# Review of the Present Status of Rotor Aerodynamics

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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 non-dimensional 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.

5 50 H. Snel Another basic problem of the momentum method (both global and annular) is that is formulated for time-independent 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 large-scale 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 so-called 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

7 52 H. Snel induced by the existing vorticity elements. These methods are typically unsteady in nature; in the initial condition the ow is vortex-free 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 self-induced 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 cut-o length (no induction for separations smaller than the cut-o length) or a viscous vortex core with regular velocity. However, the solution then becomes a function of the choice of this cut-o 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 cut-o 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 so-called 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 so-called 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

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 desert-like environment of Californian wind farms. Work on dirt-insensitive 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 (FFA-W family) and the work of Timmer and van Rooij at Delft University of Technology 53 (DU-W family). More recently, Fuglsang and Dahl 54 at Risù used optimization techniques for special-purpose 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 co-ordinates 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 divergence-free. This is expressed by equation (9). The vector equation (10) is Newton's law of conservation of momentum in the three co-ordinate directions. This is referred to as the momentum equation. The two terms on the left-hand 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, non-linear term describes the acceleration due to convection of the particle with the ow eld (convective acceleration). The right-hand 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 `no-transparency' condition (normal velocity zero) and a `no-slip' 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 non-linear 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 non-linear. 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 no-slip 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 non-conservative), 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 order-of-magnitude 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 non-dimensional 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 so-called 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 `no-transparency' 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 vorticity-free region (e.g. a region of uniform velocity), then the entire ow will be vorticity-free (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 non-transparency 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, vorticity-free 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 vorticity-free. 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 no-slip 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 high-reynolds-number 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 vorticity-free. 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. Co-ordinate 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 vorticity-free. In fact, the inviscid ow solution can be seen as the limit of the Navier±Stokes solution for Re approaching

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