The equations of the inductions of a right vortex cylinder have been derived in detail by . The most important equations and conclusions are summarized in this section. A cylindrical vortex sheet can be decomposed into tangential and longitudinal vorticity components. The strength of the tangential vorticity on the vortex cylinder is the ratio of the total vorticity strength to the helical pitch h. The total vorticity strength of the vortex cylinder ΔΓtot is equal to the trailed vorticity strength of all blades. 8γt=-ΔΓtoth

The tangential vorticity contributes to both axial and radial induced velocities. For the vortex cylinder with radius R at axial position x=0, the axial and radial induced velocity at the calculation point with radius of r and axial position of x are shown to be in the form of complete elliptic integrals . 9ua(r,x)=γt2R-r+|R-r2|R-r+xk(r,x)2πrRKk2(r,x)+R-rR+rΠk2(r,0),k2(r,x),10ur(r,x)=-γt2πRr2-k2(r,x)k(r,x)Kk2(r,x)-2k(r,x)Ek2(r,x), where 11k2(r,x)=4rR(R+r)2+x2, and K(m), E(m) and Π(n,m) are the complete elliptic integral of the first, second and third kind.

The other components of the vortex cylinder, which are the bound vortex disc, the longitudinal vorticity and the root vortex line, only have contribution to the tangential velocity. The tangential induced velocity of the entire flow field is derived by as follows: 12ut(r,x)=-ΔΓtot4πr(r<R,x=0)or(r=R,x>0)-ΔΓtot2πr(r<R,x>0)0otherwise(outsidethevortexcylinder).

Consider the superposition of the Joukowski rotor model to achieve radially varying bound circulation. There will be helical trailed vorticities emanated along each blade with the strength equal to the derivative of the bound circulation strength with respect to the radius. In Joukowski's rotor model, it is assumed that the radial distribution of the bound circulation of all blades is the same. Then consider the corresponding vortex cylinder model that is the limiting case of Joukowski's rotor model, where the number of blades tends to infinity. It is consisted of a superposition of cylindrical vortex sheets with both tangential and longitudinal vorticity. Details of the superposition of vortex cylinders have been described by . The most important aspects are summarized in this section.

With the superposition of the vortex cylinders, the bound circulation is assumed to be piecewise constant along the blade. The blade is discretized radially into n sections, and there will be one calculation point for each section. Consequently, there will be (n+1) trailing points corresponding to (n+1) vortex cylinders. For the innermost part of the rotor, which is usually the rotor hub and is defined from the center of rotation to the beginning of the first blade section, the bound circulation strength is zero since there are no blades. For the ease of notation and calculation, two ghost sections with the index of 0 and n+1 with zero circulation strength are introduced. For the system, the number of unknown variables is n, which is equal to the number of sections. A sketch of the superposition of the cylindrical vortex system is shown in Fig. .

The closure of the system determines the tangential vorticity strength of each vortex cylinder. The closure of the system is determined at the far wake, and the cylindrical vortex sheet is assumed to convect at a constant speed equal to the mean of the two far-wake velocities surrounding the vortex sheet . It can be shown that the system closure can be performed in the form of helical pitch h(r) or the annulus axial induction factor a(r) at the rotor disc, and the two formulations are equivalent to each other. The system closure in the form of the annulus axial induction factor will be used in this work and will be briefly described.

With the system closure, the axial induction factor of section i is calculated as follows: 13ai=-ua,iU0=121-1-CT,eff,i.

The effective thrust coefficient CT,eff,i is equal to the thrust coefficient from the Kutta–Joukowski analysis CT,KJ,i minus the contribution of wake rotation CT,rot,i. 14CT,eff,i=CT,KJ,i-CT,rot,i, where 15CT,KJ,i=NBfi,x*dr12ρU022πridr=ks,i1+ks,i4λri2=ks,i1+ai′,16CT,rot,i=∑j≥i+1ks,j221λRj2-1λRj+12.

The wake rotation effect increases toward the rotor rotational axis and decreases when the tip-speed ratio increases. For typical modern wind turbine designs the effect of this term is rather small and may be neglected.

