(4) BIOMECHANICS of FLIGHT
Questions:
-
How do birds, bats, and insects fly ?
-
How does size influence the mode and
speed of flight (Gliding, soaring,
flapping flight) ?
-
What determines the energy cost of
flying and how does it compare to
running and swimming ?
Aerodynamics:
-
Aerofoils, lift and drag coefficients,
Reynolds number, power.
Biomechanics of Flight 4-1
4.1
AERODYNAMICS of WINGS
1
ρS p C L v 2
2
1
FD = ρS pCD v 2
2
FL =
Lift
Drag
Where
CL
CD
Sp
=
Lift coefficient.
=
Drag coefficient.
=
Planar area of
aerofoil
( Sp ≈ Sw / 2 ).
For horizontal flight:
Weight
Drag
= Lift
= Thrust (Generated by
flapping of wings).
Biomechanics of Flight 4-2
Aerodynamic Lift
Production of lift by the Coanda effect:
-
Fluid stream tends to follow the
aerofoil surface.
-
+ve angle of attack deflects air
downward, i.e and results in a change
in momentum of air.
∴ Produces lift and drag forces
( F = dp / dt ) on wing.
-
Coanda effect due to Van der Waals
forces (small interatomic force).
-
Air molecules experience an
electrostatic "cling" to aerofoil
surface.
For effective locomotion, we require:
-
Maximisation of lift-to-drag ratio,
FL / FD , to bring about the least
energy cost for horizontal flight.
Biomechanics of Flight 4-3
Angle of attack, α
For man-made aerofoils:
Drag -
Increasing the angle of attack
also increases drag.
-
Lift
-
CD > 0
at
α = 0°.
Increasing the angle of attack
also increases lift (up to a
maximum angle of α
-
Wings stall at α
= 20°).
> 20°.
(Flow separates from upper
surface of wing, eddies form
behind the wing, pressure drag
increases).
Biomechanics of Flight 4-4
For a given angle of attack, CL is almost independent of Re.
-
Maximum lift coefficient CL(max)
CL(max) ≈ 1.0 for 103 < Re < 105 (most birds).
CL(max) ≈ 1.5 for
Re ≈ 106 (largest birds).
Biomechanics of Flight 4-5
Reynolds number:
Re = ρvl / η
-
(ηair ≈ 1.8 x 10-5 N s / m2, ρair ≈ 1.21 kg / m3)
Small fly v ≈ 1 m / s
Locust
4
Pigeon
15
l ≈ 1 mm
2 cm
12 cm
∴ Re ≈ 70
5.3 x 103
1.2 x 105
Usually Re < 106
∴ Laminar flow in boundary
layer.
Flying animals intermediate in
size spectrum:
Largest, Albatross
-
Weight: 10 kg.
Wingspan: 3.5 m.
Smallest, Insects,
- Weight: Few mg.
Biomechanics of Flight 4-6
Aerodynamic drag
Total drag force on aerofoil:
≈
Total drag
(Profile drag) + (Induced drag)
Induced drag.
Profile drag
-
Negligible pressure drag.
Mainly friction drag of aerofoil
(Even if lift is produced).
Fprofile =
Where
-
1
ρS pCDo v 2
2
Sp
=
Planar area of aerofoil.
CDo
=
Profile drag coefficient.
For laminar boundary layer:
2.6
S 1.33
CDo ≈ w . 1 / 2 ≈ 1 / 2
S p Re
Re
(decreases with increasing velocity).
Due to the production of lift.
(Force required to deflect air
downwards).
Finduced =
Where
CD(induced)
1
ρS pC D (induced )v 2
2
= Induced drag coefficient
κCL2
≈
πA
Here,
A
κ
=
Aspect ratio
=
Depends on wing shape
(κ
= π / 4 ).
∴ Finduced
FL2
≈
2 ρS p Av 2
Biomechanics of Flight 4-7
Induced drag:
At high speed:
-
Aerofoil area cuts large
air mass.
-
Lift ( FL = mg ) generated by giving a
large air mass a low velocity.
-
i.e. Low change in momentum of air.
∴ Low induced drag.
At low speed:
-
-
-
Aerofoil area cuts small
air mass.
Lift generated by giving a small air
mass a high velocity.
Total drag coefficient of aerofoil:
i.e. high change in momentum of air.
∴ High induced drag.
