AIAA AVIATION Forum
August 2-6, 2021, VIRTUAL EVENT
AIAA AVIATION 2021 FORUM
10.2514/6.2021-2544
Best Winglet of Minimum Induced Drag: Viscous and
Compressible Flow Predictions
Downloaded by NANJING UNIV OF AERONAUTICS & ASTRONAUTICS on November 3, 2021 | http://arc.aiaa.org | DOI: 10.2514/6.2021-2544
Lorenzo Russo∗ , Ettore Saetta† and Renato Tognaccini‡
Department of Industrial Engineering, University of Naples Federico II, Naples 80125, Italy
Giulio Dacome§ and Luciano Demasi¶
San Diego State University, San Diego, California 92182
The aerodynamic performance of a wing mounting a simple unconventional box winglet
are analyzed in detail by CFD simulations. The decomposition of the computed drag in its
irreversible (viscous) and reversible (lift-induced) components by a far field drag breakdown
method allows for the determination of the span efficiency, not a trivial task in case of viscous
flow. The performance are compared with the ones obtained in case of simple standard winglet
and without any tip appendices.
I. Introduction
he world of aviation is expected to become greener and sustainable in the very next future and it is a shared opinion
that, in order to accomplish this goal, the actual air-frame configuration need to be reinvented [1, 2].
In recent years, aeronautical designers have been studying the potential benefit of a drastic change in the aerodynamic
configuration of airplanes. The boundary layer ingestion (BLI) [3–5] and the distributed electric propulsion (DEP)
[6, 7], for example, are two technologies aimed to take advantage from more integrated propulsion systems to reduce the
overall power dissipation in the flowfiled and/or increase the aerodynamic performance in the more demanding mission
phases such as take-off, climb and descend.
Other configurations, named Joined Wings, are based on Prandtl’s idea of the Best Wing System [8] and are able to
drastically reduce the induced drag [9, 10] even in transonic flow condition [11], using non-planar interconnected wing
platforms.
Another valid strategy for reducing induced drag (already used for commercial air transport) is the adoption of the
wingtip devices (winglets). Introduced by Lanchester in 1897, who proposed the application of end plates at the wing
tips for aerodynamic drag reduction of lifting bodies, the potential benefit of winglets on induced drag was theoretically
showed by Negen [12] first and Mangler [13] later. Nevertheless, only with the work of Whitcomb [14] in seventies, it
was recognized the possibility to reduce induced drag without increasing the profile drag with winglets [15–17].
Thanks to the progress in both experimental and computer-aided analyses, a huge number of winglet concepts are
available today: the movable winglets [18], the blended winglet [19], the spiroid winglet [20] and the split winglet [21]
are just a few examples.
Given the interest in the induced drag performance of innovative airplane configurations, a computationally efficient
analytical procedure was introduced in Refs. [22–25]. That approach led to the definition of the Best Winglet Design,
which was shown to be represented [26] by the Box Winglet, which is the subject of the present investigation.
Box Winglets are winglets with a closed rectangular platform when seen from the front view, able to guarantee,
under the assumptions of inviscid flow, linear model, and rigid wake, the best induced drag performance among all the
infinite possible winglets included in the so-called Winglet Fundamental Rectangle (WFR): the rectangle of minimum
area which completely includes the winglet (see fig. 1).
For their potentially exceptional induced drag performance, Box Winglets are very promising concepts. Thus, other
aspects [16] should be addressed, such as weight increase and root bending moment. From a pure aerodynamic point
of view, besides induced drag, other forms of drags need to be investigated. For example, due to the extension of the
wet surface typical of winglets, viscous drag is increased. Moreover, there could also be effects on the wave drag in
T
∗ Post-doc
researcher, Department of Industrial Engineering, University of Naples Federico II, Naples 80125, Italy.
student, Department of Industrial Engineering, University of Naples Federico II, Naples 80125, Italy.
‡ Associate professor, Department of Industrial Engineering, University of Naples Federico II, Naples 80125, Italy. AIAA senior member.
