The physics and mathematical description of the Achilles heel
of stationary wind turbine aerodynamics : the tip flow
Citation for published version (APA):
Kuik, van, G. A. M. (1986). The physics and mathematical description of the Achilles heel of stationary wind turbine aerodynamics : the tip flow. (TU Eindhoven. Vakgr. Transportfysica : rapport; Vol. R-812-D). Technische Hogeschool Eindhoven.
Document status and date: Published: 01/01/1986
Document Version:
Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)
Please check the document version of this publication:
• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.
• The final author version and the galley proof are versions of the publication after peer review.
• The final published version features the final layout of the paper including the volume, issue and page numbers.
Link to publication
General rights
Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain
• You may freely distribute the URL identifying the publication in the public portal.
If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:
www.tue.nl/taverne
Take down policy
If you believe that this document breaches copyright please contact us at:
openaccess@tue.nl
providing details and we will investigate your claim.
TECHNISCHE HOGESCHOOL EINDHOVEN
Afdeling der Technische Natuurkunde
Vakgroep TRANSPORTFYSICA .
-
.... =
..
-B I -BL.
TECHN I SCHE
UNIVERSITEIT
\llr)~'II"ltl
EINDHOVEN
Titel
The physics and mathematical descriptionof the Achilles heel of stationary wind turbine aerodynamics:
Auteur
Vers 1 agno. :
Datum
Werkeenheid
the tip flow G.A.M. van Kuik R-812-D
October 1986 Wind Energy
THE PHYSICS AND MATHEMATICAL DESCRIPTION OF THE ACHILLES HEEL OF STATIONARY WIND TURBINE AERODYNAMICS:
THE TIP FLOW
G.A.M.VAN KUIK
SUMMARY
By comparison of wing theory with rotor theory the significant influence of the blade tip load and tip vortex on the non-linear flow behaviour is shown. This non-linearity is commonly neglected in wingtheory, but is not to neglect in rotor theory. All rotors are loaded not only by axial and tangential forces but also by radial forces. In the mathematical limit of the actuator disc these forces appear as a discrete edge force. Extension of the axial momentum theory with these tipforces leads to a higher maximum Cp and CT'
1. INTRODUCTION
The interest in stationary windturbine aerodynamics has decreased during the last years. There seems to be an aquiescence in the shortcomings of the current methods, with the argument that improvements are out of scope.
Furthermore it is very well possible to calculate the performance of windturbines at their design conditions: there are enough empirical factors in the
calculations which enable an adjustment of the method to the desired result. This method fails however in off-design
conditions. Fig.l [1] shows the results of some current methods in the Netherlands applied to the 25 m.HAT experimental horizontal axis turbine at Petten. It is known that the reliabilit~ of the methods in the design-point (8p=O ) is rather satisfactory but is far from satisbactory in the off-design situation (Op=lO ) which is shown in the figure. This is the more surprising as in general it could be expected that the accuracy of the methods should increase in case of diminishing loads, as all velocity perturbations are much smaller then. Apparently the relevant parameters are not adequately modelled, and maybe even there is a misunderstanding of what relevant parameters are. The
necessity of improving the aerodynamic
G.A.M. van Kuik
Technical University Eindhoven Department of Physics
Building W&S P.O. Box 513
5600MB Eindhoven the Netherlands
models not only comes from the prediction of stationary turbine performance, but far more important, from the impossibility of predicting the dynamic behaviour if the parameters are unknown. A good
understanding of the static behaviour, difficult though it may be, is a
prerequisite to a satisfactory prediction of the dynamic behaviour.
This problem is not new of course: specially helicopter rotors suffer from the same difficulties. However is is to be expected that compared with
helicopterrotors in vertical flight the behaviour of windturbine rotors depends more critically on the variation of
important parameters as the free wind speed Uo and the thrust T.
Fig.2 shows schematically the average induced velocity Uj at the rotor disc as a
function of Uo and T (measured, not
theoretically), for all rotor flow states.
