3.1. Quantifying the effects of turbulence

As mentioned in § 1, analytical wake models are powerful tools that are often used both in literature and in the field to represent the complex turbine wake in a simplified fashion. These operate by assuming the existence of a relationship between the velocity in any point of the wake and a reduced set of parameters: for instance, the widespread Gaussian wake model, developed by Bastankhah & Porté-Agel (Reference Bastankhah and Porté-Agel2014), assumes that the velocity in any point of the wake is described by the set of equations

(3.5)$$\begin{gather} \frac{\Delta{}U}{{U_\infty}} = \left( 1 - \sqrt{1 - \frac{{C_T}{}}{8 (\sigma_w/D)^2}} \right) \exp\left(-\frac{(r/D)^2}{2(\sigma_w/D)^2}\right), \end{gather}$$

(3.6)$$\begin{gather}\frac{\sigma_w}{D} = k^* \frac{x}{D} + \epsilon, \end{gather}$$

where $r$ is the radial coordinate away from the turbine axis and $k^*$ is a parameter that serves the role of the wake expansion rate $k$ in the previous section. Regarding the definition of $\epsilon$, the authors show that from considerations on the total mass flow deficit rate across the turbine disk, one must have

(3.7)\begin{equation} \epsilon = 0.25 \sqrt{\beta}, \end{equation}

where

(3.8)\begin{equation} \beta{} = \frac{1}{2} \frac{1 + \sqrt{1-{C_T}}}{\sqrt{1-{C_T}}}. \end{equation}

However, the authors show that the Gaussian model provides estimates that better match the LES data of turbines in the atmospheric boundary layer by assuming that $\epsilon = 0.2 \sqrt {\beta }$. One must note that, as the authors explain, the difference between these formulations consists in the distance necessary for the wake to equalise its pressure with the surrounding free stream: in particular, $\epsilon = 0.25 \sqrt {\beta }$ assumes that this is verified on the turbine disc, while $\epsilon = 0.2 \sqrt {\beta }$ assumes a non-zero distance is necessary to attain this.

As full knowledge of the wake geometry is known from the PIV velocity fields, one can compute the value of $k^*$ for each operating condition and inflow as outlined in figure 11: for every value of $x/D$, a velocity profile is obtained and fitted to a Gaussian profile, with the standard deviation of the fitted profile being an estimate of $\sigma _w(x)$. The value of $k^*$ is then the slope of the linear regression of this last quantity. As for the previous data on the wake diameter $D_w$ presented in figure 9, the fitting is limited to the range $0 < y/D < 1$ as the presence of the mast has impacted the symmetry of the wake; moreover, as the velocity profiles need not be Gaussian close to the turbine, especially at low values of ${I_\infty }{}$ (Medici & Alfredsson Reference Medici and Alfredsson2006), $\sigma _w$ is only computed for $x/D \ge 4$.

Figure 11. Determination of $k^*$ from PIV measurements in the mean wake. In colour, mean streamwise velocity component; in black arrows, selected velocity profiles; in dashed red, linear regression of $\sigma _w(x)$. Data for flow M1, ${\lambda} = 3.8$.

Figure 12(a) reports, for all wakes obtained at ${\lambda} = 3.8$ and thus constant ${C_T} = 0.76$, the trends of velocity deficit previously reported in figure 4 where the values on the horizontal axis are the distance from the turbine multiplied by the $k^*$ attained in each specific operating condition: it can be appreciated that none of the curves collapse on the prediction offered by Bastankhah & Porté-Agel (Reference Bastankhah and Porté-Agel2014). Figure 12(b) instead presents the velocity deficit trends where the distance from the turbine is normalised by $k^*_{0}$: this parameter is determined so that the velocity trends reported minimise, in a least-square sense, the difference with the expected trend from the Gaussian model, as it is standard in approaching this problem in industrial settings. Instead, figure 12(c) reports the same trends as in figure 12(a) having normalised the distance from the wind turbine by using two parameters: a virtual origin $x_0$ and an alternative value of $k^*$, which has been labelled ${k_{fit}}{}$. The formulation of (3.6) used in the remainder of this paper is therefore

(3.9)\begin{equation} \frac{\sigma_w}{D} = k_{fit} \frac{x - x_0}{D} + 0.25 \sqrt{\beta}. \end{equation}

Both the virtual origin $x_0$ and the alternative wake recovery rate ${k_{fit}}{}$ have been obtained as the two parameters that minimise the sum of squared residuals between each measured trend of $\Delta {}U$ and the predicted curve according to (3.5), having imposed $r/D = 0$. The use of a virtual origin to describe the evolution of bluff-body wakes in turbulence is a widely employed method for other bluff bodies such as porous disks (Rind & Castro Reference Rind and Castro2012a,Reference Rind and Castrob; Pal & Sarkar Reference Pal and Sarkar2015), spheres (Spedding, Browand & Fincham Reference Spedding, Browand and Fincham1996) or even when describing wakes of bodies in turbulent boundary layers (Sakamoto & Arie Reference Sakamoto and Arie1983); this approach has been recently shown to be valid for wakes generated by utility-scale turbines in the atmospheric boundary layer (Neunaber et al. Reference Neunaber, Peinke and Obligado2022). While the physical meaning of the wake recovery rate is immediate, the plotted trends in figure 12(c) highlight how the virtual origin $x_0$ is loosely connected to the extent of the interval of $x$ for which the velocity deficit is constant. In this case, it can be seen that a collapse, especially for large values of $x/D$, is obtained regardless of the inflow conditions, improving both on the assumption that $k^*$ can be inferred from the wake diameter growth rate, and on the current industry standard of finding the best-fit $k^*$ and employing no virtual origin. While this shows that the Gaussian wake model can be tuned to predict the wakes of turbines in a wide array of both realistic and non-Kolmogorov-like flows, this also quantifies the effect of turbulence on the wake generated by a wind turbine.

