NUMERICAL SIMULATIONS OF THE AERODYNAMIC BEHAVIOR OF LARGE HORIZONTAL-AXIS WIND TURBINES

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1 NUMERICAL SIMULATIONS OF THE AERODYNAMIC BEHAVIOR OF LARGE HORIZONTAL-AXIS WIND TURBINES Cristian Gebhardt, Sergio Preidikman, Departmento de Estructuras. Universidad Nacional de Córdoba. Casilla de Correo 916, Córdoba, Argentina Luis Ceballos, Julio Massa, Departamento de Mecánica, Facultad de Ingeniería. Universidad Nacional de Río Cuarto. Ruta 36 Km 601, Río cuarto, Argentina Abstract. In the present work, the non-linear and unsteady aerodynamic behavior of large horizontal axis-wind turbines is analyzed. The flowfield around the wind turbine is simulated with the general nonlinear unsteady vortexlattice method, widely used in aerodynamics. By using this technique, it is possible to compute the aerodynamic loads and their evolution in the time domain. The results presented in this paper help to understand how the existence of the land-surface boundary layer and the presence of the turbine support tower, affect its aerodynamic efficiency. The capability to capture these phenomena is a novel aspect of the computational tool developed in the present effort. Keywords: Large horizontal-axis wind turbines, Unsteady aerodynamics, Vortex-lattice method. 1. INTRODUCTION With increasing environmental concern, and approaching limits to fossil fuel consumption, alternative and clean sources of energy have regained interest. Among the several energy sources being explored, wind energy a form of solar energy shows much promise in selected areas of Argentina where the average wind speeds are high. The utilization of the energy in the winds requires the development of devices which convert that energy into more useful forms. Wind turbines are used to generate electricity from the kinetic energy of the wind. In order to capture this energy and convert it to electrical energy, one needs to have a device that is capable of touching the wind. This device, or turbine, is usually composed of three major parts: the rotor blades, the drivetrain (if there is one), and the generator. The blades are the part of the turbine that touches the wind and rotates about an axis. Extracting energy from the wind is typically accomplished by first mechanically converting the velocity of the wind into a rotational motion of the wind turbine by means of the rotor blades, and then converting the rotational energy of the rotor blades into electrical energy by using a generator. The amount of available energy which the wind transfers to the rotor depends on the mass density of the air, the sweep area of the rotor blades, and the wind speed. The actual amount of energy extracted from the airstream by the wind turbine strongly depends on its aerodynamic efficiency. In this respect, this paper is going to increase the capabilities in the area of large horizontal-axis wind turbines (LHAWT) design by enhancing the ability to accurately predict their aerodynamic efficiency. If the rotor blades are considered to be very thin, the speed very low subsonic, and the Reynolds number large, the boundary layers on their upper and lower surfaces can be treated as vortex sheets and merged into a single sheet, which lies on the camber (i.e., the middle) surface of the rotor blades. Although the vorticity is generated in the boundary layers by viscous stresses, there is a kinematic relationship between vorticity and the velocity field that surrounds it, which is valid whether viscous effects are explicitly modeled or not. This relationship enables one to express the disturbance velocity in terms of the vorticity. These hypotheses allow predicting the aerodynamic loads by using the unsteady and non-linear version of the vortex lattice method (UVLM). The main objective of this work is to develop a fundamental understanding of the non-linear and unsteady aerodynamic behavior of large horizontal-axis wind turbines. To accomplish this objective, the authors have developed comprehensive computational tools that can be used for predicting the uncontrolled and controlled responses of LHAWT. These numerical tools will provide the accuracy needed during the design, development, testing, and deployment of LHAWT. 2. THE AERODYNAMIC MODEL 2.1 The Mathematical Problem Consider a 3D incompressible flow of an inviscid fluid generated due to the unsteady motion of the rotor blades. The absolute velocity of a fluid particle which occupies the position R at instant t is denoted by V(R, t). Since the flow is irrotational outside the boundary layers and the wakes, the velocity field can be expressed as the gradient of a total velocity potential Ф(R, t) as follows, (, t) = Φ( R, t) V R (1)