In Eqs. () and (), ks,i is the total non-dimensional bound circulation of section i, and λr is the local speed ratio for the position with radius r. The tangential induction factor ai′ is defined as follows and can be calculated according to Eq. () with the condition of x=0. 17ks,i=ΩΓiπU0218λr=ΩrU019ai′=-ut,iΩri=∑j=inΓj-Γj+14πriΩri=ks,i4λri2

Considering Eqs. () and (), when the wake rotation effect is included, the aerodynamic loading of a section is dependent on all sections that are further outboard compared to it. The system could be solved from outboard to inboard using Eq. (), and the system closure is completed when the annulus axial induction factor of all blade sections is calculated. Then, the tangential vorticity of the vortex cylinder that is just inside section i is obtained from the annulus axial induction factor of this section and the neighboring section inside. The equation can be derived from Eq. () using the condition of x=0. 20γt,i=2U0ai-ai-1

In the vortex cylinder model, the relationship between the axial induction factor and the effective thrust coefficient is in the same form as in the momentum theory: CT,eff,i=4ai(1-ai), by inversing Eq. (). However, when the thrust coefficient (CT) is high, especially when CT>1, the momentum theory breaks down and the vortex cylinder model gives unphysical results. Then, corrections should be made for these high-thrust conditions. Different high-thrust corrections are available, such as the linear extrapolation by and the polynomial function of a and CT by . To be consistent with the BEM module implemented in the HAWC2 code , the polynomial function of a and CT by is chosen. Then, in the system closure, the equation of axial induction factor from the thrust coefficient in Eq. () should be replaced by 21ai=fa-CTMadsenCT,i=k3CT,i3+k2CT,i2+k1CT,i, where the coefficients k1 … k3 are defined: k1=0.2460, k2=0.0586 and k3=0.0883.

The system closure of the vortex cylinder model for planar rotors has been described in Sect. . It is assumed that each vortex cylinder convects with a constant velocity that is determined in the far wake. With this method of system closure, the relationship between the vortex cylinder model and the momentum theory is revealed by , and the results are generally in good agreement with higher-fidelity models. The reason is probably that the error introduced when assuming the convective velocity is constant balances the error introduced when assuming the non-expanding wake as shown for the uniformly loaded disc . In the present work, we assume the same method of system closure as for the planar rotor case: the closure is determined at the far wake and can be used for non-planar rotors with moderate out-of-plane shapes. With this assumption, the equations are in concise forms and can clearly show the connections between the model for a non-planar rotor and a planar rotor. However, this assumption does not necessarily hold for extreme cases. To investigate this, a numerical test of blades with relatively large prebend and cone angle will be shown in Sect. .

For illustration, we show the superposition of the vortex cylinders for a non-planar rotor and the corresponding planar rotor with the same radial discretization in Fig. . For each section, the corresponding planar rotor has the same total bound circulation (of all blades) Γipl as that of the non-planar rotor Γinp.

The relationship between the vortex cylinder model and the momentum theory will be described separately for the planar rotor and the non-planar rotor.

For the blade with out-of-plane shapes, it is necessary to include the unsteady airfoil aerodynamics model (usually referred to as the dynamic stall model), even for the steady-state simulation. Otherwise, the results of the tangential forces will have a visible error. The reason originates from the conclusions of unsteady 2-D aerodynamics: the correct circulatory lift can be obtained if the magnitude of the effective angle of attack is determined at the three-quarter-chord point, but the direction of it should be determined by the velocity at the quarter-chord point . For the low-fidelity model using blade element theory, the angle of attack is usually determined at only one calculation point per section. For instance, for the BEM module in the HAWC2 code, it corresponds to the three-quarter-chord point. The direction of the circulatory lift is then equivalently transformed to perpendicular to the velocity at the quarter-chord point by including an additional torsion rate drag that is proportional to the circulatory lift. In addition, the non-circulatory part of the lift force should be correctly included. For the blade with out-of-plane shapes, even when the rotational speed is constant, the mid-chord point will experience a component of acceleration that is perpendicular to the chord due to the projection of centrifugal acceleration. In addition, the angular velocity vector will also have a projection in the 2-D section that will result in an effective pitching motion of the airfoil section. The contribution of both the mid-chord acceleration and the torsion rate should be included when calculating the aerodynamic force. For details, see the reports by and .

