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

JN Sørensen, Technical University of Denmark, Lyngby, Denmark

© 2012 Elsevier Ltd All rights reserved

2.08.2.2 The Optimum Rotor of Glauert

2.08.2.3 The Blade-Element Momentum Theory

2.08.2.3.1 Tip correction

2.08.2.3.2 Correction for heavily loaded rotors

2.08.2.3.3 Yaw correction

2.08.2.3.5 Airfoil data

2.08.3.2 Numerical Actuator Disk Models

2.08.3.3 Full Navier–Stokes Modeling

2.08.4 CFD Computations of Wind Turbine Rotors

2.08.6 Rotor Optimization Using BEM Technique

References

Further Reading

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

U o u R u1

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,

Axial momentum balance for the considered stream tube results in the following equation for the thrust

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

1

Δp ¼ ρ U2 − u2

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

1

uR ¼

Introducing the axial interference factor as follows:

U

U

we obtain uR = (1 − a)Uo and u1 = (1− 2a)Uo From eqn [2]

Figure 1 Control volume for 1D actuator disk

 

Introducing the dimensionless thrust and power coefficient, respectively,

½ 

we get

Differentiating the power coefficient with respect to the axial interference factor, the maximum obtainable power is obtained as

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 Δr, eqn [2] then reads,

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,

2 where Ω is the angular velocity

Combining eqns [11] and [12], we get

1

where uθ is the azimuthal velocity behind the rotor Defining the azimuthal interference factor as,

uθ

2Ωr eqn [13] reads,

Combining eqns [11] and [15], we get

ð1− aÞa ¼ λ2

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,

P ¼ Ω 2πr2ρuuθdr ¼ 4πρΩ2

Uo a′ð1− aÞx3

or in dimensionless form,

1

Z

CP ¼ 8λ2

a′ð1− aÞx3

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′

ð1− aÞ − a′

Differentiating eqn [16] with respect to a gives,

x2ð1 þ 2a′Þ da

Combining eqns [16], [19], and [20] results in the following relationship

1 − 3a

4a − 1 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 1 The maximal power coefficient as a function

of tip speed ratio is determined by integrating eqn [18] and is shown in Table 2 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

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 1 Flow conditions for the optimum actuator disk

Table 2 Power coefficient as function

of tip speed ratio for optimum actuator disk

Wi U∞

Vrel

D

z

L

γ

V θ

Figure 2 Cross-sectional airfoil element

¼ BFn ¼ ρcBV2

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

where the induced velocity is defined as Wi = (−aU0, a′Ωr) Using the above relations, we get

Inserting these expressions into eqns [22] and [23], we get

0ð1−aÞ2

dM ¼ ρBcU0ð1− aÞΩr2ð1 þ a′Þ

Next, applying axial momentum theory, the axial load is computed as

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

where uθ = 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

1

1

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,

  



dT ¼ 4πrρU2

dr where σ = Bc/2πr An approximate formula of the Prandtl tip loss function was introduced as follows,

2r sin  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

1

4F sin2=ðσCnÞ þ 1

1

4F sin  cos =ðσCt Þ−1

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,

dT

2 ρU∞22πrdr 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

4

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],

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,

 r du

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

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

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 Lesieur et al [37], an estimate of the ratio between the largest and the smallest length scale can be given as L=ℓ ≈ Re3L =4, 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 3

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 3 Number of required mesh points for various types of computations

Airfoil Full rotor Wake RANS 105 107 105 – 10 DES 107 108 107 – 10 LES 1010 1012 107 – 10 DNS 1015 1017 107 – 10

Blade revolution

Wind turbine blade, 19.1 meter, suction side

Blade root

Leading edge Separation line

Reattchment line Trailing edge

Anomalous vortex Blade tip

Windspeed 15 m/s, 27 RPM

Figure 3 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 1 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

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