§ Visiting master student
¶ Professor, Department of Aerospace Engineering; ldemasi@sdsu.edu. Lifetime Associate Fellow AIAA.
† Ph.D.
1
Copyright © 2021 by Copyright © 2021 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
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transonic flow conditions. First and foremost the overall drag performance has to be assessed; besides the induced drag,
winglets actually increase the viscous drag, since the extension of the wet surface, and could also effect the wave drag
performance in transonic flow conditions.
The principal objective of present study is to assess the aerodynamic performance of Wing + Box Winglet (WBW)
configuration in real flow. It is very interesting to verify if the some features obtained for the Box Wing are confirmed
for the Box Winglet. Indeed, Russo et al. [11] highlighted two interesting results:
1) the theoretical optimum span efficiency is already obtained in practice by a very simple geometry (constant chord
distribution and no twist);
2) the obtained span efficiency is independent of the flight regime, it is confirmed even in the very high transonic
flow.
The first aspect is in particular analyzed in present paper. The flow around a simple Box Winglet configuration is
analyzed in subsonic flow by Computational Fluid-dynamics (CFD) simulations; the overall aerodynamic performance
is detailed applying the drag breakdown method presented in section III so that a breakdown of the total drag in induced
and viscous contributions can be performed. Finally, a comparison among the aerodynamic performance of three wing
+ winglet configurations is proposed.
II. Remarks on winglet design driven by minimum induced drag condition
Under the assumption of potential flow, the optimal load distribution for a classical cantilevered planar wing is
elliptical [27, 28] and the corresponding minimum induced drag 𝐷 ref
𝑖 is:
𝐷 ref
𝑖 =
2𝐿 2
,
𝜋𝜌∞𝑉∞2 𝑏 2
(1)
where 𝐿 is the lift, 𝜌∞ and 𝑉∞ are respectively the freestrem density and velocity of the flow and 𝑏 is the wingspan.
Referring the aerodynamic forces to 1/2𝜌∞𝑉∞2 𝑆, where 𝑆 is the reference wing surface, the aerodynamic force coefficients
𝐶 𝐿 and 𝐶𝐷 are obtained as follows:
2𝐿
𝐶𝐿 =
,
(2)
𝜌∞𝑉∞2 𝑆
𝐶𝐷 =
2𝐷
,
𝜌∞𝑉∞2 𝑆
(3)
and the well known link between lift and induced drag coefficients for the optimum wing can be derived from Eq. (1):
ref
𝐶𝐷
=
𝑖
𝐶 𝐿2
,
𝜋A
(4)
with the aspect ratio A = 𝑏 2 /𝑆.
For a given wing, the optimal aerodynamic efficiency ratio 𝜀 physically represents the ratio between its aerodynamic
efficiency (defined as 𝐸 = 𝐿/𝐷 with 𝐷 = 𝐷 𝑖 in inviscid case) and the corresponding efficiency of a reference cantilevered
wing with the same wingspan and total lift when both efficiencies are evaluated under their respective optimal conditions
[22]:
𝐸
2𝐿 2
.
(5)
𝜀 = ref =
𝐸
𝜋𝜌∞𝑉∞2 𝑏 2 𝐷 𝑖
In other words 𝜀 is a measure of the relative performance of an optimum generic wing system with respect to the
ws can be computed as
optimum cantilevered wing; therefore, the induced drag coefficient of a generic wing system 𝐶𝐷
𝑖
follow:
2
𝐶𝐿
ws
𝐶𝐷
=
.
(6)
𝑖
𝜋A𝜀
Any wing + winglet configuration can be optimized in the frame of lifting line theory using Munk’s theorems and
applying the calculus of variations techniques described in [25]. The optimization procedure of Ref. [25] led to the
Winglet Primary Properties (WPP), valid for any wing + winglet design [26]:
WPP-1. The Winglet’s Fundamental Rectangle (WFR) is the rectangle of minimum area which completely includes the
winglet. The height ℎW of the rectangle must be in the vertical direction.
