Spontaneous Symmetry Breaking of Hinged Flapping Filament Generates Lift

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Mazzino & Alessandro Bottaro (Genova University, Italy)

1 Spontaneous Symmetry Breaking of Hinged Flapping Filament Generates Lift Shervin Bagheri (KTH, Stockholm) Andrea Mazzino & Alessandro Bottaro (Genova University, Italy) Fluid & Elasticity 2012 La Jolla, San Diego, Nov , 2012

2 Bio-Inspired Flow Control Frank Fish How does non-smooth flexible surfaces, appendages affect moving bodies? 2

3 Configuration A moving body with a hinged flexible filament U Moving bluff body Flexible filament How does the filament interact with the fluid? modify the motion of the body? 3

4 Symmetry Breaking Filament flaps asymmetrically a net force/torque on body reduced drag on body 4

5 Flow Past Body Reynolds number Re = UDρ f µ Vortex shedding for Re > Re c with frequency f c U µ ρ f D (Williamson, Ann. Rev. Fluid Mech. 1996) 5

6 Flow Past Filament Reynolds number Re = UL sρ f µ mass R 1 = rigidity R 2 = ρ s ρ f L s B ρ f U 2 L 3 s µ ρ f U ρ s B L s Flexible filament 6

7 Flow Past Filament Flapping when Re > 10 3 R 1 > 0 R 2 <R 2,c U Flapping structure (Zhang & Shelley, Ann. Rev. Fluid Mech. 2012) 7

8 Numerical Treatment Flow dynamics (Navier-Stokes) Filament dynamics (Euler-Bernoulli Beam) 4 parameters 8

9 Long Filament Re = 100 R 2 =0.05 R 1 =0.1 L =3 9

10 Short Filament Re = 100 R 2 =0.05 R 1 =0.1 L =1.5 10

11 Symmetry Breaking L =0 C q =0 (drag) (lift) (torque) C L =0.18 C q =0.01 (drag) (lift) (torque) 11

12 Choice of Observable Angle of horizontal line & line connecting filament tail Consider 2 cases Rigid filament: R 2 =0.005 Flexible filament: R 2 =0.1 12

13 Bifurcation Bifurcation: (flexible filament) (rigid filament) 13

14 Beam Equation Equation governing unforced beam Eigenfrequency 14

15 Resonance Condition Free vibrations of filament f s Vortex shedding frequency If filament very slow reaction time If filament react instantaneously Thus separates two different regimes Gives resonance condition: 15

16 Filament Energy Energy E = 1 2 L 0 R 1 X t 2 + R 2 X ss 2 ds Rescaled with filament density and length Flapping synchronized with vortex shedding, time scale rescaled non-dimensional filament energy 16

17 Resonance Resonance: (flexible) (rigid) 17

18 Resonance Resonance (theoretical) Resonance (computed) Bifurcation (computed) Flexible Rigid

19 Can Filament Alter Motion? Swimming sea slug flapping of wings (Re>10) beating of cilia (Re<1) Inert cilia alter motion interaction with fluid without energy expended Clione antarctica (Childress, & Dudley, JFM 2004) 19

20 Can Filament Increase Drift? Efficient wind-borne seed dispersal Side force due to symmetry breaking may increase drift Wind Release point Rigid body with filament Rigid body Drift Dandelion plant Ground level (Burrows, New Phytol. 1975) 20

21 Thank you! Reference: Bagheri, Mazzino & Bottaro, PRL, 109, 2012 See also: Lisa Zyga, PhysORG, 22 nd Oct (

22 Outline General physics of flow past a cylinder flow past a filament Symmetry breaking of cylinder + filament resonance between fluid & structure generation of net lift, torque Immersed boundary method Conlusion & outlook 22

23 Fluid Equations Viscous incompressible fluid Momentum Continuity u t + u u = p + 1 Re 2 u u =0 Flow solver Discretize on Cartesian grid No dynamic equation for pressure Projection method (Chorin, 1968 & Temam, 1969) 23

24 Flow Past Rigid Body Viscous incompressible fluid Momentum u t + u u = p + 1 Re 2 u flow Continuity u =0 No-slip u = 0 on Γ rigid body Γ 24

25 Immersed Boundary Method Viscous incompressible fluid Momentum u t + u u = p + 1 Re 2 u Continuity u =0 No-slip u = 0 on Γ + Γ f(ζ)δ(x ζ) dζ f 1 ζ f 2 1 ζ 2 rigid body ζ 3 flow f 3 25

26 Immersed Boundary Method Immersed boundary method u t + u u = p + 1 Re 2 u+ u =0 u = 0 on Γ Γ f(ζ)δ(x ζ) dζ flow Flow field: Eulerian (Cartesian grid) Boundary: Lagrangian points Boundary force to enforce no-slip Projection method f 1 ζ f 2 1 ζ 2 rigid body ζ 3 (Taira & Colonius, JCP, 2005) f 3 26

27 Flow Past Flexible Filament Viscous incompressible fluid No-slip u t + u u = p + 1 Re 2 u u =0 + Γ f(ζ)δ(x ζ) dζ flow u(γ) = ζ f 1 Filament dynamics Inertia ρ s ζ = (T ˆτ) B 2 (Cˆn) +f Tensile Bending force force f 3 f 2 flexible filament (Peskin, 1997, 2002, Kim & Peskin 2007) 27

28 Current Work Problems to be tackled: 1. Free falling bluff body with filament 2. Interaction among particles with filament 3. Bodies with distributed, anisotropic coatings Approach: 1. Numerical (Lagrangian methods) 2. Experimental (soap film experiments) 3. Theoretical (stability/bifurcation/resonance analyses) 28

29 FSI for Multiple Moving/Flexible Bodies Developing direct numerical simulation of fluid/structure combination of vortex methods and immersed boundary methods (Gazzola et al, JCP, 2011) 29

30 Soap Film Experiments Developing experimental facilities for fluid/structure soap film, water tank et (Zhang etal, Nature, 2000) (Rutgers etal, Rev. Sci. Inst. 2001) 30

31 Discretization of Fluid Equations Viscous incompressible fluid Momentum Continuity u t + u u = p + 1 Re 2 u u =0 Discretize (Adams-Bashforth+Crank-Nicolson) Momentum u n+1 u n t N(un ) 1 2 N(un 1 )= Gp n Re L(un+1 + u n ) Continuity Du n+1 =0 31

32 Algebraic system Algebraic system Linear system LU Factorization A G D 0 A G D 0 = u n+1 p A 0 D DA 1 G = r n 0 I A 1 G 0 I Projection/Fractional step method Momentum Au = r n Pressure Poisson DA 1 Gp n+1 = Du Projection u n+1 = u A 1 Gp n+1 32

33 Symmetry Breaking (drag) (lift) C q =0 (torque) C L =0.18 C q =0.01 L =0 C q =0 33

Spontaneous Symmetry Breaking of a Hinged Flapping Filament Generates Lift

Spontaneous Symmetry Breaking of a Hinged Flapping Filament Generates Lift A. Bottaro (DICCA, University of Genova) S. Bagheri (KTH, Stockholm) A. Mazzino (DICCA, Genova) J. Favier (AMU, Marseille) A.