Hydrodynamics. Class 11. Laurette TUCKERMAN

Transcription

1 Hydrodynamics Class 11 Laurette TUCKERMAN

2 Faraday instability Faraday (1831): Vertical vibration of fluid layer = stripes, squares, hexagons

3 In 1990s: first fluid-dynamical quasicrystals: Edwards & Fauve Kudrolli, Pier & Gollub J. Fluid Mech. (1994) Physica D (1998)

4 Effect of horizontal motion ζ(x+u t,t+ t) = ζ(x,t) ζ(x,t)+ ζ ζ t+ u t = ζ(x,t) t x ζ t = ζ x u Effect of vertical motion ζ(x,t+ t) = ζ(x,t)+w t ζ(x,t)+ ζ t = ζ(x,t)+w t t ζ t = w Combined effect foru = φ: ζ t +u ζ x +v ζ y = w ζ t + φ ζ x x + φ ζ y y = φ z

5 Surface Tension Tangential force along surface= normal force if slope varies. ζ xx < 0 = F z < 0 to be counterbalanced by p > p 0 : p 0 p = σ 2 ζ x 2 Bernouilli equation (ideal fluid) satisfied at surface: φ dt φ 2 = p 0 p gζ ρ becomes: φ dt φ 2 = σ 2 ζ ρ x gζ 2

6 Bernouilli s equation at interface: t φ+ 1 2 φ 2 = σ ρ ( 2 x + 2 y)ζ Φ Oscillating frame of reference= oscillating gravity G(t) = g acos(ωt) Gravitational potential energy at interface: Bernouilli s equation at interface: [ t φ+ 12 φ 2 ] Φ = G(t)z = G(t)ζ (x,y,z=ζ(x,y)) Interface z = ζ(x, y, t) moves according to: = t ζ +u ζ = w [ ] σ ρ ( 2 x + y)ζ 2 G(t)ζ (x,y) Incompressibility: u = φ = 0

7 Base state: For small perturbations: u = 0 ζ = 0 t ζ (x,y) + u ζ = z φ (x,y,z=0+ ζ(x,y)) [ ] 1 t φ+ 2 φ 2 = (x,y,z=0+ ζ(x,y)) [ σ ρ ( 2 x + 2 y)ζ G(t)ζ Consider domain to be horizontally infinite (homogeneous) = solutions exponential/trigonometric in x = (x,y) Seek bounded solutions= trigonometric: exp(ik x) = exp(i(k x x+k y y)) ] (x,y) Height ζ(x,y,t) = k e ik xˆζk (t) Velocity φ(x,y,z,t) = k e ik xˆφk (z,t) 0 = φ = ( 2 z k 2 )ˆφ k = ˆφ k e ±kz Assume infinite depth(z ) = ˆφ k = e kzˆφk (t)

8 Drop hats and subscriptk t ζ = z φ z=0 = kφ z=0 = φ z=0 = t ζ/k t φ z=0 = σ ρ ( 2 x + 2 y)ζ G(t)ζ t t ζ/k = σ ρ ( k2 )ζ G(t)ζ tζ 2 = k 3σ ζ k(g acos(ωt))ζ ρ Defineω 2 0 = σ ρ k3 +gk â = ak ω tζ = ω 2 0(1 âcos(ωt))ζ a = 0 = Gravity-capillary waves = ζ e ±iω 0t a 0 = Linear equation forζ whose coefficients are periodic

9 Floquet theory Linear equations with constant coefficients: aẍ+bẋ+cx = 0 = x(t) = α 1 e λ1t +α 2 e λ 2t where aλ 2 +bλ+c = 0 ẋ = cx = x(t) = e ct x(0) N N c n x (n) = 0 = x(t) = α n e λ nt n=0 where n=1 N c n λ n = 0 Generalize to linear equations with periodic coefficients: a(t)ẍ+b(t)ẋ+c(t)x = 0 = x(t) = α 1 (t)e λ1t +α 2 (t)e λ 2t a(t),b(t),c(t) have period T = α 1 (t),α 2 (t) have period T n=0

10 Floquet theory continued a(t)ẍ+b(t)ẋ+c(t)x = 0 = x(t) = α 1 (t)e λ1t +α 2 (t)e λ 2t α 1 (t),α 2 (t) Floquet functions λ 1,λ 2 Floquet exponents growing solution if Real(λ j ) > 0 µ 1 e λ1t,µ 2 e λ 2T Floquet multipliers growing solution if µ j > 1 λ 1, λ 2 not roots of polynomial = calculate numerically or asymptotically ẋ = c(t)x = x(t) = e λt α(t) N N c n (t)x (n) = 0 = x(t) = e λnt α n (t) n=0 n=1

