Extracting angles of attack from blade-resolved rotor CFD simulations

The distribution of the angles of attack over the span of a rotor blade, together with blade element theory, provides a useful framework to understand forces, performance and other fluid dynamic phenomena of axial-flow rotors. However, the angle of attack is not straightforward to define for a three-dimensional rotor, where the flow is perturbed by the blade circulation, shed vorticity and wake development. This paper evaluates six methods to extract the angles of attack from blade-resolved CFD simulations of axial-flow turbines. Simulations of two different rotors are presented: a low solidity rotor designed for wind and a higher solidity rotor designed for tidal stream energy conversion. Of the analysed methods, five were obtained from the literature and are tested in terms of their internal parameters. The remaining method is named the streamtube analysis method (SAM) and is presented as an improvement on analysis methods that azimuthally average the flow data on the rotor plane, referred to as azimuthal averaging techniques (AATs). The SAM method accounts for the expansion of the streamtubes in flow-field velocity sampling and exhibits improved convergence on the internal parameters compared with AAT. The six methods are benchmarked in terms of the angles of attack, axial induction factors and the local lift and drag coefficients, identifying that most perform well and converge with each other despite the different underlying assumptions or modelling approaches. However, given the limitations and inherent dependency on internal parameters, the line averaging and SAM are suggested for general flow analysis application.

