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arXiv:1306.4148v1 [physics.flu-dyn] 18 Jun 2013

Aerodynamics of a rigid curved kite wing

G.Maneiaa, C.Tribuzib, D.Tordellac, M.Iovienoc,∗

aSequoia Automation Srl, via XXV Aprile 8, 10023 Chieri (TO), Italy

bNova Analysis Snc, via Biella 72, 10098 Rivoli (TO), Italy

cDipartimento di Ingegneria Meccanica e Aerospaziale, Politecnico di Torino,

corso Duca degli Abruzzi 24, 10129 Torino, Italy

Abstract

A preliminary numerical study on the aerodynamics of a kite wing for high altitude wind power generators is proposed.

Tethered kites are a key element of an innovative wind energy technology, which aims to capture energy from the wind

at higher altitudes than conventional wind towers. We present the results obtained from three-dimensional ﬁnite volume

numerical simulations of the steady air ﬂow past a three-dimensional curved rectangular kite wing (aspect ratio equal to

3.2, Reynolds number equal to 3 ×106). Two angles of incidence – a standard incidence for the ﬂight of a tethered airfoil

(6◦) and an incidence close to the stall (18◦) – were considered. The simulations were performed by solving the Reynolds

Averaged Navier-Stokes ﬂow model using the industrial STAR-CCM+ code. The overall aerodynamic characteristics of

the kite wing were determined and compared to the aerodynamic characteristics of the ﬂat rectangular non twisted wing

with an identical aspect ratio and section (Clark Y proﬁle). The boundary layer of both the curved and the ﬂat wings

was considered to be turbulent throughout. It was observed that the curvature induces only a mild deterioration of the

aerodynamics properties. Pressure distributions around diﬀerent sections along the span are also presented, together

with isolines of the average pressure and kinetic energy ﬁelds at a few sections across the wing and the wake. Our results

indicate that the curvature induces a slower spatial decay of the vorticity in the wake, and in particular, inside the wing

tip vortices.

Keywords: kite, wind power, aerodynamics, curved wing, numerical simulation

1. Introduction

Kites have been ﬂown in the sky for several centuries.

Nowadays, they are no longer considered just a toy for

children. The ﬂight of a kite is a complex physical phe-

nomenon and scientists have shown renewed interest in its

dynamics and have investigated new applications or im-

proved the existing ones. Till the recent past, many of such

studies have concerned the use of kites as a tool to acquire

meteorological data or as equipment for extreme sports,

but a new frontier is now appearing: wind energy con-

version. Current wind technology, based on wind towers,

has many limitations in terms of energy production, costs

and environmental impact. In fact, wind turbines not only

impact the surrounding environment with the land usage

of their installation and with the noise generated by their

blades, but the power they can provide is also limited by

the low altitude at which operate, no more than 100-150

m above the ground (see, e.g. [1]) . The possibility of col-

lecting wind energy at high altitudes could bring a ma jor

improvement in the design of next generation wind power

plants. This task can be achieved by using non-powered

ﬂight vehicles such as kites, which can provide a means to

transfer wind energy from higher altitudes, between 500

∗Corresponding author. Email: michele.iovieno@polito.it

and 1000 m above the ground, to a power conversion sys-

tem on the ground by means of tethers (see, e.g., [2]).

The design of such a high-altitude wind power gener-

ator requires a careful aerodynamic design of the kites,

integrated with automated ﬂight control. The mathemat-

ical models of the power system conﬁgurations which have

been proposed up to now – e.g. the laddermill, the yo-yo

and carousel conﬁgurations for wind energy extraction and

the towing conﬁguration for the propulsion of ships [3, 4]

— have focused on the design of non-linear predictive con-

trollers [5, 6, 7, 8, 9, 10, 11, 12]. These controllers aim

to maximize energy generation while preventing the air-

foils from falling to the ground or tethers from tangling.

In these control models, a constant lift coeﬃcient and a

constant drag coeﬃcient have always been considered for

the kite wing. However, the “engine” of such a wind gen-

erator is a power kite and model-based control systems

may not perform well without an accurate representation

of the kite dynamics. Therefore, in order to understand

how kites can convert wind energy into electric energy the

ﬁrst step is to examine the aerodynamic performance of a

kite.

