UNDERSTANDING BIRD’S FLIGHT, USING A 3-D BEM METHOD AND A TIME STEPPING ALGORITHM.

7/29/2019 UNDERSTANDING BIRDS FLIGHT, USING A 3-D BEM METHOD AND A TIME STEPPING ALGORITHM.

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4th International Conference from Scientific Computing to Computational Engineering

4th IC-SCCE

Athens, 7-10 July, 2010

IC-SCCE

UNDERSTANDING BIRDS FLIGHT, USING A 3-D BEM METHOD AND A TIME

STEPPING ALGORITHM.

Gerasimos Politis1

and Vassileios Tsarsitalidis2

1)Associate Professor, Department of Ship and Marine Hydrodynamics, School of Naval Architecture and

Marine Engineering, National Technical University of Athens, Heroon Politechniou 9, Zografos, Athens, Greece,

e-mail: [email protected]

2) Naval Architect and Marine Engineer, PHD candidate, School of Naval Architecture and Marine Engineering,

National Technical University of Athens, Heroon Politechniou 9, Zografos, Athens, Greece, e-mail:

[email protected]

Keywords: Bird flight, Biomimetics, Flapping foil propulsion; Boundary element method; Unsteady wake

rollup.

Abstract. Bird flight is a physical paradigm of how a fully unsteady flow problem, assisted by properly adjusted

biofeedback system, can lead to very well controlled flight scenarios ranging from nearly constant speed

advancement to complex flight maneuvers. To understand the flow physics of such systems, the problem of flow

around a passively (i.e. controlled by the user) flexible wing performing unsteady motions while travelling with

a given velocity in an infinitely extended fluid, is formulated and solved using a potential based 3D BEM.

Dynamic evolution of unsteady trailing vortex sheets emanating from wing trailing edges is calculated by

applying the kinematic and dynamic boundary conditions as part of a time stepping algorithm used for the

solution of the unsteady problem. Trailing vortex sheet intensity is calculated by applying at each time step a

nonlinear pressure type Kutta condition at wing trailing edges. With the proper filtering of induced velocitieswhich introduces artificial viscosity to our model, beautiful roll-up patterns emerge, indicating the main vortex

structures, by which birds wings in unsteady motion, interact by themselves and develop forces. After the

method for simulation of the geometry for bird wings and their evolution in time is described, we proceed to

some systematic calculations/simulations for a bird wing flying in a number of preselected regimes with varying

geometries, loadings and flying conditions. We show that unsteady bird fly can be comprehended by tracing a

number of well structured unsteadily moving ring vortices created in the wake of the moving bird. We present

some results showing the 3-D topology of this wake vortex pattern and how this is affected by the wing geometry,

loading and flying parameters. Some quantitative results are also presented for the developed lift and induced

drag for such systems.

INTRODUCTION

Throughout history, man has always sought to fly like birds. Many believe that we have attained such an

achievement, but we are not there yet.

While aeronautical technology has advanced rapidly over the past 100 years, nature's flying machines, which

have evolved over 150 million years, are still impressive. A simple comparison can astonish anyone. Humans

move at top speeds of 3-4 body lengths per second, a race horse runs approximately 7 body lengths per second,

and the fastest terrestrial animal, a cheetah, accomplishes 18 body lengths per second. A supersonic aircraft such

as the SR 71 Blackbird travelling near Mach 3 covers about 32 body lengths per second. Yet a common pigeon

frequently attains speeds of 90 kph; this converts to 75 body lengths per second. A European Starling (Sturnus

vulgaris) is capable of flying at 120 and various species of Swifts over 140 body lengths per second. The roll rate

of highly aerobatic aircraft (e.g., A-4 Skyhawk) is said to be approximately 720 degrees per second, while a Barn

Swallow (Hirundo rustics) has a roll rate in excess of 5000 degrees per second. The maximum positive G-forces

permitted in most general aviation aircraft is 4-5Gs and select military aircraft withstand 8-10Gs. However, many


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Gerasimos Politis and Vassileios Tsarsitalidis

Viscous forces are entered in our code in the form of simple surface drag coefficient which is a function of a

body Reynolds number.