2
Standard Winglet
𝑏w
Box Winglet
WFR
Θ
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ℎw
C
𝑏/2
Fig. 1 Sketch of the lifting lines of a Standard Winglet and a Box Winglet with relative Winglet Fundamental
Rectangle (WFR).
WPP-2. The Box Winglet identified by the curve enclosing the Winglet Fundamental Rectangle has the best induced
drag performance among all the infinite possible winglets included in the Winglet Fundamental Rectangle.
This Box Winglet represents the Best Winglet Design. That is, the optimal induced drag (maximum 𝜀) of the
corresponding system is lower than the optimal induced drag of the given system with the original winglet.
WPP-3. Given a generic simply connected closed region Θ, the Winglet of Minimum Induced Drag for a Simply
Connected Region has lifting line represented by curve C, enclosing the region Θ.
WPP-4. Under optimal conditions, a closed winglet presents the same effectiveness of a combination of multiwinglets
with an infinite number of lifting lines and a finite number of closed winglets. The closed winglets must be
tangent to the curve corresponding to the given closed winglet. All the winglets’ tips must be on the curve C of
the corresponding closed winglet.
WPP-5. Assume to have a simply connected region Θ enclosed by curve C and a closed winglet having lifting line C.
Under optimal conditions, any winglet of any shape included in the region Θ will be unloaded, unless it forms
a closed path. In that case the load can be an arbitrary constant.
III. Present Drag breakdown method
𝑆𝑓
V
Σ
𝑆𝐵
𝒆𝑧
𝑧
𝑦
𝒆𝑦
𝑥
𝒆𝑥
W
𝑆𝑠
Fig. 2
Sketch of fluid domain considered for the aerodynamic force analysis.
The standard technique for the aerodynamic force computation is known as near-field method, which relies on the
stress integration over the body surface; this method allows only for the decomposition of the total drag in its pressure
and friction contributions so that, it is impossible to identify the lift-induced, the viscous and the wave drag, if present.
On the contrary, this task is possible by some far-field methods, based on formulae derived from the integral momentum
3
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equation [29].
Far-field techniques have also 2 interesting features:
• they are able to identify the local flow structures responsible for the generation of the aerodynamic force;
• they could improve the accuracy in the calculation of total drag from a given CFD solution by removing at least
part of the so-called spurious drag implicitly or explicitly introduced by the artificial viscosity of the adopted
numerical scheme.
The drag-breakdown researches so far can be grouped into two families: thermodynamic-based integration methods
and vorticity- based force breakdown [30]. The former is based on Oswatitsch’s entropy drag concept [31] and usually
decomposes the velocity into reversible and irreversible parts; the last associated to entropy production. Thermodynamic
methods are widely adopted in industrial and research environment [32, 33] and already contributed to the design of last
generation of transonic transport aircrafts. They are limited to the computation and analysis of the force components of
irreversible nature and therefore cannot compute and analyze lift or lift-induced drag. Vorticity-based methods can
solve this problem using the so-called vortex force; see [34] for a very recent review. The vortex force is defined as the
volume integral in the flow of the Lamb vector (the cross product of vorticity times velocity); introduced by Prandtl, it is
widely discussed by Saffman [35]. Anyway, these methods are less consolidated respect the Thermodynamic ones and
never adopted in industry so far.
A. A thermodynamic method for irreversible drag computation
In present dissertation the thermodynamic method proposed by Paparone and Tognaccini and detailed in [36], is
used. Here, its fundamental formulae are recalled.
A turbulent, steady high-Reynolds number compressible flow around an aircraft configuration is considered.