Total drag force on flying animal:
CD ≈CDo + CD (induced )
2.6 κCL2
≈ 1/ 2 +
πA
Re
Total drag ≈ (drag on wings) + (drag on body)
Profile drag
Induced drag
Pressure drag
Skin friction
≈ (induced drag) + (parasitic drag)
Biomechanics of Flight 4-8
Power Required for Flight
Power required to maintain horizontal flight
P ≈ FD .velocity
-
Total power of drag versus pigeon
airspeed.
Metabolic power required for flight.
Very large birds
-
Close to limit of energy expenditure:
∴ Just able to power sustained flight.
(As a result often resort to gliding
and soaring).
Small birds
-
Surplus of power.
∴ Are able to use high energy
consumption flying techniques
(hovering)
Biomechanics of Flight 4-9
Aerofoil Shape
Flat
Cambered
Types of aerofoils:
Plate
Streamlined
Slotted
Best wing shape (highest lift / drag ratio).
Large birds:
- Streamlined wing.
( > 0.5 kg, Re > 105)
Small birds, bats and insects:
( < 0.5 kg, Re < 105)
- Flat or cambered plate wing.
Very small insects:
( < 10 mg, Re < 102)
- Impractical aerofoil, drag >> lift.
∴ Minimum size of flying animal.
Biomechanics of Flight 4-10
Aerofoil:
CL(max) ≈ 1.0 – 1.5
Slotted aerofoil:
CL(max) ≈ 2.0
∴ To prevent stalling,
fly with high
angle of attack.
Multiple slotted aerofoil:
(like biplane / triplane)
CL(max) ≈ 2.0 –4.0
The primary feathers of some birds separate to form
slotted wings.
Biomechanics of Flight 4-11
4.2
GLIDING
Gliders:
-
Birds, bats and insects.
Flying squirrels and flying
lizards.
Mechanics of Gliding
-
A bird will eventually sink to
the ground unless it resorts to
flapping or soaring.
-
No thrust
∴ Loss of PE is required to
supply work against drag.
(Gliding flight is powered by
gravity).
-
Low energy cost of locomotion.
-
However, metabolic energy is
required to maintain tension in
wing muscles.
m
v
θ
vsinθ
=
Mass (kg)
=
Airspeed, relative to air ( m / s )
=
Angle to horizontal glide ( 3-5 deg ).
=
Sinking speed ( m / s ).
Biomechanics of Flight 4-12
Equilibrium (constant velocity)
1
ρS p C L v 2
2
1
FD = mg sin θ = ρS pCD v 2
2
FL = mg cos θ =
Gliding angle is usually small ( θ
∴ v≈
Where
-
Where
CL
CD
SP
ρ
=
Lift coefficient.
=
Drag coefficient.
=
Aerofoil planar area.
=
Air density.
≈ 0, cosθ ≈ 1 ).
2mg
2N
≈
ρS p C L
ρC L
N
=
Wing loading (normal force per unit area).
N=
FL mg
≈
Sp Sp
Alter CL by adjusting angle of attack, α , therefore glide at various speeds.
Gliding modes:
(a)
v1 , Minimum glide speed, v
(b)
v2 , Travel as far as possible for given loss of height by minimising glide angle, θ
(c)
v3 , Airborne for maximum time by minimising sinking speed v sinθ
Biomechanics of Flight 4-13
(a) Minimum glide speed, v:
∴ v1 ≈
2N
ρC L
Where CL(max) = Maximum lift coefficient
-
CL(max) ≈ 1.0 – 1.5 ( at α
≈ 20° )
Example:
-
Minimum speed when landing.
Biomechanics of Flight 4-14
(b) Travel as far as possible for given loss of height by minimising glide angle, θ
-
For small angle θ , minimise tanθ
≈ sinθ ).
FD
mg
profile drag + induced drag
≈
mg
1
ρS pCDov 2
FL2
2
≈
+
mg
2 ρS p Av 2 mg
sin θ =
ρCDov 2
N
sin θ =
+
2N
2 ρAv 2
sinθ is a minimum w.r.t. v
when
d (sin θ )
=0
dv
d (sin θ ) 2 ρCDov
N -2
+
. =0
=
dv
2N
2 ρA v 3
∴ v2 ≈
2
N
ρ 2 ACDo
14
Example: Migratory birds gliding between thermals (i.e. storks).