analyse the performance of axial-flow rotors and has been widely used for turbine design and analysis (see e.g., Burton et al. and 1 Ning 2 ) despite its limitations in representing three-dimensional flow effects near the tip and the root of the blades (e.g., Glauert 3 and Shen et al. 4 ) and in misaligned flow conditions. Thus, it is desirable to analyse blade-resolved turbine simulations, experiments and even full-scale rotor flows in similar terms to those described by the blade element momentum theory.
The inverse problem is also relevant. Extracting the spanwise distribution of the lift and drag polars in order to use them with lower order models (see e.g., Wimshurst and Willden 5 ) requires resolving the local force vector, extracted by integrating pressure and shear force on the blade surfaces, in terms of a frame of reference defined by the local flow and thus by the local angle of attack.
The problem, however, is non-trivial. The flow around a turbine varies across the azimuthal and axial coordinates, with strong gradients near the blades. Furthermore, the local flow over a specific annular section is the result of the inflow characteristics, the operation of the turbine and the development of the wake, making it difficult to define an undisturbed upstream velocity such as that used for defining lift and drag characteristics in wind tunnel experiments. Finally, the concept of a zero-thickness disc cannot be directly applied to a three-dimensional rotor, so the velocities cannot be extracted at that specific plane.
Given these challenges, several flow analysis methods have been proposed in the literature. One of the first approaches developed is to use an inverse blade element momentum method that entails finding the tangential and axial induction factors to describe the local flow velocity based on the measured thrust and torque distributions and using predetermined two-dimensional lift and drag coefficients along with empirical correction factors (see e.g., Lindenburg 6 ).
Hansen et al. proposed the azimuthal averaging technique (AAT), which consists of extracting the azimuthally averaged velocities at different fixed radial coordinates upstream and downstream of the turbine and averaging these to obtain the azimuthally averaged axial and tangential velocities from which the local flow angle can be determined. 7 This method was tested by Johansen and Sørensen by using the blade-resolved CFD simulations of three different rotors (NREL Phase VI, Tellus 95 kW and LM19.1 500 kW) maintaining a fixed rotational speed and a set of inflow velocities ranging from 7 and 20 m/s. Results of their flow analysis were presented in the form of radially varying lift and drag coefficients 8 without discussing the flow analysis method details. Shen et al. introduced an iterative method to analyse the Tellus 95 turbine that considered the flow velocity near the blade to be a combination of the local incident velocity and the perturbations caused by the bound circulation on the blades. The influence from the blades was modelled through lifting-line theory, placing vortices along the blade span with a circulation distribution determined using Kutta-Jowkowski's theorem. The authors compared the method with the AAT and showed good agreement in the midspan in terms of the extracted polar coefficients and also significant sensitivity to the distance between the monitor point and the blade. 9 This method was then used to evaluate experimental data of the Model Experiments in Controlled Conditions (MEXICO) rotor by Yang et al. 10 Shen et al. proposed another method based on blade surface vortex sheets where a distributed circulation was determined directly from a pressure distribution. This method was also tested using the Tellus 95 kW rotor and proved to be more independent of the position of the monitor points than the lifting-line method through comparison of the results in terms of local lift and drag coefficients. 11 Bak et al. proposed fitting two-dimensional pressure distributions extracted from wind tunnel experiments to the pressure distributions extracted at different radial positions along a turbine blade. In their paper, the authors used the pressures extracted at four radial positions from the experiments of a DANAERO NM80 2 MW turbine, fitting the pressure surface data to determine the angle of attack. 12 The pressure-fitting method was regarded as not being reliable enough by Guntur and Sørensen as it depends on the flow not being separated, in a paper that reviewed four methods using CFD simulations of the MEXICO rotor at three different tip speed ratios (TSRs). Their review also tested the inverse Blade Element Momentum, AAT and a variation of the AAT method that extracts the velocity at the blade rather than the azimuthal average. The paper showed a relatively good agreement between the four described methods in terms of the lift and drag coefficients, but poor agreement when compared with results presented by other authors. 13 Jost et al. performed a thorough review of four different analysis methods using CFD simulations of the DTU 10 MW and the AVATAR turbines. In their paper, the AAT, the two vortex methods described by Shen  The line average method takes a closed path around each blade cross section and extracts the velocity around this path, assuming that the influence of the circulation will be negated by averaging the flow velocity components. The method showed a good agreement with AAT on the midspan and a faster response to transient changes. 14 Another method is described by Herraez et al., where the authors propose that extracting the velocities directly from the bisectrix between two blades will provide an approximation to the undisturbed flow velocity. They demonstrate the validity of the assumption with lifting-line theory, and the method is tested using experiments and simulations of the MEXICO rotor and compared with results published in the literature. 15 Rahimi et al. published another review similar to that published by Jost et al. using the same two rotors but only simulating them under steady-state conditions. As in the other review papers, the authors found a good agreement over the midspan of the blades but discrepancies at the tip and the root. They also identify limitations in the methods associated with wake expansion after the flow passes through the turbine. 16 Even though most methods have a relatively good agreement at high TSRs over the midspan, angle of attack discrepancies of about 1 or 2 can be observed in some papers. Whether this difference is acceptable or not will largely depend on the operational regime of the turbine. Tip and root sections escape also the scope for which the concept of angle of attack can be applied as these regions are significantly influenced by three-dimensional effects. Furthermore, most methods exhibit some kind of convergence limitation. For example, the AAT results are influenced by the distance between the upstream and downstream cuts and the rotor plane. The bound circulation line method described in Shen et al. requires the monitor point to be placed at least two chords from the blade, which can be complicated in high-solidity rotors, and a similar problem is observed with the line average method.
Some methods, such as the inverse Blade Element Momentum and pressure fitting, rely on two-dimensional data, which is not ideal as the lift and drag characteristics can change with radial position. 5 The circulation-based methods, on the other hand, have inviscid-flow assumptions, and despite demonstrating good performance, a method used to analyse high-quality three-dimensional turbulent CFD simulations should, arguably, be free of lower order model assumptions.
In this paper, a method based on the concept of expanding streamtubes is discussed. This is based on the work from Hunter 17 and is analysed in this paper and proposed as an improvement on the classical AAT. This method works by considering the expansion of the flow while passing through the turbine and interpolating the azimuthally averaged velocities not at a constant radial position as in AAT, but over the corresponding expanding streamtubes. The advantage of this method is an improved convergence, the lack of two-dimensional assumptions and that no smoothing of the data is required after the calculation.
For this benchmark study, two axial-flow turbines were employed: the MEXICO rotor, a widely studied wind turbine, and a higher solidity tidal turbine originally designed for blocked-flow conditions by Schluntz and Willden. 18 Blade-resolved CFD simulation data of these two rotors over a range of TSRs were used to evaluate the different methods in terms of their convergence relative to model parameters, and then, the methods are compared with each other in terms of the angles of attack, the axial induction factors and the inferred lift and drag coefficients.