Assuming a cartesian reference system with the 𝑥 axis aligned with the free-stream velocity, a straightforward application
of the momentum balance equation provides the far field drag expression (see Fig. 2):
∫
𝐷 far = − [𝜌𝑢(𝑽 · 𝒏) + ( 𝑝 − 𝑝 ∞ )𝑛 𝑥 ] dS ,
(7)
Σ
where 𝜌 is the density, 𝑽 =
is the local velocity, 𝑝 is the pressure and subscript ∞ specifies free-stream
conditions. Σ is the outer boundary of the computational domain and 𝒏 = [𝑛 𝑥 , 𝑛 𝑦 , 𝑛 𝑧 ] 𝑇 is the unit normal vector
pointing outside the computational domain. Expanding in Taylor’s series the axial velocity defect expression with
respect to the entropy variation Δ𝑠 = 𝑠 − 𝑠∞ and taking into account for second order terms at most, the entropy drag
expression is obtained:
∫
[𝑢, 𝑣, 𝑤] 𝑇
𝐷 Δ𝑠 = −𝑉∞
∇ · [𝜌𝑔(Δ𝑠)𝑽] dV ,
(8)
2
Δ𝑠
Δ𝑠
+ 𝑓𝑠2
,
𝑅
𝑅
(9)
V
where V is the flow domain and
𝑔(Δ𝑠) = 𝑓𝑠1
with 𝑅 the gas constant and the coefficients 𝑓𝑠1 and 𝑓𝑠2 given by
𝑓𝑠1 = −
1
2
𝛾𝑀∞
,
𝑓𝑠2 = −
2
1 + (𝛾 − 1) 𝑀∞
,
4
2𝛾 2 𝑀∞
(10)
(𝑀∞ is the free-stream Mach number and 𝛾 is the ratio of specific heats).
The entropy drag takes into account for the contributions
with irreversible processes: viscous and wave
Ð associated
Ð
drag. The domain V can be decomposed as V = V𝑣 V𝑤 V𝑠 𝑝 , where V𝑣 is the boundary layer and the wake
region, V𝑤 the shock wave region, and V𝑠 𝑝 the remaining part of the flow field. Therefore, the entropy drag can be
decomposed in three components:
∫
∫
∫
𝐷 𝑣 = 𝑉∞
∇ · [𝜌𝑔(Δ𝑠)𝑽] dV , 𝐷 𝑤 = 𝑉∞
∇ · [𝜌𝑔(Δ𝑠)𝑽] dV , 𝐷 𝑠 𝑝 = 𝑉∞
∇ · [𝜌𝑔(Δ𝑠)𝑽] dV .
V𝑣
V𝑤
V𝑠 𝑝
(11)
𝐷 𝑣 is the viscous drag, 𝐷 𝑤 is the wave drag and 𝐷 𝑠 𝑝 is the spurious drag component.
It is clear that the breakdown method relies on a proper selection of the three domains V𝑣 , V𝑤 and V𝑠 𝑝 . This is
performed by the definition of boundary layer and shock wave sensors discussed in [36, 37].
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B. Maskell’s formula for reversible drag computation
The major limitation of thermodynamic methods is the impossibility to obtain a direct definition of the lift-induced
drag which is not associated to dissipative phenomena.
A first approach to lift-induced drag (the reversible drag) computation consists in subtracting viscous, wave and
spurious contributions to the total (near-field) drag.
Otherwise the thermodynamic method can be applied in junction with the so-called Maskell’s formula that, using
the flow variables in the Trefftz plane W allows for the computation of the induced drag by the following expression:
∫
𝜌
(𝜓𝜁 𝑥 − Φ𝜎) dS ,
𝐷i =
(12)
2 W
where 𝜓 is the stream function, 𝜁 𝑥 is the vorticity component in free-stream direction, Φ is the flow potential and 𝜎 is
the source strength given by:
𝜕𝑣 𝜕𝑤
+
.
(13)
𝜎=
𝜕𝑦 𝜕𝑧
Generally, the contribution of the term containing 𝜎 is negligible respect to the other which means that the trailing
vorticity should fully account for induced drag and that the effect of the stream-wise gradients in the wake should not be
significant. Therefore, an approximated version of Maskell’s formula is:
∫
𝜌
𝜓𝜁 𝑥 dS ,
(14)
𝐷i ≈
2 W
with 𝜓 computed by the integration of the Poisson’s equation ∇2 𝜓 = −𝜁 𝑥 or using the Green’s function method [38].
The growth of the computational power, with the corresponding possibility to adopt very fine grids, makes the
spurious drag contribution less important to identify. However the possibility of the drag breakdown method to separate
the irreversible and reversible drag contributions allows to extract the span efficiency (a reversible property) from
viscous (irreversible) calculations.
IV. Aerodynamic performance of the Best Winglet Design in real flow
In order to gain insights on the aerodynamic performance of the Box Winglet concept, CFD simulations of a viscous
compressible flow with free-stream Reynolds number 𝑅𝑒 ∞ = 107 and free-stream Mach number 𝑀∞ = 0.3 around the
wing system (BW), shown in fig. 3, have been performed. A NACA 0012 wing section with fixed chord (𝑐 BW ) and
no twist along the wing span has been adopted. The wing section is thus rigidly translated and rotated so that a BW
configuration with wing span 𝑏 = 12𝑐 BW , winglet span 𝑏 w = 𝑐 BW and the winglet height ℎw = 2𝑐 BW is realized. The
reference wing surface used for the computation of the aerodynamic force coefficients of eqs. (2) and (3) is given by the
sum of the horizontal surfaces of the configuration: 𝑆 = 14𝑐2BW ; the aspect ratio is A = 10.28. These geometric choices
do not produce the theoretical optimum aerodynamic load presented in Ref. [26], but represents a reasonable model on
the induced drag performance. Moreover, it is justified by the observation that for the case of Box Wing of Ref. [11],
the rigid rotation of the lifting system was sufficiently close to the optimum.
Based on the minimum induced drag theorems presented in [25], if the optimal load distribution is realized, a BW
configuration characterized by a vertical aspect ratio V = ℎw /𝑏 = 1/6 as the one here analyzed, offers an efficiency ratio
𝜀 = 1.38: an induced drag reduction of nearly 40% with respect to a cantilevered elliptical wing with same wingspan
and lift.
It is then of scientific interest to verify if the BW has a similar behavior of the Box Wing (i.e., a rigid rotation
practically produces the minimum induced drag conditions) and the effects of real flow condition and deformable wake.
With that in mind, a CFD model has been carried out using the open-source software SU2 vsn. 7, developed at Stanford
University [39, 40]. Reynolds averaged Navier-Stokes (RANS) equations and Spalart-Allmaras turbulence model have
been used. The results presented in this section refers to a grid with C-C type topology with more than 9 million cells
and 256 cells along the wing section; the far field has been placed 50𝑐 distant from the wing, and 𝑦 + ≈ 1 has been
maintained in the first grid point layer all over the body (see fig. 4). The accuracy of the aerodynamic force calculation
has been checked by a grid convergence analysis and is presented in Appendix A.
The aerodynamic drag breakdown has been performed using an in-house code called BreakForce that implements
eq. (8) for the computation of the irreversible drag and eq. (12) for the reversible component [36]. For the Mach number
and the angles of attack considered no shock wave occurs in the flow field so that the irreversible drag reduces to the
5
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Box Winglet (BW)
No Winglet (NW)
Standard Winglet (SW)
Fig. 3
Views of the wing systems analyzed.
sum of the viscous drag and the spurious drag whereas the reversible drag is the induced drag. An ad-hoc boundary
layer sensor based on the turbulent viscosity and described in [36] has been adopted to partially remove the spurious
drag from the far-field computation.
In fig. 5 the drag polars obtained computing the lift and drag coefficients at four different angles of attack
(𝛼 = 0◦ , 2◦ , 4◦ , 6◦ ) are reported; the values used are collected also in table 1. Both near-field and far-field drag coefficient
values are plotted against the near-field lift coefficients. For each angle of attack a difference of almost twenty drag
counts (𝐶𝐷 × 104 ) has been found due to the presence of a spurious drag contribution. The breakdown between
irreversible and reversible components shows that the viscous drag is nearly constant for 𝛼 = 0◦ , 2◦ , 4◦ and slightly
increased at 𝛼 = 6◦ . On the other hand, the induced drag predicted by Maskell’s formula is near but not overlapped to
the induced drag polar theoretically predicted with the minimum induced drag theorems. This is an indication that, as
opposed to the Box Wing, the optimal load distribution is not realized by a simple BW configuration. Moreover, an
unexpected non-zero value for the induced drag has been found when lift is zero; this aspect needs further investigations.