Figure 12. Comparison of the wake velocity deficit trends $\Delta {}U/{U_\infty }$ with the trend predicted by Bastankhah & Porté-Agel (Reference Bastankhah and Porté-Agel2014), (a) assuming $\epsilon _0 = 0.2 \sqrt {\beta }$ and $k^*$ inferred from wake profile fitting, (b) assuming $\epsilon _0 = 0.2 \sqrt {\beta }$ and best-fitting $k^*_0$, and (c) with the trends obtained by determining the best-fit values of $k^*$ and $x_0$ assuming $\epsilon _0 = 0.25 \sqrt {\beta }$. Data for ${\lambda} = 3.8$.

Figure 13 reports the values of the two parameters used, ${k_{fit}}{}$ and $x_0$, for all test cases investigated in this study. For all the turbine operating points, it can be seen that ${k_{fit}}{}$ assumes a linear trend with ${I_\infty }{}$ if one limits the analysis to the Kolmogorov-like flows. This is in good agreement with previous literature, namely the work of Niayifar & Porté-Agel (Reference Niayifar and Porté-Agel2016) and that of Peña et al. (Reference Peña, Floors, Sathe, Gryning, Wagner, Courtney, Larsén, Hahmann and Hasager2016) on the Jensen wake model (Jensen Reference Jensen1983), where the authors show how the linear relationship between the wake recovery rate and ${I_\infty }{}$ can be recovered assuming the vertical velocity profile is well described by the Monin–Obukhov similarity theory (Monin & Obukhov Reference Monin and Obukhov1954); one must however note that no vertical velocity profile is present in our measurements, and thus the linear relationship between ${k_{fit}}{}$ and ${I_\infty }{}$ holds even outside the atmospheric boundary layer. The general trend of ${k_{fit}}{}$ mimics well what is already seen on the wake velocity deficit, for instance in figures 4 and 5, where wakes exhibiting higher values of $\Delta {}U$ far from the turbine have low values of ${k_{fit}}{}$ and vice versa.

Figure 13. Trends of (a–c) k _fit and (d–f) $x_0/D$ with the free stream turbulence properties I _∞ (horizontal axis) and ${T_0}{}$ (colour axis). Data for (a,d) ${\lambda} = 1.9$, (b,e) ${\lambda} = 3.8$ and (c, f) ${\lambda} = 4.7$.

Instead, the virtual origin $x_0$ is seen to be less affected by the free stream turbulence conditions: generally, this value is the highest for the low-turbulence case of flow L, regardless of the turbine thrust coefficient, and all other flows exhibit a value of $x_0/D \simeq 2.5$, only moderately affected by ${C_T}{}$ and free stream turbulence. It is also interesting to note that, limiting our attention to high-turbulence flows, the value of $x_0$ is higher for the non-Kolmogorov flows for the two test cases at high ${C_T}{}$ than it is for the Kolmogorov-like flow H1, while this trend appears reversed for the low-${C_T}{}$ test case of ${\lambda} = 1.9$. However, one must note that the scatter between the values of $x_0$ obtained for flows H1, H2 and H3 is comparable to the scatter between those of all flows of ${T_0} = 1$. It is therefore possible that, while the inclusion of the virtual origin $x_0$ helps in fitting the curves with the Gaussian wake model, the value of this parameter need not vary with inflow conditions beyond a certain threshold of ${I_\infty }{}$.

The parametrisation of the wake in its recovery rate ${k_{fit}}{}$ and its virtual origin $x_0$ also provides a convenient separation between two regions in the wake: in the next section, we will argue that the extent of the near-wake is loosely related to the trends of $x_0$ and to the stability of the helical vortex set enveloping the wake. Conversely, one can interpret the value of ${k_{fit}}{}$ to be the main parameter that drives the wake evolution for large distances downstream of the turbine, and thus to be representative of the far-wake evolution.

3.2. Influence on near-wake extent

Customarily, the wake of a wind turbine is often divided in two regions: a near-wake, where the rotor geometry affects the local velocity field, and a far-wake which is instead independent of the turbine geometry and self-similarity is attained (De Cillis et al. Reference De Cillis, Cherubini, Semeraro, Leonardi and De Palma2020); however, seldom a quantitative definition of their extent is given. A number of methods have been presented in the literature to estimate the location of the transition between one region and the other, based on considerations of the turbulent kinetic energy content (Wu & Porté-Agel Reference Wu and Porté-Agel2012; Sørensen et al. Reference Sørensen, Mikkelsen, Henningson, Ivanell, Sarmast and Andersen2015; De Cillis et al. Reference De Cillis, Cherubini, Semeraro, Leonardi and De Palma2020) or on the onset of wake meandering (Howard et al. Reference Howard, Singh, Sotiropoulos and Guala2015). It can be understood that these methods only provide a qualitative estimation of this transition, as there is no hard boundary between the near- and the far-wake. This however does not mean there is no merit in estimating the near-wake length, as trends of these estimations can still provide valuable information as comparison between test cases. In this work, the definition of near-wake length is slightly changed from the one presented by Wu & Porté-Agel (Reference Wu and Porté-Agel2012): having defined the turbulent kinetic energy as

(3.10)\begin{equation} \kappa = \tfrac{1}{2} \left( \overline{u'^2} + \overline{v'^2} + \overline{w'^2} \right), \end{equation}

its mean advection is

(3.11)\begin{equation} \boldsymbol{U} \boldsymbol{\cdot} \boldsymbol{\nabla}\kappa = U\frac{\partial \kappa}{\partial x} + V\frac{\partial \kappa}{\partial y} + W \frac{\partial \kappa}{\partial z}, \end{equation}

where $\boldsymbol {U} = (U; V; W)$ is the time-averaged velocity vector. Note that for steady flows, this is equal to the material derivative of $\kappa$:

