Numerical optimization of the guide vanes of a biradialself-rectifying air turbine.

André Ramos Maduroemail: andre.maduro@ist.utl.pt

Instituto Superior Técnico, University of Lisbon,Av. Rovisco Pais,1049-001 Lisboa, PortugalJuly 2016

AbstractThe focus of this paper is the design and optimization of the guide-vane system of

a self-rectifying biradial air turbine for use in oscillating water column systems for waveenergy conversion. Since the turbine is required to be self-rectifying, there are two sets ofguide-vanes, placed symmetrically on both sides of the rotor, instead of a single set (as inthe conventional turbines). The efficiency of the biradial turbine with fixed guide-vanesis affected by the losses at the entry to the downstream row of guide-vanes, due to themisalignment between the flow direction and the exit guide-vanes. A problem also sharedwith other self-rectifying axial-flow impulse turbines. The goal of this paper is to performa numerical optimization of the guide-vane system geometry so as to increase the turbineefficiency by reducing the losses at the exit guide-vane system, while ensuring the requiredflow deflection at the inlet guide-vane system.

Time dependent flow calculations are performed to assess the performance and to eval-uate the pressure losses for each guide-vane system configuration, due to the relativelylarge separated region at the exit guide-vane system. Finally, the turbine operating curvesare obtained by calculating the three-dimensional flow through the complete turbine.Keywords: Biradial turbine, guide-vane design, optimization, blade cascade, wave energy.

1. Introduction

Self-rectifying impulse turbines (includ-ing the biradial turbine) have severe lim-itations in efficiency due to low aerody-namic performance of the second stator, po-sitioned downstream of the rotor. Sincethis type of turbines are required to be self-rectifying, there are two sets of guide-vanesplaced symmetrically on both sides of therotor instead of a single set (as in conven-

tional turbines). The guide-vanes, designedto guarantee the desired flow deflection atthe impeller inlet, are the main cause of thepoor performance of such turbines in thevariant of fixed guide-vanes. When the flowexits the rotor, the existing misalignmentbetween the guide-vanes of the downstreamstator and the flow direction makes thesebehave as bluff-bodies. This effect promotesflow separation on the surface of the blades

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and increases drastically the pressure loss inthe exit stator. If this misalignment is toohigh, the flow passage between the guide-vanes is strongly obstructed. Thus furtherincreasing the loss and consequently reduc-ing the turbine performance.

In 2009, the Portuguese company Ky-maner developed a new system of guide-vanes in multiples cascades for the axial-flow turbine that bridges the effect of thedownstream stator [1, 2]. In this case,the inlet flow deflection is divided in stagesby the use of several rows of guide-vanes,strategically positioned at a given rotor dis-tance in order to increase the free spacebetween blades (see Fig. 1). This deflec-tion strategy and preferential positioningbetween sets of guide-vanes allows to in-crease the passage of outflow and conse-quently increase the efficiency of the tur-bine. Company data shows that this strat-egy guaranteed an increase of 20% in peakefficiency of the axial turbine in comparisonwith the case of a single row of guide-vanes.

The biradial turbine [3], being also a im-pulse turbine, has the same problem withthe downstream stator. Several projects re-lated with the design of guide-vanes hadbeen developed for this turbine [4, 5], butnone had addressed the effect of multiplecascades. Therefore the purpose of this pa-per is to apply this stator design concept tothe biradial turbine and evaluate the bene-fits of it implementation (see Fig. 2). Theaim is to assure the deflection of the inflow,without introducing large losses in the inletstator, and reduce the outflow obstructionwith the introduction of multiple cascades.The number of cascades it is a parameter todetermine through the downstream statorperformance evaluation for a given rotor op-eration conditions. The guide-vane sectionswere designed using a thickness and cam-

ber distribution optimization that respectsthe restriction of preferential position to as-sure minimum outflow obstruction. Thisnumerical geometry optimization problemis solved with the gradient-free Differen-tial Evolution (DE) method [6]. The DEmethod is a metaheuristic algorithm thatsolves the optimization problem by iterativeimproving a candidate solution through theuse of genetic operations. To validate theoptimized geometries, time dependent cal-culations are performed through the down-stream stator to evaluate the pressure drop.Finally a 3D flow analysis through the com-plete turbine will define the performance ofthis stator concept.

Figure 1: Stator of the self-rectifying impulse tur-bine developed by Kymaner for the ModOndas na-tional project (reproduced from [1]).

