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 you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.

Downloaded from orbit.dtu.dk on: Jul 20, 2020

A computationally efficient method for determining the aerodynamic performance ofkites for wind energy applications

Gaunaa, Mac; Paralta Carqueija, Pedro Filipe; Réthoré, Pierre-Elouan Mikael; Sørensen, Niels N.

Published in:Proceedings

Publication date:2011

Document VersionPublisher's PDF, also known as Version of record

Link back to DTU Orbit

Citation (APA):Gaunaa, M., Paralta Carqueija, P. F., Réthoré, P-E. M., & Sørensen, N. N. (2011). A computationally efficientmethod for determining the aerodynamic performance of kites for wind energy applications. In ProceedingsEuropean Wind Energy Association (EWEA).

A computationally efficient method for determining theaerodynamic performance of kites for wind energy applications

Mac Gaunaa, Pedro Filipe Paralta Carqueija, Pierre-Elouan Réthoré and Niels N. SørensenWind Energy Department, Risø National Laboratory, DTU, DK-4000 Roskilde, Denmark

macg@risoe.dtu.dk

Abstract

A new computationally efficient method for de-termination of the aerodynamic performance ofkites is proposed in this paper. The model isbased on an iterative coupling between a Vor-tex Element Method (VLM) and 2D sectionalairfoil coefficients to introduce the effect of air-foil thickness and effects of viscosity while re-taining the strength of the VLM to model phys-ically correct the effect of low aspect ratio andhighly non-planar configurations. The perfor-mance of the new method will be assessedby comparison with simulation results from thestate of the art incompressible Reynolds Aver-aged Navier-Stokes (RANS) solver EllipSys3Don a simplified kite-like geometry designedfrom lifting line theory.

1 Introduction

Several ideas of using kites as possible powersources emerged around the 70s and gradu-ally along the years, but only until recently theresearch was intensified in this topic. Projectslike the MS Beluga Skysails [1], where a con-tainer cargo ship sailed the Baltic Sea with thehelp of a secondary propulsion system consist-ing of a 160 m2 kite harvesting the energy inthe wind and reducing, therefore, the fuel con-sumption; or stationary systems for electricitygeneration, such as the one presented origi-nally by Loyd [2]. As an alternative to conven-tional wind turbines, the use of kites for har-vesting power from the wind is a topic for sev-eral research groups [3–7]. With a kite it ispossible to increase the line traction force byat least an order of magnitude compared tothe steady case by making the kite performcrosswind motion. One way to harvest the en-ergy in the wind using a kite is to generateelectricity by letting a looping kite unroll a linefrom a drum connected to a generator. At theend of the production stroke, the kite is woundback to its initial position in a low traction force

mode, from where the cycle can be repeated.The kite power generation technology is stillin its infancy, and many open questions exist.Presently, it is not possible to give a realisticdetermination of either the power production oreconomic potential of such a system becausethere are too many unanswered questions onhow to implement the basic ideas in real life.Critical key issues that have been spurred in-terest in academia include control [3, 5, 7–9], wind resources at high altitude [10], criticalparameters for the mechanical energy outputavailable [2, 4, 6, 11]. One of the areas wherework is needed to get closer to be able to as-sess the potential of a kite power system is onthe specific aerodynamic behavior/efficiency ofthe kites, including the effect of different keydesign features of the kite. Due to the rela-tively large number of inflow conditions (angleof attack and sideslip) and kite deformations(control actions and elastic deformations) thathave to be considered in such an investiga-tion, standard Computational Fluid Dynamics(CFD) methods such as Reynolds AveragedNavier-Stokes (RANS) modeling is too compu-tationally costly. Therefore, the present workdescribe a new aerodynamic model based ona coupling between a Vortex Element Method(VLM) and 2D sectional airfoil coefficients to in-troduce the effect of airfoil thickness and ef-fects of viscosity while retaining the strengthof the VLM to model physically correct the ef-fect of low aspect ratio and highly non-planarconfigurations. The performance of the newmethod will be assessed by comparison withsimulation results from the state of the artincompressible RANS solver EllipSys3D [12–15] on a reference kite-like geometry designedusing key results from classic lifting line the-ory [16]. Note that a big part of the presentwork was developed in connection with withthe master thesis project by Carqueija [17], inwhich many of the present results can also befound.