Biomechanics of Flight 4-15
(b) Travel as far as possible for given loss of height by minimising glide angle, θ
-
Derivation of minimum glide angle θmin: From previous slide:
sin θmin =
N 1
ρCDo 2
.
v +
2N
2 ρA v 2
…(1)
And since minimum velocity v2
2
v2 ≈
N
ρ 2 ACDo
14
…(2)
By substituting (2) into (1),
N2
N
ρ 2 ACDo
+
ρ 2 ACDo 2 ρA
N2
sin θmin
ρCDo
=
2N
sin θmin
ACDo
CDo
=
+
2 ACDo
2A
sin θmin
CDo
CDo
=
+
2 A 2 A
⇒ sin θmin =
CDo
A
∴ θmin = arcsin ( CDo / A )
Biomechanics of Flight 4-16
(c) Airborne for maximum time ( minimise sinking speed v
sinθ ).
v cosθ
From previous slides,
ρCDov 2
N
sin θ =
+
2N
2 ρAv 2
ρCDov3
N
⇒ v sin θ =
+
2N
2 ρAv
vsinθ is a minimum w.r.t. v
when
v sinθ
θ
v
d (v sin θ )
=0
dv
N2
v3 ≈ 2
3 ρ ACDo
14
≈0.76v2
Examples:
-
Bird hunting for prey.
-
Bird gaining height in thermal.
Biomechanics of Flight 4-17
High Induced Drag
High Profile Drag
Biomechanics of Flight 4-18
Biomechanics of Flight 4-19
Wing Loading
Wing loading is given by
mg
N≈
∝
Sp
-
(mass) / (wing area)
Glide velocities for minimum horizontal and vertical speeds v1, v2, v3 ∝
N
∴ High wing loading ( small wings ) → Fast gliding
∴ Low wing loading ( large wings )
→ Slow gliding
For geometrically similar animals:
Wing area, Sp
∝ (length)2
∝ (mass)2/3
N ≈ mg / Sp ∝ (mass) / (mass)2/3
∝ (mass)1/3
Glide velocity, v
∝ (mass)1/6
Wing loading,
∴ Large animals glide faster.
Biomechanics of Flight 4-20
Wing loading = mass1/3
Biomechanics of Flight 4-21
Control of Gliding
Glide speed:
v≈
-
2N
ρC L
Adjust glide speed by:
-
Alter CL by adjusting angle of attack of wings, α
Alter wing loading
N = FL / Sp
A bird spreads its wings
more when gliding slowly
than when gliding fast.
(Change Sp by partly folding / extending wings
and spreading tail).
Biomechanics of Flight 4-22
Gliding at low speed:
-
Involves use of the Alula (A)
-
Tuft of feathers on front edge of wing supported by
bone of index finger.
-
Alula is lifted at low speeds.
It keeps air flowing smoothly over the wings at high
angle of attack.
Therefore, the alula helps avoid stalling.
A small leading edge ‘slat’
above an aerofoil wing in a
wind tunnel demonstrates
reduced turbulence effects.
Biomechanics of Flight 4-23
Glide angle:
-
Adjust glide angle by moving wings (centre of
pressure) forward and backward.
-
Glide more steeply (for a given forward speed) by
using feet as air brakes.
Turning:
-
Is performed by rotating wings.
Gives one wing a higher angle of attack,
∴ More lift.
-
Tilt towards inside of turn.
∴ Lift has horizontal component to provide
centripetal acceleration.
Biomechanics of Flight 4-24
4.3
SOARING
-
Prolong gliding flight by using natural air movements.
-
A bird sinks relative to the air, but rises relative to the
ground if air is rising faster than the birds sinking
speed.
vbird / ground
vair / ground
vbird / air
Thermal Soaring
Thermals
-
Columns of rising air formed by irregular heating of the ground.
Warm, heated air expands, air less dense, air rises.
East African plains:
vthermal = 4 m/s > sinking speed of vulture.
-
Bird rises by gliding in a helix of small radius.
The best thermal soarers:
-
Are large birds with very low wing loading.
-
Such as vultures and storks.
Biomechanics of Flight 4-25
Slope Soaring and Wind Gradient Soaring
Slope soaring:
-
Wind can be deflected by ground or ocean
waves (used by kestrels, gulls, albatross).
