3.1. Angles and forces

The idea of the helix approach is to define a pitch angle $\boldsymbol {\varTheta }$ in the global reference frame, also referred to as the global pitch, and letting the direction of that pitch rotate. We define the global pitch as

(3.1)\begin{equation} \boldsymbol{\varTheta} = \begin{bmatrix} \varTheta_0 \\ \varTheta_{tilt} \\ \varTheta_{yaw} \end{bmatrix} = A \begin{bmatrix} 0 \\ \sin\left(\beta\right) \\ \cos\left(\beta\right) \end{bmatrix}, \end{equation}

where $A$ is the magnitude of the pitch and $\beta$ is the angle between the direction of the pitch and the $z$-axis. The multiblade coordinate transformation transforms coordinates from the rotating reference frame of the blade to the global reference frame (Bir Reference Bir2008). Frederik et al. (Reference Frederik, Doekemeijer, Mulders and van Wingerden2020) showed that applying the inverse transformation to (3.1), i.e. from the static global reference frame onto the rotating reference of each blade $b$ yields

(3.2)\begin{equation} \varTheta_b = A \sin\left(\varPsi_b +\beta\right) = A \sin\left(\varPsi_b +\omega_{e} t\right) , \end{equation}

where $\varTheta _b$ and $\varPsi _b$ are the pitch and azimuth of blade $b$, respectively. Switching the sign of $\omega _{e}$ will lead to a clockwise rotation. However, note that this case will not be discussed any further in this paper, since Frederik et al. (Reference Frederik, Doekemeijer, Mulders and van Wingerden2020) found that a counter-clockwise rotation yields higher power gains. The change in pitch angle results in a varying angle of attack $\alpha$ and therefore differences in lift within one rotation of the rotor. The differences in lift result in a radial force and moment.

There are two parameters that control the pitch angle: the amplitude $A$ and the excitation frequency $\omega _{e}$. The relation of the excitation frequency to the velocity of the flow and the turbine length scale is expressed by the Strouhal number

(3.3)\begin{equation} {St} = f_{e}D/V_{0}, \end{equation}

where by convention the frequency $f_{e}$ is used instead of the angular frequency $\omega _{e}$. Alternatively, the Strouhal number can be expressed by means of the rotor speed $\omega$ and the tip speed ratio as

(3.4)\begin{equation} St = \frac{2f_{e}\lambda}{\omega}. \end{equation}

This relation makes it possible to define the excitation frequency based on the rotor speed if Strouhal number and tip-speed ratio are known. The optimal tip-speed ratio is usually dictated by the design of the turbine (see, for example, Jonkman et al. (Reference Jonkman, Butterfield, Musial and Scott2009)) and Frederik et al. (Reference Frederik, Doekemeijer, Mulders and van Wingerden2020) found a value of ${St} = 0.25$ to yield the highest power gains. Thus, $f_{e}$ can be determined without any knowledge of the flow, as it only depends on the rotational speed of the turbine which is set by the controller. A further discussion of the Strouhal number in the case of multiple turbines can be found in § 4.

Later in this section, we examine the phase of the helix. To do so we must know in which direction the wake is deflected. We assume that the deflection is dominated by the radial force due to the helix approach. Thus, the direction of the deflection is the same as the direction of the force exerted onto the flow. We note that $\omega _e \ll \omega$ due to (3.4), hence the pitch angle varies much slower than the turbine rotates. Therefore, we assume that we can examine the blade forces in a quasi-steady state. Furthermore, we neglect any influence of the drag and only consider the lift of the airfoil, and assume that the angle of attack $\alpha$ is below the onset of stall. Finally, we ignore any changes in induced velocities and only consider changes in forces.

The normal and tangential forces acting on the fluid are proportional to $F_n \propto -C_L(\alpha )\cos \left (\phi \right )$ and $F_t \propto -C_L(\alpha )\sin \left (\phi \right )$, where $\phi$ is the angle between the relative velocity $V_{rel}$ and the plane of rotation. A diagram of the angles and forces is shown in figure 3(a). A Taylor expansion of $C_L$ around $\alpha _0=\phi$ and the geometric relationship $\alpha =\phi -\varTheta _b$ yields

(3.5)\begin{equation} C_L(\alpha) = C_L(\alpha_0) + (\alpha-\alpha_0)\left.\frac{\partial C_L}{\partial \alpha}\right|_{\alpha_0} = C_L(\phi) - \varTheta_b\left.\frac{\partial C_L}{\partial \alpha}\right|_{\phi}. \end{equation}

The forces $\boldsymbol {F}$ exerted on the flow by all blades in the global frame of reference are then proportional to

(3.6)\begin{equation} \boldsymbol{F} \propto \sum_{b=0}^{N_b-1}\begin{bmatrix} -C_L(\alpha) \cos\left(\phi\right) \\ C_L(\alpha) \sin\left(\phi\right)\cos\left(\varPsi_b\right) \\ C_L(\alpha) \sin\left(\phi\right)\sin\left(\varPsi_b\right) \end{bmatrix}.\end{equation}

By inserting (3.2) and (3.5) and after some manipulation, the total force can be shown to be

(3.7)\begin{equation} \boldsymbol{F} \propto{-}N_b\begin{bmatrix} \cos\left(\phi\right)C_L(\alpha_0) \\ C_1\sin\left(\omega_et\right)\\ C_1\cos\left(\omega_et\right) \end{bmatrix}, \end{equation}

where $C_1 = A({\partial C_L}/{\partial \alpha })({\sin \left (\phi \right )}/{2})$. Thus, the radial component, i.e. the force in the rotor plane, has a magnitude proportional to $N_bC_1$ and points in the opposite direction of the global pitch $\boldsymbol {\varTheta }$. The conservation of momentum dictates that the flow is accelerated in the direction of the force. We define the angle of the direction as $\varphi$. To keep the definition of $\varphi$ in line with the definition of the azimuth angle $\varPsi$, we define $\varphi$ to be zero when the wake is deflected in positive $z$-direction, i.e. upwards and $\varphi$ to increase counterclockwise. Therefore, $\beta$ and $\varphi$ are related as

