2.1 Regularization kernel

For the regularization kernel, three distinct Gaussian functions, shown in Fig. 1, are utilized:

1. Isotropic Gaussian (Shives and Crawford, 2013), where the forces are equally distributed in the chord and thickness direction within a cylindrical shape. The expression for this function is as follows:

   $$\begin{array}{}\text{(1)}& \mathit{\eta }\left(r\right)={\displaystyle \frac{\mathrm{1}}{{\mathit{\beta }}^{\mathrm{2}}\mathit{\pi }}}\exp \left[-{\left({\displaystyle \frac{\left|r\right|}{\mathit{\beta }}}\right)}^{\mathrm{2}}\right].\end{array}$$

2. Anisotropic Gaussian (Churchfield et al., 2017), a 2D distribution, wherein forces are distributed in an elliptical shape using distinct kernel widths in the chord and thickness directions, denoted as β_c and β_t, respectively. The expression for this function is

   $$\begin{array}{}\text{(2)}& \begin{array}{rl}\mathit{\eta }\left(r_{\mathrm{c}},r_{\mathrm{t}}\right)& ={\displaystyle \frac{\mathrm{1}}{{\mathit{\beta }}_{\mathrm{c}}\sqrt{\mathit{\pi }}}}\exp \left[-{\left({\displaystyle \frac{\left|r_{\mathrm{c}}\right|}{{\mathit{\beta }}_c}}\right)}^{\mathrm{2}}\right]\\ & \cdot {\displaystyle \frac{\mathrm{1}}{{\mathit{\beta }}_{\mathrm{t}}\sqrt{\mathit{\pi }}}}\exp \left[-{\left({\displaystyle \frac{\left|r_{\mathrm{t}}\right|}{{\mathit{\beta }}_{\mathrm{t}}}}\right)}^{\mathrm{2}}\right].\end{array}\end{array}$$

3. Anisotropic Gaussian–Gumbel (Schollenberger et al., 2020), akin to the anisotropic Gaussian but with an incorporated Gumbel function in the chordwise direction, mimicking the airfoil shape. The expression for this function is

   $$\begin{array}{}\text{(3)}& \begin{array}{rl}\mathit{\eta }\left(r_{\mathrm{c}},r_{\mathrm{t}}\right)& ={\displaystyle \frac{\mathrm{1}}{{\mathit{\beta }}_{\mathrm{c}}}}\exp \left[-\left({\displaystyle \frac{\left|r_{\mathrm{c}}\right|}{{\mathit{\beta }}_{\mathrm{c}}}}\right)\right]\cdot \exp \left[-\exp \left[-\left({\displaystyle \frac{\left|r_{\mathrm{c}}\right|}{{\mathit{\beta }}_{\mathrm{c}}}}\right)\right]\right]\\ & \cdot {\displaystyle \frac{\mathrm{1}}{{\mathit{\beta }}_{\mathrm{t}}\sqrt{\mathit{\pi }}}}\exp \left[-{\left({\displaystyle \frac{\left|r_{\mathrm{t}}\right|}{{\mathit{\beta }}_{\mathrm{t}}}}\right)}^{\mathrm{2}}\right].\end{array}\end{array}$$

In Eqs. (1)–(3), r denotes the distance between the centroid of the generic cell and the actuator line, while β represents the kernel width parameter, which is associated with a characteristic dimension of the airfoil (e.g., chord c), as described in Shives and Crawford (2013). To evaluate the anisotropic kernel functions, r and β are split into their components along the thickness-wise (subscript t) and chordwise (subscript c) directions, as shown in Fig. 1. For the isotropic Gaussian, β is set to 0.1 c, while for both the anisotropic Gaussian and anisotropic Gaussian–Gumbel, a setup with β_c=0.2 c and β_t=0.1 c is defined. These specific values are selected based on the calibration analysis conducted by the authors in Mohamed et al. (2022) for a 2D airfoil. For more comprehensive information about the ALM code, refer to Melani et al. (2021b).

Figure 1Schematic representation of the three different kernel functions used for the ALM simulations in the present work.