In this paper, we provide a preliminary set of data

concerning the aerodynamics of an arc shaped, rigid, non

twisted and non tethered wing which models an actual kite

Preprint submitted to Renewable Energy -

Figure 1: Curved wing ﬂow schematic and reference system (the direction xis the chord direction); the chords are all parallel.

Figure 2: Schematic of the curved wing and of the two airfoils analysed to validate the numerical code performances.

2

wing. The aspect ratio of the kite wing and the Reynolds

number of the air ﬂow, which gives a measure of the ratio

of inertial forces to viscous forces, we have simulated are

typical of current traction kite applications (aspect ratio

AR = 3.2, Reynolds number based on the chord length

Re = 3 ×106, see e.g. [5]). These data can be used to

improve the design of purpose-built kites for energy ex-

traction and related ﬂight control strategies. They can

also represent a benchmark for comparison with experi-

mentally collected data and with any future unsteady ﬂow

simulations carried out by means of the large eddy simula-

tion method. The present simulations rely on the Reynolds

Averaged Navier-Stokes model (RANS) and were carried

out by using the STAR - CCM+ Computational Fluid Dy-

namics code set up by CD-Adapco [13]. This code solves

the Navier-Stokes equations for an incompressible ﬂuid us-

ing a ﬁnite volumes discretization. In order to verify the

correct setting and ﬂow features of the STAR - CCM+

code, a comparison between the numerical and the labora-

tory characteristics of two archetype two-dimensional air-

foils – the NACA0012 and ClarkY proﬁles – is also shown.

One of the aims of the paper is to obtain a comparison

between the aerodynamics of a ﬂat and a curved twin-set of

rigid non-twisted wings with the same aspect ratio and the

same Reynolds number, which is a topic that has not been

discussed frequently in literature. The comparison is car-

ried out by considering the boundary layer to be turbulent

throughout. This choice slightly penalizes the prediction

of the aerodynamic drag and can be considered a sort of

systematic error we introduce a priori into the analysis to

avoid the insertion of a non rational parametrization of the

three dimensional transition on the kite, caused by the lack

of reliable information about the three dimensional transi-

tion over curved three-dimensional non axial-symmetrical

surfaces.

In the context of kite dynamics, the present study,

even though carried out by numerically simulating the

three-dimensional turbulent viscous ﬂow past a arc-shaped

curved wing, should be considered of a preliminary na-

ture. We did not address stability or optimization as-

pects, which could be considered in future works where

the present physically comprehensive numerical simula-

tions could be joined to stability and/or optimization tech-

niques. It should be noted that, in this contest, a few

papers have instead been published on simpliﬁed aero-

dynamic models of the complete system - i.e. kites and

tethers. For example, stability in a simpliﬁed simulation

where the kite is a ﬂat two-dimensional wing was consid-

ered by Alexander and Stevenson (2001) [14] , the dynam-

ics of circular trajectories of a rigid ﬂat kite was studied

by Stevenson and Alexander (2006) [15] , the optimization

of the twist spanwise distribution - in the limit of a high

wing aspect ratio - was addressed by Jackson (2005) [16]

using the inviscid lifting line theory.

The paper is organized as follows. Next section is dedi-

cated to the numerical simulation methodology. The third

section presents the comparison between the aerodynam-

ics properties of two wing sections, the Clark Y and the

NACA0012 proﬁles, obtained from our simulations and

those obtained from laboratory measurements. A fourth

section is dedicated to the aerodynamics of the kite wing

compared to the equivalent ﬂat wing. We give also infor-

mation about the mean pressure and kinetic energy ﬁelds

along many kite sections and in the wake. The concluding

remarks are in the last section.

2. Numerical method

The STAR-CCM+ industrial CFD code has been used

to carry out the simulations. This code solves the Reynolds

Averaged Navier-Stokes equations for an incompressible

ﬂuid using an unstructured, collocated ﬁnite-volume tech-

nique [17]. The convection contribution to the velocity in-

crement is predicted by an upwind scheme while a centered

spatial discretization of the convection is introduced as a

deferred correction (implicit pressure-correction method,

SIMPLE [18] and SIMPLEC [19] algorithms). The Crank–

Nicholson scheme is used for diﬀusion. The global scheme

is thus second-order in space for steady state ﬂows and,

formally, ﬁrst-order in time dependent ﬂows. This integra-

tion scheme is very stable, which is a necessary condition

for a commercial code.

