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Journal of Wind Engineering and Industrial Aerodynamics, 39 (1992) 23-3

Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands 23

A comparison between unsteady aerodynamic models

W.A.A.M. Bierbooms

Institute for Wind Energy, Delft University of Technology, Stevinweg 1, 2628 CN Delft, The Netherlands

Summary

This paper describes two methods to deal with unsteady aerodynamics. The results of the meth- ods are compared with dynamic wind tunnel measurements on a NACA 0012 aerofoil. Fur1 ~: r- more simulations of the dynamic behaviour of a flexible rotor are presented using both methods.

1. Introduction In an attempt to reduce the cost of wind energy the new generation of wind turbines tends to become more flexible. Due to the increased blade movements (in flap, lead-lag or pitch direction) the blades will experience fast fluctuations in the wind direction and velocity. These fluctuations may be so fast that a quasi-steady treatment of the aerodynamics is no longer justified. The lift coef- ficient can amply exceed the maximum quasi-steady value and hysteresis will occur. This will have consequences for the (flexible) rotor stability and the fatigue loads. One speaks of dynamic stall if the hysteresis loops occur around the stall angle. For a 2-D non-rotating wing and attached flow a theoretical derivation of the unsteady lift force is given by Theodorsen. For a 3-D rotating blade an analytical solution is not available so one has to fall back upon semi-empirical methods. In (helicopter) literature many semi-empirical methods are pre- sented. This paper starts with a short presentation of the classical theory ac- cording to Theodorsen. Next two semi-empirical methods, namely the ONERA and the Gormont method, will be described. In the ONERA method the aero- foil coefficients are described by a set of differential equations. In the method of Gormont the steady lift coefficients are used, but for another angle of attack; the momentary angle of attack is corrected by a term depending on its first derivative to account for stall delay. The two models are compared on the basis of a test case consisting of a rotor with flexibility in flap direction. The dynamic

response of this flexible rotor will be determined in case of stall, applying both methods. Finally the conclusions and references are given.

2. Unsteady profile aerodynamics according to Theedorsen

In this paper the unsteady aerodynamic phenomena with length scale in the order of a chord will be dealt with (unsteadyprofile aerodynamics). These must be distinguished from the unsteady phenomena with length scale in the order of a rotor diameter (dynamic inflow). If the fluctuations experienced by a rotor blade have high frequencies, the flow can no longer be considered as a series of succeeding steady flow patterns (quasi-steady treatment).

Theodorsen has derived (using Biot-Savart, the Kutta-Joukowski condi- tion and the unsteady Bernoulli flow equation) an analytical expression for the lift force of a non-rotating oscillating aerofoil (2D) as function of the re- duced frequency (for the derivation see e.g. \[1\] ). He made the following as- sumptions: the oscillations are harmonic, the fluid is incompressible and in- viscous (this means that the relation between the lift coefficient and the angle of attack is linear). In case of a rotating blade the situation becomes more complex and a method based on a vortex representation will lead to excessive amounts of computation time. For design purposes we therefore have to fall back upon semi-empirical methods. 3. The ONERA method

One Gf the semi-empirical methods has been developed and improved by the

French institute ONERA (Office National d'Etudes et de Recherche Aeros- patiales) since 1979. In an early version of the method the lift coefficient is given by a set of differential equations \[2-4\]. Recently the method has been extended with a description of the unsteady aerodynamic moment as well as the drag \[5 \]. Furthermore variations in the wind velocity are included. In this new version, the circulation has become the variable in the differential equa- tions instead of the lift coefficient.

One of the starting points of the ONERA method is the fact that a linear system can be represented by a transfer function. Transformation to time do- main results in a set of linear differential equations with constant coefficients. In the case of an oscillating aerofoil there is a linear relation between the aero- dynamic forces and the pitch (rotation of the blade element) and heave (trans- lation) amplitudes. The corresponding transfer function can be approximated by a simple fraction and in time domain we get a first-order differential equa- tion. This differential equation can be used for any arbitrary motion of the aerofoil except those with very high frequencies because the approximation of the transfer function is not valid then. In practice this is no drawback since motions with (very) high frequencies are not possible for a real blade. In case of stall the transfer function becomes more complex and we obtain in time domain a second-order differential equation. The non-linearity of the lift char- acteristic is incorporated in the model through an input variable ACt (the loss in lift due to stall, Fig. 1):

ACt-- Ct,~, - Ct (1)

The coefficients in the differential equations depend on ACt and on the par- ticular aerofoil. They have to be determined for each aerofoil from wind tunnel measurements on an oscillating aerofoil. From these kind of system identifi- cations ONERA has gathered ranges for the coefficients. It turns out that dy- namic stall is more a characteristic property of the flow itself than of the aero- foil, and that the most important aerofoil dependent coefficients concerns the stationary lift characteristics. This means that the ONERA method may be used even if only the steady profile characteristics are known; for the other coefficients the values for a mean aerofoil can be take~, (Table 1). Although the model is developed for a 2-D situation we will use the method for a rotating blade (3-D). In the next section the model equations for the lift are given; the equations for the moment and drag can be found in \[5\]. ." ACt ffs Fig. I. The loss in lift due to stall.