| Turbine descriptions
Two axial-flow turbines were employed in this work to test the different methods of determining radially varying angles of attack: a tidal rotor optimised for a blocked-flow condition with high solidity and high thrust 18,19 designated as Sch15B and the test rotor used during MEXICO, a wind turbine for which considerable experimental and numerical data have been published. 20,21 The tidal rotor is a three-bladed turbine designed by Schluntz and Willden 18  The MEXICO turbine is a three-bladed wind rotor with a diameter of 4.5 m designed for an FP5 project partly funded by the European Commission. Its blades were comprised of three different aerofoils (DU-91-W2-250, RISØ A1-21 and NACA64-418) with transition regions between them. The simulations used in this work consider a fixed pitch angle of -2.3 . 19 The two turbines were analysed at three TSRs, λ, covering a wide range of cases, 4.0, 5.5 and 7.0 with a variable rotational speed and a fixed inflow of 4.5 m/s for the Sch15B turbine and 4.17, 6.67 and 10.00 with a fixed rotational speed of 424.5 rpm and variable inflow speed for the MEXICO rotor.
Details of the rotors are summarised in Table 1, and the chord and twist characteristics are plotted in Figure 1. The reader should note how the tidal turbine features a larger chord over the blade span as a consequence of its blockage-oriented design. A render comparison of the two rotors is also provided in Figure 2.

| Numerical models
The data sets employed for this study come from steady-state Reynolds-averaged Navier-Stokes (RANS) CFD simulations. The blade rotation was modelled in both cases using the multiple reference frame approach (MRF), 23 and the k − ω SST turbulence model 24 with updated constants 25 was used to close the RANS equations.
The data sets were generated for different research projects, and hence, there are differences in the CFD strategies employed. The Sch15B rotor was simulated with the commercial code Fluent 19.0 with second-order upwind discretisation scheme and a coupled-pressure algorithm, whereas the MEXICO rotor was simulated with OpenFOAM 2.3.1 and the SIMPLE algorithm for pressure-velocity coupling. In both cases, the residuals were reduced by at least five orders for the velocity components and typically five orders for the turbulence scalars.
Structured meshes were employed in both cases. Non-dimensional wall distances of 30 < y + < 300 for the Sch15B and y + < 5 for the MEXICO rotors were defined, employing an enhanced wall modelling approach for the first case 26 and a wall resolved boundary layer on the second. More details of the simulations can be found in Zilic de Arcos et al. 27 and Wimshurst and Willden 5 for the Sch15B and MEX-ICO rotors, respectively.

| Azimuthal averaging technique
This method operates by obtaining the azimuthally averaged flow speeds at different radial coordinates and axial distances from the rotor plane.
Differences in the implementations stem from where the analysis sections are drawn and interpolated between to obtain the velocity at the turbine plane while avoiding the steep velocity gradients that arise on the rotor plane.
The interpolation can be done in different ways. The most straightforward way is to define an arbitrary fixed distance upstream and downstream and average the velocities. Another approach is to describe the velocities as functions of the axial coordinate and to use data at multiple points to perform a higher order interpolation.
Once the axial and tangential velocity components have been extracted, the inflow angle ϕ can be calculated as with V X and V T the axial and tangential velocity components, respectively, as obtained from the analysis method utilised.
Despite some authors assigning V T to be equal to ωr, 13,28 with ω the rotational speed and r the local radius, the present work considers the tangential induction factor a 0 such that The influence of the tangential induction factor, a 0 , is not significant in the midboard and outboard regions of the blades. However, near the root and where the rotational speed is lower, a 0 can generate some non-negligible variations in the angle of attack.
Once the inflow angle, ϕ, has been determined, the angle of attack is calculated by subtracting the twist angle β: In this work, the velocities were extracted by calculating the mean of the azimuthally averaged velocities at planes located upstream and downstream of the turbine plane, at distances x/c expressed as functions of the local chord c and the axial coordinate x.
Applying the AAT analysis process to the CFD simulations, a sensitivity analysis was performed with the results plotted in Figure 3 in terms of the angle of attack for the Sch15B and MEXICO rotors at the analysed operational cases. The MEXICO turbine was analysed with monitor distances from x/c = 0.25 to x/c = 4.0, whereas the tidal turbine was limited to a maximum x/c = 2.0 as, due to its large chord at the root sections, using any value larger than this meant extracting data at a distance that was regarded as being too far from the turbine plane.
From the convergence plots, a spread of about 0.5 is observed, but no convergence with x/c is observed. It is noteworthy that, for the high-TSR cases where the angles of attack are small, the relative error of this spread is in the order of 25%-30%. However, as they describe it to have a better performance, this method will be analysed in this study as the most complete potential flow analysis method.