2 Computational models

In this section, the computational models em-ployed in this work are described.

2.1 New coupled prediction toolusing geometry and 2D airfoilcoefficients

The new model described in this work was in-spired by the work of Horsten and Veldhuis[18], where...

Since one of the key elements in the modelis a Vortex Lattice Method (VLM), this methodis first described briefly.

2.1.1 Vortex Lattice Method

A few elements of the present method areworth mentioning but, for a full description ofthe method refer to e.g. [19].

The choice of using a VLM is due to its flex-ibility in incorporating changes of chord andtwist distribution, and also because the VLMmethods perform well at low aspect ratios. Itconsists of a distribution of vortex singularitiesover the discretized mean surface of the body(solutions of Laplace’s equation, as mentionedpreviously) which allow the calculation of liftand induced drag. Thickness is neglected inthis method which, however, incorporate thecamber of the airfoil at each section. The con-dition of flow tangency to the mean surface de-termines the strengths of the vortex singulari-ties.

The vortex singularities which, in the codeused, consist of vortex rings, are placed atthe quarter chord line of each panel. The ad-vantage of such element is on the simple pro-gramming effort that it requires and on the factthat the boundary conditions can be exactlyspecified on the actual surface, which can takemore complex shapes (ideal for kite investiga-tions) [19]. The forces are calculated on eachline element separately by applying the Kutta-Joukowsky theorem and then weighted to eachof the collocation points. The calculation of theforce is preceded by finding (i) the influencecoefficients for each line element and (ii) thecirculation, after solving the linear set of equa-tions specifying the zero normal flow boundarycondition.

In order to reduce the computational effortrequired to calculate the influence coefficientseach time the inflow angle changes, the code

incorporates an algorithm where only the coef-ficients related to the wake are updated, keep-ing constant the ones related to the geom-etry of the body. The change of inflow an-gle is incorporated in the right-hand side ma-trix (RHS), which includes the normal veloc-ity components. For fine discretizations, thisapproach becomes extremely valuable as thecalculation of influence matrices is a time con-suming process.

As it will be presented, the algorithm isbased on local geometric changes to accountfor viscosity. The problem rises due to the factthat, if geometry is changed, new influencematrices have to be computed each time thealgorithm is run. The implementation in thepresent work encircles the problem by artifi-cially changing the geometry of the body foreach iteration of the algorithm. This is doneby keeping the original geometry constant andapplying the correspondent geometric changeat each section by modifying the inflow angleat the section, changing the RHS vector. Fig-ure 1 show that as the aspect ratio of an ellip-tic wing is increased, the results from the VLMmethod tends to the results of Prandtl’s liftingline, which should be applicable for large as-pect ratios.

 

 

Figure 1: VLM method validation againstPrandtl’s classical lifting line results for ellipti-cally loaded wings. Upper: lift coefficient ver-sus aspect ratio. Lower: induced drag coeffi-cient versus aspect ratio.

2.1.2 New coupling algorithm

Several approaches using two-dimensionaldata to account for viscous effects can befound in literature. The inspiration to thepresent algorithm is Horsten and Veldhuis’ [18]formulation for wind tunnel interference cor-rection, which uses the concept of ‘morphed’wings to simulate viscosity along the lifting sur-face.

Figure 2 is needed to explain the present al-gorithm

 Figure 2: Explanation of the coupling algo-rithm’s baseline concept. The Cl is plotted ver-sus α.

The illustration shows the two-dimensionallift and drag coefficient curves for an assumedairfoil. For the effective angle of attack seenby a section - which is dependent of the down-wash created by the trailing wake - the inviscidand viscous lift coefficients can be determined.To account for viscosity, a shift on the initial ef-fective angle is performed so that, now, the air-foil works at an angle which has the same liftcoefficient as the viscous lift coefficient beforethe angle shift.

Cli(αeff∆α) = Clv (αefforiginal) (1)

This angle shift is applied directly on the ini-tial geometry of the lifting surface for the cor-responding section. Now, changing the geom-etry, gives origin to a different loading on thebody so there will be a new downwash valueand, consequently, a new effective angle. Aniteration procedure is required until the con-vergence of the angle shift value. This angleshift will, then, allow to calculate the viscouslift coefficient distribution along the body and,by integration, the total viscous lift coefficient.The total drag coefficient, on the other hand,is found by summing the induced drag coef-ficient for the updated geometry and the two-dimensional drag coefficient taken for the effec-

tive angle seen by each of the original sections(without the angle shift).