TABLE 1

The ONERA coefficients

NACA 0012 Mean aerofoil

JVd=10 ZlVd=8 a, = -- 2.29 a, = - 2.29 ro=0.2, r2=0.1 ro=0.2, r~-0.2 ao-0.25, a2=0.1 ao =0.3, a2=0.2 E2= -4.01 E2= -2.86

3.1. The differential equations for lift In the theory of Theodorsen the lift is given in terms of derivatives of the pitch and heave motions. In the ONERA method the following variables are used (Fig. 2): 1/'1, the flow velocity parallel to the aerofoil Wo, the aerofoil velocity perpendicular to the aerofoil (positive downwards) - W~ = Ob, the rotational velocity of the aerofoil (positive clockwise; with O the pitch angle) The original equations of ONERA for the lift are somewhat modified cor- responding to comments by Peters \[6\]; this concerns a generalization of the equations to larger angles of attack. This leads to

(c c ) dLx=-½Pc dr SL ~ Wosin O+kL-~ ~VI sinO+ Vp(FI +/'2) ,

( c~¢oc°s°+ c ) dLy= ½pcdr SL-2 kL ~ ~Vl cos O+ Vt(Fl +r2) , (2) with p the air density, Vt the tangential velocity, c the chord, Vp the parallel velocity and dr the incremental radius. In the above equations SL and kL are coefficients depending on the particular aerofoil. In practice the values for a flat plate can be used (for small Mach number: SL--~ and kL=n/2). The vari- able F is the circulation divided by b (half chord). The total circulation is the sum of/'1 and F2. FI is the linear circulat;,on variable and concerns attached flow. Ft is described by a first-order differential equation:

+ (oL~.C~ +dL) We, +aLaL W~, (3) with ~:=b/V~ the time constant, t~o the zero lift angle, C~ the lift gradient and

~.L, aL, C~L, dL aerofoil dependent coefficients. The values for a flat plate may be used: ~L=0.17, aL=2~, C~L=0.53; for dL see (5).

I C1~' Otln~

 vl~ i- ~ _

Ii Vl e

- Rotorplane

Fig. 2. The definition of the velocities.

The non-linear circulation variable r2 concerns stall and is described by a second-order differential equation:

1 1 f2+~L~F2+rL~f'2=--rL-~VIACz--EL~Wo (4) and aL, rL EL aerofoil dependent coefficients. These coefficients depend on the loss in lift ACl:

aL-'a° +a2( ACl)2' ~/-r-L-'r° +r2( ACz)2' (5) EL-'E2(ACI) 2, dL-~allACtl.

The coefficients in (5) have to be determined for each aerofoil from wind tunnel measurements (flat plate values cannot be used). If the values for a particular aerofoil are not available the values for a mean aerofoil may be taken instead (see Table 1 ).

Wind tunnel measurements show that there is no need to include more de- rivatives of Wo or W1 in Eqns (2)- (4). The unsteady lift is given by Eqns (2)- (4), which should be incorporated in a rotor simulation model. This is done in case of a rotor with flap flexibility. The simulations are given in Section 5. First we will state some remarks on the method.

The actual appearance of stall takes place at a larger angle of attack than according to quasi-steady theory. Expressed in dimensionless time (7= tV~/b) this stall-delay appears to be 8 constant A~d \[ 7 \]; for most aerofoils A~d is about

8. The coefficients in the differential equations (3) and (4) have a physical meaning. The coefficient ~t L determines a time constant:

½c (6) = = for c=0.3 m and V1-80 m/s: ~1 ~0.01 s.

The coefficients rL and aL from (4) are related to an eigenfrequency and damping:

a)n_ ~L , (7) T

~L/~ aL 5-- 20)n -- 2~.~/rL (8)

Depending on the angle of attack (or AC~) and reduced frequency k the fre- quency of a (forced) oscillation can be close to this eigenfrequency. In case of forced harmonic pitch and heave motion the velocities are (Fig. 3):

V1 = V cos O-/l b sin O, Wo = -/~ b cos O+O b(2+ ½ ) - V sin O, W~ =-Ob.

For the first-order differential equation we get

2L/~'bi (_9 aL (Cla Wo jl. O.L W1 ) =Ck(Clc~ Wo _~aL Wl).