Method description
The method requires the velocities at different measuring points located on the turbine plane, as well as the pressure distribution over the corresponding sections. Assuming the pressure to be constant through the boundary layer, the velocity at its edge U τ is calculated using Bernoulli's equation: with p−p ∞ the pressure at a point on the blade surface and U ∞ = ðV 2 ∞ + ω 2 r 2 Þ 1=2 the relative flow speed at a certain radial position r and V ∞ the undisturbed flow velocity. Then, the edge velocity is calculated as which is used to calculate the local bound circulation density at different points around the cross section as a velocity jump over the boundary layer: where δU| wall is the velocity change through the boundary layer and e r ! is a unitary radial vector aligned with the blade spanwise axis. With the circulation defined, the induced velocity is computed using the Biot-Savart law, integrating numerically in both the chordwise and spanwise directions for each blade: with Z the number of blades, S the blade surface, y ! the coordinates of a point on the blade section, x ! the coordinates of the flow analysis point where the velocity is measured, dτ the chordwise and dr the radial directions.
Then, the undisturbed velocity used to calculate the angle of attack is obtained by subtracting the induced velocity from the velocity at the Finally, the inflow angle and the angle of attack can be calculated according to Equations (1) and (4) λ=4.17, the separated region occupies most of the suction surface, as can be seen in Figure 4.
The assumption of the circulation changing sign after separation occurs results from the observation that the surface flow direction reverses after the separation point and beneath the separated shear layer. However, as the edge velocity, defined previously, only applies outside the shear layer, which is above the separation bubble, we consider that this assumption is not valid. Considering the change in tangential velocity from the wall to above the separated shear layer, the net circulation should still remain positive despite the circulation being negative immediately adjacent to the wall. Hence, there should be no change in the sign of the circulation density.
Furthermore, it is worth noting that the sign change makes no significant difference in the results for other cases except for the MEXICO rotor at the lower TSR analysed and, to a lesser extent, to the λ=6.67 case. Figure 5 shows As an improvement, the monitor points can be placed at a constant radius and the chordwise distance projected over the constant-radius arc ζ. This limits the distance at which the monitor points can be placed near the root for a high-solidity turbine, as the points quickly fall into the next blade. Hence, the analysis was performed placing the monitoring points at ζ/c values of 0.5, 0.75, 0.95 and, as a special case, over the bisectrix between two consecutive blades (thus, at varying ζ/c with r/R). The results in Figure 6 show good agreement at the root section for the Sch15B turbine, eliminating the problem seen using a linear chordwise extrapolation to the monitoring points, with some spread in results for ζ/c=0.5 associated with the local flow perturbation caused by the blade, which is particularly apparent for the low λ case. For the MEXICO rotor, similar divergence of results are seen for low sampling distances and also for analysis using the bisectrix as this is now a large number of chords from the blade for this lower solidity rotor.

| Pressure fitting
This method was originally described by Bak et al. and entails fitting two-dimensional pressure coefficient distributions to the surface pressure fields extracted from three-dimensional simulations. The method was developed as an approach to process experimental data but can also be used to analyse blade-resolved CFD simulation data. Despite being disregarded by Guntur and Sørensen, 13 this method is included in the analysis.
The original authors state that, to implement the method, the pressure should be fitted only to the pressure side of the blades, where separation is unlikely to occur. Then, the least squares method can be applied to minimise the value of a residual S by varying the angle of attack α and the inflow velocity W: with χ/c the non-dimensional chordwise coordinate, n the number of points along the chord where pressure data are available at a given radius r, P i the pressure as measured at the ith point, ρ the flow density, W the inflow velocity and C pres the pressure coefficient as a function of the position of the point χ i /c and the angle of attack α.
In this analysis, it is worth noting that the Sch15B turbine consists of a single RISØ A1-24 aerofoil, whereas the MEXICO rotor comprises three different aerofoils with corresponding transition regions. The analysis presented herein covers the entire Sch15B turbine, but was limited to the NACA 64-418 aerofoil, which is used from 74.4% of the span up to the tip on the MEXICO rotor.
The two-dimensional pressure coefficient distributions C pres were extracted from two-dimensional simulations following the CFD configura- The results of the pressure fitting process can be seen, for two typical blade sections, in Figure 7. Good agreement is observed between the two-dimensional CFD data and the pressure field extracted from the blade-resolved rotor CFD. This method, despite showing promise across the blade midspan where the flow is well attached and largely two-dimensional, has significant problems where the flow separates and due to the assumption that the flow is two-dimensional, where three-dimensional effects distort the blade pressure fields from those of two-dimensional aerofoil sections (Wimshurst and Willden 5 ). The main virtue of this approach is its inherent simplicity: it just requires sampling the axial and tangential velocities from the flow (V X and V T ) and employing these to directly calculate the angles of attack (Equations 1 and 4) without any further manipulation. However, a limitation is that any non-homogeneity in the flow, such as environmental turbulence or boundary-layer shear effects, is neglected, which would negatively impact the extracted results.

