Modeling of Three-dimensional Unsteady Wake Past a Large Migratory

Bird during Flapping Flight

BEAUMONT F., BOGARD F., MURER S., POLIDORI G. MATIM

University of Reims Champagne Ardenne

Faculty of Exact and Natural Science

FRANCE

Abstract: This preliminary study aimed to model the aerodynamic behavior of a large migratory bird during a

forward flapping flight. Computational Fluid Dynamics (CFD) was used to model the flow around and in the

wake of a Canada Goose flying at an altitude of 1000m and a speed of 13.9m/sec. Flapping of the wings was

modeled through dynamic meshing and subroutines implemented in a computational code using the Finite

Volumes method. Monitoring of the flow quantities during the unsteady calculation revealed a close

relationship between the wing-flapping dynamics and the cyclic variation of the forces acting on the bird. Post-

processing of the 3D results revealed a complex flow pattern mainly composed of two contra-rotating vortices

developing at the wingtip. In a perpendicular plane to the main flow direction, we demonstrated that the bird's

wake can be divided into two distinct zones: the downwash zone and the upwash zone. The latter is used by

birds flying in formation to reduce their energy expenditure. We have also shown that when the bird flaps its

wings, the trail of upwash left by the wingtips moves up and down in a wave-like motion. Further studies,

which will include several birds, will be necessary to understand all the aerodynamic implications related to the

flight of migratory birds in formation.

Key-Words: Computational Fluid Dynamics (CFD), Vortex, wake, flapping wings, migratory birds, upwash,

downwash

Received: April 12, 2021. Revised: January 8, 2022. Accepted: January 20, 2022. Published: March 1, 2022.

1 Introduction to visualize the whirling wake of a Columba livia

pigeon and discovered that when a bird flew slowly,

the wake consisted of vortex loops. Although the

published photographs are quite blurred and difficult

to interpret, it was the first demonstration that the

sheet of vortices curled into discrete structures

associated with the wing flapping cycle. More

recently, some authors [15-16] have applied a digital

particle image velocimetry (DPIV) technique to bird

flight. However, this experimental technique

requires both the use of a low-turbulence wind

tunnel and a cooperative and well-trained bird. For

these reasons, no quantitative studies on the wakes

of large high-speed flying birds have been published

to date [17]. An alternative to experimental

techniques is computational fluid dynamics (CFD).

Numerical simulations are often less expensive and

less constraining than laboratory experiments while

providing all flow quantities, including those that

are generally not accessible from measurements.

CFD has been used by scientists to model wing

flapping and improve understanding of the

mechanics of bird flight. Maeng et al. [12]

conducted a numerical study of the energy savings

of Canada geese in flight. More recently, Song et al.

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DOI: 10.37394/232013.2022.17.2

Beaumont F., Bogard F., Murer S., Polidori G. Matim

E-ISSN: 2224-347X

Volume 17, 2022

Many animal species travel in highly organized

groups [1-3]. According to many studies, the

primary interest of such formations would be to

reduce energy expenditure and improve the

locomotor performance of the animals in the group

[4-11]. Among the organized groups, the V-shaped

formation IS so characteristic of bird flights THAT

has always intrigued researchers and continues to

interest not only scientists but also many people

who enjoy this lively spectacle [4, 7, 9-12].

Research on the topic shows that when birds are

flying in a group, there is an energetic advantage to

birds flying behind and next to another bird by using

the areas of eddies generated by the wings of the

bird preceding it [4, 7, 9-11]. However, a definitive

understanding of the aerodynamic implications of

these formation flights is still difficult to establish.

Aerodynamic models are nevertheless of great

interest to ecologists who are trying to understand

the strategies and constraints related to the

migration of wild birds. For many years, scientists

have been striving to study the mechanics of bird

flight, whether from a biomechanical or

aerodynamic point of view. As precursors, Magnan

et al. [13] used tobacco smoke

[18] studied the aerodynamics of the flapping wings

of a calliope hummingbird using a dynamic meshing

method to reproduce the wing-flapping motion.

These studies show that CFD is an interesting and

complementary alternative to classical flow

visualization methods and can be applied to the

study of complex vortex wakes.

