Category Principles of Helicopter Aerodynamics Second Edition

Computational Methods for Helicopter Aerodynamics[49]

The ability to design rotorcraft with confidence requires a new order of aerodynamic predictive technology that is both true to the basic flow physics and readily usable by industry.

F. X. Caradonna (1990)


This chapter reviews computational methods that have been applied or are being applied to the analysis of various problems in helicopter aerodynamics. The intention is not to review the details of such methods and their associated numerical algorithms per se, but to give a general overview of the underlying principles of the various approaches along with a summary of their capabilities and limitations for the various helicopter problems alluded to in previous chapters. The approaches are described with ample reference to original methodologies and results, as appropriate. A range of methods is covered in this chapter, from classical thin airfoil theory and surface singularity potential flow methods, to advanced computational methods that solve numerically the vorticity transport, Euler and Navier-Stokes equations. Several examples of the application of advanced aerodynamic methods to helicopter problems are also shown and discussed, with suggestions for areas of new research where appropriate.

While historically the helicopter industry has relied extensively on more parsimonious aerodynamic methods such as various forms of “momentum theory” combined with signif­icant empiricism for helicopter design work, this is fast changing. See Gessow (1985) for a good historical review of helicopter predictive methodologies through 1980. Since then, rapid advances in computer technology (both in terms of speed and memory) have allowed much more ambitious numerical methods to be used in helicopter aerodynamics, and this has spawned a great deal of new fundamental work, both in fluid flow modeling and in algorithmic development. Aerodynamic problems that were once considered intractable, or were relegated to the fastest super-computers, are now solvable on desktop workstations. The tremendous recent advances in computer capabilities have played a large part in mo­tivating and accelerating the development of more complete computational techniques for helicopter aerodynamic analysis, including vortex methods. This has helped to improve the rigor of the overall analysis and even to remove some of the more sweeping levels of empiricism – an important factor especially when new helicopters begin to depart from legacy designs.

It is the modeling of the rotor wake and its effects on the rotor and airframe that proves key to solving ultimately many of the outstanding problems in predicting helicopter performance

and improving helicopter design. The ability to better predict and understand rotor wake related problems continues to challenge the helicopter analyst, however, and it is here that further research must be focused. Advances in computer power have made possible the better integration of traditionally separate disciplines of helicopter analysis and will ultimately allow better rotor systems and more capable helicopters to be designed. Yet, bigger problems and more ambitious numerical techniques continue to push the limits of computer resources, with the ultimate (albeit elusive) goal of predicting the behavior of an entire helicopter in an arbitrary flight condition. Yet there is still much work to be undertaken before predictive models can be improved to the level that the aerodynamics of a new helicopter can be adequately predicted before its first flight.

Over the past two decades, advances have been made in the understanding of problems in helicopter aerodynamics using so-called computational fluid dynamics or CFD. Here, finite-difference or finite-volume approximations to the governing flow equations are used to model, from first principles, the complex flow field about the helicopter rotor and its airframe. The general field of helicopter CFD is reviewed by Caradonna (1990), Landgrebe (1988,1994), McCroskey (1995), and Conlisk (2002). These works contain many references documenting past, present, and potentially future capabilities of CFD as it might be applied to helicopter problems. The choice of which governing equations to use affects the level of physics captured by the CFD scheme, as well as the computational effort and time taken to solve the problem. Furthermore, the choice of numerical method affects accuracy, stability, and cost. To some, the development of CFD by itself is held out to be the “Holy Grail” for the helicopter aerodynamicist, yet this is a very misleading perspective because CFD has its limitations. These limitations include grid dependent solutions, numerical issues, the ability to model turbulence, and so on. CFD methods must also be validated against measurements if they are to realize the high predictive confidence levels needed for design work. Therefore, CFD does not, by itself, hold the answer to all the various aerodynamic problems on helicopters. The real answer lies in the successful integration of advanced forms of aerodynamic analysis into other disciplines of engineering analysis. It is also unwise for other approaches to be abandoned in the shorter term while “CFD” matures to an accepted level of capability, given that this could still be decades off.

Chapter Review

This chapter has introduced the aerodynamics of the wind turbine, mainly to show both the differences between and similarities to helicopter rotors. The idea that the wind contains energy has been outlined and the ability of a wind turbine to extract that energy has been explained. It is apparent that many strides have been made in the understanding and modeling of wind turbine aerodynamics, especially over the last two decades where environmental concerns have driven more research into renewable energy resources. For the future there is a need to refine and expand existing models and in some cases to develop new aerodynamic models that can encompass a wider range of operating conditions and wind turbine now states. This will allow adverse effects to be properly predicted early in the design cycle and more efficient and reliable wind turbines to be designed for the future.

It has been shown how predictive approaches for wind turbine aerodynamics have paral­leled the methods used for helicopter analysis. The fundamental performance of the machine has been examined using momentum theory and blade element momentum theory (BEMT). Despite its relative simplicity and limitations, the BEMT allows a clear examination of the design factors that influence wind turbine performance and design. However, the BEMT is limited in terms of generality to a less than desirable range of operating conditions.