2 The spatial/temporal evolution of the total velocity potential is governed by the continuity equation for incompressible flows. ( t) 2 Φ = r ; 0 (2) A set of boundary conditions (BCs ) must be added (Gebhardt et al., 2008a, 2008b ). The location of the body s surface is known, possibly as a function of time, and the normal component of the fluid velocity is prescribed on this boundary. The first BC requires the normal component of the velocity of the fluid relative to the body to be zero at the boundaries of the body. This BC, commonly called the no penetration or impermeability BC, becomes: ( ) ( ) V V nˆ = Φ V nˆ = 0 (on the surface of the solid body), (3) S S where V S is the velocity of the boundary surface S, and ˆn is the unit normal vector. In general, V S and ˆn vary in space and time. A regularity condition at infinity must also be imposed. This second BC requires that the flow disturbance, due to the motion of the body (or bodies) through the fluid, should diminish far from the body. This is usually called the regularity condition at infinity and is given by ( ) ( R ) lim V R, t = lim Φ, t = 0 (4) R R Since the disturbance velocity field is computed according to the Biot Savart law, the regularity condition at infinity is satisfied identically. For incompressible potential flows, the velocity field is determined from the continuity equation, and hence, it may be established independently of the pressure. Once the velocity field is known, the pressure is calculated from the unsteady Bernoulli equation. Moreover, since the speed of sound is assumed to be infinite, the influence of the BCs is immediately radiated across the whole fluid region; therefore, the instantaneous velocity field is obtained from the instantaneous BCs. In addition to the BCs, the Kelvin Helmholtz theorems and the unsteady Kutta condition are used to determine the strength and position of the wakes. The integral representation of the velocity field V(R, t) in terms of the vorticity field Ω(R, t)= V(R, t), is an extension of the well known Biot Savart law. For three dimensional flows, it takes the following form: (, t) V R (, t) ( ) 1 Ω R R R = 4π R S( 0, t) R R0 ds ( R ) 0, (5) where R 0 is a position vector on the compact region S(R 0, t) of the fluid domain. The integrand in the surface integral (5) is zero wherever Ω(R,t) vanishes. Thus, the region where the flow is irrotational does not contribute to V anywhere. V can be evaluated explicitly at each point, i.e., independently of the evaluation of V at neighboring points. As a consequence of this feature, which is absent in finite difference methods, the evaluation of V can be confined to the viscous region; the vorticity distribution in the viscous region determines the flow field in both, the viscous and inviscid regions. In order to formulate the no penetration BC given by Eq. (3) it is convenient to divide the total velocity potential Ф(R, t) into two parts, one due to the bound vortex sheet Ф B and another due to the free vortex sheet Ф W. Hence, Eq. (3) can be rewritten as: ( Φ + Φ V ) nˆ = 0 (6) B W S 2.2 The Unsteady Vortex-Lattice Method In the UVLM, the continuous bound-vortex sheets are discretized into a lattice of short, straight vortex segments of constant circulation Γ i (t). These segments divide the surface of the body into a number of elements of area. The model is completed by joining free vortex lines, representing the continuous free-vortex sheets, to the bound-vortex lattice along the edges of separation; such as the trailing edges and tips of the rotor blades. Experience with the vortex-lattice method suggests that the geometric shape of the elements in the lattice affects the accuracy and the rate of convergence. It was found that rectangular elements work better than other shapes. Consequently, as much as possible we use rectangular, or nearly rectangular, elements everywhere except in those places where we are forced to use triangular elements: for example, at the hub of the windmill. Each element of area in the lattice is enclosed by a loop of vortex segments. To reduce the size of the problem, we can consider each element to be enclosed by a closed loop of vortex segments having the same circulation. Then the requirement of spatial conservation of circulation is automatically satisfied. These loop circulations are denoted by G(t). Because the vortex sheets are approximated by a lattice, the no penetration condition given by Eq. (3) or (6) can be satisfied at only a finite number of points, the so called control points. The control points are the centroids of the corner points (aerodynamic nodes). The problem consists of finding the circulations G i (t) around the discrete vortices on the

