Volume 2 wind energy 2 08 – aerodynamic analysis of wind turbines

Volume 2 wind energy 2 08 – aerodynamic analysis of wind turbines Volume 2 wind energy 2 08 – aerodynamic analysis of wind turbines Volume 2 wind energy 2 08 – aerodynamic analysis of wind turbines Volume 2 wind energy 2 08 – aerodynamic analysis of wind turbines Volume 2 wind energy 2 08 – aerodynamic analysis of wind turbines Volume 2 wind energy 2 08 – aerodynamic analysis of wind turbines Volume 2 wind energy 2 08 – aerodynamic analysis of wind turbines

2.08 Aerodynamic Analysis of Wind Turbines JN Sørensen, Technical University of Denmark, Lyngby, Denmark © 2012 Elsevier Ltd All rights reserved 2.08.1 2.08.2 2.08.2.1 2.08.2.2 2.08.2.3 2.08.2.3.1 2.08.2.3.2 2.08.2.3.3 2.08.2.3.4 2.08.2.3.5 2.08.3 2.08.3.1 2.08.3.2 2.08.3.3 2.08.4 2.08.5 2.08.6 2.08.7 References Further Reading Introduction Momentum Theory One-Dimensional Momentum Theory The Optimum Rotor of Glauert The Blade-Element Momentum Theory Tip correction Correction for heavily loaded rotors Yaw correction Dynamic wake Airfoil data Advanced Aerodynamic Modeling Vortex Models Numerical Actuator Disk Models Full Navier–Stokes Modeling CFD Computations of Wind Turbine Rotors CFD in Wake Computations Rotor Optimization Using BEM Technique Noise from Wind Turbines 225 226 226 227 228 229 230 230 230 231 231 231 232 232 233 234 236 238 239 240 2.08.1 Introduction The aerodynamics of wind turbines concerns, briefly speaking, modeling and prediction of the aerodynamic forces on the solid structures of a wind turbine and in particular on the rotor blades of the turbine Aerodynamics is the most central discipline for predicting performance and loadings on wind turbines The aerodynamic model is normally integrated with models for wind conditions and structural dynamics The integrated aeroelastic model for predicting performance and structural deflections is a prerequisite for design, development, and optimization of wind turbines Aerodynamic modeling may also concern design of specific parts of wind turbines, such as rotor blade geometry or performance predictions of wind farms Using simple axial momentum theory and energy conservation, Lanchester [1] and Betz [2] predicted that even an ideal wind turbine cannot exploit more than 59.3% of the wind power passing through the rotor disk A major breakthrough in rotor aerodynamics was achieved by Betz [2] and Glauert [3], who formulated the blade-element momentum (BEM) theory This theory, which later has been extended with many ‘engineering rules’, is today the basis for all rotor design codes in use by industry From an outsider’s point of view, aerodynamics of wind turbines may seem simple as compared to aerodynamics of, for example, fixed-wing aircraft or helicopters However, there are several added complexities Most prominently, aerodynamic stall is always avoided for aircraft, whereas it is an intrinsic part of the wind turbines operational envelope Stall occurs when the flow meets the wing at a too high angle of attack The flow then cannot follow the wing surface and separates from the surface, leading to flow patterns far more complex than that of nonseparated flow This renders an adequate description very complicated, and even for Navier–Stokes simulations, it becomes necessary to model the turbulent small-scale structures in the flow, using Reynolds-averaging or large eddy simulations (LESs) Indeed, in spite of the wind turbine being one of the oldest devices for exploiting the energy of the wind, some of the most basic aerodynamic mechanisms are not yet fully understood Wind turbines are subjected to atmospheric turbulence, wind shear from the ground effect, wind directions that change both in time and in space, and effects from the wake of neighboring wind turbines These effects together form the ordinary operating conditions experienced by the blades As a consequence, the forces vary in time and space and a dynamical description is an intrinsic part of the aerodynamic analysis At high wind velocities, where a large part of the blade of stall-regulated turbines operates in deep stall, the power output is extremely difficult to determine within an acceptable accuracy When boundary layer separation occurs, the centrifugal force tends to push the airflow at the blade toward the tip, resulting in the aerodynamic lift being higher than what it would be on a nonrotating blade When the wind changes direction, misalignment with the rotational axis occurs, resulting in yaw error Yaw error causes periodic variation in the angle of attack and invalidates the assumption of axisymmetric inflow conditions Furthermore, it gives rise to radial flow components in the boundary layer Thus, both the airfoil characteristics and the wake are subject to complicated three-dimensional (3D) and unsteady flow behavior Comprehensive Renewable Energy, Volume doi:10.1016/B978-0-08-087872-0.00209-2 225 226 Aerodynamic Analysis of Wind Turbines In the following, a brief introduction is given to wind turbine aerodynamics It is not possible in a short form to introduce to all aspects of rotor aerodynamics and the scope is on conventional aerodynamic modeling, as it is still used by industry in the design of new turbines, and on state-of-the-art methods for analyzing wind turbine rotors and wakes Specifically, the basics of momentum theory, which still form the backbone in rotor design of wind turbines, are introduced Next, state-of-the-art advanced aerodynamic models is presented This includes vortex models, generalized actuator disk/line models, and computational fluid dynamics (CFD) Finally, a short introduction is given to rotor optimization and modeling of aerodynamically generated noise 2.08.2 Momentum Theory The basic tool for understanding wind turbine aerodynamics is the momentum theory in which the flow is assumed to be inviscid, incompressible, and axisymmetric The momentum theory consists basically of control volume integrals for conservation of mass, axial and angular momentum balances, and energy conservation In the following, we will give a brief introduction to momentum theory for design and analysis of wind turbines, starting by the simple, albeit important, one-dimensional (1D) momentum theory, from which the Betz limit can be derived, and ending with the practical BEM theory, which forms the basis for all rotor design codes in use by industry 2.08.2.1 One-Dimensional Momentum Theory We first revisit the simple axial momentum theory as originated by Rankine [4], Froude [5], and Froude [6] Consider an axial flow of speed Uo passes through an actuator disk of area A with constant axial load (thrust) T Denoting by uR the axial velocity in the rotor plane, and let u1 be the axial velocity in the ultimate wake where the air has regained its undisturbed pressure value, pw = po, and let ρ denote the density of air We now consider a 1D model for the stream tube that encloses the rotor disk (see Figure 1), and denote by Ao and A1 the cross-sectional area of the flow far upstream and far downstream of the rotor, respectively The equation of continuity requires that the rate of mass flow, m˙ , is constant in each cross-section Thus, m˙ ¼ ρUo Ao ¼ ρuR A ¼ ρu1 A1 ½1Š Axial momentum balance for the considered stream tube results in the following equation for the thrust T ¼ m˙ ðUo u1 ị ẳ uR AUo u1 ị ẵ2 Applying the Bernoulli equation in front of and behind the rotor, we find that the total pressure head of the air in the slipstream has been decreased by Á p ẳ Uo2 u21 ẵ3 The pressure drop takes place across the rotor and represents the thrust, T = AΔp Combining eqns [2] and [3] shows the well-known result that uR ẳ u1 ỵ Uo Þ ½4Š Introducing the axial