The VLM code incorporates two-dimensional viscous considerations onthe solution, after external input of the span-wise drag coefficient vector, Cd. In otherwords, the code integrates along the spanthe two-dimensional Cd vector found with thecoupling algorithm and adds the result to thetotal induced drag.

From thin airfoil theory we have

Csli = Clα · (αseff ) (2)

with,

αseff = α∞ + αstwist − αs0 − αsi (3)

Clα =∂Cl∂α

= 2π (4)

αsi introduced already due to three-dimensional effects. As soon as the mentionedangle shift start to be applied on the liftingsurface’s initial geometry, the section lift co-efficient changes until it converges. This finalvalue, referred here as Csli,final , is expressedby the following

Csli,final = Clα ·(α∞ + αstwist − αs0 − αsi,final −

∆Csl

(αseff,final

)Clα

)(5)

In equation 5, αsi,final is the final induced an-gle, for the converged angle shift given by thelast term of the equation. Subtracting equa-tions 2 and 5 results in,

Csli,original − Csli,final =

Clα ·(αsi,final − αsi,original

)+∆Csl (α∞ + αstwist − αs0 − αsi,final) (6)

= Clα ·(αsi,final − αsi,original

)+∆Csl (α∞ + αstwist − αs0

−αsi,original − (αsi,final − αsi,original))

(7)

Csli,original being the lift coefficient taken fromthe first calculation with no angle shift applied.Reorganizing yields,

∆αsi = αsi,final−αsi,original =Csli,original − Csli,final

2π−∆αs

(8)

∆αs =

∆Csl

(α∞ + αstwist − αsi,original −

(αsi,final − αsi,original

))Clα

(9)

The difference between induced angles, ∆αsi ,in equation 8 can then be determined withoutcomputing the induced angles themselves.

As for the two-dimensional drag coefficients,the values are taken for

Csd = Cd(αefforiginal

)= Cd

(Csli,original

Clα+ αs0

)(10)

From the relations above, the algorithm canbe structured as follows:

1. VLM is called and the Csl,i,original for eachsection along the span is saved. This firstvalue is considered as being the baseline,original value.

2. Initialize ∆αsi as equal to zero

3. Find αseff for each section, from:

αseff =Csli,original

Clα+ αs0 − ∆αsi (11)

4. Calculate ∆Csl = ∆Cl(αseff )

5. Calculate ∆αs, from:

∆αs =∆CslClα

(12)

6. Call VLM with the artificial angle correc-tion, ∆αs, and save for each section alongthe span the new lift coefficient, Csl,i,final

7. Calculate ∆αsi using Equation 8

8. Return to step 3 until convergence

Convergence is controlled through the resid-ual value between the ∆αsi ’s of the two last it-erations. The iteration process runs until thecondition

εs = max(∣∣∆αsi,k − ∆αsi,k−1

∣∣) ≤ 10−3 (13)

is satisfied, for iteration k.For a certain predetermined number of iter-

ations, if the results have not converged, a Su-cessive Over Relaxation method is applied tofasten up convergence.

Several differences can be pointed betweenthe approach in [18] and the present algorithm:

1. Iteration is introduced in the present algo-rithm to ensure that the induced drag isbased on the correct loading.

2. The computation of the induced angles isavoided. As a note, it is known from theorythat the wake should be force free. How-ever, not computing it properly can leadto wrong calculations of the induced dragforce on the body and, consequently, theinduced angles [19]. The potential flowcode used assumes a prescribed wakemodel and, therefore, it is better to not cal-culate induced angles.

3. The two-dimensional form drag coeffi-cients are calculated for the initial effectiveangle corresponding to the original lift dis-tribution, and not to the shifted effectiveangle, as in [18]. The reason is that theactual angle seen by the lifting surface isstill the original effective angle, Cl,original,without any angle shifts. The shift in ge-ometry serves only the purpose to matchthe viscous lift distribution to the inviscidone.