(3.8)\begin{equation} \varphi = {\rm \pi}-\beta = {\rm \pi}- \omega_et. \end{equation}

The relation of force, moment, global pitch and their respective angles are shown in figure 3(b,c). A more detailed derivation of the forces can be found in Appendix A. There, we also derive the radial component of the moment induced by the helix approach. As stated previously, we will not consider the effect of the moment in the remainder of this paper. We assume that the force causes the wake deflection and that the moment only affects the wake mixing.

Figure 3. Schematics of angles, velocities, forces and moments acting on the flow. (a) Top view of a cross-section of a blade at azimuth $\varPsi =0$, showing the local angles and forces. (b) View of the rotor looking downstream, showing azimuth and global pitch as well as force and moment due to the helix approach. (c) Top view of the rotor showing the total moment and force on the flow. Gray shaded area illustrates the deformed wake.

3.2. Tip vortices

An important aspect of the wake recovery is the breakdown of the tip vortices. The tip vortices shield the wake from the ambient turbulence. Only after the breakdown of the tip vortices does the wake recovery begin, as discussed, for example, in Lignarolo et al. (Reference Lignarolo, Ragni, Scarano, Simão Ferreira and van Bussel2015). Therefore, it is of great importance to the process of wake recovery, when the breakdown occurs. The vorticity contours of a snapshot of the flow field of the high-resolution case with greedy and helix operation are shown in figure 4. Recall that the high-resolution case has a resolution of $\Delta x=D/64$, in order to resolve the tip vortices. The resolution corresponds, e.g. , to the coarse resolution used by Ivanell et al. (Reference Ivanell, Mikkelsen, Sørensen and Henningson2010), where tip vortex instabilities were studied. Compared with the tip vortices in greedy operation, the tip vortices of a turbine in helix operation exhibit some important differences. Due to the changing pitch of the blade, the vorticity generated by the blade changes with time. This is also visible in the tip vortices. Furthermore, the deformation of the wake stretches and squishes the vortices. This also changes the distance between the vortices and thus the strength of their interaction see, e.g. , Ivanell et al. (Reference Ivanell, Mikkelsen, Sørensen and Henningson2010). Both aspects lead to an earlier breakdown, as can be seen in figure 4. In the helix wake, the vortices are merged where the blades were pitched to stall and thinned where the blades were pitched to feather. Furthermore, the deformation of the wake is clearly visible. While the classical pairing and leapfrogging of vortices, as reported, for example, by Tophøj & Aref (Reference Tophøj and Aref2013) or Sarmast et al. (Reference Sarmast, Dadfar, Mikkelsen, Schlatter, Ivanell, Sørensen and Henningson2014), and the following breakdown is apparent in the greedy case, the breakdown of the helical wake is not as orderly. A pairing of vortices is barely visible since the vortices have already begun to merge. The deformation also leads to changing distances between the vortices. In summary, we found that the tip vortices of the helix approach vary in thickness due to the varying pitch angle, which, in turn, leads to the merging of vortices. Furthermore, the helix approach leads to a clear helical deformation of the wake. Finally, we could not observe the leap-frogging of vortex pairs, which is a typical feature of tip vortices in regular operation. A more thorough investigation of the interplay of deformation and vortices is omitted here for the sake of brevity.

Figure 4. Vorticity contour of a single turbine in (a) greedy and (b) helix operation.

3.3. Wake deflection

The helix approach leads to a rotating radial force exerted onto the flow as shown in § 3.1. This force must also lead to an acceleration of the fluid in the radial direction, causing the helical shape of the wake illustrated in figure 4. However, we show that this radial force not only deforms but also deflects the wake. Recall, that we define deflection as an imposed translation of the mean wake centre due to the forces exerted by the turbine, whereas meandering is the stochastic fluctuation of the wake centre position due to the ambient turbulence, and deformation is a change in wake shape without translation of the mean wake centre position. The differentiation between deformation and deflection is of high relevance since deflection potentially reduces the overlap of the wake and the downstream turbine, while deformation may lead to increased wake mixing but does not necessarily reduce overlap. In established yaw-based wake-steering approaches, the force exerted onto the wake is stationary and so is the deflection of the wake. However, in the helix approach, the radial force rotates, thus the deflection also rotates. Therefore, we have to extend the concept of deflection to a rotating frame of reference, in order to analyse the rotating wake of the helix. Thus, we describe deflection as a mean translation of the wake centre in coordinates rotating with the frequency of rotation of the helix $f_e$. We want to point out that the distance between the wake centre and the cross-stream position of the turbine is the same in the static and rotating frames of reference and is therefore easily computed from the static frame of reference, which is why we will focus our analysis on this quantity. To quantify the deflection, we compute the wake centres from the instantaneous streamwise velocity in cross-stream planes downstream of the turbine, by fitting a two-dimensional Gaussian velocity deficit with the width of the rotor (Quon, Doubrawa & Debnath Reference Quon, Doubrawa and Debnath2020). For the latter, we employ the samwich toolbox (https://ewquon.github.io/waketracking). The wake centre distance $d$ is then defined as the distance of the wake centre to the centre of the rotor plane.