The problem symmetry, see ﬁgures 1-2, allows us to

consider the inﬁnite half-space by the side of the plane

of symmetry of the wings as the computational domain.

The domain boundaries are located three chord lengths

upstream and six chord lengths downstream from the lead-

ing edge. The upper and lower boundaries are placed at

ﬁve chord lengths each from the leading edge. The lateral

boundary is located at three chord lengths from the wing

tip.

Velocity and pressure are imposed on the domain in-

let and uniformity conditions are imposed on the lateral

boundaries and on the outlet. This includes the symme-

try condition on the symmetry plane. The wing surface

is treated like a rigid, non porous, wall where a no-slip

condition applies.

The grid mesh is composed of an inner layer surround-

ing the wing surface with a thickness that is suitable to

capture the boundary layer. The mesh is particularly re-

ﬁned around the leading and trailing edges of the wing,

while it is coarser on the remaining wing part of the sur-

face. The outer mesh is composed of tetrahedral elements

that become coarser towards the external boundaries of the

computational domain, see ﬁgure 3. The optimal cell den-

sity has been estimated by running several two-dimensional

cases with a progressively increasing number of cells until

a good agreement with laboratory data, obtained from the

existing literature, has been reached. The mesh has been

extended along the wing span, while maintaining a similar

density, until a ﬁnal cell count close to 2.5×106has been

obtained, see ﬁgure 3.

The simulations were carried out using a commonly

employed eddy-viscosity turbulence model, the turbulent

3

Figure 3: On the left: Naca0012 airfoil mesh (Re = 3 ×106). On the right: the half ﬂat wing volume mesh (Clark Y airfoil, Re = 3 ×106,

AR=6)

viscosity transport equation model by Spalart and All-

maras [20]. This model was built using heuristics and

dimensional analysis arguments. The transport equation

is local, which means that the equation at one point does

not depend on the solution at the neighbouring points,

and it includes a non-viscous reduction term that depends

on the distance from the wall. This property makes the

model compatible with grids of any structure. A laminar-

turbulence transition was not imposed, the boundary layer

was considered turbulent throughout. This choice was

motivated, on the one hand, by the desire to avoid the

inclusion of uncertain parameters linked to the as yet un-

known transition dynamics on curved three-dimensional

non axial-symmetrical surfaces, and, on the other, because

of the awareness that the associated overestimation of the

drag coeﬃcients implies results on the safer side.

3. Validation of the numerical results.

The NACA0012 and Clark Y airfoil test cases.

The numerical results produced using the STAR-CCM+

code have been validated through a comparison with re-

sults produced in the laboratory for two important two-

dimensional test cases, the incompressible ﬂows past the

NACA0012 and the Clark Y proﬁles, at various angles of

attack. In order to obtain a set of numerical results that

are consistent with laboratory results in a range of dif-

ferent ﬂow conditions it is necessary to achieve suﬃcient

conﬁdence in the use of the code parameter setting. This

setting is of utmost importance as it speciﬁes the physical

model of the problem under study.

For this reason, we analysed the ﬂow around two wing

sections for which a large amount of laboratory literature is

available. The NACA 0012 is a symmetrical airfoil with a

maximum thickness of 12% of the chord. This is probably

the most extensively studied wing section. The Clark Y

is an asymmetrical proﬁle with a ﬂat bottom. This proﬁle

is the section proposed for the kite wing. An extended

laboratory database was also available for this section.

The ﬂow past the two wing sections has a Reynolds

number Re of 3 ×106. The angle of incidence αis varied

from 0oto 20o. The Re is based on the chord length, c= 1

m, and on the freestream velocity, U∞= 43,86 m/s. The

air density is set to ρ= 1,225 kg/m3, the pressure to p∞=

101325 Pa, the dynamic viscosity to µ= 1,79 ×10−5poise

and the temperature to T∞= 288 K. The boundary layer

is considered turbulent throughout, a condition which is

physically close to the ﬂow conﬁguration experimented for

sections with a non zero surface roughness.

The domain extends three chords in front of the pro-

ﬁle, six chords downstream from the leading edge and ten

chords in the transversal direction. The mesh consist of

30105 cells. The mesh is reﬁned through prism layers in

the proximity of the airfoil. Approaching the stall con-

dition, the number of iterations has been increased, from

about 1000 to about 1800, to grant solution convergence.

