Low-order modeling for transition prediction applicable to wind-turbine rotors

This work aims at developing a low-order framework to predict the onset of transition over wind-turbine blades without requiring three-dimensional simulations. The effects of three-dimensionality and rotation on the transition location are also analyzed. The framework consists of a model to approximate the base-flow and another to predict the transition location. The former is based on the quasi-three-dimensional Euler and boundary-layer equations and only requires the pressure distribution 5 over an airfoil to provide an approximation for the base-flow over the blade. The latter is based on the envelope of N factors method, where this quantity is computed using the parabolized stability equations (PSE) considering rotational effects. It is shown that rotation accelerates the flow towards the tip of the blade in the fully developed flow region and towards the opposite direction close to the stagnation point. The database method embedded in the EllipSys3D RANS code indicates overly premature transition locations, matching those obtained with a PSE analysis of a two-dimensional base-flow. The consideration 10 of the spanwise velocity, as carried out in the developed model, has a stabilizing effect, delaying transition. Conversely, rotation plays a destabilizing role, hastening the transition onset. Moreover, airfoils with lower pressure gradients are more susceptible to its effects. The increase in the rotation speed makes transition occur through increasingly oblique disturbances from the middle to the tip of the blade, whereas the opposite happens for lower radial positions. Tollmien-Schlichting (TS) waves seem to trigger transition. However, highly oblique critical modes that may be intermediates between TS and crossflow ones occur 15 for low radii. The developed framework allows transition prediction with reasonable accuracy using chordwise cp distributions as input, such as those provided by XFOIL.


Introduction
In wind-turbine design, accurate determination of aerodynamic loads is of importance as they are related to properties, such as performance and structural loads. Since the aerodynamic loads can be influenced by the boundary-layer character, an accurate 20 determination of the transition location can be significant to obtain a successful wind-turbine design. This has long been recognized by aerodynamiscists, and significant efforts have been devoted to the development of transition models (Saric et al., 2003;Langtry et al., 2006;Pasquale et al., 2009;Krumbein, 2009;Colonia et al., 2017).

Boundary-layer equations
There are several integral formulations of the boundary-layer equations (BLE) (Du and Selig, 2000; van Garrel, 2004;Dumitrescu and Cardos, 2011;Drela, 2013;Garcia et al., 2014). However, a differential formulation is expected to be more accurate than its integral counterpart, and, based on experience, it appears that an accurate base-flow is needed to obtain correct results in a subsequent stability analysis. For this reason, a differential formulation is chosen in the present case. 70 When expressed in the coordinate system described in Sect. (2.1), the differential form of the BLE can be written as (Warsi, 1999;Schlichting and Gersten, 2017) In these equations, c p , γ, κ, µ, M, Re, and P r denote specific heat capacity at constant pressure, ratio of specific heats, thermal conductivity, dynamic viscosity, Mach number, Reynolds number based on a reference length l 0 , and Prandtl number, respectively. Moreover, ρ, p, and T denote density, pressure, and temperature, whereas u, Ω, and h represent velocity, rotation, and metric vectors, respectively. The subscripts 1, 2, and 3 indicate components in the respective x 1 , x 2 , and x 3 directions. r is 85 the radial position.
In the BL model, the chordwise curvature of the wing model is neglected, while the radial curvature is considered. Thus, the metric vector becomes h 1 = x 2 + r 0 r 0 , h 2 = 1, h 3 = 1.
Since the code is intended for analysis of laminar flows, turbulent fluctuations and statistics need not be considered. In order 90 to obtain a well-conditioned system which solution is compatible with the subsequent PSE analysis, the terms in the system of Eqs.
(1) to (4) are normalized by the reference quantities given in Table 1. The value of l 0 is set to c 0 , the chord of the airfoil at the radial position r 0 , where the analysis is performed.