The distribution of $d$ at $3D$, $5D$ and $7D$ downstream of the turbine for all single-turbine cases is shown in figure 5. Without the helix, the median of the wake centre distance increases downstream due to the naturally occurring wake meandering, which is in line with other experimental and numerical studies (Kang, Yang & Sotiropoulos Reference Kang, Yang and Sotiropoulos2014; Howard et al. Reference Howard, Singh, Sotiropoulos and Guala2015; Bastankhah & Porté-Agel Reference Bastankhah and Porté-Agel2017). The distribution is heavily skewed to the right. We find a slight increase in spread with a higher inflow velocity (i.e. lower tip-speed ratio), in (a), but that finding does not persist further downstream. If the helix approach is applied, the mode of the distribution is shifted to the right and the distribution has a higher variance. In the case with a lower tip-speed ratio, the shift to the right is less pronounced, indicating that the helix approach does not deflect the wake as much as when the turbine is operated at the optimal tip-speed ratio. In figure 5(d–f) we can examine the influence of increased turbulence intensity on the wake centre distance. We see that mode of the distribution of G1I$10$ is shifted to the right and that the distribution has a much longer tail compared with G1I$5$. For case H1I$10$, the mode is similar to that of G1I$5$, yet the density of occurrences in the range of $0.2D$ to $0.4D$ is larger in the first plane. However, in the planes further downstream the mode is increased if the helix is applied. Overall, we observe that the influence of the helix approach is stronger at lower turbulence intensities. This is in line with other studies, which have found a reduced effect of wake-steering approaches at high turbulence intensities (Kheirabadi & Nagamune Reference Kheirabadi and Nagamune2019). Wake meandering is dominated by ambient turbulent structures larger than the rotor diameter (Larsen et al. Reference Larsen, Madsen, Thomsen and Larsen2008; España et al. Reference España, Aubrun, Loyer and Devinant2011). Therefore, the wake meandering, that is, the spread of the distribution, should increase with an increase in turbulence length scale, shown in figure 5(g–i). Indeed we find that the distribution has a significantly higher spread in all three planes, with and without the helix approach. Nevertheless, the helix approach shifts the mode of the distribution to the right. Thus, the density of samples with a low wake centre distance is decreased. Therefore, we can deduce that the helix approach is still able to deflect the wake centre effectively. It should be noted that with stronger turbulence intensity, the identification of the wake centre tends to become more difficult (Doubrawa et al. Reference Doubrawa2020). Yet, since the results of the cases without the helix match the expectations, we take our findings to be of reasonable quality.

Figure 5. Distributions of wake centre distance at cross-stream planes $3D$ (a,d,g), $5D$ (b,e,h) and $7D$ (c,f,i) downstream of the turbine. Panels (a–c), (d–f) and (g–i) show different tips speed ratios, turbulence intensities and turbulent length scales, respectively.

Due to meandering, the position of the wake centre is a symmetric bivariate normal random variable. If deflection is present, the wake centre will not be zero in the mean, if we take the mean in the rotating frame of reference. Therefore, the wake centre distance $d$ should be distributed according to a Rice distribution:

(3.9)\begin{equation} f(d\mid\nu,\sigma) = \frac{d}{\sigma^2}\exp\left(\frac{-(d^2+\nu^2)} {2\sigma^2}\right){\rm I}_0\left(\frac{d\nu}{\sigma^2}\right), \end{equation}

where $f$ is the probability density function with two parameters $\nu$ an $\sigma$ and ${\rm I}_0$ denotes the modified Bessel function of the first kind of order zero (Rice Reference Rice1944, Reference Rice1945). The magnitude of the meandering is characterised by $\sigma$ whereas the deflection determines $\nu$. In the case without the helix, there is no deflection, thus $\nu =0$, in which case the Rice distribution becomes a Rayleigh distribution. If the ratio of deflection to meandering is high, the Rice distribution approximates a normal distribution, which explains the results observed for the helix cases in figure 5.

It is also possible to estimate the parameters $\sigma$ and $\nu$ from the computed wake centre distances. The results of such an estimation, performed according to Koay & Basser (Reference Koay and Basser2006), are shown in figure 6. Note that this method for estimating parameters has much higher accuracy than general methods for estimating parameters of distributions, but does not always converge. The method is prone to failure for very small values of $\nu$. First, we show the meandering parameter $\sigma$ in figure 6(a–c). In all these cases, $\sigma$ grows with downstream distance. Overall, the cases with and without helix behave somewhat similarly, while the meandering without helix is slightly lower. The comparison of the deflection parameter $\nu$, however, shows a clear distinction between the helix and non-helix cases. The non-helix cases have low values for $\nu$ throughout the wake whereas the deflection in the helix cases increases with downstream distance. Finally, figure 6(g–i) compares deflection with meandering. If the helix is applied, the ratio of $\nu /\sigma$ is almost constant after $3D$ downstream at a value around 2. Without the helix, the ratio is significantly lower and a slight decrease with distance is observable. As pointed out before, in theory, without the helix $\nu =0$, which is not the case in our results. We attribute this offset to the difficulty of determining the wake centre correctly and the low sensitivity of the Rice distribution to changes in $\nu$ if $\nu$ is close to zero. Therefore, we have included complementary computations of the meandering parameter based on a Rayleigh distribution, that is, assuming that $\nu =0$. It is shown in figure 6(a–c). If $\nu =0$ is enforced, the meandering parameter of the wake is estimated to be very similar between helix and non-helix cases and only slightly higher if the helix is applied.

Figure 6. Estimated parameters of the Rice distribution of the wake centre distances. Missing data are due to the lack of convergence of the method to estimate the parameters, which is the case for low ratios of $\nu /\sigma$. Dotted lines represent estimations of $\sigma$ based on a Rayleigh distribution.