3 bound lattice such that the velocity field V satisfies conditions (3) or (6) at the control points. In order to find these circulations, we construct a matrix of aerodynamic influence coefficients A ij for i,j =1,2,,NP where NP is the number of elements (closed loops of constant vorticity) in the bound lattice in the bound lattice. The coefficient A ij represents the normal component of the velocity at the control point of the i th element associated with a unit circulation around the vortex of element j th, and is in general a function of time. In terms of the coefficients A ij ( Konstandinopoulos et al., 1981; Preidikman, 1998; Katz and Plotkin, 2001), the no penetration condition given by Eq. (6) can be written as follows: NP j = 1 () = [ Φ V ] nˆ ( = 1, 2,, ) A G t i NP (7) ij j W B i i The linear algebraic system of equations given by Eq. (7) is used to compute the unknown circulations G j (t). At the end of each time step, to satisfy the Kutta condition, vorticity is shed into the flowfield and become part of the grids that approximate the free vortex sheets of the wake. Because the vorticity in the wake now was generated on, and shed from, the body at an earlier time, the flowfield is history-dependent and so the current distribution of vorticity on the surface of the body depends to some extent on the previous distributions of vorticity. The vorticity distribution in and the shape of the wake are determined as part of the solution so the history of the motion is stored in the wake. We say that the wake is the "historian" of the flow. As time passes and the vorticity in the wake convects far downstream, its associated velocity field does not have any appreciable influence on the flow around the body; thus, the historian has a fading memory. In the numerical method, this means that only the wake near to the body is important; the rest can be safely neglected. The method developed in this effort treats the position of, and the distribution of vorticity in, the wakes as unknown and they are determined as part of the solution. The present method employs an explicit routine for generating the unsteady wake (instead of the iterative scheme that was used previously by some investigators), providing efficiency without a loss of accuracy and even providing solution for some cases where the iterative methods did not converge. To generate the wakes the discrete vortex segments at the trailing edge an the tip of each rotor blade are convected at the local particle velocity, V[R(t)], calculated from the Biot Savart law. The updated positions, R(t+Δt), of the vortex points are computed according to ( ) () () () ( τ ) R t +Δ t = R t +ΔR t R t + V R Δ t (8) This approximation for the value of ΔR(t), does not need iterations and is stable. 2.3 Loads Computation The aerodynamics loads acting on the lifting surfaces (rotor blades) are computed as follows: i) the pressure jump at the control point of each element is computed from the unsteady version of Bernoulli Eq. (9); ii) the force at each area element is computed as the product among the pressure jump times the area of the element times the unit normal vector; iii) the resultant force and moment are computed as the vector addition of the forces and moments produced by each element, respectively. Φ 1 p 1 p + V V + = V V + (9) t 2 ρ 2 ρ The details of each term in Eq. (9) are shown in Konstandinopoulos et al. (1981), Preidikman (1998) and Preidikman and Mook (2005). 3. RESULTS The results obtained using the computational tool developed in this effort are presented in this section. First, in Section 3.1 the influence of the turbine support tower is presented; then, in Section 3.2, the incidence of the land-surface boundary layer is shown. 3.1 Influence of the turbine support tower During the rotational cycle, the blades of a horizontal axis wind turbine encounter a region of disturbed inflow when they pass near the azimuthal position of the turbine support tower. For upwind rotor configurations this effect is due to the slowdown and deflection of the flow upstream of the tower and so its severity depends on the proximity of the rotor disk to the support tower. If the rotor is far enough upstream of the tower, the interference effect becomes negligible. The time and spatial evolution of the wakes, neglecting and including the presence of the turbine support tower is presented in Fig. 1. In Fig. 1b, it is possible to note how the wakes brake when they impact the turbine support tower.