Spanwise-derivative approximations
As they stand, the BL equations are dependent on all three coordinate directions so that their numerical solution requires a full 95 volume discretization. Such a discretization can easily result in a solution procedure that is very costly from a computational perspective. By employing approximate models for the derivative terms in the x 2 direction, instead of exact expressions, one can obtain a quasi-three-dimensional model requiring discretization in the x 1 and x 3 directions only. The reduced dimension of the discretization typically results in significant savings in computational cost and meshing effort. Furthermore, a judicious selection of the model for the x 2 derivative can provide accurate base-flows. These beneficial properties lead a quasi-three-100 dimensional model to be employed in the present work. Similarity solutions for rotating flows suggest that the velocity in the x 1 direction can be assumed to depend on the x 2 coordinate linearly (Greenspan, 1968;Hernandez, 2011). This approximation is employed in the present work, together with the further assumption that the velocity in the x 2 direction, pressure, and temperature does not depend on x 2 . Thus,

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The subscript 0 denotes evaluation at the radial location r 0 . This choice can result in a momentum imbalance in the x 2 direction at the boundary-layer edge, as pointed by Sturdza (2003) for swept-wing flows. Sturdza argued that the imbalance could be compensated by defining an additional source term A that accounts for the momentum difference. The extra source term is then multiplied by a blending function f (x 3 ) and added to the right-hand side of the spanwise momentum equation (Eq. (3)). A is found by considering momentum balance at the boundary-layer edge. With the current approximation of spanwise 110 derivatives and curvature terms, A becomes where the subscript e denotes evaluation at the boundary-layer edge. The blending function is selected to linearly depend on the wall-normal distance inside the boundary layer, i.e., 115 5 https://doi.org/10.5194/wes-2020-107 Preprint. Discussion started: 12 November 2020 c Author(s) 2020. CC BY 4.0 License.

Discretization and solution
The spanwise approximations described in Sect. (2.1.2) make the system of the BLE (Eqs. (1) to (4)) include only derivatives in the x 1 and x 3 directions. The derivatives in the x 3 direction are evaluated using a second-order central finite-difference scheme, whereas the derivatives in the x 1 direction are evaluated using a second-order backward Euler finite-difference scheme.
The BLE can be expressed as where Φ = (u 1 , u 2 , T ) T denotes the vector of primary variables. Pressure can be obtained from those variables by using the constitutive relations for isentropic flow. The components of the matrices A 1 , A 2 , A 3 , A 4 , and A 5 are found by collecting terms in Eqs.
The solution is computed by space marching in the x 1 direction. Uniform boundary conditions are assumed at the inflow.

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The attachment-line equations (Cebeci, 1999) are solved at the first inflow node, since the BLE are ill-conditioned when u 1 is equal to zero. Because of the boundary-layer singularity (Goldstein, 1948), the system of equations can become strongly ill-conditioned if flow separation is encountered. However, the present code is intended to be used for transition prediction, and separation within a laminar-flow region typically causes transition. Therefore, the separation point can be taken as a reasonable approximation of the transition location, and the issue is circumvented.