In conclusion, we find that the helix approach induces a significant deflection of the wake centre, which reduces the overlap of the wake with potential downstream turbines and consequently increases the kinetic energy available to the downstream turbines. We also find that the helix increases meandering in the far wake, although this effect is not very pronounced.

3.4. Spectra

In figure 7 we show the premultiplied spectra in streamwise, azimuthal and radial direction at three locations in the wake for cases G1V9 and H1V9. For the other cases, the results yield no substantial differences. The spectra are obtained at four locations in each plane, half a rotor diameter away from the centre of the plane along the $y$- and $z$-axis in positive and negative direction. The signals are transformed using Welch's method, with 50 segments and an overlap ratio of 0.4 (Welch Reference Welch1967). All resulting spectra from a plane are then averaged. We find that the presence of the turbine increases energy in the fluctuations across the spectrum, especially in the streamwise direction. Furthermore, we observe that the presence of the turbine leads to higher energy contained in azimuthal fluctuations at frequencies at ${St}\approx 0.7$. Other studies, such as Howard et al. (Reference Howard, Singh, Sotiropoulos and Guala2015) and Heisel, Hong & Guala (Reference Heisel, Hong and Guala2018), have found the meandering of the root vortex to fall into this frequency range. The effect is strongest in the near wake and decays further downstream. Finally, if the helix approach is used, a notable peak can be found near the helix frequency of $St=0.25$. In almost all cases it contains the most energy of all frequencies. However, in case G1V9, we can also observe a maximum of energy at that frequency in the streamwise fluctuations at $x=5D$, although not as distinct as that of case H1V9. A variety of studies such as Okulov et al. (Reference Okulov, Naumov, Mikkelsen, Kabardin and Sørensen2014) and Howard et al. (Reference Howard, Singh, Sotiropoulos and Guala2015) have found a frequency in that range to be the naturally occurring frequency of wake meandering. Thus, we conjecture that the helix approach is most efficient if applied with the same frequency since it amplifies the naturally occurring instability of the wake and thus leading to a more rapid breakdown. This agrees with the observation in figure 6, where we observed slightly higher meandering if the helix is applied. The other frequencies show a slight decrease in energy. This is due to the position of the probes used to measure the spectrum. Due to the deflection of the wake, the probes are outside of the wake, thus reducing the turbulence intensity and leading to a decrease in the energy contained in the fluctuations.

Figure 7. Premultiplied spectra of velocity fluctuations in planes $1D$, $3D$ and $5D$ downstream of the turbine. Spectra are measured at 4 points, 1 radius up, down, left and right from the centre in the plane, and then averaged. The grey dashed line marks the frequency of the helix.

3.5. Wake mitigation

Although § 3.3 showed that the helix approach also leads to wake steering, Frederik et al. (Reference Frederik, Doekemeijer, Mulders and van Wingerden2020) stressed the improvement in wake mixing, which is also suggested by the earlier tip-vortex breakdown observed in § 3.2. In this section, we examine the effect of wake mixing and wake steering on the mitigation of wake effects. Wake effects include a reduction in kinetic energy in the flow and increased turbulence intensity. Wake mixing leads to entrainment of kinetic energy into the wake from the surrounding flow, while wake steering reduces the overlap of the wake with a potential downstream turbine. To distinguish the effect of reduced overlap and wake mixing, we utilise a meandering frame of reference. The meandering frame of reference follows the wake. The mean flow is obtained by translating the cross-stream planes so that the wake centre determined in § 3.3 is in the centre of the plane. The resulting flow field is then averaged. Since this frame of reference can only be established a posteriori, we are limited to first-order statistics, i.e. averages.

To distinguish between the energy available due to wake mixing and due to wake deflection and meandering, we show the kinetic energy in the rotor area, $\bar {E}_{K,R}$, both in the static frame of reference and the meandering frame of reference in figure 8. By comparing G1V$9$ and H1V$9$, we find that the application of the helix leads to a faster increase in kinetic energy downstream of the turbine in both frames of reference. In the static frame of reference, the flow has regained approximately 60 % of the kinetic energy of the undisturbed flow at $x=5D$ if the helix is applied, whereas we observe only little above $40\,\%$ in case G1V$9$. A comparison of the kinetic energy between the two reference frames shows that a large amount of the additional kinetic energy present in the static frame of reference in case H1V$9$ is due to the reduced overlap due to wake meandering and deflection. In contrast, the difference between the two frames of reference is much smaller in case G1V$9$. The comparison with the cases with lower tip-speed ratio shows that the turbines extract significantly less energy from the flow, as to be expected. Interestingly, the helix approach leads to a higher extraction of energy compared with G1V$13$. Yet, due to increased entrainment of energy, the rotor disk contains more energy at $x=3D$, if the helix is applied. We can also see that the increased mixing does not have so much of an effect in the region up to $x=5D$. Since the difference between the two frames of reference is negligible in case G1V$13$, we conclude that the reduced overlap has little to no effect in the case without the helix. However, in H1V$13$ we consistently find a difference between the two frames of reference. Since we know from figure 6, that the meandering is virtually the same in G1V$13$ and H1V$13$, this difference in the cases must stem from the deflection of the wake. In figure 8(b) we observe stronger increases in kinetic energy in the region $2D$ to $4D$ downstream of the turbine in cases G1I$10$ and H1I$10$ compared with the respective baseline cases. In addition, the helix approach leads to more kinetic energy due to the reduced overlap in the static frame of reference. However, with higher ambient turbulence, the wake is not deflected as much. Therefore, the flow in case H1I$5$ contains more kinetic energy in the rotor area far downstream. The two effects balance each other at around $x=6D$. Finally, we examine the influence of the length scale of turbulence in figure 8(c). We can also observe more available kinetic energy in the region between $x=2D$ and $x=5D$. However, in comparison with H1I$10$, the helix approach is able to deflect the wake better, as was also shown in figure 6, and thus increases the amount of kinetic energy in the rotor area in the static frame of reference even more than in case H1L$40$.