I I I - MEASUREMENTS 8 pilch = 10· --+---I ______ ~HO - - PHATAS FDO : -- -- HELIX - - - PREDICHAT O~r---~--~----+---~--~----+----+-;
I~
~
\\
0.10 1--+ft
-/-H--+---r----'''''40:r-=-''1-\k-, f----1 j f---I---1~~!
I
~
j
i
o
2 4 6 8m
U K- "
Fig. 1. Comparison of measured and calculated performance in off-design conditions.
.///,
OPTIMALLOAD U
O UNDISTURBED WINO SPEED
Ui PERTURBATION OF U AT ROTOR DISC
T T01Al THRUST EXERTED UPON FLOW
Fig. 2. Survey of all rotor flow states, expressed 1n the most important parameters U
o Ui T .
Once Uj is known, it is not difficult to calculate power, torque etcetera: the determination of Uj is the main difficulty of each aerodynamic model. Fig.2 shows that in case of heavily loaded
windturbines, Ui is most sensitive to small variations of T and Un: the slope is the steepest there. This implies that any small variation in the windspeed (e.g. a gust) and any small change of thrust
(e.g. by change of blade pitch), causes large changes of induced velocity.
Mathematically spoken: the problem is very non-linear. Helicopter rotors in hover show a more gentil behaviour, while this flow situation commonly is considered as one of the severest non-linear ones. A heavily loaded windturbine exceeds the rotor in hover as far as this is
concerned.
Nevertheless, improvements can be made in the aerodynamic model, by a careful examination of the more or less known non-linear effects of ordinary wing theory, and an appropriate tranformation of these to rotor theory. It appears that specially the tip flow is incorrectly modelled. This incorrect modelling is admissable in wing theory, but is important in rotor theory, as will be shown.
2. COMPARISON OF THE NON-LINEARITY OF WINGS AND ROTORS
The physical desciption of the action of a wing or rotorblade can be done in terms of forces acting on the wing/blade, or in terms of velocities. Forces and velocities are coupled of course, but a separate explanation may improve the clarity of the problem. The theoretical support of this section is not reflected extensively, but can be found in (2).
2.1. The vortex sheet of a wing and a rotorblade.
The kinematic disturbance left in the flow by any lifting device of finite dimensions is the vortex sheet. Fig.3a shows the well known vortex sheet of a single, rectangular wing, characterized by two tip vortices and a nearly flat sheet in between. The influence of these
tipvortices on the flow around the wing itself is very small, exept at the edges. Therefore, in linearized wing theory, one deals with a flat vortex sheet, without rolled up edges (Fig.3b). For aspect ratio's smaller than 1 this isn't allowed anymore: two wing tips are too large relatively, and the non-linearity becomes
;
-,
Fig. 3a The real vortex sheet of a wing
Fig. 4a The real vortex sheet of a rotor blade (propeller) .
I
~'---#
Fig. 4b The linearized vortex sheet.
significant. Fig.4a shows the vortex
sheet for a helicopter rotor in hover; due to the asymmetry the tipvortex is much stronger than the root vortex, and the vortex sheet itself is far from being flat. Furthermore it is transported to a position below the blade instead of behind the wing as in Fig.3a. Fig.4b shows a linearized version of Fig.4a: again the tip and root vortices are neglected, and the vortex sheet is assumed to be shaped as a rigid cork screw. There are different versions of the linearized vortex sheet, but all of them neglect the tipvortex. The vortex sheet of a windturbine b1ad cannot be sketched yet, because its shape is simply not known. What is known, is that at the rotor disc, the axial velocity is certainly not constant, and the radial velocity is certainly not zero. Specially at the blade tips the actual flow pattern
Fig. 5a The real load on a wing.
Fig. 5b The linearized load.
will be quite different from the linearized pattern. Therefore sketcht<'. with the complexity of Fig.4a may be expected too for a windturbine rotor.