11 for exponent λ Region of stability for multipliere λt Imaginary part non-unique = choose Im(λ) ( πi/t, πi/t] = ( iω/2, iω/2]

12 2 tζ = ω 2 0(1 acos(ωt))ζ Temporal Floquet problem, witht = 2π/ω ζ(t) = c 1 e λ 1t f 1 (t mod T)+c 2 e λ 2t f 2 (t mod T) Two Floquet exponentsλ j and Floquet functions f j (t) for each k Real(λ) 0 for some j, k = flat surface unstable = Faraday waves with spatial wavenumberk and temporal frequency Im(λ) Im(λ) e λt waves period 0 1 harmonic T = 2π/ω (same as forcing) ω/2 1 subharmonic 2T = 4π/ω (twice forcing period)

13 Instability Tongues

14 Floquet functions λ (ζ-< ζ >)/ t / T λ (ζ-< ζ >)/ t / T (ζ-< ζ >)/ t / T within tongue 1 /2 within tongue 2 /2 within tongue 3 /2 subharmonic harmonic subharmonic µ = 1 µ = +1 µ = 1 λ -0.03

15 Inclusion of viscosity ρ t u = p+µ u u = 0 ê z ρ t u = ê z p+ê z µ u Assuming v = 0, then ρ t w = µ 2 w τ ij = pδ ij +µ( xj U i + xi U j ) As before, for linear stability analysis, evaluate atz = 0 using flat interface with normal in z direction. Continuity of tangential stress = 0 = τ xz = µ( x w + z u) 0 = τ yz = µ( y w + z v) 0 = x τ xz + y τ yz = µ( 2 xw + 2 yw + xz u+ yz v) = µ( 2 x + 2 y 2 z)w Normal stress is not zero at interface: counterbalanced by surface tension σ( 2 x + 2 z)ζ = τ zz = (p ρg(t)ζ)+2µ z w z=0

16 Instability tongues for viscous fluids Thresholds are finite instead of zero. Tongues are rounded Minima of tongues rise with frequency (1/2, 2/2, 3/2,... tongues)

17 Square patterns in Faraday instability

18 Hexagonal patterns in Faraday instability

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21 Ideal flow Cylinder wake with downstream recirculation zone Von Kármán vortex street (Re 46) Laboratory experiment (Taneda, 1982) Off Chilean coast past Juan Fernandez islands

22 Stability analysis of von Kármán vortex street 2D limit cycleu 2D (x,y,tmod T) obeys: Add 3D perturbation t U 2D = (U 2D )U 2D P 2D + 1 Re U 2D t ( U 2D +u 3D ) = (U 2D (t) )u 2D (U 2D (t) )u 3D (u 3D )U 2D (t) (u 3D )u 3D ( P 2D +p 3D )+ 1 Re ( U 2D +u 3D ) Subtract 2D equation from 3D equation and neglect quadratic terms to obtain equation governing perturbation u 3D : t u 3D = (U 2D (t) )u 3D (u 3D )U 2D (t) p 3D + 1 Re u 3D Linear equation which is homogeneous in z and periodic in t

23 von Kárman vortex street: Re = U d/ν 46 spatially: two-dimensional (x,y) (homogeneous in z) temporally: periodic,st = fd/u appears spontaneously U 2D (x,y,t mod T)

24 Infinitesimal perturbation u 3D obeys linear equation: t u 3D = (U 2D (t) )u 3D (u 3D )U 2D (t) p 3D + 1 Re u 3D Equation is linear and homogeneous in z. Seek solutions which are bounded in z, hence periodic u 3D (x,y,z,t) e iβz with coefficients which are periodic in t with period T : Floquet form u 3D (x,y,z,t) e λt f(t mod T) Therefore u 3D (x,y,z,t) e iβz e λ βt f β (x,y,t mod T) Fix β, calculateλ β and µ β e λ βt. Real part ofλ β > 0 µ β > 1 (For each β value, there are actually many eigenvalues λ β. Selectλ β with largest real part.)

25 From Barkley & Henderson, J. Fluid Mech. (1996) mode A:Re c = β c = = 2π/β c 4 mode B:Re c = 259 β c = 7.64 = 2π/β c 1

26 mode A atre = 210 mode B atre = 250 From M.C. Thompson, Monash University, Australia ( mct/mct/docs/cylinder.html)