Figure 8. Kinetic energy of the mean flow in the rotor area at cross-stream planes with a distance of $1D$. Lines marked with crosses indicate averaging in the meandering frame of reference while the lines marked with a plus sign represent energy averaged in the static frame of reference.

To gain a more thorough understanding of the wake recovery process the energy transport is analysed in the following. Based on the Reynolds-averaged momentum equation of an inviscid fluid (see, e.g., Stull Reference Stull1988, p. 90),

(3.10)\begin{equation} \bar{u}_j\frac{\partial{\bar{u}_i}}{\partial{x_j}} ={-}\frac{1}{\rho}\frac{\partial{\bar{p}}}{\partial{x_i}} - \frac{\partial{}}{\partial{x_j}}\overline{u_i' u_j'}, \end{equation}

a transport equation for the kinetic energy of the mean flow, $\overline {E_K} = \tfrac {1}{2}\bar {u}_i^2$ can be derived by multiplying each component with the respective mean velocity and summing over all three equations, as used in Hamilton et al. (Reference Hamilton, Suk Kang, Meneveau and Bayoán Cal2012) and Lignarolo et al. (Reference Lignarolo, Ragni, Ferreira and van Bussel2014):

(3.11)\begin{equation} \bar{u}_j\frac{\partial{\overline{E_K}}}{\partial{x_j}} ={-}\frac{\partial{\bar{p} \,\bar{u}_i}}{\partial{x_i}}-\left(-\overline{u'_i u_j'}\right)\frac{\partial{\bar{u}_i}}{\partial{x_j}}-\frac{\partial{}}{\partial{x_j}}\left[\bar{u}_i\overline{u_i'u_j'}\right]. \end{equation}

The first term of the right-hand side corresponds to a source or sink of $\overline {E_K}$ due to a pressure gradient, the second term corresponds to the conversion of kinetic energy to turbulent kinetic energy and the last term corresponds to the turbulent flux of kinetic energy. Since the problem is axisymmetric, a cylindrical coordinate system can offer some simplifications. In a cylindrical coordinate system, the kinetic energy is transported towards the wake centre by the term $-({\partial }/{\partial r})[\bar {u}_i\overline {u_i'u_r'}]$. Among others, Cal et al. (Reference Cal, Lebrón, Castillo, Kang and Meneveau2010) showed that this transport dominates the wake recovery, which was confirmed in preliminary examinations conducted for this study. The total transport of kinetic energy to the wake via radial turbulent transport, $\varepsilon (x)$, is given by

(3.12)\begin{equation} \varepsilon(x) ={-}\int_{x_0}^x\iint_{A_{R}} \frac{\partial{}}{\partial{r}}\left[\bar{u}_i\overline{u_i'u_r'}\right] \,\mathrm{d}A \,\mathrm{d}\kern0.7pt x, \end{equation}

where the rotor-swept area $A_{R}={\rm \pi} D^2/4$. The effect of the helix approach on $\varepsilon (x)$ is shown in figure 9. The entrainment is calculated in a vertical plane in the centre of the rotor plane. The results are normalised with the mean kinetic energy per rotor area of the undisturbed flow, $E_{K,u}=1/2\,V_{0}^2 A_{R}$ multiplied with the inflow velocity. In the case G1V$9$ we find that little to no energy is entrained in the area of the near wake, as was to be expected due to the presence of the tip vortices. However, with the onset of the transition and more pronounced wake meandering more energy is entrained. The figure also shows clearly that the total transport increases earlier and faster if the helix approach is applied. The cases G1V13 and H1V13 show much lower entrainment compared with the baseline cases. However, as before, the application of the helix approach leads to an increase in $\varepsilon$. In figure 9(b) we can observe the influence of an increase in turbulence intensity. Generally, the entrainment is notably higher. Yet, the relative increase in entrainment due to the helix approach is smaller when compared with the lower ambient turbulence intensity. A similar observation can be made in figure 9(c), which shows the influence of the turbulence length scale. Entrainment increases even more compared with G1L$40$. However, the helix approach also improves entrainment more than in case H1I$10$. Similarly to H1I$10$, the helix approach mostly improves entrainment in the region of $1D$ to $6D$. Thus, we conclude that the helix approach generally increases entrainment. Even in cases of high turbulence intensity, which naturally have a higher entrainment due to the ambient turbulence, the earlier transition leads to an increase in the crucial section of the wake where a downstream turbine would be placed in a farm.

Figure 9. Total kinetic energy entrainment into wake in a vertical plane in the centre of the rotor. Entrainment normalised with the streamwise transport of kinetic energy in the undisturbed flow.