2.2. Forces on a wing and a rotorb1ad~ The forces on a wing can be divided in the lift force (perpendicular to the undisturbed velocity Uo ) the drag force
(parallel to Uo ) and the lateral force in
span direction. The latter one is always neglected, but in [2] it has been shown that these lateral forces are inherent to the production of a vortex sheet. only in case of an ellipting wing load, these forces on the upper and lower side of the wing are equal and cancel each other. The forces on the wing tips are due to the low pressure in the core of the tipvortex. For a measured lift distribution like Fig.Sa, see [3J, page 163. In linearized wing theory, the force distribution of Fig.5b is assumed, according to the vortex sheet shape of Fig.3b. The lateral forces are neglected. The corresponding blade force distributions, again for helicopter rotors in vertical flight, are shown in Fig.6a,b. The lift distribution of Fig.6a has been measured often (e.g.[4]), the drag and edge forces are estimates, according to Fig.4a.
As with the isolated wing, the distribution of the radial forces is connected to the distribution of the vorticity sheet in the wake, and to the direction of the radial velocity at the rotor disc.
Fig. 6a The real load on a rotor blade (propelle.r).
~
i
Fig. 6b The linearized load.
2.3. The origin of the non-linearity of rotor flow
Comparison of Fig.5a with Fig.6a shows an increased "tip force" on the rotor blade. In wing theory, these tip forces are significant only for very small aspect ratio's: in rotor theory the effective aspect ratio is difficult to define: only the outermost 1/3 of the blade is really effective, so the apparant aspect ratio "felt" by the blade tip is smaller than the geometric aspect ratio. This effect, together with the complicated shape, and the interference with the other turns of
the helical vortex sheet, determines kinematically the non-linearity of rotor flow. In terms of forces there is a clear . parallel between wings and rotors:the
non-linearity of wing flow becomes
significant when the force parallel to the main flow, the induced drag, is of the same order of magnitude as the lift. The induced drag is the drag inherently connected to the generation of lift, and is responsible for the change of energy level of the flow.
If the rotor is seen as a whole,it is also a device generating a main force in the direction of
vo,
namely the thrust T, which takes care of the energy conversion.In the application of wings the aim is to minimize the induced drag; in the
application of rotors the thrust is optimized w.r. to the energy extraction, which results in a high thrust. This results in a severe non-linearity. 3. THE ACTUATOR DISC
The classical way of obtaining power limits for windturbine rotors, propellers and helicopter rotors, is to combine the concept of the actuator disc with the axial momentum theory. Lanchester ([5], 1915) was the first to determine in this way the limits for an optimal energy
extraction by a windturbine rotor, and for the induced power of a rotor in hover. The actuator disc is the limit of a rotor with N blades, and a total thrust T, for N + 00
with T=constant (the force on each blade tends to zero). The result is an axially symmetric disc which is porous. In its most elementary form the load on the disc is constant and all azimuthal forces are zero. It can be shown that this actuator disc is the above described limit for rotorblades with constant axial load, while the tip speed ratio A + 00 •
This actuator disc concept is assumed to define the maximum power extraction by a windturbine, the well know Lanchester-Betz maximum. Chapter 2 however has illuminated that radial forces and .tip forces are a necessary part of the load distribution on real rotor blade. Hence it might be
necessary that this concept should be completed. This is the subject of the next section. In section 3.2 the momentum
theory is applied to the new actuator disc concept, leading to a higher maximum C •
P 3.1. The edge forces on an actuator disc/strip
The actuator disc with constant, normal surface load is only a mathematical idea: physically it is not realisable. It has
its compeer in wing theory, which is in the same way only a mathematical idea: the wing with constant load, Fig.7a. In
general such a lifting line is used to calculate "far field" flow situations, as in the near field this mathematical
concept is insufficient. The edge forces and the radial forces are degenerated to a force distribution at the edge: the
beginning of the two "legs" of the horseshoe vortex are bound to the wing, and therefore carry these forces. In the limit of vanishing chords, this becomes a discrete force at the edge, which is directed perperdicular to the local velocity. In the actuator disc situation
Fig. 7a The lifting line with constant load as the mathematical limit of a wing.