This criterion was also applied to the three-dimensional

simulations, where the number of iterations necessary to

converge resulted to be of the same order of magnitude.

The results concerning the polar curve for the NACA0012

airfoil were compared with the experimental measures re-

ported by Abbott & Doenhoﬀ [21]. Figure 4 shows the

4

0

0.004

0.008

0.012

0.016

0.02

0.024

0.028

0.032

0.036

-1.6 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 1.6

Section lift coeﬃcient CL

Section drag coeﬃcient CD

Exp. Re =3×106[11]

Exp. Re =6×106[11]

Exp. Re =6×106,

NACA 0012 airfoil

standard roughness [11]

present simulation,

Re =3×106

0

0.004

0.008

0.012

0.016

0.02

0.024

0.028

0.032

0.036

-0.8 -0.4 0 0.4 0.8 1.2 1.6 2

Section lift coeﬃcient CL

Section drag coeﬃcient CD

Exp. Re =3×106[23]

Exp. Re =8.37 ×106[22]

present simulation,

Re =3×106

Clark Y airfoil

Figure 4: Comparison between the polar curves of the NACA0012

and Clark Y proﬁles obtained from the present numerical simulations

(STAR-CCM+ code with the Spalart-Allmaras turbulence model)

and from laboratory measurements in literature: Abbott and von

Doenhoﬀ [21], Silverstein [23] and Jacobs and Abbott [22].

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

-8 -4 0 4 8 12 16 20 24

Angle of attack α[◦]

Lift coeﬃcient CL

straight wing AR =6

straight wing AR =3.367

kite wing AR =3.197

Re =3×106

Clark Y airfoil

0

0.04

0.08

0.12

0.16

0.2

0.24

0.28

0.32

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Lift coeﬃcient CL

Drag coeﬃcient CD

straight wing AR =6

straight wing AR =3.367

kite wing AR =3.197

Optimum loading

(Jackson 2005 [16])

Re =3×106

Clark Y airfoil

Figure 5: Aerodynamics characteristics of the curved wing contrasted

with the equivalent ﬂat wing. The Reynolds number, based on the

chord length, is equal to 3 ×106. All simulations have been carried

out with the STAR-CCM+ code using the Spalart-Allmaras turbu-

lence model. The optimum loading for a tension kite from [16] is

shown as a reference.

5

numerical prediction of the lift and drag coeﬃcients of

the airfoils compared with the laboratory data. These

coeﬃcients are deﬁned as Cl=L/(ρU 2

∞c/2) and Cd=

D/(ρU 2

∞c/2) where Land Dare the lift and drag forces

per unit length and cis the chord length. As expected,

since the boundary layer was considered turbulent along

the entire proﬁle, the agreement is excellent with the data

relevant to the airfoil with standard roughness. The agree-

ment becomes more qualitative with regards the data rel-

evant to smooth proﬁles. As previously explained, in this

last case, the numerical prediction of the drag coeﬃcient is

biased. It can in fact be observed that the setting for the

boundary layer of a turbulence ubiquity condition induces

an over-estimation of the drag of about a 40%. However,

it is important to note that this bias does not spoil the

parallelism between the laboratory and numerical polar

curves.

For the Clark Y airfoil, the numerical prediction of the

aerodynamic characteristics was compared with the wind

tunnel measurement carried out in the Langley variable-

density tunnel (Re = 8.37 ×106, [22]) and in the full-scale

tunnel (Re = 3 ×106, [23]). It should be noted that, for

the Clark Y proﬁle, data for rough surfaces are not present

in literature. Even though old, we decided to consider

these experimental data because they were not produced

in a low-turbulence tunnel. As a consequence, in princi-

ple, these data can oﬀer closer a priori estimate of the

ﬂow conﬁguration over a proﬁle where the boundary layer

is turbulent throughout. Fig ure 4 shows a comparison

between the numerical and laboratory experiments. As

above, it can be seen that the numerical polar is parallel to

the laboratory one. Again in this case, the numerical polar

over-predicts the drag coeﬃcient by about a 40÷50%.

In conclusion, we can observe a good trend of the polar

curves, which are parallel to the experimental ones. The

drag is over-estimated, as it should be, since we decided

not to enlarge this preliminary study of the kite wing aero-

dynamics with other parameters to ﬁx the transition line

on the wings. The drag over-estimation should therefore

be considered a consistent result.