The effect of wake mitigation is a reduction in the velocity deficit. In figure 10, the cross-stream velocity profiles at $1D$, $3D$ and $5D$ are shown. The transition from the near-wake region to the far-wake region, where the wake recovers, is indicated by the typical switch from a double-Gaussian to a single-Gaussian profile (Sørensen et al. Reference Sørensen, Mikkelsen, Henningson, Ivanell, Sarmast and Andersen2015; Porté-Agel et al. Reference Porté-Agel, Bastankhah and Shamsoddin2020). The profile of G1V$9$ exhibits the usual features, i.e. a double-Gaussian profile at $1D$, followed by the transition and reaching a single-Gaussian profile at $5D$. Furthermore, the entrainment of momentum leads to an expansion of the wake. The reduced tip-speed ratio and, thus, lower thrust in case G1V$13$ leads to a noticeably smaller velocity deficit, while the transition shows a similar progression. The application of the helix approach only has little to no effect at the first plane. In the second plane shown we observe an earlier transition in case H1V$9$ in comparison with G1V$9$, indicated by a lack of a local maximum of mean velocity at the centre of the wake. In the case with a lower tip-speed ratio the velocity deficit profile has the same shape with and without the helix, however, the deficit is reduced if the helix is applied. At $x=5D$ all wakes shown in figure 10(c) have transitioned, but the helix leads to a visible reduction in velocity deficit, regardless of tip-speed ratio. The increase in turbulence intensity from 5 % to 10 % leads to a faster transition and faster recovery, indicated by the lack of a local maximum at the centre of the profile already at $x=3D$ visible in figure 10(e). However, at $x=5D$, the cases H1I5 and H1I10 exhibit little difference, whereas we find a significant reduction in velocity deficit from G1I5 to G1I10. This indicates that the helix approach increases wake mixing more effectively at lower turbulence intensities. The effect of increasing the turbulence length scale is shown in figure 10(g–i). At $x=1D$, the difference between the cases is small, only the local minima are increased at higher $L$, and the application of the helix has a negligible effect. The larger turbulence length scale accelerates the transition to a single-Gaussian profile, and the application of the helix approach leads to a reduced velocity deficit, similar to the effect of the higher turbulence intensity. Further downstream, at $x=5D$, the larger turbulence length scale leads to an even faster recovery than the increased turbulence intensity. Furthermore, the helix approach leads to an additional reduction in the velocity deficit. Thus, we conclude that the helix approach enhances the wake recovery most effectively in weak ambient turbulence. As we found in the previous section, it also deflects the wake most effectively at weak turbulence. This indicates that the approach might be most suitable in conditions of low turbulence and that, if applied to a wind farm, its effectiveness will decrease further downstream in the farm as turbulence levels increase.

Figure 10. Mean velocity profiles in the vertical line through the centre of cross-stream planes at (a) $1D$, (b) $3D$ and (c) $5D$ downstream of the turbine.

Since we have found a strong dependence of the helix approach on ambient turbulence, we also want to examine its effect on the wake turbulence levels. Therefore, we show the turbulence intensity at $x=1D$, $x=3D$ and $x=5D$ in figure 11. Note that the turbulence intensity is computed in a static frame of reference and therefore also includes contributions due to the helical deflection of the wake. As mentioned in § 3.4, the deflection of the wake leads to an apparent velocity fluctuation due to the translation of the wake and a translation of the turbulent region of the wake. Strictly speaking, the first effect should not be qualified as turbulence intensity. However, a separation of these effects is not possible here. Furthermore, the effect on potential downstream turbines would also be similar to that of large turbulent structures. To use familiar terminology, we continue to call the measured quantity turbulence intensity. In figure 11(a–c) we display the comparison of the two tip-speed ratios. In the vicinity of the rotor, the turbulence levels are very high near the root and the tips of the blades, as could be expected. The application of the helix approach leads to even higher turbulence intensities in these areas. Without the helix, the root and tip areas of high $Ti$ appear to be separated by a local minimum with a magnitude close to the ambient $Ti$, which is significantly weakened or does not exist if the helix is applied. Further downstream we only find a local minimum of turbulence intensity at the centre of the plane, which is significantly larger without the helix. Due to our previous observation that the transition happens earlier if the helix is applied, it is also reasonable to assume that the turbulence intensity profile transitions earlier. In the last plane shown, $Ti$ still has a local minimum without the helix, however, the application of the helix approach leads to a single maximum of turbulence intensity at the centre of the plane. The cases with lower tip-speed ratio, i.e. G1V$13$ and H1V$13$, exhibit much lower levels of turbulence at the root and tip vortices and consequently in all planes downstream as well. Furthermore, at $x=3D$, G1V$13$ still exhibits separate peaks at the tip and the root, indicating a slower transition process. This finding is supported by figure 11(c), which shows a local minimum even if the helix is applied. The maximum value quickly decreases downstream and the maximum values in cases G1I10 and H1I10 are very similar to G1I5 and H1I5, however, the local minima at the centre of the plane are significantly higher. This can be attributed to a combination of faster wake decay as well as increased meandering. The results presented in figure 11(g–i) show the effect of the turbulence length scale. The larger turbulence length scale leads to a slight increase in ambient turbulence levels. As with cases G1I10 and H1I10, we observe an increase in the overall turbulence level. At $x=3D$, the local minimum of turbulence intensity in the centre of the profile is reduced compared with the respective case with a lower turbulent length scale. We observe that the helix approach leads to an even further reduction in the local minimum, although the effect is not as strong compared with the effect it had at lower turbulent length scales. At $x=5D$, H1L120 overlaps with G1L120, showing again that the effectiveness of the helix approach is reduced at larger turbulent length scales. Overall, we find that the helix approach leads to even higher turbulence intensity in the root and tip vortices, however, the effect is stronger at low turbulence intensities and length scales, a trend that has also been observed in all previous sections.

Figure 11. Turbulence intensity profiles in the vertical line through the centre of cross-stream planes at $1D$ (a,d,g), $3D$ (b,e,h) and $5D$ (c,f,i) downstream of the turbine.

In conclusion, we have shown that both increased wake mixing, as well as reduction of wake overlap, contribute to the increase in kinetic energy in the rotor area downstream of the turbine. The reduction of overlap is most effective at low turbulence intensities when we can also observe a strong deflection of the wake. A larger length scale of the ambient turbulence seems to amplify the effect of the helix approach.