For blades with in-plane or out-of-plane shapes, the curved blade length in an elementary annulus (ds) is different from the change of the radius (dr). Then, for the momentum analysis in a stream tube, it is necessary to multiply the local thrust and torque coefficient with the term dsdr to account for this difference . For the Kutta–Joukowski analysis of the non-planar rotor in Eq. (), the term dsdr is already included in the equation of force. So, it is not necessary to include this term again in the thrust coefficient in Eq. () during the convergence calculation using the vortex cylinder model.

According to the relationship between the momentum theory and the vortex cylinder model described in Sect. , there are two possible methods of using the vortex cylinder model when coupling with the blade element theory. Firstly, the vortex cylinder model can be used as a modification to the existing BEM model, which is named and labeled as the BEM-VC model. Otherwise, the vortex cylinder model can be considered as the complete replacement of the momentum theory in the BEM method. With the blade element theory coupled with the vortex cylinder model, we have the blade element vortex cylinder (BEVC) model, which does not include any momentum theory results. The reader may argue that the BEVC model should completely replace the BEM model for the prediction of the aerodynamic loads in an existing aeroelastic code. As a first step, the authors recommend the use of the vortex cylinder model as a correction to the BEM method (BEM-VC). This is because the framework of the BEM method with many sub-models and corrections has been implemented in the aeroelastic codes and has been intensively tested.

Prandtl's tip-loss factor is commonly applied to the BEM method to account for the difference between a finite number of blades and the assumption of an infinite number of blades in the momentum theory . Similarly, the vortex cylinder model also assumes infinite number of blades. Also, considering the relationship between the momentum theory and the vortex cylinder model discussed in the previous sections, a tip-loss correction should be applied to the vortex cylinder model. The tip-loss factor F presented by  was implemented in the BEM module in the HAWC2 code : 35F=2πcos⁡-1exp⁡-NB2Rtot-rrsin⁡φ, where φ is the inflow angle.

The tip-loss correction is applied by scaling the thrust coefficient with the inverse of the tip-loss factor when calculating the blade axial induction. 36aBpl=fa-CTCT,eff/F, where the subscript “B” indicates the induction at the blade.

As has been described previously in this section, the proposed vortex cylinder model can be used in two ways: either used as a correction to the BEM method (BEM-VC) or solely used and coupled with the blade element theory (BEVC). Details of the implementation of both methods have been described previously in this work and are summarized in Algorithm 1.

The pressure-based incompressible three-dimensional solver EllipSys3D was used to solve the Reynolds-averaged Navier–Stokes equations, using a finite-volume discretization. An inlet/outlet strategy was followed for the boundary conditions of the outer limit of the CFD domain. The flow was assumed to be fully turbulent, and the k-ω SST model was employed. These higher-fidelity simulations are labeled in the present work as CFD.

Several rotor-resolved meshes were built. They were generated in two consecutive steps, which were fully scripted in order to ensure a similar resulting grid quality. First, a structured mesh of the blade surface was generated with the openly available Parametric Geometry Library (PGL) tool . A total of 128 cells were used in the spanwise direction, and the chordwise direction was discretized with 256 cells. Secondly, the surface mesh was radially extruded with the hyperbolic mesh generator HypGrid to create a volume grid. A total of 256 cells were used in this process, and the resulting outer domain was located at approximately 11 rotor diameters. A boundary layer clustering was taken into account, with an imposed first cell height of 1×10-6 m, in order to target y+ values lower than unity. The resulting volume meshes accounted for a total of 14.2 million cells. The grid topology is illustrated in Fig. , through the particular case of the baseline straight blade.

While a steady solver was used, unsteady separation is expected near the root of the wind turbine blade in operation. To mitigate the effects that this can have on the conclusions of the present work, all the CFD quantities were averaged for the last 350 iterations.