1st row of guide-vanes

Rotor2nd row of guide-vanes

Radial flow

Figure 2: Flow representation for exit guide-vanesystem of the biradial turbine.

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Periodic surface

Periodic surface

1st row of guide vanes

2nd row of guide vanes

Figure 3: Computational domain and blades preferential position for purely radial exit flow.

LE1LE2

TE1

TE2

1st row2nd row

Figure 4: Blades parameter for a guide-vane system of multiple cascades.

2. Blade cascades design

The positioning of the blades is one ofthe most important geometrical parametersin this stator design. The guide-vanes areplaced in a configuration which maximizesthe flow free passage between the blades ofthe downstream stator (see Fig. 3). Thisconfiguration is based on the shadow effectprovided by the upstream blades when theflow exit the impeller. Thus, the bladesrows are placed within the same angularwindow ∆β relative to the rotor outflow di-rection. This strategy ensures that down-stream blades are in the wake of upstreamones (see Fig. 2).

To control the flow free path in thedownstream stator it’s defined the blockagefactor parameter ζ, given in percentage by

ζ(%) = ∆β∆Θ × 100% , (1)

where ∆Θ corresponds to the row space an-

gle between blades, controlled by the num-ber of blades in the circumferential direc-tion (fixed to 57 blades). The blockage fac-tor ζ represents the obstructed path causedby the exit guide-vanes (symmetrical of theinlet) and it’s an indicator of how large isgoing to be the pressure drop in the exitstator.

For a given ζ, the leading and trail-ing edges of each cascade blade are coin-cident with the lines of constant ∆β (seeFig. 4). Therefore the blades stagger an-gle and chord are managed simultaneouslyby the design parameters ∆R that controlthe radial position between blades edges,making this two variables implicit in theoptimization algorithm. The blades thick-ness distribution is scaled form the NACA63A012 [7], using a scale factor parame-ter for each blade. The camber line is setfrom Bézier curves, defined with 2 controlpoints, to reduce the number of variables

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per blade. The first and the last point de-fine the beginning and the end of the curve(see Fig. 5), while the intermediate pointscoordinates (2 and 3) define the design vari-ables for the camber line. A preliminaryexploratory work has shown that the use of2 intermediate points, with a fixed radialposition (rLE, rT E), generate a wide rangeof geometries for the camber line curvature.The angular position of intermediate points(δLE, δT E) controls the blade inlet/exit an-gles (εLE, εT E) when placed in the preferen-tial position.

However if the exit flow is not purelyradial, as shown in Fig. 3, the first blade’swake does not provide a full shadowing ef-fect to the downstream ones. In this case(α2 6= 90o), the blades are positioned be-tween two logarithmic spirals assuming con-servation of angular momentum (see Fig. 6).This means that the stator must be opti-mized with a given rotation λ between cas-cades, function of the angle α2, defined by

λ = cotα2 ln(RLE1

RLE2

). (2)

This rotation between blade cascadesdoes not interfere with the way the bladesections are parameterized.

Since the design method repeats it-self at each DE optimization step, theparametrization process should be generic,free of discontinuities and also able to keepthe number of design parameters as low aspossible. In this way, each cascade bladeis designed with the same number of op-timization parameters (four). Two designvariables for the inlet/exit blade angles, onefor the thickness scale factor and anotherone for the chord and stagger angle (∆R).When the number of cascades increases onetime there are five more variables to opti-mize (four from the new cascade and one for

the radial relative position between blades).In order to reduce even more the numberof parameters, the stator to rotor diameterratio is fixed to RT E/Rrotor = 4.65, whereRT E is the radial coordinate of the trailingedge of last stator blade, the closest one tothe rotor, and Rrotor = 244 mm is the rotorradius.

Camber line

LE

LE

TE

TE

Figure 5: Bézier curve with 2 control points usedto define the camber line. The parameters δLE andδT E are related to εLE and εT E respectively, thatare use as design variables.

Logarithmic spiral

TE

Figure 6: Relative rotation between radial cas-cades.

3. Computational domain

The computational domain used for theflow simulations is set to bi-dimensional andit represents the stator midplane relative tothe axial direction. Periodic boundaries areused to reduce the domain size (see Fig. 3)and consequently the computational time.This domain is used for the inflow and out-flow calculations, due to the turbine sym-

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metry condition. The only difference be-tween the two domains is the radial size ofthe upstream block. In the outflow simula-tion, the exit block is extended due to thelarge wake and vortices release associated tothe flow separation at the blades surfaces.