interference factor as follows: aẳ Uo uR Uo ẵ5 we obtain uR = (1 − a)Uo and u1 = (1 − 2a)Uo From eqn [2], we get the following expressions for thrust and power extraction: Uo Figure Control volume for 1D actuator disk uR u1 Aerodynamic Analysis of Wind Turbines 227 T ẳ 2AUo2 a1 aị ẵ6 P ẳ uR T ẳ 2AUo2 a1 aị ẵ7 Introducing the dimensionless thrust and power coefficient, respectively, CT ≡ T 2 ρAUo ; CP ≡ P ½8Š ρAUo we get CT ẳ 4a1 aị; CP ẳ 4a a ị ẵ9 Differentiating the power coefficient with respect to the axial interference factor, the maximum obtainable power is obtained as CPmax ¼ 16 ¼ 0:593 27 a¼ for ½10Š This result is usually referred to as the Betz limit or the ‘Lanchester–Betz–Joukowsky limit’, as recently proposed by van Kuik [7], and states the upper maximum for power extraction which is no more than 59.3% of the kinetic energy contained in a stream tube having the same cross-section as the disk area can be converted to useful work by the disk However, it does not include the losses due to rotation of the wake and therefore it represents a conservative upper maximum 2.08.2.2 The Optimum Rotor of Glauert Utilizing general momentum theory, Glauert developed a simple model for the optimum rotor that included rotational velocities In this approach, Glauert treated the rotor as a rotating axisymmetric actuator disk, corresponding to a rotor with an infinite number of blades The main approximation in Glauert’s analysis was to ignore the influence of the azimuthal velocity and pressure in the axial momentum equation For a differential element of radial size r, eqn [2] then reads, T ẳ 4Uo2 aịarr ½11Š Applying the Bernoulli equation in a rotating frame of reference across the rotor plane, we get the following equation for the pressure drop over the rotor, where Ω is the angular velocity Combining eqns [11] and [12], we get p ẳ ru ỵ u2 ẵ12   T ẳ p 2rr ẳ 2u r ỵ uθ rΔr ½13Š where uθ is the azimuthal velocity behind the rotor Defining the azimuthal interference factor as, a′¼ u 2r ẵ14 eqn [13] reads, T ẳ 42 ỵ aịar r ẵ15 aịa ẳ x2 ỵ aịa ẵ16 Combining eqns [11] and [15], we get where x = r/R and λ = ΩR/Uo is the tip speed ratio This equation can also be derived by letting the induced velocity be perpendicular to the relative velocity in the rotor plane Introducing Euler’s turbine equation on differential form, we get the following expression for the useful power produced by the wind turbine, ZR P¼Ω Z1 2πr uu dr ẳ Uo 2 a1 aịx3 dx ẵ17 or in dimensionless form, Z1 CP ẳ 82 a1 aịx3 dx ẵ18 228 Aerodynamic Analysis of Wind Turbines By assuming that the different stream tube elements behave independently of each other, it is possible to optimize the integrand for each x separately (see Glauert [3] or Wilson and Lissamann [8]) This results in the following relation for an optimum rotor, da aị a ẳ da ½19Š Differentiating eqn [16] with respect to a gives, 2a ẳ x2 ỵ 2aị da da ½20Š Combining eqns [16], [19], and [20] results in the following relationship a′ ¼ − 3a 4a − ½21Š The analysis shows that the optimum axial interference factor is no longer a constant but will depend on the rotation of the wake and that the operating range for an optimum rotor is 1/4 ≤ a ≤ 1/3 The relations between a, a′, a′x2λ2, and λx for an optimum rotor are given in Table The maximal power coefficient as a function of tip speed ratio is determined by integrating eqn [18] and is shown in Table The optimal power coefficient approaches 0.593 at large tip speed ratios only It shall be mentioned that these results are valid only for a rotor with an infinite number of blades and that the analysis is based on the assumption that the rotor can be optimized by considering each blade element independently of the remaining blade elements 2.08.2.3 The Blade-Element Momentum Theory The BEM method was developed by Glauert [3] as a practical way to analyze and design rotor blades In the BEM theory, the loading is computed using two independent methods, that is, by a local blade-element consideration using tabulated two-dimensional (2D) airfoil data and by use of the 1D momentum theorem First, employing BEM, axial load and torque are written as, respectively, Table Flow conditions for the optimum actuator disk a a′ a′x2λ2 λx 0.25 0.26 0.27 0.28 0.29 0.30 0.31 0.32 0.33 0.333 1/3 ∞ 5.500 2.375 1.333 0.812 0.500 0.292 0.143 0.031 0.003 01 0 0.0296 0.0584 0.0864 0.1136 0.1400 0.1656 0.1904 0.2144 0.2216 0.2222 0.073 0.157 0.255 0.374 0.529 0.753 1.150 2.630 8.58 ∞ Table Power coefficient as function of tip speed ratio for optimum actuator disk λ CP max 0.5 1.0 1.5 2.0 2.5 5.0 7.5 10.0 0.288 0.416 0.480 0.512 0.532 0.570 0.582 0.593 Aerodynamic Analysis of Wind Turbines z U∞ Wi 229 L Vrel α φ γ –θ D –Ωr Vθ Figure Cross-sectional airfoil element dT Cn ẳ BFn ẳ cBVrel dr ẵ22 dM ¼ BrFt ¼ ρcBrVrel ⋅ Ct dr ½23Š where c is the blade chord, B is the number of blades, Vrel is the relative velocity, Fn and Ft denote the loading on each blade in axial and tangential direction, respectively, and Cn and Ct denote the corresponding 2D tabulated force coefficients From the velocity triangle at the blade element (see Figure 2), we deduce that sin  ẳ U aị ; Vrel cos  ẳ r ỵ aị Vrel ẵ24 where the induced velocity is defined as Wi = (−aU0, a′Ωr) Using the above relations, we get ¼ Vrel U∞2 ð1 − a ị U aịr1 ỵ aị ẳ sin  cos  sin  ½25Š Inserting these expressions into eqns [22] and [23], we get dT ρBcU02 ð1− aÞ ẳ Cn dr sin  ẵ26 dM BcU0 aịr ỵ aị ẳ Ct dr sin  cos  ½27Š Next, applying axial momentum theory, the axial load is computed as dT ¼ ρðU0 uwake ị2ruR ẳ 4U02 a1 aị dr ẵ28 where uR = U0(1 − a) is the axial velocity in the rotor plane and uwake = U0(1 − 2a) is the axial velocity in the ultimate wake Applying the moment of momentum theorem, we get dM ¼ ρruθ 2πruR ¼ 4r U0 a1 aị dr ẵ29 where u = 2Ωra′ is the induced tangential velocity in the far wake Combining eqns [26] and [27] with eqns [28] and [29], we get after some algebra sin =ðσCn ị ỵ ẵ30 sin  cos =Ct ị ẵ31 aẳ aẳ 2.08.2.3.1 Tip correction Since the above equations are derived assuming azimuthally independent stream tubes, they are only valid for rotors with infinite many blades In order to correct for finite number of blades, Glauert [3] introduced Prandtl’s tip loss factor In this method, a correction factor, F, is introduced that corrects the loading In a recent paper by Shen et al [9], the tip correction is discussed and various alternative formulations are compared However, here we limit the correction to the original form given by Glauert [3] In this model, the induced velocities are corrected by the tip loss factor F, modifying eqns [28] and [29] as follows, 230 Aerodynamic Analysis of Wind Turbines dT ẳ 4rU2 aF1 aị dr ẵ32 dM ẳ 4r U aF1 aị dr ẵ33 where = Bc/2r An approximate formula of the Prandtl tip loss function was introduced as follows, Fẳ  ! BR rị cos − exp − π 2r sin  ½34Š where  = (r) is the angle between the local relative velocity and the rotor plane The coefficients (Cn, Ct) are related to the lift and drag coefficients (Cl, Cd) by Cn = Cl cos  + Cd sin  and Ct = Cl sin  − Cd cos , respectively (Cl, Cd) depend on local airfoil shape and are obtained using tabulated 2D airfoil data corrected with 3D rotating effects Equating eqn [26] to eqn [32] and eqn [27] to eqn [33], the final expressions for the interference factors read 4F sin2 =Cn ị ỵ ẵ35 4F sin  cos =Ct ị1 ẵ36 aẳ aẳ 2.08.2.3.2 Correction for heavily loaded rotors By putting eqn [32] into dimensionless form, we get the following expression for the local thrust coefficient, CT ¼ dT 2 ρU∞ 2πrdr ¼ 4aF1 aị ẵ37 For heavily loaded rotors, that is, for a values between 0.3 and 0.5, this expression ceases to be valid as the wake velocity tends to zero