The lifting-line module in the aerodynamic solver MIRAS is implemented as a time-marching approach and uses the 2-D airfoil data. This work uses a modified version of the lifting-line module that includes the influence of the curved bound vortex on the induced velocity as described by , which is labeled as LL-mod in that work. The bound vorticity is located at the quarter-chord line, and the calculation points are placed on the three-quarter-chord line. The influence of the curved bound vortex is modeled by adding the difference of the induced velocity due to the 3-D bound vorticity and an imaginary 2-D bound vorticity (infinitely long line vortex) evaluated at the three-quarter-chord point to the induced velocity of the blade section . The curved bound vortex influence is assumed to be constant along the chord. The angle of attack to determine the circulatory lift and drag coefficient is the angle of attack at the three-quarter-chord point. The flow environment to determine lift and drag, especially the direction of the lift and drag force, is at the quarter-chord point. In addition, as the discussion in Sect. , the non-circulatory lift is included in the lifting-line model when calculating the aerodynamic forces. For the setup in this study, each blade is discretized into 50 sections with cosine spacing. Each simulation is calculated for 20 000 time steps and each step correspond to 1.5∘ of azimuthal angle, resulting in a total of 83.3 revolutions. The airfoil data are from 2-D fully turbulent CFD results . The vortex core size in the calculations is 0.1 % of the local chord length. The first row of trailed vorticities begins from the trailing edge of the blade.

Three low-fidelity aerodynamic models are used for the comparison. The first one is the BEM method implemented in the HAWC2 code version 12.8 . The second one is the BEM method with radial induction correction in Eq. () (BEM-ur) in the same version of HAWC2 code and is described by . The third one is the BEM-VC method proposed in this work and is implemented in a test version of the HAWC2 code based on version 12.8. As has been discussed in Sect. , the proposed BEM-VC method utilizes the vortex cylinder model as a correction to the existing BEM method. The results should be identical to the BEVC model when the drag is excluded in the momentum balancing. So, the results from the BEVC model, which solely uses the vortex cylinder model and does not directly use the momentum theory, are not shown. For the low-fidelity models in the HAWC2 code, each time step corresponds to 0.01 s and each simulation is calculated for 700 s to get the steady-state value. Each blade is discretized radially into 80 sections. The airfoil data are also from 2-D fully turbulent CFD results and are identical to those used in the lifting-line method. As has been described in Sect. , the flow angle seen by the airfoil section is used to calculate the tip-loss factor for the BEM-VC method. For the BEM method and BEM-ur method, the original implementation of the tip-loss factor in the HAWC2 code using the inflow angle is applied. Since all three low-fidelity models are implemented in the HAWC2 code, it is then guaranteed the transformation of the velocity and force between different coordinate systems during the computation is consistent.

There are five different wind turbine blades used for the comparison; all of them are based on the IEA-10.0-198 10 MW reference wind turbine (RWT) . The baseline straight blade is modified by aligning the half-chord line to a straight main axis. For the upwind dihedral blades, the main axes that determine the planforms are obtained from modified Bézier curves which are parameterized with dihedral ratio r‾s, dihedral magnitude Δd and tip dihedral angle Λtip, which are illustrated in Fig. . In addition, some cases with large cone angles are applied to these dihedral blades to exploit the range of capability of the models. The radius of the unconed rotor is 99 m, of which the hub radius is 2.8 m. The parameterization of the dihedral blades is very similar to that for the previous study of swept blades . The dihedral blades W-1 to W-4 correspond to Blade-1 to Blade-4 in the previous study but with out-of-plane shapes (dihedral) instead of in-plane shapes (sweep). The blades are assumed to be stiff, which means the effect of elastic deformation is not included. In addition, the pitch angle is zero for all test cases.

The parameters of these upwind dihedral blades are summarized in Table . The main axes of these dihedral blades are illustrated in Fig. . The purpose of having dihedral blades with different dihedral magnitude and different tip dihedral angle is to represent different possible shapes of the dihedral blades.