(3.12)\begin{equation} \frac{{\rm D} \kappa}{{\rm D}t} = \frac{\partial \kappa}{\partial t} + \boldsymbol{U} \boldsymbol{\cdot} \boldsymbol{\nabla}\kappa = \boldsymbol{U} \boldsymbol{\cdot} \boldsymbol{\nabla}\kappa, \end{equation}

as ${\partial }/{\partial t} = 0$ in steady-state conditions. The near-wake is then defined as the location where ${{\rm D}\kappa }/{{\rm D}t} > 0$ and vice versa for the far-wake. Physically, the interpretations are as follows: as a flow parcel traverses the rotor-swept plane, its turbulent kinetic energy increases under the effect of both the circulation generated by the blades and the vorticity these shed, either as a vortex sheet or as the system of tip- and root-vortices. As the parcel travels downstream, it will cede this to its surrounding; therefore, a positive value of this material derivative denotes that the evolution of the particle is driven by its interaction with the turbine and vice versa.

As the data collected consist of the two in-plane components of velocity $u$ and $v$, it is impossible to compute the full turbulent kinetic energy as defined in (3.10) as no information is available on $w'$; similarly, without $W$, the term relative to the advection of $\kappa$ in the $z$-direction cannot be computed. For this reason, the definitions in (3.11) and (3.10) have been implemented by assuming that ${v}/{w} = O(1)$ and ${\partial {}}/{\partial {}y} \simeq {} {\partial {}}/{\partial {}z}$; thus, the turbulent kinetic energy is estimated as

(3.13)\begin{equation} \kappa = \tfrac{1}{2} \left(\overline{u'^2} + 2 \overline{v'^2} \right), \end{equation}

and its material derivative is estimated as

(3.14)\begin{equation} \frac{{\rm D}\kappa}{{\rm D}t} = U \frac{\partial\kappa}{\partial x} + 2 V \frac{\partial\kappa}{\partial y}, \end{equation}

which is an assumption that holds true particularly close to the wake centre, as the wake is somewhat axisymmetric, and gets progressively less so farther from that, as an azimuthal velocity component is introduced as a result of the torque generated by the blades and the assumption that $v \simeq w$ need not hold (Medici & Alfredsson Reference Medici and Alfredsson2006).

Figure 14 reports the trends of the material derivative of $\kappa$ with distance from the turbine; this is adimensionalised by the term ${D}/{U_\infty ^3}$. To reduce the effect of experimental noise on the measurements, the trends reported have been obtained by averaging the value of $\kappa$ for $|y/D| < 1/2$, equivalent to limiting this analysis only to particles that have traversed the turbine rotor. Moreover, each trend is low-pass filtered to remove all components having a wavelength smaller than $D/2$. The trends highlight a clear distinction between the two regions at positive and negative ${{\rm D}\kappa }/{{\rm D}t}$, as well as showing that this value does not trend to a zero value at long distances from the turbine. This is expected and is due to the natural decay of turbulence with free stream distance, proper of both grid-generated turbulence (Kistler & Vrebalovich Reference Kistler and Vrebalovich1966; Hearst & Lavoie Reference Hearst and Lavoie2016) and bluff-body wakes (Wygnanski, Champagne & Marasli Reference Wygnanski, Champagne and Marasli1986). Defining the value of $x$ at which ${{\rm D}\kappa }/{{\rm D}t}$ changes sign as $x_\kappa$, one can observe the trends of this parameter with free stream turbulence characteristics in figure 15. Data for ${\lambda} = 1.7$ are not reported as ${{\rm D}\kappa }/{{\rm D}t}$ is negative along the whole domain and no change in sign is observed; for this operating condition, the torque generated by the blades is marginally lower than that at ${\lambda} = 4.7$ and considerably lower than that at ${\lambda} = 3.8$ (Gambuzza & Ganapathisubramani Reference Gambuzza and Ganapathisubramani2021): the circulation generated by the blades is therefore lower and so is the intensity of the tip- and root-vortices generated by the blades. This, paired with a lower ${\lambda}{}$ and therefore more spaced vortices, might have led to a less intense interaction between the turbine and the flow traversing the rotor-swept plane.

Figure 14. Trends of mean material derivative of turbulent kinetic energy ${{\rm D}\kappa }/{{\rm D}t}$ for $|y/D| < \tfrac {1}{2}$ as a function of the streamwise distance from the turbine for inflows M1, H1 and H3. Turbine operating at ${\lambda} = 3.8$.

Figure 15. Location of the change of sign $x_\kappa /D$ in the spatial mean of ${{\rm D}\kappa }/{{\rm D}t}$ as a function of the free stream turbulence intensity ${I_\infty }{}$ (horizontal axis) and integral time scale ${T_0}{}$ (colour axis), shown for (a) ${\lambda} = 3.8$ and (b) ${\lambda} = 4.7$.

The trends of $x_\kappa$ are somewhat similar to those of the virtual origin $x_0$ reported in figure 13(e, f) for the two high-${\lambda}{}$ test cases: in particular, it can in both cases be seen that the low-${I_\infty }{}$ inflow conditions result simultaneously in a high $x_0$ and a high $x_\kappa$. Moreover, among the three flows at ${I_\infty } \approx {} 11.5\,\%$, the Kolmogorov-like conditions of flow H1 are the ones that result in the lowest values of both $x_0$ and $x_\kappa$. However, focusing one's attention only to the Kolmogorov-like flows at ${T_0} \le 1$, $x_\kappa$ shows a clear decreasing trend that is not readily seen in the values of the virtual origin. This suggests that the changes in value of the virtual origin are broadly affected by the same mechanisms that regulate the near-to-far-wake transition, although $x_0$ cannot be taken as representative of the transition distance.