Edge velocity model
The velocity in the x 2 direction at the boundary-layer edge is required as input to the quasi-three-dimensional BL model. In order to avoid the necessity of a costly simulation to obtain it, a model for u 2e is devised with inspiration from the conical-wing approximation (Cebeci, 1999;Sturdza, 2003) . An approximation for u 2e is obtained by combining the Euler equation in the x 2 direction with an approximation for the variation of the pressure coefficient in this direction. The Euler equation in the x 2 135 direction can be written as (Warsi, 1999) where We assume that u2 h2 ∂u2 ∂x2 ≈ 0, based on the fact that the flow and the variations in the x 2 direction have a small magnitude. A 140 second hypothesis is that u3 h3 ∂u2 ∂x3 ≈ 0, built on the evidence that the flow and variations in the normal direction at the boundarylayer edge are small. Since u 3 ≈ 0 and Ω 1 ≈ 0, the term 2u 3 Ω 1 is neglected in Eq. (11). However, the terms u3 h3 ∂u2 ∂x3 and 2u 3 Ω 1 6 https://doi.org/10.5194/wes-2020-107 Preprint. Discussion started: 12 November 2020 c Author(s) 2020. CC BY 4.0 License. may be relevant close to the stagnation point because u 3 ≈ ||u|| and Ω 1 ≈ ||Ω||. Therefore, Eq. (10) should be valid only after a slightly downstream distance from the stagnation point. Moving all terms except the one containing ∂u2 ∂x1 to the right-hand side, dividing both sides of the equation by ρ u1 h1 , and including the scale factors given by Eq. (5) yield All terms on the right-hand side are known except for the x 2 pressure gradient. An approximation for this term can be found by rewriting the definition of the pressure coefficient with the reference speed equals to the rotational one, i.e., and assuming that where c p0 is the pressure coefficient at the radial position r 0 and r = x 2 + r 0 . Equation (14) models the variation in c p due to the change of the reference velocity with r, as well as a first-order variation in c p due to the change of the angle of attack α.
The latter is defined as 155 with w ∞ and θ representing the incoming-flow velocity and the geometric twist angle, respectively. Note that Eq. (14) is singular for α 0 = 0 and may not be very accurate for small values of α 0 . Therefore, some other approximations may be more suitable for these cases. With inspiration from the conical-wing approximation (Cebeci, 1999;Sturdza, 2003), c p0 is assumed to be constant along conical lines. These lines as well as other parameters related to the conical-wing approximation are illustrated in Fig. 2. With this assumption, the derivative of c p0 in the x 2 direction can be related to its derivative in the x 1 direction by The angles β 1 and β 0 are defined as where x 1c denotes the x 1 coordinate of point C, where the line connecting the center of rotation O and the cone apex A 165 intersects the arc with radius r 0 . These assumptions lead to an expression for the pressure derivative, given by Inserting Eqs. (13), (14), and (18) in Eq. (12) provides an expression that can be integrated along x 1 to obtain the distribution of u 2e in this direction. However, it is necessary to obtain an approximation for u 2e at the initial point of integration. In order to do that, we use as inspiration the swept-wing approximation (Cebeci, 1999) and assume that u 2e can be approximated by 170 the velocity over a conical line (see Fig. 2). This approximation yields where 2ωr 0 is a reference velocity. However, Eq. (19) is not very accurate if u 1e is small, as is the case near the attachment line. As such, it is advisable to start the integration at a position x 10 downstream of the attachment line, where u 1e has a value that is comparable to the freestream velocity. An approximate initial value for u 2e at the corresponding x 10 location can be 175 found from 3 PSE model The coordinate system employed in the PSE analysis is the one in Fig. 1. The PSE is derived from the continuity, Navier-Stokes, energy, and state equations (Hanifi et al., 1994;Kundu et al., 2016), as shown in Eqs. (21) to (24). Because of the complexity 180 of performing a full three-dimensional analysis, periodicity is assumed in the x 2 direction. Moreover, rotation terms are added to the momentum equations.
where λ = − 2 3 µ denotes the second viscosity coefficient under the Stokes hypothesis. The quantities in these equations have been normalized with the reference values given in Table 1, with l 0 selected as the local boundary-layer thickness µ∞x1 ρ∞u∞ 1 2 .

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The curvature terms are evaluated numerically using standard geometrical relations (Råde and Westergren, 2004).
The flow can be decomposed as where t denotes time,q = ū 1 ,ū 2 ,ū 3 ,T,ρ T stands for the vector of variables of the base-flow from the BL model or the mean-flow from RANS (assumed O(1)), andq is the vector of the perturbation of these variables (assumed O( )) (Hanifi 200 et al., 1994). The perturbation part has the form whereq (x 1 , x 3 ) denotes the slowly varying part of the perturbation, i the imaginary unit, and Θ is found from where α and β are the wavenumber in the x 1 and x 2 directions, respectively, whereas γ denotes the temporal angular frequency 205 of the disturbance. Including these relations in Eqs. (21) to (24), assuming that the variation in the x 1 direction is weak compared to the variation in the x 3 one (there is a scale of 1/Re between them), neglecting terms of order 2 , and collecting terms that are multiplied byq, ∂q ∂x3 , ∂ 2q ∂x 2 3 , ∂q ∂x1 , respectively, into matrices B 1 , B 2 , B 3 , B 4 , we obtain a system of the form Equation (30) can be rewritten as an equivalent first-order system given by withr being a vector consisting of the amplitude functionq and its derivatives in the x 3 direction (Hanifi et al., 1994).
The computer algebra software Maple (Maplesoft, 2016) is used to obtain Eq. (31). In addition, the following normalization