Wake rupture (a) neglecting the presence of the tower (b) considering the presence of the tower Figure 1. Wake evolution neglecting and considering the presence of the turbine support tower.

merger of two streams with different velocities.

4 Wake rupture (a) neglecting the presence of the tower (b) considering the presence of the tower Figure 1. Wake evolution neglecting and considering the presence of the turbine support tower. Vorticity cannot be created or destroyed in the interior of a homogeneous fluid under normal conditions, and is produced (or destroyed) only at the boundaries, where normal conditions excludes the merger of two streams with different velocities. For an inviscid fluid, vorticity is convected with the fluid in the sense that the flux of vorticity associated with each surface element moving with the fluid remains constant for all times. When the wakes impact the turbine support tower, they break because they can not penetrate the solid. A readjustment of the circulation occurs at the solid surface. Figure 2 shows a detailed view of the rupture of the wakes. Z Wake rupture Turbine support tower Figure 2. Detailed view of the process of wake rupture. Figure 3a shows the time history of the axial force that acts in a direction perpendicular to the rotor; this load is responsible for bending effects on the turbine support tower. It is possible to note that the presence of the turbine support tower produces an alternating variation in the axial force (dashed blue line) respect to the value obtained with the model where the presence of the tower is neglected (solid black line). In both curves, after the transient, the value of the axial reaches a constant value. When the turbine support tower is included, the variation of the axial force undergoes three periods per revolution of the rotor. This fact is explained because the blades pass in front of the tower three times

per revolution of the rotor. Figure 3b shows the time history of the power produced by the windmill. The situation is similar to that of the axial force.

5 per revolution of the rotor. Figure 3b shows the time history of the power produced by the windmill. The situation is similar to that of the axial force. When designing, we must take these load variations into account because they can either produce fatigue in some of the LHAWT components and/or produce significant dynamic effects that can compromise the structural integrity of the wind turbine structure. Axial force [ N] 3.8 x (a) Axial force considering the boundary layer One rotor revolution Produced power [W] 4.8 x (b) Produced power considering the boundary layer Ignoring Neglecting the tower the turbine support tower. Considering the the tower turbine support tower Ignoring Neglecting the tower the turbine support tower. Considering the the tower turbine support tower Azimut [ ] Azimut [ ] Figure 3. Time evolution of the axial force and the produced power including the influence of the turbine support tower. 3.2 Incidence of the land-surface boundary layer Figure 4 show the time history of the wakes neglecting and including the presence of the of the land-surface boundary layer. Comparing Fig. 4a and Fig. 4b, it is possible to note the difference in the shape of the wakes. When the land-surface boundary layer is neglected, the wake moves uniformly almost parallel to the land-surface. On the other hand, when the land-surface boundary layer is included, the wake moves and deforms in the vertical direction copying the velocity profile associated to the land-surface boundary layer. Presence of the tower Uniform free stream velocity profile (a) neglecting the land-surface boundary layer Free stream velocity profile associated to the land-surface boundary layer (b) considering the land-surface boundary layer Figure 4. Wake evolution neglecting and considering the land-surface boundary layer. Figure 5a shows the time evolution of the axial force. It is clear that the existence of the land-surface boundary layer produces a reduction in the magnitude of that load (dashed blue line) with respect to the case where the effect of the land-surface boundary layer is neglected (solid black line). When the steady state is reached, the reduction in the magnitude of the axial force is around 3 %. The time history of the produced power is shown in Fig. 5b. The situation is similar to that of the axial force. When the steady state is reached, the reduction in the produced power is around 2 %. Figure 5 show the variations originated by the presence of the turbine support tower. This case, which includes the presence of both the turbine support tower and the land-surface boundary layer, was the most complex situation analyzed in the present work. The adopted wind profile to model the land-surface boundary layer follows the CIRSOC 102 standard (1982) where the wind velocity is a function of altitude and terrain ruggedness.