Nevertheless, a span efficiency higher than one (𝜀 ≈ 1.17) has been found for this simple BW configuration.
6
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Fig. 4
BW configuration at 𝑅𝑒 ∞ = 107 , 𝑀∞ = 0.3 and 𝛼 = 0◦ . 𝑦 + distribution over the wing surface.
𝛼
0◦
2◦
4◦
6◦
𝐶 𝐿near
0
0.1794
0.3578
0.5360
𝐶𝐷near
0.0145
0.0154
0.0182
0.0234
𝐶𝐷far
0.0125
0.0133
0.0159
0.0204
𝐶𝐷irr
0.0122
0.0122
0.0123
0.0129
𝐶𝐷rev
0.0003
0.0011
0.0036
0.0076
𝐶 𝐿2 /(𝜋A𝜖)
0
0.0007
0.0029
0.0064
Table 1 BW configuration at 𝑅𝑒 ∞ = 107 and 𝑀∞ = 0.3. Lift and drag coefficients at different angles of attack.
Drag breakdown based on eq. (8) for the irreversible component and eq. (12) for the reversible contribution.
Theoretical induced drag computed with 𝜀 = 1.38.
7
0.5
0.4
0.3
𝐶𝐿
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0.6
0.2
near
far
irr
rev
0.1
0
𝐶𝐿2
𝜋A 𝜖
0
50
100
𝐶𝐷 ×
150
200
250
104
Fig. 5 BW configuration at 𝑅𝑒 ∞ = 107 and 𝑀∞ = 0.3. Drag polars and breakdown based on eq. (8) for the
irreversible component and eq. (12) for the reversible contribution. Theoretical optimum induced drag (dashed
line) computed with 𝜀 = 1.38.
8
0.6
0.5
0.4
0.3
𝐶𝐿
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A. Box Winglet vs. No Winglet configurations
The same analysis performed for the BW configuration has been repeated for a rectangular wing without winglet
(NW configuration - see fig. 3). A NACA 0012 airfoil rigidly translated along the wing span without any twist and
rotated at the wing tip was used for this configuration. A chord 𝑐 NW = 1.167𝑐 BW was chosen in order to maintain the
same wing span 𝑏 = 12𝑐 BW , reference surface 𝑆 = 14𝑐2BW and aspect ratio A = 10.28 of the previous configuration.
The results showed in terms of drag polars in fig. 6 and aerodynamic coefficients in table 2 refer to a C-C type
topology grid with 3 million total cells and 256 along the wing section (see section V for the grid convergence analysis).
Also in this case, the comparison between the far-field and the near-field values highlights a contribution of almost
20 drag counts for all the angles of attack considered; the viscous drag (𝐶𝐷irr ) changes only for the highest value of 𝛼
tested, whereas, the induced drag (𝐶𝐷rev ) overlaps the theoretical drag polar obtained using 𝜀 = 0.97.
0.2
near
far
irr.
rev.
0.1
0
𝐶𝐿2
𝜋A 𝜖
0
50
100
𝐶𝐷 ×
150
200
250
104
Fig. 6 NW configuration at 𝑅𝑒 ∞ = 107 and 𝑀∞ = 0.3. Drag polars and breakdown based on eq. (8) for the
irreversible component and eq. (12) for the reversible contribution. Theoretical induced drag computed with
𝜀 = 0.97.
𝛼
0◦
2◦
4◦
6◦
𝐶 𝐿near
0
0.1756
0.3510
0.5253
𝐶𝐷near
0.0099
0.0109
0.0142
0.0197
𝐶𝐷far
0.0074
0.0084
0.0114
0.0168
𝐶𝐷irr
0.0074
0.0074
0.0075
0.0081
𝐶𝐷rev
0
0.0010
0.0039
0.0087
𝐶 𝐿2 /(𝜋A𝜖)
0
0.0010
0.0039
0.0088
Table 2 NW configuration at 𝑅𝑒 ∞ = 107 and 𝑀∞ = 0.3. Lift and drag coefficients at different angles of attack.
