This work aims to develop a simple framework for transition prediction over wind-turbine blades, including effects of the blade rotation and spanwise velocity without requiring fully three-dimensional simulations. The framework is based on a set of boundary-layer equations (BLEs) and parabolized stability equations (PSEs), including rotation effects. An important element of the developed BL method is the modeling of the spanwise velocity at the boundary-layer edge. The two analyzed wind-turbine geometries correspond to a constant airfoil and the DTU 10-MW Reference Wind Turbine blades. The BL model allows an accurate prediction of the chordwise velocity profiles. Further, for regions not too close to the stagnation point and root of the blade, profiles of the spanwise velocity agree with those from Reynolds-averaged Navier–Stokes (RANS) simulations. The model also allows predicting inflectional velocity profiles for lower radial positions, which may allow crossflow transition. Transition prediction is performed at several radial positions through an “envelope-of-envelopes” methodology. The results are compared with the

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 aerodynamic loads can be influenced by the boundary-layer character, an accurate determination of the transition location can be significant to obtain a successful wind-turbine design. This has long been recognized by aerodynamicists, and significant efforts have been devoted to the development of transition models.

There are several transition models available

There are also more advanced transition-prediction methods, such as those based on direct numerical simulations (DNSs) and parabolized stability equations (PSEs)

In two-dimensional flow fields, the waves causing instability are typically of the Tollmien–Schlichting (TS) type

The present work aims to develop a simple model for transition prediction applicable for wind-turbine blades and to understand the effects of blade rotation on the boundary-layer flow and its stability. Firstly, a model to compute the boundary-layer profiles over the wind-turbine blades is developed. This model is based on the quasi-three-dimensional boundary-layer equations (BLEs)
and accounts for effects of the blade rotation and the three-dimensional outer flow. A technique to obtain an approximation for the spanwise velocity is also provided, such that the only required inputs are the chordwise distribution of pressure or streamwise velocity and the blade geometry. Secondly, the

This section describes the boundary-layer (BL) model developed in this work.

The coordinate system of the BL model is illustrated in Fig.

Coordinate system on the wind-turbine blade.

There are several integral formulations of the boundary-layer equations (BLEs)

In the BL model, the chordwise curvature of the wing model is neglected, while the radial curvature is considered. Thus, the metric vector becomes

Since the code is intended for analysis of laminar flows, turbulent fluctuations and statistics need not be considered. In order to obtain a well-conditioned system whose solution is compatible with the subsequent PSE analysis, the terms in the system of Eqs. (

Reference values.

As they stand, the BL equations are dependent on all three coordinate directions so that their numerical solution requires a full volume discretization. A three-dimensional discretization can result in a solution procedure that is costly in terms of computational capacity and CPU time. By employing approximate models for the derivative terms in the

Similarity solutions for rotating flows suggest that the velocity in the

The spanwise approximations described in Sect.

The BLEs can be expressed as

The solution is computed by space marching in the

The velocity in the

Conical parameters.

With this assumption, the derivative of

The coordinate system employed in the PSE analysis is the one in Fig.

The flow can be decomposed as

In the

Wind-turbine blades with radial sections of analysis. The surface is colored with a normalized measure of the axial position of the mesh point. The radial coordinate

The results of the proposed approach are compared to those from the EllipSys3D RANS code. This solver is based on the incompressible Navier–Stokes equations and employs a block-structured, finite-volume discretization, including a second-order 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 shear stress transport (SST)

Two different full-scale wind-turbine rotors are investigated. Both have three blades, and their geometries are illustrated in Fig.

The main parameters of the two cases are given in Table

Physical parameters of the wind turbines.

For the benefit of the reader, the abbreviations of the methods used in the following sections are summarized in Table

Abbreviations of the employed methods.

The pressure distributions from RANS and XFOIL are shown in Fig.

Comparison between XFOIL and RANS pressure distributions for the suction side of the airfoils of geometries 1 and 2 at three radial positions.

Here, we compare the chordwise distributions of spanwise velocity at the edge of the boundary layer

Figure

Spanwise edge velocity.

The results for Geometry 2 are presented in Fig

We present the chordwise and spanwise velocity profiles obtained with RANS simulations and the boundary-layer model as a function of the normal coordinate

Boundary-layer profiles for Geometry 1.

Boundary-layer profiles for Geometry 2.

The spanwise velocity at the inner radial position of Geometry 1 is directed towards the root of the blade as portrayed in Fig.

The BLR and RANS spanwise velocity profiles are in close agreement at the middle and outer radial positions of Geometry 1 as presented in Fig.

The BLR and BLX results for the spanwise velocity at the inner and middle radial parts of Geometry 2 (Fig.

The effects of rotation on the spanwise velocity are investigated using the approach of

Analysis of the data for Geometry 1 shows that the inviscid flow is accelerated in the

Figure

Spanwise velocity profiles for Geometry 1 for several rotation speeds.

Spanwise velocity profiles for Geometry 2 for several rotation speeds.

The boundary-layer profiles for Geometry 2 are presented in Fig.

The quasi-three-dimensional PSE model is applied to analyze the disturbance growth inside the boundary layer. The onset of transition is assumed to occur when the amplification factor

The transition locations for Geometry 1 as a function of the radial position are presented in Fig.

Figure

Transition locations.

The PSER contours of the

Figure

Figure

PSE results for the mode leading to transition.

The results for Geometry 2 are presented in Fig.

Transition locations for several rotation speeds.

In the following, we analyze the effects of rotation on the transition locations. Figure

Figure

PSEX results for the mode leading to transition for several rotation speeds.

The PSEX contours of the

Figure

A framework for transition prediction applicable to flows over wind-turbine blades is developed. The method, which comprises a boundary-layer model and the PSE, accounts for effects of the quasi-three-dimensional flow and the blade rotation. It aims to provide more reliable transition predictions without requiring three-dimensional simulations. Using the developed method, we have analyzed the role of flow three-dimensionality and rotation on the transition onset over two geometries.

The developed method provides accurate chordwise velocity profiles and, for locations not too close to the root of the blade and stagnation point, spanwise velocity. The flow is highly three-dimensional close to the root of the blade, reducing the accuracy of a quasi-three-dimensional approach. The spanwise velocity obtained with the model better agrees with RANS for geometries respecting the conical-wing approximation. Some of the spanwise velocity profiles contain inflection points, which may allow crossflow instability, not considered in two-dimensional transition models. Rotation was shown to accelerate the flow towards the tip of the blade in the developed flow region, while the opposite occurs near the stagnation point.

Transition locations from the

In order to better understand the transition process over the rotating blades and validate the prediction of the presented approach, in-depth investigation through DNS simulations and detailed experimental works are desired.

Part of the codes and data employed/developed is available upon direct request to the corresponding author.

TF implemented the models, performed the analysis, and wrote the final version of the manuscript. ML developed the model, obtained part of the results, and wrote the first version of the document. NS and FZ performed the RANS simulations using the EllipSys3D code. AH and DH developed the NOLOT PSE code (among other researchers), supported the analysis, provided useful discussions, and contributed with critical feedback. All authors reviewed the manuscript.

The authors declare that they have no conflict of interest.

This research has been supported and funded by the Strategic Research Area initiative STandUP for Energy.

This research has been supported by the STandUP for Energy.

This paper was edited by Sandrine Aubrun and reviewed by three anonymous referees.