The airfoils are aligned perpendicular to the curved main axis, which is the half-chord line. The chord and twist distribution of the dihedral blades remain unchanged compared to the baseline straight blade. The radius of the dihedral blades is identical to that of the baseline straight blade, but the curved blade length is increased due to the dihedral. For the simulations in this section, the uniform inflow of 8 m s-1 with no yaw error is applied to the rotor with a constant rotational speed of 0.855 rad s-1. For the unconed rotors, the tip-speed ratio is 10.58. At this operational condition, the thrust coefficient of the unconed rotor with baseline straight blades is 0.90 and the rotor power coefficient is 0.46, as predicted using the BEM method. At a radius of 70 m, the angle of attack predicted by the BEM method is 5.76∘.

For the test cases described in Sect. , the distributed aerodynamic loads calculated from different aerodynamic models are summarized and are compared in this section. For the calculation of the aerodynamic loads, both lift and drag force are included. In this study, the focus is on the influence of the blade dihedral on the loads. The near-root region (i.e., up to an approximate radius of 20 m) experienced flow separation in the CFD solution, and it is not the focus of this study.

The integrated aerodynamic loads, which are the aerodynamic power and thrust from different models, are compared in this section. Please note that when comparing the integrated aerodynamic loads, errors in the distributed loads may cancel out. So, it is important to bear in mind that the performance of the different aerodynamic models is not fully represented by their abilities to predict the total aerodynamic power or thrust of the rotor. The aerodynamic force (per unit length of radius) F* on each blade section is composed of the axial force Fa*, the tangential force Ft* and the radial force Fr*. They are defined to be positive when aligned with x, z and y coordinate respectively as defined in Fig. . For the calculation of the aerodynamic load, both lift and drag force are included. The position of applying the force on the blade section is p. For simplicity, we use the half-chord point coordinate as p. This means we neglect the distance between the half-chord point and the quarter-chord point and also the contribution of the twist to the vector p. In addition, the contribution of the sectional airfoil aerodynamic moment (calculated from Cm) to the aerodynamic momentum of the rotor is also neglected. Then, the distributed aerodynamic moment from each blade section is 42M=p×F*=xy0×Fa*Fr*Ft*=yFt*-xFt*xFr*-yFa*.

The x component of the aerodynamic moment M is the contribution to aerodynamic torque. Then, the aerodynamic power of the rotor is the integrated contribution of Mx of all NB blades at the rotational speed of Ω: 43P=NBΩ∫0RtotyFt*dr.

The aerodynamic thrust of the rotor is the total contribution of the axial force of all NB blades: 44T=NB∫0RtotFa*dr.

The aerodynamic power and thrust of the rotor with baseline straight blades without cone are summarized in Table . For the rotors with dihedral blades without cone, it is difficult to directly draw conclusions from the absolute value of power and thrust. To better illustrate and compare the integral effects of the rotor dihedral represented by the aerodynamic power and thrust predicted using different methods, the relative difference of the aerodynamic power and thrust with respect to the baseline rotor from each method are calculated and are summarized in the bar plots in Figs.  and .

For the aerodynamic power, the relative change predicted by LL is underestimated compared to the prediction by CFD, but the results are showing similar trends. One of the reasons could be the use of the 2-D airfoil data in the lifting-line method. The ordinary BEM method predicts almost no influence of blade dihedral on power, except for W-1, which predicts the correct direction in which the power increases but underestimates the magnitude. The BEM-ur method predicts the same direction as CFD and LL in which the power of the dihedral rotors are increased compared to the baseline. However, the magnitude of the increment is overestimated by a factor of approximately 2. Comparing to the other two BEM methods, the relative increment of power predicted by the proposed BEM-VC method is in better agreement with the predictions by CFD and LL.

For the aerodynamic thrust, the magnitude of the relative decrement is overestimated by approximately 20 % by LL compared to CFD. The BEM method predicts the correct trend that the thrust decreases but the magnitude is underestimated compared to LL and CFD. The BEM-ur method predicts a small increase in aerodynamic thrust instead of a decrease in thrust as predicted by the higher-fidelity models. The relative change of the thrust predicted by the proposed BEM-VC method is very similar to the results predicted by LL, and the relative difference of the predicted relative change is less than 20 %.