This preliminary study is an essential step in

understanding the aerodynamic mechanisms related

to the cooperative flight of migratory birds. In this

work, we have developed a computational method

based on three-dimensional numerical simulations

that integrate realistic wing flapping kinematics. For

these purposes, Computational Fluid Dynamics

(CFD) code was used to model the flow around and

in the wake of a Canada goose flying at an altitude

of 1000 m and a speed of 13.9 m/sec. Wing flapping

dynamics were modeled through a dynamic mesh

and sub-program implementation. To our best

knowledge, this study is the first to provide a

detailed and realistic representation of the unsteady

three-dimensional wake of a large migratory bird

during a flapping flight.

2 Methods

2.1 Geometry and Computational Domain

The simplified geometry (Figure 1b) was designed

using CAD software (ANSYS Workbench Design

Modeler®) following the actual shape of a Canada

goose [19] in flight position. The wings were

modeled using a two-joint arm model [19] that

provides a simplified reproduction of the wing-

flapping dynamics. Many studies consider that the

overall shape of a bird's wing is that of a NACA

profile [20-21] because the wings function like an

airfoil, i.e. a curved surface that produces lift while

minimizing turbulence in its wake. We, therefore,

used a NACA 4412 profile whose aerodynamic

performance was evaluated in the study by Malik et

al. [22]. The distance between the geometry and the

limits of the domain was defined to respect the

recommendations published in the CFD best

practice guidelines [23], which resulted in a

blocking ratio lower than 3. The dimensions of the

computational domain, as well as the boundary

conditions, are shown in Figure 1(c).

Fig. 1: (a) A Canada goose in flight (Photo Alan

D.Wilson — Wikimédia); (b) Simplified geometry

of the Canada goose. (c) Computational domain size

and boundary conditions.

2.1.1 Boundary Conditions

At the entrance of the computational domain, we

imposed a speed of 50 km/h (13.9 m/s) which

corresponds to the average speed of the Canada

geese during a migratory flight [24]. At the exit of

the computational domain, we imposed a pressure

exit condition with the ambient static pressure. A

symmetry condition was imposed on the upper and

lateral surfaces of the computational domain.

Table 1. Numerical parameters and dimensional

features of a Canada goose [24-25]. Numerical

values were determined assuming that the bird is

flying at an altitude of 1000 m [26].

Variable

Value

Chord length (c)

0.21 m

Wing size

0.82 m

Flight speed (U∞)

13.9 m/s

Wingbeat frequency

4 HZ

Wingbeat amplitude

0.6 m

Wingspan

1.8 m

Flight altitude

1000 m

An air density of 1.11 kg/m3, corresponding to an

altitude of 1000m, was used as a reference value for

the calculation [26]. The CFD parameters, as well as

the dimensional parameters of the wing, are

summarized in Table 1.

2.1.2 Flapping Motion Modeling

We modeled the wing motion based on the wing-

flapping dynamics of a bird in flight [19] (Fig. 2).

Birdwing flapping is complex and involves several

rotational movements, each of which influences the

flight mechanics. For simplification, this study

focuses only on the main rotational movement of the

wings while neglecting the rotation of the leading

edge of the wing which influences the thrust force.

(a)

(b)

(c)

(b)

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Beaumont F., Bogard F., Murer S., Polidori G. Matim

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Volume 17, 2022

The average wing flapping frequency of a Canada

goose is about 4 Hz [24-25, 27-30]. To describe the

real-time evolution of the wings’ motion, a dynamic

mesh has been adopted allowing to update of the

wings’ position at each time step (Δt=0.001s). The

updating of the volumetric mesh is automatically

performed by the FLUENT solver and consists of a

local remeshing that mainly uses an interpolation

method to regenerate the mesh in the computational

area, which is suitable for large and complex

movements. Considering the amplitude of the

wing’s motion, we chose to use the spring-based

smoothing and local re-meshing methods. The

trajectory of the wings, which consists of an up and

down rotational movement, has been implemented

in the computational code through User Defined

Functions (UDF).

Fig. 2: Kinematics of a complete cycle of wing

beats. Black arrows pointing upwards represent the

upstroke phase while the black arrows pointing

downwards represent the downstroke phase. The

half-arrow represents the change in wing-flapping

direction during the wingbeat cycle.