with an unrealistic large expansion as a result It is therefore common to replace it by a simple empirical relation Following Glauert [3], an appropriate correction is to replace the expression for a ≥ 1/3 with the following expression:  a  CT ẳ 4aF 3aị ẵ38 As discussed in, for example, Spera [10] or Hansen [11], other expressions can also be used 2.08.2.3.3 Yaw correction Yaw refers to the situation where the incoming flow is not aligned with the rotor axis In this case, the wake flow is not in line with the free wind direction and it is impossible to apply the usual control volume analysis A way of solving the problem is to maintain the control volume and specify an azimuth-dependent induction In practice, it works by computing a mean induction and prescribe a function that gives the azimuthal dependency of the induction The following simple formula has been proposed by Snel and Schepers [12],    r wi ẳ wi0 ỵ tan cos blade ị ẵ39 R where wi0 is the annulus averaged induced velocity and χ is the wake skew angle, which is not identical to the yaw angle because the induced velocity in yaw alters the mean flow direction in the wake flow In the notation used here, θblade denotes the azimuthal position of the blade and θ0 is the azimuthal position where the blade is deepest in the wake For more details, the reader is referred to the text book by Hansen [11] 2.08.2.3.4 Dynamic wake Dynamic wake or dynamic inflow refers to unsteady flow phenomena that affect the loading on the rotor In a real flow situation, the rotor is subject to unsteadiness from coherent wind gusts, yaw misalignment, and control actions, such as pitching and yawing When the flow changes in time, the wake is subject to a time delay when going from one equilibrium state to another An initial change creates a change in the distribution of trailing vorticity which then is convected downstream and first can be felt in the induced velocities after some time However, the BEM method in its simple form is basically steady; hence, unsteady effects have to be included as an additional ‘add-on’ In the European CEC Joule II project ‘Dynamic Inflow: Yawed Conditions and Partial Span Pitch’ (see Schepers and Snel [13]), various dynamic inflow models were developed and tested Essentially, a dynamic inflow model predicts the time delay through an exponential decay with a time constant corresponding to the convective time of the flow in the wake As an example, the following simple model was suggested, Aerodynamic Analysis of Wind Turbines Rf  r  du T i ỵ 4ui U0 ui ị ẳ R dt 2πrΔr 231 ½40Š where the function f(r/R) is a semiempirical function associated with the induction The equation can be seen to correspond to the axial momentum equation, eqn [28], except for the time-term that is responsible for the time delay 2.08.2.3.5 Airfoil data As a prestep to the BEM computations, 2D airfoil data have to be established from wind tunnel measurements In order to construct a set of airfoil data to be used for a rotating blade, the airfoil data further need to be corrected for 3D and rotational effects A simple correction formula for rotational effects was proposed by Snel and van Holten [14] for incidences up to stall For higher incidences (>40°), 2D lift and drag coefficients of a flat plate can be used These data, however, are too big because of aspect ratio effects and here the correction formulas of Viterna and Corrigan [15] are usually applied (see also Spera [10]) Furthermore, since the angle of attack is constantly changing due to fluctuations in the wind and control actions, it is needed to include a dynamic stall model to compensate for the time delay associated with the dynamics of the boundary layer and wake of the airfoil This effect can be simulated by a simple first-order dynamic model, as proposed by Øye [16], or it can be considerably more advanced, taking into account also attached flow, leading edge separation and compressibility effects, as in the model of Leishman and Beddoes [17] 2.08.3 Advanced Aerodynamic Modeling Although the BEM method is widely used and today constitutes the only design methodology in use by industry, there is a big need for more sophisticated models for understanding the underlying physics Various numerically based aerodynamic rotor models have in the past years been developed, ranging from simple lifting line wake models to full-blown Navier–Stokes-based CFD models In the following, the most used models will be introduced 2.08.3.1 Vortex Models Vortex wake models denote a class of methods in which the rotor blades and the trailing and shed vortices in the wake are represented by lifting lines or surfaces At the blades, the vortex strength is determined from the bound circulation which is related to the local inflow field The global flow field is determined from the induction law of Biot–Savart, where the vortex filaments in the wake are advected by superposition of the undisturbed flow and the induced velocity field The trailing wake is generated by spanwise variations of the bound vorticity along the blade The shed wake is generated by the temporal variations as the blade rotate Assuming that flow in the region outside the trailing and shed vortices is curl-free, the overall flow field can be represented by the Biot–Savart law Utilizing the Biot–Savart law, simple vortex models can be derived to compute quite general flow fields about wind turbine rotors The first example of a simple vortex model is the one due to Joukowsky [18], who proposed to model the wake flow by a hub vortex plus tip vortices represented by an array of semi-infinite helical vortices with constant pitch (see also Margoulis [19]) However, this model contains inherent problems due to the singular behavior of the vortices, and as an axisymmetric approximation, one may represent the tip vortices as a series of ring vortices To compute flows about actual wind turbines, it becomes necessary to combine the vortex line model with tabulated 2D airfoil data This can be accomplished by representing the spanwise loading on each blade by a series of straight vortex elements located along the quarter chord line The strength of the vortex elements are determined by employing the Kutta–Joukowsky theorem on the basis of the local airfoil characteristics When the loading varies along the span of each blade, the value of the bound circulation will change from one filament to the next This is compensated for by introducing trailing vortex filaments whose strengths correspond to the differences in bound circulation between adjacent blade elements Likewise, shed vortex filaments are generated and advected into the wake whenever the loading undergoes a temporal variation While vortex models generally provide physically realistic simulations of the flow structures in the wake, the quality of the obtained results still depends on the input airfoil data In vortex models, the flow structure can either be prescribed or computed as a part of the overall solution procedure In a prescribed vortex technique, the position of the vortical elements is specified from measurements or semiempirical rules This makes the technique fast to use on a computer, but limits its range of application to more or less well-known steady flow situations For unsteady flow situations and complicated flow structures, free wake analysis becomes necessary A free wake method is more straightforward to understand and use, as the vortex elements are allowed to advect and deform freely under the action of the velocity field The advantage of the method lies in its ability to calculate general flow cases, such as yawed wake structures and dynamic inflow The disadvantage, on the other hand, is that the