Various parameters used in the axisymmetric boundary layer.
1. Introduction
In the design process of improved rotor blades the need for accurate aerodynamic predictions is very important. During the last years a large effort has gone into developing CFD tools for prediction of wind turbine flows (Duque et al., 2003; Fletcher et al., 2009; Sørensen et al.,2002). However, there are still some unclear aspects for engineers regarding the practical application of CFD, such as computational domain size, reference system for different computational blocks, mesh quality and mesh number, turbulence, etc. Thus, in the design process and in the power curve prediction of wind turbines, the aerodynamic forces are calculated with some form of the blade element method (BEM) and its extensions to the threedimensional wing aerodynamics. The results obtained by the standard methods are reasonably accurate in the proximity of the design point, but in stalled condition the BEM is known to underpredict the forces acting on the blades (Himmelskamp, 1947). The major disadvantage of these methods is that the airflow is reduced to axial and circumferential flow components (Glauert, 1963). Disregarding radial flow components present in the bottom of separated boundary layers of rotating wings leads to alteration of lift and drag characteristics of the individual blade sections with respect to the 2D airfoils (Bjorck, 1995).
Airfoil characteristics of lift (
During the development of stallregulated wind turbines, there were several attempts to predict 3D poststall airfoil characteristics (Corrigan & Schlichting, 1994; Du & Schling, 1998; Snel et al., 1993), but these methods predicted insufficient delayed stall in the root region and tended to extend the delayed stall region too far out on the blade.
The present work aims at giving a conceptualization of the complex 3D flow field on a rotor blade, where stall begins and how it progresses, driven by the needs to formulate a reasonably simple model that complements the 2D airfoil characteristics used to predict rotor performance.
Understanding wind turbine aerodynamics (Hansen & Butterfield, 1993) in all working states is one of the key factors in making improved predictions of their performance. The flow field associated with wind turbines is highly threedimensional and the transition to twodimensional outboard separated flow is yet not well understood. A continued effort is necessary to improve the delayed stall modeling and bring it to a point where the prediction becomes acceptable. In the sequel, based on previous computed and measured results, a comprehensible model is devised to explain in physical terms the different phenomena that play a role and to clarify what can be modeled quantitatively in a scientific way and what is possible in an engineering environment.
In 1945 Himmelskamp (Himmelskamp, 1947) first described through measurements the 3D and rotational effects on the boundary layer of a rotating propeller, finding lift coefficients much higher moving towards the rotation axis. Further experimental studies confirmed these early results, indicating in stalldelay and poststalled higher lift coefficient values the main effects of rotation on wings. Measurements on wind turbine blades were performed by Ronstend (Ronsten, 1992), showing the differences between rotating and nonrotating pressure coefficients and aerodynamic loads, and by Tangler and Kocurek (Tangler & Kocurek, 1993), who combined results from measurements with the classical BEM method to properly compute lift and drag coefficients and the rotor power in stalled conditions.
The theoretical foundations for the analysis of the rotational effects on rotating blades come at the late 40’s with Sears (Sears, 1950), who derived a set of equations for the potential flow field around a cylindrical blade of infinite span in pure rotation. He stated that the spanwise component of velocity is dependent only upon the potential flow and it is independent of the span (the socalled independence principle). Then, Fogarty and Sears (Fogarty & Sears, 1950) extended the former study to the potential flow around a rotating and advancing blade. They confirmed that, for a cylindrical blade advancing like a propeller, the tangential and axial velocity components are the same as in the 2D motion at the local relative speed and incidence. A more comprehensive work was made once more by Fogarty (Fogarty, 1951), consisting of numerical computations on the laminar boundary layer of a rotating plate and blade with thickness. Here he showed that the separation line is unaffected by rotation and that the spanwise velocities in the boundary layer appeared small compared to the chordwise, and no large effects of rotation were observed in contrast to (Himmelskamp, 1947). A theoretical analysis done by Banks and Gadd (Banks &Gadd, 1963), focused on demonstrating how rotation delays laminar separation. They found that the separation point is postponed due to rotation, and for extreme inboard stations the boundary layer is completely stabilized against separation.