Fig. 7b The actuator disc with constant load as the mathematical limit of a rotor.
(Fig.7b) this discrete edge force is also present. In the following one futher simplification will be made: the
two-dimensional analogon of the actuator disc is considered: the actuator strip.
The proof of existence and the
determination of the edge forces is done most easily in two dimensions; the step to three dimensions then is a small one. Now the problem has been reduced to a 2-D problem, it is possible to treat this edge problem completely independent of the proceeding arguments. These are necessary only in order to explain physically the existence of the discrete edge force in the mathemaical limit of the actuator strip/disc.
The procedure is as follows. The
equation for incompressible, inviscid flow is the Euler equation, which can be
written as:
av
VH ,. f + p
<y
x ~ - d; }in which H is the Bernoulli constant
( 1)
p + ~ P
y.y,
1
is the force density,Y is
the velocity vector, w is the vorticity vector V xY,
and p is the specific mass of the fluid. .Taking the curl of (l) gives
dU>
.!.
V x f - V x ('i x ":I)p -
-ar-
~- (2)The right-hand side determines the velocity field, which apparently is a function only of 1 x ! instead of the forcefield ! itself.
Line distributions of constant and normal forces, such as an actuator strip, have 1 x
!
= 0 everywhere, except at the edges. Therefore, any actuator disc with the same constant normal force, and having the same edges, creates the same velocity field! For the half-infinite actuator strip in the absence of an external flow, symmetry arguments combined with this statement, yield as solution for this flow field a discrete vortex growing linearly in time. For, at the actuator strip edge y= 0 by symmetry, but w ~ 0, while on all other positions y ~ O-but ~ = 0, so the second right-hand side term of (2) is
identically zero. During the start of an actuator strip flow with the undisturbed flow at rest, the velocity at the edges remains zero, as long as the mutual
interference of the edge vorticity can be neglected. The instationary discrete vortices which are expected to occur then, are indeed observed during the initial phases of a 2-D actuator strip flow ([6J,
flow visualization of experiments in a shallow water tank).
If the velocity at the edge is non-zero, the vorticity created at the edge is transported by the flow as a vortex sheet, and a stationary flow with
(y.V}w - 0 is the result.
In-(2] it is shown that the vortex sheets, emanating from the edges, are characterized by
(3)
in which Vs is the velocity of the sheet itself, y is the strength of the sheet and
A p is the pressure jump across the
actuator strip. If s denotes the distance from the edge, measured along the sheet, the leading edge of the sheet is a flow singularity:
lim y s->{J
lim V 0
s->{J s
and the shape of the sheet is characterized by
lim r: (0) = ~
e->«> S em
(4)
(5)
Here c and m are constants, and r,O are cylinder coordinates with the edge as origin. (5) denotes a spiral, with 0 -+ co
as core. The shape becomes more and more circular. In the core of the spiral, the vorticity y = 00 is bound to the edge as
Vs=O, and therefore carries a discrete
force Ee' This edge force is
perpendicular to -the ()cal at
edge. In (2] expressions are derived .for
the limit behaviour of the sheet velocity Vs and strength y. The most important conclusion is that the edge forces are
inherently connected to actuator strip. A formal proof that the same holds for actuator discs has not been attempted yet, but there is no doubt about this.
3.2. The extended actuator disc momentum
theory
Momentum theory including edge forces is not new: it has been developed for tipvane wind turbines already. In [7] this momentum theory has been applied to the new actuator disc concept. Here only the main results are mentioned.