2.2 Meshing Methods

The three-dimensional grid of the computational

domain was created with the ANSYS Workbench

Meshing software (Figure 3). The unstructured

mesh of the fluid domain is composed of

approximately    prismatic and hexahedral

elements. The boundary layer grid requires a very

fine mesh close to the wall to completely resolve the

viscoelastic and laminar sub-layer. To correctly

reproduce the boundary layer separation, an

inflation layer composed of 15 layers of a

progressive thickness (growth ratio: 1.2) was

generated around the body wall (Figure 3c). To

obtain a Y* value lower than 1, necessary for a

precise resolution of the boundary layer, the size of

the cells adjacent to the wall around the bird's body

has been adjusted to a value of 10 μm. In addition, a

mesh refinement zone was created around and in the

wake of the bird. In this zone, the average size of

the elements is less than or equal to 0.05 m. Outside

of this area, a Cartesian mesh is used, allowing to

reduce the number of elements and consequently the

calculation time.

Fig. 3: Surface mesh on the bird's body and in a 2D

vertical plane intersecting a wing of the bird (a);

detail of the mesh around the wing in the shape of a

NACA profile (b) and detailed view of the inflation

mesh (c) around the wings.

2.3 Numerical Methods

The calculations were performed using the CFD

was used to solve the Reynolds Averaged Navier-

Stokes (RANS) equations in 3D [31]. The simple

algorithm was used for pressure-velocity coupling

with second-order discretization schemes and

gradients computed using the least-squares cell-

based method. The SST turbulence model k-ω is a

two-equation model that is used for many

aerodynamic applications. This turbulence model is

fully adapted to unfavorable pressure gradients and

separation flow and is therefore ideally suited for

aerodynamic studies. [32-33]. The drag and lift

coefficients were monitored throughout the iterative

calculation. The calculations were performed on a

Dell Workstation Precision 7920 workstation and

were parallelized on 48 Xeon Gold 3.2 GHz

processors. Note that the methodology used in this

paper is fully described in a previous paper as well

as the validation of the numerical results [34].

3 Results and Discussion

Four forces act on a flying device, whether it is a

bird, bat, insect, or airplane: lift, thrust, drag, and

gravity (Figure 4). Thrust must be equal to drag and

lift must be equal to gravity in straight and level

flight. In the study of flapping wings, several

parameters can be used to quantify the flow

characteristics. The most representative ones are

presented below. Two parameters that provide

important information in the study of flapping wings

propulsion are the drag and lift coefficients, they are

defined as follows:

0.025s

0.150s

0.175s

0.125s

0.1s

0.075s

0.05s

0.250s

0.225s

0.2s

Half wing beat cycle

Downstroke

Upstroke

0.125s

0.250s

Half-wing flapping cycle

Downstroke

Upstroke

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



 (1)





 (2)

,  and  are the forces of drag, lift, and thrust

(identical to the forces of drag but of opposite sign),

respectively; ρ is the density of the fluid, U is the

forward velocity, and  is the projected area (m²).

With the sole exception of the hummingbird,

birds generate lift and thrust by flapping their wings,

a complex, unstable, three-dimensional wing

movement that changes with each new wing

position. The flapping is performed in two phases:

the downstroke, or power stroke, which produces

much of the thrust, and the upstroke, or recovery

stroke, which, depending on the bird's wing,

produces a certain amount of lift. Of course, wings

not only generate lift and thrust, but they also induce

drag. Lift and drag are two components of the

resulting aerodynamic force acting on the wing. The

effective magnitude of the resultant aerodynamic

force depends mainly on the magnitude of the

flapping speed of the wing [35]. The lift can act in

any direction relative to gravity since it is defined as

the direction of flow rather than the direction of

gravity. When a bird flies in a straight line, most of

the lift is opposite to gravity [36]. However, when a

bird climbs, descends, or tilts in a turn, the lift is

inclined concerning the vertical.

Fig. 4: Aerodynamic forces acting on a bird in

flight.

Figure 5 depicts the variation of the lift and drag

coefficients throughout a wing-flapping cycle. As

shown in Figure 5, the lift coefficient is maximum at

mid-downstroke and minimum at mid-upstroke. The

drag coefficient is minimal at mid-downstroke and

maximal between mid-upstroke and the start of the

downstroke. On the other hand, the drag coefficient

becomes negative during the downstroke, indicating

that reverse drag forces are generated and then lift

forces are increased during the downstroke. We can

also see that the lift coefficient becomes negative in

the upstroke, indicating that inverse lift forces are

generated, and then drag forces are increased during

the upstroke. On the other hand, we can also see that

the negative lift coefficient is larger than the

negative drag coefficient.