Chapter Review

These limitations are largely eliminated with the use of aerodynamic methods based on vortex theory. Today, most engineering methods used in wind turbine analyses are based on blade element theory combined with either inflow models or vortex models to represent the nonuniform induced velocity associated with the vortical wake trailed from the turbine. One advantage of this class of predictive methods is the flexibility to include a wide range of validated subcomponent models representing various physical effects that are difficult to model from first principles. This approach also allows the subcomponent models to be vali­dated against idealized laboratory experiments, and the flexibility to progressively upgrade the models as a deeper understanding of the physics is obtained. They can also be com­bined with structural dynamic models of the blades, tower, and power generation systems to produce powerful aeroelastic tools for the detailed structural design of wind turbines. As in the case of all predictive models, however, sustained validation of the approach against experimental measurements is essential, despite the inherent difficulties and lengthy nature of the process.

It is clear, however, that there are two key areas that need continued consideration if the aerodynamic design of wind turbines is to be further improved. These are the modeling of the turbine wake and the modeling of the unsteady aerodynamics of the blade sections. Inflow models have attractive mathematical forms and low computational overheads that will always be useful for certain types of aerodynamic and turbine performance analyses. Vortex wake methods are attractive because of their appealing physical nature and their flexibility to handle a broad range of steady and transient operating conditions. While “prescribed” vortex wake models have seen some use in wind turbine applications, they have limited scope in practical applications, and today they are fast being surpassed by free-vortex wake approaches. It is still up to the wind energy practitioner to use these modeling tools, to explore their capabilities, and to determine the limitations in their use through validation studies with measurements. Only when this is done will the analyst be in a position to decide which models require further development. While it has been emphasized that significant unsteady effects may be produced on wind turbines even with the absence of dynamic stall, the particular problems produced by dynamic stall remain a serious concern for wind turbines, even for pitch-controlled machines. In higher winds when much of a wind turbine blade can be stalled, existing performance methods tend to predict power outputs that are considerably lower than those actually measured. Some of the potential mechanisms contributing to the 3-D aspects of the problem have been discussed, although the exact mechanisms at play still require a much deeper analysis of the flow. While a full understanding and modeling of these problems is the subject of ongoing research, they can, in most cases, be traced to the development of the 3-D boundary layer on the rotating blades.






Chapter Review


Chapter Review

Advanced Aerodynamic Modeling Requirements

There are a number of comprehensive models that have been developed for the analysis of wind turbines. Many of these models encompass the principles of the BEMT and are coupled to the structural dynamic analysis of the wind turbine and its tower. Such approaches parallel the philosophy of the comprehensive models used for helicopter analy­ses (Section 14.11) – see Manwell et al. (2002) for a discussion of the capabilities of these models. Yet although the fundamentals of wind turbine design are fairly well understood, the success at predicting blade loads and power output does not yet seem as great as would be desired. This is reflected in a recent “blind” prediction of the loads and performance of a comprehensively instrumented wind turbine that was tested under controlled conditions in the 80 x 120 ft (24.4 x 36.6 m) wind tunnel at NASA – see Fingersh et al. (2001). The primary objective of these experiments was to create a definitive set of turbine airloads and performance measurements, over a wide range of operating conditions, that was free of the uncertainties caused by the various atmospheric effects that are always found in field tests
with turbines. These wind tunnel results provided the analyst with an opportunity to really understand better the physics of wind turbine aerodynamics and perhaps gives a definitive data resource for validating predictive methods and resolving outstanding modeling issues. The results from the NREL blind comparisons [Simms et al. (2001)] showed considerable deficiencies between the various predictions for blade loads and turbine power output, even for unyawed, unstalled operating conditions. Such results underline the need for more fun­damental research in aerodynamic subcomponent methodologies and their interdependent coupling if overall predictions are to be improved.

As in helicopter work, advanced computational aerodynamic methods based on numer­ical solutions to the Euler and Navier-Stokes equations (see Chapter 14) have begun to see some use in wind turbine analysis. This class of CFD methods has the potential to provide a consistent and physically realistic simulation of the turbine flow field. The huge compu­tational costs, large memory requirements, and numerous numerical issues associated with such CFD methods (see the detailed discussion in Chapter 14) means that they have not yet seen significant use in wind turbine applications. In particular, problems involving flow separation, such as dynamic stall, have proved extremely challenging for Navier-Stokes based methods, in part because of a need to develop better turbulence models. Also, the prediction of the 3-D vortical wake behind a turbine has proved just as daunting for CFD as for any other type of method. This is because vortical wake formation is a result of complex 3-D, viscous, separated flow effects and also because the numerical methods have difficulty in preserving concentrated vorticity as it is convected downstream.

While it is clear that some have begun to take up the challenge of modeling wind turbine problems using CFD – see Duque et al. (1999,2000,2003) and Iida (2001) – the capabilities of these methods have not yet been validated sufficiently to be assigned the confidence levels that are necessary for wind turbine design purposes. With faster computers and with improved numerical algorithms, CFD methods will ultimately prevail and will become increasingly used in the design of better and more efficient wind turbines.

Dynamic Stall and Stall Delay

It has been explained in Chapter 9 how the phenomenon of dynamic stall can give rise to large unsteady blade airloads that can limit the performance of a rotor system. While dynamic stall is a well-known unsteady flow problem found on wind turbines, the foregoing discussion has emphasized that unsteady airloads will be produced even in the absence of dynamic stall. These effects manifest as both amplitude and phase changes in the blade airloads compared to what would be obtained under quasi-steady conditions. Both circulatory and noncirculatory contributions to the airloads are, of course, always involved (see Chapter 8).