/"1= 2L/T+ito (9) (10)

In (10) the fraction is replaced by the Theodorsen function Ck (Fig. 4). Under h.b

- 0 I L-- A I _l

Fig. 3. An oscillating aerofoil.

4.0~ i -0.1 -0.15

-0.

| -0.21 0 0,1 O~ a~mfnr f~tlom !O~ k-lO i

 • ,,,,- / ,-o.,/

// ~ ,e

03 OA 0.~ 0,6 0.7 Oil 0,9 mdt~ Fig. 4. The transfer functions according to the Theodorsen function and the ONERA method.

the assumptions of small angles, no stall (thus/'2--0 and dL= 0), incompres- sible flow and C~ = 2~, C~o = 0 the ONERA equations transform into the clas- sical expression according to Theodorsen:

dL=p b dr{ xb\[-Ft" b+ ~b(2+ ½ )- VO\]

-(~/2)bOb+ VCh 2~ \[-/~ b+O b(2+ ½ )- VO\]- V Ck 2xOb}. (1)

4. The Gormont method

The modelling of the unsteady lift according to the Gormont method

(Boeing-Vertol Gamma Function method) is relatively simple. This method is widely used for VAWT performance prediction. The lift coefficient is given as function of the angle of attack and its first derivative through h~terpolation of the steady lift characteristic (Fig. 5):

Ct(o~,&)= o~-O~o Cz(o~-a). (12) ~-C~o - A

For the shift in angle of attack A the following relation is valid:

A=~(cJ&\[ ) '/2 2 W sign(&). (13)

The parameter 7 depends on the aerofoil thickness (and Mach number); for most aerofoils 7 lies between 0.5 and 1.5. For a NACA 0012 aerofoU: 7=0.62. In case of harmonic motion Eqn. (13) e~_uals A=Tv/k. This means that stall- delay is assumed to be proportional to x/k in contradiction to \[ 7 \] which states that it is proportional to k.

l c,(e~ A) I./ o ¢t-b ¢ -(X

~:~o

Fig. 5. The determination of the unsteady lift coefficient according to the Gormont method.

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qmsi..stcady j~ "",.,,. ~;"'',%,,.,..." ,' i*| 0'810 12 13 14 15 16 17 18 angle of amuck alfa (dcsn~0

Fig. 6. Hysteresis loops of the lift coefficient of a NACA 0012 aerofoil. (a) Pitch motion, (b) heave motion.

I *~) 8LAO(

Fig. 7. A schematic representation of the rotor. 5. Simulations

In Fig. 6 hysteresis loops of an oscillating aerofoil (NACA 0012) are shown.

The calculated lift coefficients according to both methods correspond well with the measurements; for larger values of the reduced frequency the agreement is less. Figure 6b shows some peculiar features of the Gormont method which are physically incorrect: the lower part of the loop follows the shape of the static lift curve too closely and the left and right part of the loop are straight lines.

Now both methods will be compared on the basis of a simulation of a flexible rotor. In Fig. 7 the rotor is shown schematically. The only degree of freedom in the rotor is flap (perpendicular to the rotor plane). This degree of freedom is represented by a hinge together with a hinge-spring. The blades themselves are rigid and have linear taper and twist. The rotor angular velocity is constant. Figure 8 shows the response on a step in the wind velocity (from 10 to 1 m/s at t-0.5 s) in case of stall (tip speed ratio is 4). Limit cycle behaviour is observed in the quasi-steady and unsteady simulations. However, the ampli- tude and frequency differ in the unsteady case. This will influence the fatigue loads. The decrease in frequency in the unsteady case is in agreement with \[8 \]. Measurements are not available, so it is not clear which of the two unsteady methods gives the best results.

6, Conclusions

The unsteady aerodynamic forces are modelled in two ways. Simulations show differences between employing quasi-steady and unsteady aerodynamics. Unsteady aerodynamic effects mav have an important influence on the fatigue life of the wind turbine and its stability. The ONERA method gives better results in hysteresis than the Gormont method on the expense of more com-

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Fig. 8. Response of the rotor to a step in the wind velocity. (a) Flapping angle, (b) lift coefficient puter time. It can be concluded that for both load and stability calculations unsteady aerodynamics must be incorporated in the modelling. Future mea- surements on a rotor have to decide which of the unsteady methods is the most suitable.

References

1 A. Bramwell, Helicopter Dynamics (Arnold, London, 1976). 2 D. Petot, Progress in the semi-empirical prediction of the aerodynamic forces due to large amplitude oscillations of an airfoil in attached or separated flow, 9e European Rotorcraft Forum, Stresa, 1983. 3 W. Bierbooms, A dynamic model of a flexible rotor including unsteady aerodynamics, Euro- pean Wind Energy Conference, Glasgow, 1989. 4 W. Bierbooms, A dynamic model of a flexible rotor, report IW-89034-R, Delft University of

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