Fig. 5: Evolution of the lift and drag coefficients

during one flapping cycle.

3.1 Pressure

Pressure is the normal force per unit area exerted by

the air on itself and on the surfaces it hits. The lift

force is transferred by the pressure, which acts

perpendicular to the surface of the wing. Thus, the

net force is expressed in the form of pressure

differences. The direction of the net force implies

that the average pressure on the upper surface of the

wing is lower than the average pressure on the lower

surface [37].

Fig. 6: Pressure coefficients on the bird's body and

wings at t=0.125s (Mid-downstroke) and at t=0.25s

(Mid-upstroke).

Figure 6 shows the distribution of pressure on the

bird's body and wings at t=0.125s (Mid-downstroke)

and t=0.25s (Mid-upstroke). Note that the pressure

range has been deliberately limited to the interval [-

0.5-0.5] to highlight the slightest change in pressure.

During the downstroke, the air collides with the

underside of the wing, generating lift. During this

Lift

Weight

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phase, the pressure is positive on the underside of

the wing and negative on the upper phase. During

the upstroke phase of the wing, the opposite

happens, the pressures are globally positive on the

top of the wing (extrados) while they are negative

on the bottom (intrados). Moreover, we can see that

the positive and negative pressures are maximum at

t=0.125s (Mid-downstroke), which corresponds to

the maximum lift (see Fig. 5). It is also remarkable

that the highest pressures are at the wingtips, where

the vortices develop. The lowest pressure occurs on

the upper surface of the wing and the highest

pressure under the wing.

Fig. 7: Streamlines displayed in a vertical plane at

0.5m, 1m, 1.5m, and 2m respectively in the wake

behind the bird and for the mid-upstroke position.

This pressure differential causes the airflow to turn

back behind the wing, which in turn causes the

airflow to rotate downstream of the wingtips. Once

the reversal is completed, the wake consists of two

counter-rotating cylindrical vortices.

These vortices are generated during the downstroke

phase of high aspect ratio wings [38] and will be

discharged with the downstream airflow. This

couple of vortices pulls the air downwards,

generating lift in the opposite direction. The wake

develops while the wing is flapping up and down

and especially during the downstroke, the induced

vortices then play a considerable role.

Green [37] identified three mechanisms that

describe the origin of wingtip vortices. The first and

most common mechanism is the pressure imbalance

at the tip, during the lift generation process. The

pressure difference accelerates the flow from the

pressure side (lower surface) to the suction side

(upper surface), leading to the formation of a vortex

at the wingtip (or simply at the tip). The strength of

the vortex depends on the magnitude of the pressure

difference, which in turn depends on the angle of

attack. For more information, figure 7 shows the

streamlines of the flow in the bird's wake, drawn in

4 parallel vertical planes equidistant of 0.5m. The

bird's wake is made up of two main structural parts:

a pair of wingtip vortices and a pair of tail vortices.

We can clearly distinguish the two counter-rotating

vortices that developed at the wingtips. We can also

see that the vortices eventually dissipate, their

energy being transformed by viscosity. In addition,

we used a volume rendering method that provides

an overview of the three-dimensional structure of

the vortex wake while giving quantitative

information on vorticity. Figure 8 shows the

volumetric representation of the wake behind the

bird at two instants of the upstroke and downstroke

phases. Note that the range of vorticity plotted along

the x-axis (in the direction of the bird's movement)

has been deliberately limited to the interval [-100;

100] to show the slightest change in vorticity

intensity. The colors red and blue indicate that the

vortices have an opposite direction of rotation, one

vortex at the wingtip circulates clockwise while the

other circulates counterclockwise. The two wingtip

vortices do not merge because they flow in opposite

directions. The wingtip vortices come up from the

wingtips and tend to dive and roll over each other

downstream of the wing. Again, the tip vortices

eventually dissipate, their energy being transformed

by viscosity. We can also see that when the bird

flaps its wings, the trail of upwash left by the

wingtips moves up and down in a wave-like motion

related to the flapping of the wings.

Fig. 8: Reconstruction of the approximate shape and

movement of the vortex wake by the volume

rendering method for two different moments during

the Upstroke and Downstroke phases. Color refers

to x-vorticity intensity.