Let also v denote the velocity ofA with respect to an inertia reference frame. Consider also a parallel vector

field d with d a unit vector. Then, according to a well known vector identity:

( ) ( )v d v d d v d (2)

The differential power done from the fluid to the body at A is given by:

( )( ) (( ) )pow F v F d d v F d v d (3)

and the net power from a flexible body is given by:

_ ( )( ) (( ) )S S S

net pow F v dS F d d v dS F d v d dS (4)

In birds fly there is always a preferable instantaneous direction in which the bird intends to move. Take

d along this direction. Then the first term in the right hand side of equation (4) is the instantaneous ( )EHP t and

the second term is the instantaneous ( )DHP t . The ratio ( ) / ( )EHP t DHP t defines the instantaneous efficiency.

Efficiencies for finite time flight intervals are calculated by:

0

0

( )

( )

T

T T

EHP t dt

DHP t dt

(5)

where T denotes the time interval. Notice that during birds fly, there are instances where the bird wing absorbspower from the flow. At those points ( )DHP t becomes eventually small (smaller than ( )EHP t ) giving thus rise

to instantaneous efficiencies greater than one. Another case of interest to birds fly, to be considered in a futurepaper, is when they fly in formations. In this case the wake of the former bird affects the later and ther are

formations of optimum efficiency as a system.

PHENOMENOLOGICAL CONSIDERATIONS REGARDING GEOMETRY AND MOTION OF BIRD

FLIGHT.

In order to simulate the motion of a bird's wings, advice has to be taken from biologist investigations. As seen

in the picture below taken from biology clipart, the motion of a bird in flight is asymmetrical. Additionally, the

wing outline and all other geometric/motion details are time dependent and in general allow for a large variety of

alternatives.

Figure 1. A schematic view of a bird in flight (biology clipart)

This geometric complexibility, which nature has introduced to the flying creatures by natural selection, is

dependent from the creature operational objectives such as the ability of sucesfull hanting, the ability to cope

with their enemies or the peculiarities in the operational environment.

As a result, a multitude of flying creature wings can be met with different complexities. Simpler wing

geometries are usually present in insects. For example a nearly elliptical wing is present in the case of agrasshopper. The grasshopper wing has not joints and is mainly stiff, allowing span wise and chord wise


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deformations induced by the elastic properties of its construction. In many insects, wings are operating in tandem,

allowing thus a balancing of developed aerodynamic forces. Birds are almost always equipped with wings of

more complex motion capabilities and have joints. Not only can the wing as a whole change its position with

time, but also its geometric details (such as the wing outline, the twist, the camber etc) can be time dependent.

This wing motion flexibility is intimately connected with its physical anatomy and the existence of a spinal

column with a muscle system. Biologists have systematically investigated the anatomy of bird wings and arrangethem in groups according to their characteristics. From these investigations it is shown that there are species that

employ a simpler wing outline and motion. For example the wing of a hummingbird in nearly elliptic without

joints and its motion can be described by a combination of a flapping and a twisting motion, figure 2.

Unfortunately (for the scientists) the more complex wing motions are the rule for the flying creatures. For

example a seagulls wing in simultaneous acceleration and climbing condition (high thrust and lift) presents ahighly asymmetric wing motion with very strong wing shape variations with time. This is connected with the

existence of a joint in seagulls wing, which allows independent control by the birds brain, of the two parts ofthe wing.

A challenge for the scientist which attempts to numerically simulate the birds flight is the introduction of aminimal, yet richly enough, group of geometric and motion parameters to control the variations of the geometric

and kinematic characteristics of a birds wing in flight.

Figure 2. X,Y flapping and twisting motions of a birds wing.

HANDLING BIRDS WING GEOMETRY AND MOTION

Regarding simulation of geometry and kinematics of bird wings, our proposal is lent from the anatomy of the

real wing that is we propose the use of a spine-rail combination. Thus the instantaneous position of a birds wingcan be defined by: (a) a spine (a line tracing the wing in the span-wise direction), (b) the rails (a number of lines

tracing the camber line in the chord-wise direction) and (c) a thickness distribution which is superimposed on the

spine-rail surface (otherwise termed the reference wing surface). The time dependent geometry of abirds wingcan then be reproduced by defining the successive positions of the reference surface as well as the thickness

distribution which, if necessary, can also be time dependent.

More specifically the spine is discritized by a variable (user determined and bird dependenet) number ofstraight line spine segments of given length, not necessarily equally spaced. Each segment is connected to a

next and previous one by a start and end joint respectively. Instantaneous spine geometry can then bedefined by giving the position of the starting node of the first segment and the two rotation angles for each (and

all) segments, defining their orientation in space with respect to a global XYZ coordinate system. Assuming the

X axis to be along birds instantaneous velocity and the XY plane as the plane of instan taneous flight symmetry(we currently limit our investigation to cases where bird wings are moving with a XY transverse plane of

symmetry), the spine rotation around the X axis is termed Y-flapping while the spine rotation around the Y axis is

termed the X-flapping. Figure 2 shows schematically those notions.

