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VISCOUS MEANS FLOWS IN NEARLY INVISCID FARADAY WAVES

VISCOUS MEANS FLOWS IN NEARLY INVISCID FARADAY WAVES. Elena Martín Universidad de Vigo, Spain - Drift instabilities of spatially uniform Faraday waves. - clean free surface - slightly contaminated free surface - Mean flow effects in the Faraday internal resonance.

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VISCOUS MEANS FLOWS IN NEARLY INVISCID FARADAY WAVES

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  1. VISCOUS MEANS FLOWS INNEARLY INVISCID FARADAY WAVES Elena Martín Universidad de Vigo, Spain - Drift instabilities of spatially uniform Faraday waves. - clean free surface - slightly contaminated free surface - Mean flow effects in the Faraday internal resonance

  2. Nearly inviscid Faraday waves Weakly nonlinear dynamics of nearly inviscid Faraday waves is coupled to the associated viscous mean flow (streaming flow) This coupling has effect in the dynamics beyond threshold small parametric forcing (wforcing ~ 2w0)

  3. Drift Instabilities • Reflection symmetry breaking of the mean flow • Drift instabilitiesof spatially constant and spatially modulated drift waves in annular containers. Douady, Fauve & Thual(Europhys. Lett. 10, 309, 1989) Drift modes Compresion modes

  4. Usual amplitude equations Weakly damped + spatially uniform + monochromatic Faraday wave SW

  5. Simplified model = 2D + x-Periodic, no wave modulation • Nondimensional model free surface:

  6. Formulation (Martin, Martel & Vega 2002, JFM 467, 57-79) x-periodic functions, period L

  7. Matching with the bulk region Boundary layers and bulk regions Limit Singular perturbation problem

  8. Inviscid modes Viscous modes water Infinite non-oscillatory modes exist for each k, whose damping grow with the wave number k Slow non-oscillatory mean flow Linear analysis (Martel & Knobloch 1997) )

  9. Weakly nonlinear analysis: bulk expansions

  10. Amplitude equations Weakly damped + spatially uniform + monochromatic Faraday wave Drifted SW

  11. usual Navier Stokes equations + const. Mean flow equations

  12. Coupled spatial phase-mean flow equations

  13. Numerical results: SW(L/2),basic solution Surface waves: Standing waves Mean flow: Steady counterrotating eddies (obtained by Iskandarani & Liu (1991)) Symmetries: x-reflexion, periodicity (L/2) Stability: Depends on the mean flow Mean flow stream function Mean flow vorticity Re = 260, k = 2.37, L = 2.65 (kL=2p)

  14. k = 4 Numerical Results: PrimaryInstability of SW • Hopfbifurcacion SW(L/2)

  15. Numerical results: bifurcation diagrams (depend strongly on k, L) k = 4, L =p/2

  16. k = 2.37 Numerical Results: PrimaryInstability of SW • Hopfbifurcacion SW(L/2)

  17. Numerical results: bifurcation diagrams (depend strongly on k, L) k = 2.37, L = 2.65

  18. Numerical results: Oscillating SW, no net drift Surface waves: Oscillating standing waves with no net drift Mean flow: array of laterally oscillating eddies whose size also oscillates Symmetries: x-reflexion after half the period of the oscillation, periodicity (L/2) Stability: k = 2.37, L = 2.65 (kL=2p)

  19. Numerical results: Oscillating SW, no net drift Surface waves: Oscillating standing waves with no net drift Mean flow: laterally oscillating eddies whose size also oscillates (different size for each pair of eddies) Symmetries: x-reflexion after half the period of the oscillation Stability: k = 2.37, L = 2.65 (kL=2p)

  20. Numerical results: TW Surface waves: Drifted standing waves, constant drift Symmetries: None, Stability: ´ k = 2.37, L = 2.65 (kL=2p)

  21. Numerical results: SW Surface waves: Standing waves Symmetries: x-reflexionStability: k = 2.37, L = 2.65 (kL=2p)

  22. Numerical results: 2L SW Surface waves: Standing waves Symmetries: x-reflexion, periodicity (L/2) k = 2.37, L = 5.3 (kL=2p 2)

  23. Numerical results: 2L SW Surface waves: Standing waves Symmetries: x-reflexion k = 2.37, L = 5.3 (kL=2p 2)

  24. Numeric results: 2LTW Surface waves: Drifted standing waves, constant drift Symmetries: None k = 2.37, L = 5.3 (kL=2p 2)

  25. Numerical results: chaotic solutions k = 2.37, L = 5.3 (kL=2p 2)

  26. Formulation with surface contamination (Marangoni elasticity+surface viscosity), Martin & Vega 2006, JFM 546, 203-225

