### Abstract

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. FSI MODELING
- 3. DISCRETE SOLUTION PROCEDURES
- 4. SIMULATION OF THE NREL 5MW OFFSHORE BASELINE WIND TURBINE ROTOR
- 5. CONCLUSIONS
- Acknowledgements
- REFERENCES

In this two-part paper, we present a collection of numerical methods combined into a single framework, which has the potential for a successful application to wind turbine rotor modeling and simulation. In Part 1 of this paper we focus on: 1. The basics of geometry modeling and analysis-suitable geometry construction for wind turbine rotors; 2. The fluid mechanics formulation and its suitability and accuracy for rotating turbulent flows; 3. The coupling of air flow and a rotating rigid body. In Part 2, we focus on the structural discretization for wind turbine blades and the details of the fluid–structure interaction computational procedures. The methods developed are applied to the simulation of the NREL 5MW offshore baseline wind turbine rotor. The simulations are performed at realistic wind velocity and rotor speed conditions and at full spatial scale. Validation against published data is presented and possibilities of the newly developed computational framework are illustrated on several examples. Copyright © 2010 John Wiley & Sons, Ltd.

### 1. INTRODUCTION

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. FSI MODELING
- 3. DISCRETE SOLUTION PROCEDURES
- 4. SIMULATION OF THE NREL 5MW OFFSHORE BASELINE WIND TURBINE ROTOR
- 5. CONCLUSIONS
- Acknowledgements
- REFERENCES

Coupled fluid–structure interaction (FSI) simulations at full scale are essential for accurate modeling of wind turbines. The motion and deformation of the wind turbine blades depend on the wind speed and air flow, and the air flow patterns depend on the motion and deformation of the blades. In recent years, stand-alone 3D fluid mechanics simulations with simplified wind turbine configurations were reported in 1–4, some at reduced scale and some with limitations in terms of the representation of the exact geometry and prediction of the FSI involved. Structural analyses of the individual turbine blades under assumed load conditions or loads coming from separate computational fluid dynamics simulations were also reported (see, e.g. 5–8). To the best of our knowledge, no coupled fluid–structure simulations of the full-scale wind turbine blades were attempted. This problem presents a significant computational challenge because of the high wind speeds, complex and sharp geometric features and sizes of the wind turbines under consideration. This in part explains the current, modest nature of the state-of-the-art in wind turbine simulation. In order to simulate the coupled problem, the equations governing air flow and blade motions and deformations need to be solved simultaneously, with proper kinematic and dynamic conditions coupling the two physical systems. Without that the modeling cannot be realistic.

In this work, we use isogeometric analysis based on non-uniform rational B-Splines (NURBS) 9 for FSI modeling of wind turbine rotors. In Part 1 10 of this paper, the wind turbine geometry modeling and aerodynamics simulation procedures were described in detail and the validation results were presented. In this work, we focus on the details of structural and FSI modeling.

The blade structure is governed by the isogeometric rotation-free shell formulation with the aid of the bending strip method 11. The method is appropriate for thin shell structures comprised of multiple *C*^{1}- or higher-order continuous surface patches that are joined or merged with continuity no greater than *C*^{0}. The Kirchhoff–Love shell theory that relies on higher-order continuity of the basis functions is employed in the patch interior as in 12. Strips of fictitious material with unidirectional bending stiffness and with zero mass and membrane stiffness are added at patch interfaces in the overlapping fashion. The direction of bending stiffness is chosen to be transverse to the patch interface. This choice leads to an approximate satisfaction of the appropriate kinematic constraints at patch interfaces without introducing additional stiffness to the shell structure. Furthermore, as the functional representation of the structural patches is enriched, the thickness of the overlap region goes to zero. Although NURBS-based isogeometric analysis is employed in this work, other discretizations such as T-Splines13, 14 or Subdivision surfaces 15–17 are also well suited for the proposed structural modeling methodology.

The FSI formulation presented in this paper assumes matching discretization at the fluid–structure interface. We adopt a strongly coupled solution strategy and employ Newton linearization to solve the nonlinear coupled equations for the fluid, structure and fluid mesh motion. However, the fluid, structure and mesh linear solves are decoupled at the Newton iteration level, leading to a block-iterative FSI procedure 18. The approach is robust due to the relatively large rotor mass. We note that the lack of rotational degrees of freedom in the structural discretization facilitates the strong FSI coupling.