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2.2 Numerical setup

ALM simulations are carried out with the steady Reynolds-averaged Navier–Stokes (RANS) CFD solver available in Ansys^® Fluent^® (v. 20.2). The corresponding setup follows a consolidated numerical approach developed by some of the authors for airfoil simulation (Balduzzi et al., 2021), which features the coupled algorithm for pressure–velocity coupling and the second-order upwind scheme for both RANS and turbulence equations. The k–ω shear stress transport (SST) is used for turbulence modeling.

Figures 2a and 3a illustrate the adopted computational domain, whose dimensions – length L=60 c, width W=40 c, and height H_D=3 H – are selected to minimize blockage effects and allow the blade wake to develop properly. At the boundaries, the standard far-field boundary conditions for external flows are applied: uniform velocity at the inlet, ambient pressure at the outlet, and symmetry on the other surfaces (including the bottom one). This way, the spanwise symmetry of the problem is exploited, thus halving the global number of elements.

Figure 2Overview of the grid used for ALM simulations: (a) computational domain, (b) side view of the mesh corresponding to the wing, (c) detail of the ALM region at the wing midspan, and (d) detail of the surface mesh along the ALM region in the spanwise direction.

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Figure 3Overview of the grid used for blade-resolved simulations: (a) computational domain, (b) side view of the mesh corresponding to the wing, (c) detail of the surface mesh at the wing tip, (d) detail of the rotating region at the wing midspan, and (e) detail of the prismatic grid used for the boundary layer discretization at the blade leading edge.

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The discretization strategy is selected to ensure the optimal resolution of the vorticity distribution along the blade span as well as in the wake, i.e., the tip vortex, at a minimum computational cost, following the results of a sensitivity analysis performed by some of the authors in a previous study (Melani et al., 2022). Details of this setup are reported in Table 1.

Table 1Characteristics of the grids used for the BR-CFD and ALM simulations in the present work.

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A uniform Cartesian grid is used for the ALM region normal to the wingspan (blue square in Fig. 2c), as required by the ALM method, scaling the local cell size h_ALM on the kernel dimension $\mathit{\beta }/c$ under the constraint $h_{\mathrm{ALM}}<\mathrm{0.4}\cdot \min ({\mathit{\beta }}_{\mathrm{t}},{\mathit{\beta }}_{\mathrm{c}})$ for stability reasons. The criterion adopted for the dimensioning of the smearing radius β varies with the kernel shape. For the standard isotropic function, a value of $\mathit{\beta }c=\mathit{\beta }t=\mathrm{0.1}$ c, ensuring a correct description of the tip vortex core, is selected (Melani et al., 2022). Differently, the anisotropic and Gauss–Gumbel functions are tuned instead to minimize the error in terms of velocity field with respect to BR-CFD in a 2D environment (Mohamed et al., 2022). As this process does not consider tip effects, the setup of these two functions might not be optimal for the scope of this study. Therefore, possible conclusions about their application must be verified in future work. The ALM region is progressively expanded to the domain boundaries via an unstructured, quadrilateral mesh, always scaling the local cell size on h_ALM. The bottom grid is then extruded along the blade span, optimizing the grid density at the tip by distributing the elements according to an exponential bias, as shown in Fig. 2b–e (Melani et al., 2022). For further details, refer to Melani et al. (2021b).

4.1 Tip vortex tracking metrics

The various modeling strategies analyzed in this work, in turn, affect the tip vortices shed from the airfoil. Hence, the effect of the applied methodology on the vortex structure is investigated through multiple metrics, namely vortex center position, core radius, and circulation. These properties are affected by viscous decay, as the vortex is convected downstream, so the analysis is performed over four vertical sampling planes with varying distance from the airfoil (1, 2, and 5 chords), starting from the aerodynamic center (see Fig. 4).

Figure 4Schematic representation of the sampling setup used in the present work: (a) the sampling planes, (b) vortex sketch and definition of core radius, and (c) velocity profile of the vortex.

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As summarized in Fig. 4, the tip vortex metrics are computed following the methodology commonly used in the literature (van der Wall and Richard, 2006; Soto-Valle et al., 2022):

• Vortex center C. The vortex center defines the axis of rotation of the vortical structure, and it is used to track the position of the tip vortex as it is convected downstream. In the present work, its position is computed from the resolved velocity field by calculating the λ₂ scalar field (Jeong and Hussain, 1995) and locating the position of its minimum on the sampling plane. This methodology is selected among the available vortex identification methods as it can distinguish the contributions of viscous stresses and irrotational straining.