4. Aerodynamics of the curved kite wing and mod-

iﬁcation of the ﬂow past the wing. Comparison

with the equivalent ﬂat wing

We present two main sets of results. The ﬁrst is asso-

ciated to the polar curves CL=CL(α), CD=CD(α) and

the pressure distribution on the upper and lower surfaces

of both the ﬂat and curved wings. These data specify the

aerodynamic characteristics of the two kinds of wings. We

adopted the Clark Y airfoil as wing section for both the

curved kite wing and the straight wing. The aerodynamic

lift and drag coeﬃcients are formed using the reference

force 1/2ρSU2

∞, where Sis half the net surface (or the

surface formed by the set of wing chords). The Reynolds

number, based on the chord length and the free stream air

velocity, is ﬁxed and is equal to 3 ×106. The polar curves

are shown in ﬁgure 5, while the pressure distribution in

part (a) - the ﬂat wing, and part (b) - the curved wing -

are shown in ﬁgures 6 and 7.

As explained before, see the introduction and following

sections, the drag coeﬃcients are biased (overestimated by

about 40-50 %)) due to the turbulence ubiquity ﬂow con-

ditions that were adopted in the boundary layer in the

absence of a reliable criterion to approximate the three-

dimensional separation transition on the curved surfaces

of the kite wing. The polars were obtained by varying the

angle of attack in the [−8oto 20o] range. Figure 5 shows

the polar curves for two ﬂat wings with aspect ratios equal

to 3.37 and 6, and for the kite wing with an aspect ratio

of 3.2. It can be observed that the characteristics are very

close in the angle of attack range [−8o,4o], to which a

range of lift coeﬃcients from −0.2 to 0.6 corresponds. As

expected, the slope of the lift coeﬃcient curve deteriorates

(decreases) a little moving from the ﬂat wing with AR = 6

to the ﬂat wing with AR = 3.37, and from the latter to the

curved wing with AR = 3.37. Beyond an angle of attack

of about 4o, the lift coeﬃcient curve for the curved wing

starts to bend. Stall is reached at an angle of attack of

18o, where CL= 1.1 and CD= 0.18. It should be noted

that the ﬂat wings have not stalled yet at 18oof incidence.

An analogous behaviour is shown by the CD=CD(α) po-

lar curve, which deteriorates by reducing the aspect ratio

and by switching from the ﬂat to the curved conﬁguration.

This curve contains the information on the aerodynamic

eﬃciency. For the ﬂat wing with AR = 6, the maximum

eﬃciency is 40. This value reduces to 26 for both the ﬂat

and the curved wings (AR ∼3).

The set of pressure distributions on the lower and up-

per wing surfaces also describes the lift distribution on the

wing. The pressure is normalized in the form of pressure

coeﬃcients Cp(Cp=p−p∞/(1/2ρU2

∞), where the suf-

ﬁx ∞means the asymptotic upstream condition, ρis the

density and U∞the free stream speed). The two wing dis-

tributions are compared in ﬁgures 6 and 7 at α= 6oand

α= 18◦, respectively. The distributions agree with the be-

haviour observed in the laboratory on a two-dimensional

wing with a Clark Y section [24]. In particular, with the

exclusion of the wing tip region, a slightly negative pres-

sure value is observed at the trailing edge, which is a typ-

ical feature of the Clark Y proﬁle. The position along the

span on the curved and on the ﬂat wing is measured in

terms of the angle θand of the value of the zcoordinate,

respectively in ﬁgures 6 and 7, see ﬁgures 1 and 2. The

correspondence is made in such a way that the correspond-

ing sections have the same value as the curvilinear (for the

curved wing) or rectilinear (for the ﬂat wing) coordinates

running along the wing span. It can be noted that, at the

lower angle of attack of 6◦, see Figure 6, the ﬂat wing and

the curved wing show similar pressure distributions at the

corresponding sections. The distributions are almost equal

on the lower surface. On the higher surfaces, but only in

the region near the leading edge, a 10 −15% of diﬀerence

is noted in the central part of the wing, a value which in-

creases to 40-50% in the wing tip region. The same trend

6

Figure 6: Pressure coeﬃcient distributions at diﬀerent z=const sections along the ﬂat wing (part a). Pressure coeﬃcient distributions at

diﬀerent θ=const sections along the curved wing (part b). The angle of incidence is 6o;Re = 3 ×106based on the chord length, AR = 3.2.