method is far more computing expensive than the prescribed wake method, since the Biot–Savart law has to be evaluated for each time step taken Furthermore, free-vortex wake methods tend to suffer from stability problems owing to the intrinsic singularity in induced velocities that appears when vortex elements are approaching each other This can to a certain extent be remedied by introducing a vortex core model in which a cut-off parameter models the inner viscous part of the vortex filament In recent years, much effort in the development of models for helicopter rotor flow fields have been directed toward free wake modeling using advanced pseudo-implicit relaxation schemes, in order to improve numerical efficiency and accuracy (see Leishman [20]) A special version of the free-vortex wake methods is the method by Voutsinas [21] in which the flow modeling is taken care of by vortex particles or vortex blobs 232 Aerodynamic Analysis of Wind Turbines A generalization of the vortex method is the so-called Boundary Integral Element Method (BIEM) Where the rotor blade in a simple vortex method is represented by straight vortex filaments, the BIEM takes into account the actual finite thickness geometry of the blade The theoretical background for BIEMs is potential theory where the flow, except at solid surfaces and wakes, is assumed to be irrotational In a rotor computation, the blade surface is covered with both sources and doublets, while the wake only is represented by doublets (see, e.g., Katz and Plotkin [22] or Cottet and Koumoutsakos [23]) The circulation of the rotor is obtained as an intrinsic part of the solution by applying the Kutta condition on the trailing edge of the blade The main advantage of the BIEM is that complex geometries can be treated without any modification of the model Thus, both the hub and the tower can be modeled as a part of the solution Furthermore, the method does not depend on airfoil data and viscous effects can, at least in principle, be included by coupling the method to a viscous solver 2.08.3.2 Numerical Actuator Disk Models The actuator disk denotes a technique for analyzing rotor performance In this model, the rotor is represented by a permeable disk that allows the flow to pass through the rotor, at the same time as it is subject to the influence of the surface forces The ‘classical’ actuator disk model is based on conservation of mass, momentum, and energy, and constitutes the main ingredient in the 1D momentum theory Combining it with a blade-element analysis, we end up with the BEM model In its general form, however, the actuator disk might as well be combined with a numerical solution of the Euler or Navier–Stokes equations In a numerical actuator disk model, the Navier–Stokes (or Euler) equations are typically solved by a second-order accurate finite difference/volume scheme, as in a usual CFD computation However, the geometry of the blades and the viscous flow around the blades are not resolved Instead, the swept surface of the rotor is replaced by surface forces that act upon the incoming flow This can either be implemented at a rate corresponding to the period-averaged mechanical work that the rotor extracts from the flow or by using local instantaneous values of tabulated airfoil data In the simple case of an actuator disk with constant prescribed loading, various fundamental studies can easily be carried out The generalized actuator disk method resembles the BEM method in the sense that the aerodynamic forces has to be determined from measured airfoil characteristics, corrected for 3D effects, using a blade-element approach For airfoils subjected to temporal variations of the angle of attack, the dynamic response of the aerodynamic forces changes the static aerofoil data and dynamic stall models have to be included The first computations of wind turbines employing numerical actuator disk models in combination with a blade-element approach were carried out by Sørensen and Myken [24] and Sørensen and Kock [25] This was later followed by different research groups who employed the technique to study various flow cases, including coned and yawed rotors, rotors operating in enclosures, and wind farm simulations For a review on the method, the reader is referred to Vermeer et al [26], Hansen et al [27], or the VKI Lecture Series [28] The main limitation of the axisymmetric assumption is that the forces are distributed evenly along the actuator disk; hence, the influence of the blades is taken as an integrated quantity in the azimuthal direction To overcome this limitation, an extended 3D actuator disk model has been developed by Sørensen and Shen [29] The model combines a 3D Navier–Stokes solver with a technique in which body forces are distributed radially along each of the rotor blades Thus, the kinematics of the wake flow is determined by a full 3D Navier–Stokes simulation, whereas the influence of the rotating blades on the flow field is included using tabulated airfoil data to represent the loading on each blade As in the axisymmetric model, airfoil data and subsequent loading are determined iteratively by computing local angles of attack from the movement of the blades and the local flow field The concept enables one to study in detail the dynamics of the wake and the tip vortices and their influence on the induced velocities in the rotor plane A model following the same idea has been suggested by Leclerc and Masson [30] A main motivation for developing such types of model is to be able to analyze and verify the validity of the basic assumptions that are employed in the simpler more practical engineering models Reviews of the basic modeling of actuator disk and actuator line models can be found in the PhD dissertations of Mikkelsen [31], Troldborg [32], and Ivanell [33] 2.08.3.3 Full Navier–Stokes Modeling During the past four decades, a strong research activity within the aeronautical field has resulted in the development of a series of CFD tools based on the solution of the Navier–Stokes equations Within aerodynamics, this research has mostly been related to flows around fixed-wing aircraft and helicopters Looking specifically on the aerodynamics of horizontal-axis wind turbines, we find some striking differences as compared to usual aeronautical applications First, as tip speeds generally never exceed 100 m s−1, the flow around wind turbines is incompressible Next, the optimal operating condition for a wind turbine always includes stall, with the upper side of the rotor blades being dominated by large areas of flow separation This is in contrast to the cruise condition of an aircraft where the flow is largely attached Some of the experience gained from the aeronautical research institutions has been exploited directly in the development of CFD algorithms for wind turbines Notably is the development of basic solution algorithms and numerical schemes for solution of the flow equations, grid generation techniques, and the modeling of boundary layer turbulence These elements together form the basis of all CFD codes, of which some already have existed for a long time as standard commercial software Today, there exist two main paths to follow when conducting CFD computations; either the equations are solved by using Reynolds averaging or by introducing space filtration The most popular method is based on solving the Reynolds-averaged Navier–Stokes (RANS) equations, closing the system by introducing a suitable one-equation or two-equation turbulence model, such as the Spalart–Allmaras [34] or the k − ε [35] model By using this kind of model, only the time-averaged flow field is Aerodynamic Analysis of Wind Turbines 233 computed, whereas the unsteady