• Vortex core radius r_C. The core radius defines the size of the inner part of the vortex, where the fluid rotates as a rigid body. As the tip vortex is convected downstream, the vortex structure is dissipated due to viscous decay, and the size of the core increases (vortex aging). In this work, the core radius is calculated from the induced velocity field as the distance between the vortex center C and the location of maximum induced velocity V_ind (Mauz et al., 2019) (see Fig. 4). V_ind is calculated by subtracting the convection velocity (u_c, v_c) of the vortex from the velocity field as follows:

  $$\begin{array}{}\text{(4)}& {\displaystyle }& {\displaystyle }{u}_{\mathrm{ind}}=u-u_{\mathrm{c}},\text{(5)}& {\displaystyle }& {\displaystyle }{v}_{\mathrm{ind}}=v-v_{\mathrm{c}}.\end{array}$$

  The convection velocity is assumed to be equal to the velocity in the vortex center, where the vortex-induced velocity is 0 (Yamauchi et al., 1999; van der Wall and Richard, 2006):

  $$\begin{array}{}\text{(6)}& {\displaystyle }& {\displaystyle }{u}_{\mathrm{c}}=u\left(x_{\mathrm{c}},y_{\mathrm{c}}\right),\text{(7)}& {\displaystyle }& {\displaystyle }{v}_{\mathrm{c}}=v\left(x_{\mathrm{c}},y_{\mathrm{c}}\right).\end{array}$$

  The core radius is calculated from horizontal and vertical slices of the induced velocity field, passing by the vortex center. In this way, the velocity profile of the vortex (see Fig. 4) is obtained along the two directions. The results are averaged to provide a more representative value. Additionally, the aspect ratio (AR) of the vortex, defined as the ratio between the average core radii in the horizontal and vertical directions, is calculated as to account for possible asymmetry of the vortical structure.

• Vortex circulation Γ. Circulation is a measure of the vortex intensity and is used alongside the core radius to measure its aging in the wake. In this work, it was computed as the integral of the in-plane vorticity ω_x. To avoid the inclusion of spurious contributions, the integration domain is centered on the vortex center and limited to a radius of 2 c from the vortex center.

  $$\begin{array}{}\text{(8)}& \mathrm{\Gamma }=\underset{A}{\int }\mathit{\omega }\mathrm{d}A\end{array}$$

4.2 Angle of attack sampling

The LineAverage method was originally introduced by Jost et al. (2018) for HAWTs, in an attempt to increase the accuracy of previously available methods that capture the effects of shed and trailing vorticity on the measured angle of attack on turbine blades. To this end, the undisturbed velocity vector V is computed as the integral average of the flow velocity field along a closed line around the airfoil (see Fig. 5):

where v_j is the local velocity, tangential to the sampling plane, and s_j the arc length at the node j along the sampling circle. According to its creators, this sampling strategy should be able to completely remove the effect of bound circulation on the local inflow velocity, since in the averaging process the induced velocity components on any pair of opposite points on the closed path are leveled out, still accounting for the net distortion associated with shed and trailing vorticity.

Figure 5Schematic representation of the sampling setup used in the present work.

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The choice of this method for the present investigation is justified by the accuracy and robustness shown in (i) a previous study of some of the authors on blade-resolved simulations of vertical-axis wind turbines (Melani et al., 2020), in which it has been validated against high-fidelity numerical and experimental data, and (ii) its application to ALM simulations of both horizontal- (Bergua et al., 2023) and vertical-axis machines (Melani et al., 2021a, b).

The analysis carried out in the present work uses a sampling radius r_s=1 c and N=80 evenly distributed sampling points, as recommended by Jost et al. (2018) and Rahimi et al. (2018).

5.1 Test case

For the simulation campaign, a constant-chord NACA0018 wing is selected, whose geometrical characteristics are reported in Table 2. The choice of such a simple test case is justified by the necessity of reducing the number of governing parameters involved as much as possible, thus increasing the generality of the analysis and minimizing the possibility of biases. For instance, having a constant chord c along the span ensures that the observed load reduction towards the tip was only related to the presence of the tip vortex, without the spurious contribution of the blade tapering as in many studies on the subject (Jha et al., 2014; Churchfield et al., 2017; Martínez-Tossas and Meneveau, 2019; Meyer Forsting et al., 2019).