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where the superscript * denotes the complex conjugate (Hanifi et al., 1994). The following boundary conditions are employed The derivatives in the x 3 direction are computed with a fourth-order compact finite-difference scheme, whereas the derivatives in the x 1 direction are computed with a second-order compact finite-difference scheme. Given initial values of α and β, the growth of the disturbances along x 1 is evaluated by marching Eq. (31) in the x 1 direction. In order to avoid restrictions on 220 the step size, the stabilization method described in Andersson et al. (1998) is employed.

Results
The accuracy of the proposed quasi-three-dimensional edge velocity, BL, and PSE models is investigated by comparison with results from the EllipSys3D RANS code (Michelsen, 1992(Michelsen, , 1994Sørensen, 1994Sørensen, , 2009). This solver is based on the incompressible Navier-Stokes equations and employs a block-structured, finite-volume discretization, including a second-order 225 upwind scheme for the discretization of convective terms and a central difference scheme for the discretization of the viscous ones. Turbulence is modeled using the SST k − ω turbulence model (Menter, 1993) and transition predictions are performed using a database method combined with a model for the turbulence intermittency factor γ (Michelsen, 1992(Michelsen, , 1994Sørensen, 1994Sørensen, , 2009. Two different full-scale wind-turbine rotors are investigated. Both have three blades, and their geometries are illustrated in Fig. 3. The shaded colors show a normalized measure of the axial position of each mesh point on the blade surface. The first geometry (Geometry 1) has a tapered and twisted blade with a symmetric NACA 63-018 airfoil profile along its entire span. It was mainly designed to allow the investigation of the accuracy of the conical-wing-based edge velocity model when applied to a geometry respecting its geometrical assumptions. The second geometry (Geometry 2) has a tapered and twisted blade with spanwise-varying cross-sectional properties. This enables the evaluation of the quasi-three-dimensional model when applied 235 to a general wind-turbine blade geometry. It is assumed that the flows over the three blades are similar so that it is sufficient to analyze one blade. We focus on the suction side of the blade since transition often occurs earlier there.
(a) Geometry 1 (b) Geometry 2 The main parameters of the two cases are given in Table 2. Both were computed using a temperature of 287.5 K, density of 1.225 kg · m −3 , dynamic viscosity of 1.784 · 10 −5 kg · m −1 · s −1 , ratio of specific heats of 1.4, and gas constant of 287 J · kg −1 · K −1 . The meshes used for the RANS computations of Geometries 1 and 2 have 15.5 · 10 6 nodes, of which 240 118 · 10 3 are surface ones. The boundary layer is discretized with approximately 50 nodes in the wall-normal direction. The corresponding meshes for the BL and PSE models have 200 and 500 points in this direction, respectively. This level of discretization provided spatially converged results for test cases. However, a lower number of grid points could be used for increased performance when computing the envelope of N factors with the PSE.

Spanwise edge velocity 245
In this section, we compare the chordwise distributions of spanwise velocity at the edge of the boundary layer obtained with numerical simulations and the edge velocity model (EVM). The streamwise velocity at the inviscid streamline u 1e required as input to the EVM is obtained from RANS and XFOIL simulations. Then, EVMR and EVMX refer to the EVM results with inputs from RANS and XFOIL, respectively. The analyses are performed in the inner (0 < r 0 /R ≤ 1/3), middle (1/3 < r 0 /R ≤ 2/3), and outer (2/3 < r 0 /R ≤ 1) parts of the blade, where r 0 is the radial position of analysis and R is the radius of 250 the wind-turbine rotor.   These results suggest that the edge velocity model can provide a reliable approximation for u 2e for radial positions not too close to the root of the blade and stagnation point. The results are expected to be more accurate for geometries respecting the swept-wing approximation, such as Geometry 1.