6 Axial force [ N] 4 x (a) Axial force considering the tower One rotor revolution Produced power [W] x 105 (b) Produced power considering the tower Ignoring Neglecting the boundary the land-surface layer boundary layer 4.3. Ignoring Neglecting the boundary the land-surface layer boundary layer. Considering the boundary land-surface layer boundary layer. Considering the the boundary land-surface layer boundary layer 4.2 Azimut [ ] Azimut [ ] Figure 5. Influence of the land-surface boundary layer existence on the axial force and the produced power. 4. CONCLUSIONS Several results obtained by using the computational tool developed in this work were presented. The main objective of this effort was to simulate, in the time domain, the non-linear and unsteady aerodynamic behavior of large horizontal-axis wind turbines. Some important conclusions can be drawn from these results. Though the aerodynamic behavior is usually not fully understood, the results presented here help to understand the aerodynamic behavior associated to LHAWTs, whose complexity is well-known and accepted. The aerodynamic interference due to the presence of the turbine support tower has been satisfactorily captured and in agreement with results from wind tunnel experiments. Although the presence of the turbine support tower does not change the aerodynamic performances of the windmill, this interference originates alternating load components, which can produce fatigue in the LHAWT components and/or non-desirable dynamic effects. These aspects were no studied in detail in the present effort. The effects produced by the land-surface boundary layer were also satisfactorily captured. It was shown, that the presence of the land-surface boundary layer reduces the aerodynamic efficiency of the windmill. Although the computational tool developed in this effort establishes a good starting point towards a better understanding of the aerodynamic behavior of LHAWTs, it will be necessary to carry out simulations that include structural dynamics, control systems, and highly complex environmental conditions that usually take place in the regions where these machines are installed. 5. REFERENCES Gebhardt, C., Preidikman, S. and Massa, J., 2008a. Simulaciones Numéricas de la Aerodinámica no Estacionaria de Generadores Eólicos de eje Horizontal y Gran Potencia, Memorias del Primer Congreso Argentino de Ingeniería Mecánica - I CAIM. Gebhardt, C., Preidikman, S., Massa J. and Weber, G., 2008b. Comportamiento Aerodinámico y Aeroelástico de Rotores de Generadores Eólicos de Eje Horizontal y de Gran Potencia, Mecánica Computacional, Vol. 27, pp Katz, J. and Plotkin, A., Low-Speed Aerodynamics, Cambridge University Press. Konstandinopoulos, P., Mook, D.T. and Nayfeh, A.H., A Numerical Method for General, Unsteady Aerodynamics, AIAA Preidikman, S., Numerical Simulations of Interactions Among Aerodynamics, Structural Dynamics, and Control Systems, Ph.D. Thesis, Virginia Polytechnic Institute and State University. Preidikman S. and Mook, D.T., Modelado de fenómenos aeroelásticos lineales y no-lineales: los modelos aerodinámico y estructural, Modelización Aplicada a la Ingeniería, I, Regional Bs. As, UTN, pp Reglamento CIRSOC 102, Acción Dinámica del Viento Sobre las Construcciones, INTI-CIRSOC. 6. RESPONSIBILITY NOTICE The authors are the only responsible for the printed material included in this paper.

ANALYSIS OF HORIZONTAL AXIS WIND TURBINES WITH LIFTING LINE THEORY

ANALYSIS OF HORIZONTAL AXIS WIND TURBINES WITH LIFTING LINE THEORY Daniela Brito Melo daniela.brito.melo@tecnico.ulisboa.pt Instituto Superior Técnico, Universidade de Lisboa, Portugal December, 2016 ABSTRACT