3.2 Influence of Wake Flow on Migratory Bird

Flight

When a bird is flying, a rotating air vortex separates

from each wingtip. These vortices imply that the air

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inside the vortex system, in the immediate wake of

the bird, moves downwards (downwash) while the

air on the sides, outside the vortex system, moves

upwards (upwash).

Fig. 9: schematic diagram of the vortex flow

developing in the wake of a large bird flying at high

speed (a). Radial velocity profile Uy plotted along

an axis passing through the core of the vortex flow

at x=1.5m and for Mid-downstroke (b).

Figure 9 (a) shows a schematic diagram of the

airflow in the wake of a bird. Figure 9 (b) shows the

vertical velocity plotted along a horizontal line

passing through the core of the vortex flow at x=0.5

m in the wake of the bird (for mid-downstroke).

From the graph, we can evidence two distinct zones:

the upwash zone, in which the velocities are

positive, and the downwash zone, in which the

velocities are negative. Some studies suggest a close

relationship between the position of birds in groups

during formation flights and the flow topology in

the wake of birds in flight [39-40]. Concretely, this

means that in specific flight formations (such as V

formations), the bird that flies in the wake of

another bird could take advantage of upward

currents and improve its lift, which would reduce its

energy cost. This is the reason why many species of

large birds fly in a V formation because the interest

of collaborative flight is to reduce energy

expenditure and improve the locomotor

performance of the birds in the group [41-42]. The

study by Portugal et al. [41] suggests that birds

flying in V formation would have phasing strategies

to cope with the dynamic wakes produced by wing

flapping. Our results seem to confirm this

observation. The oscillation of the vortex flow in the

wake of the bird suggests that phasing of the wing

flapping of the following bird with the preceding

bird is necessary to reduce energy expenditure. In

future numerical studies, it will be possible to study

the aerodynamic interaction between several birds

flying in formation but also the influence of wing

flapping phasing on the forces acting on the birds in

flight.

4 Conclusion

In this study, we modeled the unsteady flow around

and in the wake of a large flying bird by including

realistic wing flapping kinematics. A CFD method

was used to highlight the three-dimensional vortex

structures developing in the wake of a Canada goose

flying at an altitude of 1000m and a speed of

13.9m/s. Post-processing of the 3D results revealed

a complex vortex flow whose main structure is

composed of two contra-rotating vortices

developing at the wingtip. From flow velocity

fields, we highlighted two distinct zones: the

upwash zone, in which the flow is ascending and the

velocities are positive, and the downwash zone, in

which the flow is descending and the velocities are

negative. It seems that migratory birds use the area

of upward currents to improve lift and save energy

costs during a migration. Furthermore, it has been

shown that when the bird flaps its wings, the trail of

upwash left by the wingtips also moves up and

down in a wave-like motion. The unsteady wake

dynamics suggest that when migratory birds fly in

formation, phasing of the wing-flapping between the

follower and the lead bird is necessary to benefit

from a reduction in energy expenditure. To our best

knowledge, this study is the first to provide a

detailed and realistic representation of the unsteady

three-dimensional wake of a large migratory bird

during a flapping flight. Further studies, which will

include several birds flying together, will be

necessary to understand the full aerodynamic

implications of migratory birds flying in formation.

References:

[1]. Couzin I. D., Krause J., Franks N.R. & Levin

S.A., Effective leadership and decision

making in animal groups on the move.

Nature, Vol. 433, 2004, pp. 513–516.

[2]. Nagy M., Akos Z., Biro D. & Vicsek T.,

Hierarchical group dynamics in pigeon flocks.

Nature, Vol. 464, 2010, pp. 890–894.

[3]. May R.M., Flight formations in geese and

other birds. Nature, Vol. 282, 1979, pp. 778–

780.

WSEAS TRANSACTIONS on FLUID MECHANICS

DOI: 10.37394/232013.2022.17.2

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[4]. Lissaman P.B. & Schollenberger C.A.,

Formation flight of birds. Science, Vol. 168,

1970, pp. 1003–1005.

[5]. Liao J.C., Beal D.N., Lauder G.V. &

Triantafyllou M.S., Fish exploiting vortices

decrease muscle activity. Science, Vol. 302,

2003, pp. 1566–1569.

[6]. Bill R.G. & Hernnkind W.F., Drag reduction

by formation movement in spiny lobsters.