Velocity profiles
The streamwise and spanwise velocity profiles for Geometry 1, obtained from RANS and the proposed model, are shown in predicted by the EVMX model being higher than that from RANS. Figures 6e and 6f show that, at the outer radial position, there is close agreement between BLR and RANS results. The BLX results for u 2 display higher values than the BLR and RANS profiles, but the same shape. The results show that the BL model accurately predicts the profiles of streamwise velocity. Concerning the spanwise velocity, the agreement between the model and RANS profiles improves with the radial position. The disagreements are larger at the inner radial location, probably because of the influence of three-dimensionality generated by the root of the blade. The results are more accurate for Geometry 1 since it better agrees with the conical-wing approximation and has a constant airfoil geometry.

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The XFOIL-based results present a higher spanwise velocity than those from RANS. However, this ensues from the higher u 2e  values obtained with the EVMX model, due to differences in the c p distributions, and not from the BL model. The magnitude of the spanwise velocity is low, which might indicate a small influence on transition.
We investigate the effects of rotation on the spanwise velocity. Analysis of the EVMX data for Geometry 1 shows that the inviscid streamline is accelerated in the −x 2 direction near the stagnation point due to a negative x 2 pressure gradient and the 310 Coriolis force to a lesser extent. The dominant term of the latter is −2ρu 1 Ω 3 in Eq. (11), pointing in the −x 2 direction. After roughly 10 % of the chord, where the flow reaches its maximum streamwise velocity (fully developed flow), the spanwise pressure gradient vanishes. Hence, the centrifugal force, with leading term ρΩ 2 3 x 2 in Eq. (11), and the inertial term with ρu 2 1 in Eq. (12) overcome the Coriolis force and accelerate the flow in the +x 2 direction. For low radii, the Coriolis force tends to increase faster with the rotation speed than the centrifugal and inertial ones, impelling the flow in the −x 2 direction. For the 315 middle and outer parts of the blade, the centrifugal and inertial forces tend to grow faster with ω, forcing the flow in the +x 2 direction. Figure 8 presents the profiles of spanwise velocity obtained with the BLX approach at several rotation speeds for Geometry 1. The selected speeds are 5 %, 50 %, 100 %, and 150 % of that used in RANS (0.64 rad · s −1 ). One can observe that, compared to an almost translatoric situation (0.032 rad · s −1 ), rotation tends to accelerate the flow in the x 2 direction, driven 320 by the centrifugal and inertial forces. At the inner radial position, the spanwise velocity decreases for ω rising from 0.32 to 0.96 rad · s −1 because the Coriolis force grows faster than its counterparts. Considering r 0 /R = 0.58 and 0.89, the spanwise velocity increases with ω since the centrifugal and inertial forces have higher growth rates for larger radii.
The same analysis is carried out for Geometry 2, for which the rotation speed used in RANS is 0.9 rad · s −1 , and the results are presented in Fig. 9. The airfoils of Geometry 2 sustain negative streamwise and spanwise pressure gradients over a larger