The paper is outlined as follows. In Section 2 we describe the individual constituents of the FSI problem. We recall the air modeling approach from Part 1 of this paper. We give details of the structural formulation for wind turbine blades that are based on the bending strip method. We also briefly describe our composite material modeling procedures for wind turbine blades. We then focus on the problem of the fluid domain motion. We develop a formulation in which the rotating part of the fluid domain motion is handled exactly, whereas the rest is computed using linear elastostatics. We conclude the section with a statement of a fully coupled FSI problem. In Section 3 we present our discrete solution procedures for the coupled FSI problem. We also introduce a new class of time integration procedures for structures dominated by large rotational motions. In Section 4 we simulate the NREL 5MW offshore baseline wind turbine rotor 19 and present the computational results. In Section 5 we draw conclusions and outline future research directions.

### 3. DISCRETE SOLUTION PROCEDURES

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. FSI MODELING
- 3. DISCRETE SOLUTION PROCEDURES
- 4. SIMULATION OF THE NREL 5MW OFFSHORE BASELINE WIND TURBINE ROTOR
- 5. CONCLUSIONS
- Acknowledgements
- REFERENCES

In this section, we briefly summarize our space discretization approach of the coupled FSI problem given by Equation (41). We also present an adaptation of a class of time integration procedures for structures dominated by large rotational motions.

The solid and fluid mesh motion equations are discretized using the Galerkin approach. The fluid formulation makes use of the residual-based variational multiscale method 28, 29, which was presented in detail for moving domain problems in Part 1 of this paper. The coupled FSI equations are advanced in time using the Generalized-alpha method 21, 30, 31. Within each time step, the coupled equations are solved using an inexact Newton approach. For every Newton iteration, the following steps are performed: 1. We obtain the fluid solution increment holding the structure and mesh fixed; 2. We update the fluid solution, compute the aerodynamic force on the structure, and compute the structural solution increment. The aerodynamic force at control points or nodes is computed using the conservative definition given in Part 1 of this paper; 3. We update the structural solution and use elastic mesh motion to update the fluid domain velocity and position. We recall that only the deflection part of the mesh motion is computed using linear elastostatics, whereas the rotation part is computed exactly. This three-step iteration is repeated until convergence to an appropriately coupled discrete solution is achieved. The proposed approach, also referred to as ‘block-iterative’ (see 18 for the terminology), is stable because the wind turbine blades are relatively heavy structures.

Because the structural nonlinearity is stronger than that of the fluid, it may be beneficial to take additional inner iterations on the structure to improve its convergence.

In the proposed FSI framework, the fluid and structural solves are decoupled. This gives us the flexibility of adjusting the structure time integration procedures to better capture the important features of the solution. In particular, we note that the bulk of the structural displacement comes from rotation of the blades about the horizontal axis. To better approximate rotation, we separate the structure nodal or control point degrees of freedom into rotation and deflection as follows. Let **U**, and be the vectors of nodal or control point displacements, velocities and accelerations, respectively. We set

- (42)

- (43)

- (44)

where **U**_{θ}, and are given by

- (45)

- (46)

- (47)

The above Equations (45)–(47) present an exact relationship between the nodal or control point displacements, velocities and accelerations corresponding to the rotation. To relate the deflection degrees of freedom between time levels *t*_{n} and *t*_{n + 1}, we make use of the standard Newmark formulas 32

- (48)

- (49)

where γ and β are the time integration parameters chosen to maintain second-order accuracy and unconditional stability of the method, and Δ*t* = *t*_{n + 1} − *t*_{n} is the time step size.

Combining exact rotations given by Equations (45)–(47) and time-discrete deflections given by Equations (48) and (49), we obtain the following modified Newmark formulas for the total discrete solution:

- (50)

- (51)

We employ Equations (50)–(51), in conjunction with the Generalized-alpha method, for the time discretization of the structure.

In the case of no rotation, for which **R** is an identity tensor, Equations (50) and (51) reduce to the standard Newmark formulas. In the case of no deflection, pure rotation is likewise recovered.