Table 2Main geometrical parameters of the test wing.

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This test wing is simulated in the operating conditions reported in Table 3. The freestream, chord-based Reynolds number is selected for this airfoil to obtain a behavior of the lift curve that is as linear as possible at the considered loading conditions, i.e., low (pitch of 6°), middle (pitch of 8°), and high loads (pitch of 10°), without excessively raising the computational cost as it would happen in a high-Reynolds case. Accordingly, turbulence intensity is higher than in standard wind tunnel conditions to stabilize laminar transition in BR-CFD simulations. The inlet Mach number M is kept at a minimum to avoid compressibility effects.

Table 3Freestream values for the simulations.

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5.2 Validation of BR-CFD simulations

The setup from Sect. 3 is validated, at least for the 2D case, against the experimental measurements of Timmer (2008), as shown in Fig. 6. It is observed how the matching between 2D numerical results and experiments is good until AoA = 10°, with a nearly perfect matching in terms of slope of the lift curve and drag value at the zero-lift point. Approaching the static stall point, the two datasets start diverging, as the stall predicted by BR-CFD occurs earlier than the measured one. This issue does not affect the present study, as the analysis is limited to the attached flow region, with a maximum tested angle of attack of 10° (see Table 3), thus justifying the use of 2D BR-CFD polar data for both the reconstruction of the equivalent angle of attack from BR-CFD simulations of the finite wing (see Sect. 5.3) and the standard ALM simulations (see Sect. 5.4).

Figure 6Comparison in terms of lift (C_L) and drag (C_D) coefficients between the BR-CFD simulations, both 2D and 3D, and the experimental measurements from Timmer (2008), for $Re=\mathrm{500}\times {\mathrm{10}}^{\mathrm{3}}$.

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On the other hand, the accuracy of 3D BR-CFD simulations cannot be verified, as no reliable data for this test case are currently available in the literature. The confidence in the adoption of these results indeed derives from (i) the experience of some of the authors in these kinds of simulations (Balduzzi et al., 2017) and (ii) the physical soundness of the results presented in Fig. 6, where the lift computed at specific sections of the wingspan (here called “local”) decreases going towards the tip, while the drag presents an abrupt increase in the last 1 % of the wing. The corresponding 3D curves, obtained from pressure integration over the wing surface, lie somewhere in between and are closer to the local ones at midspan. (iii) And, finally, confidence is also derived from the nature of the current analysis, whose aim is not to provide a benchmark for this test case but rather some insights into the physical mechanism underlying tip losses and the capability of the ALM to reproduce tip effects.

5.3 Investigation of the flow mechanism

A preliminary and fundamental step of the present investigation was the identification of the physical mechanism underlying the spanwise load degradation observed in the blade-resolved (BR-CFD) simulations taken as reference. It was indeed a priority to understand if this mechanism actually falls in the spectrum of the aerodynamic phenomena interpretable as a dynamically equivalent variation in a 2D angle of attack, as prescribed by the lifting line theory (LLT) (Prandtl and Tietjens, 1934), or if it is characterized by inherent 3D characteristics. In fact, only in the first case would the ALM have the possibility to capture the effect of tip vortices on the blade loads without the need for semi-empirical corrections, as it is based on the use of tabulated polar data. This work would then have a solid theoretical foundation. In the second case, oppositely, the only way to improve the ALM accuracy would be to rely on dedicated corrections such as the ones currently used for the simulation of wind turbines (e.g., Glauert, 1935; Shen et al., 2014).

For this purpose, a 2D angle of attack, α_2D,eq, was computed for each spanwise section of the blade. As for blade-resolved simulations, it is not possible to use the LineAverage method (see Sect. 4.2) for radii lower than the airfoil chord. Since the sampling line would intersect the airfoil surface, a different strategy was adopted, following the work of Soto-Valle et al. (2020, 2021). In greater detail, α_2D,eq was selected as the angle of attack that, when imposed onto a 2D calculation, would minimize the error of the predicted pressure coefficient C_P (Eq. 10) distribution along the chord with respect to the one of 3D BR-CFD. It was a priority in this process to match the minimum C_P value on the suction side (SS), as in the attached flow regime it is the main variable regulating lift production.