In the NASA report done by McCroskey and Dwyer (McCroskey & Dwyer, 1969), the socalled secondary effects in the laminar incompressible boundary layer of propeller and helicopter rotor blades are widely studied, by means of a combined, numerical and analytical approach. They showed that approaching the rotational axis, the Coriolis force in the cross flow direction becomes more important. On the other hand the centrifugal pumping effect is much weaker than was generally thought before, but its contribution increases particularly in the region of the separated flow. The last two decades have known the rising of computational fluid dynamics and the study of the boundary layer on rotating blade has often been carried on through a numerical approach. Sørensen (Sørensen, 1986) numerically solved the 3D equations of the boundary layer on a rotating surface, using a viscousinviscid interaction model. In his results the position of the separation line still appears the same as for 2D predictions, but where separations are more pronounced a larger difference between the lift coefficient calculated for the 2D and 3D case is noticed. A quasi 3D approach, based on the viscousinviscid interaction method, was introduced by Snel et al (Snel et al., 1993) and results were compared with measurements. They proposed a semiempirical law for the correction of the 2D lift curve, identifying the local chord to radii (
Recently, with the advent of the supercomputer, computational fluid dynamics (CFD) tools have been employed to investigate the stalldelay for wind turbines (Duque et al., 2003; Fletcher et al., 2009; Sørensen et al., 2002). These calculations in addition to Narramore and Vermeland’s results (Narramore & Vermeland, 1992) showed that the 3D rotational effect was particularly pronounced for the inboard sections, but the genesis of this phenomenon is a problem still open. The present concern aims at giving explanations in physics terms of the different features widelyobserved experimentally and computationally in wind turbine flow.
2. Flow at low wind speeds
At low wind speed conditions, i.e. at high speed ratios (TSR>3), the visualization of the computed flow indicates that the flow is wellbehaved and attached over much of the rotor,
Fig. 1
.
Figure 1
shows the separated area and radial flow on the suction side of a commercial blade with 40 m length at the design tip speed ratio (TSR=5); the secondary flow is strongest at approximately 0.17
Therefore, at low wind speed, the main rotational effect is due to the Coriolis force which delays the occurrence of separation to a point further downstream towards the trailing edge, and by this the suction pressures move towards higher levels as
The pressure field created by the presence of the turbine is related to the incoming flow field around the blade, taken as being composed of the free wind velocity and the socalled induction velocity due to the rotor and its wake. Thus, the incoming field results from a weak interaction between two different flows: one axial and the other rotational
the radial independence principle is applied to flow effects, i.e. induction velocities used at a certain radial station depend only on the local aerodynamic forces at that same station;
the mathematical description of the air flow over the blades is based on the 2D flow potential independent of the span, and on corrections for viscosity and 3D rotational effects.
These assumptions suitable to BEM methods reduce the complexity of the problem by an order of magnitude yielding reliable results for the local forces and the overall torque in the proximity of the design point, at high tip speed ratios. In order to estimate the 3D rotational effects, usually neglected in the traditional BEM model, the flow around a hypothetic blade with prestall/stall incidence and chord constant along the whole span is considered in the sequel.
2.1. Representation of flow elements
The set of equations including a simplified form of the inviscid flow and the full threedimensional boundarylayer equations are used to identify the influence of the3D rotational effects at low wind speeds.
where
where
where the nondimensional velocity
where
Because a wind turbine can operate a long time at low tip speed ratios, the inboard regions of the blade are stalled, and there leadingedge separation bubbles can occur (Dumitrescu & Cardoş, 2010). Behind the mild separation it is assumed that the flow relaxes with the vanishing skin friction. Therefore, the following flow is applied to solve
where
Figures 5 and 6 illustrate such chordwise inviscid velocity distributions and the corresponding variations of the peripheral skinfriction coefficient
Fig 6. Variation of a) the peripheral skinfriction coefficient; b) the boundarylayer shape parameter.
Continuity
Momentum, θ component:
Momentum,
where
with
The equations (7)(9) are integrated with respect to z (normal to blade) from 0 to δ (boundary layer thickness) and the integral forms of equations are obtained as
where (
An order of magnitude analysis shows that
where
The Mager cross flow profile is assumed (Mager, 1951)
where
Equations (11) and (12) can now be written in terms of the parameters
The skinfriction relation for flows with pressure gradients and rotation effects is based on the experimental data for a turbulent boundary layer in a rotating channel (Lakshminarayana & Govindan, 1981),
This correlation is a modified version of the correlation developed by Ludwieg and Tillmann (Ludwieg & Tillman, 1949), which includes the effect of rotation. In this relation
The pressure gradients cause large changes in velocity profiles and, consequently, in the shape parameter
The entrainment equation is chosen to model the rotor boundary layer, which in the coordinate system used here can be written as
where the entrainment coefficient
The entrainment function
Also, the similarity solutions have shown that
Then, it is assumed that the variation of the entrainment rate with
Equation (19) is written in a form similar to Eqs. (15) and (16)
Equations (15), (16) and (22) are to be solved for
2.2. Application
A hypothetic blade, with constant incidence and chord along the span is chosen to analyze the rotational effects on the boundary layer of a wind turbine blade. In this example, an external flow with the distribution of velocity on the surface of blade given by Eqs. (5) and (6) is imposed.