| Line average
Jost et al. introduced this method, which works with a similar circulation-based framework as others previously discussed. It assumes that each blade section produces a circulation effect that can be removed from the flow field to obtain the inflow angles. 14 Despite using a similar underlying assumption to other methods discussed, this method does not directly model the blade as a set of bound vortices to obtain the induced velocity, but averages the velocity components V X and V T over a symmetric closed pathline centred on the blade quarter chord for different radial positions, as shown schematically in Figure 8. As the average is taken around the blade, not only does the circulation influence vanish, but the method should also be capable of capturing local and even transient effects.
The major drawback of the method is associated with trailing vorticity, which, if captured around the pathline averages, can negatively influence the accuracy of the results. For this reason, the method was tested with circular pathlines projected over constant radii, and a convergence study was performed extracting the angles of attack for the two rotors under different conditions, using different radii pathlines expressed as a function of the local chord c. The results can be seen in Figure 9, showing that the method is almost invariant to variations in the pathline radius used to enclose the blade sections in the midspan sections, with some small divergence of the results observed at the tip and root of the blades.

| Streamtube analysis method
The concept of using the SAM to extract the angles of attack and the flow velocity components at the turbine plane was originally developed by Hunter 17 in an attempt to improve the convergence of the traditional AATs by additionally considering the expansion of the streamtube in the velocity extraction process. In common with the line average and bisectrix methods, this approach has the advantage of taking the data directly from the flow and not requiring any form of post-processing or smoothing. However, the definition of a streamtube in a viscous flow is not as straightforward as in momentum theory. When a viscous flow passes through a turbine, it develops fast local gradients, vortices of different scales and a wake that will experience mixing and exchange energy with the surrounding fluid.
In the viscous-flow scenario, the behaviour of the flow past a turbine can be analysed using sets of independent streamlines seeded over the azimuth at a constant radius. In Appendix B1, we provide more details on the streamlines behaviour, tracking their radial position and showing a low spread from the average radial position. The streamlines analysis, expensive in computational terms, shows that the streamtubes can be approximated to have a circular cross section of radius R S (x) tangent to the azimuthally averaged flow speed. This approach is introduced in the next section as a simplified and more practical method, as well as less expensive in computational resources, to build the tube surface using the azimuthal averages of the axial and radial flow speeds, in a similar way to the AAT but integrating these velocities in the axial direction to obtain the expanding streamtube surface. A comparison of both the streamlines and streamtubes approach is also provided in Appendix B1, showing an excellent agreement except near the tip where strong gradients and three-dimensional flow effects dominate.

| Azimuthally averaged streamtubes
Assuming the streamtubes to maintain a circular cross section, as detailed in Appendix B1, we can numerically integrate the azimuthally averaged radial and axial velocities to construct a set of streamtube surfaces. For the case of steady-state simulations, we can define the streamlines as a function of the axial coordinate x. Defining V X and V R to be the azimuthally averaged axial and radial velocity components, respectively, and as introduced in Section 2.3.1, and noting the streamtube of radius R S will be tangent to the local flow, we may write which can be solved numerically as a function of x for any given value of r, as shown schematically in Figure 10 for a first-order numerical integration method.
Once the streamtubes have been reconstructed, the averaged velocity components over the streamtube R S (x) (i.e., at varying radial positions) can be used to calculate the angle of attack as described in Section 2.3.1. This requires interpolating upstream and downstream along the identified streamtube to extract the velocities either side of the turbine plane. The same first-order interpolation scheme as used in Section 2.3.1 is employed to interpolate along the streamtube. The full approach is summarised in Algorithm 1.
The streamtubes reconstruction approach is tested by integrating the velocities using first-and fourth-order Runge-Kutta methods, each with axial steps of Δx/R = 1/30 and Δx/R = 1/60, for x/R 2 [−0.5, + 0.5]. The results from this analysis are shown in Figure 11 in the form of a set of streamtubes for the Sch15B rotor at two operational conditions, showing little variation between the four different cases except in zones near the tip, where the strong gradients are better captured by the higher order integration scheme. In Figure 12, the results for the different integration schemes and Δx/R values are presented for the MEXICO and Sch15B rotors in terms of the angle of attack distribution α, also showing a good agreement between the cases except near the tip.
Finally, the streamtube method sensitivity to x/c is analysed on the same basis as the AAT method in Section 2.3.1, that is, at the same values that were used, and the results are plotted in Figure 13. From these plots, it can be observed that the spread of α is reduced in the inboard sections and root proximities when compared with the AAT, whereas some variations with x/c can still be observed towards the tip. The reduction in spread of α, and therefore insensitivity to x/c, is particularly apparent for the lower solidity MEXICO rotor for higher TSRs where the flow remains attached.

