docInstabilities in Hydrodynamics 2012 Laurette TUCKERMAN laurette
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Instabilities in Hydrodynamics 2012 Laurette TUCKERMAN laurette
Instabilities in Hydrodynamics 2012 Laurette TUCKERMAN [email protected] Fluid Dynamical Examples Extreme Multiplicity in Cylindrical Rayleigh-Bénard Convection (K. Borońska & L. Tuckerman) Hof, Lucas & Mullin, Phys. Fluids (1999) Results from Time-Dependent Simulations two-tori torus mercedes four rolls pizza dipole two rolls three rolls CO asym three rolls Ra (U, V, W, T ) = 4 × Nr × Nθ × Nz = 4 × 40 × 120 × 20 = 384 000 Complete Bifurcation Diagram Thresholds from Conductive State m=1 m=2 m=0 m=3 Pizza: Bifurcations from m = 2 mode ....... ..... . .....1860 . .... .|. 1849 sssssssssssssssssssss sssss s s s ss sss ss 2500 sss s ssss ss ssss s .. .. .. .. .. .. .. .. 2500 .. .. .. .. .. .. .. .. .. .. .. .. . . 1879 2353 2700 2700 Trigonometric 3000 3000 =⇒ 5000 5000 10 000 Rolls .....................Four . .. .... ... .. .. 19 500 . . .. .. .. 23 130 .. .. .. .. Pizza .. .. .. .. .. .. .. 19 440 .. Conductive .. . 22 660 ..................... .... ... 10 000 Rolls Tori: Bifurcations from m = 0 mode .......... .. ...... ........... .. ..... 1870 .... . ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 2360 ................... ..... ..... 2330 1862 2300 2328 3100 5000 3100 5000 ....... . . . . . .... .... ..... ... . ... ..... ..... ..... . 3076 ........... . ... . | 3076 .. .. .. .. .. .. .. 4918 5000 5000 .. ... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . 5438 9000 Two-tori ........ ...... .... .. . . . . ... ......... 12 711 9000 9000 12 700 18 000 30 000 One-torus 9000 12 700 | 12 711 18 000 Conductive Stability of the m = 0 branches Flowers and Automobiles: Bifurcations from m = 3 Mercedes . ....... . . . . . ... ..... ... ..... .... 4634 ........ . 5000 5000 ................... ..... .. 2100 .... .| 1985 ............ .. ... . 4650 .... .. ..... 4103 | 4103 4650 | 4634 5000 5000 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 5503 7000 7000 7000 10 000 20 000 ........ ...... .... 10 000 .. . . . . .... ........ 18 762 30 000 Cloverleaf Mitsubishi 10 000 Marigold 7000 10 000 20 000 | 18 762 Conductive Stability of the m = 3 branches Dipoles: Bifurcations from m = 1 mode Asymmetric Three Rolls Three rolls ............. .............. ...... . . . . . .. ... .. 1832 2100 ... . ..... ............................ . .. ........ 1832 2100 ... . 1828 2300 2500 2300 2500 .. .. .. .. 3000 8000 .. .. .. ... .. .. .. .. 3000 8000 .. .. . 3762 ..... ..... .. ..... 21h 100 .. .h h.... .. .... .. ... .. . 30 000 .. .. .. .. .. .. .. .. 10 000 . 20 000 .. 30 000 .. .. .. .. ... .. .. Tiger .. .. .. .. .. ... .. .. . Conductive .. .. .. .. 21 078 22 125 Isolated Branches: two rolls and CO Two Rolls ...... . . . . ..... 10 000 ... .... .. 8677 ...... ... 10 000 20 000 30 000 20 000 30 000 CO 7200 9100 10 300 Many roads lead to two rolls four rolls asym. three rolls two rolls A XISYMMETRIC S PHERICAL C OUETTE F LOW WITH σ = 0.18 C.K. Mamun & L.S. Tuckerman A XISYMMETRIC S PHERICAL C OUETTE F LOW WITH σ = 0.18 Faraday instability Faraday (1831): Vertical vibration of fluid layer =⇒ stripes, squares, hexagons In 1990s: first fluid-dynamical quasicrystals: Edwards & Fauve J. Fluid Mech. (1994) Kudrolli, Pier & Gollub Physica D (1998) Oscillating frame of reference =⇒ “oscillating gravity” G(t) = g (1 − a cos(ωt)) G(t) = g (1 − a [cos(mωt) + δ cos(nωt + φ0)]) Flat surface becomes linearly unstable for sufficiently high a Consider domain to be horizontally infinite (homogeneous) =⇒ solutions exponential/trigonometric in x = (x, y) Seek bounded solutions =⇒ trigonometric: exp(ik · x) X Height ζ(x, y, t) = eik·xζ̂k (t) k Oscillating gravity =⇒ temporal Floquet problem, T = 2π/ω X j j ζ̂k (t) = eλk tfk (t) j Height ζ(x, y, t) = X eik·xζ̂k (t) k Ideal fluids (no viscosity), sinusoidal forcing =⇒ Mathieu eq. ∂t2ζ̂k + ω02 [1 − a cos(ωt)] ζ̂k = 0 ω02 combines g, densities, surface tension, wavenumber k ζ̂k (t) = 2 X e j λk t fkj (t) j=1 Re(λjk ) > 0 for some j, k =⇒ ζ̂k ր =⇒ flat surface unstable =⇒ Faraday waves with wavelength 2π/k Im(λjk ) eλT waves period 0 1 harmonic same as forcing π/ω −1 subharmonic twice forcing period Floquet functions 0 -0.001 -0.002 -0.003 0 within tongue 1 /2 subharmonic µ = −1 0.006 (ζ-< ζ >)/ λ (ζ-< ζ >)/ λ 0.001 XX XX X X X X X X X X XX XXXXXXXXXX X X X X X X X X X X 1 X 3 t/T X 2 X X X X X XX XX XX X X X X XXX XX X XX X X X X X XXXX X XXX XXX XX XX X X X X X X X X X X X X X X X X X X X X X X X X X X X 2 3 X 1X X X X X X X t / T X X X X X X X X X XX XX X XX XXX XXX XX XXX XX X X X 0.004 X X X X X 0.002XXXXXXX X 0 -0.002 0 within tongue 2 /2 harmonic µ = +1 From Périnet, Juric & Tuckerman, J. Fluid Mech. (2009) 0.03 XX XXX XX XX XX X X X X X X X X X X X X 0.01 X X X X XXXX X XX XX X XXXX XX X XXXXXX X XXXXXXXXXXX X X X X X XXXXX X XX XXXXX X X 0 XXXX X XXXXXXX2 X X 3 0 X X X 1 XXXX t / T X X X X X XXX X X XX -0.01 X X X X X X -0.02 X XX XXX 0.02 (ζ-< ζ >)/ λ XXX XX X X X X X 0.002 X XXXXXXXXXX X 0.003 -0.03 within tongue 3 /2 subharmonic µ = −1 Hexagonal patterns in Faraday instability From Périnet, Juric & Tuckerman, J. Fluid Mech. (2009) Cylinder wake Ideal flow with downstream recirculation zone Von Kármán vortex street (Re ≥ 46) Laboratory experiment (Taneda, 1982) Off Chilean coast past Juan Fernandez islands von Kárman vortex street: Re = U∞d/ν ≥ 46 spatially: two-dimensional (x, y) (homogeneous in z) temporally: periodic, St = f d/U∞ appears spontaneously U2D (x, y, t mod T ) Stability analysis of von Kármán vortex street 2D limit cycle U2D (x, y, tmod T ) obeys: ∂tU2D = −(U2D · ∇)U2D − ∇P2D + 1 Re ∆U2D Perturbation obeys: ∂tu3D = −(U2D (t)·∇)u3D −(u3D ·∇)U2D (t)−∇p3D + 1 Re ∆u3D Equation homogeneous in z, periodic in t =⇒ u3D (x, y, z, t) ∼ eiβz eλβ tfβ (x, y, t mod T ) Fix β, calculate largest µ = eλβ T via linearized Navier-Stokes From Barkley & Henderson, J. Fluid Mech. (1996) mode A: Rec = 188.5 βc = 1.585 =⇒ λc ≈ 4 mode B: Rec = 259 βc = 7.64 =⇒ λc ≈ 1 Temporally: µ = 1 =⇒ steady bifurcation to limit cycle with same periodicity as U2D Spatially: circle pitchfork (any phase in z) mode A at Re = 210 mode B at Re = 250 From M.C. Thompson, Monash University, Australia (http://mec-mail.eng.monash.edu.au/∼mct/mct/docs/cylinder.html) transition to mode A initially faster than exp =⇒ subcritical transition to mode B initially slower than exp =⇒ supercritical Describe via complex amplitudes An, Bn amplitude and phase of mode A, B at fixed instant of cycle 2 4 An+1 = µA + αA|An| An − βA|An| An 2 Bn+1 = µB − αB |Bn| Bn Numerical and experimental frequencies From Barkley & Henderson, JFM (1996) Solution to minimal model From Barkley, Tuckerman & Golubitsky, PRE (2000) Interaction between A and B: An+1 = Bn+1 = µA + αA|An|2 + γA|Bn|2 − βA|An|4 An µB − αB |Bn|2 + γB |An|2 Bn As Re is increased: mode A =⇒ A + B =⇒ mode B