Transition
The quasi-three-dimensional PSE model is applied to analyze the disturbance growth within the boundary layer. The stability analyses are performed with BLR 2D, BLR, BLX, and RANS base-flows. These analyses will be referred to as PSER 2D, 335 PSER, PSEX, and PSE RANS. Transition is assumed to occur when the amplification factor N based on the integral disturbance energy (Hanifi et al., 1994) reaches N crit . It is assumed N crit = 9 in the current work. In the EllipSys3D code, used to perform the RANS simulations, the intermittency factor γ is zero in the laminar region and one in the fully turbulent flow. γ starts to grow when the database method embedded in the solver indicates that transition occurs. Therefore, it is reasonable to select The transition locations as a function of the radial position are presented in Fig. 10 for Geometry 1. The results indicate that transition is delayed as the radial position increases. This is in agreement with observations from the literature that report stabilization effects of rotation for increasing radii (Du and Selig, 2000). At the inner part of the blade, up to r 0 /R = 0.40, PSER and RANS transition locations are close to each other. For the middle and outer parts, the RANS database method 345 indicates earlier transition locations than the PSER results with a maximum difference of 10 % at r 0 /R = 0.89. Moreover, the RANS and the PSER 2D results are close to each other, which possibly indicates that the RANS transition locations disregard stabilizing effects of three-dimensionality and are thus overly premature. The PSE RANS results (not shown) support this claim because they presented only modes that do not reach N crit . This fact means that the RANS base-flow becomes turbulent (stable) too early, before a mode could reach N crit . The later transition locations obtained with the PSER approach seem to 350 be a consequence of the stabilization provided by considering the velocity and gradients in the spanwise direction. The PSEX results indicate transition locations generally lying between those from the PSER and RANS. These differences arise from the pressure distributions from XFOIL not exactly matching those from RANS, although they are close to each other. The maximum difference between the PSEX and PSER results is 12 % at r 0 /R = 0.26. Figure 11 presents the transition locations for Geometry 2. The PSER and PSEX results are close to each other and indicate 355 later onsets of transition than the other methods. The maximum difference between PSER and RANS transition locations is 27 % at r 0 /R = 0.40. The discrepancies between PSER and PSEX results occur because the pressure distributions from XFOIL do not exactly reproduce those from RANS despite being close to each other. The RANS and PSER 2D transition locations lie near one another and indicate earlier transition onsets. It is possible to infer that RANS converges to a two-dimensional transition mechanism and that the three-dimensionality, as considered in the PSER and PSEX results, has a stabilizing effect.

