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04_biomechanics_of_flight1-1_cm_219.pdf
(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
Lift
Drag
Where CL = Lift coefficient. CD = Drag coefficient. Sp = Planar area of
aerofoil ( Sp | Sw / 2 ).
For horizontal flight:
Weight = Lift Drag = Thrust (Generated by
flapping of wings).
2
2 1
= vCSȡF LpL
2
2 1
= vCSȡF DpD
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, D
For man-made aerofoils:
Drag - Increasing the angle of attack also increases drag.
- CD > 0 at D = 0°.
Lift - Increasing the angle of attack also increases lift (up to a maximum angle of D = 20°).
- Wings stall at D > 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 = Uvl / K (Kair § 1.8 x 10-5 N s / m2, Uair § 1.21 kg / m3)
- Small fly v § 1 m / s l § 1 mm ? Re § 70 - Locust 4 2 cm 5.3 x 103 - Pigeon 15 12 cm 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)
Profile drag
- Negligible pressure drag. - Mainly friction drag of aerofoil
(Even if lift is produced).
Where Sp = Planar area of aerofoil.
CDo = Profile drag coefficient.
- For laminar boundary layer:
(decreases with increasing velocity).
Induced drag.
- Due to the production of lift. (Force required to deflect air downwards).
Where CD(induced) = Induced drag coefficient
Here, A = Aspect ratio N = Depends on wing shape
(N = S / 4 ).
2 )(2
1 = vCSȡF inducedDpinduced
Aʌ Cț L
2
Ĭ
2
2
2 Ĭง
AvSȡ F
F p
L induced
2
2 1
= vCSȡF Dopprofile
2/12/1 Re 2.6
Ĭ Re
33.1 .Ĭ
p
w Do S
S C
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 Total drag coefficient of aerofoil: mass a high velocity.
- i.e. high change in momentum of air. ? High induced drag.
Total drag force on flying animal:
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
)(+Ĭ inducedDDoD CCC
Aʌ Cț L
2
2/1 +Re 6.2
Ĭ
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
Types of aerofoils: Plate
Streamlined
Slotted
Flat Cambered
Best wing shape (highest lift / drag ratio).
Large birds: ( > 0.5 kg, Re > 105) - Streamlined wing.
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: CL(max) | 2.0 –4.0 (like biplane / triplane)
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 = Mass (kg) v = Airspeed, relative to air ( m / s ) T = Angle to horizontal glide ( 3-5 deg ). vsinT = Sinking speed ( m / s ).
Biomechanics of Flight 4-12
Equilibrium (constant velocity)
Where CL = Lift coefficient. CD = Drag coefficient. SP = Aerofoil planar area. U = Air density.
Gliding angle is usually small ( T | 0, cosT | 1 ).
Where N = Wing loading (normal force per unit area).
- Alter CL by adjusting angle of attack, D , 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, T
(c) v3 , Airborne for maximum time by minimising sinking speed v sinT
2
2 1
=cos= vCSȡșmgF LpL
2
2 1
=sin= vCSȡșmgF DpD
LLp Cȡ N
CSȡ mg
v 2
Ĭ 2
Ĭง
pp
L
S mg
S F
N Ĭ=
Biomechanics of Flight 4-13
(a) Minimum glide speed, v:
Where CL(max) = Maximum lift coefficient
- CL(max) | 1.0 – 1.5 ( at D | 20° )
Example:
- Minimum speed when landing.
LCȡ N
v 2
Ĭง 1
Biomechanics of Flight 4-14
(b) Travel as far as possible for given loss of height by minimising glide angle, T
- For small angle T , minimise tanT | sinT ).
sinT is a minimum w.r.t. v when
Example: Migratory birds gliding between thermals (i.e. storks).
mg F
ș D=sin
mg drag induced+drag profile
Ĭ
mgAvSȡ FvCSȡ
p
L Dop
2
2 2
2 +
mg 2 1
Ĭ
2
2
2 +
2 =sin
Avȡ N
N vCȡ
ș Do
0= )(sin
dv șd
41
2
2
2 Ĭง DoACȡ
N v
0= 2-
. 2
+ 2
2 =
)(sin 3vAȡ
N N
vCȡ dv șd Do
Biomechanics of Flight 4-15
(b) Travel as far as possible for given loss of height by minimising glide angle, T
- Derivation of minimum glide angle Tmin: From previous slide:
(1)
And since minimum velocity v2
(2)
By substituting (2) into (1),
2 2
min 1
. 2
+ 2
=sin vAȡ
N v
N Cȡ
ș Do
41
2
2
2 Ĭ DoACȡ
N v
2
2
2
2
min 2 +
2 =sin
N ACȡ
Aȡ N
ACȡ N
N Cȡ
ș Do Do
Do
A AC
AC C
ș Do Do
Do
2 +
2 =sin min
A C
ș Do=sinය min A
C A
C ș DoDo
2 +
2 =sin min ( )ACș Do /arcsin=ง min
Biomechanics of Flight 4-16
(c) Airborne for maximum time ( minimise sinking speed v sinT ). v cosT
From previous slides, T
v sinT
v
vsinT is a minimum w.r.t. v when
Examples:
- Bird hunting for prey.
- Bird gaining height in thermal.
2
2
2 +
2 =sin
Avȡ N
N vCȡ
ș Do
Avȡ N
N vCȡ
șv Do 2
+ 2
=sinය 3
0= )sin(
dv șvd
2
41
2
2
3 0.76Ĭ3 Ĭ v
ACȡ N
v Do
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
v (mass) / (wing area)
- Glide velocities for minimum horizontal and vertical speeds v1, v2, v3 v
? High wing loading ( small wings ) o Fast gliding
? Low wing loading ( large wings ) o Slow gliding
For geometrically similar animals:
Wing area, Sp v (length)2 v (mass)2/3
Wing loading, N | mg / Sp v (mass) / (mass)2/3 v (mass)1/3
Glide velocity, v v (mass)1/6
? Large animals glide faster.
pS mg
N Ĭ
N
Biomechanics of Flight 4-20
Wing loading = mass1/3
Biomechanics of Flight 4-21
Control of Gliding
Glide speed:
- Adjust glide speed by:
- Alter CL by adjusting angle of attack of wings, D
- Alter wing loading N = FL / Sp
(Change Sp by partly folding / extending wings and spreading tail).
LCȡ N
v 2
Ĭ
A bird spreads its wings more when gliding slowly than when gliding fast.
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.
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.
vair / ground vbird / air
vbird / ground
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 = Weight. Thrust = 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 v (Length)2 v (mass)2/3
- 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.
Wing beat frequency v (wing length)-1 2 Hz condor 20 Hz hummingbird 260 Hz mosquito
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 - Negligible drag.
- High muscle tension required to counteract effect of gravity. ? Intermediate cost of transport.
Swimming - High drag in viscous medium (water).
- Buoyancy negates effect of gravity. ? Low cost of transport.
Flying - 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