### 4. SIMULATION OF THE NREL 5MW OFFSHORE BASELINE WIND TURBINE ROTOR

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. FSI MODELING
- 3. DISCRETE SOLUTION PROCEDURES
- 4. SIMULATION OF THE NREL 5MW OFFSHORE BASELINE WIND TURBINE ROTOR
- 5. CONCLUSIONS
- Acknowledgements
- REFERENCES

The wind turbine rotor is simulated at prescribed steady inlet wind velocity of 11.4 m/s and rotor angular velocity of 12.1 rpm. This setup corresponds to one of the cases reported in 19. The problem setup is illustrated in Figure 2. The dimensions of the problem domain and the NURBS mesh employed are the same as in Part 1 of this paper. The properties of air are taken at standard sea-level conditions. The time step is chosen to be Δ*t* = 0.0003s.

As in Part 1, rotationally periodic boundary conditions for the fluid are imposed in order to reduce computational cost. However, because the rotor blades are subject to gravity forces, a fully rotationally periodic structural solution is not expected in this case. Nevertheless, we feel that the use of rotationally periodic boundary conditions for the fluid domain is justified due to the fact that the fluid periodic boundaries are located sufficiently far away from the structure and are not expected to affect the structural response. We note that rotationally periodic boundary conditions were employed earlier in 33, 34 for parachute simulations.

A symmetric fiberglass–epoxy composite with [ ± 45/0/90_{2}/0_{3}]_{s} lay-up, which enhances flap-wise and edge-wise stiffness is considered for the rotor blade material. The 0^{∘} fiber points in the direction of a tangent vector to the airfoil cross-section curve. The orthotropic elastic moduli for each ply are given in Table I. For simplicity, the entire blade is assumed to have the same lay-up. The resulting **A**, **B** and **D** matrices from Equations (18)–(20) are

- (52)

- (53)

- (54)

The total laminate thickness distribution is shown in Figure 3(a). The blade shell model together with the bending strips covering the regions of *C*^{0}-continuity is shown in Figure 3(b).

Table I. Material properties of a unidirectional E-glass/epoxy composite taken from 35.*E*_{1} (GPa) | *E*_{2} (GPa) | *G*_{12} (GPa) | ν_{12} (–) | ρ (g/cm^{3}) |
---|

39 | 8.6 | 3.8 | 0.28 | 2.1 |

The computations are advanced in time until a statistically stationary value of the aerodynamic torque is obtained. The rigid rotor under the same wind and rotor speed conditions is simulated for comparison. Contours of the pressure on the flexible blade in the current configuration are shown in Figure 4. The large negative pressure on the suction side creates a lift force vector with a component in the direction of the blade rotation, which generates a favorable aerodynamic torque.

The aerodynamic torque (for a single blade) is plotted in Figure 5 for both rigid and flexible blade simulations. Both cases compare favorably to the data reported in 19 for this setup obtained using FAST 36, which is a widely used software in wind turbine aerodynamics simulation. Computational modeling in FAST makes use of look-up tables to obtain steady-state lift and drag data for airfoil cross-sections and incorporates empirical modeling to account for the rotor hub, blade tips and trailing-edge turbulence. In our simulations, we are able to capture this important quantity of interest using 3D FSI procedures, which do not rely on empiricism and are 100% predictive.

Rotor blade deflected shape at the point of maximum tip displacement is shown in Figure 8. As expected, the blade mostly displaces in the flap-wise direction, although some edge-wise deflection is also present. Time histories of the flap-wise and edge-wise displacements are shown in Figure 7. The maximum flap-wise tip deflection reaches nearly 6 m, which is significant, and is consistent with the data reported in 19. There is a sudden decrease in the edge-wise deflection around *t* = 1.2 s. At that time, the blade tip passes its lowest vertical position (see Figure 6 for blade location at different time instances) and the direction of the gravity force vector reverses with respect to the direction of the lift force vector.

Note that the aerodynamic torque for the flexible blade exhibits low-magnitude, high-frequency oscillations, whereas the rigid blade torque is smooth (see Figure 5). To better understand this behavior, we examine the twisting motion of the wind turbine blade about its axis. Figure 9 provides a definition of the twist angle for a given blade cross-section. Time histories of the twist angle at four different cross-sections are shown in Figure 10. The twist angle increases with distance from the root and reaches almost 2^{∘} near the tip in the early stages of the simulation. However, starting at *t* = 1.2 s, when the blade tip reaches its lowest vertical position, the magnitude of the twist angle is reduced significantly. The reversal of the gravity vector with respect to the lift direction clearly affects the edge-wise bending and twisting behavior of the blade. The blade twist angle undergoes high frequency oscillations, which are driven by the trailing-edge vortex shedding and turbulence. Local oscillations of the twist angle lead to the temporal fluctuations in the aerodynamic torque.