2.2 Computational Fluid Dynam-ics: EllipSys3D

2.2.1 Method

The in-house flow solver EllipSys3D [12–15]is used in all CFD computations presentedin the following. The EllipSys3D code isa multiblock finite volume discretization ofthe incompressible Reynolds-averaged Navier-Stokes (RANS) equations in general curvilin-ear co-ordinates. The code uses a collocatedvariable arrangement, and Rhie/Chow interpo-lation [20] is used to avoid odd/even pressuredecoupling. As the code solves the incom-pressible flow equations, no equation of stateexists for the pressure, and the SIMPLE algo-rithm of Patankar and Spalding [21] is used toenforce the pressure/velocity coupling. The El-lipSys3D code is parallelized with MPI for exe-cution on distributed memory machines, usinga non-overlapping domain decomposition tech-nique. The solution is advanced in time us-ing a second-order iterative time-stepping (ordual time-stepping) method. In each globaltime step the equations are solved in an iter-ative manner, using underrelaxation. First, themomentum equations are used as a predictorto advance the solution in time. At this pointin the computation the flow field will not fulfilthe continuity equation. The rewritten continu-ity equation (the so-called pressure correctionequation) is used as a corrector to make the

predicted flow field satisfy the continuity con-straint. This two-step procedure correspondsto a single subiteration, and the process is re-peated until a convergent solution is obtainedfor the time step. When a convergent solutionis obtained, the variables are updated and thecomputation continues with the next time step.For steady state computations the global timestep is set to infinity and dual time stepping isnot used. This corresponds to the use of localtime stepping. To accelerate the overall algo-rithm, a three-level grid sequence is used inthe steady state computations. The convec-tive terms are discretized using a third-orderupwind scheme, implemented using the de-ferred correction approach first suggested byKhosla and Rubin [22]. In each subiteration,only the normal terms are treated fully implic-itly, while the terms from non-orthogonality andthe variable viscosity terms are treated explic-itly. Thus, when the subiteration process isfinished, all terms are evaluated at the newtime level. The three momentum equationsare solved decoupled using a red/black Gauss-Seidel point solver. The solution of the Pois-son system arising from the pressure correc-tion equation is accelerated using a multigridmethod. In the present work the turbulence inthe boundary layer is modelled by the k − ω

SST model of Menter [23]. The equations forthe turbulence model are solved after the mo-mentum and pressure correction equations inevery subiteration/pseudo time step. In thepresent work, all computations are performedusing a γ − R̃eθ Laminar-turbulent transitionmodel [24].

2.2.2 Mesh

The central part of the blades have a span-wise discretization of the mesh points followinga tangent hyperbolic distribution. The roots andthe tips surfaces of each blades are meshedusing the commercial software Pointwise togenerate the surface fitted domains. The 3Dmesh generation is done with a 3D version ofhypgrid [25] an in-house hyperbolic mesh gen-eration code. Some illustrations of the meshgeneration on mesh are illustrated in Fig.3.

2.2.3 Boundary Conditions

A zero gradient is enforced normal to the outletof the downstream end of the spherical domain

Figure 3: Details of the computational mesh.

where the flow leaves the domain. At the up-stream part of the spherical domain the undis-turbed wind speed is specified. The surface ofthe blades are set as wall (no-slip) boundaryconditions.

3 Reference Kite Geometry

In order to develop a reference geometry withwhich to test the developed code, it was cho-sen to make a shape with the crossectionalsection consisting only of one single airfoiltype. The sectional shape of this is the NACA64-418 section [26]. In order to have a plan-form which performs well in at least one point,a design was based on the classical lifting lineresults of Munk [16]. Munk’s analysis showedthat the solution that leads to the system oftrailed vorticity for which the induction in the di-rection perpendicular to the projection of thewing on the Trefftz plane is proportional to thecosine of the wing angle minimizes the induceddrag.