field is modeled through the turbulence model If the flow is dominated by a broad spectrum of time scales, the low frequencies may be simulated partly by maintaining the time-term in the RANS equations In this case, it is sometimes referred to as URANS (unsteady RANS) The advantage of RANS or URANS is that a fully resolved computation can be established with some few million mesh points, which makes it possible to reach a full 3D solution even on a portable computer In the past years, refined one- and two-equation turbulence models have been developed to cope with specific flow features In particular, the k − ω SST model developed by Menter [36] has shown its capability to cope with lightly separated airfoil flows, and today this model is widely used for wind turbine computations The accuracy of the computations, however, is restricted by the turbulence model’s lack of ability of representing a full unsteady spectrum Thus, for attached flow the accuracy is fully adequate, whereas for stalled flows, it may degenerate completely This is further rendered complicated by the laminar–turbulent transition process that also has to be modeled in order compute the onset of turbulence An alternative to RANS/URANS is LES In LES, the Navier–Stokes equations are filtered spatially on the computational mesh and only the subgrid scale (SGS) part of the turbulence is modeled using a so-called SGS model The advantage of LES is that all the dynamics of the flow field is captured and that accurate solutions can be obtained even under highly separated flow conditions The computational price, however, is often prohibitive, even when solving parallelized computing algorithms on large cluster systems, because of the large number of mesh points that are needed to resolve practical flows at high Reynolds numbers As compared to direct numerical simulation (DNS), where the Navier–Stokes equations are solved directly without any modeling of the turbulence, LES is, however, still several orders of magnitude faster To give an estimate of computing expenses and the number of mesh points required to resolve a turbulent flow field, one can use the Kolmogorov length scale, ℓ, as the smallest scale and the length of the considered object, L, as the largest length scale According to 3=4 Lesieur et al [37], an estimate of the ratio between the largest and the smallest length scale can be given as L=ℓ ≈ ReL , where the Reynolds number ReL = UL/υ, with U denoting a characteristic wind speed and υ is the kinematic viscosity For an airfoil of a wind turbine blade, a typical value is ReC ≅ 5⋅106 where index C denotes the chord length Thus, for a DNS computation of an airfoil section, we need in the order of 105 mesh points in each direction, resulting in a total of approximately 1015 mesh points For a corresponding LES computation, this may be reduced to about 1010 mesh points, if we assume that the SGS covers about 1.5 decades A main difference between RANS and LES is that RANS computations may be carried out in a pure 2D domain, for example, when studying or designing airfoils, whereas LES is always intrinsically unsteady and 3D As a compromise between the fast computing time of RANS methods and the accuracy of LES, Spalart et al [38] developed the detached eddy simulation (DES) technique This technique is a hybrid approach in which the flow near boundaries is solved using a traditional RANS turbulence model and the outer flow is modeled using a SGS model However, this technique puts severe bounds on the grid, since very high aspect ratios are needed near the boundaries, whereas the grid is required to be as isotropic as possible in the LES domain When computing wakes, the number of mesh points need not depend on the Reynolds number, if for example, the influence from the surface is ignored In this case most of the flow can be simulated by using LES technique to simulate the dynamics of the main vortex structures and model the smaller scales by an SGS model However, if one wishes to include the surface-bounded boundary layer in the computation the number of mesh points is mainly determined by the Reynolds number, which for a modern wind turbine of a diameter of about 100 m is about ReD ≅ 107 An overview of the required number of mesh points for different approaches is given in Table 2.08.4 CFD Computations of Wind Turbine Rotors The research on CFD in wind turbine aerodynamics was initiated through European Union-sponsored collaborate projects, such as VISCWIND [39], VISCEL [40], and KNOW-BLADE in Europe The first full Navier–Stokes simulation for a complete rotor blade was carried out by Sørensen and Hansen [41] and later followed by Duque et al [42] and Sørensen et al [43] in connection with the American NREL experiment at NASA Ames and the accompanying National Renewable Energy Laboratory/ National Wind Technology Center (NREL/NWTC) aerodynamics blind comparison test [44] This experiment has achieved a significant new insight into wind turbine aerodynamics and revealed serious shortcomings in present-day wind turbine aerodynamics prediction tools First, computations of the performance characteristics of the rotor by methods based on the BEM technique were found to be extremely sensitive to the input blade section aerodynamic data The predicted values of the distribution of the normal force coefficient deviated from measurements by as much as 50% Even at low angles of attack, model predictions differed from measured data by 15–20% [44] Next, the computations based on Navier–Stokes equations convincingly showed that CFD had matured to become an important tool for predicting and understanding the flow physics of Table Number of required mesh points for various types of computations RANS DES LES DNS Airfoil Full rotor Wake 105 107 1010 1015 107 108 1012 1017 105 – 108 107 – 1010 107 – 1014 107 – 1019 234 Aerodynamic Analysis of Wind Turbines Wind turbine blade, 19.1 meter, suction side Blade root Leading edge Wind Separation line Reattchment line Trailing edge tion Blade revolu Anomalous vortex Blade tip Windspeed 15 m/s, 27 RPM Figure Sketch of flow topology and limiting streamlines on a wind turbine blade modern wind turbine rotors The Navier–Stokes computations by Sørensen et al [43] generally exhibited good agreement with the measurements up to wind speeds of about 10 m s−1 At this wind speed, flow separation sets in and for higher wind speeds it dominates the boundary layer characteristics Hence, it is likely that the introduction of a more physically consistent turbulence modeling and the inclusion of a laminar/turbulent transition model will improve the quality of the results (Sørensen [45]) A large number of full 3D Navier–Stokes computations have later been carried out by different research groups The computations include RANS and DES simulations of full rotor systems, the hub, studies of tip flows, blade–tower interaction, and wind turbine blades under parked conditions Reviews can be found in Hansen et al [27] and Sørensen [46], and various contributions were published in the proceedings from TWIND2007 [47] To illustrate the degree of complexity one obtains using a full 3D Navier–Stokes methodology in Figure 3, we show a computation of a rotating 19.1 m long wind turbine blade It is clearly seen here that a complicated flow topology results, including a large separated area, which could not be obtained using the BEM technique or inviscid computations 2.08.5 CFD in Wake Computations Modern wind turbines are often clustered in wind parks in order to reduce the overall installation and maintenance expenses Because of the mutual interference between the wakes of the turbines, the total power production of a park of wind