A prerequisite to predicting dynamic stall and its onset is to predict adequately the unsteady airloads under attached flow conditions. Despite its apparent simplicity compared to the stalled case, however, this is a nontrivial problem for which the development of general models that are valid for the turbine environment is still a major challenge. Currently, dynamic stall must be modeled using more parsimonious, semi-empirical models, for which a number of different approaches have been developed for helicopter applications (see Section 9.5) and adapted for wind turbine use. Many modem wind turbine analyses use the method developed by Leishman & Beddoes (1989), which has been modified and popularized by Pierce & Hansen (1995). However, while often giving good results, these models are not strictly predictive tools and can really only be used confidently for conditions that are not too much different from the conditions under which the models were originally validated.

The problem of dynamic stall is of particular importance on wind turbines because the large unsteady airloads that are produced (see Fig. 13.19) can cause structural damage – see Shipley et al. (1994). However, the power output from a wind turbine can be regulated deliberately by promoting stall, either by pitch regulation or through airfoil design (see Section 13.9). An example is shown in Fig. 13.28 where the onset of stall creates much turbulence over the blade and in the downstream wake. Unfortunately the prediction of dynamic stall has not yet been very successful on wind turbines. While part of the problem is the imprecise modeling of the delayed onset of flow separation resulting from unsteady aerodynamic effects, there are also subtle 3-D effects unique to the rotating environment that seem particularly pronounced on wind turbines.

In summary, the effects underlying the problem of dynamic stall can be divided into two areas: 1. Unsteady pressure gradient reduction effects on the blade, which give rise to delays or lags in the development of the 3-D boundary layer compared to that obtained under quasi-steady conditions; and 2. Coupled influences of the centrifugal and Coriolis effects acting on the boundary layer in the rotating flow environment (i. e., a radial flow effect). In general it is fair to say that a better integrated modeling capability for dynamic stall depends on gaining a much better understanding of the physics governing the 3-D boundary layer developments on rotating blades, both with wind turbines and helicopter rotors.

Several experimental and modeling studies have provided some insight into 3-D ef­fects on rotating blades operating near stall – see Young and Williams (1972), Madsen &

Christensen (1990), Snel (1991), Narramore & Vermeland (1992), Dwyer & McCroskey (1971), Robinson et al. (1999), and Schreck et al. (2000, 2001). The first observations of the phenomenon are often attributed to Himmelskamp (1950). The motivation is clear, in that when stall occurs existing performance methods tend to predict power outputs that are lower than those actually measured. From an order of magnitude analysis of the 3-D bound­ary layer equations applied to a rotating flow environment, Snel (1991) has speculated that the Coriolis acceleration terms can act to alleviate adverse pressure gradients and so may delay the onset of flow separation and stall, but see also Corten (2000) for a critique of this analysis. The Coriolis and centripetal acceleration terms can be seen in a modified form of the boundary layer equations (see Section 7.3.2) first given by Fogarty (1951):

Подпись:Подпись: (continuity),3 и dw

——- b — =0

Эх 3 z

Подпись:Подпись: (13.69)

Dynamic Stall and Stall Delay

ди ди 9 1 dp 3 2u

и—– b v——- fi x =—————- b v—- (x momentum),

Эх 3z p Эх dzl

3 и dv 2 1 fy? d2v

и—– 1- w—— H 2fiw — fi x =———— b v—- (y momentum),

Эх 3z P dy 3zz

Dynamic Stall and Stall DelayПодпись: and

where x is chordwise, у is spanwise, and z is normal to the blade. The Q2x terms are centripetal accelerations and the 2Q, u term is a Coriolis acceleration. Fogarty (1951) has calculated the radial or spanwise development of the boundary layer flow on the blade and has shown that for helicopter rotors these effects are small, except near the inboard part of the blade. It is here, however, where the blades of wind turbines (because of their design) tend to exhibit much higher values of lift coefficients compared to a helicopter rotor, and so the effects of radial flow can be more important here. Experimental results have shown increases in sectional maximum lift coefficients significantly beyond what would be expected based on using 2-D static airfoil characteristics. The helicopter rotor experiments of Dwyer & McCroskey (1971) also suggest favorable effects on the spanwise development of the boundary layer, which tend to delay the onset of flow separation to a higher blade section AoA and thus serve to increase the maximum thrust of the rotor system.

Similar observations have been found using CFD methods such as in the work of Narramore (1992), although confidence in these types of analyses to represent fully stalled flows is not yet high enough. As a result, approximate methods have been developed to model the observed 3-D “stall delay” effects – see Corrigan (1994) and Du & Selig (1998). A rigorous approach is still lacking because these models have not been validated under unsteady conditions or for situations where dynamic stall may occur. It has been shown in Chapter 9 that dynamic stall is fundamentally different to the phenomenon of static stall, so this is clearly an area where much further research is necessary if better 3-D models of unsteady airfoil behavior representing stall are to be developed and used confidently for rotor airloads predictions in either helicopter rotor or wind turbine work. To model these more general 3-D effects, the complexity of the models must usually be increased and parameters added. One must be cautious though that modeling accuracy is not ob­tained at the expense of the parsimony that is necessary to use these models in design analyses, or to destroy confidence in predictions – see page 808 for a discussion of this point.