7

Figure 7: Pressure coeﬃcient distributions at diﬀerent z=const sections along the ﬂat wing (part a). Pressure coeﬃcient distributions at

diﬀerent θ=const sections along the curved wing (part b). The angle of incidence is 18o;Re = 3 ×106based on the chord length, AR = 3.2

8

can be observed at the angle of attack of 18◦, see ﬁgure 7.

However, the diﬀerence in the pressure values at the lead-

ing edge now rises to values close to 15% in the central

part of the wing and close to 100% in the wing tip region,

see also the ﬂow visualization in ﬁgures 12 and 13.

The second set of results concern information on the

structure of the ﬂow ﬁeld, in particular, on the pressure

and kinetic energy ﬁelds, see ﬁgures 8 - 13. The pressure

ﬁeld visualization in ﬁgures 8 and 9 shows that the pressure

distributions are qualitatively similar for the two kinds of

wings. The pressure drop above the wing and inside the

wing tip vortices is more intense at the higher angle of at-

tack. At a three chord length behind the leading edge, for

both angles of attack, the pressure becomes almost uni-

form, with the exclusion of the traces of the tip vortices.

A similar overall behaviour can also be observed for the

kinetic energy ﬁeld, see ﬁgures 10 - 11. Most of the av-

eraged kinetic energy is concentrated above the wing and

just outside the wing tip vortex cores. Above the curved

wing, for both angles of attack, a comparatively small ki-

netic energy, with regards to the ﬂat conﬁguration, can be

seen. It means that, in this section (3/4 of the chord), the

ﬂow on the curved wing has already separated, a fact that

can be also deduced from the pressure distribution near

the wing ends in ﬁgures 6 and 7.

Furthermore, if we consider the vorticity in the wake

and, in particular, that in the tip vortices, interesting ob-

servations can be made. For instance, in the 3/4 chord

section, the vorticity produced by the ﬂat wing is 1.3 times

that produced by the curved wing, at α= 6o, a value that

decreases to 1.075 at α= 18o. However, if we compare

the vorticity of the wing tips at 6oto that at 18o, a ra-

tio of 0.78 can be observed for the ﬂat wing and of 0.64

for the curved wing. By moving downstream the section

at a 3 chord lengths the vorticity produced by the curved

wing becomes slightly higher than that produced by the

ﬂat wing (ﬂat/curved yields ratios of 0.99 at 6oand 0.85

at 18o). This is in agreement with the fact the kinetic en-

ergy at 3 chords downstream is about 22% of that on the

wing for the ﬂat conﬁguration, and is about 28% for the

curved conﬁguration. These observations mean that the

curvature is less eﬃcient in increasing the vorticity than

the increase of angle of incidence, but it is capable, due

to non-linear convective eﬀects, of inducing slower spatial

decay in the near wake.

Another interesting point is that, by changing the an-

gle of incidence, the convergence of the tip vortex axes

remains almost constant (2.1o) for the ﬂat wing, while it

increases for the curved wing (1.9oat 6oof angle of at-

tack, 2.5oat 18oof angle of attack). This can also be

seen also in ﬁgures 12 and 13 observing the visualization

of the streamlines of the tip vortices. Thus, it can be con-

cluded that the curvature induces more intense non linear

eﬀects (convection and stretching) on the vorticity and, as

a consequence the vortices keep their identity for longer

distances.

We conclude this section by citing the work of Jackson

[16], where it can be noted that a possible design point for

a kite made by a ﬂexible membrane, like the ones available

on the market, could be CL= 0.55 and CD= 0.1. This op-

timized result has been obtained under several hypothesis:

inviscid ﬂow, lifting line theory (asymptotically accurate

for large aspect ratios) and last, but not least, the need

to maintain a constant tension in the kite canopy. In our

case, a rigid wing has been considered, showing that light

but stiﬀ structures, such as, for instance, inﬂatable wings,

could be preferably employed to obtain a higher eﬃciency.

In fact, from the polar curve in ﬁgure 4, it can be seen that

the curved wing at the same lift coeﬃcient CL= 0.55 has

a deﬁnitely lower drag coeﬃcient CD= 0.04. However, the

design of similar structures requires the analysis of com-

plex ﬂuid structure interactions due to the small bending

stiﬀness of the wing and the huge deformations that occur

under aerodynamic loads. This kind of analysis was not

the aim of the present work.

