Review of the Present Status of Rotor Aerodynamics


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1 WIND ENERGY Wind Energ., 1, 46±69 (1998) Review Article Review of the Present Status of Rotor Aerodynamics H. Snel,* Netherlands Energy Research Foundation, Petten, The Netherlands Key words: aerodynamics; wind turbines; computational uid dynamics; boundary layers; stall A review is presented of the state of the art in wind turbine rotor aerodynamics. Both the `engineering methods', used in aeroelastic computer programmes, and more fundamental Euler and Navier Stokes are described and reviewed and problem areas in urgent need of a solution are addressed. An attempt is made to minimise the use of equations in the main text, to access a wider public. The basic equation systems are treated separately in an Appendix. *c 1998 John Wiley & Sons, Ltd. Introduction Fluid mechanics, and speci cally aerodynamics, is one of the core disciplines of wind energy. It is needed to describe the wind eld from which the rotor extracts energy. It is needed for rotor design, both with respect to energy yield (performance) and the dynamic loads that develop in interaction with the elastically deforming structure (aeroelasticity). It is needed to understand and reduce aeroacoustic noise production and transmission by the rotor blades. It is needed to understand the development of the wake behind the rotor, which (in wind farm situations) can determine the in ow eld for the next downstream rotor. The basic equations believed to describe the ow of uids (including air) in general, the Navier±Stokes equations, have been known for a long time. However, their character is such that solutions for the ow conditions and geometries of interest, even by numerical methods, cannot be obtained as yet. For practical (industrial) design and analysis purposes a number of approximating models are being used that permit solutions in a routine fashion. These methods, often referred to as engineering methods, have been signi cantly improved over the last decade. At the same time, simpli ed Navier±Stokes solvers for wind turbine applications (with limitations such as `Reynolds averaging', turbulence modelling and global steadiness assumptions) are now being developed and tested in research settings. This article describes the present state of the art and science of rotor aerodynamics. Both the developments in engineering models and in Navier±Stokeslike solutions are described, as well as the evergrowing body of experimental data that is becoming available. Actual problem areas, in urgent need of solution, are also addressed. Needless to say, the article leans on other recent review articles such as those by Hansen and Butter eld, 1 Snel and van Holten 2 and Sùrensen. 3 The present article, however, apart from being an update in time, is wider in scope than the aforementioned reviews, which are limited to particular areas. At attempt is made to minimize the use of equations in the text in order to make the article accessible to a wider public. In the Appendix an overview is given of the underlying systems of ow equations and approximations used, including the pertinent physical interpretation. The mathematically more initiated reader is referred to this. Rotor Aerodynamics Rotor aerodynamics refers to the interaction of the wind turbine rotor with the incoming wind. It can be separated intuitively into a global ow eld that extends from far upstream of the turbine to far *Correspondence to: H. Snel, Netherlands Energy Research Foundation, Petten, The Netherlands. # 1998 John Wiley & Sons, Ltd.
2 Review of Rotor Aerodynamics 47 downstream, and a local (rotor blade) ow eld which refers to the (viscous) ow about the rotor or even the individual blades. This local ow eld can be regarded as a box within (but separated from) the global eld, while boundary conditions (pressures, velocities continuous) on the common boundary unite the two regions. This separation is used to advantage in the classical blade element momentum (BEM) method, where a balance is considered between changes in momentum and energy ow rates in the global part and aerodynamic forces on the blades which are inferred from local ow conditions. The BEM method, although in a number of details modi ed from its original form, still forms the backbone of rotor design and aeroelastic computer programs and will receive due attention. Also in `modern' computational treatments the methods used for the global ow eld usually di er from those used in the rotor blade region. In view of this, the following discussion of rotor aerodynamic methods will be separated into methods to analyse the global ow eld and methods for blade ow analysis. Nevertheless, it is of utmost importance to realize that these regions are strongly interconnected and cannot quantitatively be determined without taking into account their interaction:. the ow in the global region determines the in ow conditions for the rotor blades;. the forces on the blade (possibly expressed as a pressure change) determine the ow in the global region. Finally, the development of new aerodynamic pro les speci cally for use in wind turbine rotor blades will be reviewed. The Global Flow Region The Momentum Method The classical analysis method for the global ow region is the axial momentum method, in which the rotor is modelled as an actuator disc. Momentum and energy ows are considered in a control volume consisting of the stream tube that encloses the actuator disc. This analysis has its roots in propeller theory as developed by Froude 4 and Lanchester. 5 Betz 6 applied this analysis successfully to the wind turbine situation, considering the axisymmetric and steady ow about a uniformly loaded actuator disc. The situation is shown in Figure 1. Although the method and its results are part of the standard knowledge of the rotor aerodynamicist, some observations will be made here for future reference and background. For the conditions outlined above, Betz derived the famous limit that bears his name, stating that the maximum amount of energy extraction from the wind equals the 16/27th part of the kinetic energy content in the wind, i.e. C P;max ˆ 16=27 ˆ where C P denotes the power or performance coe cient. Basic to this result are the concepts that there is no net axial pressure force due to the pressure distribution on the external stream tube (S in Figure 1) and that there are no external radial forces on the ow. Both concepts are made acceptable by Betz. It must be noted that according to van Kuik 7 the radial force assumption does not hold true owing to an edge singularity of the actuator disc ow, and that the `real' maximum of the power coe cient is expected to be slightly higher than the Betz limit. Apart from this, ow concentrators such as solid di users 8 or `tip Figure 1. Basic ow eld for global axial momentum analysis
3 48 H. Snel vanes' 9 can in principle augment the C P,max value, but up to date this does not seem an economically or technologically attractive option. For future reference we state some fundamental results and relations pertaining to the axial momentum method for a uniformly loaded disc. For a more complete discussion the reader is referred to the excellent exposition by de Vries. 10 First it is noted that upstream of the disc the stream tube enclosing the control volume will progressively widen on approaching the disc, as the braking e ect of the disc lowers the velocity from its original free stream value of V to a value denoted by V D, i.e. V D ˆ V u i ˆ 1 a V; a ˆ ui V 2 where u i is the induced velocity in the rotor plane and the nondimensional factor a is known as the axial induction factor. Across the actuator disc a pressure drop occurs to below ambient pressure. In the downstream part of the ow, the wake region, a gradual pressure recovery occurs until ambient pressure is reached again. In this process the velocity further decreases until a level of V w is reached where the pressures are equalized again. The method assumes no viscous interaction and a completely incompressible ow. The total power extracted can be written as the product of the mass ow and the di erence in speci c kinetic energy in the free wind and the far wake, i.e. P ˆ ra R V D 1 2 V2 V 2 w 3 and likewise as the product of the axial force D ax on the actuator disc and the local ow velocity, i.e. P ˆ D ax V D 4 Finally, the axial force can be written as the di erence in momentum ux in the incoming and outgoing ows: D ax ˆ ra R V D V V w 5 Combining (4) and (5) and equating the power extraction as expressed by (3) and (4), it follows that the velocity V D at the disc equals the algebraic average of the free wind velocity and the far wake velocity, i.e. or, using the notation of (2), V D ˆ 1 2 V V w 6 V w ˆ 1 2a V 7 Note that basic to this result is the assumption that all the `organized' kinetic energy that is removed from the own eld is extracted at the disc and not, for example, converted into turbulent or recirculation kinetic energy. Betz's result is straightforwardly obtained from the foregoing equations by maximizing the power output in terms of the induction factor. Glauert 11 applied the momentum method on an annular level for concentric annuli of radial extension Dr. This enables the matching of the results of momentum analysis with the blade element properties and geometry within the speci c annulus. In this way the method can be converted into a real rotor design and analysis tool. At the same time an expression for the angular (moment of) momentum balance was added by Glauert, in which changes in the angular momentum (from the free stream value of zero) are equated to the torque exerted by the rotor on the air. This involves the introduction of a `tangential' or angular induction velocity u t. In more recent years, Wilson and Lissaman 12 updated the method to include a tip correction factor (derived by Prandtl) in terms of the induction velocities and in general cast the theory in a form that lends itself to computer solution; this work forms the basis on which most modern design tools have been constructed.