The curves plotted in
Fig. 7
are calculated on the prestall (
Secondly, in the stall incidence condition, leading edge separation bubbles are formed on the upper surface of the blade, at the root area, which delays the occurrence of massive separation. Beginning at the root of the blade, the bubble continues to stretch towards the trailing edge and at approximately
3. Flow at high wind speeds
At high wind speeds conditions (
This behavior, which is characterized by significantly increased lift coefficients as compared to the corresponding 2D case, and by a delay of the occurrence of flow separation to higher angles of attack, is caused by the rotational augmentation at the root area of the blade. The knowledge extracted from experimental and computational visualizations of these rotational effects on the flow field along the blade can be used to develop a model for their prediction or the socalled stalldelay phenomenon.
The wind turbine blade sections, often operate with stall at low tip speed ratios and can undergo a delayed stall phenomenon on the inner part of span. As the wind velocity increases, the inboard regions of the blades are stalled, having
a form that suggest an outer rotational flow with angular velocity higher than that of the constantspeed rotor,
In contrast with the Glauert weak wake (Glauert, 1963), the strong potential vortexlike wake is responsible for all the 3D rotational effects, affecting the boundary layer on the turbine blades once the phenomenon of stall is initiated. The enhanced rotational outer flow and the bottom boundary layer are the cause of the highlift effect of the airfoils at high angle of attack, i.e. the stall delay.
3.1. Boundary layer beneath a Rankinelike vortex
The axisymmetric angular velocity (Ekman) boundary layer developing on the suction side of the rotor disk is used to illustrate the contribution of the suction effect of a strong wake on the chordwise surface pressure distribution and volume flux in the radial direction. Thus, consider an axisymmetric rotating flow of a viscous incompressible fluid over a disk of radius
The angular velocity of the disk Ω_{
b
} is constant and that of the fluid far away from the disk Ω_{
v
} may be an arbitrary function of the radius
where
This vortex model avoids the discontinuity of the velocity derivatives at the point of transition from free to forced modes and gives the possibility of the description of the flow near the hub.
For the inertial system of cylindrical coordinates
The boundary conditions at the surface of the disk and the edge of boundary layer are
The outer flow radial momentum equation becomes then
where
The tangential and radial momentumintegral equations may be derived from equations (26) and (27) for radial and angular velocity profiles which satisfy the boundary conditions on the disk surface and the smoothness requirements at the edge of the boundary layer
Outside the boundary layer, the tangential velocity
where λ_{1}, λ_{2}, λ_{3} and λ_{4} are profile form parameters defined in Table 1 (Rott & Lewellen, 1966)
Flow 








Laminar  6.75 

2.5  1.372  0.467  0.667  2  12 
Turbulent  1.69 

4.93  1.63  0.222  0.250  0.0225  0.0513 
The Blasius shear laws for threedimensional flow (Schlichting, 1979) cover both the laminar and turbulent case
where
This is the final result of the analytical formulation and the equations (32) and (33) must be integrated numerically for a particular circulation function Γ_{ v } (Eq. (24)) with an appropriate choice of the boundary layer form parameters.
The onset of vortical flows like tornadoes can be visible on the constantrotational speed rotor at low speed ratios (λ < 3.0). At the root area of the blade these flows generally behave like a vacuum pump which is featured by volume flow rate (
Figures 12 and 13 show such characteristics induced by the wake modeled as a Rankinelike vortex with its axis normal to the rotor disk for different tip speed ratios. The suction characteristics show that the effects of the disk defined by
On the other hand, large radial volume fluxes close to the hub along with the previous results (Fig. 7a) indicate a 3D attached boundary layer with small leadingedge separation bubbles at high angles of attack.
Therefore, at low tip speed ratios and close to the hub, there is essentially a 3D attached flow field and a theoretical description in this region can be performed by viscousinviscid interactive flow models.