turbines is reduced as compared to an equal number of stand-alone turbines Thus, the total economic benefit of a wind park is a trade-off between the various expenses to erect and operate the park, the available wind resources at the site, and the reduced power production because of the mutual influence of the turbines A further unwanted effect is that the turbulence intensity in the wake is increased because of the interaction from the wakes of the surrounding wind turbines As a consequence, dynamic loadings are increased that may excite the structural parts of the individual wind turbine and enhance fatigue loadings The turbulence created from wind turbine wakes is mainly due to the dynamics of the vortices originating from the rotor blades The vortices are formed as a result of the rotor loading To analyze the genesis of the wake, it is thus necessary to include descriptions of the aerodynamics of both the rotor and the wake Although many wake studies have been performed over the past two decades, a lot of basic questions still need to be clarified in order to elucidate the dynamic behavior of individual as well as multiple interactive wakes behind wind turbines When regarding wakes, a distinct division can be made between the near- and the far-wake region The near wake is normally taken as the area just behind the rotor, where the properties of the rotor can be discriminated, so approximately up to rotor diameter downstream Here, the presence of the rotor is apparent by the number of blades, blade aerodynamics, including stalled flow, 3D effects, and the tip vortices The far wake is the region beyond the near wake, where the focus is put on the influence of wind turbines in park situations; hence, modeling the actual rotor is less important The near wake research is focused on the performance and the physical process of power extraction, while the far wake research is more focused on the mutual influence when wind turbines are placed in clusters or wind farms The far wake has been a subject of extensive research both experimentally and numerically Semianalytical far wake models have been proposed to describe the wake velocity after the initial expansion (e.g., Ainslie [48]) Detailed numerical studies of the far wake have been carried out by Crespo and Hernández [49] using methods based on the UPMWAKE model in which the wind turbine is supposed to be immersed in an atmospheric boundary layer This model uses a finite difference approach and a parabolic approximation to solve the RANS equations combined with a k − ε turbulence model As illustrated in Table 3, prohibitively many mesh points are needed if one wishes to carry out LES or DNS of the wake in an atmospheric boundary layer However, employing the actuator line technique and representing the ambient turbulence and shear Aerodynamic Analysis of Wind Turbines 235 flow by body forces, the number of mesh points become affordable even for a high Reynolds number LES computation Using this technique, near-wake computations have been carried out by Sørensen and Shen [29], Ivanell et al [50, 51], and Troldborg et al [52, 53] In a recent survey by Vermeer et al [26], both near-wake and far-wake aerodynamics are treated, whereas a survey focusing solely on far-wake modeling was earlier given by Crespo et al [54] To illustrate the type of results that may be achieved combining LES and the actuator line technique, results obtained from simulations by Troldborg et al [53] of a stand-alone wind turbine will be presented The computations were carried out using airfoil data from the Tjæreborg wind turbine The blade radius of this turbine is 30.56 m and it rotates at 22.1 rpm, corresponding to a tip speed of 70.7 m s−1 The blade sections consist of NACA 44xx airfoils with a chord length of 0.9 m at the tip, increasing linearly to 3.3 m at hub radius m The blades are linearly twisted 1° per m Figure shows instantaneous vortex structures in the near wake of a rotor operating at a wind speed of W0 = m s−1, corresponding to a tip speed ratio of about 12 It is seen here that the wake flow collapses into small-scale turbulence about diameter behind the rotor The actual position depends on the loading and the ambient turbulence level In the simulation in Figure the inflow conditions were pure laminar and the collapse is due to intrinsic instabilities of the flow Figure shows the contours of the instantaneous absolute vorticity in the x/R = plane for three different cases Regions of high vorticity appear as light colors Note that the rotor is located to the left in the plots and that only the downstream development of the wake is shown The Figure Vortex structures in the near wake after the Tjæreborg wind turbine (Troldborg et al [53]) Figure Downstream development of wake behind wind turbine Upper figure: W0 = m s−1; middle figure: W0 = 10 m s−1; lower figure: W0 = 14 m s−1 In all figures, the rotor is located to the left (Troldborg et al [53]) 236 Aerodynamic Analysis of Wind Turbines bound vorticity of the blades is seen to be shed downstream from the rotor in individual vortex tubes A closer inspection of the vorticity contours at W0 = m s−1 revealed that the distinct tip-vortex pattern is preserved about 0.5 rotor radii downstream, where after they smear into a continuous vorticity sheet In the case where the free stream velocity is W0 = 10 and 14 m s−1, distinct tip vortices can be observed about 1.5 and rotor radii downstream, respectively In all cases, the structures might have been preserved even further if a finer grid had been used Moreover, it should be noted that using the absolute value of the vorticity as a means of identifying vortices is limited by its strong dependence on the chosen contour levels, and therefore, vortex structures might very well be present even though they are not immediately visible For the rotor operating at the highest tip speed ratio, instability of the tip vortices are observed only and rotor radii downstream where the entire wake flow completely breaks up In the case where W0 = 10 m s−1, the tip vortices are observed to undergo a Kelvin–Helmholtz instability approximately rotor radii downstream The root vortex also becomes unstable at this position Further downstream the root and tip vortices interact, which causes the flow to become fully turbulent Instability of the tip vortices is also observed in the last case where W0 = 14 m s−1, but as expected it takes place even further downstream (ca 10 rotor radii downstream) and is not as strong due to the generally higher stability and persistence of the tip vortices, when the tip speed ratio and thus also the thrust is low 2.08.6 Rotor Optimization Using BEM Technique In the past three decades, the size of commercial wind turbines has increased from units of about 50 kW in the early 1980s to the latest multi-MW turbines with rotor diameters over 120 m In spite of repeated predictions of a leveling off at an optimum mid-range size and periods of stagnation, the size of commercial wind turbines has steadily increased with about a 5-doubling in installed generator power over a period of one decade The overall goal is to reduce the cost price of the produced energy, and as long as increasing the size results in a reduction of the cost price, it is likely that the wind turbines will increase in size also for many years to come There are obviously factors that may bring this trend to an end, such as problems related to the handling and manufacturing of the large blades A more sophisticated way of capturing more energy from the wind, however, is to improve the aerodynamic efficiency of the energy conversion by using optimization techniques in the initial design In the development