4 Review of Rotor Aerodynamics 49 The axial momentum method has many de ciencies both from the theoretical and the practical viewpoint. To start with, the application of the momentum and energy relations on an annular level assumes that the ow in a certain annulus has no relation with the remaining annuli, i.e. annular independence (this will be referred to as the `annular' momentum method). It is well known that this is not correct. Goorian 13 noted that instead of the momentum analysis on an annular level, a Euler solution of the ow eld should be done, of course, after specifying the force distribution on the rotor (actuator) disc. In fact, in view of the process of vorticity mixing and di usion in the wake, a complete Navier±Stokes solution is necessary for a correct description. The next subsection contains a more detailed discussion on this subject and its present state. Here, it is noted that for axial symmetric ow and rotor loading conditions that are not too far from uniform, the results from annular momentum theory are in practice quite acceptable. However, the annular independence assumption becomes completely inadequate for the case of yaw misalignment, i.e. when there exists an angle of appreciable magnitude between the wind direction and the rotor axis. This is a very important condition in practice, since wind direction variations that result from the 3D character of turbulence are too rapid to be followed by the yawing system of the turbine (either active or passive), and instantaneous yaw misalignments of 208±308 are frequent. An important part of fatigue lifetime consumption of rotor blades can be attributed to these conditions, hence a good prediction of the resulting loads is of utmost importance. For that reason, recently a number of improvements have been proposed and implemented by di erent groups, as described in References 14 and 15. In these improved methods a mean speed is calculated in the annuli by applying basic annular momentum theory to the axial component of the wind speed, but an azimuthal distribution is subsequently applied which depends on the yaw angle. The form of this azimuthal distribution is taken from an early model by Glauert 11 in some cases, while others base the form on the results of Goankar and Peters. 16 These modi cations, although certainly not exact, a ord reasonable comparison with measurements, as reported in References 15 and 17 and shown in Figure 2 adapted from Reference 17. In this gure, calculated yaw moments for the Tjñreborg 2 MW turbine are compared with the yaw moments derived from blade root bending measurements. The conditions are indicated in the gure. The calculations were performed with yawed ow models implemented in di erent aeroelastic computer programs. Models commonly used before the improvements discussed above resulted in zeromean yawing moments. The new models give the correct sign of the (restoring) moment but underestimate the azimuthal variations. Moreover, it should be stressed that the methods have not been validated for the dynamically changing yaw angles that occur in practice. Figure 2. Results of improved yaw models within BEM method
5 50 H. Snel Another basic problem of the momentum method (both global and annular) is that is formulated for timeindependent ow: a possible change with time of the energy or momentum content within the control volume shown in Figure 1 is not considered. On the contrary, the modelling assumes an instantaneous equilibrium between momentum ux di erence far upstream and downstream and the aerodynamic forces on the actuator disc (or rotor blades), even for changing conditions. This is known as the equilibrium wake assumption. In practice, the ow eld about the rotor is ever changing in time, as a result of wind speed and direction changes, wind shear e ects, blade control and dynamic deformation. Hence there is no global equilibrium, but an evolving wake situation. These e ects have been thoroughly studied both experimentally and theoretically in the framework of the Joule `dynamic in ow' projects. Experiments on the Tjñreborg 2 MW turbine with fast pitch changes revealed large overshoots of the equilibrium loads. The e ects can now be modelled by rewriting the equations for the annular momentum method in the form of dynamic di erential equations in time instead of the algebraic equilibrium relations. This type of modelling is presently used in all main design and analysis programs. Descriptions of various forms of implementation can be found in Reference 17. Sample results of calculations with these dynamic e ects and the comparison with measurements are shown in Figure 3, taken from Reference 17. Again, this gure compares results of calculations of blade root bending moments using the dynamic models with measured bending moments on the Tjñreborg turbine. The measured overshoots are reasonably well predicted. Finally, the momentum method breaks down for very high rotor loading, as has been known since the pioneering work of Betz and Glauert. This condition will typically arise for operation at values of the tip speed ratio l above 1.3 or 1.4 times the value for which C P,max is attained (usually referred to as l des ). For the constant rotational speed wind turbine this means that it occurs at low wind speed, much lower than rated wind speed, at which energy production is relatively low. For a variable speed turbine it may not occur at all during normal operation. The physics of the ow in this operating mode is such that a considerable amount of kinetic energy is converted into largescale turbulent recirculation modes. For this reason, this condition is known as the `turbulent wake state'. This means that the equality between the two expressions (3) and (4) for P no longer holds. The usual way of resolving the calculational problem that results is by using an empirical relation between annular (axial) forces and the induction factor a, e.g. the socalled Glauert relation. A discussion of di erent methods used can be found in Reference 18. It is generally accepted that the current methods tend to give an underestimation of the extracted power at low wind speeds. No recent work has been done in this respect, although for a xed speed wind turbine it Figure 3. Results of calculations with dynamic in ow models compared with measurements
6 Review of Rotor Aerodynamics 51 can be estimated that 15%±20% of the yearly energy generation takes place in this condition. Exact values depend on the matching between design wind speed and annual mean wind speed. If the prediction error were 20%, then the total error in annual energy yield would be of the order of 4%. There are no important implications for blade loads, as the phenomenon occurs at low wind speed, i.e. low blade loads. This is possibly the reason that the problem has not been regarded of high priority. With the recent adaptations as explained in the preceding paragraphs, the adaptations concerning blade ow (see page 53) and many more that undoubtedly will come, annular blade element momentum theory can be expected to remain as the aerodynamic model of choice for aeroelastic response calculations for at least the next 5 years. Euler and Navier±Stokes Solvers for the Global Flow Field As mentioned in the preceding subsection, the global ow eld should in principle be analysed by more sophisticated methods in order to overcome the de ciencies of momentum theory. In this context it must rst be stated that because of the ow speeds involved (practically below 25 m s 71, which is the cutout wind speed for almost all turbines), the assumption of incompressibility is completely justi ed. With regard to the blade ow, with blade tip speeds well below 100 m s 71, the incompressibility assumption is still justi able. The most rigorous way of analysing the ow eld would then be by using the timedependent incompressible Navier±Stokes equations. However, this leads to the problem of a set of equations that cannot be solved in a practical sense (see Appendix). The next best solution method is possibly the use of the Reynoldsaveraged Navier±Stokes (RANS) equations with a suitable turbulence model. Although this statement introduces the problem of deciding which of the many existing turbulence models is the `suitable' one, the problem can in practice be solved. Still, a precautionary note should be made regarding the fact that some experts feel that, precisely because of the basic unsteady nature of turbulence and the need for closure models, RANS methods will not lead to completely satisfactory solutions. These ideas are expanded somewhat on pages 56 and 58. One step more down the line of complexity would be the use of the Euler equations instead of the Navier±Stokes equations. The Euler equations are in fact the nonviscous form of the Navier±Stokes equations. The processes of creation, di usion and dissipation of vorticity are not considered by the Euler equations, but vorticity transport is. There seems to be good justi cation for this approximation in view of the large Reynolds numbers involved in the global ow and the fact that there are no solid boundaries in this region. Hence vorticity will be created only in the local (rotor) ow box owing to the presence of the blades, or owing to nonuniform disc loading when this part is considered to be an actuator disc. Still, it may be expected that downstream of the rotor the vorticity created is very much concentrated in the blade wake and that di usion may have some e ects. More seriously, the concentrated vortices that form especially towards the edge of the rotor (re ected as a velocity discontinuity (!) across S in the momentum method) will create turbulence and mixing with the outer ow. This e ect is not considered in the momentum method (although in principle it is possible) and cannot be accounted for in any way by the Euler equations. The importance of this has recently been investigated, as discussed later in this subsection. Euler solutions have been obtained by several authors. First of all, a number of publications have been devoted to solutions for the axisymmetrical case, where the incoming ow is supposed to be vorticityfree and only the wake contains vorticity, created at the actuator disc. A thorough discussion can be found in Reference 19. Although interesting from the theoretical point of view, e.g. to check the seriousness of the annular independence assumption of the momentum method, this application does not represent a condition in which momentum theory is very much in error. A special class of Euler solutions is formed by the socalled vortex wake methods. Usually these methods are combined with a lifting line or lifting surface representation of the rotor blades. Vorticity formed at the blades trailing and shed vorticity (see pages 54±56) is convected into the wake with the local total velocity, calculated as the vectorial sum of the free stream velocity V and the relevant velocities
7 52 H. Snel induced by the existing vorticity elements. These methods are typically unsteady in nature; in the initial condition the ow is vortexfree and the evolution of the vortex wake is calculated in time. As for vortex elements, either vortex line elements or vortex particles are used by di erent groups. Typical applications with vortex line elements can be found in References 20 and 21 while the group of the National Technical University of Athens specializes in the vortex particle method, e.g. Reference 22. Line element approaches have been applied to helicopter aerodynamics also. The aim of calculations with this method is the prediction of the ow eld in the rotor plane, resulting from the undisturbed wind and the induction of all (or a speci c part) of the wake vorticity. Among the methods that can be considered as operational at this moment, two have been exercised to a large extent in Joule projects, i.e. the ROVLM method of the University of Stuttgart 21 and the GENUVP method of the National Technical University of Athens. 22 The important advantage of the vortex wake methods is that they lend themselves straightforwardly to the calculation of unsteady situations, in the in ow or with regard to the rotor blade pitch angles, and can be applied straightforwardly to the yaw misalignment situation. However, a theoretical problem exists with regard to the stability of the calculations, especially for the vortex line element case. This problem is basic to all curvilinear vortex elements and their discretized version (rectilinear elements that connect to each other at an angle di erent from 1808): the selfinduced velocity `has a logarithmic singularity. Physically, the vorticity is not concentrated in (singular) lines of zero thickness, so the problem is only a mathematical one. The consequence is that without precautionary measures the method does not converge when the element size is reduced to ever smaller values, since the `collocation' points where velocities are calculated (usually the midpoints of vortex line elements) come too close to neighbouring vortex line elements. The same problem occurs when during the convective process being simulated a vortex collocation point moves too close to a line element and the point is ejected at very high speed. In that case the result becomes something like `vortex spaghetti' and the computation breaks down. One way to overcome this problem is to prescribe either a cuto length (no induction for separations smaller than the cuto length) or a viscous vortex core with regular velocity. However, the solution then becomes a function of the choice of this cuto or viscous core length, which certainly is not a part of the Euler solution. Fortunately, it can be made probable by simple model studies 23 that this dependence is very small indeed, as long as the cuto distance has a value that can be estimated on the grounds of either empirical information (e.g. Reference 24 concerning vortex core size) or the action of viscosity. A practical disadvantage of the free vortex wake methods is the very large amount of computer CPU time needed for a calculation. In order to obtain a realistic ow situation in the rotor plane (needed for the calculation of the blade ow), the calculated wake must extend to at least two rotor diameters behind the rotor plane. This represents some 15 rotor revolutions and a large number of vortex elements that have to be traced. Calculation times involved in free wake calculations are some tens of thousands times larger than those needed for momentum methods. Because of this, many socalled prescribed wake methods have been developed, in which the geometry of the wake is either xed or described by a few parameters which determine its shape. Hybrid methods have also been developed, in which the near wake (e.g. one rotor diameter downstream) is treated as a free wake and the rest as prescribed wake. The ROVLM program of the University of Stuttgart o ers these possibilities and has checked the results against calculations with a completely free wake. It was reported by Bareiû et al. 21 that a 75% reduction of the computer time can be obtained by accepting 5% di erences with the free wake calculations. It should be noted that for the hybrid wake the downstream prescribed wake part does have an in uence on the development of the near (free) wake. Although much computational e ort can be saved by prescribed wake methods, the advantages of such a computation over the (annular) momentum method are not so clear. In fact, annular independence is now replaced by a `prescribed' dependence, without certainty about the correctness of the prescription other than the comparison with free wake calculations for some conditions. Finally, yet another approach which e ectively is a Euler solver is represented by the socalled asymptotic acceleration potential method. This method has been developed and used mainly at Delft University of Technology. It was rst formulated 25 in order to extend Prandtl's lifting line theory to the
8 Review of Rotor Aerodynamics 53 case of rotating wings with unsteady ow for helicopter applications. In Prandtl's theory the induction due to the trailing vorticity system from a stationary 3D wing is used to adapt the local in ow velocity eld. The method was expanded and implemented by van Bussel 26 for application to wind turbines. A cautionary note should be made that the method assumes small perturbations of the main ow, which is justi ed in helicopter applications but much more doubtful for wind turbines. Because of its linearity, the method is much more e cient in computer time than straightforward vortex wake methods but still belongs to a category that does not compete with momentum methods. Like vortex wake methods, its applicability is mainly in examining ow elds and inferring improvements to momentum methods. It has been used as such in the Joule `dynamic in ow' projects, and a recent example for yawed ow can be found in Reference 27. Navier±Stokes solutions for the global ow eld are not often used but can yield interesting information. For example, Madsen 28 applied a Navier±Stokes solver with a k±e turbulence model to axially symmetric ow across an actuator disc. For a disc loading corresponding to the maximum energy extraction condition, (i.e. a ˆ 1 3 ; D ax ˆ 4 9 rv2 A R ) the solutions shown very little e ect of turbulent wake mixing on the conditions in the rotor plane. The most interesting results are obtained for high rotor loading, corresponding to D ax rv2 A R, where the momentum theory breaks down and Glauert's correction is often used. For this case the Navier±Stokes solution does show a remarkable in uence of the turbulence mixing. For a turbulence mixing of 10% the numerical solution follows the Glauert correction reasonably well for values of the axial force coe cient C D,ax up to 1.15, but beyond that value the induction obtained by the Navier±Stokes solutions is appreciably higher than that indicated by Glauert. Hence turbulent mixing of the wake seems to be important, but only for high axial loading. In e ect, the important use of Euler methods or Navier±Stokes solutions at this stage of development is in providing information on which to base corrections to the momentum method which is used in aeroelastic design codes. This is especially true for situations such as yawed ow or high rotor loading. Applied in this manner and used with su cient precaution, the methods have the (far) future potential to be used as a substitute for experimental work. Of course it must be stressed that before this can be the case, the methods themselves must be subjected to a long process of validation against results from measurements (with wind tunnel quality and of su ciently high Reynolds number) in order to be trustworthy. Blade Flow Analysis The ow conditions just upstream of the rotor plane, as determined from the analysis of the global ow eld, form the `in ow' condition for the rotor blades and determine the aerodynamic forces on the blades. It is repeated that the magnitude of these forces (e.g. in terms of pressure drop across an actuator disc with nonuniform loading) in turn forms the needed input for the analysis of the global ow eld. The actual matching of these ow regions depends very much on the method of analysis that is chosen. 2D Blade Element Theory Blade element theory is the method of choice for the analysis of blade properties when the annular momentum method is used for the global ow region. In fact, the application of the momentum (axial and angular) equations results in analytical relations between the induced velocity components and the aerodynamic forces exerted by the elements of the blade in this region. In 2D blade element theory, it is assumed that these forces are equal to the forces on the same aerodynamic pro le taken from twodimensional wind tunnel aerofoil tests. Hence the force characteristics in terms of the nondimensional lift and drag coe cients C i (a) and C d (a) can be taken from wind tunnel measurements, as long as the Reynolds number is roughly the same in the measurements and in the turbine situation. This approach is based on Prandtl's slender wing (lifting line) approximation, in which, for an aerodynamic wing with a large spantochord ratio, the forces on an element are taken equal to the 2D forces for an equivalent angle of attack that are formed by the mean ow plus the velocities induced by the 3D trailing vortex
9 54 H. Snel system. In the case of the wind turbine this 3D induction is the result of the helical trailing vortex system in the rotor wake and can be equated to the induction velocities of momentum theory. Relating the aerodynamic forces (calculated in the manner) in the annulus to the momentum loss in the same, a closed set of nonlinear algebraic equations is obtained which can be solved numerically. This is the feature that makes the combination of momentum theory and 2D blade element theory computationally very e cient. In the stateoftheart models used for aeroelastic analysis, a number of corrections are applied to this strictly 2D method, such as the socalled tip correction and, more recently, a correction for 3D e ects in stall and models for dynamic stall. The correction known as the `tip' correction accounts for the fact that induction is not uniform over the annulus under consideration owing to the nite number of blades and the resulting nonuniform vorticity distribution in the wake. In fact, the tip correction is a nonuniformity correction. There are several ways in which this correction can be applied; see for instance Reference 10 for a discussion of the various possibilities. The most common implementation is the one introduced by Wilson and Lissaman, 12 based on an analysis by Prandtl. This correction, which depends on the tip speed ratio, the number of blades and the radial position, is only important in the blade tip region and e ectively changes the local angle of attack, which tends to the zerolife angle approaching the tip. The 3D correction for stall is essentially unlike the tip correction in the sense that it is a correction to the 2D (wind tunnel determined) aerofoil characteristics, mainly to account for the e ects that radial ow has in a rotating system. This e ect is completely di erent from that in a nonrotating system, since Coriolis forces are introduced that act as additional pressure gradients on the ow about the blade section. This e ect is explained in more detail on page 57. It has been in uential in improving BEM results, compared with measurements, for stalled ow. Finally, when analysing an unsteady situation (turbulent wind, dynamically responding blades, etc.), the measured steady C l (a) and C d (a) data are usually replaced by a model for dynamic lift and drag coe cients. This is especially important in stall, as important hysteresis loops form. In fact, generally speaking, the dynamic stall loops of the lift coe cient contribute to stable ap (outofplane rotor blade bending) dynamics, whereas the use of steady coe cients in stall sometimes leads to unstable results of the computation. Stable ap behaviour is observed experimentally. There are a relatively large number of socalled engineering models in use for the simulation of dynamic stall phenomena. These models have the form of ordinary di erential equations in time and are completely empirical in nature. A recent overview of methods presently in use can be found in Reference 29. There is still considerable ongoing activity in this eld, among others towards the modelling of the selfexcited vortex shedding that causes higherfrequency unsteadiness in stall. Much of this is based on the approach pioneered by Truong. 30 Accurate predictions of 3D stall delay and of dynamic stall characteristics are still the most challenging problems for aerodynamic analysis. Although considerable progress has been made, both in understanding and in modelling, the results are not yet su ciently accurate when compared with measurements. The prediction of the maximum power for a stallcontrolled turbine is within 15% accurate, while power curve guarantees must be given by manufacturers of 5% accuracy. Also, there are still serious discrepancies in the simulation of some dynamic problems in stall, especially the inplane dynamics of large blades. This will be discussed on page 58. Lifting Line and Lifting Surface Theory Lifting line or lifting surface theory is usually applied to blade ow analysis when vortex wake methods are used for the global ow. In the lifting line method the rotor blades are modelled as `bound vortex lines', i.e. as a line (geometrically coinciding with the blade quarterchord line) along which a vortex strength is de ned that will vary with radial (span wise) position. The local vortex strength G is related to the local angle of attack, often through a (potential ow) relation between circulation and lift: G ˆ cc l a V ef 2 8
10 Review of Rotor Aerodynamics 55 where c is the local chord, C l is the lift coe cient as a function of the angle of attack a, and V ef is the e ective in ow velocity. V ef is vectorially composed of the wind speed, the relative speed Or due to rotation and the induction due to the wake (see next paragraphs). The local angle of attack is the angle between the blade chord and the V ef vector. When applying (8) and using C l (a) from wind tunnel measurements, the lifting line method is equivalent to the 2D blade element method as far as blade ow analysis is concerned. When combined with vortex wake methods, there will be vorticity `trailing' from the vortex line, in principle in the form of a vortex sheet, with local strength equal to the spanwise derivative of the `bound' lifting vortex line strength G. In the computational methods this is sheet discretized, e.g. in line element vortices for the free vortex wake described in Section The trailing vortex elements which have an initial direction (of the rotation vector) normal to the blade axis are convected into the wake and contribute to the induction in the rotor plane. In the case of unsteady conditions, apart from trailing vorticity associated with changes in the circulation along the lifting line, socalled `shed vorticity' will be produced, associated with changes in the bound vortex strength with time. This is comparable with the wellknown starting vortex that is created when an aircraft wing builds up its lift. Shed vorticity has an initial direction parallel to the lifting line. Its contribution to `induced velocities' must be very carefully considered. In fact, in blade element theory (or lifting line theory) the induced velocities are used to modify the free stream velocity magnitude and direction in such a way that the ow situation becomes comparable with the 2D situation and hence 2D theory or experiments can be applied locally to obtain sectional forces. Consequently, only the vorticity of 3D type must be used to determine the induced velocity. This includes all trailing vorticity, but not all of the shed vorticity. Indeed, in the unsteady 2D situation there will be a shed vorticity system also, but this is included in the 2D theory or experiment. Hence, for this system, only the di erence between the 2D and 3D situations should be used in accounting for the induced velocity. This is mathematically formulated in Reference 25. In the developing wake this is very di cult to trace and it is not always clear from the publications in this eld whether the implementations are consistent in this manner. In lifting surface theory the blade is represented in more detail. Instead of a bound vortex line, a distribution of vortex elements over a surface, usually the blade's camber surface is employed. In this case no 2D information is used. Instead, for each vortex element a boundary condition is used for ow alignment with the surface. All vorticity, wake and bound, must be used to nd the ow velocity vector, and a system of equations in terms of the bound vortex strength of the surface elements ensues. The Stuttgart ROVLM program makes use of the lifting surface method. Obviously, this method is much more demanding in computation time than the lifting line method. In the lifting surface representation the thickness of the blade sections is ignored. This can be amended by representing the real blade geometry in the calculations and by the use of a surface singularity distribution to resolve the inviscid and incompressible ow equations. Usually, source distributions on the outer blade surface are used together with a vorticity distribution along the camber surface. This method, usually referred to as the `panel' method, has been used quite extensively in aircraft aerodynamics, but only very few publications for wind turbine applications can be found. Reference 31 is one of the few examples. From the results it may be concluded that the added computational complexity is in no way compensated by the improved quality of the results. The main advantage of the lifting surface and panel methods is that they are 3D in character. However, 3D e ects are less important in attached ow. For separated ow the fact that lifting surface and panel methods are based on the inviscid ow equations turns into a signi cant disadvantage. In this respect the lifting line method has a relative advantage in that it allows a simple introduction of separated ow characteristics by using empirical C l (a) data in (8), possibly expanded with 3D stall delay models and dynamic stall models. Another disadvantage of the inviscid ow theory used in the lifting surface and panel methods is that pro le drag is not predicted, even in attached ow. Still, this quantity is very important for the prediction of the turbine's main shaft torque and power. In order to get acceptable predictions of these quantities in computer codes based on lifting line methods, the pro le drag must be