Science, Vol. 193, 1976, pp. 1146–1148.

[7]. Weimerskirch H., Martin J., Clerquin Y.,

Alexandre P. & Jiraskova S., Energy saving in

flight formation. Nature, Vol. 413, 2001, pp.

697–698.

[8]. Fish F.E., Kinematics of ducklings swimming

in formation: consequence of position. J. Exp.

Zool., Vol. 273, 1995, pp. 1–11.

[9]. Badgerow J. P. & Hainsworth F.R., Energy

savings through formation flight? A

reexamination of the vee formation, J. Theor.

Biol., Vol. 93, 1981, pp. 41–52.

[10]. Cutts C.J. & Speakman J. R., Energy savings

in formation flight of pink-footed geese, J.

Exp. Biol., Vol. 189, 1994, pp. 251–261.

[11]. Hummel D., Aerodynamic aspects of

formation flight in birds, J. Theor. Biol., Vol.

104, 1983, pp. 321–347.

[12]. Maeng J.S. et al., Park J.H., Min Jang S., Han

S.Y., A modelling approach to energy savings

of flying Canada geese using computational

fluid dynamics, J. Theor. Biol., Vol. 320,

2013, pp. 76–85.

[13]. Magnan A., Perrilliat-Botonet C., Girerd H.,

Essais d’enregisterements cinémato-

graphiques simultanées dans trois directions

perpendiculaires dexu à de l’écoulement de

l’air autour d’un oiseau en vol, C. r. hebd.

Séanc. Acad. Sci. Paris, Vol. 206, 1938, pp.

462–464.

[14]. Spedding G.R., Hedenström A. & Rosén M.,

Quantitative studies of the wakes of

freelyflying birds in a low-turbulence wind

tunnel, Experiments in Fluids, Vol. 34, 2003,

pp. 291–303.

[15]. Spedding G.R., Rosén M. & Hedenström A.,

A family of vortex wakes generated by a

thrush night ingale in free flight in a wind

tunnel over its entire natural range of flight

speeds, Journal of Experimental Biology, Vol.

206, 2003, pp. 2313–2344.

[16]. Nafi A.S., Ben-Gida H., Guglielmo C.G.,

Gurka R., Aerodynamic forces acting on birds

during flight: A comparative study of a

shorebird, songbird and a strigiform,

Experimental Thermal and Fluid Science,

Vol. 113, 2020, p. 110018.

[17]. D. Michael., Animal locomotion: a new spin

on bat flight. Curr. Biol., Vol. 18, 2008, pp.

468-470.

[18]. Song J, Tobalske BW, Powers DR, Hedrick

TL, Luo H., Three-dimensional simulation for

fast forward flight of a calliope hummingbird,

R. Soc. Open sci., Vol. 3, 2016, p. 160230.

[19]. Liu T., Kuykendoll K., Rhew R.D., Jones S.,

Avian Wings, AIAA, 2004, pp. 2004-2186.

[20]. Shyy W., Berg M., Ljungqvist D., Flapping

and flexible wings for biological and micro air

vehicles, Prog. Aerosp. Sci., Vol. 35, 1999,

pp. 455-505

[21]. Usherwood J.R., Hedrick T.L., Biewener

A.A., The aerodynamics of avian take-off

from direct pressure measurements in Canada

geese (Branta canadensis), J. Exp. Biol., Vol.

206, 2003, p. 4051.

[22]. Malik K., Aldheeb M., Asrar W., Erwin S.,

Effects of Bio-Inspired Surface Roughness on

a Swept Back Tapered NACA 4412 Wing, J

Aerosp Technol. Manag., Vol. 11, 2019, p.

e1719.

[23]. Blocken B., Computational Fluid Dynamics

for urban physics: Importance, scales,

possibilities, limitations and ten tips and tricks

towards accurate and reliable simulations,

Building and Environment, Vol. 91, 2015, pp.

219-245,

[24]. Tucker V.A., Schmidt-Koenig K., Flight

speeds of birds in relation to energetics and

wind directions, The Auk, Vol. 88, 1971, pp.

97-107.

[25]. Funk G. D., Milsom W.K., Steeves J.D.,

Coordination of wingbeat and respiration in

the Canada goose. I. Passive wing flapping.

Journal of applied physiology, Vol. 73, n°3,

1992, pp. 1014–1024.

