Inviscid flow through vertical axis wind turbines is modeled at high tip speed ratios using actuator disk theory with theconstraint of conserving mass as well as momentum and energy in the flow. Cylindrical and spherical disks are considered.Conservation of mass flow in a stream tube requires that the magnitude of the normal force exerted on this flow be the samethrough the upwind and downwind segments of the actuator disk. Since this force sequentially slows and deflects the flowaway from, and then back towards, the undisturbed wind direction in the wake of the disk, it plays a primary role indetermining the onset of choked flow (i.e., no flow) through outboard segments of the disk. The effect is to reduce thecapture area of inflow wind from which energy can be extracted to produce power. This effect appears to be morepronounced with the cylindrical than the spherical actuator disk. (Author)
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AIAA-97-0972-
CONSERVATION OF MASS EFFECTS ON FLOW THROUGHVAWT ACTUATOR DISKS ROTATING AT
HIGH TIP SPEED RATIOS
By
A. J. Eggers, Jr., Ken Chaney and Holt AshleyRANN.INC.
Palo Alto, California
Abstract
Inviscid flow through VAWTs is modeled at high tipspeed ratios using actuator disk theory with theconstraint of conserving mass as well as momentumand energy in the flow. Cylindrical and sphericaldisks are considered. Conservation of mass flow in astream tube requires that the magnitude of the normalforce exerted on this flow be the same through theupwind and downwind segments of the actuator disk.Since this force sequentially slows and deflects theflow away from and then back towards theundisturbed wind direction in the wake of the disk, itplays a primary role in determining the onset ofchoked flow (i.e., no flow) through outboard segmentsof the disk. The effect is to reduce the capture area ofinflow wind from which energy can be extracted toproduce power. This effect appears to be morepronounced with the .cylindrical than the sphericalactuator disk.
Predicted power coefficients differ little from thoseobtained with the DMST CODIF code (noconservation of mass constraint) for disks having lowsolidity and hence low induction effects at high tipspeed ratios. With increasing solidity or tip speedratio, choking constrains power coefficients tomaximum values approaching 0.5, after which theydecrease substantially with the spread of choked flow.This effect occurs earlier and is more pronounced withthe cylindrical actuator disk. In contrast, the DMSTcode generally predicts increasing power coefficientswith increasing solidity or tip speed ratio, withmaximum values in the neighborhood of 0.6. It isconcluded that the constraint of conservation of masscan more severely influence the performance ofcylindrical than spherical actuator disks. These diskssimulate VAWTs with height-to-diameter ratios(H/D) of oo and 1, respectively. A model of flowthrough the equatorial region of actuator disks with
intermediate values of H/D indicates that performanceapproaches that of the cylindrical disk for values ofH/D much in excess of 2.
Introduction and Background
The use of actuator disk theory to model flow throughpropellers and horizontal axis wind turbines(HAWTs) has long been appealing because of itsrelative simplicity compared to vortex models. Thisuse was pioneered by Joukowski, Glauert and others1,and today it is a mainstay in predicting theperformance and loads generated by HAWT rotorblades.2 In the simplest application, the rotor isidealized as a planar actuator disk with the axis ofrotation parallel to the undisturbed wind direction.Conservation of mass, momentum and energy issatisfied in the normal flow through the disk, usuallywith the assumption of locally two-dimensional flowabout each blade section.