Tower Shadow

The effects of the tower wakes or “shadow” can be seen in the flow visualization of the downstream turbine shown in Fig. 13.20 where the bluff body-type flow from the upstream tower creates a velocity deficiency in the wind as it approaches the turbine disk. Experience suggests that there may be a 30% reduction in the local wind speed behind the tower. At a fundamental modeling level this problem is a good example where there is a need to distinguish properly the effects on the airloads on the blades arising from an assumed uniform change in a uniform AoA across the blade element from the effects resulting from a nonuniform velocity field. In an essentially incompressible flow, the former can be viewed as the net response to changes in AoA using superposition with the Wagner function (see Section 8.10). This is equivalent to assuming that at any instant the AoA is constant over the chord of the airfoil. This assumption is typical of what is used in many unsteady aerodynamic models for all modes of forcing. The latter can be viewed as the net response to a nonuniform vertical component of the velocity field using superposition with the KUssner function (see Section 8.11), which is the more accurate and correct assumption.

Подпись: 0.8 7 0.7 ^ ' і і ■ ■ і ■ і ■ -і-! і і ■ ■ і ■ » ■ і ■ ■ ■ і ■ ■ ■ і ■ ■ ■ і ■ ■ і ■ ■ i 140 180 220 260 300 340 Blade azimuth - deg. Figure 13.27 Prediction of the unsteady lift during a simulated tower shadow encounter using 2-D unsteady airfoil theory.

The velocity deficit in the flow behind the support tower can be modeled as a spatial variation in the velocity normal to the surface of the blade. In a 2-D model, the airfoil section moves rapidly through a velocity field, resulting in an attenuation in lift and a phase lag compared to the quasi-steady case. For the purposes of simple illustration, the problem can be modeled by assuming a 2-D airfoil traveling at unit velocity in an assumed velocity field normal to its chord given by w = 0.08 + 0.02 cos(5ty) where ф is in radians defined for the range 144° < js < 216° and w = 0.1 elsewhere. The results in Fig. 13.27 show

the differences in unsteady airloads produced using the steady and unsteady predictions. Clearly there are significant differences and the results emphasize again the importance of modeling unsteady effects if reliable quantitative predictions of the airloads are a goal. See also Munduate et al. (2003) for an analysis of the tower shadow problem and a comparison with measured airloads on the turbine blades.

Unsteady Aerodynamic Effects on Wind Turbines

It will be apparent from Fig. 13.24 that in the yawed (out of wind) case the component of the wind speed produced in the plane of the turbine produces a Nbtrev unsteady aerodynamic excitation at the blades and so the net power output must fluctuate. Because of the relatively low rotational velocity of wind turbines (about 20 rpm for large turbines and up to 800 rpm for small turbines) and low tip speeds compared to a helicopter, changes in wind speed (or turbulent fluctuations) can result in significant changes in AoA at the blade elements compared to the velocity induced by blade rotation alone. There are also blade flapping effects to consider, assuming that the blades are indeed free to flap about a teetering or gimbaled hub (see Chapter 4). The resulting changes in local AoA and onset velocity contribute significantly to the unsteady flow environment at the blades. While unsteady aerodynamic effects occur under other operating conditions and as a result of other sources, the out of wind (yaw misalignment) performance of the wind turbine is of fundamental importance in wind turbine design.

Wind turbines operate for most of their time in a relatively unsteady flow environment – see Robinson et al. (1995), Huyer et al. (1996), Schreck et al. (2000), and Leishman (2002). Figure 13.26 summarizes the various sources of unsteady airloads on a wind turbine. It is apparent that, like the helicopter, these sources can be decomposed into a variety of essen­tially periodic and aperiodic contributions. The airloads on each blade element vary in time because of yawed flow, shear in the ambient wind, ambient turbulence, blade flapping and vibratory displacements, and other factors such as “tower shadow” (see Section 13.12.1). Despite the unsteady flow environment, however, wind turbines tend to be relatively quiet

Flowfield Structure


Mostly Periodic Mostly Aperiodic


Unsteady Aerodynamic Effects on Wind Turbines

Unsteady Aerodynamic Effects on Wind Turbines

quasi-steady aerodynamics. There is a further lag in the development of the inflow in response to the changing downstream wake structure; such wake adjustments may take place over as many as 10 rotor revolutions.

2. Velocity gradients in the wind: The wind turbine operates in an atmospheric bound­ary layer with notable velocity gradients – see the schematic in Fig. 13.8. The effects of this gradient can also be seen to some extent in Fig. 13.20 where it is apparent that the lower part of the wake nearer the ground convects more slowly downstream than the upper part. This produces another source of nonuniformity in AoA over the turbine disk, producing a further source of unsteadiness at the blades.

3. Nonsteady velocity fluctuations in yawed flow: Large nonsteady vari ations in the local velocity at the blade elements will be produced whenever the wind turbine is misaligned (yawed) relative to the wind. This is a common occurrence because yaw control mechanisms usually cannot track the wind to sufficient accuracy for perfect alignment to be assured at all times. The low tip speeds of wind turbines means that large excursions from axial flow may occur, which can amplify un­steady effects even though the blade pitch and AoA may be relatively constant. Under these conditions the amplitudes of the excursions involved may often stretch the limits of the unsteady aerodynamic models, which are often derived under the assumption of small perturbations (see Section 8.13.)