5. Conclusions

In this work we have compared the aerodynamics of

two rigid non twisted non tethered wings that are alike in

all aspects (i.e. shape and proﬁle section, aspect ratio and

Reynolds number) but which diﬀer in their curvature: an

arc shaped curved wing which models a kite wing and ref-

erence straight wing. Given the lack of information on the

transition on curved wings, we carried out a comparison

between the ﬂat and curved conﬁguration by modelling

the boundary layer on the wings as turbulent from the

leading edge. The results were obtained through the com-

putation of the numerical solutions of the Reynolds aver-

aged Navier-Stokes equations (STAR-CCM+ code) for the

mean ﬂow. We observed a slight deterioration of the over-

all aerodynamic performances of the curved wing (non-

tethered kite) with respect to the ﬂat conﬁguration. To-

wards the wing tips, the lift on the curved wing was com-

paratively lower than that of the ﬂat wing, due to a more

extended separation region above the airfoil.

A non trivial behaviour was observed in the vorticity

dynamics in the near wake, up to ﬁve chords downstream

from the trailing edge. The curved wing did not generate

more intense wing tip vortices than the ﬂat wing, how-

ever, the downstream decay in the near wake was slower.

At the higher angle of incidence, α= 18o, the curved wing

induces a higher convergence of the wing ends-vortices.

The ﬂat wing instead maintained a constant convergence.

Such information could be useful for the design of a system

conﬁguration where a set of kites ﬂy under mutual inter-

ference (in a ladder or carousel conﬁguration), as proposed

for wind generators systems.

Moreover, the data outlined in this study have three

other implications. Firstly, they can be a ﬁrst step for

more advanced, unsteady simulations, namely large eddy

simulation of the ﬂow ﬁeld near the wing and inside the

wake. Furthermore, they could be used by designers of

9

Figure 8: Pressure levels in ﬂow sections (y, z) across the wing and the wake at 3/4, 1, 2 and 3 chord lengths. Part a) ﬂat wing. Part b)

curved wing. The angle of incidence is 6o;Re = 3 ×106based on the chord length, AR = 3.2, p∞= 101325 Pa.

Figure 9: Pressure levels in ﬂow sections (y, z) across the wing and the wake at 3/4, 1, 2 and 3 chord lengths. Part a) ﬂat wing. Part b)

curved wing. The angle of incidence is 18o;Re = 3 ×106based on the chord length, AR = 3.2, p∞= 101325 Pa.

10

Figure 10: Averaged kinetic energy levels in ﬂow sections (y, z ) across the wing and the wake at 3/4, 1, 2 and 3 chord lengths downstream

from the leading edge. Part a) ﬂat wing. Part b) curved wing. The angle of incidence is 6o;Re = 3 ×106based on the chord length,

AR = 3.2, E∞= 247.45 Pa.

Figure 11: Averaged kinetic energy levels in ﬂow sections (y, z ) across the wing and the wake at 3/4, 1, 2 and 3 chord lengths downstream

from the leading edge. Part a) ﬂat wing. Part b) curved wing. The angle of incidence is 18o;Re = 3 ×106based on the chord length,

AR = 3.2; E∞= 247.45 Pa.

11

Figure 12: Flow streamlines for the curved wing at an incidence of 6o; the streamline visualization is associated to the pressure levels obtained

on the wing surface, the symmetry plane and outlet boundary. Re = 3 ×106based on the chord length, AR = 3.2. Note that the xdirection

is the chord direction, see ﬁgure 1, and it is parallel to the lateral domain boundaries.

Figure 13: Flow streamlines for the curved wing at an incidence of 18o; the streamline visualization is associated to the pressure levels

obtained on the wing surface, the symmetry plane and outlet boundary. Re = 3 ×106based on the chord length, AR = 3.2. Note that the x

direction is the chord direction, see ﬁgure 1, and it is parallel to the lateral domain boundaries.

12

kites for wind power plants to improve the set up of au-

tomated non linear control systems. Lastly, they could

represent a basic mean ﬂow to be used as an equilibrium

starting point for perturbative stability analysis.

References

[1] D.J.Milborrow, “Wind energy a technology that is still evolv-

ing”, Journal of Power and Energy 225, 539–547, 2011.