In this case, the viscid panel method used in the original approach of Soto-Valle et al. (2021) was replaced with 2D BR-CFD, using the same airfoil meshing strategy and turbulence modeling of 3D simulations (see Sect. 3). In this way, the bias in the computation of α_2D,eq, related to the inherent differences between CFD and panel methods, is avoided.

Figure 7 presents the results of the workflow described above, along with the corresponding velocity field, for the low-load case (pitch of 6°). For the sake of brevity, only a few relevant blade sections are shown in the picture. The spanwise load degradation associated with the tip vortex can be reasonably approximated with a reduction in the equivalent 2D angle of attack α_2D,eq until 80 % of the blade span. Consequently, the deviation between the section lift coefficient computed at α_2D,eq and the 3D one is limited, as shown in Fig. 9. Therefore, this flow region is here called the 2D region. This effect is also visible from the non-dimensional velocity field, with the simultaneous rotation of the pressure side (PS) stagnation region around the airfoil leading edge and reduction in the SS suction peak.

Figure 7Extraction of the pressure coefficient C_P profile and non-dimensional velocity field $V/V_{\mathrm{0}}$ from 3D BR-CFD from different sections along the blade span, for a pitch of 6°. The 3D pressure data are compared with those coming from the 2D BR-CFD simulations for the computation of the equivalent angle of attack α_2D,eq.

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The same trend is found between 80 % and 97 % of the span. Going towards the tip, however, a pressure deviation of increasing intensity arises between 3D and 2D BR-CFD in the rear of the blade ($\mathrm{0.5}\le x/c\le \mathrm{0.9}$) towards the trailing edge, despite the good correspondence between the two datasets in the leading edge region. This phenomenon, called the decambering effect, is well known in the literature (Sørensen et al., 2016) and derives from the radial outflow generated by the tip vortex structure. Traces of this pattern are visible in Figs. 7d–e and 8d–e (bottom) as a deviation of the freestream velocity vectors towards the SS wall. Since in this part of the blade the flow progressively becomes 3D and the deviation between 2D and 3D lift coefficients is increasing (see Fig. 9), the corresponding blade section is called the transition region.

Figure 8Extraction of the pressure coefficient C_P profile and non-dimensional velocity field $V/V_{\mathrm{0}}$ from 3D BR-CFD from different sections along the blade span, for a pitch of 10°. The 3D pressure data are compared with those coming from the 2D BR-CFD simulations for the computation of the equivalent angle of attack α_2D,eq.

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Figure 9Normalized plot of the lift coefficient computed from α_2D,eq vs. the one directly extracted from the 3D BR-CFD simulations. The amount of deviation from the ideal situation in which the two are the same is used to identify the different flow regimes developing along the blade span.

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Differently, in the region between 97 % of the span and the tip, the flow becomes fully 3D (3D region) so that 2D theory and the concept of angle of attack itself lose validity (Branlard, 2017). Indeed, it is not possible anymore to approximate the blade loads using polar data, as testified in Fig. 9 (right). The extension of this region and the intensity of the corresponding loads are small enough though for the hypothesis of an α_2D,eq to still be adopted, confirming the prescriptions of the lifting line theory.

As the same considerations can be made for the high-load scenario (pitch of 10°) presented in Fig. 8 without losing coherence, it is possible to conclude, at least for this specific case, that the effect of tip vortices on a resolved flow field can be reasonably approximated as an angle of attack reduction. A corollary of this conclusion is that the ALM can, in theory, reproduce this effect without resorting to additional corrections. Whether this is actually feasible or not is investigated in Sect. 5.4. A feature of the blade spanwise flow that is apparent from Figs. 7 and 8 and should not be ignored is that the reduction in α_2D,eq does not happen in the freestream, which remains basically undisturbed, but at the blade chord scale. This is key when deciding on the velocity sampling strategy, as is later explained in Sect. 6.