4. Nonsteady wake induction effect: Time-varying aerodynamic conditions at the ro­tor will have an effect on the strength and positions of vorticity shed and trailed into the downstream wake. This evolutionary process has a hereditary effect de­pending on the prior time-history of the rotor airloads and appears as a temporal lag in the development of the inflow at the rotor disk. This problem has already been discussed in Section 13.11.

5. Local sweep effects: Another source of unsteady effects can be traced to the direc­tion of the incident flow velocities on the rotating blades. Because of the relatively low rotational velocity of wind turbines, the local sweep angle of the flow at any blade element can be very large when the turbine is yawed. With attached flows the independence principle applies and no significant changes in airloads occur. How­ever, the existence of a radial flow component can affect the development of the 3-D boundary layer on the blade and the onset of stall (see Section 13.12.2), lead­ing to somewhat more complicated aerodynamic behavior than would be predicted under nominally 2-D conditions. (See Section 9.7 for further details.)

6. Tower shadow effects: For downwind machines the passage of the blade through the tower wake or “shadow” results in transient changes in AoA on the blade el­ements. Upstream machines also suffer from the effects of the tower, this effect being similar to the blade passage effect for the helicopter rotor/airframe problem (see Section 11.2.3). This interaction effect may involve effective reduced frequen­cies that exceed 0.2. The resulting blade loads cannot thus be predicted accurately using quasi-steady assumptions. Such problems also involve steep velocity gradi­ents in the flow. At a modeling level the nonuniformity of the velocity field across the blade chord must be considered carefully. Such effects are modeled fairly well using the unsteady airfoil methods with imposed gust fields (see Section 8.16.3), although downstream wake effects may also have to be considered – see Munduate et al. (2003) and Wang & Coton (2001).

In general, the blade structural loads can be nonsteady because of additional factors, such as periodic variations of gravity forces on the blade in the rotating frame of reference,

transient rotational speed and blade flexibility. To calculate blade loads, coupled nonlin­ear equations involving rotational, inertial, and unsteady aerodynamic forces need to be solved for prescribed operating conditions. This is a major challenge for the wind energy community and parallels the issues known already for helicopters – see Section 14.11.

Vortex Wake Considerations

Despite the various assumptions and approximations used with the BEMT, val­idation studies have shown it can give good preliminary predictions of turbine loads and performance [see Hansen (1993)]. It also offers considerable insight into basic design pa­rameters affecting the turbine performance, including blade geometric parameters such as planform and twist. Under yawed conditions, however, the BEMT is less able to model the physics of the turbine aerodynamics because of the strongly nonaxisymmetric flow and the 3-D nature of the airloads produced over the turbine disk; see Schreck et al. (2000) for a comprehensive discussion. A major source of this three-dimensionality is the vortical wake system produced behind the turbine, a visualization of which is shown in Fig. 13.20. Like a helicopter, the turbine wake is comprised of strong vortices that trail from each of the blade tips and these are rendered visible in this photograph by the use of smoke generators. Because a wind turbine extracts energy from the wind, the wake expands downstream of the turbine. It may also distort as a result of the wind gradient and turbine yaw angle. While flow visualization of the wake downstream of a wind turbine is relatively rare, a good review

Vortex Wake Considerations

Figure 13.20 Photograph of the vortical wake behind a horizontal axis wind turbine ren­dered visible using smoke injection. Notice the skewed wake resulting from the wind gradient and the effects of the tower shadow. Source: Photo courtesy of NREL.

of the field and survey of existing results is given by Vermeer et al. (2003). See also Grant et al. (2000) and Vermeer (2001) for a discussion of the general characteristics of wind turbine wakes.

Various modifications to the blade element approaches have been used to represent these 3-D wake effects, for instance, by using inflow models, some of which have been derived from helicopter applications (see previously in Section 3.5.2). While improvements in predictions of net power are usually obtained using these methods, their ability to predict blade loads over an acceptably wide range of operating conditions or for new wind turbine designs is not very satisfactory; see Fingersh et al. (2001). A more rigorous treatment of the turbine wake requires a method that can represent the strengths (circulation) and spatial locations of the vortical elements that are trailed by each blade and convected into the downstream wake. This can be done using vortex methods, which have become very popular tools for the analysis helicopter wake problems. (See Section 10.7 for the fundamental theory of vortex wake methods.) The development of a vortex wake model is usually based on the assumption of an incompressible potential flow, with all vorticity assumed to be concentrated within vortex filaments. From the strengths of the vortex filaments (determination of which requires proper coupling to the blade lift distribution), the induced velocity field can be obtained through the application of the Biot-Savart law (Section 10.7.1).

The idea of a vortex model applied to the wake of a wind turbine is shown schematically in Fig. 13.21. Vortex methods applied to turbine wake problems can be categorized into “prescribed” or “free” vortex techniques, just as they are for helicopters (see Section 10.7). A prescribed vortex technique is v/here a discrete representation is made of the vorticity field, but the positions of the vortical elements are specified a priori based on semi-empirical rules – see Robison et al. (1995). Prescribed vortex methods can be viewed more as postdic – tive rather than predictive methods because they use experimental results for formulation purposes and so are strictly limited in scope to the conditions that encompass the range of measured operating conditions for which they were originally formulated. The free-vortex method (FVM) is a predictive method because the elements are allowed to convect and deform freely under the action of the local velocity field. While a disadvantage of all types of vortex methods for modeling the wake is their relatively higher computational overhead (mainly because of the need to evaluate the Biot-Savart law a very large number of times), they are an inherently more appealing physical approach for modeling the wake than the use of inflow models.