The edge forces are perpendicular to the local velocity, and cannot perform work. They do of course change the
momentum balance, but do not occur in the energy balance. As in the augmented wind turbine systems, the effect of the edge forces is to increase the mean velocity at the disc Vd with a factor oVd' compared with the classic result. with all indices explained in Fig.s, and using y as the axial component of Y, the results are:
u dUd
!(I
+ ...:) +-U U
o 0
(6)
The dimensionless coefficients are defined as:
c
p T edge, aX1.8 • 1 2 pu • A o p (7)in which P denotes the extracted power and A the disc area. The edge forces generate
Uo
===l>
a (moderate) mass flow augmentation, leading to higher Cp values. Once the relation between CTe and CT t:.H is known, (6) can be solved numerically. Preliminary estimates indicate as the new maximum Cp~
0,65. This is also the estimation of the non-linear calculations of the "actuator cylinder" flow in [8]. Fig.9 shows the new performance diagram, compared with the classic one. It is clear that the total thrust coefficient might become ~ 1,2 which is in close agreement with
experimental evidence «(7). The disc load coefficient C T AH still has 1 as maximum value. Fig.9 indicates that a certain
c
p value is realized at a lower C T value then expected previously. This might be the explanation of the systematicoverestimation of the axial thrust by the present calculation methods (1].
- - - T
Fig. 9. The performance diagram according to the extended momentum theory.
4. CONCLUSIONS
By two independent methods, the rotor blade edge flow and edge load are shown to be inadequately modelled in the present theories. The "ideal" actuator disc carries a discrete force at the edge, having an axial and radial component. In all other situations this edge force appears as a distributed force along the blade, with a concentration near the tip. The radial forces increase the mass flow as in augumented systems, while the axial edge forces restore the momentum balance. The radial forces at the rotorblades are of course physically realized by viscous stresses in the boundary layer. Due to the strong flow asymmetry w.r. to the blade, these radial forces are much more
significant than in ordinary wing theory. This might be the explanation of the remarkable changes in aerofoil
characteristics due to radial effects, as treated in (9). still to be done is the numerical confirmation of the predicted higher value of Cp , and the translation of the actuator d1maxsolution back to the real rotor situation. Probably the current methods have to be extended by adding a vortex ring with accompanying force at the rotor disc edge. The strength of the ring is to follow from the actuator disc
solution.
BIBLIOGRA!?B:Y
[lJ G.J .W.van Bussel . , .. "1 25 m.HAT as testcas'9 for
t::'"
calculation methods of fastrunning windturbines, currently in use in the Netherlands", in Dutch - TUE-repolt R762D - Technical
Eindhoven, Dept. of Physics, ~:and
Energy Group - december 1985.
[2] G.A.M.van Kuik, "The relation between force fields and vorticity in
inviscid, incompressible flow, with emphasis on actuator disc problems" -TUE report, edited as in [1] - to appear at the end of 1986.
(3) D.Kuchemann, "The aerodynamics design of aircraft" - Pergamon Press - 1978. [4] R.H.Miller, "A simplified approach to
the free wake analysis a hovering rotor" - Vertica vol.6, pp.89 to 95, 1982.
[5] F.W.Lanchester itA contribution to the theory of propulsion and the screw propeller" - Transactions of the Institution of Naval Architects, vol. 57 - 1915.
[6] M.D.Greenberg, J.H.W.Lee "Line
momentum source in shallow, inviscid fluid" - Journal of Fluid Mechanics, vol.145 - 1984.
(7) G.A.M.van Kuik itA revision of the actuator disc concept and momentum theory" - TUE report R732D - edited as in (1] - August 1985.
(8] H.A.Madsen "On the real and ideal energy conversion in a straight bladed, vertical axis wind turbine" -Institute of Industrial Constructions and Energy Technology - Aalburg
University Centre Denmark - 1983. [9] D.J.Milborrow "Changes in aerofoil
characteristics due to radial flow on rotating blades" 7th BWEA workshop -Oxford - March 1985.