Since the trailed vorticity is equal to thechange in bound vorticity along the wing it is astraightforward task to determine the inducedvelocities in the direction perpendicular to thetrailed vortex sheet at the Trefftz plane. Dueto the linearity of the problem result in a linear

system which can be written as

~A~Γb = ~Vp (14)

Here, the ~A matrix will depend only on the ge-ometry of the lifting line. ~Γb is a column vectorholding the bound vorticity, and ~Vp is the in-duced velocities in the Trefftz-plane normal tothe intersectional curve. Munks condition forminimum induced drag can be written

~Vp = cos(~Θ)K (15)

where ~Θ is the vector with the inclination of thewing, and K is a constant. Upon combiningEquations (14) and (15), we see that the boundvorticity of the case which minimizes induceddrag for a given lift will be

~Γb = ~A−1

cos(~Θ)K (16)

Here we see that K is simply the scaling factorthat determines the level of the bound circula-tion. We also see that the distribution shapeof it is otherwise constant, given by the shapeof the wing. When the definition of the 2D liftcoefficient

Cl =l

0.5ρV 2∞c

(17)

is combined with the locally lifting part of theJoukowski equation

l = ρV∞Γb (18)

we get an expression for the chordlength onthe wing as

c =2ΓbV∞Cl

(19)

Combining this local expression for thechordlength with the expression for the optimalbound circulation, Equation (16), we can there-fore get the expression for the chordlengths forthe whole wing as

~c =2~A−1

cos(~Θ)

V∞ClK (20)

Again, we see that if we choose a design liftcoefficient, the constant K simply scales thechordlengths on the rotor. This way once thegeometry of the line that defines the span inspace is determined, we are now able to de-termine the distribution of chordlengths alongthat span using Equation (20). The constantK which corresponds to the desired mid chordlength, or wing aspect ratio can then be picked.

OnceK is chosen, the corresponding boundvorticity can be evaluated using Equation (16),

and from this and the layout of the wing liftingline in space, the induced velocities from thetrailed vortices at the location of the lifting linecan be determined. This enables the determi-nation of the direction of the chordlengths inspace using the design angles of attack corre-sponding to the design lift coefficient and theairfoil section.

Using the wing layout procedure describedabove with the elliptic shape (half-ellipse of to-tal with 1 and height 0.4) of the span in spacedepicted in Figure 4. The figure also showchord distribution (mid-chord length of 0.3) andlocal twist of the wing, which has the NACA 64-418 airfoil as crossection.

−0.5 0 0.5

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

Y coordinates

Z c

oord

inat

es

−1 −0.5 0 0.5 1

−0.5

0

0.5

y/(b/2)

Cho

rd ,

c [m

]

−1 −0.5 0 0.5 12

2.5

3

3.5

4

4.5

y/(b/2)

α twis

t [º]

Figure 4: Layout of the reference kite geome-try. Upper: layout of the wing span in space(x-coordinate=default wind direction, is zero).Middle: Chordlength. Lower: Local twist of thewing.

4 Results

CFD computations were carried out on the ref-erence kite at a range of inflow conditions,both pitch angles and sideslip angles. Figure5 show examples of the predicted flowfields,where the complex nature of the flow situationsare visible.

Figure 5: Visualizations of the predicted flow-field around the kite using CFD. Upper andmiddle: αpitch = 120, βsideslip = 00. Lower:αpitch = 00, βsideslip = 80

4.1 Zero sideslip angle

Comparison of the integral computational re-sults obtained with the raw VLM, the new cou-pled method and the CFD results for the ref-erence kite at zero sideslip angle is shown inFigure 6.

It is seen that the performance of the newcoupled model is very good. The model cap-tures the beginning of stall well on lift, and thedrag for which the flow is attached is predictedin very close agreement with the CFD results.The underprediction of the drag does not set inbefore αpitch = 8O. Please bear in mind that

−5 0 5 10 15 200

0.5

1

1.5

αpitch

, [º]

CL [−

]

VLMAlgorithmCFD

−5 0 5 10 15 200

0.02

0.04

0.06

0.08

0.1

0.12

αpitch

, [º]C

D [−

]

VLMAlgorithmCFD

0 0.05 0.1 0.15 0.20

0.5

1

1.5

CD

[−]

CL [−

]

VLMAlgorithmCFD

Figure 6: Comparison of lift and drag coeffi-cients simulated on the reference kite at zerosideslip angle. Upper: CL versus pitch angle.Middle: CD versus pitch angle. Lower: CL ver-sus CD.

this angle is the inclination of the onset flow rel-ative to the design point of the reference kite,so the flow angle relative to the kite is morethan zero when αpitch = 00. Overall the per-formance of the model is found to be very goodfor these cases.