Drag breakdown based on eq. (8) for the irreversible component and eq. (12) for the reversible contribution.
Theoretical induced drag computed with 𝜀 = 0.97.
Figure 7 shows the comparison between the aerodynamic performance of the BW and the NW configurations for the
far-field total drag, the viscous drag and the induced drag. For each angle of attack tested the overall far-field drag of the
BW configuration is higher than the one of the NW configuration. This is mainly caused by the increase of the viscous
9
0.4
0.4
0.4
0.2
0.2
0.2
0
0
0
𝐶𝐿
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drag component for the higher extension of the wet surface: the presence of the vertical joints of enlarges the total wet
surface of the BW configuration of almost 72% respect to the cantilever wing.
On the contrary, a relevant reduction of the induced drag happens and increases with the angle of attack. This
behaviour suggests the possibility to obtain values of the BW overall drag lower than the ones of the NW configuration
when 𝐶 𝐿 > 0.6.
Finally, it is worth noting that, at the same angle of attack, the BW configuration realizes an higher lift coefficient
respect to the NW configuration.
100
150
𝐶𝐷far ×
200
80
100
104
𝐶𝐷irr ×
120
0
104
20
40
𝐶𝐷rev ×
60
80
104
Fig. 7 Comparison of the aerodynamic performance for a Box Winglet BW (solid line) and a No Winglet NW
(dashed line) configuration (𝑅𝑒 ∞ = 107 , 𝑀∞ = 0.3).
B. Box Winglet vs. Standard Winglet configuration
Finally, the aerodynamic drag analysis has been repeated for a rectangular wing with a standard winglet shape as
the one shown in winglet fig. 3 (SW configuration). Also in this case a NACA 0012 airfoil has been rigidly translated
along the wing span without any twist. A chord 𝑐 NW = 1.167𝑐 BW was chosen in order to maintain the same wing span
𝑏 = 12𝑐 BW , reference surface 𝑆 = 14𝑐2BW and aspect ratio A = 10.28 of the previous configurations.
A mesh with 7.5 million cells (256 along the wing section) and C-C type topology has been used to obtain the results
showed in terms of drag polars in fig. 8 and aerodynamic coefficients in table 3 (see section V for the grid convergence
analysis).
A spurious drag contribution of almost 10 drag counts has been found for these computations and the induced drag
(𝐶𝐷rev ) polar overlaps the theoretical one obtained using 𝜀 = 1.27.
𝛼
0◦
2◦
4◦
6◦
𝐶 𝐿near
0
0.1970
0.3924
0.5853
𝐶𝐷near
0.0114
0.0124
0.0155
0.0208
𝐶𝐷far
0.0106
0.0117
0.0148
0.0199
𝐶𝐷irr
0.0105
0.0106
0.0110
0.0117
𝐶𝐷rev
0.0001
0.0011
0.0038
0.0082
𝐶 𝐿2 /(𝜋A𝜖)
0
0.0009
0.0037
0.0083
Table 3 SW configuration at 𝑅𝑒 ∞ = 107 and 𝑀∞ = 0.3. Lift and drag coefficients at different angles of attack.
Drag breakdown based on eq. (8) for the irreversible component and eq. (12) for the reversible contribution.
Theoretical induced drag computed with 𝜀 = 1.27.
As in previous section, figure 9 shows the comparison between the aerodynamic performance of the BW and the SW
configurations. As could be expected, the overall far-field drag of the BW configuration is higher than the one of the SW
configuration at each angle of attack analysed. As for the NW configuration, there is an increase of the viscous drag
component for the higher extension of the wet surface. The drag values are nearer respect to the NW case since the
total wet surface of the BW configuration is just 34% higher than SW one. Anyway, despite the theoretical higher span
10
0.6
0.5
𝐶𝐿
0.3
0.2
near
far
irr.
rev.