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The fact that the PSE RANS results (not shown) presented no mode reaching N crit also supports the claim that transition is triggered too early in RANS and the validity of the later PSER and PSEX transition locations. The increase in the radial 20 https://doi.org/10.5194/wes-2020-107 Preprint. Discussion started: 12 November 2020 c Author(s) 2020. CC BY 4.0 License. position has the effect of delaying the transition onset. However, this effect is less marked in Geometry 2 because the relative importance of the rotation effects compared to the spanwise pressure gradient is smaller.   The results for Geometry 2 are presented in Fig. 14. There are differences between the modes in the vicinities of the airfoil 385 for the inner and middle radial locations. These variations are probably caused by the spanwise velocity, which is higher in the base-flow of the PSEX analyses. At the outer radial position, the PSER and PSEX modes converge since the spanwise velocity profiles are closer to each other. The modes causing transition in Geometry 2 also bear a resemblance to TS waves.
In the next, we analyze the effects of rotation on the transition location. Figure 15 presents the PSEX transition locations as a function of the radial position and rotation speed for Geometry 1. The analyses were performed with rotation speeds 390 corresponding to 5 %, 50 %, 100 %, and 150 % of that from RANS for Geometry 1 (0.64 rad · s −1 ). The trend shown in the picture indicates that the increase in the rotation speed accelerates transition. In particular, the rise in ω from 0.32 to 0.96 rad · s −1 makes transition occur 37 % earlier. The fact that the case corresponding to 5 % of the RANS rotation speed (not shown) did not present any mode reaching N crit further indicates the destabilizing effect of rotation. These effects take place through the rotation terms in the PSE and the variation of the spanwise velocity. The former seems to be preponderant since, 395 at r 0 /R = 0.89, there is no significant variation in the spanwise velocity with ω, but transition occurs earlier regardless. There is a delay in transition for increasing radius up to r 0 /R = 0.47. In this region, the Coriolis force is prevalent. Further increases in radius do not produce significant changes in the transition locations, indicating a balance between the rotation effects.    However, the variation of ω does not play a role as important as for Geometry 1. For instance, transition occurs 8 % earlier on average for an increase in ω from 0.45 to 1.35 rad · s −1 . The smaller sensitivity of transition to variations in the rotation speed ensues from the fact that the airfoils of Geometry 2 maintain favorable pressure gradients over a larger chordwise extent, which makes the rotation effects have smaller relative importance. Although the changes in the spanwise velocity with the rotation speed may affect the transition locations, the rotation effects embedded in the PSE seem to be the driving force of the variation in the transition onsets. This is because the spanwise velocity of Geometry 2, especially at the middle and outer radial locations, varies more with the rotation velocity than in Geometry 1, but the transition locations present smaller changes.
Transition is delayed when increasing the radius up to r 0 /R = 0.58, a range along which the Coriolis force is dominant. Only slight variations in the transition locations occur after this radial position, pointing to a balance in the rotation effects. This region lies mostly in the −η half-plane, meaning that the waves causing transition propagate in the −x 2 direction (towards the root of the blade). These modes present η = −27 • , −20 • , and −20 • for ω = 0.32, 0.64, and 0.96 rad · s −1 . Considering Geometry 2, in Figs. 17b, 17d, and 17f, we also observe the displacement of the critical region to lower x 1 /c with the increase in ω. Moreover, the flat critical region extending from η = −60 • to 40 • obtained with ω = 1.35 rad · s −1 shows that the higher 415 rotation velocity allows transition to be triggered by a broader range of disturbances. The critical region is located mostly in the −η half-plane, indicating stronger susceptibility of transition to waves traveling towards the root of the blade. The modes causing transition present η = −7 • , −15 • , and −28 • for ω = 0.45, 0.9, and 1.35 rad · s −1 . The analysis of the full geometry indicates that the increase in ω reduces the critical |η| in the region 0 ≤ r 0 /R ≤ r, where r = 0.58 and 0.5 for Geometries 1 and 2. After this position, the opposite occurs, i.e., rising ω leads to increasingly oblique critical modes. This inference is in 420 agreement with the observation of earlier transition at lower radii, since modes with smaller |η|, tending to two-dimensional waves, have a tendency to be more unstable.
The PSEX profiles of the modes leading to transition in Geometry 1 are displayed in Fig. 18. The variable being plotted is the disturbance in the streamwise velocity. All modes collapse at the inner radial location, indicating that the change in ω does not alter the transition mechanism. At r 0 /R = 0.58 and 0.89, the modes for ω = 0.64 and 0.96 rad · s −1 are in close 425 agreement, while the mode for ω = 0.32 rad · s −1 differs from the previous ones mainly close to the wall. The shape of the modes seems to be closely related to their propagation angles, with higher |η| modes tending to have only one peak like those at r 0 /R = 0.26. Figure 19 shows the results for Geometry 2. At the inner radial location, the modes for ω = 0.9 and 1.35 rad · s −1 are in agreement except close to the wall. The mode for ω = 0.45 rad · s −1 has a single-peaked structure and is associate with a high |η|. At r 0 /R = 0.58, the modes for ω = 0.45 and 0.9 rad · s −1 have double peaks and agree, whereas 430 the one for ω = 1.35 rad · s −1 does not present a pronounced peak close to the wall since its associated |η| is high. At the outer radial location, the modes present similar shapes and propagation angles. The critical modes in the two geometries resemble TS waves. The single-peaked modes observed at r 0 /R = 0.26 for Geometry 1 and r 0 /R = 0.40 for Geometry 2 (ω = 0.45 rad·s −1 ) might represent an intermediate stage between a TS and crossflow transition. These modes have η ≤ −50 • , and the spanwise velocity reaches its highest values at these locations.

Conclusions
A framework for transition prediction applicable to wind-turbine rotors comprising a model for the base-flow and a version of the PSE is developed. The technique accounts for quasi-three-dimensional and rotational effects. It aims at providing more reliable transition predictions than database methods at a computational cost lower than those requiring three-dimensional simulations. This work also analyzes the role of three-dimensionality and rotation on the transition onset.

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The developed method provides accurate profiles of streamwise velocity, and, for locations not too close to the root of the blade and stagnation point, spanwise velocity. The use of c p distributions from XFOIL as input to the model leads to an  Regarding the transition onset, three-dimensionality displays a stabilizing role. The quasi-three-dimensional effects considered in the developed model, such as the velocity and gradients in the spanwise direction, delay transition. This is true even though the spanwise velocity has a low magnitude in most of the blade. Conversely, considering a two-dimensional base-flow leads to earlier transition locations. These results are close to those from the database method in the EllipSys3D RANS code,