3.6. Wake shape

The previous subsection showed that the helix approach exhibits properties of wake-steering approaches since it significantly deflects the wake. A feature consistently found in studies examining yaw misalignment is the curled shape of the wake. We want to determine what effect the helix approach has on the wake shape. We show the velocity contours averaged in the meandering frame of reference at the cross-stream planes downstream of the turbine in figure 12 as well as the vector field of the cross-stream velocity components. The average velocity is computed by first translating the wake centre to the centre of the plane. In the case of the helix approach, the velocity field is also rotated with the helix frequency to obtain a ‘frozen’ helix. The mean is then computed by averaging the shifted and, if necessary, rotated velocity fields. The wake of the greedy case exhibits the expected rotational symmetry. In contrast, the helical wake has a shape similar to that of the curled wake found for yawed turbines and reported, for example, in Howland et al. (Reference Howland, Bossuyt, Martínez-Tossas, Meyers and Meneveau2016) or Hulsman et al. (Reference Hulsman, Wosnik, Petrović, Hölling and Kühn2022). Furthermore, we can observe the counter-rotating vortex pair observed in curled wakes. The topic of wake deformation is theoretically analysed in Zong & Porté-Agel (Reference Zong and Porté-Agel2020). They conclude that the asymmetric kidney shape originates from an uneven distribution of cross-stream velocity, leading to transport and shear of the wake, in turn causing deflection and deformation, respectively. There are three main differences between the deformation of the helical wake and the wake of a yawed turbine. First, due to the helical shape of the wake, the velocity deficit rotates, as can be clearly seen in figure 12. Second, the shapes of the helical and yawed wake have different chiralities. A tentative explanation could be found in the direction of rotation of the helix, which is clockwise. Thus, a negative vorticity is added, possibly having the opposite effect of the positive vorticity due to the hub vortex, which was shown in Zong & Porté-Agel (Reference Zong and Porté-Agel2020) to be the cause for the asymmetric shape. Finally, the vortices greatly differ in strength, the vortex with the same rotational direction as the wake is stronger than the vortex in the opposite direction. Still, the similarity to the curled wake of a yawed turbine could be a starting point for the development of engineering wake models for the helix approach.

Figure 12. Mean velocity contours in the cross-stream plane at $4D$, $5D$ and $6D$ downstream of the wake of cases G1V$9$ (a–c) and H1V$9$ (d–f) in the meandering frame of reference. In (d–f) the velocity fields are also rotated with the helix. Arrows show the direction of the cross-stream velocity field and are colored according to the velocity deficit. Line of sight is downstream.

3.7. Kinematics of a helical wake

To understand how the helical wake behaves and how it will interact with downstream turbines, the wake first has to be described in more detail. In particular, the phase of the helix when it reaches the turbine downstream will be of interest, which will be referred to as the helix angle later on. For the following discussion, we assume that all fluid particles in a cross-stream slice of the wake are advected by the same velocity, a common assumption for dynamic wake meandering models (Larsen et al. Reference Larsen, Madsen, Thomsen and Larsen2008). Furthermore, we assume that the force exerted due to the pitching is the primary cause of deflection, thus ignoring the effects of naturally occurring wake meandering and the radial moment due to the helix approach, as mentioned at the beginning of § 3. Based on the remarks at the beginning of this section, we define an average transport velocity,

(3.13)\begin{equation} \bar{u}_{helix}(x) = \frac{x-x_{0}}{t-t_{0}}, \end{equation}

where $t_{0}$ is the point in time the fluid particle passed the rotor-swept area of the turbine at $x_{0}$. Given the assumption of a wake that is passively advected by the mean flow, the direction of deflection at point $x$ and time $t$ is the same as when the fluid particle passed the rotor disc at $x_{0}$ and $t_0$. Hence, (3.13) can be rearranged for $t_{0}$. After inserting (3.8) and (3.3), the angle of deflection at an arbitrary position $x$ downstream of the turbine, $\varphi (x,t)$, can be described as

(3.14)\begin{align} \varphi(x,t) &= \varphi\left(t-\frac{x-x_{0}}{\bar{u}_{helix}}\right) = {\rm \pi}- \omega_e\left(t-\frac{x-x_{0}}{\bar{u}_{helix}}\right) \nonumber\\ &= 2{\rm \pi} {St}\frac{x-x_{0}}{D}\frac{V_{0}}{\bar{u}_{helix}}+\varphi(x_0,t). \end{align}

We can eliminate the time dependency by looking at the difference between the angle of deflection at the turbine and the angle in the wake, the phase shift of the helix:

(3.15)\begin{equation} \varphi(x,t)-\varphi(x_{0},t) = 2{\rm \pi}{St}\frac{x-x_{0}}{D}\frac{V_{0}}{\bar{u}_{helix}} = \Delta \varphi(x). \end{equation}

From (3.15) we can see that the helix turns clockwise when looking downstream. This is also shown in figure 13. In addition, (3.15) shows that the phase shift scales with the size of the turbine and that it only depends on the ratio of the free-stream velocity to the helix transport velocity for a given turbine. To assess, whether this relationship holds in a simulated wake, the phase difference of the helical wake is shown in figure 14. We compute $\Delta \varphi$ by rotating the position of the wake centre, which was determined in § 3.3, by $-\varphi (x_0)$. Via (3.8), $\varphi (x_0)$ is calculated for which $\beta$ was recorded during the simulation. The angle of the averaged wake centre is then computed and unwrapped, i.e. $2{\rm \pi}$ is added whenever the angle jumps more than ${\rm \pi}$ between two consecutive planes. Since the maximum increase in $\Delta \varphi$ between two planes is ${\rm \pi} /2$, this procedure yields the correct angle. From these angles the local mean helix transport, $\bar {u}_{helix}$ is computed by rearranging (3.15), to compare the results to the assumption of a constant transport velocity.

Figure 13. Schematics of a helical wake including a representation of deflection $d$ and helix angle $\phi$. The dotted line represents the centre of the wake.