Prescribed vortex wake models have seen some use for wind turbines, e. g., Kocurek (1987), Robison et al. (1995), and Coton & Wang (1999). The development of more gen­eralized prescribed wake models for wind turbine applications is a significant undertaking because not only do the positions of the wake vortices need to be documented, but the experimental conditions must encompass a wide range of turbine geometric (e. g., blade shape and twist) and operating states (wind speed, yaw angle, etc.). Therefore, a trade-off must be drawn between the costs of doing more experiments and collecting more data to build the model, and the costs of developing a better and more flexible model that can be validated against existing data. Prescribed wake models are also only strictly applicable when the operating conditions are nominally steady-state, that is, in a steady wind (which never exists in practice). Free-vortex methods have fewer potential limitations when applied to new turbine designs and/or more general operating states. Such methods are yet to see significant use for wind turbine applications, but offer much more rigor than the ad hoc application of “inflow” models derived from helicopter rotor models, especially because these have not yet been validated for all of the flow operating states of a wind turbine.

Vortex Wake Considerations

Figure 13.21 Modeling the downstream wake of a wind turbine using a free-vortex fila­ment method, with vortices shown trailing from the tips of each blade.

Free-vortex methods are based on a discretized, finite-difference representation of the governing equations for the wake and, when solved, track the evolution of discrete vortex ele­ments through the flow. The number of discrete elements can be in the thousands, making the tracking process memory intensive and computationally demanding, although still consider­ably less demanding than when using the various types of CFD methods solved on Eulerian grids (described in Chapter 14). The vortex wake method must be supplied with boundary conditions that relate the lift (bound circulation) on the blades to the strengths of the trailed vortex filaments. This is usually done by using a lifting line or lifting surface model, such as discussed in Section 14.9. Experience suggests that the FVM generally gives better pre­dictions of blade loads than is possible with prescribed or rigid wake models. It can also, in principle, deal with transient problems such as when the turbine yaws into and out of the wind – all the wake dynamics can be accounted for in a time-accurate manner without having to make assumptions about the effects on the wake geometry or the time constants of the flow. Under transient conditions, the plane of the turbine may yaw momentarily into its own wake, raising the complexity of the problem of defining the induced velocity field even further.

Vortex Wake Considerations

Representative examples of time-accurate FVM wake calculations for a HAWT are shown in Fig. 13.22. In both cases, the wake geometry is shown for a two-bladed HAWT operating in a steady wind speed at different tip speed ratios. Figure 13.22 shows top views of the wake geometry for zero yaw angle, with predictions of the corresponding power output in Fig. 13.23. At high tip speed ratios (low wind speeds) the induced velocity at the turbine is a large fraction of the wind speed and so the turbine operates in the vortex ring state. This can be seen in Fig. 13.22(a) by the accumulation of wake vorticity near the turbine disk. Increasing the wind speed causes the wake to develop into the turbulent wake state [see Fig. 13.22(b)], which is characterized by a wake instability and turbulence downstream

Vortex Wake Considerations

Figure 13.23 Power coefficient as a function of tip speed ratio using vortex wake and BEMT theories.

of the disk. As the turbine approaches the point of maximum efficiency for XTSR « 6 notice that the wake expands downstream of the disk because at this condition the turbine is now extracting maximum energy from the wind and slowing (braking) the flow. As the vortices age, they diffuse under the action of viscosity and turbulence (see Section 10.6.2) and the wake slowly begins to contract again. Further decreases in XTSR cause the wake to convect more quickly downstream, and then the wake expansion decreases as the power output drops.

Figure 13.24 shows time-dependent FVM calculations where the turbine quickly yaws 30° out of wind from an initial operating condition near to its peak efficiency. Such a situation can also be produced when the wind direction changes. This serves to illustrate several points about wind turbine performance, in particular the nonuniformity of the inflow and the unsteady aerodynamic effects produced in the wake when the turbine continuously yaws into and out of the wind. Initially, just after the yaw starts, the turbine blades move into their own wake. This is a situation where there can be relatively powerful interactions between the blades and discrete vorticity in the wake, leading to highly unsteady local blade loads. However, these blade-wake interactions are not as severe as those found on helicopters. Notice that the adjustments to the developing wake take place relatively slowly, with the older part of the wake disintegrating and the new part near the rotor continuously evolving. Only after about ten turbine revolutions does the solution become essentially quasi-steady and periodic once more. In this condition, the component of wind velocity parallel to the plane of the disk begins to skew the wake, which induces a velocity field that causes the wake to roll up along its top and bottom edges. This is another source of the gradient in the induced velocity across the plane of the disk that inflow models (such as Eq. 13.63) attempt to mimic.

