A parameter study of the influence of struts on theperformance of a vertical-axis marine current turbine

Anders Goude1, Staffan Lundin1 and Mats Leijon11Swedish Centre for Renewable Electric Energy Conversion,

The ngstrm Laboratory,Uppsala University,

Box 534, SE-751 21 Uppsala, Sweden

AbstractMarine currents are an important offshore source of

renewable energy. A lot of effort is spent on the devel-opment of technology for, for example, electricity gener-ation from tidal currents. In the present paper, the perfor-mance of a vertical axis marine current turbine is exam-ined numerically under the variation of certain parame-ters.

The turbine is modelled with an in-house code, basedon the double multiple streamtube model. Correctionsare made due to a finite aspect ratio and tip losses for theblades. Published experimental data for the lift and dragcoefficients of the blades for different Reynolds numbersare used in the model.

Structural integrity is a major concern of any under-water machinery due to the considerable hydrodynamicforces involved. Special attention is paid to the impor-tance of struts and related supporting structure for theturbine blades. As a rule of thumb, the efficiency of theturbine may be expected to rise with increased aspectratio of individual blades. However, with leaner bladesmore support structure is required, which carries a costin terms of a negative effect on efficiency. We study howthe level of acceptable stress on a turbine blade influ-ences the total turbine efficiency depending on the num-ber of struts required to support the blade.

Keywords: Turbine, blades, struts, simulation

NomenclatureNb = number of bladesNs = number of strutsc = chord for the bladecs = chord for the strutc0 = reference chord length = stressD = stress proportionality constanth = turbine heightR = turbine radius

c Proceedings of the 8th European Wave and Tidal EnergyConference, Uppsala, Sweden, 2009

L = distance between strutsCD = drag coefficientCL = lift coefficientFD = drag forceFL = lift forceFNH = normal (radial) force on bladeFT = tangential force on bladeFNC = centrifugal force on bladeFNl = total normal force on blade per unit lengthT = torque = rotational speed of the turbineV = flow velocityVb = flow velocity at the bladeVe = flow velocity at the centreVc = relative velocity at a point on the strutV1 = free-stream velocity = water densityx, y = cartesian coordinates in the turbine planer = coordinate along the direction of the strut = angle in the turbine planexblade = x coordinate for a given bladeyblade = y coordinate for a given bladeblade = coordinate for a given bladePloss = losses due to drag of struts

1 IntroductionTurbines for marine current power generation come in

many shapes and sizes. In a recent review, Khan et al. [1]considered several turbine and non-turbine systemsfor hydrokinetic energy conversion. From an engineer-ing point of view, different concepts entail various prosand cons, and what technological solution is objectivelybest in a given situation is rarely very clear.

One main category of turbine is the cross-stream axisturbine, characterized by having its axis of rotation per-pendicular to the main direction of fluid flow. There aresub-groups within this category, and for wind power ap-plications several variants of the Darrieus and Savoniustype turbines have been used [2]. For applications inwater, however, it is mainly the straight-bladed Darrieusturbine either in a squirrel-wheel configuration [3] orin an open-ended configuration [4] and the Gorlov

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(a) (b)

Figure 1: Two types of cross-stream axis turbine. (a) Open-ended Darrieus. (b) Squirrel-wheel Darrieus.

turbine which resembles the Darrieus turbine but hashelical blades [5] which have been used (Fig. 1). Dueto the orientation of the axis, these turbines are often re-ferred to as vertical-axis turbines, but obviously the axismay be cross-stream regardless of its orientation relativeto the horizon.

In the case of the open-ended Darrieus turbine, theturbine blades are connected to the turbine axis by meansof struts. Depending on the geometry of the turbine length, radius, solidity, number of blades , the numberof struts required to fully support the blades will vary,as will the geometric specifics of individual struts. Thestruts carry a penalty in terms of drag, which will ad-versely affect the power coefficient of the turbine. It istherefore of some interest to study the impact of bladestruts on the performace of a cross-stream axis turbine.

1.1 Problem considered in present paper

The purpose of the present investigation was to studythe impact in terms of drag loss of adding support strutsto turbine blades as the blades become leaner due to thenumber of blades being increased.

Consider a straight-bladed, open-ended Darrieus tur-bine with vertically oriented axis of rotation. For sim-plicity, we assume that the blades are not tapered (i.e.they have a constant chord length and hydrofoil thick-ness). For a given solidity, the blade chord will be in-versely proportional to the number of blades, and, con-sequently, the aspect ratio of a single blade will be pro-portional to the number of blades. Higher aspect ratioof the blade gives a lower induced drag, which will in-crease the lift-to-drag ratio. For the same total blade sur-face area, it might then be expected to see lower draglosses for the same amount of lift when the surface areais divided among a greater number of blades. A smallerchord does however decrease the Reynolds number ofthe blade, which usually gives a lower lift-to-drag ratio.This means that the hydrodynamical benefits of increas-ing the number of blades are lower for turbines whichhave a large height compared to their radius.

A leaner blade also possesses less structural strength.Depending on the material from which it was made, ablade can only withstand a certain amount of stress. Thestress will depend on the forces experienced by the bladeand its geometrical properties, especially the distancesbetween support points. As the aspect ratio of a bladeis increased, thresholds will be passed where additional

support struts for the blade will be required. These sup-ports cause drag without adding to the lift, and this dragpenalty may or may not exceed the gain in terms of in-duced drag reduction from the increase in aspect ratio.

The limiting factor for the number of struts to a bladeis the stress on the blade due to the radial hydrodynamicforce. Depending on the thickness of the blade, the prop-erties of the material it is made of in combination withthe construction method will allow a maximum distancebetween support points. Given the blade length, thistranslates to a minimum number of struts to support theblade.

The dominating force for sizing the struts is theweight of the blade. The stresses due to gravitationalforces dominate over those due to hydrodynamic forcesfor the struts. This is again in part due to the choice ofmaterial and method of construction, which highly influ-ence the blade weight.

In the following section, the expressions for calcu-lating the maximum distance between struts and the re-quired minimum number of struts per blade will be de-rived. These equations are then used to illustrate the im-pact of strut losses on the performance of two exampleturbines designed for different flow cases.

2 Theory2.1 Hydrodynamic model

The simulation model chosen for the hydrodynamicalsimulations was the double multiple streamtube modeldeveloped by Paraschivoiu [6], which has shown goodresults for simulating Darrieus wind turbines. The modelseparates the turbine into one upstream part and onedownstream part, where both parts are solved separatelyusing momentum conservation and the velocity is calcu-lated at blade positions and at the centre of the turbineonly. Further details about the method may be found inreferences [7] and [8]. The present model is based onexperimental data for lift and drag coefficients obtainedfrom [9]. For dynamic stall modelling, the method devel-oped by Gormont [10] and later modified by Mass [11]and Berg [12] was chosen. For the calculation of forceson the blades to be used in structural mechanical cal-culations, corrections due to a finite aspect ratio wereneglected in order to get a small overestimation on theforces. For power coefficient calculations, tip correc-tions were applied according to Paraschivoiu [8] withthe modification that the lift and drag coefficients wererecalculated with the dynamic stall model using the re-duced angle of attack, and induced drag was applied af-ter the dynamic stall calculations. This modification wasmade to prevent overestimations of the power coefficientat high tip speed ratios.

2.1.1 Calculation of loss due to struts

The losses due to struts can either be determined bycalculating the force due to drag over the strut and in-tegrating to obtain the losses for the whole turbine, or

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by using an equivalent drag coefficient for the entire tur-bine as suggested by Moran [13], where the losses areproportional to the cube of the rotational speed. Here,the force will be integrated for higher flexibility and be-cause the drag coefficients for the selected strut profileare known. It is assumed that the changes in flow veloc-ity due to the struts can be neglected, and that the strut isapproximately horizontal giving zero angle of attack.

To determine the drag forces, it is necessary to knowthe local velocity at the strut. Consider a blade positionon the upwind disc. Symbols are illustrated in Fig. 2.To determine the velocity, start by noting that when theblade is located at angle blade, the y coordinate of theblade is given by

yblade = Rsinblade. (1)

Now, since we are interested in a point on the strut, letthe coordinate r denote the position on the strut. We have

0 < r < R. (2)

This gives the local coordinates as

y = r sinblade and x = r cosblade. (3)

We want to find out in which streamtube this point islocated. If streamtube expansion is neglected as donein [7], this will correspond to the streamtube with value

= arcsin( r

Rsinblade

). (4)

In the double multiple streamtube model, the velocitiesat the blade and at the centre are known. To approximatethe velocity at the strut, it is assumed that the velocityvaries linearly between the boundary of the turbine andthe centre. By using the velocities Vb at the blade andVe at the centre for the streamtube obtained from Eq. (4),linear interpolation gives the velocity as

V =Ve+VbVe

R2 r2 sin2 blader cosblade. (5)

Following the streamtube model, it is assumed that thevelocity only has x-components. Therefore, in the localframe of reference of the blade, the velocity in the direc-tion of the chord is given by

Vc =r+V sin blade. (6)The corresponding expression can be derived for thedownwind part.

When the velocity is known, the force can be calcu-lated according to

dFD =CDcsV 2c2

dr. (7)

The torque is given by

T (blade) =R

0

CD (r,blade)cs(r)Vc (r,blade)2

2r dr (8)

r

R

xblade

yblade

x

y

cs

blade

V1Ve V

Figure 2: Symbols used in derivations.

and the power loss by

Ploss = NbNs1

2pi

2pi0

T (blade)dblade, (9)

where Ns is the number of struts and Nb is the number ofblades.

2.2 Support struts geometryIf the normal forces on the blades are balanced to give

a low torque on the joint between the blade and the strut,the bending moment in the blade will mainly depend onthe distance between the strut and the force applied. Fora given profile, the bending resistance will increase withthe cube of the chord, giving the expression for the ten-sion in the blade as

DL2

c3FNl (10)

where FNl is the total normal force per unit length, andD is a constant (see Sec. 2.2.3). FNl is given by

FNl =FNH +FNC

h . (11)

Eq. (10) assumes that the distance L >> c and that thedisplacements are small enough for linear theory to hold.

2.2.1 Distance between struts as a function of num-ber of blades

For a turbine, the tip speed ratio that gives the highestpower coefficient is primarily a function of the solidityof the turbine. This means that for a turbine to have thesame optimum tip speed ratio, the solidity of the turbineshould remain constant. By denoting the chord and num-ber of blades of one turbine as c1 and Nb1 and of a secondturbine as c2 and Nb2, a constant solidity gives

c1Nb1 = c2Nb2. (12)If it is assumed that the power coefficient is approxi-mately the same for both turbines, the torque has to be

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the same (due to the assumption of same tip speed ratio),giving the relation between the tangential forces of theblades as

FT 1Ns1 FT2Ns2 (13)and, since the power coefficient is assumed to be thesame, the flow field through the turbine should be sim-ilar, giving approximately equal angles of attack forthe blades, which means that the relations betweenthe hydrodynamical normal forces should be similar toEq. (13), i.e.

FNH1Ns1 FNH2Ns2. (14)The centrifugal force will depend on the mass of theblade. For a solid blade, the mass is proportional to thesquare of the blade chord, and the relation between thecentrifugal forces is

FNC1c22 = FNC2c21 (15)

for a constant rotational speed. By combining Eqs. (10),(11), (12), (14) and (15) and assuming that the maximumallowed stress is the same, one obtains the expression

L2 L1 Nb1Nb2

FNH1 +FNC1

FNH1 +FNC1 Nb1Nb2. (16)

It should be noted that by changing the chord, both theReynolds number and the aspect ratio of the blade willchange, effects of flow curvature will be different andthe behaviour in dynamic stall will change. Therefore,the assumption of constant power coefficient, and henceEqs. (13) and (14), can be questioned.

2.2.2 Distance between struts as a function of designflow velocity

Using the expression for the hydrodynamic lift force,

FL = chCLV 2

2, (17)

and the assumptions that the variation in lift coefficientdue to the change in velocity is small and that the angleof attack remains constant, one will obtain the relationbetween the forces at different velocities as

F2 F1(V2

V1)2. (18)

By combining Eqs. (10) and (18) and assuming constantchord, one obtains the relation between the maximumdistance between the struts and the design flow velocityas

L2 L1 V1V2 . (19)

2.2.3 Number of struts per bladeThe constant D in Eq. (10) can be determined from

FEM calculations. This was carried out assuming thatthe force was evenly distributed over the entire blade,which is the force distribution obtained due to the as-sumptions outlined in Sec. 2.1. The joints between the

struts and the blades were considered as fixed surfaces.For these boundary conditions, the maximum stress ona blade is found close to the joints. For the NACA0021blade, the constant was determined to be D 18. Foran evenly distributed force and a constant chord, thedistance between two struts should be approximately2.5 times the distance between the outer strut and theblade tip in order to obtain the same stress on both sidesof the junction of the outermost strut. (In the real case,the force is smaller close to the blade tip due to the tipvortices; hence the struts should be a little closer to eachother in the real case to create the same stress on bothsides of the outermost junction.) By using this result,once the distance between the struts has been calculated,it is possible to determine the necessary number of strutsas

Ns =hL+0.2 (20)

where Ns should be rounded to the nearest larger integerand should never be smaller than 2.

2.2.4 Calculation of the size of the strutsThe forces on the struts can be separated into three

parts: the normal (or radial) force, the tangential forceand the gravitational force. The normal force has twocontributions, the hydrodynamic force and the centrifu-gal force. Since its direction is parallel to the strut, themaximum tension will be obtained at the point where thestrut chord is smallest. At the strut root, this force willbe negligible compared to the other forces, which causebending moments. The tangential force is an oscillatingforce, and will cause a bending moment that is largest atthe strut root.

The last force is gravity, which also causes a bendingmoment. One difference between the tangential forceand gravity is that in the tangential direction, the bladecan move freely, while in the vertical direction, the bladecannot move freely since there are several struts con-nected to the blade, preventing it from rotating. This cancause additional bending moments in the strut tip, andalso in the blades. For the case of blades with high den-sity, the contribution from gravity will be the dominatingterm at the strut tip.

Based on FEM calculations, the strut chord was cho-sen as 0.8 times the blade chord at the tip and 1.2 timesthe blade chord at the root for a blade with chord 0.2 m,which causes the tension in the blade to remain withinthe chosen limits. Since gravity gives a constant load,fatigue will not be a major concern here, since the pointwhere the stress from gravity is located is where strut isat its maximum thickness, while the tangential force willgive its largest contribution at the trailing edge. How-ever, it is desired that the deflection of the struts is lim-ited, since it can affect the performance of the turbineand can cause additional tension in the blades. The de-flection increases approximately with the chord of thestrut to the power of four, while it decreases linearly withthe force, which for gravity decreases with the square ofthe blade chord. Considering this, the chord of the strut

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2.5 3 3.5 4 4.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Tip speed ratio

Chor

d [m

]

3 blades6 blades

Figure 3: Chord lengths of the 2.5 m/s turbines.

was chosen ascs = 0.8

c0c (21)

for the strut tip and

cs = 1.2

c0c (22)for the strut root, where c0 = 0.2 m as in the FEM calcu-lation.

Since the bending resistance decreases with the cubeof the strut chord, while the drag on the strut only de-creases linearly with the chord, small optimisations ofthe strut chord will not lead to any major reduction inthe drag from the struts.

Different thicknesses of the hydrofoil were investi-gated, but the possible reduction in strut chord obtainedby choosing a thicker profile were basically compen-sated by the higher drag coefficient of the thicker chord,giving approximately the same drag losses. The samewas seen for a thinner profile. Therefore, the same pro-file was chosen for the struts as for the blades.

3 ApplicationIn order to illustrate the effect of the equations derived

in Sec. 2, two theoretical turbines were studied; one with3 blades, the other with 6 blades, but besides that essen-tially equivalent.

The size of the turbine was taken as 5 m in length witha radius of 2.5 m, and the maximum flow speed was as-sumed to be 2.5 m/s. These admittedly arbitrary figures

2.5 3 3.5 4 4.50.3

0.35

0.4

Pow

er c

oeffi

cient

Tip speed ratio

3 blades6 blades

Figure 4: Power coefficients of the turbines optimised for flowspeed 2.5 m/s. The stair-step shape of the curves due to the in-clusion of more struts with increasing tip speed ratio is clearlyvisible.

would give an expected maximum power output on theorder of 60 kW depending on the power coefficient ofthe turbine and the efficiency of the system as a whole;not a very large station but quite sufficient for our presentpurposes. Both the blades and the struts were given theNACA0021 profile, and all components were assumedto be constructed from solid steel with a mass density of8 times that of water. Using data from ship propeller de-sign [14], the allowable stress in the blades was taken as100 MPa, which gives reasonable safety factors.

Neglecting for the time being the strut losses, the op-timum turbine solidities for tip speed ratios from 2.5through 4.5 were calculated (Fig. 3). Given these solidi-ties, the maximum normal (radial) force on the bladeswas calculated, neglecting three-dimensional blade tipeffects in order not to underestimate the force, but in-cluding centrifugal effects due to the mass of each blade.Based on the normal force, the minimum number ofstruts for operation at each speed vas computed, and fi-nally the losses due to the struts as well as the blade tipeffects were included. This process was carried out fortwo flow speeds, 1.5 m/s and 2.5 m/s.