5.4 ALM

As discussed, Sect. 5.3 demonstrates that, at least for this test wing, the main mechanism responsible for the spanwise load degradation observed in the BR-CFD results is a progressive reduction in the local equivalent angle of attack α_2D,eq. This confirmed the assumptions of lifting line theory. Starting from there, the question naturally arose whether the ALM can reproduce the flow field observed in the BR-CFD simulations, both along the blade span and in the wake, and how this information can be properly extracted from the resolved velocity field for the computation of blade loads. To make the analysis as general as possible, three different kernel shapes from the literature are considered: the standard isotropic Gaussian and two new formulations, namely anisotropic Gaussian and Gauss–Gumbel (see Sect. 2.1). The standard LineAverage sampling line at r_s=1 c is also reported for clarity.

5.4.1 Wing loads

The analysis starts from the benchmarking of the LineAverage method (see Sect. 4.2) in its standard setup (r_s=1 c), i.e., the one that is commonly found in the literature and provides good results for 2D flows. The angle of attack sampled from the flow field with this method will from now on simply be referred to as α and must be distinguished from the equivalent 2D angle of attack α_2D,eq that was reconstructed from the BR-CFD simulations by comparing the airfoil pressure data.

Figure 10 reports the α spanwise profile for BR-CFD, frozen ALM (from here on abbreviated as F-ALM), and standard ALM, for the three load conditions considered in this work, i.e., low (pitch of 6°), middle (pitch of 8°), and high loads (pitch of 10°). For the sake of clarity, only the curves of the isotropic case are reported, as changing the kernel shape provided no relevant difference. The three datasets are in good agreement with each other, although both standard ALM and F-ALM tend to filter out some of the flow oscillations observed in the BR-CFD simulations, and they predict an angle of attack that increases going towards the tip.

Figure 10Comparison between BR-CFD, frozen ALM (F-ALM), and standard ALM for the three operating conditions under consideration in terms of angle of attack sampled with the standard LineAverage setup (r_s=1 c). As the three kernel shapes yield the same results, only the data for the isotropic function are reported.

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As this trend does not follow the one observed in Sect. 5.3 and, more generally, the common understanding of the phenomenon, relevant discrepancies arise when the sampled angle of attack is cross-compared with the corresponding blade forces (see Fig. 11). The crosswise force coefficient C_y predicted by the standard ALM, which scales linearly with α, starts deviating from the 3D BR-CFD one – computed by integrating the blade pressure distribution – already at 60 % of the span, keeping a constant profile before increasing at 90 % of the span. In contrast, the BR-CFD simulations show a decreasing trend up to the tip of the blade. Looking at the streamwise force coefficient C_x, the difference seems smaller for most of the blade span but only because the drag coefficient C_D is approximately constant in the attached flow region (see Fig. 6).

Figure 11Comparison between BR-CFD, frozen ALM (F-ALM), and standard ALM for the three operating conditions under consideration in terms of (a) force coefficient along the crosswise direction and (b) force coefficient along the streamwise direction. As the three kernel shapes yield the same results, only the data for the isotropic function are reported.

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At the tip, where the flow is fully 3D (see Sect. 5.3), the 3D BR-CFD and ALM profiles deviate abruptly due to the induced drag associated with the shedding of the tip vortex. However, this phenomenon lies outside of the range of validity of 2D airfoil theory, upon which the ALM and all methods based on polar data are built, and therefore outside of the scope of the present study.

5.4.2 Spanwise flow

To gain a deeper insight into the apparently unphysical behavior observed in Sect. 5.4.1 and, more generally, into how the flow field along the blade develops for the different simulation methods, Figs. 12 and 13 report the comparison in terms of non-dimensional velocity $V/V_{\mathrm{0}}$ between BR-CFD, F-ALM, and ALM, along different spanwise locations, for a pitch of 6° and a pitch of 10°, respectively. The standard LineAverage sampling line at r_s=1 c is also reported for clarity.

Figure 12Comparison in terms of the non-dimensional velocity magnitude $V/V_{\mathrm{0}}$ between BR-CFD, frozen ALM (F-ALM), and standard ALM (ALM) at different spanwise sections, for a pitch of 6°.

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Figure 13Comparison in terms of the non-dimensional velocity magnitude $V/V_{\mathrm{0}}$ between the BR-CFD, frozen ALM (F-ALM), and standard ALM (ALM) different spanwise sections, for a pitch of 10°.