The corresponding time-history of the power output from the turbine under these condi­tions is shown in Fig. 13.25. Notice that the power output drops rapidly by an amount that is proportional to the cube of the change in yaw angle, as would be suggested by Eq. 13.2. In this case, an initial 35% reduction in power output is caused by the 30° yaw misalignment to the wind. Thereafter there is some recovery in average power output as the wake structure

Vortex Wake Considerations

(a) time = 0 (b) time = 2 revs


Vortex Wake Considerations

(c) time = 5 revs


(d) time = 10 revs


Vortex Wake Considerations

Figure 13.24 Free-vortex wake calculations of a three-bladed wind turbine yawing 30° out of wind. Top views of evolving wake: (a) Time = 0. (b) Time = 2 revs, (c) Time — 5 revs, (d) Time = 10 revs.


Vortex Wake Considerations

Figure 13.25 Power coefficient as a function of time for the condition when the turbine is suddenly yawed out of the wind.


Vortex Wake Considerations

reorganizes and the inflow velocities over the turbine disk reach an equilibrium. These non­steady inflow effects are often modeled in wind turbine analyses using variations of dynamic inflow theory, an approach discussed previously in Section 10.9. The idea is to consider the unsteady aerodynamic lag of the inflow development over the turbine disk in response to changes in blade pitch inputs or changes in turbine thrust. The equations describing the distribution of inflow are written in the form of a set of ordinary differential equations, with a time constant (or constants) representing the dynamic lag in the buildup of the inflow. Dynamic inflow models mathematical form and relative numerical efficiency, and their predictive success for various applications in the helicopter field are attractive because of their simplicity (see Section 10.9). However, the models have not been rigorously validated for rotors operating in the windmill state, although see Houston & Brown (2003) for some further insight into this issue. The main advantage of representing the aerodynamic model as sets of ordinary differential equations is that this form is appealing for many structural dynamic and aeroelastic analyses of the wind turbine, and this formulation of the inflow model allows the entire coupled problem to be solved simultaneously using the same numer­ical methods. For a discussion and background of dynamic inflow models applied to wind turbine problems see Bierbooms (1991), Snel & Schepers (1991), Hansen & Butterfield (1993), and Snel (2001). Bierbooms (1991) and Snel & Schepers (1991) suggest specific values of the wake time constants for wind turbine work, based on empirical evidence of the “induction lag.” Based on the results from the FVM the complexity of the physics of the unsteady wake problem cannot be underestimated. Yet, if the time constant(s) of a dynamic inflow model can be obtained, say using time-accurate vortex-wake calculations suitably validated with controlled experiments, then this approach provides a particularly useful form of parsimonious model that can relate changes in turbine loads to the lag in the inflow development.

Yawed Flow Operation

While ideally the wind turbine operates with the wind direction normal to its plane of rotation, this is never achieved in practice and there is always some yaw misalignment to the wind flow. This produces a velocity component parallel to the plane of rotation of the turbine disk, which leads to unsteady aerodynamic forces and also to a skewed wake. This is analogous to the helicopter rotor in forward flight, where the wake is blown back behind the disk. This skewed wake is responsible for an inflow gradient across the wind turbine disk. While the gradients are particularly strong parallel to the direction of the wake skew, a gradient is also produced in the other direction because of the asymmetry in aerodynamic loads between the effective advancing and retreating sides of the disk.

While the origins of this induced velocity distribution are complex (see Section 13.11 for a detailed discussion), a number of mathematical models for the effects have been derived, mostly following the linear inflow models developed for the helicopter case (see Section 3.5.2). Snel (2001) gives a review. The general approach is to correct the value of the induction ratio predicted from the BEMT by using an equation of the form

aY = a{ + Ks(y)r sin jr + Kc(y)r cos jf), (13.63)

where у is the yaw misalignment angle of the turbine disk with respect to the wind and aY is the corrected value of the induction ratio. The correction technique can be applied a posteriori using the values for a in the unyawed case, although the validity of this approach is perhaps questionable. Another variant of the technique is to correct the value of a based on the yaw angle and then to perform an iterative series of momentum balances at each of the elements until the values for a converge. This is a more pleasing approach.

The coefficients Ks and Kc used in practice have been based on a number of assumptions and approaches, including results for the helicopter case (Section 3.5.2) as well as from inflow measurements made on subscale wind turbines in wind tunnels. For instance, the skewed cylindrical wake model of Coleman et al. (1945) seems to have received some

attention for use in wind turbine analyses. In this case, the Ks is related to the wake skew

angle using

Ks = tan(x/2), (13.64)

where, as shown by Burton et al. (2001), x can be related approximately to the yaw angle and induction factor using

X = (0.6a + 1) y. (13.65)

In another example, inflow measurements were made using a 1.2-m-diameter subscale wind turbine placed in the open jet wind tunnel and used to determine the inflow coefficients, including higher harmonic variations – see Schepers (1999).

Подпись: 1The mathematical form of the inflow model given in Eq. 13.63 has been compared to measurements of blade airloads. It would seem if these inflow effects are properly accounted for then considerable improvements in the prediction of the loads under yawed operations are possible, at least for small yaw misalignment angles. Nevertheless, the approach of correcting the induction ratio in this manner is not rigorously valid because it violates the principles under which the BEMT equations were derived in the first place, including the

need for axisymmetric flow. Approaches based on vortex theory are probably the next best practical level of analysis of this problem.