To understand the conditions that drive the evolution of the near-wake, it is useful to observe the behaviour of the tip-vortices shed by the turbine. Vortex identification from an instantaneous velocity snapshot can be carried out with a number of techniques; in this work, the $Q$-criterion (Hunt, Wray & Moin Reference Hunt, Wray and Moin1988; Jeong & Hussain Reference Jeong and Hussain1995; Haller Reference Haller2005) has been employed due to the simplicity of the underlying equations and the ease of adaptation to planar data. For an instantaneous velocity field $\boldsymbol {u}(t) = (u(t), v(t))$, the in-plane value of $Q$ is computed as

(3.15)\begin{equation} Q ={-}\frac{1}{2} \left(\left(\frac{\partial u}{\partial x}\right)^2 + 2 \frac{\partial u}{\partial y} \frac{\partial v}{\partial x} + \left(\frac{\partial v}{\partial y}\right)^2\right), \end{equation}

and vortices are individuated as continuous regions in the flow where $Q>0$. Figure 16 reports the map of $Q$ for a representative velocity snapshot obtained for the operating conditions of ${\lambda} = 3.8$ and inflow L, along with the methodology used to individuate the location and intensity of individual vortices. Given a computed map of $Q$, contiguous regions of positive $Q$ are individuated as the regions bounded by iso-lines of $Q$ equal to a certain threshold, which in this case has been chosen as $0.3 (D/{U_\infty })^2$; the iso-lines are then approximated with the best-fitting ellipse, with the centre of the ellipse being used as the location of the vortex and the value of $Q$ at the centre being an indication of the vortex intensity.

Figure 16. Identification of the tip-vortices location (white plus signs) from isocontours of $Q$-criterion (black dotted) approximated with best-fitting ellipses (white dashed). Data displayed are those of a representative snapshot obtained at ${\lambda} = 3.8$ and inflow L.

The position of the vortices is shown in figure 17 for four edge-cases, all having the turbine operating at ${\lambda} = 3.8$. From these, it can be seen how the low turbulence intensity of flow L results in a very coherent and stable train of vortices mostly concentrated around the tip-height line at $y/D = 0.5$, with some small instability setting on at $x/D > 3.5$. For the two Kolmogorov-like inflows at high ${I_\infty }{}$, namely M1 and H1, shear layer instability is favoured by the high amount of free stream turbulence and the vortices positions are more erratic: this shear layer instability in turn drives the wake meandering seen in figure 6 for the same two test cases. The test case of the non-Kolmogorov flow H3 is instead seen, despite the same value of ${I_\infty }{}$, to result in an initially more stable shear layer, with the tip-vortices occupying a region close to $y/D = 0.5$, larger than that generated under inflow L but visibly bounded, unlike that of the two previous flows M1 and H1; this initial shear layer stability is possibly connected to the less intense wake meandering observed in figure 6. The mechanisms that drive the breakdown of the tip-vortex set are well known and extensively studied in the literature. Ivanell et al. (Reference Ivanell, Mikkelsen, Sørensen and Henningson2010), Sarmast et al. (Reference Sarmast, Dadfar, Mikkelsen, Schlatter, Ivanell, Sørensen and Henningson2014) and Lignarolo et al. (Reference Lignarolo, Ragni, Scarano, Simão Ferreira and van Bussel2015) show how, at a certain distance from the turbine, the phenomenon of leapfrogging, that is, the interaction between a vortex and the preceding or following one, contributes to the breakdown of this structure; in particular, Ivanell et al. (Reference Ivanell, Mikkelsen, Sørensen and Henningson2010) shows that the additional perturbations due, for instance, to free stream turbulence favour the onset of leapfrogging closer to the rotor, which is further confirmed by Sarmast et al. (Reference Sarmast, Dadfar, Mikkelsen, Schlatter, Ivanell, Sørensen and Henningson2014). In the case of the results reported in figure 17, the leapfrogging cannot be directly observed; however, it can be reasonably assumed that the more erratic trajectory of these vortices in the case of inflows M1 and H1 does indeed favour interaction between adjacent vortices closer to the rotor.

Figure 17. Instantaneous position of the tip-vortices for inflows (a) L, (b) M1, (c) H1 and (d) H3, coloured by their peak $Q$. Turbine operating at ${\lambda} = 3.8$. For clarity, a random subset of 1200 individual vortices is shown for every panel.

It has been previously reported in the literature that the presence of the helical vortex set envelopes the wake and prevents transfer of momentum between the wake and the surrounding free stream, thus delaying the wake evolution (Lignarolo et al. Reference Lignarolo, Ragni, Krishnaswami, Chen, Simão Ferreira and van Bussel2014, Reference Lignarolo, Ragni, Scarano, Simão Ferreira and van Bussel2015; De Cillis et al. Reference De Cillis, Cherubini, Semeraro, Leonardi and De Palma2020): this suggests that the initial wake evolution is hampered by the presence of a stable shear layer and thus by a train of coherent vortices, which effectively shield the near-wake from the free stream. This can be easily visualised, for the test cases here reported, with quadrant analysis to characterise the events on the mean vortices trajectory. At these locations, a $v'<0$ denotes flow crossing the vortices trajectory from the high-momentum free stream to the wake and $v'>0$ denotes conversely motion from the wake outwards; simultaneously, $u'>0$ indicates motion of high-momentum flow and $u'<0$ indicates that of low-momentum flow. Wake evolution is therefore promoted either by the sweep of high-momentum flow inside the wake, for which $u'>0$ and $v'<0$, or by ejection of low-momentum flow from the wake, for which $u'<0$ and $v'>0$; vice versa, wake evolution is hampered by interaction events, for which $u'$ and $v'$ have the same sign. Thus, the individual contributions to the in-plane Reynolds shear stress $\overline {u'v'}$ in each quadrant can be defined as

(3.16)\begin{equation} \overline{u'v'}_j = \int_{Q_j} u'v' \, P(u', v') \, {\rm d}u'\, {\rm d}v', \end{equation}

where $Q_j$ denotes the $j$th quadrant and $P(u', v')$ is the joint probability density function for an event in the $(u', v')$-space. Moreover, one can define

(3.17)\begin{equation} \overline{u'v'}_{1+3} = \overline{u'v'}_1 + \overline{u'v'}_3 \end{equation}

and likewise for $\overline {u'v'}_{2+4}$.