It should be noted that until a time steppingsimulation of a kite system, using for instancetabulated data from the present model, hasbeen performed, it is not clear what the typi-cal operational conditions is for a kite. It is verylikely, however, that situations with stall will beavoided, because the lowered lift to drag val-ues in this case results in much lower kite ve-locities, which again results in lower than opti-mal kite traction forces.

4.2 Sideslip angle

Figure 7 show the performance of the mod-els for cases with sideslip angles different fromzero.

0 5 10 15 200.25

0.3

0.35

0.4

0.45

β, [º]

CL, [

−]

VLMAlgorithmCFD

0 5 10 15 200.01

0.02

0.03

0.04

0.05

0.06

β, [º]

CD

, [−

]

VLMAlgorithmCFD

0.01 0.02 0.03 0.04 0.05 0.060.25

0.3

0.35

0.4

0.45

CD

, [−]

CL, [

−]

VLMAlgorithmCFD

Figure 7: Comparison of lift and drag co-efficients simulated on the reference kite forsideslip angles. Upper: CL versus angle ofattack. Middle: CD versus angle of attack.Lower: CL versus CD.

As in the previous case, the agreement be-tween CFD and the new algorithm in the regionwhere the flow is not stalling is excellent. Asthe sideslip angle is increased the agreementdeteriorates somewhat. As was mentionedpreviously, the typical operational conditions isfor a kite are unknown, but likely to remail at-tached for the majority of the time. Thereforeit is likely that the present model could provideaerodynamic data useful for detailed analysisof kite energy systems.

4.3 Computational time

The ratio between computational times usedfor the same number of cases for the new al-gorithm and the CFD method is approximately1/400.

5 Conclusions and furtherwork

The present report contains description of anew, computationally light, algorithm, whichcan determine the aerodynamic loading on akite for wind energy applications. The modelcouples a Vortex Lattice Method with 2D airfoildata iteratively to take into account effects ofairfoil thickness and effects of viscosity.

The computational time of the new coupledalgorithm is approximately 1/400 of the time ofthe state of the art CFD prediction tool Ellip-Sys3D.

The agreement between the present modeland the CFD results is excellent for caseswhere the flow remains attached over the kite.The agreement deteriorates as the flow entersthe stalled state.

As the typical operational conditions is for akite in a kite power system are unknown, butlikely to remain attached for the majority of thetime, it is likely that the present model can pro-vide aerodynamic data useful for detailed anal-ysis of kite energy systems.

Further work Further work include

• Further validation of the new model withcrossection shapes closer to what is foundon real kites.

• Development of a time stepping tool whichbuilds on a database of results producedwith the present method.

• Investigation of the effect of line drag,control strategies, etc. using a databasewith aerodynamic results from the presentmethod

• Investigations of what a ’good’ kite designis (effect of design choices of the kite lay-out)

• More realistic determination of the powerproduction capabilities of a kite power sys-tem using performance characteristics fora realistic kite simulated using the presentcoupled method

• Investigation of the effect of extremeevents (shear, gusts).

• Fatigue analysis of specific key elementsin a kite power system.

Based on the results presented in the pa-per, the envisaged further work is thereforea detailed investigation of design choices ofthe kite, control system design (sensor, actu-ator, control algorithms for both generator andflight path), line specifications, effect of ex-treme events (shear, gusts) and fatigue analy-sis of specific key parts of the kite energy sys-tem.

Acknowledgements

It is gratefully acknowledged that the work inthis paper was heavily helped by all the inspi-rational discussions we had with the Kitemillguys: Olav Aleksander Bu, Thomas Hårklau,Eric Beaudonnat and Bruno Legaignoux.

References

[1] SkySails. http://www.skysails.info/

[2] M.L. Loyd. Crosswind Kite Power. Energy,Vol 4, 3, p. 106-111. 1980.

[3] P. Williams and B. Lansdorp and W. Ock-els Optimal Crosswind Towing and PowerGeneration with Tethered Kites. Journal ofGuidance, Control and Dynamics, Vol 31,1. 2008.