  27. Matching with the bulk region Surface contamination: upper boundary layer changes

  28. Coupled spatial phase-mean flow equations

  29. g =1,d =1 g =1,d =0.001 g =0.001,d =1 g =0.001,d =0.001 Surface contamination parameter

  30. Standing wave solutions SW(L/2) k = 2.37, L = 2.65 Surface contamination G = 0.1 G = 0.5 G = 0.9

  31. Primary instability of SW(L/2) k =2.37, L =2.65

  32. Primary instability of SW(L/2) k =2.37, L =2.65

  33. Bifurcation diagram G = 0.9, k =2.37, L =2.65

  34. Complex attractors k =2.37, L =2.65 • = 0.371 Re =274 • = 0.371 Re =276.4

  35. = 0.35 Re =1440 • = 0.65 Re =780 More complex attractors k =2.37, L =2.65

  36. Forcing (e1, 2w) w(k) 3w(3k) excites nonlinear interaction Forcing (e3, 6w) 3w(3k) w(k) Mean flow effects in the Faraday internal resonance

  37. Faraday internal resonance 1:3 (Martin, Proctor & Dawes) Four counterpropagating surface waves A(t), B(t), C(t), D(t), n=3

  38. Faraday internal resonance 1:3 Amplitude equations

  39. Faraday internal resonance 1:3 Mean flow equations

  40. Results: forcing frequency 2w0

  41. Results: forcing frequency 2w0

  42. Results: forcing frequency 2w0 PTW CPTW Chaotic

  43. Bifurcation diagram: forcing frequency 2w0 with mean flow without mean flow

  44. Results: forcing frequency 6w0 with mean flow without mean flow The mean flow seems to stabilize the non resonant solution |A|=|B|=0. The standing wave |C|=|D| destabilizes as in the non-resonant case Resonant solution Non-resonant solution

  45. Results: forcing frequencies 2w0, 6w0 with mean flow without mean flow Competition between the resonant basic state |A|=|B|, |C|=|D| obtained for 2w0frequency and the non resonant solution |A|=|B|=0, |C|=|D| obtained for 6w0 frequency For the case m1=m3, both states coexist and loose stability through a parity-breaking bifurcation. Not qualitatively new results Resonant solution Non-resonant solution

  46. Conclusions • The usual amplitude equations for the nearly inviscid problem are faulty. It is necesary to take into account the mean flow term. • The results indicate that the usually ignored mean flow plays an essential role in the stability of the surface waves and in the bifurcated wave patterns • The presence of the surfactant contamination at the free surface enhances the couplingbetween the mean flow and the surface waves, specially for moderately large wave numbers. • The new states that appear, caused by the coupling with the mean flow, include travelling waves, periodic standing waves and some more complex and even chaotic attractors. • In spite of the 2D simplification, no lateral walls and no spatial modulation the model explains the drift modes observed by Douady, Fauve & Thual (1989) in annular containers

  47. Related references • Martín, E., Martel, C. & Vega, J.M. 2002, “Drift instabilities in Faraday waves”, J. Fluid Mech. 467, 57-79 • Vega, J.M., Knobloch, E. & Martel, C. 2001, “Nearly inviscid Faraday waves in annular containers of moderately large aspect ratio”, Physica D 154, 147-171 • Martín, E. & Vega, J.M. 2006, “The effect of surface contamination on the drift instability of standing Faraday waves”, J. Fluid Mech. 546, 203-225 • Martel, E. & Knobloch, E. 1997, “Damping of nearly inviscid Faraday waves”, Phys. Rev. E 56, 5544-5548 • Nicolas, J.A. & Vega, J.M. 2000, “A note on the effect of surface contamination in water wave damping”, J. Fluid Mech. 410, 367-373 • Martín, E., Martel, C. & Vega, J.M. 2003, “Mean flow effects in the Faraday instability”, J. Modern Phy.B 17, nº 22, 23 & 24, 4278-4283 • Lapuerta, V., Martel, C. & Vega, J.M. 2002 “Interaction of nearly-inviscid Faraday waves and mean flows in 2-D containers of quite large aspect ratio, Physica D, 173 178-203 • Higuera, M., Vega, J.M. & Knobloch, E. 2002 “coupled amplitude-mean flow equations for nearly-inviscid Faraday waves in moderate aspect ratio containers” J. Nonlinear Sci. 12, 505-551

  48. 3D problem with clean surface (Vega, Rüdiger & Viñals 2004, PRE 70, 1)

  49. Conclusions • For deep water problems, the destabilization of the SW takes place through a pitchfork bifurcation that leads to TW. The same happens for small K and high Marangoni or surface viscosity numbers. • For small K and small Marangoni and surface viscosity numbers, the SW destabilize through a Hopf bifurcation. This bifurcation and the appearance sequence of the secondary bifurcations depend strongly on the values of the Marangoni elasticity and surface viscosity. Complex attractors appear

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