Airfoils for Wind Turbines

The importance of airfoil sections for any type of rotating wing cannot be underes­timated, to the point that Chapter 7 has been devoted to an understanding of the subject. As has been done for helicopter rotors, entire families of airfoil sections have been developed for wind turbines. Many commercial wind turbines use standard NACA airfoil sections such as the NACA 4-digit series. In particular, the NASA LS-1 (low-speed) airfoil seems a common choice because of its known insensitivity to surface finish. Figure 13.18 shows some representative airfoils that have been used for wind turbines. The influence of airfoil

Airfoils for Wind Turbines

Airfoils for Wind Turbines

Figure 13.18 Examples of airfoil sections used for wind turbines: (a) NACA 4415. (b) LS(1)-0417. (c) NREL S809.

section on the power output of a wind turbine is relatively small, but the use of an efficient low drag airfoil will always help maximize energy production. A catalog of airfoil sections for wind turbine applications has been compiled by Miley (1982). See Tangier (1987,1995) for more recent wind turbine airfoil section developments.

There are, however, some special considerations in the design, selection, and modeling of the characteristics of airfoil sections for wind turbines compared to those for a helicopter. The most important is that wind turbines operate at much lower Reynolds numbers than helicopters, typically at or below 106 even at the blade tip. This means that Reynolds num­ber sensitivity is a more important issue and the accurate modeling of Reynolds number effects on the aerodynamic characteristics of the blade will be critical if the blade loads and power output from the wind turbine are to be predicted accurately. Again, these effects are readily implemented using look-up table techniques or equations fitted to the mea­sured airfoil characteristics (Section 7.11.3). As previously mentioned, also important is a knowledge of the sensitivity of the airfoils to surface roughness. The normal operation of wind turbines in the low atmosphere tends to cause the accumulation of dirt and dead insects and other foreign matter at the leading edge of the blades, and this can act to de­grade airfoil performance and reduce turbine power output – see Clark & Davis (1991) and Eggleston (1991). Representative effects of surface roughness on airfoil performance have been shown previously in Fig. 7.37 and appear as increased drag, decreased maximum lift and perhaps as a change in stall characteristic, usually leading to a very gradual or “soft” stall.

For variable blade pitch (feathering) turbines, airfoil sections are designed for high values of maximum lift, in much the same way as is done for helicopters – see Section 7.9. The pitch control system is used to adjust the AoA so that the best energy extraction efficiency is obtained over a wide range of wind speeds up to maximum power. For fixed pitch stall, controlled wind turbines, the airfoil sections must be designed to begin stall at lower values

Подпись: -"Q 1 Д .--І--Г І' І 'Ч j ■ . J ■ і ■ I ■ I -5 0 5 10 15 20 25 30 Figure 13.19 Lift and drag characteristics of the NREL S809 wind turbine airfoil under steady and unsteady conditions with dynamic stall (a = 10° 4- 10°tut, к — 0.077). Data sources: Ramsay et al. (1995) and Hand et al. (2001).

Angle of attack – deg.

of lift coefficient and then to maintain that value over a relatively wide range of angles of attack. This requires airfoil sections with appropriate pressure distributions that initially promote trailing edge separation and thus make this prestall behavior relatively insensitive to AoA. One example is the NREL S809 airfoil [see Tangier (1987)], the geometric shape of which is shown in Fig. 13.18(c) and the aerodynamic characteristics are given in Fig. 13.19. Notice that under steady (static) conditions the maximum lift is obtained at a moderate angle of attack, the lift then plateaus for about an 8 degree AoA range as trailing edge separation develops and stabilizes. This helps regulate the turbine’s power output. Over this range there is only a modest increase in the sectional drag. Gross breakdown of the flow and static stall occurs at about 15°. These characteristics are also carried forward somewhat into the dynamic regime, with lift overshoots but with relatively wide flat lift peaks compared to those found on helicopter airfoil sections under dynamic stall conditions (see, for example, Fig. 9.15). In this case, the lift overshoots are a result of the suppression of trailing edge

flow separation and not the formation of dynamic stall per se, and “classic” dynamic stall with leading edge vortex shedding is not obtained until a much higher AoA.

Effects of Stall

In addition to the reduction in power output caused by tip losses and viscous drag, the effects of stall on the performance of the turbine must be considered. Stall generally occurs at high wind speeds (low tip speed ratios) and/or with high blade pitch angles. Static airfoil characteristics, such as the nonlinear changes in the lift with the onset of flow separation, can be incorporated into the БЕМТ theory using a look-up table or other such strategy, such as described in Section 7.11.3 for the helicopter rotor. Approaches specifically for the wind turbine problem are also described by Tangier (2002) and Coton et al. (2002). Representative results of power output showing the effects of nonlinear aerodynamics, and the effects of stall in particular, are shown in Fig. 13.17. While stall is often used on wind turbines as a means of power regulation (these are called “stall-regulated” turbines), it generally has a deleterious effect on power output if it occurs at other operating conditions. Notice that after peak efficiency is attained the effect of stall tends to cause the power output to drop much more sharply with increasing wind speed than if no stall was present. This is consistent with experimental measurements of power output on wind turbines and illustrates the need for good stall models if the power output is to be predicted accurately over a wide range of wind speeds. The numerous difficulties of representing nonlinear aerodynamics and stall effects has historically limited the accuracy of predictions of blade loads and the power output from wind turbines.