| BENCHMARK AND DISCUSSION
The concept of α is non-trivial to define for a rotating blade in three dimensions, and hence, it is not possible to conclusively validate the methods with reference to any metric. Instead, this analysis evaluates how the different methods converge with each other in terms of three different variables: the spanwise distribution of the angle of attack α, the spanwise distribution of the axial induction factor a and the local lift and drag coefficients at two radial positions (from the computed relationship between angle of attack and normal and tangential force coefficients), one in the  Table 2.
Notice in Table 2 that two different variables for the MEXICO and Sch15B rotors were used to assess the vortex sheet method. This is because we considered the selected variables and their respective values that achieve the best performance of the method for each rotor, as presented in Section 2.3.2.

| Angles of attack
The angles of attack are the first set of results to be analysed as they are the key focus of the current work. In Figure 14, the reader will note that most methods converge with each other quite well even though they are based on very different assumptions and theories, with the pressure method being the outlier.  The pressure method had already been reported by Guntur and Sørensen 13 as not being reliable, and the present analysis agrees with their conclusion. The pressure method displays good agreement in the midspan region for the TSRs of 7.0 and 5.5 for the Sch15B rotor, where the flow behaves mostly as two-dimensional, with no separation and at relatively low angles of attack. However, outside of this well behaved region, the results diverge from other methods considerably. A similar trend is observed for the MEXICO rotor, even though the data are more limited; recall that only the NACA profile occupying the outer region of the blade is analysed here.
The vortex sheet method diverts slightly from the rest of the methods in some of the cases, without observing a specific pattern, but even when running on a completely different theoretical basis, it still displays a good agreement with the rest of the analysed methods. The bisectrix method also shows some minor differences, especially for the Sch15B case at higher angles of attack and between the midspan and blade root.
This is due to the higher local blade solidity of the inboard region, which is more pronounced for this turbine. This high local solidity is sufficient to perturb the flow as far as the local bisectrix, rendering the assumption of undisturbed flow, once lifting line effects mutually cancel along the bisectrix, invalid.

| Axial induction factors
Blade angles of attack are generally small and subject to small angle differences between analysis methods, partially due to the dominance of the rotational velocity component in determining the inflow angle. It is therefore instructive to examine the different analysis method calculations of the axial induction factor, defined according to Equation (13) and plotted in Figure 15.
The induction factors uncover more significant differences in the calculations. They show a significant divergence of the vortex sheet method from the other methods, which was masked in the angle of attack analysis. This is likely due to the inviscid nature of the vortex sheet method considering the circulation around the blades and does not fully capture viscous drag effects. The drag contributes significantly to overall thrust and thus flow deceleration. The lack of drag considerations in the vortex sheet model leads to an overestimation of through rotor flow speed and thus an underestimation of the induction factor. The error is higher for the Sch15B rotor than the MEXICO rotor, due to the higher thrust loading of the former.
The figure also shows how the bisectrix method diverges near the root, departing from the other solutions as the bisectrix gets closer to the blades and their influence becomes stronger, as discussed previously. Note also how the AAT, SAM and line average methods maintain relatively good agreement with each other, with the exception of the tip regions, where three-dimensional flow effects dominate, and at higher TSRs where thrust tends to be higher, increasing the wake expansion and the three-dimensional effects at the outboard blade sections. Those three methods also appear to capture some of the separation-induced effects on the MEXICO rotor at λ = 4.17 around the midspan, which manifest as wiggles or rapid spanwise changes in induction factor.

