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

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

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.

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

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

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)

a { CT CT cos a CT CT = 3.5 sin a CT sin a CD = CT sin a CL = CT cos a 1 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

total force lift a drag U Surface area, S

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

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

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

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

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

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.

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

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