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It is evident how all ALM variants produce a distortion of the velocity field along the span that is similar, at least qualitatively and with all the well-known limitations of the ALM method, to that of BR-CFD. Even the SS high-velocity bubble's progressive elongation, related to the presence of the tip vortex, is captured. The only relevant deviation is visible at 99 % of the span, where the flow becomes fully 3D, thus invalidating the assumption of 2D sectional flow that underlies all polar-based methods like the ALM (see Sect. 5.3). At the wing's midspan, the magnitude of this velocity distortion is correctly predicted by both F-ALM and ALM, regardless of the selected kernel shape. Moving towards the sections close to the wing tip (e.g., 90 % and 95 % of the span), however, the two approaches progressively diverge from one another. The F-ALM, despite having the same force distribution of BR-CFD (see Fig. 11), systematically underestimates the corresponding velocity distortion. This issue is well known in the scientific community (Jha et al., 2014; Dağ and Sørensen, 2020; Martínez-Tossas and Meneveau, 2019), where it has been justified by the loss of circulation related to the spreading of aerodynamic forces into the computational grid. The presented results further confirm this theory.

Notably, the use of non-conventional kernel shapes such as an anisotropic shape and Gauss–Gumbel exacerbates the situation by spreading forces over a wider area compared to their isotropic counterpart. The standard ALM instead predicts a spanwise evolution of the velocity field much closer to that observed in BR-CFD simulations. In this case, the discussed loss of circulation induced by the regularization kernel is compensated for by the higher lift force, i.e., higher circulation magnitude, given by the ALM in the last 40 % of the blade span (see Fig. 11).

To better quantify the differences in the spanwise circulation distribution predicted by each simulation method and link them to the corresponding sampled angle of attack profile (see Fig. 10), Fig. 14 reports the comparison between BR-CFD, F-ALM, and ALM in terms of non-dimensional crosswise velocity $V_y/V_{\mathrm{0}}$: this quantity is equivalent to the non-dimensional downwash velocity induced by the tip vortex on the wing plane. Only the high-load case (pitch of 10°) is considered here for the sake of brevity.

Figure 14Comparison in terms of the non-dimensional downwash velocity $V_y/V_{\mathrm{0}}$ between the BR-CFD, frozen ALM (F-ALM), and standard ALM (ALM) at different spanwise sections, for a pitch of 10°.

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Focusing at first on the midspan plane, it is observed that the induced velocity field is dominated by the airfoil bound circulation, which is connected to the generated lift. In BR-CFD, its effect, although distributed along the airfoil chord, can be reduced to upwash ($V_y/V_{\mathrm{0}}>\mathrm{0}$) and downwash ($V_y/V_{\mathrm{0}}<\mathrm{0}$) bubbles positioned at the leading and trailing edges, respectively.

This pattern is well approximated in the far field by F-ALM and ALM, using the $V_y/V_{\mathrm{0}}$ distribution typical of a Lamb–Oseen vortex – anti-symmetric with respect to the flow normal direction – as prescribed by classic ALM theory (Shives and Crawford, 2013). It must be noted that a small difference in size between the upwash and downwash bubbles is still present due to the influence of the tip vortex. In this situation, the standard LineAverage setup (r_s=1 c) works as intended, giving an α estimation that is coherent between BR-CFD, F-ALM, and ALM (see Fig. 10) and close to the reference α_2D,eq from Sect. 5.3. The sampling line lies, in fact, in a relatively undisturbed flow region (LineAverage was indeed first applied in ALM for wind turbine simulation (Melani et al., 2021a) to capture the local deceleration of the flow without including the contribution of bound vorticity).

Moving towards the tip, due to the simultaneous decrease in airfoil bound circulation and increase in the tip vortex strength, the effect of the vortex-like pattern previously observed at the midspan is progressively concentrated around the airfoil aerodynamic center (AC). In the process, the balance in size between the upwash and downwash bubbles is progressively broken, as the tip vortex slows down the size reduction in the downwash bubble in the rear part of the airfoil and stretches it in the wake direction.