3.2. Bladewake interaction
Flow at low tip speed ratios is dominated by a vortexlike wake which triggers the stall delay phenomenon on the inboard halfrotor disk. The dual working mode of the rotor blades as a stationary disk in a fast rotating flow near the hubsucking mode, and as the rotating disk in the outboard separated flowpumping mode, are clearly visible from their pressure distributions when the wind turbine operates at
On the upper side of the inner blade sections there are two suction pressure fields: one is azimuthally uniform generated by circulation augmentation of wake and the other is the aerodynamic non uniform pressure field with a suction spike at the leadingedge (Fig. 4). These two pressure fields interact and give rise to the radial effects which contribute to stall delay, as well as to a higher lift coefficient at the root area of the blade. There are two main effects: superimposed pressure fields of wake (
Fig. 12
) and of blade (Fig. 4), which decrease the suction spikes at the leading edge until they cancel, and a radial flow effect induced by the volume flow merger in the boundary layers of wake and blade, resulting in velocity profile skewing and boundarylayer energizing (no separation). Assuming the conservation of circulation once the phenomenon of stall is initiated (here at
As we will show in the next section, the phenomenon of stalldelay can be described as a three step process: generation of the strong wake by the clustering of shed vortices from the inboard blade segments at
3.3. Application
The sucking suction effect is affected by two key nondimensional parameters
distribution for the previous hypothetic blade, with constant poststall incidence and chord along the span
4. Model for delayed stall regime
Stall occurs at an airfoil when flow separates at high angles of attack, typically>15^{0}, beyond this angle the 2D lift coefficient drops significantly. It has long been known (Himmelskamp, 1947; Ronsten, 1992; Snel et al., 1993), however, that the observed power curve under conditions where most parts of the blade are stalled is consistent with significantly higher lift coefficients. The lack of a conceptual model for the complex 3D flow field on the rotor blade, where stall begins, and how it progresses, has hindered the finding of an unanimously accepted solution. This section aims at giving a better understanding of the delayed stall events.
To interest an engineer in stalldelay one might cite the fact that the inboard regions of the blades are stalled for much of their operational time. Thus predicting blade loads and the power output during stall is very important in making good predictions of wind turbine performance. With a physicist one could reason that rotational stall admits interesting transient states from 3D inboard delayed stall in partially separated flow to 2D outboard massively separated flow. Here, the detailed physical understanding of the viscous flow surrounding the hub area is dominated by the interaction between the vorticity and pressure fields and their dynamics. However, all the existent models have been developed more or less intuitively without a basic fluid dynamics support. Six different models mostly used to correct the airfoil characteristics for stall delay, including a wide range of different assumptions, were recently analyzed (Breton et al., 2008). The conclusion was that none of the six models studied correctly represented the flow physics, and that this was ultimately responsible for their lack of generality.
4.1. New physicsbased model
The key to the stalldelay phenomenon is the rotational flow surrounding the root area of the blade, i.e. the wake rotation. At low tip speed ratios (or high wind speeds), just as a vortex is shed from each tip blade, a vortex is also shed beginning from the section
There are two common characteristics which most strong potential vortices exhibit, like those produced at the blade root; the first is that the vortex velocity field above the surface boundary layer is always dominated by the tangential velocity component, while the second shows that the balance of forces between the flow outside the boundary layer and vortex core is governed by a balance between the centrifugal force and radial pressure gradient
Then, as the blade section moves through the air a circulation
a starting vortex
The wake’s circulation, which is constant up to the hub induces a suction pressure along the upper side of blade (Fig. 16) stabilizing the boundary layer against the separation at poststall angles of attack (angles of 30 degrees exist currently at the blade root), followed by forming of a leadingedge separation bubble. The bubble at the innermost sectional airfoil has almost no effect on integrated loads, because it is never more than a few percent of the chord in length, and the airfoil acts as in an ideal fluid flow.