Application of actuator disk theory to flow throughvertical axis wind turbines (VAWTs) is complicatedby the fact that it must model the flow through boththe upwind and downwind segments, with thedownwind disk operating in the wake of the upwinddisk. These disks are circular sections in plan view,not planar sections as with a HAWT. With Darrieustype VAWTs, which are of most practical interest,these disks are also curved in profile view. It followsthat, even with the assumption of locally two-dimensional flow about each blade section, modelingof the disturbed flow through a VAWT is nontriviallycomplicated due to its obvious nonuniformities, whichinclude both spatial and time variations in local winddirection and speed approaching the downwind disk.Much effort has been devoted to seeking simplifiedactuator disk models of flow through VAWTs whichprovide useful predictions of performance and loads.Excellent summaries of this work are provided byBerg3 and Paraschivoiu.4'5 The focus has been on
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double multiple stream tube (DMST) models, whichretain the assumption of locally two-dimensional flowabout each blade section. These models satisfyconservation of momentum and energy for the flowthrough the upwind and downwind disks in theundisturbed wind direction, but angular deflection offlow is neglected. Berg3 notes that the DMST CODIFcode is among the more accurate for predicting theperformance of lower height-to-diameter ratio (H/D)VAWTs representative of those currently in operationin wind farms. Paraschivoiu4 modified hisCARDAAV code to the version CARDAAX, whichaccounts for stream-tube expansion and lateraldisplacement, but not angular deflection in the wakeflow approaching the downwind disk. Thisapproximation for conserving mass was found to havea more significant effect on reducing powercoefficients at high tip speed ratios (TSR), but aboveTSR=8 the reduction was less than 4%. For VAWTswith low solidity and H/D (e.g., the SANDIA 5- and17-meter research turbines) the reductions were,therefore, regarded as second-order. Currently,simplified actuator disk models like the DMSTCODIF code are employed to predict the performanceof higher H/D VAWTs (e.g. H/D > 2) like theFlo Wind EHD class of experimental machines.6
Uncertainty remains in the basic actuator disk modelsof flow through VAWTs.7 The purpose of this paperis to revisit this issue in the simplest case of inviscidflow through VAWT actuator disks rotating at hightip speed ratios in a uniform wind. Flow about bladesections can then be taken as unstalled, andconservation of mass, momentum and energy in thedisturbed flow, including angular deflections, can bemore easily satisfied.
Actuator Disk Models
The actuator disks of this study are idealized' as aninfinite number of symmetrical blade elements havingequal infinitesimal chords and gaps aligned with theircircular paths of rotation. Thus the overall disturbedflow is quasi-steady. Inviscid flow through cylindrical(H/D = oo) and spherical (H/D = 1) disks is firstexamined using the classical methods ofdiscontinuous potential flow theory. In both cases theflow can be transformed to that for the disk having afinite number of blades with the same total solidity,same induction effects, and therefore generating thesame average power per revolution.
velocity. Continuity of flow in a stream tube passingthrough the upwind and downwind disks requires that
Nu u = VND
(D
where VNu and VM> are the flow velocities normal toand through each disk. The differential normal forcesare given by
FN = Bc Rd0
(2)
Here Be is the aggregate chord of the blades in thedisk, and
CM = RQ(3)
It follows from Eqs. (l)-(3) that conservation of massrequires
FNU = F,ND
(4)
in any stream tube encountering both disks. Tangentand normal force coefficients are related by
CT = 2(5)
Thus for RO/V0 » 1 (i.e., angle of attack is small),CT « CN, and components of stream velocity tangentto the disk are altered only slightly compared tocomponents normal to the disk. It follows to theaccuracy of this model that conservation ofmomentum and energy are confined to the normalcomponents. Actuator disk theory then yields
It follows that the validity of the model is restricted tovalues of a,, < 1/3, which are required to avoid chokedflow (i.e. VHP = 0) through the downwind disk.Choked flow through the outboard portion of the
actuator disk may occur at even lower values of a,,when either ap or au =1 (see Fig. 1 and Eqs. (6) and(7))-
The local power coefficients per unit of arc lengthalong the disk are due to the tangential forces andgiven by
*Note that equation (14) includes only upwind anddownwind disk forces acting on a stream tube, andthis is strictly valid only for the total flow through theactuator disk.