This effect is visible from the BR-CFD, F-ALM, and ALM simulations but with relevant differences in the magnitude of the phenomenon. As a consequence of the tendency to overestimate the reduction in local circulation along the span (already observed in Figs. 12 and 13), all F-ALM simulations exaggerate the size difference between the front positive and the rear negative induction regions with respect to BR-CFD. Once again, the use of non-standard kernel shapes accentuates this trend, especially at the sections at 95 % and 99 % of the span. When using the standard ALM, on the other hand, the balance between bound circulation and the trailing vorticity related to the tip vortex is partially restored, better capturing the $V_y/V_{\mathrm{0}}$ distribution observed in BR-CFD than the F-ALM approach. A relevant improvement in the prediction of the lateral extension of the downwash region in the rear part of the wing is also achieved. This time, the standard LineAverage setup (r_s=1 c) is not able to follow the progressive concentration of the induction pattern around the airfoil AC (see Fig. 14), missing most of the downwash induced by the shed vortex and thus yielding the unphysical values for the angle of attack observed in Fig. 10.

5.4.3 Tip vortex structure

Figure 15 shows the comparison in terms of non-dimensional downwash velocity $V_y/V_{\mathrm{0}}$ between blade-resolved CFD (BR-CFD), frozen ALM (F-ALM), and standard ALM (ALM) for three different load/pitch conditions. The downwash velocity is sampled in the near wake, along a line one chord away from the blade AC.

Figure 15Comparison in terms of the spanwise non-dimensional downwash velocity between BR-CFD; frozen ALM (F-ALM) for isotropic, anisotropic, and Gauss–Gumbel kernel shapes; and standard ALM at the three operating conditions under consideration.

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It is apparent that the F-ALM gives a fair estimation of the downwash immediately downstream of the wing trailing edge (x=1 c) up to 80 % of the span, then rapidly loses accuracy in the last 20 %. In fact, the magnitude of the velocity induced by vortices in the tip region is heavily underestimated, consistent with the fact observed in Sect. 5.4.2 that F-ALM tends to overspread the computed forces over a wider area compared to high-fidelity simulations. The effect of the kernel shape is not as marked as on the spanwise velocity field (see Figs. 12–14), although small discrepancies between different functions are still present, especially with the Gauss–Gumbel kernel. On the other hand, when the standard ALM is used, the corresponding improvement in the description of the spanwise flow field positively reflects on the wing wake structure: the predicted downwash peak is closer in magnitude to the BR-CFD one for all loading conditions, although it extends over a spanwise region ($\mathrm{0.9}<z/\left(\mathrm{2}H\right)<\mathrm{1}$), i.e., wider than the one observed in F-ALM.

Shifting the focus on the tip vortex structure in the near and far wake reported in Fig. 16, the differences between the F-ALM and ALM approaches are reduced, as the flow behavior is dominated by the integral balance between the bound vorticity along the blade and the one shed into the wake. In fact, the circulation Γ of the tip vortex is well predicted, proving that the ALM is conservative, although a small deviation is observed at the higher loads, e.g., a pitch of 10°. Regarding the shape of the tip vortex, it can be inferred from Fig. 16 that the F-ALM and ALM approaches, when the kernel width is properly tuned (see Sect. 2.1), provide a satisfying estimation of the BR-CFD vortex characteristics, i.e., core radius r_C and aspect ratio (AR), especially in the near wake ($x/c\le \mathrm{1}$). However, the standard ALM tends to slightly underestimate the vortex core AR. In the far wake, both F-ALM and ALM overestimate the vortex aging speed with respect to the BR-CFD, leading to the r_C deviations up to +100 % at $x/c=\mathrm{5}$ in the case of F-ALM. The results are improved when standard ALM is used, consistent with its better description of the spanwise flow evolution (see Sect. 5.4.2). In the authors' view, this accelerated vortex aging observed in the ALM simulation might be related to the absence of the turbulence generation normally occurring in the presence of a physical blade tip. A minor role might also be played by the inevitable difference in grid resolution with BR-CFD.

Figure 16Comparison in terms of tip vortex core radius, aspect ratio, and intensity up to five chords downstream of the airfoil between BR-CFD, frozen ALM (F-ALM) with three different kernel shapes (isotropic, anisotropic, Gauss–Gumbel), and standard ALM for a pitch of 6, 8, and 10°.

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