of new wind turbines, aerodynamic and structural optimization has become an important issue for optimizing the energy yield and thereby minimizing the cost price of the produced energy How to reduce the cost of a wind turbine per unit of energy is an important task in modern wind turbine research Classical models for aerodynamic optimization of rotors can be found in the text books of Glauert [3] and Theodorsen [55] and in revised form by Okulov and Sørensen [56, 57] However, an aerodynamic optimal rotor may not necessarily be the most cost-effective, as the target is to reduce the price of the produced energy Since an optimization technique works together with aerodynamic and structural models, results from an optimization procedure will often be influenced a great deal by the models used Thus, accurate and efficient models for predicting wind turbine performance are essential for obtaining reliable optimum designs of wind turbine rotors The first multidisciplinary optimization method for designing horizontal-axis wind turbines is due to Fuglsang and Madsen [58] The objective used in their method was to minimize the cost of energy employing multiple constrains Generally, multiobjective optimization methods are employed in which the blades are optimized by varying blade structural parameters such as stiffness, stability, and material weight Site specifics from sites comprising normal flat terrain, offshore, and complex terrain wind farms can also be incorporated in the design process of the wind turbine rotors [59] To illustrate the basics of design optimization of wind turbine rotors, in the following we show the features of an optimization model developed at Technical University of Denmark and ChongQing University [60] for optimizing the geometry of wind turbines to maximize the energy yield The method is based on combining an aeroelastic model containing 11 degrees of freedom with the BEM technique The most important issue when performing optimizations is to locate the main parameters and a suitable object function In the model, the object function is defined as the minimum cost price of the produced energy, determined by computing the annual energy production (AEP) and the production cost of the turbine In the following, the cost model and the design variables used in the optimization model are presented As design variables we choose chord length, twist angle, relative thickness, and tip pitch angle Estimating the cost of a wind turbine is an important and difficult task, but also crucial for the success of an optimization The cost model includes the capital costs from foundation, tower, rotor blades, gearbox, and generator plus the costs from operation and maintenance The total cost of a wind turbine can be expressed as C¼ N X i ¼1 Ci ¼ N X Ri bi ỵ bi ịwi ị ẵ41 i ẳ1 where Ci is the cost of the i-th component of the wind turbine and N is the number of main components, Ri is the initial cost of the i-th component determined from a reference rotor, bi is the fixed part of the i-th component that counts for manufacturing and transport, (1 − bi) is the variable part of the i-th component, and wi is the weight parameter of the i-th component The weight parameter in Fuglsang and Madsen [58] was dependent on the design loads of extreme forces and moments and lifetime equivalent fatigue forces and moments To get more information about the cost of a whole wind turbine, the reader is referred to Fuglsang and Madsen [58] As the costs from operation and maintenance often can be counted as a small percentage of the capital cost, reduction of the capital cost becomes the essential task for the design Further, a well-designed wind turbine with a low energy cost always has an aerodynamically efficient rotor Therefore, the rotor design plays an important role for the whole design procedure of a wind turbine In the current study, we restrict our objective to the cost from the rotor Thus, the objective function is defined as, Aerodynamic Analysis of Wind Turbines f xị ẳ COE ẳ Crotor AEP 237 ẵ42 where COE is the cost of energy of a wind turbine rotor and Crotor is the total cost for producing, transporting, and erecting a wind turbine rotor In the current study, the fixed part of the cost for a wind turbine rotor brotor is chosen to be 0.1 Therefore, the total cost of a rotor, Crotor, is a relative value defined as Crotor ¼ brotor ỵ brotor ịwrotor ẵ43 where wrotor is the weight parameter of the rotor In the present study, the weight parameter is calculated from the chord and mass distributions of the blades Dividing a blade into n cross-sections, wrotor is estimated as wrotor ¼ n X mi ⋅ ci ; opt Mtot ci ; orig i ẳ1 ẵ44 where mi is the mass of the i-th cross-section of the blade, ci,opt is the mean chord length of the i-th cross-section of the optimized blade, ci,orig is the mean chord length of the i-th cross-section of the original blade, and Mtot is the total mass of the blade The power curve is determined from the BEM method In order to compute the AEP, it is necessary to combine the power curve with the probability density of wind speed (i.e., the Weibull distribution) The function defining the probability density can be written in the following form !   k ! Vi ỵ k Vi ẵ45 exp f Vi < V < Vi ỵ ị ẳ exp − A A where A is the scale parameter, k is the shape factor, and V is the wind speed In the current study, the shape factor is chosen to be k = 2, corresponding to the Rayleigh distribution If a wind turbine operates the full 8760 h yr−1, its AEP is computed as AEP ¼ M −1 X iẳ1 PViỵ1 ị ỵ P V ị i ị f Vi < V < Viỵ1 ị 8760 ½46Š where P(Vi) is the power at wind speed Vi and M denotes the number of wind speeds considered As an example of the optimization model, we here show how the performance of the Tjæreborg MW rotor may be improved using optimization The optimization is based on the original rotor; thus, the rotor diameter and the rotational speed are chosen to be the same, whereas chord length, twist angle, relative thickness, and tip pitch angle are chosen as design variables A cubic polynomial is used to control the chord distribution and a spline function is used to control the distributions of twist angle and relative thickness Since the cost of a rotor depends on the lifetime of the blades, power output, shaft torque, and thrust are constrained in the optimization process The values are here constrained not to exceed the values of the original design As a usual procedure for optimization problems, we have one objective function and multiple constraints To achieve the optimization, the fmincon function in Matlab is used The Tjæreborg turbine is equipped with a three-bladed rotor of radius 30.56 m In the BEM computations, 20 uniformly distributed blade elements are used The optimization design is performed from a radial position at a radius of 6.46 m to the tip of the blade In the optimization process, the lower limits for chord, twist angle, and relative thickness are m, 0°, and 12.2%, respectively, and the upper limits are 3.3 m, 8°, and 100%, respectively To reduce the computational time, four points along the blade are used to control the shape of the blade The outcome of the optimization is shown in Figure 6, in which the chord and twist distributions of the original and the optimized Tjæreborg rotor are compared (a) 3.5 (b)10 Optimized rotor Orginal rotor Optimized rotor Orginal rotor Twist angle (°) Chord (m) 2.5 1.5 0.5 0 10 15 20 Radius (m) 25 30 10 15 20 Radius (m) 25 30 Figure (a) Chord and (b) twist angle distributions of the original and the optimized Tjæreborg MW rotor Reproduced from Xudong W, Shen WZ, Zhu WJ, et al (2009) Shape optimization of wind turbine blades Wind Energy 12(8): 781–803 