The discretization scheme used forthis computational domain is hybrid (seeFig. 7). Around each blade there is a struc-tured block of mesh, in "O" shape, whoseconstruction is based on the surface curva-ture of the blade [8]. This type of construc-tion assures that the density of points on thesurface of the blade is higher in the zones ofhigh curvature, witch is perfect to define theblade trailing edge (see Fig. 8). Involving allthe "O" shape meshes there is one unstruc-tured block of mesh whose construction isbase on Delaunay triangulations [9]. Thisblock leaves plenty of room for the bladesections to change shape every optimiza-tion step, and it’s responsible for makingthe connection between the upstream andthe downstream blocks of structured mesh.

X

Y

Z X

Y

Z

Unstructured mesh

Structured mesh

Figure 7: Mesh blocks.

X

Y

Z X

Y

Z

Figure 8: Detail of the computational mesh nearthe blade trailing edge.

4. Viscous flow computation

The viscous flow simulations wereperformed using the commercial soft-ware FLUENT, that solves the Reynolds-Averaged Navier-Stokes equations (RANS)coupled with a transitional flow k − ω SSTturbulence model. In this calculations theflow is considered incompressible (Ma<0.3at Tamb = 300K), and the equations aresolved using the pressure-based coupled al-gorithm. The pressure is computed with the2nd order scheme and the other equationswith the 2nd order upwind scheme. The in-flow conditions are assumed to be steady,while the outflow conditions are consideredto be unsteady, due to the relatively largeseparated flow area at the downstream re-gion of the blades, making this calculationstime dependent.

The inlet boundary condition for the in-flow and outflow calculations is the same.It is imposed as mass flow with turbulenceintensity and hydraulic diameter as closurequantities. The turbulence intensity is dif-ferent for each case, being a higher value es-timated for the exit stator due to the pas-sage through the rotor. For both calcula-tions an outflow condition is imposed at thedomain exit with the target mass flow equalto the inlet. The blade walls are set to wallswith no-slip boundary condition.

The working fluid is air with ρ =1.225 kg/m3 and viscosity µ = 1.7894 Pa s.The reference pressure is the standard at-mospheric pressure pref = 101325 Pa. Thedesign mass flow rate used for this calcula-tions results from the design conditions atthe rotor entrance (V1t = 2U and α1 = 30o).Combining this conditions with model sizeReynolds number Re = ΩRrotor

2/ν = 2.0×105, where rotation speed Ω = 500 rpm, re-sults the design mass flow rate of 1.49 kg/s.

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5. Optimization strategy

The inlet stator blades are optimizedthrough the DE algorithm for a given block-age factor parameter ζ (Eq. 1). This pa-rameter is not a design optimization vari-able because it is intended, in this paper,to evaluate how it influences the multiplecascades. Therefore blades geometry andposition parameters are optimized to guar-antee the minimum angle at the rotor en-trance (α1) and low pressure losses at theinlet stator.

This multi-objective problem can besolved using an objective function F (z) witha weight factor, as the one used in [5]. Thisweight factor choice is always a difficult de-cision due to the parameters different na-ture and because it influences directly thefinal optimization result. To avoid usinga weight factor the pressure drop is esti-mated through the blade’s boundary layerseparation. If the flow separates from thesurface of any blade, the pressure losses inthe inlet stator are considered too high andthe tested geometry is discarded from theoptimization process. If the flow does notseparate, the pressure loss is considered toosmall to be optimized, and the algorithmis responsible to compute the mean value ofthe flow angle at the impeller entrance (α1).This pressure losses evaluation strategy al-lows to set a single term objective function,defined by

F (z) = α1 . (3)Since the blades have a rounded trail-

ing edge there is always flow separation inthat region. Therefore the flow separationis evaluated in 90% of the line length thatdefines the suction side of the blade, start-ing from the leading edge. This means thatseparation can occur in the last 10% of the

line. An acceptable value that assures lowpressure losses in the inlet stator.

After each geometry calculation the flowseparation is evaluated before computingthe objective function. To evaluate this, thevelocity profiles are analyzed using a localcoordinate system on the suction side of theblade (see Fig. 9).

Inflection pointFlow separation point

Reversed flow

Zero velocity line

Figure 9: Local coordinate system (sx, sy) on thesuction side of a blade.