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= 0, andit is dictated by choking of the outflow through thedownwind disk as will be noted later. The extent ofthis choking increases with a, , and at the highestvalue of 3o = 0.3 (Fig. 3), it appears to be so extensivethat flow through the upwind disk is restricted to asmall portion near the x/R axis of symmetry.! Theonset of this choking in the flow through thedownwind disk is clarified by Figs. 4 and 5. It isevident that choking occurs when ac -> 1 and hence
Some elaboration on this phenomenon is in order. Itis noted that at the inner boundary of choked flowthere is obtained from equation (16)
The variations of local power coefficients withazimuth angle from Eqs. (12) and (13) are shown inFigs. 6 and 7 for two values of a,,. As expected, thesepower coefficients fall to zero in the regions of chokedflow. They are very small at best on the downwinddisk at the highest a<, = 0.3. The integrated total
*Thus with the constraint of conservation of mass, V]cos 0] (rather than Vi) is critical to determining thepower generated by the downwind disk for a given a^(seeEq.(13)).(The straight lines in these figures should not beconstrued as streamlines.JGiven the boundary conditions on flow direction andstatic pressure which must be satisfied.
power coefficients are shown in Fig. 8 as functions ofa«. The maximum total power coefficient is about0.48, occuring near ao = 0.15. The figure shows thatthe effect of choked flow is to markedly reduce totalpower coefficients at higher values of ao.
Spherical Disk (H/D = 1)
Blade chord is held constant over the span, so thatflow around blade elements is everywhere unstalled athigh tip speed ratios except near the poles (blade rootregions) of the downwind portion of the sphericaldisk. These regions are very small and they mergewith those where no flow can pass through the diskbecause the gaps between the blade elements vanish.Both of these effects are neglected in this study.
Modeling of the flow through the spherical disk issimplified because it is axisynunetric. It follows thatFig. 1 also depicts flow in an axisynunetric plane of aspherical disk. With one exception, the equations arethe same as those previously discussed for acylindrical disk. This exception enforces continuity offlow through a stream tube and reads
Equation (18) differs from that for a cylindrical disk(Eq. (16)) in that it accounts for the lateral expansionof flow approaching the downwind disk in any planeof symmetry. Choking through the disk is thereforedelayed. A typical solution of Eq. (18) shown in Fig.9 verifies this perception. Thus when 3o = 0,3, chokedflow is very limited with the spherical actuator disk,whereas it is quite extensive with the cylindrical disk(Fig. 3). This contrast is rooted in the shift fromchoking being initiated in flow through the downwindcylindrical disk (HD™* - 1) to this initiation occuringin flow through the upwind spherical disk (a^ux = 1).That this is true can be noted by comparing Fig. 10with Fig. 5.
The predicted power coefficients for the spherical andcylindrical actuator disks are compared in Fig. 11.These coefficients are higher for the cylindical VA WTat lower values of a,,, and vice versa at higher values.The superiority of the spherical VAWT at higher a,, isdue to the reduced choking of flow through theactuator disk discussed above. The peak Cp's of thetwo VAWT models are essentially the same.
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It is also informative to compare DMST CODIF andCOMME (Conservation of Mass, Momentum andEnergy model) predictions of total power coefficientsfor cylindrical and spherical VAWTs. Thiscomparison for cylindrical VAWTs is shown in Fig.12, and one notes that there is good agreement for a<,'sup to about 0.1. The CODIF code predictsincreasingly higher power coefficients at values ofa, = 0.15 and above. The comparison for sphericalVAWTs shown in Fig. 13 indicates good agreementup to a,, = 0.15, and the overprediction by the CODIFcode increases more slowly at higher values of a^,.This trend towards improved agreement with thespherical VAWT (H/D = 1) is consistent with Berg'sfinding noted earlier that the CODIF code predictsmeasured power with useful accuracy for lower H/DVAWTs.