The power coefficient curves of the turbines opti-mised for 2.5 m/s are plotted in Fig. 4. The curves as-sume a stair-step character, due to the discrete additionof strut losses. As predicted by Eq. (16), for the 6 bladedturbine with the thinner, leaner blades, the steps occurmore frequently than for the 3 bladed turbine, and asa consequence the power coefficient drops off consid-erably more quickly as the tip speed ratio increases.

Since the turbines were optimised for 2.5 m/s, theymight be expected to perform less well at 1.5 m/s. InFig. 5, the power coefficients of all turbines are plottedat the lower speed. The 6-bladed turbines again exhibitmore steps in the curves and a quicker drop-off with in-creasing tip speed ratio. Not surprisingly, the turbinedesigned for 1.5 m/s performs consistently better thanits higher-speed counterpart. For the 3 bladed turbinesthe difference is less pronounced, and the two curves be-have almost identically until the slightly leaner-bladed2.5-m/s turbine takes a step down due to extra struts be-ing added at an approximate tip speed ratio of 3.2.

At almost any site considered for establishment of a

2.5 3 3.5 4 4.50.3

0.35

0.4

Tip speed ratio

Pow

er c

oeffi

cient

3 blades (2.5)6 blades (2.5)3 blades (1.5)6 blades (1.5)

Figure 5: Power coefficients of turbines optimised for flowspeeds of 1.5 m/s and 2.5 m/s, all run at 1.5 m/s.

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marine current power station be it a tidal channel, ariver reach, or some other location the strength of thewater current will vary with time. The main point illus-trated by Fig. 5 is that it is far from obvious that a tur-bine should be optimised for the peak flow speed. Thedetails all depend on the specific flow characteristics ofthe particular site, but on the whole it can be expectedthat the peak flow speed will occur during a rather smallfraction of the operating life of the power station. Look-ing at the 6-bladed turbines of the present example, itis clear that the turbine designed for 2.5 m/s will not bethe best choice if it spends most of its service life op-erating at 1.5 m/s. The other turbine, on the other hand,might not be structurally strong enough to handle the hy-drodynamic forces at 2.5 m/s, and so would have to beshut down when the flow speed exceeds some predeter-mined value. The choice of design speed would have tobe based on statistics of the flow at the site under con-sideration, so that it might be determined which optiongives the best output over time.

4 DiscussionIt is hardly controversial to claim that strut losses are,

in some sense, significant for the performance evalua-tion of an open-ended Darrieus turbine. The mechanismsgoverning the losses are however quite complex, and thefact that strut losses are added in steps with additionalstruts makes it far from obvious what will be the optimaldesign choice for a given case.

In theory, more blades means leaner blades, whichmeans higher blade aspect ratio, which means lower in-duced drag. However, as is clearly seen in Fig. 4, leanerblades also entails higher sensitivity to hydrodynamicforces and thus an increased requirement for struts which brings drag. Fig. 5 indicates that the 3-bladed de-sign has a significanly smoother strut requirement withvarying tip speed ratio, which clearly ought to be desir-able in many cases.

That being said, there are other factors to take intoaccount. As mentioned previously, gravitational effectson the struts were largely disregarded in this study. Inreality, the struts have to carry the weight of the blades,which is proportional to the square of the blade chordlength (assuming solid blades). This means that the totalweight of the turbine blades on the 3 bladed turbine istwice that of the 6 bladed turbine, since the turbine so-lidity is approximately the same and the blade chords ofthe two turbines consequently vary by a factor of 2. Theweight of the struts will somewhat even out the differ-ence, but on the whole a turbine may be expected to beheavier the lower the blade number.

The turbines were assumed to be constructed out ofsolid steel. This is simple and not unreasonable. Manyother materials and construction techniques might how-ever be considered. It could, for instance, be an optionto make the larger blades (such as on the 3 bladed tur-bine) hollow, in order to save weight and material. Var-ious composite materials might also be an option. Such

changes would influence the mechanical properties ofthe blades and struts, and might serve to mitigate the ef-fects of strut losses. These considerations are howeveroutside the scope of the present study.

AcknowledgmentsThe work reported was financially supported by Stat-

kraft AS, Vattenfall AB, ngpannefreningens Founda-tion for Research and Development, The J. Gust. RichertFoundation for Technical Scientific Research and CFsEnvironmental Fund.

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