11 56 H. Snel added `arti cially' to the load calculations, based on an estimated value of the angle of attack or based on sophisticated boundary layer calculations. The only way in which more sophisticated blade ow models can contribute notably to the understanding of real phenomena and improvement of the accuracy of design methods is by including the viscous e ects, as discussed in the next subsections. General Aspects for Viscous Flow Modelling Before entering into a more detailed discussion of the present possibilities of viscous ow modelling, it is important to realize the fundamental problems inherent to this. Basically, these problems can be related to the fact that the ow about the blade will, at least partly, be turbulent in nature. In laminar ow at low Reynolds number the viscous ow equations can be solved numerically without too much di culty. However, owing to the large Reynolds numbers involved (a typical value is Re ˆ for a megawattsize wind turbine), the adverse pressure gradient on the blade suction side, blade roughness, radial ow due to yaw error and in ow turbulence (1%±2% turbulence intensity is typical for the blade ow problem), smallscale turbulence will develop in the shear layer on the blade surface. It is of importance to know at what location the transition from laminar to turbulent ow will occur, but in view of the large number of parameters involved, this is next to impossible. In fact, Madsen et al. 32 suggest that the actual problem of double stall* can be traced to di erent transition locations, in uenced perhaps by the above parameters. In general the aerodynamic characteristics of the blade will be in uenced by the location of transition. In current practice of the application of boundary layer theory the general procedure is either to have the user x the transition point or to let the computer program decide where transition takes place, the latter based on the value of some boundary layer speci c quantities. For 2D ow the socalled spatial ampli cation theory of van Ingen 33 and Smith 34 is often used, but the extension to 3D ows is not well established. Moreover, even in 2D it is questionable to what extent the above parameters can be taken into account for the determination of the transition location. The situation is worse in the case of Navier±Stokes solvers, since many commercial packages do not even permit transition but perform only completely laminar or completely turbulent calculations. Another fundamental problem is turbulence modelling. For highreynoldsnumber ows the only practical way of treating turbulence is through a procedure known as `Reynolds averaging' (see Appendix), which leads to a larger number of unknowns (the socalled Reynolds stresses) than equations. In order to resolve this di culty, closure relations must be introduced, the socalled turbulence model. A large number of such models exist in the literature, which are all semiempirical. The most widely used turbulence model is the socalled k±e model, 35 in which k represents the turbulent kinetic energy and e its dissipation rate. This model is usually applied together with a version of the turbulent wall law. This is convenient in the sense that the region directly adjacent to the solid surface, where the largest gradients occur, is solved in an analytical manner. However, it is known that this model is not su ciently accurate for ows with important adverse pressure gradients and with large separated regions, producing too high values of the shear stress. Possibly the k±o model developed by Wilcox 36 together with an adaptation proposed by Menter 37 improves the situation. Indications to this e ect can be found in Reference 38. 3D Boundary Layer Methods A relatively e cient way of introducing the e ects of viscosity into calculations for highreynoldsnumber ows is by separating the ow domain into an inviscid potential ow part and a thin viscous boundary layer on the surface of the blade. The equations to be analysed in each of these regions are far simpler than the complete Navier±Stokes equations and for this reason it has been the method of choice for aircraft wing aerodynamics (see Appendix). For the outer region a Euler or even a potential ow solver can be used. The pressure eld determined with this method is used as input to the boundary layer *Double or multiple stall is the phenomenon in which a turbine can operate at distinct power levels under e ectively the same external conditions. The cause of this phenomenon is not known with certainty.
12 Review of Rotor Aerodynamics 57 equations. In the boundary layer approximation the spatial pressure derivatives in the two surface directions are used as driving forces, while variations across the layer (normal to the surface) can be neglected. If desired, a quantity known as the displacement thickness can be obtained from the boundary layer solution and used to obtain a correction on the potential ow solution of the pressure eld. A basic problem with the boundary layer equations when used with a prescribed pressure eld is the fact that the solution becomes singular at a ow separation. This causes numerical divergence problems with the corresponding computer programs. The problem can be solved by using what is called a strong interaction technique, in which the boundary layer equations and the outer ow equations are solved simultaneously or in a more sophisticated iterative manner than described above. Details about this process for an application to rotor blades can be found in the thesis of Sùrensen, 39 who pioneered the application of these techniques in a complete 3D fashion for wind energy use, obtaining very interesting results clarifying the importance of radial ows in the rotating case. The computational results showed the development of radial ow especially when separation occurs. Radial transport of retarded boundary layer material towards the tip (in the separated case travelling essentially with the chordwise blade velocity) has the e ect of a favourable (i.e. accelerating) chordwise pressure gradient. The result is that the separated wake for the rotating blade is thinner than for the corresponding 2D case, which means higher lift and lower drag forces. The e ects are especially noticeable close to the blade root, decreasing in magnitude for stations closer to the tip. Recently, boundary layer measurements (for a Reynolds number of ) with LDA techniques showed radial ow components of the same order of magnitude as the (reversed) chordwise velocities in separation. 40 Calculations with the boundary layer approximation, even in strong interaction with the outer ow, usually break down in deep stall. Quasi3D Boundary Layer Methods, the Case of Stall Although 3D boundary layer methods are an order of magnitude less demanding in computational time compared with Navier±Stokes solutions, they are still of a magnitude that precludes their use outside the research environments. Also, for the design of aerofoils for use in stallcontrolled wind turbines there is a need for a less expensive and quicker method. This was made possible by the development of a quasi3d boundary layer method, which took place mainly in the Netherlands. Based on physical insight obtained from the complete 3D boundary layer calculation results, Snel 41 presented an orderofmagnitude analysis of the di erent terms in the 3D boundary layer equations for a rotating slender blade in terms of the local chordtoradius ratio c/r. It was shown that for the case of separated ow the radial `parts' of the convective acceleration are of order (c/r) 2/3 compared with the main terms (relatively of order one). In the chordwise boundary layer momentum equation the Coriolis force term remains O(1) and contains the radial velocity component. Hence the radial momentum equation has to be resolved also. Neglecting all terms of O(c/r) 2/3 and smaller, both the momentum equations and the continuity equation do not contain radial `convective' derivatives and can be solved in a 2D (stripwise) fashion. This is the crucial advantage of the method and enables the solution of the equations to be done on a fast personal computer in a tolerable turnaround time. The method has been implemented in (originally 2D) boundary layer strong interaction codes, namely in the NLR ULTRANV code 42 and later in the commercial XFOIL code. In comparison with measured results, the calculated results show qualitative agreement, but they need improvement in the quantitative sense. However, used with judgement, the method o ers a tool for pro le design and analysis that can be used on a PC. An important drawback of the method is its application in the blade root region, where the radial ow e ects are important but where c/r has a value of about Neglecting terms of order (c/r) 2/3 then becomes doubtful. Another drawback of boundary layer methods in general is that the conditions under which the boundary layer approximation is valid (boundary layer thickness small compared with the chord) perhaps include the initial stall regime, but certainly not deep stall, where the wake thickness is of the order of the chord length. In fact, this limits the applicability to an approximately 208 angle of attack. Stallcontrolled (or partial span pitchcontrolled) wind turbines will operate with angles of attack close to