[4] G.M. Dadd and D.A. Hudson and R.A.Shenoi. Comparison of Two Kite ForceModels with Experiment. Journal of Air-craft. Vol 47, 1. 2010.

[5] M. Canale and L. Fagiano and M. Milanese.Power Kites for Wind Energy Generation:Fast Predictive Control of Tethered Airfoils.IEEE Control Systems Magazine. 2007

[6] I. Argatov and P. Rautakorpi and R. Sil-vennoinen. Estimation of the mechanicalenergy output of the kite wind generator.Renewable Energy. Vol 34, p 1525-1532.2009.

[7] B. Houska and M. Diehl. Optimal Controlfor Power Generating Kites. Proceedings ofthe 45th IEEE Conference on Decision andControl. p. 2693-2697. 2006.

[8] P. Williams and D. Sgarioto and P.M.Trivailo Constrained Path-planning for anAerial-Towed Cable System. EuropeanConference for Aerospace Sciences (EU-CASS). 2006

[9] A.R. Podgaets and W.J. Ockels. Three-dimensional simulation of a Laddermill.Proceedings of the 3. Asian Wind PowerConference. 2006.

[10] C.L. Archer and K. Caldeira Global As-sessment of High-Altitude Wind Power. En-ergies. Vol 2, p. 307-319. 2009.

[11] I. Argatov and R. Silvennoinen. Energyconversion efficiency of the pumping kitewind generator. Renewable Energy. Vol.35. p 1052-1060. 2009.

[12] Michelsen J.A. Basis3D - a Platform forDevelopment of Multiblock PDE Solvers.Technical Report AFM 92-05, TechnicalUniversity of Denmark, 1992

[13] Michelsen J.A. Block structured Multigridsolution of 2D and 3D elliptic PDE’s. Tech-nical Report AFM 94-06, Technical Univer-sity of Denmark, 1994

[14] Sørensen N.N. General Purpose FlowSolver Applied to Flow over Hills. Risø-R-827-(EN), Risø National Laboratory,Roskilde, Denmark, June 1995

[15] Sørensen N.N., Michelsen J.A., Schreck,S. Navier-Stokes predictions of the NRELphase VI rotor in the NASA Ames 80 ft X120 ft wind tunnel. Wind Energy. 2002;5(2-3):151-169.

[16] M.M. Munk. The Minimum Induced Dragof Aerofoils. NACA Technical Report R-121. 1923.

[17] P.F.P. Carqueija. Aerodynamic Investiga-tions of a High-altitude Wind Energy Ex-traction System. MSc Thesis. Instituto Su-perior Tecnico, Universidade Tecnica deLisboa. October 2010.

[18] B.J.C. Horsten and L.L.M. Veldhuis A NewHybrid Method to Correct for Wind Tun-nel Wall and Support Interference On-line.World Academy of Science, Engineeringand Technology, Vol 58. 2009.

[19] J. Katz and A. Plotkin Low-Speed Aerody-namics - From Wing Theory to Panel Meth-ods. Second Edition. Cambridge UniversityPress. 2001.

[20] Rhie, C.M. A numerical study of the flowpast an isolated airfoil with separation. PhDthesis, Univ. of Illinois, Urbana-Champaign,1981.

[21] Patankar, S.V. and Spalding, D.B., A Cal-culation Prodedure for Heat, Mass andMomentum Transfer in Three-DimensionalParabolic Flows. Int. J. Heat Mass Transfer,15:1787, 1972.

[22] Khosla, P.K. & Rubin, S.G. A diago-nally dominant second-order accurate im-plicit scheme. Computers Fluids, 2:207–209, 1974.

[23] Menter, F.R. Zonal Two Equation k-ω Tur-bulence Models for Aerodynamic Flows.AIAA Paper 93-2906, 1993

[24] Sørensen, N.N. Cfd modelling of laminar-turbulent transition for airfoils and rotors us-ing the gamma-(re)over-tilde (theta) model.Wind Energy, 12(8):715–733, 2009.

[25] Sørensen, N.N., HypGrid2D, a 2-Dmesh generator. Technical report, Risø-R-1035(EN), Risoe National Laboratory,1998.

[26] Abbott, I.H. & von Doenhoff, A.E. Theoryof Wing Sections, including a summary ofairfoil data Dover, 1949.