The trends of the quadrant contributions on the mean vortices trajectory are reported in figure 18, divided into events that hamper the wake evolution (figure 18a) and those that favour it (figure 18b). By observing the trends of the second and fourth quadrant, for which a negative value indicates more intense sweep and ejection events and therefore a faster-evolving wake, one can see that the fast-evolving wake under flow H1 is indeed the one that results in the largest value of $\overline {u'v'}_{2+4}$ and thus in the most intense interaction between free stream and wake. For the Kolmogorov-like flows, the intensity of these events is roughly proportional to the free stream turbulence intensity ${I_\infty }{}$; this relationship however does not hold for the non-Kolmogorov test case of inflow H3, which is seen instead to have a value of Q2 and Q4 events slightly lower than those of flow M1, despite the larger ${I_\infty }{}$. By instead observing the trends of Q1 and Q3 events, it can be seen that the relationship between events intensity and ${I_\infty }{}$ need not necessarily hold close to the wind turbine: in fact, for small values of $x/D$, flow L results in interaction events that are more intense than those developed for inflow M1 and H, despite the much less intense inflow turbulence; this however decays rapidly and by $x/D = 2$, flow L is once again the one for which the events intensity is the smallest. It is instead more interesting to observe the intensity of interaction events for flow H3, which are seen to be more intense than those of flow M1 along the whole of the near-wake, and larger than those of flow H1 close to the turbine until $x/D \approx 2.25$.

Figure 18. Sum of the contributions to the Reynolds shear stress in (a) the first and third quadrant and in(b) the second and fourth quadrant, on the mean tip-vortices trajectory. Data for inflows L, M1, H1 and H3, and ${\lambda} = 3.8$.

From the data here presented, it is evident that the main cause of the delayed onset of wake evolution has to be found in the robustness of the train of tip-vortices, which is in turn driven by the shear layer instability. A stable shear layer inhibits sweep and ejection events, and favours interaction events. The effect of free stream turbulence intensity ${I_\infty }{}$ on the shear layer stability is non-trivial, as it has been seen that a non-Kolmogorov-like inflow with energy content skewed towards low-frequency contributions results in a more stable shear layer than a Kolmogorov-like flow.

3.3. Far-wake evolution

While the stability of the enveloping helical vortex structure is consistent with the near-to-far-wake transition, this phenomenon does not provide an explanation for the rate at which the wake recovers, and thus for the different values of the wake recovery rate ${k_{fit}}{}$ as seen in figure 13. To understand the drivers of wake evolution in the far-wake region, it can be useful to approach the Reynolds-averaged Navier–Stokes equations, in particular, the one that describes the evolution of the streamwise momentum component: for a steady three-dimensional flow, this can be expressed as

(3.18)\begin{equation} U \frac{\partial{}U}{\partial{}x} + V \frac{\partial{}U}{\partial{}y} + W \frac{\partial{}U}{\partial{}z} + \frac{\partial{}\overline{u'^2}}{\partial{x}} + \frac{\partial{}\overline{u'v'}}{\partial{y}} + \frac{\partial{}\overline{u'w'}}{\partial{z}} ={-}\frac{1}{\rho{}} \frac{\partial{}P}{\partial{}x}, \end{equation}

where the contributions due to viscosity have been neglected as the Reynolds number is high. The momentum equation in the $y$-direction can be simplified (Townsend Reference Townsend1999) to the expression

(3.19)\begin{equation} \frac{\partial{}\overline{v'^2}}{\partial{}y} + \frac{\partial{}\overline{v'w'}}{\partial{}z} ={-}\frac{1}{\rho{}} \frac{\partial{}P}{\partial{}y}. \end{equation}

As for the turbulent kinetic energy computation in § 3.2, one can remedy the lack of information on the out-of-plane components of velocity by assuming that ${\partial {}}/{\partial {}y} \simeq {} {\partial }/{\partial {}z}$ and $\overline {v'^2} \simeq {} \overline {w'^2}$. While this is a strong assumption, it can be thought as reasonable especially close to the wake centreline: the wake generated by the wind turbine being roughly axisymmetric, both $y$ and $z$ represent radial directions from the wake axis outwards. With these assumptions, (3.18) near the wake centreline can be expressed in the simplified form:

(3.20)\begin{equation} U\frac{\partial{}U}{\partial{}x} ={-}2 V\frac{\partial{}U}{\partial{}y} - \frac{\partial{}(\overline{u'^2} - 2 \overline{v'^2})}{\partial{}x} -2 \frac{\partial{}\overline{u'v'}}{\partial{}y} + R, \end{equation}

where $R$ is a residual term that takes into account the differences between the left- and right-hand sides of the equation due to the simplifying assumptions employed in this formulation, and the $-2\overline {v'^2}$ in the second term of the right-hand side derives from (3.19): with the simplifying assumption employed, one has