238 Aerodynamic Analysis of Wind Turbines From Figure 6(a), it is seen that the optimized blade attains a remarkable reduction in chord length in the region between 10 and 23 m, as compared to the original rotor At a radius of 15 m, the chord reduction reaches a maximum value of about 16% From a position at a radius of 23 m to a position at a radius of 28 m, the optimized chord has almost the same value as the original distribution of the chord, whereas it decreases significantly in the tip region The twist angle, Figure 6(b), is slightly smaller than the original distribution The performance of the optimized rotor is computed using the aerodynamic/aeroelastic code and compared to the original rotor Since the Tjæreborg rotor is a pitch-controlled rotor, the output power of the rotor is set to be the rated power of MW when the wind speed is larger than the rated wind speed of 15 m s−1 The AEP of the optimized rotor is reduced about 4%, whereas the cost of the optimized rotor is reduced by about 7.1% Thus, the cost of energy of the Tjæreborg rotor is reduced about 3.4% 2.08.7 Noise from Wind Turbines Although offshore wind energy is evolving fast, most wind turbines are still placed in rural environments, where wind turbine noise is of great concern since it may be the only major noise source Machinery noise is generally not as important as aerodynamic noise, as it has been reduced efficiently by well-known engineering techniques, such as proper insulation of the nacelle As a rule of thumb, aerodynamic noise from a wind turbine blade increases with the fifth power of the relative wind speed, as seen from the moving tip of the blades With the tip speed being the most significant parameter, aerodynamic noise has been controlled by lowering the tip speed to a maximum of about 60 m s−1 However, in recent years, the biggest development of wind turbines has taken place offshore, with the result that the latest generation of wind turbines operate at tip speeds up to 80 m s−1 Thus, for turbines erected near the shore or for offshore turbines tested at land sites, noise has again become a subject of great concern with respect to public acceptance This is best illustrated by the increasing number of conferences concerning wind turbine noise; for example, in 2005 the Initiative for Noise Control Engineering in Europe (INCE/Europe) initiated a biannual conference series on wind turbine noise, which in 2009 took place in Aalborg, and in 2011 is scheduled to take place in Rome Through the years, several models have been proposed to explain and predict wind turbine noise Some of the models are somewhat simplistic, whereas others make use of complex CFD solvers that have not yet matured to be applied to compute noise emission for realistic rotors (see Wagner et al [61] for a thorough review of various models) As a compromise between computing speed and accuracy, the most commonly used models are based on semiempirical relations As a basis, most models employ the experimental results and scaling laws on airfoil self-noise by Brooks et al [62] together with the turbulence inflow model proposed by Amiet [63] This includes, for example, the models of Fuglsang and Madsen [64] and Zhu et al [65, 66] and the model employed in the SIROCCO project [67], as well as further developments by Moriarty et al [68] and Lutz et al [69] In the following, we show some of the features of a typical semiempirical model, such as those referred to above In the model, only aerodynamic noise is considered (i.e., mechanical noise is not considered) Aerodynamic noise can be divided into ‘airfoil selfnoise’ and ‘turbulence inflow noise’ The former is a result of the interaction of the boundary layer of the airfoil with the trailing edge and the latter results from the interaction of the existing turbulence in the wind with the airfoil In the model, the airfoil self-noise prediction is based on the functions given by Brooks et al [62] In total, five airfoil self-noise mechanisms were identified and studied separately: • • • • • turbulent boundary layer trailing edge noise, separation-stall noise, laminar boundary layer vortex shedding noise, tip vortex formation noise, and trailing edge bluntness vortex shedding noise As a result, scaling laws were proposed, yielding the sound pressure level at the observer position as a function of frequency for the 1/3 octave band spectrum The scaling laws for the different mechanisms are all of similar form:  SPLi ¼ 10 log  δÃi Mfði ị LD h r2  ỵ Fi Stị ỵ Gi Reị ẵ47 where i is the boundary layer displacement thickness, M is the Mach number, f(i) is the raised power that depends on  is a sound directivity function, and r is the distance the particular noise mechanism i, L is the airfoil section semi-span, D h to the observer The additional terms Fi(St) and Gi(Re) are functions of the ‘Strouhal number’ St = fδ*/U and the ‘Reynolds number’ Re The nature of dependency is different for each noise mechanism but it is impressive that all the formulas look so much alike For turbulent inflow, the prediction equation is normally based on the work of Amiet [63] This model takes the following form:      − 7=3  ΔL Kc Lp ¼ 10 log ρ20 c02 l M3 I2 ^k3 ỵ ^k ỵ 58:4 þ 10 log þ Kc r ½48Š Aerodynamic Analysis of Wind Turbines 239 where l is turbulence length scale, I is turbulence intensity, ρ0 is density, c0 is speed of sound, ΔL is blade segment semi-span, k^ is corrected wave length, and Kc is low-frequency correction Taking into account all the variable dependencies, the problem of predicting the noise spectrum at a given observer position for a given airfoil reduces to identifying the following quantities: • The boundary layer thickness δ* at the trailing edge of the airfoil • The relative wind speed defining M and Re • The boundary layer transition type (forced or natural), leading to tripped or untripped flow • Miscellaneous input parameters to the turbulence inflow noise model, such as turbulent length scale and intensity, in the model reduced to the knowledge of the height from the ground z and the roughness length zo Here we not go into the theory behind the empirical correlations, and for details about the nature of each of the modeled noise mechanisms, we refer the reader to the original work of Brooks et al [62] and Amiet [63] or the text book of Wagner et al [61] As mentioned above, an important parameter for the calculation of airfoil self-noise is the boundary layer thickness at the trailing edge This can be calculated by use of the viscous–inviscid interactive computing program XFOIL [70] It is important to note that the scaling laws shown above are deduced from experiments based only on the NACA 0012 airfoil For this reason, an independent calculation of δ* for each airfoil type is vital This was carried out for different values of the Reynolds number and angle of attack and the computed boundary layer thickness was stored in a database and subsequently determined by interpolation Essentially, the code consists of a ‘traditional’ BEM code, to compute the relative velocities along each blade element defining the rotor, coupled with the routines to predict the noise contribution for each noise source along the span of the rotor blades In short, the prediction code works as follows First, the relative velocities seen by the blade elements are computed, just like in an ordinary BEM computation Next, a table looking up in the boundary layer thickness database is made 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H (1998) Review of the present status of rotor aerodynamics Wind Energy 1: 4 6–6 9 Aerodynamic Analysis of Wind Turbines 24 1 [9] Sørensen, JN and Sørensen, JD (eds.) (20 11) Wind Energy Systems: