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Lecture 27: Lift Many biological devices (Biofoils) are used to create Lift. How do these work? First, some definitions… wing section,c (chord) wing length, R chord section analysis…. (normal to U) lift wing area, S total (normal to wing) force angle of attack = a drag (parallel to U) wing velocity = U Two ways to derive lift: 1) mass deflection total force a U Surface area, S Force dtd (mU ) ~ SU U Force 12 Ctotal SU 2 Ctotal f (Re,a ) 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. 1) Mass deflection lift a U Force 12 Ctotal SU 2 2 1 Lift 2 Clift SU Drag 12 Cdrag SU 2 Clift Ctotal cosa Cdrag Ctotal sin a Cviscous total force Lift and drag are defined as components perpendicular and parallel to direction of motion. drag Surface area, S air deflected downward by wing RoboFly dimensionless scaling parameters amplitude · length2 Reynolds number = frequency · viscosity reduced frequency = forward velocity length · angular velocity total force CL Fs a q CD 3.5 90o 80 total force orientation q (degs) 3.0 total force coefficient CT 100 2.5 2.0 1.5 1.0 60 40 20 0 -20 0.5 -40 0.0 -60 -9 0 9 18 27 36 45 54 63 72 81 90 angle of attack a (degs) -9 0 9 18 27 36 45 54 63 72 81 90 angle of attack a (degs) 4 3 CT CT CT cos a a 2 CT = 3.5 sin a 1 CT sin a 0 0 15 30 45 60 75 90 angle of attack (a) 3 4 CD = CT sin a CL = CT cos a 3 2 CL CD 2 1 1 viscous drag { 0 0 0 15 30 45 60 75 angle of attack (a) 90 0 15 30 45 60 75 angle of attack (a) 90 lift a total force drag Surface area, S U Lift Clift SU 1 2 2 Drag 12 Cdrag SU 2 Clift k sin a cosa Cdrag k sin a sin a Cviscous k ~ Polar plot of lift and drag: 3 highest lift:drag ratio 2 force coefficients 3.0 CD 2.5 2.0 1.5 CL lift coefficient 3.5 a=45 a=22.5 1 1.0 0.5 a=-9 0 0.0 -0.5 a=90 a=-9 -9 0 9 18 27 36 45 54 63 72 81 90 angle of attack (degs) -1 0 1 2 drag coefficient 3 4 2. Circulation fluid travels faster over to of biofoil Flow is tangential at trailing edge U Flow separates at leading edge Law of continuity applies to streamline Difference in velocity across surface is equivalent to net circular flow around biofoil = Circulation, G mathematically: G U dS G dA U Kutta-Joukowski Theorem: Lift UG (lift per unit span) 1 combine with 2 previous definition: CL SU 2 / R UG G CL Uc R=biofoil length c= biofoil width Consider 2D biofoil starting from rest: G=0 G=0 G starting vortex bound vortex -G Required by Kelvin’s Law 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 Area = A Circulation, G Ring momentum = mass flux through ring= GA Force = d/dt (GA) = G d/dt(A) = G R U Force/R = GU = Kutta-Joukwski Therefore, elongation of vortex ring is manifestation of force on biofoil. Three important descriptors of fluid motion: 1. velocity, u(x,y) ux uy 2. vorticity, (x,y) Duy Dx x y u(x,y) 3. circulation, G Dux Dy GS Fslap = m U / t Fstroke = G A /t Momentum of vortex ring G A where m is bolus of accelerated water, moving at velocity, u impulse (F x t) = mass x velocity A G = circulation