13 58 H. Snel 358 or 408 in the blade root sections at high wind speed. For these conditions a complete Navier±Stokes solution is the only possibility. Navier±Stokes Solutions Navier±Stokes solutions, both 2D and 3D are a current theme for many research groups in the eld of wind energy. It must be stressed, however, that all published work considers the Reynoldsaveraged Navier±Stokes (RANS) equations, which need an empirical turbulence closure model. This is the only practical possibility for external highreynoldsnumber ows at present. There are some important problems regarding the usefulness of the method. One fundamental question regards the applicability of RANS, which averages the turbulent uctuations (by either time or ensemble averages) of turbulent ow elds which are unsteady by their very nature into a ow eld of statistical averages. A growing number of experts seem to believe that there is no way around direct numerical solutions (DNSs) of the Navier± Stokes equations or at least largeeddy simulations (LESs), which do represent (at the larger scale) the unsteadiness in a direct manner. If this opinion is shown to be correct, then useful Navier±Stokes solutions are even much further away, as such solutions for 3D ows around blades at high Reynolds numbers are very far removed from what is possible at this moment. Even with the RANS method, practical problems regarding transition and turbulence modelling exist (see page 56). Wolfe and Ochs 43 report their use of a commercial CFD code (ACE) to compute the ow about an S809 pro le and an NACA 0012 pro le, comparing the computed results with measurements of Ohio State University. 44 They observe that no commercial Navier±Stokes code includes a transition criterion. Upon their speci c request, a possibility of modelling the e ects of transition was built into the code, and only with this aid was a good solution obtained for attached ow conditions. A distinct disadvantage is that a change in the prescribed transition location results in the necessity of constructing a new grid. A dependable and practical solution of the transition prediction problem is urgent. Also they argue that the less than satisfactory result for deep stall conditions, and indeed the poor prediction of the maximum life coe cient for the S809 aerofoil, is possibly caused by the de ciency of the k±e model for the condition of stalled ow. Chaviaropoulos 45 describes recent work at CRES (Greece) in the realm of 2D unsteady and quasi3d Navier±Stokes modelling. Here also the problem regarding the turbulence modelling is noted. The code developed at CRES (research code) does allow the possibility to use xed or free (determined during the computation) transition, but di erences arising from this are not discussed. For the quasi3d modelling, the expansion in terms of c/r, as discussed on page 57, is used. Qualitatively, the e ect of increased lift in separated ow is reproduced. Finally, some of the work by Sùrensen and colleagues at the Technical University of Denmark (TUDk) in cooperation with Risù must be noted. Within this cooperation the EllipSys3D steady Navier±Stokes solver was developed. In Reference 46 this program is applied in a complete CFD approach for the ow about a wind turbine rotor. The global ow eld is resolved by an axisymmetric Euler solver, with the rotor represented by an actuator disc. The rotor ow is solved by the use of EllipSys3D. In fact, the blade ow is rst solved as if no induction occurs. Next the blade forces calculated are introduced as pressure jumps across the actuator disc. The resulting induction is calculated and with this the blade in ow is reformulated. This process is repeated until convergence is reached. With regard to turbulence modelling, the Menter 37 adaptation of the k±o model is used with good results. When arguing that 3D Navier±Stokes solutions are important especially in the case of stalled ow, it should be stressed that up to date no nonstationary timerealistic 3D Navier±Stokes solutions for wind turbine blades have been reported. Although correct modelling of steady stall (or rather timeaveraged stall) is important for the prediction of power production, it is well known from wind tunnel measurements that stall is intrinsically unsteady in character, even in stationary external conditions, and to an even much larger extent in the everchanging conditions of the real wind turbine. This is of utmost importance for load uctuations and even for the aeroelastic stability of the blade. With regard to outofplane ( apwise) blade oscillations, it is now generally accepted that dynamic stall phenomena are
14 Review of Rotor Aerodynamics 59 responsible for an e ective damping, whereas the stationary C l ±a curve would give rise to negative damping. Stateoftheart `engineering' dynamic stall models are able to model this relatively well. However, for inplane (lead±lag) blade oscillations, aerodynamic damping may not be improved by dynamics. In fact, recent experiences with large stallcontrolled rotor blades have shown extreme inplane load uctuations (up to ve times the gravitational amplitude), which are held responsible for some recent blade ruptures. Although the exact cause is not known and it is hypothesized that part of the problem is of a structural dynamics nature, aerodynamic damping or lack of it must play a role. This underlines the importance of improved understanding and modelling. It is nally observed that even blades of pitchcontrolled turbines may experience dynamic stall phenomena through wind gusts around the rated wind speed. Available Experimental Data and Need for Controlled Measurements The validity of analysis and calculation methods can only be shown by comparison with experimental results. At the same time, these results add to physical insight that is essential to further improve the analysis methods. Although experiments on rotating blades are essential, nonrotating measurements can be helpful in comparing results with rotating data and establishing di erences, to separate rotational e ects from other 3D e ects. Early eld tests were done by Hales 47 on a small rotor mounted behind a car, both in yaw and in stall. The pressure distribution was measured through pressure taps on a few sections. These were the rst tests that showed the particular behaviour of stall on a rotating blade in detail. Still, the Reynolds number (around ) of these data was relatively low. Nevertheless, it marked the beginning of a series of eld tests, also for larger machines. Recently, quite a number of institutes have been engaged in aerodynamic eld tests on ve di erent rotors of di erent sizes. Results of these are now available in a database within an IEA activity known as Annex XIV. 48 The rotor diameters of the turbines used range from 10 m (NREL test facility and Delft University test turbine) to 27.5 m (ECN HAT 25 test facility). On the largest of these machines, Reynolds numbers of typically can be attained, which are su ciently high to make the results of direct interest. The objective of the di erent test programs was to measure aerodynamic forces on the blades, which in almost all cases was done with the aid of pressure taps on a number of sections. Only the Risù turbine measures the forces by way of threecomponent force balances in which three di erent sections of small radial extent are suspended. A detailed description of the test facilities, the data acquisition systems and the measured data is contained in Reference 48. Special attention was given to stalled operation and yaw misalignment, since the most urgent unresolved problems lie in this area. The typical triangular pressure distribution for rotating blade stall at inboard blade sections, as opposed to the at distribution in the separated region that is measured in the wind tunnel, was measured for all rotors. For yaw conditions, much of the information present in the database can be used undoubtedly to verify yaw models in wind turbine response programs. This not only concerns the blade aerodynamic forces, but also measured load signals on the blade. Much of this type of analysis remains to be done. Present plans are to establish a new IEA Annex in which the `owners' of the database will make structured use of the wealth of data available there. Nevertheless, it must be realized that the comparison of eld measurements with computational model data is a di cult task. The largest problem is that the in ow in the rotor plane cannot be determined in a deterministic sense, since the only information is given by the registration of wind speed and direction at (at best) a number of stations at some distance (hopefully upstream) from the rotor. Also, wind speed and direction will be nonuniform over the rotor and the distribution will vary stochastically. Basically, only statistical data values can be obtained with some degree of accuracy, both for wind speed and wind direction. A di erent problem underlies the determination of the angle of attack pertaining to the measured pressure distribution on a section. Even if the in ow were completely determined, still the angle of attack
15 60 H. Snel as de ned in the modelling includes the induction of the wake vorticity (or the relevant part of it). More information regarding the in ow conditions and the angle of attack could be obtained if the ow eld in the rotor plane could be measured. This, however, is not a practical proposition for eld measurements. Moreover, even these data should be treated with considerable care, as the bound blade vorticity and the 2D part of the shed vorticity in uence the measurements but should not be included in the de nition of the angle of attack. In the end the di culties are related to the fact that the angle of attack is a 2D concept that is de ned in a wind tunnel environment. Its generalization to the case of 3D rotating blades is by no means trivial, and the practical translation to measured quantities is a matter of further research. Notwithstanding the enormous value of eld measurements at representative Reynolds numbers, the above observations state a clear case for sophisticated wind tunnel measurements. At present, valuable pressure data on rotating blades exist for small Reynolds numbers (between and ) in the form of FFA measurements on a twobladed rotor in the Chinese CARDC tunnel. These data, however, were obtained with a slow measurement system only capable of taking data averaged over a rotation. Nevertheless, these data have been used extensively by many groups for validation of stall models. Flow eld information has been obtained in the past mainly in the Delft University open jet wind tunnel, again