In summary, the proposed BEM-VC model is in better agreement with higher-fidelity models when predicting the integrated aerodynamic power and thrust of the dihedral rotor, compared to the ordinary BEM method. The BEM method with radial induction correction (BEM-ur) predicts worse compared to the BEM method for both aerodynamic power and thrust.

The results shown up to this point in this work all correspond to fairly high thrust coefficients around approximately 0.9. In this section, the results for operational conditions corresponding to lower thrust coefficients are shown. For simplicity, three of the operational conditions defined in the IEA Wind TCP Task 37 report that anticipated lower thrust coefficients are used. The operational conditions of these lower loading cases are summarized in Table , by means of wind speed, tip-speed ratio and pitch angle θp. For each case, the rotational speed is 0.909 rad s-1, and uniform inflow with no yaw error is applied to the rotor. For each operational condition, the baseline straight blade is pitched with θp, and the thrust coefficient predicted by BEM is also included in Table . In order to allow the comparability of the results of this straight blade with the blade accounting for a dihedral angle, the effects of pitching were introduced as an additional twist instead of a rotation around the pitching axis. The motivation for this was related to the introduction of in-plane geometry components by pitching the blade. The twisted blade is named W-1-U*, where * is the wind speed.

For the results shown in the previous sections, the lifting-line (LL) results were in good agreement with the CFD results. Therefore, only the LL method is used to generate the higher-fidelity results for the comparison in this section.

First, the differences of the axial load and tangential load of the dihedral blade W-1-U12 and the pitched straight blade are shown in Fig. . As for the high loading case in Fig. , both the ordinary BEM method and the BEM method with radial induction correction (BEM-ur) are not able to correctly predict the influence of the blade dihedral. In comparison, the proposed BEM-VC method correctly predicts the shape and magnitude of the load redistribution for both axial and tangential loads. The decrease in the load further inboard compared to where the blade starts to become dihedral is also well predicted. As for the high loading case, the largest difference between the LL and the BEM-VC results is mainly for the tip-most part. As previously mentioned, this could be due to the use of Prandtl's tip-loss correction in the model.

The results for the dihedral blades W-1-U15 and W-1-U20, at wind speeds of 15 and 20 m s-1 respectively, are shown in Figs.  and . The same conclusion can be made from these lower loading results. However, with the decrease in the thrust coefficient, the difference between the prediction by the BEM method and the LL method decreases. This is because the relative importance of the induced velocities decreases in these cases, making the performance more directly dictated by the velocity components from the inflow and the rotor rotation and less from the induced velocities.

The computational efforts to obtain the steady-state results used in the present work, measured in CPU time, are summarized in this section. The CFD computations using EllipSys3D were performed on DTU's high-performance computing (HPC) cluster Jess, in which each node has 20 cores running at 2.8 GHz. All the CFD simulations of the present work required a wall clock time of approximately 3.5 h when using 216 cores. The lifting-line (LL) computations using the MIRAS code were performed on the Sophia HPC cluster, in which each node has 32 cores running at 2.9 GHz. Each of the LL simulations in the present work required a wall clock time of approximately 100 h when using 32 cores. Note that the computational time for the LL method in the MIRAS code in this study is relatively high, because the settings were chosen to achieve the highest possible fidelity irrespective of the computational cost. Therefore the computational effort for the MIRAS calculations in this work is not indicative of the performance for normal use of the tool. Settings that increased the computational effort in this work are small time steps, not using far-wake cutoff, etc. The computational time is expected to be largely decreased if efforts are dedicated to improving the simulation setup. However, this is beyond the scope of the present work.

The computations using the HAWC2 code were performed on a single core of a 2018 workstation at 4.8 GHz. The simulations were performed with structural properties included and with large stiffness to approximate stiff structures. The simulations were run for 600 s in the simulation time to reach steady state. The simulations required a wall clock time of approximately 600 and 650 s for the BEM method and the BEM-VC method, respectively. For a stand-alone version of the BEM method or the BEM-VC/BEVC method, one steady-state computation can be done in much less than 1 s using a single CPU core.