(3.21)\begin{equation} 2\frac{\partial{}\overline{v'^2}}{\partial{}y} ={-}\frac{1}{\rho{}} \frac{\partial{}P}{\partial{}y} \Rightarrow -\frac{1}{\rho{}} \frac{\partial{}P}{\partial{}x} = 2\frac{\partial{}\overline{v'^2}}{\partial{}x} + \frac{1}{\rho{}} \frac{\partial{}P_0}{\partial{}x}, \end{equation}

with the terms at the right-hand side of the implication sign being obtained by integrating the left-hand side in $y$ and then taking the derivative in $x$. In particular, $P_0$ is the integration coefficient that stems from the integration of ${\partial {}P}/{\partial {}y}$ in $y$ and thus is constant in the vertical direction: for this reason, its evolution in $x$ is due to the presence of a potential pressure gradient due to the facility, which is arbitrarily assumed to be negligible here to simplify the analysis. The variation of the terms of (3.20) in $x/D$ is reported, for four test cases having ${\lambda} = 3.8$, in figure 19, along with the value of the virtual origin computed for each wake. It can be seen that for the cases of high ${I_\infty }{}$ (inflows M1, H1 and H3), the dominant term of those at the right-hand side of (3.20) is indeed the one relative to the in-plane Reynolds shear stress, as it is often assumed for far-wakes (Tennekes & Lumley Reference Tennekes and Lumley1972), and that for $x/D > x_0/D$, the residual $R$ is often small enough to be negligible; this is however not true for the case of the low ${I_\infty }{}$ of flow L, for which instead there is a non-negligible contribution of the mean advection term $V {\partial {}U}/{\partial {}y}$. This is consistent with results previously published in the literature, where higher Reynolds shear stresses generated in the turbine wake result in a swifter evolution of the streamwise velocity component (Chamorro & Porté-Agel Reference Chamorro and Porté-Agel2010), or base flows exhibiting a larger Reynolds shear stress upstream of the wind turbine result in a faster wake evolution despite similar values of free stream ${I_\infty }{}$ (Zhang, Markfort & Porté-Agel Reference Zhang, Markfort and Porté-Agel2013).

Figure 19. Terms of (3.20), made adimensional by multiplication with $D/{U_\infty }^2$, for the inflows (a) L, (b) M1, (c) H1 and (d) H3, (black and red lines), along with the value of $x_0$ (dashed vertical line); turbine operating at ${\lambda} = 3.8$.

Assuming therefore that, at least for the high-${I_\infty }{}$ flows,

(3.22)\begin{equation} U \frac{\partial{}U}{\partial{}x} \simeq{-}2 \frac{\partial{}\overline{u'v'}}{\partial{y}} \end{equation}

holds true for $x/D > x_0/D$ and on the wake centreline, one can obtain an analytical relationship between the value of the wake recovery rate $k^*$ and the in-plane Reynolds shear stress at the centreline. In fact, noting that

(3.23)\begin{equation} U \frac{\partial{}U}{\partial{}x} = \frac{1}{2} \frac{\partial{}U^2}{\partial{}x}, \end{equation}

one can integrate (3.22) to yield

(3.24)\begin{equation} \int_{x_0}^{x_{fw}} \frac{\partial{}U^2}{\partial{}x} \, {{\rm d}x} = U_{fw}^2 - U_{x_0}^2 ={-}4 \int_{x_0}^{x_{fw}} \frac{\partial{}\overline{u'v'}}{\partial{y}} \, {{\rm d} x}, \end{equation}

where $x_{fw}$ is a far-downstream station, $U_{x_0}$ is the value of $U$ at the virtual origin and likewise $U_{fw}$ is the value of $U$ downstream of the turbine. The modified formulation of Bastankhah & Porté-Agel (Reference Bastankhah and Porté-Agel2014) presented in this paper can be used to estimate the values of $U$ in the wake: arbitrarily choosing $x_{fw} = x_0 + nD$, one has

(3.25)$$\begin{gather} \frac{U_{fw}^2}{{U_\infty}^2} = 1 - \frac{{C_T}}{8(\epsilon_0 + nk^*)^2}, \end{gather}$$

(3.26)$$\begin{gather}\frac{U_{x_0}^2}{{U_\infty}^2} = 1 - \frac{{C_T}}{8 \, \epsilon_0^2}, \end{gather}$$

which, substituted in (3.24), yield

(3.27)\begin{equation} \frac{{C_T}}{8 \, \epsilon_0^2} - \frac{{C_T}}{8(\epsilon_0 + nk^*)^2} ={-}4 \int_{x_0}^{x_0+nD} \frac{1}{{U_\infty}^2} \frac{\partial{}\overline{u'v'}}{\partial{}y} \, {{\rm d} x}. \end{equation}

This is a quadratic equation in $k^*$: its positive solution is found as

(3.28)\begin{equation} k^*_{est} = \frac{\epsilon_0}{n} \left( \sqrt{\frac{{C_T}{}}{{C_T}{} + 32 \, I_{RSS} \, \epsilon_0^2}} - 1\right), \end{equation}

where $I_{RSS}$ refers to the integral of the Reynolds shear stress derivatives at the right-hand side of (3.27):

(3.29)\begin{equation} I_{RSS} = \int_{x_0}^{x_0+nD} \frac{1}{{U_\infty}^2} \frac{\partial{}\overline{u'v'}}{\partial{}y} \, {{\rm d} x}. \end{equation}

