, immediately at the inlet a part of the turbulence falls in the sub-grid range. Due to the numerical dissipation caused by the differencing schemes and turbulence modelling, the energy cascade hands down its energy to lesser scales than the resolved ones.

It should be kept in mind, that the turbulence intensity the rotor model is experiencing through velocity sampling is the resolved turbulence intensity TIres, and the sub-grid turbulence intensity TIsgs is therefore only felt indirectly, by an augmentation of the effective viscosity. When looking at the fraction of the resolved turbulent kinetic energy over the total turbulent kinetic energy, it can be seen that the resolved scales exceed 96 %, which lies well above the criterion of 80 % proposed by . For the flow case with U∞=15 m s-1, the total simulation run-time ttot results in roughly ttot⋅U∞/(10R)≈5.5 flow through times. The synthetic turbulence field is large enough that it does not need to be recycled during one simulation.

In Fig.  the effects of the ambient turbulence on the turbine wake are shown. While there are no noticeable impacts for the low-turbulence case with TIsyn=0.1 %, the beginning of strong non-linear interactions can be observed for TIsyn>0.1%. For TIsyn=15 % the inflow turbulent structures seem to outgrow the structures created by the wind turbine.

In Fig.  it can be seen how the strength of the vortical structures of the ambient fluid increases with higher turbulence intensity up to the point for Tsyn=15 % where its amplitude almost equals the one emitted by the rotor model. In Fig.  the impact of the rotor presence on the energy spectrum can be seen. The wavenumber κp relating to the frequency of a blade passage (3 times rotor frequency) obtained by κp=2π3f/U∞ shows a very distinct peak and its higher harmonics at the multiples of κp.

As the velocity time series obtained from the simulations do not exhibit periodicity, the Welch method is used to generate the energy spectra.

It is interesting to note that the distinct peaks in the spectra occur at the wavenumber relating to the frequency of the blade passage and its harmonics. The harmonics are caused by the strong excitement of the fluid by the blade passage and its interaction with the non-linear term in the NS equations. As the blade forces and hence the strength of the tip vortices are very comparable, the peaks are very similar among the different cases for -0.4≤x/R≤0.2. The higher the turbulent kinetic energy content stemming from the ambient flow, the faster the peaks are dampened and blend into the ambient flow. For example there is almost no discernible effect by the blade at x/R=0.4 for TIsyn=15 % while for TIsyn=0.1 % the velocity oscillations are still very noticeable. Although it is of lesser amplitude, the upstream region is also under the influence of the distinct blades to a certain extent.

When looking at the vertical planes in Fig. , the influence of the sheared flow can be seen by a higher velocity deficit in the wake on the lower half of the rotor. In the vertical plane in Fig. , it can be seen that there is an increase in the vorticity magnitude towards the ground. While the increase appears to be rather subtle, it is shown in that the TI increases significantly towards the ground as expected in a shear layer flow. Looking at the energy spectra in Fig.  reveals a similar picture as shown above for the case of homogeneous isotropic turbulence. As the blade forces and hence the strength of the tip vortices are very comparable, the peaks are very similar among the different cases for -0.4≤x/R≤0.2. Due to the dissipation caused by the ambient turbulence, these peaks dampen at a different pace as seen at x/R=0.4.

While before and at the rotor position for -0.4≤x/R≤0 the peaks remain very distinct, vortical structures by the ambient fluid and emitted by the blade cause the injected peaks to dampen and distribute energy to adjacent wavenumbers as seen clearly for x/R≥0.2. Depending on the level of the ambient turbulence the peak gets attenuated up to a point where it almost completely blends in with ambient turbulence as seen for TIx,z/R=0=15 % at x/R=0.4. This relates to the observation made earlier that ambient structures are almost as important as the structures emitted by the blade.

This means that in the near wake in a turbulent flow with an ambient turbulence intensity of TIx,z/R=0≥10 %, the velocity fluctuations at x/R=0.4 already seem to have only a weak relation to the injected turbulence by the rotor but a much stronger one to the ambient turbulence. This means that for this kind of flow, an actuator disk method would probably also be sufficient when looking at the flow characteristics beyond x/R=0.4.