The phenomenon of stalldelay can be described as a three step process: rise of strong wake by clustering of vortices shed from the inboard blade span at
The specific mechanism for circulation (or lift) decay is not presently known. However, a measure of the degree of radial instability of the boundary layer and implicitly the lift decay can be obtained from the sucked volume flow rate (Fig. 13) into the boundary layer beneath the Rankine vortexlike wake. Thus, the maximum sucked flow rate would correspond to the minimum volume of the separation bubble and
boundary layer, namely larger values of the flow rate signify smaller bubble volumes and thinner boundary layers for reducing tip speed ratios. Further, it can be easily found that a thicker boundary layer with distributed vorticity (
Point  Flow Structure  Forces 
1  The attached 3D boundary layer with leadingedge separation bubble  Exceed 2D CLmax, extrapolate linear regime 
2  The separation bubble is sucked and its volume reduces  Maximum lift, CL,INV at 
3  Thick skewed 3D boundary layer with stretching bubble  Gradual decay of lift for 
4  The bubble breaks away from the trailing edge and forms a free shear layer  Readjust to stall regime 
The various stages of the delayed stall process are summarized schematically in Fig. 17. and Table 2. Stage 1 represents the delay in the onset of separation in response to reduction in adverse pressure gradients produced by the influence of the strong vortex wake at the root. The phenomenon is initiated at
Stage 2 of the delayed stall process involves the suction pressure redistribution of the start pressure distribution with the conservation condition of its integrated load and the simultaneous move of pressure center to the midchord; as the suction/tip speed ratio increases/decreases the volume of the leadingedge separation bubble is reduced. The start pressure distribution is known from the inviscid aerodynamic loading solution over the chord; the pressure redistribution continues up to no suction peak exists at the leadingedge and even more no suction surface pressure gradient exists. On the other hand, the pressure redistribution is associated with a vorticity concentration process, which influences the stability of boundary layer.
Stage 3 is the sectional circulation (lift) decay which occurs at greater radii than the start location, where the separation bubble surface stretches all the way to the trailing edge. Beyond that zero flow rate location the flow is separated over the whole airfoil so that full stall occurs. By reason of the boundary layer stability it is inferred that the stalldelay effects are present up to a 50% of span for all the regimes.
Stage 4 is the relaxation to stall regime, which begins just as the integrated normal load of 2D stall, C_{NS}, is reached (end of delayed stall) and continues as far as the recovery of the specific pressure distribution (suction peak at the leading edge) is achieved.
4.2. Application
The combined experimental rotor (Schepers et al. 1997) of the Renewable Energy Laboratory (NREL) is chosen to illustrate the stalldelay model proposed for a wind turbine blade. The rotor uses the NREL S 809 airfoil and a simpler BEM method. In this example, the one parameter pressure distribution on the upper surface of blade at the root with
The comparison of predicted and available measured
5. Chapter review
The phenomenon of stalldelay has been shown to be an important consideration in wind turbine design because, in the presence of turbulence, it ultimately generates the loads with the highest peaktopeak fatigue cycles for both blade and rotor shaft bending (Tangler, 2004). It has been shown that stalldelay is characterized by a favorable delay in onset of flow separation to higher angles of attack for the inboard regions of the blades. This is followed by the less favorable event of leadingedge separation bubble stretching. As long as the bubble surface stays over airfoil, it acts to decay the lift gradually. Then, the leadingedge separation occurs. The stalldelay begins at the hub and crosses varied partially separated flow regions over the inboard halfrotor disk, depending on the wind speed. Therefore, the consideration of stalldelay phenomenon represents a necessary refinement in the rotor design process which will more accurately define the performance and power control at high wind speeds.
While the prediction of the conditions for stalldelay onset and their subsequent effects clearly forms an essential part of any rotor design process, it has been shown that this is a problem not yet fully understood, nor easily predicted. For engineering analyses, the modeling of stalldelay still remains a particularly challenging problem. This is mainly because of the need to balance physical accuracy with computational efficiency and/or the need to formulate a model of stalldelay in a particular mathematical form. To this end, a number of semiempirical models have been developed for use in wind turbine design work. A discussion of these semiempirical methods has been recently presented, along with the conclusion that none of the six most known models correctly represented the physics of flow (Breton et al., 2008). Generally, predictions are good when measurements are available for validation or empirical refinement of the model, but their capabilities for general blade parameters (twist, taper and airfoil shape) are less certain.
A conceptualization of the complex 3D flow field on a rotor blade, where stall begins, and how it progress, has been proposed, along with a reasonably simple model that complements the 2D airfoil characteristics used to predict rotor performance. This concept considers that the 3D flow field near the hub behaves like the rotational flow over a stationary disk, inducing a strong suction pressure towards the center. Here, the centrifugal force has a strong stabilizing effect on the boundary layer, in which the rotor is acting as a vacuum pump on the separated volume of bubble. The rough model has shown encouraging results, but some uncertainties remain in the prediction of the sectional circulation decay and also in the proper validation of predictions with more measured and CFD computed airloads on the rotor. The more correct physical description provided by this present model should give more coherence and consistence of future experimental and computational studies on the stalldelay phenomenon.
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