The rail, which is attached to each end joint in a plane normal to the instantaneous position of the spinesegment, is discritezed similarly by a number of straight line segments tracking the local camber distribution.

Rails obtain their position on this normal plane by determining a twist angle relative to an initial position. By

superimposing to this plane a thickness distribution, a section of the final wing surface has been constructed.Birds wing camber and thickness distributions are lent from traditional 2-D data of the NACA family, which ischaracterized by analytic descriptions with a minimum number of defining free parameters. For example we can

Y-flapping

X-flappingTwisting


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From relation (8) we observe that the combination0

( , , )St determines fully the ( )a t and consequently the

maximum angle of attack for the 2-D case. Since this is related to the maximum attainable lift as well as

phenomena like viscous separation and stall, it is common to characterize an unsteady flight condition for a 2-D

airfoil by the non-dimensional variables0

( , , )St . In the sequel we shall use the previous non-dimensional

variables based on a single birds wing section (for example in the first two cases we use the section at 70% ofthe birds wing semi-span) although it is clear that only a crude relation between the flow variables of the 3-D

flow case with the 2-D case exists.

CASES SIMULATED

We decide to investigate three different modes of flight. The first is a motion where the wing flaps around the X-

axis and twists harmonically. This is the simplest case of motion of a birds wing and can be considered toresemble the motion of a hummingbird. The second case is an alteration of the first, with the difference that the

flapping around X-axis has a mean value larger than zero. Such a motion is employed by most birds when they

maintain speed in steady flight (level flight). This type of asymmetry produces an additional amount of lift in

level flight. The third case is a motion with strong deformations due to wing joints that simulates the motion of

seagulls wings when accelerating and climbing at the same time (high thrust and lift). The wing outline is thesame for all cases for reasons of comparison figure 3. Similarly a NACA 0012 section is used in all cases

considered.

X

Y

Z

Figure 3. Wing geometry

Case 1 (hummingbird in vertical hover)

Figures 4, 5 show the time evolution of the wing geometry over one period. The motion is characterized by a

strouhal number: 0.15St andmax

10a (max

max( ( ), (0... ))a a t t T at the spanwise section located at

70% of semi-span. Figures 6,7 and 8 show the 3-D pattern of the shear layer emanating from the wing trailing

edge at various perspectives. By tracking the shear layer deformation it is possible to identify regions where wellstructured and intensive ring vortices evolve. For the non-experienced observer figures 9,10 and 11 present an

artistic addition which shows explicitly the shape and direction of such vortices. From those figures we observe

that in the considered flight a continues strip of ring vortices is generated with axis inclined with respect to the

axis of bird parallel movement. Four vortices are created in one period, two during upstroke, with axis inclined

upward, and two other during downstroke, with axis inclined downward. In the case considered all the vortices

have the same intensity and their vertical projection sums to zero in one period. This flight is realistic only when

applied to a bird in entirely vertical hover. In level flights the bird needs both thrust and lift which is the subject

of the next case. Figure 12 presents the calculated thrust and lift. Notice the symmetry of the lift, by which its

average value is zero. From the calculated unsteady thrust it is observed that there is a substantial average thrust.

Finally the mean (one period) efficiency of the flapping wing for this case is 63%.


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Figures 4,5. Sucessive wing positions over one period

Figures 6,7,8. Shear layer dynamics. Colour represents dipole intensity.

Figures 9,10,11. Artistic addition of vortex rings and corresponding flow jets.

its

Forces

50 100 150 200 250 300-0.4

-0.2

0

0.2

0.4

Thrust

Lift

Figure 12. Instantaneous thrust and lift as a function of time.