Figure 20 reports the values of the derivative of the in-plane Reynolds shear stress on the wake centreline, here approximated as the horizontal axis having $y/D = 0$. It can be seen that this quantity is, for the Kolmogorov-like flows, increasing with increasing ${I_\infty }{}$. This relationship however does not hold for the non-Kolmogorov-like test case of flow H3, for which the derivative of the Reynolds shear stress is the lowest of all test cases here analysed. Figure 21 reports the fields of the Reynolds shear stress in the turbine wake for the test cases corresponding to the test cases presented in figure 20: from these, it can be appreciated that the effect of the free stream conditions on the quantity plotted in figure 20 are indeed representative of a more general and less localised phenomenon. Indeed, it can be appreciated that flow H1 is the test case for which the Reynolds shear stress is the strongest for low values of $x$, and that the stress in the wake generated by the test cases M1 and H3 are comparable in absolute values. Moreover, one can notice how, for all test cases, the value of this derivative is similar once a large enough distance from the turbine has been covered, such as for $({x-x_0})/{D} > 4$, while all differences in these trends are concentrated at small distances from the turbine. This is in accordance with (3.22) and the data reported previously in figure 12, which shows that all streamwise velocity deficit trends tend to the same overall shape (and therefore, the same spatial gradient in the streamwise direction) sufficiently far from the turbine.

Figure 20. Distribution of the derivative of Reynolds shear stress on the wake centreline from the location of the virtual origin onwards. Data for ${\lambda} = 3.8$.

Figure 21. Fields of Reynolds shear stress $\overline {u'v'}$ in the turbine wake for the test cases presented in figure 20.

The values of $k_{fit}$ obtained from fitting the velocity deficit trends have been reported as square markers in figure 22. Alongside these, we present the values of $k^*_{est}$ obtained by multiplying the result of (3.28) by 4, as this equation appears to underestimate the actual values of $k_{fit}$ by such a factor. Focusing on the medium- and high-${\lambda}{}$ cases, it can be appreciated that, with the single exception of the low-${I_\infty }{}$ test case at ${\lambda} = 3.8$, this equation is able to capture both the linear trend of $k^*$ with ${I_\infty }{}$ and the drop in the wake recovery rate for the non-Kolmogorov-like inflow conditions, regardless of the thrust generated by the turbine. However, as it can be seen from figure 22(a), this relationship need not hold for the low-${C_T}{}$ case. In this case as well, however, the linear trend of $k^*$ with ${I_\infty }{}$ and the drop for high-${T_0}{}$ flows are recovered. Therefore, it is reasonable to approximate the wake recovery rate $k_{fit}$ as

(3.30)\begin{equation} {k_{fit}} \simeq 4 \frac{\epsilon_0}{n} \left( \sqrt{\frac{{C_T}{}}{{C_T}{} + 32 \, I_{RSS} \, \epsilon_0^2}} - 1\right) \end{equation}

for high-thrust operating conditions and high values of ${I_\infty }{}$.

Figure 22. Estimation of ${k_{fit}}{}$ from integration of the Reynolds shear stress on the wake trajectory according to (3.30) (round markers) and comparison with the values found by fitting the wake velocity deficit trends (square markers). Data from (a) ${\lambda} = 1.9$, (b) ${\lambda} = 3.8$ and (c) ${\lambda} = 4.7$.

Following this, it can be understood that the driver of the mean wake velocity evolution is the Reynolds shear stress: knowing this, it can be expected that flows with higher values of free stream turbulence intensity result in faster-evolving wakes as they favour a larger content of Reynolds shear stress in the wake, which explains the trend of $k^*$ with ${I_\infty }{}$, seen to be linear in the literature. A large body of recent literature (Chamorro et al. Reference Chamorro, Lee, Olsen, Milliren, Marr, Arndt and Sotiropoulos2015; Tobin et al. Reference Tobin, Zhu and Chamorro2015; Deskos et al. Reference Deskos, Payne and Gaurier2020; Gambuzza & Ganapathisubramani Reference Gambuzza and Ganapathisubramani2021) has shown a relationship between the spectrum of the incoming velocity fluctuations and the power harvested by a turbine from said incoming velocity field, with the turbine harvesting more power for spectra biased towards lower frequency. It can therefore be assumed that as the turbine harvests power from low-frequency velocity contributions, these will not be present in the wake: in other words, if the turbine acts as a low-pass filter from the point of view of harvested power, it must act as a high-pass filter from the point of view of its wake; this is indeed observed by Tobin & Chamorro (Reference Tobin and Chamorro2019), in which the velocity spectra downstream of the turbine are biased towards higher-frequency contributions than those upstream of it. A similar observation is reported by Heisel et al. (Reference Heisel, Hong and Guala2018) for data acquired in the wake of a utility-scale wind turbine. The idea that a turbine acts as to remodulate the spectral content in its wake has been also shown from the LES data by Chatterjee & Peet (Reference Chatterjee and Peet2018, Reference Chatterjee and Peet2021), where the authors show that for an infinite wind farm, the presence of large scales in the free stream favours the transfer of mean kinetic energy from the free stream to the wake, thus hastening the wake recovery and changing the spectral content of the wake. While this would suggest that flows H2 and H3 should result in a faster-evolving wake, a number of differences are present between these works and the present study. For instance, in the case of these cited works, the authors present their conclusions in the frame of an infinite turbine; it is possible that, for an isolated turbine, the mechanism of power extraction dominates over that of mean kinetic energy entrainment. Moreover, the length scales of the eddies generated by Chatterjee & Peet (Reference Chatterjee and Peet2018) are at most of ${O}(10D)$, or equivalently corresponding to inflows with high-energy content at $f \, D/{U_\infty }{} = 10^{-1}$. The two high-time scale flows here investigated, namely H2 and H3, instead show peaks in the power spectral distribution at $f \, D/{U_\infty }{} = 10^{-2}$, that is, one order of magnitude lower than what was investigated in the cited works, further suggesting that the mean kinetic energy entrainment mechanics are affected by the free stream content of turbulence in a non-trivial way.