Figure 14. Angle of deflection and mean helix transport velocity downstream of a single turbine in helix operation.

The examination of figure 14(a,d) reveals that the transport velocity increases with downstream distance. The shape is reminiscent of the centre line velocity deficit in a wake. We also observe that the transport velocity is significantly higher in case H1V$13$. The helix transport velocity is linked to the velocity in the wake, which is higher in H1V$13$ compared with the baseline case, see figure 8. The effect is also visible as a decrease in $\Delta \varphi$. The influence of turbulence intensity, shown in figure 14(b,e), is negligible. Again, this observation is closely in line with figure 8(b), though the differences in $\bar {u}_{helix}$ are even smaller. In figure 14(c,f), where the dependency on turbulence length scale is shown, we find a slightly higher $\bar {u}_{helix}$ throughout the wake in the case with higher turbulence length scale. Again, this matches the earlier findings in the examination of the kinetic energy. Overall, the helix transport velocity is not constant throughout the wake. Instead, the helix accelerates, like the fluid in the wake. Comparing all cases, we see that the change in ambient turbulence leads to negligible differences in the helix angle. However, the change in tip-speed ratio and consequently thrust coefficient and velocity deficit results in a large difference.

4.1. The phase difference between two helices

According to (3.15), the helix of the first turbine has an angle $\varphi _0$ at the location of the second turbine, $x_1 = x_0+S_x$, of

(4.1)\begin{equation} \varphi_0(x_1,t) = \Delta \varphi_0 (S_x) + \varphi_0(x_0,t) = {\rm \pi}+ 2{\rm \pi} {St}\left(\frac{S_x}{D}\frac{V_0}{\bar{u}_{helix,0}}-\frac{V_0t}{D}\right). \end{equation}

We have added the index 0 to the helix transport velocity to indicate that it is the helical wake of the first turbine that is being transported with that velocity. There are two ways of applying the helix approach at the second turbine, depending on whether the Strouhal number is based on the free stream or local inflow velocity. If the Strouhal number is based on the local inflow velocity $V_1$, the angle of the helix of the second turbine is given by

(4.2)\begin{equation} \varphi_1(x,t) = {\rm \pi}+ 2{\rm \pi} {St}\left(\frac{x-x_1}{D}\frac{V_1}{\bar{u}_{{helix},1}}-\frac{V_1t}{D}\right). \end{equation}

This implies that the phase difference between the helix of the first and the second turbine is described by

(4.3)\begin{equation} \varphi_1(x_1,t)-\varphi_0(x_1,t) = 2 {\rm \pi}{St}\frac{(V_0-V_1)t}{D} - \Delta \varphi_0(S_x). \end{equation}

Therefore, the phase difference between the two helices changes with time and can not be set to a constant value determined by controllable parameters. However, based on the preliminary findings in Korb et al. (Reference Korb, Asmuth, Stender and Ivanell2021), we know that the phase difference is very important for the power production of the downstream turbines. Thus, a definition of the Strouhal number based on the local inflow velocity does not allow for an analysis of the influence of the phase difference and is therefore disregarded for the rest of this study. Instead, we choose a definition based on the free-stream velocity. With that definition, the phase of the second helix is described by

(4.4)\begin{equation} \varphi_1(x,t) = \varphi_0(x_0,t) + \Delta \varphi_1(x-x_1) = 2{\rm \pi} {St}\frac{x-x_1}{D}\frac{V_0}{\bar{u}_{{helix},1}} + \varphi_0(x_0,t). \end{equation}

By evaluating (4.4) and (4.1) at the location of the second turbine, $x_1=x_0+S_x$, the phase difference between the helices can be found to be

(4.5)\begin{equation} \varphi_1(x_1,t)-\varphi_0(x_1,t) ={-}\Delta\varphi_0(S_x) , \end{equation}

which is constant in time. By replacing $\beta$ with

(4.6)\begin{equation} \beta' = \beta - (\varPhi_1 + \Delta\varphi_0(S_x)), \end{equation}

in (3.2) at the second turbine, the effective phase difference can be set to a value of choice $\varPhi _1$:

(4.7)\begin{equation} \varphi'_1(x_1,t) - \varphi_0(x_1,t) = \varphi_0(x_0,t) + \varPhi_1 + \Delta\varphi_0(S_x) - (\varphi_0(x_0,t)+\Delta\varphi_0(S_x))) = \varPhi_1. \end{equation}

Thus, adding a constant shift to the sinusoidal variation of the pitch allows for the exact control of the phase difference. The phase difference between the helices exerted by the two turbines will influence the interaction and affect the total efficiency of the park. Thus, it is of interest how to choose $\varPhi _1$.

4.2. The angle of attack

As discussed previously, the force exerted on the wake will deflect the wake in the direction of said force. Therefore, while ignoring any dissipative processes, the fluid will have a velocity component $v_{helix}$ in the direction of the deflection, $\varphi _0(x,t)$. This velocity vector at the second turbine has the form

(4.8)\begin{equation} \begin{bmatrix} u_{helix} \\ -v_{helix} \sin\left(\varphi_0(S_x,t)\right) \\ v_{helix} \cos\left(\varphi_0(S_x,t)\right) \end{bmatrix}. \end{equation}

By examining the velocities at the blade and including the varying pitch angle of the downstream turbine we can find an expression for the angle of attack at the blade. To minimise the deviation of the angle of attack, the pitch angle has to be in phase with the variation of the angle of the relative velocity. In Appendix B we show that this is the case when $\varPhi _1=0$. However, a closer examination of the equations also shows that the influence is strongest near the root of the blade and becomes negligible near the tip. Thus, from a theoretical point of view, the change in the angle of attack likely has only little influence on the performance.