[26]. Fergus C., Canada goose. Wildlife Notes-20,

LDR0103, Pennsylvania Game Commission,

2010.

[27]. Gould L.L., Heppner F., The vee formation of

Canada geese, The Auk, Vol. 91, n°3, 1974,

pp. 494-506.

[28]. Hainsworth F., Induced drag savings from

ground effect and formation flight in brown

pelicans, J. Exp. Biol., Vol. 135, 1988, pp.

431-444.

[29]. Hedrick T.L., Tobalske B.W., Biewener A.A.,

Estimates of circulation and gait change based

on a three-dimensional kinematic analysis of

flight in cockatiels (Nymphicus hollandicus)

WSEAS TRANSACTIONS on FLUID MECHANICS

DOI: 10.37394/232013.2022.17.2

Beaumont F., Bogard F., Murer S., Polidori G. Matim

E-ISSN: 2224-347X

Volume 17, 2022

and ringed turtle-doves (Streptopelia risoria),

J. Exp. Biol., Vol. 205, 2002, p. 1389.

[30]. Tennekes H., The Simple Science of Flight:

From Insects to Jumbo Jets, MIT Press,

Cambridge, MA, 1996.

[31]. Menter F.R., Zonal two-equation k-to

turbulence model for aerodynamic flows,

AIAA Paper 1993–2906. AIAA 1993-2906,

23rd Fluid Dynamics, Plasma dynamics, and

Lasers Conference, 1993.

[32]. Polidori G., Legrand F., Bogard F., Madaci F.,

Beaumont F., Numerical investigation of the

impact of Kenenisa Bekele’s cooperative

drafting strategy on its running power during

the 2019 Berlin marathon, Journal of

Biomechanics, Vol. 107, 2020, p. 109854.

[33]. Fintelman D.M., Hemida H., Sterling M., Li

F.-X., CFD simulations of the flow around a

cyclist subjected to crosswinds, Journal of

Wind Engineering and Industrial

Aerodynamics, Vol. 144, 2015, pp. 31-41.

[34]. Beaumont F., Murer S., Bogard F., Polidori

G., Aerodynamics of a flapping wing as a

function of altitude: New insights into the

flight strategy of migratory birds, Physics of

Fluids, Vol. 33, 2021, p. 127118.

[35]. Dvořák R., Aerodynamics of bird flight, EPJ

Web of Conferences, Vol. 114, 2016, p.

01001.

[36]. Hoerner S.F. & Borst H.V., Fluid-dynamic

Lift: Practical Information on Aerodynamic

and Hydrodynamic Lift, L.A. Hoerner, 1985.

[37]. Green S.I., Wing Tip Vortices, Fluid Vortices,

Kluwer Academic Publishers, Netherlands,

1995.

[38]. Hedenström A., Spedding G.R., Beyond

robins: aerodynamic analyses of animal flight,

J. Royal Soc.: Interface, Vol. 5, 2008, pp.

595-601.

[39]. Weimerskirch H., Martin J., Clerquin Y.,

Alexandre P., Jiraskova S., Energy saving in

flight formation, Nature, Vol. 413, 2001, pp.

697–698.

[40]. Hummel D., Aerodynamic aspects of

formation flight in birds, J. Theor. Biol., Vol.

104, 1983, pp. 321–347.

[41]. Portugal S. J., Hubel T. Y., Fritz J., Heese S.,

Trobe D., Voelkl B., Hailes S., Wilson A. M.,

Usherwood J. R., Upwash exploitation and

downwash avoidance by flap phasing in ibis

formation flight. Nature, Vol. 505, 2014, pp.

399–402.

[42]. Lissaman P.B.S., Shollenberger C.A.,

Formation Flight of Birds, Science, Vol. 168,

n°3934, 1970, pp. 1003–1005.

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

Fabien Beaumont performed the simulations and

optimization and wrote the original draft.

Guillaume Polidori performed the investigation,

analysis, and validation.

Sébastien Murer and Fabien Bogard provided

supervision and review.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

There are no sources of funding.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

_US

WSEAS TRANSACTIONS on FLUID MECHANICS

DOI: 10.37394/232013.2022.17.2

Beaumont F., Bogard F., Murer S., Polidori G. Matim

E-ISSN: 2224-347X

Volume 17, 2022