6. Results

In order to determine the effect of thenumber of cascades ZR in the blockage fac-tor ζ, several optimizations were performed,see Fig. 10. It is chosen to evaluate this ef-fect for three configurations (ZR = 1, 2 and3 cascades) with the design outflow angleα2

∗ = 90o. This means that the outflowis purely radial and it corresponds to theminimum kinetic energy at the rotor exit.For each optimized geometry (point in thegraphic), the angle at the rotor entranceα1 represents the maximum flow deflectionthat respects the preferential position andseparation condition on the surface of theblades, guaranteeing low pressure losses inthe inlet stator.

As predicted, the blades blockage factordecrease with the increasing number of cas-cades. For all configurations the flow deflec-tion decreases with the decreasing blockagefactor due to the blade load reduction.

To check the performance of each con-figuration, time dependent calculations areperformed to evaluate the pressure drop in

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15 20 25 30 35 40 45 50

α1 [ ]

40%

50%

60%

70%

80%

90%

ζ

ZR =1 ZR =2 ZR =3

Figure 10: Blockage factor for systems of 1, 2 and3 cascades optimized for α2

∗ = 90o.

the downstream stator. The selected ge-ometries for this calculations match the ro-tor entrance operating condition (α1

∗ =30o). The outflow angle is set to the sta-tors design condition (α2 = 90o). Fig. 11presents the dimensionless pressure drop(Eq. 4) for each selected configuration asfunction of the Reynolds number (Eq. 5),

Kp = ∆p0112ρVRTE

2 , (4)

Restat = cVRTE

υ. (5)

The dimensionless pressure drop Kp isdefined as the total pressure variation acrossthe blades rows ∆p01 normalized by the dy-namic pressure based on the radial compo-nent of the flow velocity at the last bladetrailing edge radial section (the same for allconfigurations). In this case, the Reynoldsnumber is based on the chord c of the lastblade and the radial velocity at the sameblade trailing edge radial section.

Due to the extremely turbulent flowverified at the downstream section of theblades, the pressure losses are completelyindependent from the Reynolds number.

The high blockage factor observed forthe ZR = 1 configuration (70%) severely

104 105 106 107

Re

2.5

3.0

3.5

4.0

Kp

ZR =1 ZR =2 ZR =3

Figure 11: Outflow pressure loss for systems of 1,2 and 3 cascades optimized for α2

∗ = 90o.

penalizes this turbine’s stator performance.In this configuration the pressure drop is al-most twice the verified in the other testedconfigurations. Since the 2 and 3 cascadesconfigurations present almost the same exitlosses and both fulfill the inlet stator oper-ating condition, it is preferred to adopt theZR = 2 configuration. This one brings morebenefits from the economical and manufac-turing point of view.

However, in order to increase the rangeof stator candidates the 2 cascade config-uration is optimized for two new differentangles: α2

∗ = 65o and 115o. Which meansa rotation of ±λ between the two cascades(see Fig. 6). The optimization process isreplicated from the previous one, used forthe design angle α2

∗ = 90o, and is presentedin Fig. 12. In this two new configurationsthe blockage factor presents the same evo-lution as before, but in the particular caseof the α2

∗ = 65o configuration, the block-age factor is significantly smaller for thesame rotor entrance condition (α1). Thereason behind this huge reduction relays onthe way the blades are placed within thelines of constant ∆β. Where the configura-tions optimized for design angles α2

∗ closerto the rotor entrance design angle (α1

∗) are

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the most benefited.Finally, a set of calculations for differ-

ent outflow angles were performed to ver-ify if the design angle α2

∗ of each config-uration fulfills the purpose of it optimiza-tion (see Fig. 13). This result shows thateach optimized configurations assures thebest performance for their own design an-gle (α2 = α2

∗), and on the other hand showsthat all configurations are very sensitive tovariations of the outflow angle.

15 20 25 30 35 40 45 50

α1 [ ]

10%

20%

30%

40%

50%

60%

70%

80%

90%

ζ

α2∗ = 65

α2∗ = 90

α2∗ = 115

Figure 12: Blockage factor for ZR = 2 optimizedfor α2

∗ = 90± 25o.

60 70 80 90 100 110 120

α2 [ ]

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

Kp

α2∗ = 65

α2∗ = 90

α2∗ = 115

Figure 13: Influence of the angle α2 in the outflowpressure loss, for ZR = 2 optimized for α2

∗ = 90±25o.

7. Turbine performance

7.1. Modeling the three-dimensional flowTo have more reliable values of each

stator performance, three-dimensional flowcalculations through the complete turbinewere performed. The aim of this calcula-tions is the validation of the rotor operatingconditions, namely the angle at the rotorentrance, and find out which configurationprovides more benefit to the turbine.