As no time-resolved data are available in the wake of the turbine, the hypothesis that spectra in the wake are biased towards high-frequency contributions cannot be verified; however, one can quantify the dominant scales in the wake by means of two-point correlation. Defining the two-point correlation coefficient between a fixed point $(x_f, y_f)$ and a generic point $(x, y)$ in the domain as

(3.31) \begin{equation} R_{uu} = \frac{\overline{u'(x_f, y_f) u'(x, y)}}{\left({\overline{u'^2(x_f, y_f)}}\,\, {\overline{u'^2(x, y)}}\right)^{1/2}}, \end{equation}

the value of $R_{uu}$ acts, in space, as analogous to the classical autocorrelation in time for a time-resolved, single-point measurement. Figure 23 reports the value of $R_{uu}$ for the far-wake generated by the turbine under inflows H1, which is Kolmogorov-like, and H3, which is not; the points that are chosen as fixed are $(x_f/D = 5.5, y_f/D = 0)$ as representative of the flow in the wake, and $(x_f/D = 5.5, y_f/D = 1)$ for the free stream; moreover, the figure also reports the iso-line of $R_{uu} = 0.75$ for all cases. It can be seen that for both the Kolmogorov-like inflow conditions of flow H1 and those of flow H3, the correlation decreases by moving from the free stream to the far-wake: the characteristic lengths of the presented iso-line reduce both in $x$ and $y$ showing a bias towards lower wavelengths in the wake spectral composition, which can be thought of as analogous to the bias towards high-frequency contributions observed by Tobin & Chamorro (Reference Tobin and Chamorro2019). Moreover, it can also be observed that while the changes in $R_{uu}$ are limited in the vicinity of the fixed point for the case of flow H1, the difference between the correlation maps for flow H3 is instead more evident: this does further point towards a clear difference between the spectra of the free stream and the wake for non-Kolmogorov flows, as the turbine has harvested the low-frequency, high-wavelength contributions in the free stream.

Figure 23. Two-point correlation coefficient $R_{uu}$ maps in the far-wake of the turbine operating under base flows (a,c) H1 and (b,d) H3, and iso-line of $R_{uu} = 0.75$ (black line); fixed point (a,b) $x_f/D = 5, y_f/D = 1$ and (c,d) $x_f/D = 5, y_f/D = 0$ (white plus sign). Turbine operating at ${\lambda} = 3.8$.

To quantify this phenomenon, one can attempt to define an equivalent turbulence intensity by purposefully filtering out the low-frequency contributions in the free stream velocity power spectral density:

(3.32)\begin{equation} {I_{filt}}{}({f_{filt}}) = \left( \frac{1}{{U_\infty}^2} \int_{{f_{filt}}}^\infty \phi_{u}(f) \, {\rm d}f \right)^{1/2}, \end{equation}

where ${f_{filt}}$ is the cut-off frequency of an idealised high-pass filter. Tobin et al. (Reference Tobin, Zhu and Chamorro2015) suggests, from comparison between the power spectral densities of the free stream turbulence and the mechanical power generated by the turbine,

(3.33)\begin{equation} {f_{filt}} = \frac{2Q}{J\omega}, \end{equation}

where $Q$ is the torque exerted by the turbine on the shaft, $J$ is the rotor moment of inertia and $\omega$ is the rotor angular velocity. Additionally, Li, Dong & Yang (Reference Li, Dong and Yang2022) suggest that side-to-side motions of a turbine excite wake meandering if these come at a Strouhal number $St = f\,D/{U_\infty }$ greater than 0.1; there might thus be merit in using a constant

(3.34)\begin{equation} {f_{filt}} = 0.1 \frac{{U_\infty}}{D} \end{equation}

for all test cases. Lastly, if one assumes that the turbine fully converts the low-frequency contributions to the free stream velocity into mechanical power, one could use

(3.35)\begin{equation} {f_{filt}} = \underset{0.05< St<20}{\text{argmin}}\left(f \phi_u(f)\right) \end{equation}

for the non-Kolmogorov-like flows H2 and H3, and ${f_{filt}} = 0$ otherwise: this means that ${I_{filt}}{} = {I_\infty }$ for the Kolmogorov-like inflow conditions. Note that, for both flows H2 and H3, (3.35) returns $f_{filt} \approx 0.1 ({{U_\infty }}/{D})$: for this reason, the only difference between the approaches of (3.34) and (3.35) is on whether filtering is applied to the Kolmogorov-like test cases.

Figure 24 shows the values of $k_{fit}$ obtained by fitting the velocity deficit trends of figure 4 as a function of the filtered, equivalent free stream turbulence intensity ${I_{filt}}{}$: the top row employs (3.33), the middle row uses (3.34) and the bottom row uses (3.35) for the non-Kolmogorov-like inflows only, without any filtering of the inflow for the more canonical turbulence cases. The figure shows that, independently of the definition that one employs for ${f_{filt}}{}$, ${I_{filt}}{}$ is lower for the non-Kolmogorov-like flows than it is for the more canonical test cases. In particular, using the arbitrary definition of (3.35), one recovers the linear trend of $k$, this time against ${I_{filt}}{}$, which is the current state-of-the-art in the application of analytical wake models in turbulent inflows (Niayifar & Porté-Agel Reference Niayifar and Porté-Agel2016; Peña et al. Reference Peña, Floors, Sathe, Gryning, Wagner, Courtney, Larsén, Hahmann and Hasager2016): this is an additional indicator of a bias towards the remodulation of the wake spectral content by the presence of the turbine, previously seen in the literature.

Figure 24. Trends of ${k_{fit}}{}$ with the filtered turbulence intensity ${I_{filt}}$, for (a,d,g) ${\lambda} = 1.9$, (b,e,h) ${\lambda} = 3.8$ and (c, f,i) ${\lambda} = 4.7$, using three different definitions of ${f_{filt}}$.