Lecture 27: Lift

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

First, some definitions…

wing section,c(chord)

totalforce(normal to wing)

drag

lift(normal to U)

(parallel to U)

chord section analysis….

wingvelocity = U

angle ofattack =

winglength, R wing

area, S

Two ways to derive lift:1) mass deflection

totalforce

U

air deflecteddownward by wing

USUmUForce dtd ~)(

Surface area, S

)(Re,

221

fC

SUCForce

total

total

Pressure always acts normal to the surface of an object.Therefore, this mass deflection force acts roughly perpendicular to surface of biofoil.

1) Massdeflection

totalforce

drag

lift

U

air deflecteddownward by wing

Surface area, S

221 SUCForce total

Lift and drag are defined ascomponents perpendicularand parallel to direction of motion.

viscoustotaldrag

totallift

drag

lift

CCC

CC

SUCDrag

SUCLift

sin

cos

221

221

RoboFly

amplitude · length2

frequency · viscosityReynolds number =

reduced frequency =forward velocity

length · angular velocity

dimensionless scaling parameters

totalforce

-9 0 9 18 27 36 45 54 63 72 81 90

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

tota

l for

ce c

oeff

icie

ntC

T

angle of attack (degs)

-9 0 9 18 27 36 45 54 63 72 81 90

-60

-40

-20

0

20

40

60

80

10090o

tota

l for

ce o

rien

tati

on

degs

angle of attack (degs)

CL

CD

Fs

0 15 30 45 60 75 90

0

1

2

3

4

CT

angle of attack ()

CT = 3.5 sin

0 15 30 45 60 75 90

0

1

2

3

4

CD

angle of attack ()

CD = CT sin

0 15 30 45 60 75 90

0

1

2

3

angle of attack ()

CL = CT cos

CL

{viscous

drag

CT sin

CT

CT cos

totalforce

drag

lift

U

Surface area, S

~

sinsin

cossin

221

221

k

CkC

kC

SUCDrag

SUCLift

viscousdrag

lift

drag

lift

-9 0 9 18 27 36 45 54 63 72 81 90-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

angle of attack (degs)

forc

e co

effi

cien

ts

CL

CD

Polar plot of lift and drag:

0 1 2 3 4-1

0

1

2

3

drag coefficient

lift

coe

ffic

ient

=-9

=-9

=22.5

=45

=90

highestlift:drag ratio

U

Flow is tangentialat trailing edge

Flow separatesat leading edge

2. Circulation

Law of continuity applies to streamline

fluid travels fasterover to of biofoil

U

Difference in velocityacross surface is equivalent

to net circular flow around biofoil = Circulation,

dSUmathematically:

dA

Kutta-Joukowski Theorem:

UtLif (lift per unit span) Uc

C

URSUC

L

L

/221

combine withpreviousdefinition: R=biofoil length

c= biofoil width

Consider 2D biofoil starting from rest:

=0

=0

startingvortex

boundvortex

Required byKelvin’s Law

Consider 3D biofoil starting from rest:

startingvortex

tipvortex

boundvortex

Downward flowthrough centerof vortex ring

Circulation, , is constantalong vortex ring

Helmholtz’ Law requires that a vortex filament cannot end abruptly:

How is structure of vortex ring related to lift on biofoil?

Circulation, Area = A

forward velocity, U

Ring momentum =mass flux through

ring=A

Force = d/dt (A)

= d/dt(A)

= R U

Force/R = U = Kutta-Joukwski

R

Therefore, elongation of vortex ring is manifestation of force on biofoil.

Three important descriptors of fluid motion:

2. vorticity, (x,y)

1. velocity, u(x,y)

u(x,y)

ux

uy

x

y

uxyx

uy

3. circulation,

Fslap = m U / t

where m is bolus of accelerated water, moving atvelocity, u

impulse (F x t) = mass x velocity

Fstroke = A /t

Momentum of vortex ring A

= circulation

A