The rotor used for this calculation, aswell as the respective meshes, are reported[10]. This rotor was optimized to maximizethe total-static efficiency for the design con-dition α1 = 30o, the same angle used toanalyze the turbine stators. All the 5 opti-mized stator configurations were simulatedwith this rotor.

The computational domain is definedby: 2 stators (inlet and exit) and 1 rotor,assuming periodic boundaries (see Fig. 14).The periodicity of the rotor and stators do-mains are different due to number of blades(11 rotor blades to 57 circumferential sta-tor blade sets). The Mixing Plane model[11] was then used to compute the quanti-ties between domains interfaces. Two rotorconnection block were used to avoid intensegradients at the mixing plane. A movingreference frame [11] was used to computethe flow through the rotor.

The rotor rotation speed Ω was set to538.4 rpm, which respects the full scaleReynolds number of Re = 2.02 × 106 fora rotor diameter of 1.5 m [12] and turbineflow rate Q = 37.1 m3/s. The atmosphericstatic pressure is imposed as boundary con-dition at the exit surface of the domain.All computations are performed assumingsteady incompressible flow.

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Inlet stator

Outlet stator

Rotor blade

Rotor connection block

Rotor

Periodic surface

1st cascade 2nd cascade

Figure 14: Turbine computational domain.

7.2. ResultsAs expected, Fig. 15 shows that bound-

ary layer on the nozzle walls has a signifi-cant effect on the axial distribution of theangle of the absolute flow at the rotor entry.Nevertheless the results indicate that flowincidence to the rotor blades is small in theapproximately 60% of the channel height.

50 45 40 35 30 25 α1 [ ]

0%

10%

20%

30%

40%

50%

Channelheight

Figure 15: Profile of angle α1, for the system of 2cascades optimized for α2

∗ = 90o.

The turbine total-static efficiency η andthe dimensionless flow coefficient Φ plottedin Figs. 16-17 are defined as

η = P

Q∆p , (6)

Φ = Q

ΩD3 . (7)

Here, Q is the volume flow rate, P is theturbine power output (P = ΩT , T as the

turbine torque), Ω is the rotation speed, Dis the rotor diameter and ∆p is the pressurehead (difference between the turbine inlettotal pressure and the exit average staticpressure).

0.0 0.1 0.2 0.3 0.4 0.5Φ

50%

55%

60%

65%

70%

75%

80%

η

ZR =1 ZR =2 ZR =3

Figure 16: Turbine efficiency versus dimensionlessflow rate, for systems of 1, 2 and 3 cascades opti-mized for α2

∗ = 90o.

0.0 0.1 0.2 0.3 0.4 0.5Φ

50%

55%

60%

65%

70%

75%

80%

η

α2∗ = 65

α2∗ = 90

α2∗ = 115

Figure 17: Turbine efficiency versus dimensionlessflow rate, for systems of 2 cascades optimized forα2

∗ = 90± 25o.

For all configurations, the peak efficien-cies are very close to the rotor design con-dition (Φ = 0.19). The large efficiency dropverified for smaller flow rate coefficients iscaused by the effect of high negative flowincidence to the rotor blades, and due tothe large misalignment between the flow di-rection and the downstream stator blades(see Fig.18). The smooth efficiency dropfor the higher flow rate coefficients is causedby the combined effect of the positive flow

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incidence to the rotor blades, and the pres-sure loss in the downstream stator, which isproportional to the square of the flow ratecoefficient.

0.0 0.1 0.2 0.3 0.4 0.5

Φ40

60

80

100

120

140

α2[]

ZR =1; α2∗= 90

ZR =2; α2∗= 90

ZR =3; α2∗= 90

ZR =2; α2∗= 65

ZR =2; α2∗= 115

Figure 18: Rotor exit angle versus dimensionlessflow rate, for all configurations.

8. Conclusions

A new stator blades design was pre-sented for the biradial turbine. Several sta-tor configurations were tested in order toevaluate the effect of the number of cas-cades.

The implementation of multiple cas-cades increased the turbine performancedue to the reduction of the downstream sta-tor flow obstruction. Results of numericalflow calculations indicate a total-static peakefficient of 76% for the ZR = 2 configura-tion, optimized for α2

∗ = 65o. This may becompared with the corresponding numericalresults for the axially movable guide-vanesbiradial turbine [12], which gave 83%.

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