Equatorial Disk (1 < H/D £ QQ)
A similar study has been conducted of the effect ofH/D on flow in the equatorial plane of a VAWTactuator disk operating at high tip speed ratios (againsee Fig. 1 and Eqs. (12)-(15)). In this event continuityof flow is satisfied when
where RE is the equatorial radius of the VAWT disk,and RCE is its radius of curvature in profile at theequator. RE/RCE is the key parameter which definesthe profile shape and H/D of an inertial TroposkienVAWT.8 Note that when RE/RCE = 0, Eq. (19) reducesto (16) for a cylindrical actuator disk (H/D = oo), andwhen RE /RCE = 1 it reduces to Eq. (18) for aspherical disk (H/D = 1). When RE/RCE = 0.5, H/D issomewhat over 2. Solutions to Eq.(19) have beenobtained over a range of 3o , and for values of RE/RCEfrom 0 to 1 increased incrementally by l/8th. Figure14 shows the total power coefficients predicted withthe aid of these solutions. The effects of blade profiledrag on these coefficients can be estimated with thesimple approximation
(20)
where Cf is the blade section friction drag coefficientat high tip speed ratios, and S is disk solidity. Forrotors in the range of interest, Cf/S2 is the order of 1;in this event profile drag reduces equatorial powercoefficients as shown in Fig. 15. It is clear that theCODIF code increasingly overestimates thesecoefficients at higher 3o with decreasing RE/RCE • At3o = 0.25 and RE/RCE = 0.5, this overestimationappears to be near 50% or more.
Concluding Remarks
The VAWT actuator disk models of this studyconsistently indicate that conservation of mass effectsmay substantially reduce performance in operation athigher tip speed ratios. This reduction is exacerbatedwith increases in solidity and/or H/D. The mostsignificant uncertainty in these models may concernthe stability of choked flow (e.g., trapped vortices) inthe interior of the disks. If this flow is unstable, ascould readily occur with a finite number of blades,then associated nonsteady flow effects must also beconsidered. Obviously variable speed operation tolimit tip speed ratio would improve performance atlower wind speeds.
Acknowledgments
This work was supported by the DOE NationalRenewable Energy Laboratory under SubcontractTAO-3-12241-OM04813.
References
1. Joukowski, N.E., et al, "Aerodynamic Theory,"W.F. Durand, Editor in Chief, pp 180-181. 1934.
2. Tangier, James L., "A Horizontal Axis WindTurbine Performance Prediction Code for PersonalComputers," SERI, January, 1987.
3. Berg, Dale E., "An Improved Double-MultipleStreamtube Model for the Darrieus-Type VerticalAxis Wind Turbine." Paper presented at SixthBiennial Wind Energy Conference and Workshop,Minneapolis, MN, June 1-3, 1983.
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4. Paraschivoiu, I., et.al, "Streamtube ExpansionEffects on the Darrieus Wind Turbine," AIAA Journalof Propulsion and Power, Vol. 1, Number 2, March-April 1985.
5. Paraschivoiu, I., "Double-Multiple StreamtubeModel for Studying Vertical-Axis Wind Turbines,"AIAA Journal of Propulsion and Power, Vol. 4,Number 4, July-August 1988.
6. Bell, Benjamin, "HoWind's Advanced EHD SeriesWind Turbine." Paper presented at ASME WindEnergy Symposium, Houston, TX, January 29-February 1, 1995.
7. Ashley, Holt, et al, "Some FundamentalConsiderations of the Aerodynamic and StructuralComplexities of Intermediate Size VAWT RotorSystems of the Darrieus Type, Vols. 1 and 2," DOEContract No. DE-AC04-86AL33181, June 24, 1988.
8. Eggers, A.J., Jr., et al, "Considerations of GravityEffects on VAWT Rotor Configurations WhichMinimize Flatwise Moments and Stresses." Paperpresented at 10th ASME Wind Energy Symposium,Houston, TX, January 20-23,1991.
Figure l. Flow Through cylindrical Actuator Disk (Forcesshown are exerted by the disk on the fluid.)
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