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Aerodynamics of a rigid curved kite wing

Authors:
  • Kitenergy Srl

Abstract and Figures

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 3x10^6). Two angles of incidence -- a standard incidence for the flight of a tethered airfoil (6{\deg}) and an incidence close to the stall (18{\deg}) -- 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.
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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 ×105poise
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 coecient CL
Section drag coecient 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 coecient CL
Section drag coecient 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 coecient 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 coecient CL
Drag coecient 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=pp/(1/2ρU2
), where the suf-
fix means the asymptotic upstream condition, ρis the
density and Uthe 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.
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13
... As a matter of fact, studies on kite have increased significantly during the last decade. The literature provides numerous articles that started to treat flight dynamics [11,23] flight control [9], structure deformation [4] or aerodynamic forces modeling [16,17,24]. One of the first studies on kites and their ability to produce energy was achieved in 1980 [15]. ...
... Dadd et al. [6] enhanced 2D airfoil predictions by taking into account the three-dimensional effects with the Prandtl formula for an elliptical wing, while Naaijen and Koster [18] enhanced it with the classical lifting line method. Direct calculations on a 3D geometry were also performed either under inviscid flow assumption [3,5,10] or through Navier-Stokes simulations as performed by Maneia [16], Maneia et al. [17] or Wachter [24]. For such calculations, the geometry of the kite is typically a defined reference shape [16] or measured by wind tunnel experiments [24]. ...
Chapter
Full-text available
The use of kites for auxiliary propulsion reduces oil consumption for vessels. But the complexity of the kite numerical simulation induces the development of computationally effi cient models based on lifting line theory to evaluate the aerodynamic characteristics of the kite. The presented 3D lifting line model takes into account the three-dimensional shape of the kite and the viscosity of the fl uid. The proposed model was applied to a F- one Revolt Leading Edge Infl atable kite to predict its lift-to-drag ratio. Finally, this method is in very good agreement with CFD simulations in the case of a paragliding wing, but needs a much smaller computational effort.
... However, this choice is more relevant in more advance stage of the design. These data can be evaluated experimentally [9,16,23] or numerically by means of methods like lifting line [24,25,26], VLM [27]., panel methods or RANSE simulation [23,25,28]. ...
Conference Paper
Full-text available
A Design Of Experiment method was applied combined with a performance prediction program to assess the influence of four design parameters on the propulsive capacity of kites used as auxiliary propulsion for merchant vessels. Those parameters are the lift coefficient, the lift to drag ratio or drag angle, the maximal load bearable by the kite and the ratio of the tether length on the square root of the kite area. These parameters are independent from the kite area and, therefore, they could be used with various kite ranges and types. The maximum wing load parameter is the one that shows the most influence on the propulsive force. Over 50% of the gains obtained through this study are directly attributable to it. Then the ratio of the tether length on the square root of the kite area comes as the second greatest influence factor for true wind angles above 70°. While the drag angle is more influential for the narrower angles. In fact, the most substantial gains are made upwind.
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Chapter
Full-text available
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