[2] J.Breukels, W.J.Ockels,“Past, present and future of kites and

energy generation”, PES 2007 Clearwater, FL, USA, 2007.

[3] SkySails GmbH & Co., 2010: http://www.skysails.info

[4] L.Fagiano, M.Milanese, V.Razza, M.Binansone “High-Altitude

Wind Energy for Sustainable Marine Transportation”, IEE

Transaction on Intel ligent Transportation Systems,13(2), 781–

791, 2012

[5] M.Canale, L.Fagiano, M.Milanese, “Power kites for wind energy

generation.”, IEEE Control System Magazine,27, (6), 25–38,

2007.

[6] M.Canale, L.Fagiano, M.Milanese, “High Altitude Wind Energy

Generation Using Controlled Power Kites”, IEEE Transactions

on Control Systems Technology 18(2), 279–293, 2010.

[7] L.Fagiano, M.Milanese, D.Piga, “Optimization of airborne wind

energy generators”, International Journal of Robust and Non-

linear Control 22(18), 2055–2084, 2012.

[8] L.Fagiano, M.Milanese, D.Piga, “High-Altitude Wind Power

Generation”, IEEE Transactions On Energy Conversion 25(1),

168–180, 2010.

[9] C.Novara, L.Fagiano, M.Milanese, “Direct Data-Driven Inverse

Control of a Power Kite for High Altitude Wind Energy Conver-

sion”, IEEE Multi-Conference on Systems and Control, Denver,

USA 2011.

[10] A.Ilzhoer, B.Houska, M.Diehl, “Nonlinear MPC of kites un-

der varying wind conditions for a new class of large-scale wind

power generators”, International Journal of Robust and Non-

linear Control,17(17), 1590-1599, 2007;

[11] P.Williams, B.Lansdorp, W.J.Ockels, “Optimal Trajectories for

Tethered Kite Mounted on a Vertical Axis Generator”, AIAA

Modelling and Simulation Technologies Conference and Exhibit,

20 - 23 August 2007, Hilton Head, South Carolina

[12] M.L.Lyod, “Crosswind kite power”, Journal of Energy,4(3),

pp. 106-111, 1980.

[13] CD-Adapco Inc. STAR-CCM+ User Guide, 2007.

[14] K.Alexander, J.Stevenson, “Kite Equilibrium and Bridle

Length”, Aeronautical Journal,105, No. 1051, 535-541. 2001.

[15] J.C.Stevenson, K.V.Alexander, “Circular ﬂight kite tests: con-

verting to standard results”, The Aeronautical Journal,110,

No. 1111, 605–614, 2006.

[16] P.S.Jackson, “Optimum Loading of a Tension Kite”, AIAA

Journal 43, No. 11, 2273–2278, 2005.

[17] J.H.Ferziger, M.Peric, Computational methods for ﬂuid dynam-

ics, Springer, Berlin, 1999.

[18] L.S.Caretto, A.D.Gossman, S.V.Patankar, D.B.Spalding, “Two

Calculation procedures for steady, three-dimensional ﬂows with

circulation”, Proceedings of Third International Conference on

Numerical Methods in Fluid Dynamics, Paris, 1972.

[19] J.P.Van Doormal, G.D.Raithby, “Enhancements of the SIMPLE

method for predicting incompressible ﬂuid ﬂows”, Numerical

Heat Transfer 7, 147–163, 1984.

[20] P.R.Spalart, S.R.Allmaras, “A One Equation Turbulence Model

for Aerodynamic Flows”, La Recherche A´erospatiale,1, 5–21,

1994.

[21] I.H.Abbott, A.E. von Doenhoﬀ, Theory of wing sections: in-

cluding a summary of airfoil data, Dover, New York, 1959.

[22] E.N.Jacobs, I.H.Abbott, “Airfoil section data obtained in the

NACA variable-density tunnel as aﬀected by support interfer-

ence and other corrections”, NACA-TR-669, 1939.

[23] A.Silverstein, “Scale eﬀect on Clark Y airfoil characteristics

from NACA full-scale wind-tunnel tests”, NACA-TR-502, 1935.

[24] C.J.Wenzinger, W.B.Anderson, “Pressure distribution over air-

foils with fowler ﬂaps”, NACA Report no. 620, 1937.

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