| Force coefficients
The lift and drag coefficients at different radii can be used as a metric to evaluate different methods (see e.g., Johansen and Sørensen, 28 Shen 9 and Guntur and Sørensen 13 ). First, the distributions of thrust and tangential force per unit length, F X and F T , respectively, are extracted from the CFD simulations by integrating the pressure and shear forces over a finite number of blade slices normal to the blade axis. 5,27 These forces are then resolved into components parallel and normal to the local flow to yield blade sectional lift and drag coefficients: where L 0 and D 0 are the sectional lift and drag forces, respectively.
In this analysis, we consider two radial positions; one at the midspan region (r/R = 0.65), where a small spread is seen between the angles of attack provided by the analysed methods, and one at the near tip region (r/R = 0.97) where the spread is larger. Each simulation is analysed to yield single lift and drag coefficients at the corresponding radial position using the local angle of attack. Simulations at multiple TSRs are used to generate dependency with angle of attack. The results of this analysis can be seen in Figures 16 and 17 for the Sch15B and MEXICO rotors, respectively.
Disregarding the pressure-fitting method for the moment, we note that the spread between the computed coefficients is low even for the r/R = 0.97 cases on the two rotors. As with the previous analysis, it can be seen that some divergence is present for the vortex sheet method but still exhibits the same general trend as the other methods.
From this analysis technique, it is difficult to draw further conclusions, as force coefficient analysis masks the error sources. Recalling Equations (14) and (15), it is noted that the flow analysis methods differ in the calculation of W, which is dominated by the V T ≈ ωr component, and by the trigonometric relationship between the rotor-based frame of reference (thrust and torque) and the flow-based frame of reference (lift and drag). The trigonometric relationship between the two different frames of reference depends on the inflow angle ϕ and not directly on α. Despite the absolute error for α and ϕ are the same (the difference between α and ϕ being the local blade twist angle β), ϕ is generally larger than α especially at high twist (inboard) regions of the blade, and thus, the relative error passed to the force calculations will be lower.
Comparison of the local lift and drag coefficients inferred from each model demonstrates that in certain applications, the variations in coefficients determined by the different flow analysis methods may be negligible.

| CONCLUSIONS
This work has analysed six different methods based on different underlying assumptions to extract the angle of attack from three-dimensional blade-resolved CFD simulations. These methods are based variously on circulation theory, streamtube interpolation and least squares fitting of two-dimensional surface pressures. Five of the six methods analysed showed good agreement with each other in calculating the spanwise distribution of angle of attack, particularly over the midspan of the two rotors analysed in this work. The pressure method was the only exception, which has also been reported to provide inaccurate results in other studies. 13 Although the remaining methods generally performed well, a number of limitations arising from their underlying assumptions and the physics of rotor blades were identified. The strong radial-flow velocity components and trailing vortices, which arise near blade tips, render twodimensional concepts such as angle of attack less appropriate, and hence, these regions should not be treated in the same way as the rest of the blade. However, as the angle of attack calculation is dominated by the tangential speed, the spread between different models in the tip region tends to be small. The differences between methods were highlighted by comparing the axial induction factor.
The vortex sheet method is developed from potential flow hypotheses and thus models the turbine under inviscid assumptions. It was found that, after correcting the post-separation assumption in the original method, 11 better alignment of the predicted axial induction factor was achieved between this and the other flow analysis methods for the MEXICO rotor than it was achieved for the Sch15B rotor. The lower solidity and thrust characteristics of the former rotor result in a smaller flow perturbation that in turn requires smaller flow correction.
The line average and bisectrix methods showed good agreement in all of the analysed flow metrics. Of the two methods, we consider that the first is preferable as the latter could be significantly affected by ambient flow conditions such as environmental turbulence and shear. Both methods are influenced by the shed vortices and other blade-generated flow phenomena, especially near the root. This can be especially significant for high solidity rotors.
The SAM provided improved performance over the AAT in terms of the internal parameter x/c, reducing uncertainty in the computed flow field values. Furthermore, the analysis of the streamtubes can also provide a valuable tool in exploring the flow phenomena around a rotor more generally, such as understanding near-rotor wake development.
Although it is not possible to conclusively validate the accuracy of the methods with respect to an independent model of angle of attack variation, we consider that the SAM and the line average methods are the most appropriate for general applications. They both have low sensitivity