0.1
0
𝐶𝐿2
𝜋A 𝜖
0
50
100
150
200
250
𝐶𝐷 × 104
Fig. 8 SW configuration at 𝑅𝑒 ∞ = 107 and 𝑀∞ = 0.3. Drag polars and breakdown based on eq. (8) for the
irreversible component and eq. (12) for the reversible contribution. Theoretical induced drag computed with
𝜀 = 1.27.
efficiency, the BW configuration shows an induced drag higher than the SW geometry in this analysis.
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0
0
0
𝐶𝐿
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0.4
100
200
150
𝐶𝐷far ×
104
110
120
𝐶𝐷irr ×
104
130
0
20
40
𝐶𝐷rev ×
60
80
104
Fig. 9 Comparison of the aerodynamic performance for a Box Winglet BW (solid line) and a Standard Winglet
SW (dashed line) configuration (𝑅𝑒 ∞ = 107 , 𝑀∞ = 0.3).
V. Conclusions
A performance comparison of three simple wing geometries has been proposed in subsonic viscous regime: straight
wing with box winglet, standard winglet and no winglet. The adoption of a far field drag breakdown procedure allowed
for the separation of the irreversible and reversible (lift-induced) drag components thus allowing for the computation,
11
even in case of viscous flow, of the theoretical span efficiency. The computed span efficiency of the BW configuration is
higher than the NW one but lower than the SW configuration. It seems that, as opposed to the already studied case of
box wing, the span efficiency obtained is far from the theoretical optimum value, indicating that for the BW a more
complex distribution of chords and/or angle of attack may be needed to achieve the theoretical optimum conditions.
0.6
6◦
6◦
400
4◦
2◦
0.2
0◦
2◦
LVL0.5 LVL2
LVL1
0.4
𝐶𝐿
𝐶𝐷 × 104
𝐶𝐿
0.6
4◦
0.4
200
0.2
0◦
LVL4
0
0
0.5 1
2
LVL
0
4
0.5 1
2
LVL
4
100
200
300
400
𝐶𝐷 × 104
Grid convergence analysis for the Box Winglet (BW) configuration (𝑅𝑒 ∞ = 107 , 𝑀∞ = 0.3).
Fig. 10
0.6
6◦
LVL2
6◦
400
4◦
0.6 LVL1
4◦
2◦
0.2
0◦
2◦
200
0.4
𝐶𝐿
𝐶𝐷 × 104
0.4
𝐶𝐿
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Appendix A
In the following pages the results of the grid convergence analyses performed for the configurations analyzed are
collected.
The number of cells of the fine grid level (LVL1), whose results were described in the previous sections was devided
2 and 4 times to obtain a medium (LVL2) and a coarse (LVL4) grid level. The near-field lift and drag coefficients against
the grid levels are plotted in fig. 10 for the BW configuration, fig. 11 for the NW configuration and fig. 12 for the SW
configuration. Fig. 11 clearly shows that the results for the simplest NW geometry are already converged at LVL2.
On the contrary, this result is not clear for the more complex geometries. Therefore, in case of BW configuration, an
additional superfine grid level (LVL0.5) has been built-up doubling the number of cells of LVL1; the results of LVL0.5
show that a satisfactory convergence is obtained with LVL1.
0.2
0◦
0
LVL4
0
1
2
4
0
1
2
LVL
Fig. 11
4
LVL
100
200
300
𝐶𝐷 × 104
Grid convergence analysis for the No Winglet (NW) configuration (𝑅𝑒 ∞ = 107 , 𝑀∞ = 0.3).
12
400
6◦
6◦
300
𝐶𝐷 × 104
0.2
0◦
LVL2
200
2◦
100
0◦
0.4
0.2
LVL4
0
0
1
2
4
0
1
2
LVL
Fig. 12
LVL1
4◦
0.4
2◦
0.6
𝐶𝐿
4◦
𝐶𝐿
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0.6
4
LVL
100
200
300
𝐶𝐷 ×
104
400
Grid convergence analysis for the Standard Winglet (SW) configuration (𝑅𝑒 ∞ = 107 , 𝑀∞ = 0.3).
13
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15