for very low Reynolds numbers. This information has been and will be valuable in the improvement of yaw modelling. Another type of measurement that is needed is that of unsteady stall characteristics, rst in a 2D tunnel environment, for the aerofoils to be used in wind turbines and for the typical values of reduced frequency for this application (below 0.1). One step in this direction was set by the very useful measurements at Ohio State University 50 which were used extensively in the Joule `dynamic stall and 3D e ects' project. Again, extension of 2D knowledge to the case of dynamic stall on rotating blades is very di cult. In the eld measurements discussed above, dynamic stall conditions have certainly been encountered, but the measured signals are very much in uenced by stochastic wind speed and direction changes, on which only some statistical information can be obtained. Perhaps with the aid of careful ltering, and by concentrating on regions such as blade tower passing, some information can be obtained from the existing measurements. In fact, work in this direction is done within the Joule Stallvib project and will no doubt be reported in the near future. However, it is likely that only controlled wind tunnel measurements can give su cient detail in information. Such measurements should be done on a su ciently large rotor to obtain Reynolds numbers of at least approaching 10 6, since for the typical reduced frequencies of wind turbine rotor blades, viscous e ects in the boundary layer dominate the dynamics and hence Reynolds e ects must be expected. Also, the model should be equipped with fast pressure sensors in order to capture fast changes in pressure distributions. The importance of this knowledge was discussed on pages 58±59. If ow eld quantities (velocities in or directly behind the rotor plane) could also be measured, together with the known and controllable in ow, de nitive information could be obtained on many aspects. Also, such measurements should yield data against which to evaluate the results of 3D Navier±Stokes solvers that are becoming available. Plans to do such measurements are in development, both in the United States (NREL) and in the context of European collaboration. It is this author's conviction that this type of measurement presents the only way to remove, at least to some extent, the main remaining areas of uncertainty in aerodynamic knowledge. Aerofoil Design A speci c item in the application of aerodynamics is the development of new aerofoil sections speci cally designed for use in wind turbine rotors. In the early stages of development the pro les used more often were those of the familiar NACA and NASA section families, also known as general aviation pro les. In fact, the NACA 44xx, NACA 230xx and NASA LS1mod sections were among the popular aerofoils in the very beginning, later being replaced by the NACA 63 AND NACA 64 families, especially for stallcontrolled rotors. The latter are still being used in the outboard part of many blades, but the continuous
16 Review of Rotor Aerodynamics 61 quest for lighter blades has led to the introduction of thicker aerofoils in the root section. This has been one motivation for the development of speci c wind turbine aerofoils. Another important driving force has been the desire to use aerofoils with very low susceptibility to the e ects of dirt accumulation and/or fabrication irregularities. This aspect is especially important for the operation of wind turbines in the desertlike environment of Californian wind farms. Work on dirtinsensitive aerofoils started in the United States, where Tangler and Somers 51 have been active in the development of the S pro le family, both thick and thin. These pro les have been utilized exclusively in the United States. In Europe the work of BjoÈ rck at FFA 52 must be mentioned (FFAW family) and the work of Timmer and van Rooij at Delft University of Technology 53 (DUW family). More recently, Fuglsang and Dahl 54 at Risù used optimization techniques for specialpurpose aerofoil design. With respect to the use of thick aerofoils, it has become clear that upscaling in thickness of the general aviation aerofoils gives very poor performance. Special pro les of relative thickness of up to 30% have been designed at both FFA and DUT and are presently used by the main blade manufacturers. An inherent property of these thick aerofoils is the small a range between the design value and the C l,max value. On the other hand, thinner pro les have a much wider a range in this context, especially the general aviation families. Since, moreover, the change in a with wind speed (for a xed speed, xed pitch turbine) is much larger in the root than at the tip, it follows that a thick root section will stall too soon and a thin tip section will stall too late. This is resolved by some manufacturers by using vortex generators in the root section (up to 50% span) and using stall strips in the tip section. Another approach is to design thin aerofoils speci cally for a smaller a range. This also is one item of research and development. In general, since blade design details are of a competitive nature, not much information is present in the open literature with regard to these items. However, it is clear that the application of vortex generators or other boundary layer manipulators on rotating blades is done in an empirical manner and that much could be learned by systematic (wind tunnel) investigations. Acknowledgements The author wishes to express his gratitude to many of his colleagues who have given information and comments that have improved the quality of this review. In particular, thanks go to Gustave Corten and Gerard Schepers. Appendix: System of Flow Equations Introduction The derivation of the equations describing the ow of uids can be found in many textbooks and monographs, e.g. Reference 55. In this appendix the equations are stated and interpreted physically to aid in the understanding of the basis of certain approximations that are discussed in the main text. Space coordinates will be given in a Cartesian system x j, j ˆ 1, 2, 3, and the velocity eld will be given by the three velocity components u j, which are functions of position x j and time t. Other important ow quantities are the pressure p and the mass density r. Throughout this appendix, incompressible ow of uniform density will be assumed. The approximation of incompressibility is valid for ow elds in which the local velocities are small compared with the speed of sound propagation (small Mach number). In that case, density variations resulting from pressure variations can be neglected. The speed of sound a in an ideal gas (e.g. air at atmospheric conditions) is given by p a ˆ grt ; g ˆ cp c v ; R ˆ c p c v
17 62 H. Snel where c p and c v are the gas (air) speci c heat coe cients for contrast pressure and constant volume processes respectively, R is the gas constant and T denotes absolute temperature (K). For air, g ˆ 1.4 and R ˆ 287 J kg 71 K 71. Hence for normal atmospheric temperatures a will be equal to approximately 340 m s 71. For local ow speeds up to 100 m s 71 the approximation of incompressibility is acceptable. For the density to be uniform, the additional requirement exists that temperature and strati cation e ects on the density must also be neglected. Navier±Stokes Equations Under the conditions outlined above, the Navier±Stokes equations describing the ow of a uid j ˆ 0 i j ˆ u j 2 f i ; i ˆ 1; 2; 3 10 j The usual convention is used that a repeated index in a single term implies summation over the index values. The scalar equation (9) is known as the continuity equation, expressing mass conservation. For the constant density case this is equivalent to volume conservation. Hence the net volume ow across a closed surface must be zero or mathematically expressed: the velocity eld is divergencefree. This is expressed by equation (9). The vector equation (10) is Newton's law of conservation of momentum in the three coordinate directions. This is referred to as the momentum equation. The two terms on the lefthand side (IHS) of the equation express the acceleration of a uid particle. The partial time derivative is known as the local acceleration and the second, nonlinear term describes the acceleration due to convection of the particle with the ow eld (convective acceleration). The righthand side (RHS) of the equation contains the forces (per unit mass) that are responsible for the accelerations. The rst term is the force due to pressure di erences and the second term represents the viscous force. The quantity v is the kinematic viscosity of the uid. Its value is a slowly varying function of the temperature; for `normal' atmospheric air its value is approximately equal to m 2 s 71. Finally, the term f i represents a possible external (body) force per unit mass. The solutions to these equations must satisfy certain boundary conditions. On solid surfaces within the ow eld the ow velocity relative to that surface must be equal to zero. This can be conveniently decomposed into a `notransparency' condition (normal velocity zero) and a `noslip' condition (tangential velocity zero). It should be noted that these conditions are not prescribed on an actuator disc. Instead, external forces applied at such a surface are to be prescribed, but the ow can pass through and by an actuator disc. The total system consists of four scalar equations in four unknowns, namely the velocity vector components u i and the (scalar) pressure p. It is nonlinear in the velocity components through the convective acceleration terms. Analytical solutions are known for very few special cases. Direct numerical solution of the equations for ows with large Reynolds numbers (see below) has been impossible for one main reason: the ow develops instabilities in regions of large shear and uctuations with time and position occur on such small scales that no computer power is able to handle these.