-
Best slope soarers have fast gliding
speed (faster than wind).
-
Large birds with high aspect ratio wings
with high wing loading.
Wing gradient soaring:
-
Gradient in horizontal wind speed in
boundary layer.
Close to the surface of the sea (12m).
Biomechanics of Flight 4-26
4.4
FLAPPING FLIGHT
Horizontal flight:
Forces are similar to gliding with exception
that thrust is exerted.
Lift and thrust are supplied by muscles.
Work is done against gravity and drag.
Fast Flight
Downstroke
-
Power stroke, +ve angle of attack
produces lift, thrust and small amount
of drag.
Upstroke
-
Acts to reposition the wing. There is a
small angle of attack that produces a
small amount of lift and also a small
amount of drag.
Biomechanics of Flight 4-27
Over a complete cycle of flapping:
Lift
Thrust
=
=
Weight.
Drag on wings and body.
Wing is folded on upstroke
-
Reduces wing area.
-
Drives less air.
-
Reduced aerodynamic forces (lift and
drag).
Biomechanics of Flight 4-28
Vortex rings:
-
Exploit circulation of air around wings
to increase lift.
-
Vortex ring is produced by moving air.
Fast flapping flight
-
Continuous vortex ring is produced.
Biomechanics of Flight 4-29
V-formation flying
-
Large migrating birds (geese).
-
Reduce metabolic energy cost by
reducing the number of vortices.
Biomechanics of Flight 4-30
Slow Flight
-
Lift is generated on down stroke
(vortex ring is produced).
-
No lift is generated on upstroke.
Biomechanics of Flight 4-31
Bounding Flight
-
Beat wings for a few cycles, fold wings
against the body, followed by beat
wings again.
-
Minimise metabolic energy
consumption.
The optimum flapping rate at which
muscles are most efficient and produce
the most power
-
Type of flight used by birds with plenty
of power in reserve.
Biomechanics of Flight 4-32
Undulating Flight
-
Beat wings for a few cycles, glide, beat
wings again.
-
Used by large birds (i.e. crows, gulls).
Hovering
-
Slow forward flight.
-
The velocity of the wingtip
>> the velocity of the COG.
-
Several techniques are used by insects,
bats and small birds.
-
Much of the lift is generated by
“Unsteady effects”.
-
Features a high wing beat frequency
(for hummingbird, 15-60 Hz).
Biomechanics of Flight 4-33
Scaling
Geometrically similar birds:
∝
∝
Wing area
-
(Length)2
(mass)2/3
Wing beat frequency
2 Hz
20 Hz
260 Hz
∝ (wing length)-1
condor
hummingbird
mosquito
Is observed for petrels, albatrosses.
Different wings for different flying habits.
-
Size (mass, wingspan, wing area)
Wing loading (force / area).
Aspect ratio (length / span).
Seabirds
-
Gannet, albatross, gull, tern (High aspect ratios).
Thermal soarers -
Vulture, pelican, stork (Low wing loading, low aspect ratio).
Birds of prey
-
Owl, hawk, kite, eagle (fly with load of prey)
Low wing loading
Game birds
-
Grouse, pheasant, peacock, turkey (not fly well).
Low aspect ratio, small and broad wings.
Biomechanics of Flight 4-34
Principal component analysis of wing dimensions
separates birds with different flying habits
Biomechanics of Flight 4-35
Metabolic Energy Cost
Rate of metabolic energy consumption for flying:
-
Greater power is required for flight than for swimming and running.
-
Wing muscles must produce lift to stay aloft (overcome gravity) and thrust for
forward motion.
-
There is a minimum power consumption at optimum flying speed.
Biomechanics of Flight 4-36
Energy cost of transport [ J / (kg m)].
-
Larger flying animals have higher flying velocity
∴
Travel given distance with less energy consumption
∴
Low energy cost of transport.
Comparison of cost of transport:
Running
Swimming
Flying
-
Negligible drag.
-
High muscle tension required to counteract effect of gravity.
∴ Intermediate cost of transport.
-
High drag in viscous medium
(water).
-
Buoyancy negates effect
of gravity.
∴ Low cost of transport.
-
Considerable drag in
sparse medium (air).
-
Produce lift to counter
effect of gravity and thrust
for locomotion.
∴ High cost of transport.
Biomechanics of Flight 4-37