We note that in the computations presented here, the structure is modeled as a shell with a smooth thickness variation. Structural members, such as spar caps and shear webs, which provide additional bending and torsional stiffness for improved blade response, are not considered here and will be added to the blade structural model in the future.

Figure 11 shows the blade cross-section twist angle as a function of cross-section distance from the root at different time instances. After the blade passes its lowest point, the distribution of the twist angle changes drastically.

Isosurfaces of the air speed at different time instances are shown in Figure 12. Note that, for visualization purposes the rotationally periodic 120^{∘} domain was merged into a full 360^{∘} domain. Fine-grained turbulent structures are generated at the trailing edge of the blade along its entire length. The vortex forming at the tip of the blades is convected downstream of the rotor with little decay.

Figure 13 shows the isocontours of air speed at a planar cut superposed on the spinning rotor. Note the high-intensity turbulence in the blade aerodynamic zone, which is a segment of the blade where the cylindrical root rapidly transitions to a thin airfoil shape. This suggests that the blade trailing edge in this location is subjected to high-frequency loads that are fatiguing the blade. The blade displacement under the action of wind forces is also clearly visible.

Figure 14 shows the isocontours of relative wind speed at a 30 m radial cut at different time instances. For every snapshot, the blade is rotated to the reference configuration to better illustrate the deflection part of the motion. On the pressure side, the air flow boundary layer is attached to the blade for the entire cord length. On the suction side, the flow detaches near the trailing edge and transitions to turbulence.

At *t* = 0.7 s, the composite blade experienced maximum flap-wise tip deflection. At this time instant we found that the magnitudes of the stress components (in the basis corresponding to the material axes) for every ply are below the composite strength. The most critical stress component of the entire blade is σ_{22} (0^{∘} fiber orientation) in ply number 14. The maximum value of σ_{22} reaches 22.63 MPa, whereas the corresponding failure strength is 39 MPa 35. This indicates that the proposed blade design can withstand the simulated operating conditions. The isocontours of σ_{22} are plotted in Figure 15, and show strong tension on the front and compression on the back of the blade.

### 5. CONCLUSIONS

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. FSI MODELING
- 3. DISCRETE SOLUTION PROCEDURES
- 4. SIMULATION OF THE NREL 5MW OFFSHORE BASELINE WIND TURBINE ROTOR
- 5. CONCLUSIONS
- Acknowledgements
- REFERENCES

In this paper, we presented our computational FSI procedures for the simulation of wind turbine rotors at full scale. The air flow is modeled using the residual-based variational multiscale formulation of turbulent flow and the structure is governed by the rotation-free Kirchhoff–Love shell theory with the aid of the bending strip method. NURBS-based isogeometric analysis is employed for spatial discretization. The fluid and solid are strongly coupled at their interface. The strong coupling is in part facilitated by the fact that the structure has only displacement degrees of freedom. The coupled system is solved in a block-iterative fashion, which is a robust procedure for the present application due to the relatively high structural mass of the wind turbine blades.

For wind turbine rotors, the structural motion is dominated by rotation about the horizontal axis. For this we found it advantageous for overall accuracy of the computations to separate the structural displacement into rotation and deflection parts. With this decomposition, we modified the Newmark formulas to treat the rotation part of the structural motion exactly. In addition, only the deflection part of the mesh motion makes use of the partial differential equations of linear elastostatics, whereas the mesh rotation is computed exactly.

We applied our computational framework to the simulation of the NREL 5MW offshore baseline wind turbine rotor. The rotor blades are modeled as symmetric composite laminates homogenized in the through-thickness direction. The computational results give good prediction of the aerodynamic torque and blade tip deflection. To our knowledge, this is the first application of the fully coupled FSI procedures to wind turbine rotor simulation at full scale.

This work is only a first step in the direction of FSI modeling of wind turbines. In the future, we plan to enhance our blade structural modeling to include spar caps, shear webs and other structural components not considered in this work.

We also feel that the effect of the wind turbine tower is important. The presence of the tower will affect the aerodynamics and, consequently, wind loading on the blades. As a result, the rotor–tower interaction needs to be taken into account. For this, we plan to adopt procedures developed in 37 for the coupling of rotating and stationary domains that are particularly well suited for isogeometric discretizations.

In the long run, we plan to combine FSI and structural optimization to devise better blade designs and understand the sensitivity of power generation to changes in wind conditions, blade geometry and material properties.