18 Review of Rotor Aerodynamics 63 The Vorticity Transport Equation An alternative form of the momentum equation can be obtained by casting it in terms of the vorticity vector o k, which is de ned mathematically as the curl of the velocity vector u i : o j u i 11 The vorticity is directly related to the rotational velocity of uid particles, the rotational axis being along the direction of the vorticity vector. Taking the curl of equation (10) and making use of (9), an equation is obtained which is usually called the vorticity transport j ˆ o j o i k j j 12 The IHS of the equation only describes the rate of change in vorticity following a material particle; the RHS describes the causes of the change. The rst term on the RHS is the most di cult to describe physically. It expresses the redistribution of vorticity by deformation of vortex lines, such as stretching and rotation. Note that this term is equal to zero in the 2D case, since then the vorticity vector is normal to the plane of ow, while the velocity derivative in that normal direction equals zero. The second term describes the viscous di usion of vorticity, as is clear from its form, which is that of a Laplace operator. It is similar to a conduction term if the quantity is a scalar, e.g. temperature. It should be kept in mind, however, that vorticity is a vector quantity and a function of the velocity eld u i, so equation (12) is strongly nonlinear. The last term describes the creation of vorticity by the body forces. In fact, vorticity can be created at solid boundaries as a result of the noslip condition or through the action of external forces. With respect to these external forces the reservation should be made that they can only create vorticity if they are not derived from a force potential (i.e. if they are nonconservative), since in that case the curl would be equal to zero. This in fact made the pressure term disappear on taking the curl of (10). Physically, this becomes clear by observing that pressure forces are normal (to the particle surface) and cannot cause rotation. The external forces at an actuator disc will usually create vorticity. The Reynolds Number As a result of the small value of viscosity, the viscous forces expressed in equation (10) will only be important in regions where the spatial derivatives of the velocity (velocity gradients) are large, or more precisely, where the variations in these gradients are large. Equivalently, the di usion of vorticity will only be important in regions of high vorticity gradients. This can be expressed conveniently through the use of the Reynolds number, as will be discussed here. In fact, let U be a representative value of the velocity and let L be a representative length scale of the problem. If spatial derivatives are of the order of U/L, then the orderofmagnitude relation between the convective acceleration and the viscous forces (per unit mass) can be inferred from (10) to be " O u i j " # ˆ O u 2 j U 2 L v U ˆ UL ˆ Re 13 v L 2 The Reynolds number Re is an important nondimensional quantity that characterizes the type of ow. For the ow about a wind turbine the characteristic velocity is of order Or and for the length scale it is
19 64 H. Snel customary to use the aerofoil chord length c. A typical value of the Reynolds number for a large turbine (e.g. c ˆ 1.5 m, Or ˆ 50 m s 71 ) will be around This means that the accelerations are very large in comparison with the viscous forces, so the latter cannot be responsible for the accelerations. Hence the pressure forces must be dominating the ow eld. This considerations leads to the approximation of ow without viscosity. The Euler Equations Neglecting the viscous terms in equation (10), the socalled Euler equations are obtained i j ˆ j i ; i ˆ 1; 2; 3 14 i together with the unchanged equation (9). These equations are of lower order in the spatial derivatives than the Navier±Stokes equations and cannot be made to obey the same boundary conditions. In fact, solutions to the Euler equations are only supposed to satisfy the `notransparency' boundary condition. This also implies that no vorticity is generated in the ow eld, apart from the possible action of the external body forces. However, this is in disagreement with physical reality, as will be discussed below. If the ow is emanating from a vorticityfree region (e.g. a region of uniform velocity), then the entire ow will be vorticityfree (or irrotational) under the assumptions that the Euler equations are a correct model and that no vorticity is created by body forces. In this case the velocity eld can be described even more simply by noting that a vector quantity whose curl equals zero can be expressed as the gradient of a scalar function. Hence u i F 15 where F is known as the velocity potential. Since the divergence of u i equals zero, F satis es the Laplace 2 2 i ˆ 0 16 which has to satisfy the nontransparency condition on solid surfaces, i.e. ˆ u n ˆ 0 17 where n is the unit direction vector normal to the surface. This ow model is known as potential ow (inviscid, vorticityfree and incompressible). It is possible to resolve the velocity eld by solving the Neumann problem for the Laplace equation. The pressure can then be obtained from the momentum equation (14). When applying the Euler equations to the global ow eld about a turbine rotor, the ow upstream of the rotor may be regarded as vorticityfree. However, vorticity is created at the actuator disc which models the rotor, through the external forces prescribed there. In the wake, downstream of the rotor, this vorticity is transported, but the Euler equation does not model its di usion. Because of the large Reynolds number involved and the absence of solid surfaces, this is a valid approximation in general. However, when applying inviscid ow theory to the ow about the solid blade surface, the following problem arises. Vorticity is formed physically on the actual blade surface as a result of the noslip condition, but this condition is not applicable to the inviscid equations (14) and (16). In reality a thin region on the boundary will exist where the velocity gradients are so high that the viscous terms are of the same order of magnitude as the pressure terms and the acceleration. This region, known as the boundary
20 Review of Rotor Aerodynamics 65 layer, contains all the vorticity and is extended into the wake. It is thin because of the fact that the convectional velocity of the vorticity in the direction parallel to the surface is very much higher than the `di usion velocity' in the direction normal to the surface. A x to the problem of vorticity formation and di usion in highreynoldsnumber ows is found in Prandtl's boundary layer theory. Boundary Layer Equation In the boundary layer approximation the ow eld is modelled in two parts. One is the boundary layer directly adjacent to a solid boundary and its ensuing wake, containing vorticity. The other part is usually called the `outer ow', which is modelled as vorticityfree. Within the boundary layer, let x 2 denote the direction normal to the solid surface and x 1 and x 3 be directions along the surface; see Figure 4 (surface curvature can be neglected usually). Figure 4. Coordinate directions used for boundary layer description In the boundary layer approximation the pressure change across the boundary layer is neglected, i.e. the momentum equation for direction x 2 2 ˆ 0 18 For the momentum equations parallel to the surface the viscous terms containing derivatives in directions x 1 and x 3 are neglected (in these directions, di usion is very small compared with convection) and only the second derivative in the direction normal to the surface is retained, since in that direction the velocity gradient is very large indeed. Hence these equations i j ˆ u i ; i ˆ 1; 3; j ˆ 1; 2; 3 j 2 2 Note that the convective terms of the LHS of the equation remain completely 3D in character. The boundary layer equations are of parabolic type, as opposed to the elliptic character of the Navier± Stokes equations. This means that the boundary layer equations can be solved by a marching procedure in the streamwise direction, starting from `initial' conditions and applying boundary conditions on the body (zero relative velocity) and on the interface between the body and the outer ow (velocity equality). There is no ow of information in the direction opposed to the velocity direction. The Navier±Stokes equations, however, need boundary conditions all around the ow eld, and information can travel in all directions. This results in far more e cient numerical solutions schemes for the boundary layer equations in comparison with the Navier±Stokes equations. The boundary layer equations are solved with known pressure gradient terms along the surface. These latter are obtained from an inviscid solution of the outer ow. In this solution, however, the boundary layer vorticity must be present by introducing a quantity of vorticity, which is placed along the boundary or within a surface surrounded by the boundary. The outer ow eld is assumed to be vorticityfree. In fact, the inviscid ow solution can be seen as the limit of the Navier±Stokes solution for Re approaching
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