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Case 2 (hummingbird maintaining steady level flight)

The next case considered is similar to the first in all aspects, with the exception of introducing a mean valuein the Y-flapping angle. This introduces an asymmetry to the upstroke with respect to the down stroke motion of

the wing which results in the development of a lift force usefull for a level flight. Again 0.15St and

max10a at the spanwise section located at 70% of semi-span. Figures 13, 14 show the time evolution of the

wing geometry over one period. Figures 15,16 and 17 show the 3-D pattern of the shear layer emanating from the

wing trailing edge at various perspectives. Figures 18, 19 and 20 present an artistic addition which shows

explicitly the shape and direction of created ring vortices. From those figures we observe that the considered

flight is maintained by a continues creation of ring vortices with inclined axis with respect to the axis of bird

parallel movement. Four vortices are created in one period, two of them with axis inclined upward and two other

with axis inclined downward. In oposition to the the previous case, the vortices have not the same intensity and

their vertical projection does not sum up to zero in one period. Thus a mean lift is obtained. This flight is realistic

for a level flight, where the bird needs both thrust and lift. Figure 21 presents the calculated thrust and lift. Notice

the assymmetry of the lift, by which its average value is different from zero. From the calculated unsteady thrustit is observed that there is a substantial average thrust. Finally the mean (one period) efficiency of the flapping

wing for this case is 60%.

Figures 13,14. Sucessive wing positions over one period

Figures 15,16,17. Shear layer dynamics. Colour represents dipole intensity.

Figures 18,19,20. Artistic addition of vortex rings and corresponding flow jets.


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its

Forces

0 100 200 300-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Thrust

Lift

Figure 21. Instantaneous thrust and lift as a function of time.

Case 3 (seagulls wings in highly deformed motion)The next case considered is entirely different from the previous two cases. More specifically the inner part of

the wing breaks harmonically with amplitude 35o

and mean value -25o

while the phase lapse between breaking

and flapping is 90o

. This leads to breaking the wing during upstroke and extending it during downstroke. For this

case 0.4St andmax

12a at the spanwise section located at 55% of semi-span. Figures 22, 23, 24 and 25

show the time evolution of the wing geometry over one period for the upstroke and the downstroke movements

respectively. Figures 26,27,28,29,30 and 31 show the 3-D pattern of the shear layer emanating from the wing

trailing edge at various perspectives. More specifically figures 26, 28 and 30 present an artistic addition which

shows explicitly the shape and direction of created ring vortices. From those figures we observe that the

considered flight is again characterized by a creation of ring vortices with inclined axis with respect to the axis of

bird parallel movement. Notice that, the topology and number of vortices created by this flight in one period is

different from that observed in the previous cases. There is also a ground effect between the wings during

upstroke, the effect of which can be seen in the time evolution of shear layer geometry. Figure 32 presents thecalculated thrust and lift. Notice the effect of wing breaking in the instantaneous forces , which results in both

serious lift and thrust forces. The mean thrust is 0.21N while the mean lift is 0.19N and the mean propulsive

efficiency is 46%, lower than that of the cases considered previously.

Figures 22,23. Sucessive wing positions over one periodUpstroke


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Figures 24, 25. Sucessive wing positions over one period - Downstroke

Figures 26, 27. Shear layer dynamics. Colour represents dipole intensity - Side view

Figures 28,29. Shear layer dynamics. Top view


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Figures 30, 31. Shear layer dynamics. Perspective view

Note that, except for the big difference in position and arrangement of vortex rings, there is also a groundeffect between the wings during upstroke, the effect of which can be seen in the time evolution of results. The

mean thrust is 0.21N while the mean lift is 0.19N and the mean propulsive efficiency is 46%, lower than cases

before, but more than acceptable, as the lift is also very high.

its

Forces

0 50 100 150 200

-0.5

0

0.5

1

1.5

2

2.5

3

Thrust

Lift

Figures 32. Instantaneous thrust and lift as a function of time.

CONCLUSIONS:

Bird flight is a very complex phenomenon much more difficult than that of fish swimming. In this paper the

problem of simulating the birds wing geometry and kinematics has been attacked and a solution has beenproposed using the spine-rail concept. A numerical method has then been developed based on this approach

capable of producing unsteady BEM grids simulating bird wings in various flight regimes. The method has been

applied to three realistic bird flight cases and the dynamics of the free shear layers and the unsteady forces have

been calculated. From the calculated shear layer patterns it is shown that in all cases bird dynamics is maintained

by considering the conservation of linear momentum with a number of strong well formed ring vortices (and

corresponding jet flows) with proper directions with respect to the instantaneous axis of bird flight. From the

preliminary investigation it is clear that the determination of the proper range of the parameters controllingbirds

flight with best efficiency and or maneuvering characteristics should be a difficult subject for further research fora biomemetic system designer.


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UNDERSTANDING BIRD’S FLIGHT, USING A 3-D BEM METHOD AND A TIME STEPPING ALGORITHM.

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