1. Lecture 27: Lift
Many biological devices (Biofoils) are used to create Lift.
How do these work?

2. chord section analysis….
First, some definitions…
wing section,c
(chord)
wing length, R
wing area, S
total
force
(normal to wing)
drag
lift
(normal to U)
(parallel to U)
chord section analysis….
wing
velocity = U
angle of
attack = a

3. air deflected downward by wing
Two ways to derive lift:
1) mass deflection
total
force a U
Surface area, S
air deflected downward by wing
Pressure always acts normal to the surface of an object.
Therefore, this mass deflection force acts roughly perpendicular to surface of biofoil.

4. air deflected downward by wing
1) Mass deflection
total
force
lift
Lift and drag are defined as components perpendicular and parallel to direction of motion. a
drag U
Surface area, S
air deflected downward by wing

5. RoboFly dimensionless scaling parameters amplitude · length2
Borf_silent
amplitude · length2
frequency · viscosity
Reynolds number =
reduced frequency =
forward velocity
length · angular velocity
dimensionless scaling parameters

6. Fs a q total force 90o CL CD total force coefficient CT- 9 1 8 2 7 3 6 4 5 .
total force coefficient CT
angle of attack
a (degs) - 9 1 8 2 7 3 6 4 5
90o
total force orientation
q (degs)
angle of attack
a (degs)

7. a { CT CT cos a CT CT = 3.5 sin a CT sin a CD = CT sin a CL = CT cos a1 5 3 4 6 7 9 2 CT
angle of attack (a)
CT = 3.5 sin a
CT sin a a CT
CT cos a 1 5 3 4 6 7 9 2
angle of attack (a)
CL = CT cos a CL 1 5 3 4 6 7 9 2 CD
angle of attack (a)
CD = CT sin a {
viscous
drag

8. total
force
lift a
drag U
Surface area, S

9. highest lift:drag ratio
Polar plot of lift and drag: 1 2 3 4 -
drag coefficient
lift coefficient
a=-9
a=22.5
a=45
a=90
highest lift:drag ratio - 9 1 8 2 7 3 6 4 5 .
angle of attack (degs)
force coefficients CL CD

10. 2. Circulation U fluid travels faster over to of biofoil
Flow is tangential at trailing edge
Law of continuity applies to streamline
Flow separates at leading edge U

11. Kutta-Joukowski Theorem:
Difference in velocity across surface is equivalent to net circular flow around biofoil = Circulation, G
mathematically: U
Kutta-Joukowski Theorem:
(lift per unit span)
combine with previous definition:
R=biofoil length c= biofoil width

12. +G -G G=0 G=0 Consider 2D biofoil starting from rest:
starting vortex
bound vortex
Required by Kelvin’s Law

13. Circulation, G, is constant along vortex ring
Consider 3D biofoil starting from rest:
Helmholtz’ Law requires that a vortex filament cannot end abruptly:
bound vortex
Downward flow through center
of vortex ring
starting vortex
tip vortex
Circulation, G, is constant along vortex ring

15. How is structure of vortex ring related to lift on biofoil?
forward velocity, U R
Circulation, G
Area = A
Ring momentum = mass flux through ring= GrA
Force = d/dt (GrA)
= Gr d/dt(A)
= Gr R U
Force/R = GrU = Kutta-Joukwski
Therefore, elongation of vortex ring is manifestation of force on biofoil.

17. ux uy u(x,y) Three important descriptors of fluid motion:
1. velocity, u(x,y)
u(x,y) ux uy
2. vorticity, w(x,y)
3. circulation, G Dx
Duy
w =
G = S w x y
Dux Dy

20. Fslap = m U / t
where m is bolus of accelerated water, moving at velocity, u
impulse (F x t) = mass x velocity
Fstroke = r G A /t
Momentum
of vortex ring
r G A
G = circulation A