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
53
Embed
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. - PowerPoint PPT Presentation
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
VISCOUS MEANS FLOWS INNEARLY INVISCID FARADAY WAVES
Four counterpropagating surface waves A(t), B(t), C(t), D(t), n=3
Faraday internal resonance 1:3
Amplitude equations
Faraday internal resonance 1:3
Mean flow equations
Results: forcing frequency 2
Results: forcing frequency 2
Results: forcing frequency 2
PTW
CPTW
Chaotic
Bifurcation diagram: forcing frequency 2
with mean flow without mean flow
Results: forcing frequency 6
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
Non-resonant solution
Resonant solution
Results: forcing frequencies 2 6
with mean flow without mean flow
Competition between the resonant basic state |A|=|B|,
|C|=|D| obtained for 2frequency and the non resonant solution |A|=|B|=0,
|C|=|D| obtained for frequency
For the case m1=m3, both states coexist and loose stability through a parity-breaking bifurcation. Not qualitatively new results
Non-resonant solution
Resonant solution
Conclusions
• 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 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.
• The usual amplitude equations for the nearly inviscid problem are faulty. It is necesary to take into account the mean flow term.
• The presence of the surfactant contamination at the free surface enhances the coupling between the mean flow and the surface waves, specially for moderately large wave numbers.
• 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
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
3D problem with clean surface (Vega, Rüdiger & Viñals 2004, PRE 70, 1)
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
General Conclusions
• 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 new states that appear, caused by the coupling with the mean flow, include limit cycles, drifted standing waves and some more complex and even chaotic attractors.
• The destabilization of the simplest steady state takes place through a Hopf bifurcation, while the appearance sequence and even the stability of the other described solutions depend strongly on the parameter values. Hysteresis phenomena is also obtained.
• It is inconsistent to ignore a priori in the amplitude equations the effect of the mean flow and retain the usual cubic nonlinearity.
Conclusions
• The results indicate that the presence of the surfactant contamination at the free surface enhances the coupling between the mean flow and the surface waves, specially for moderately large wave numbers.
• 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
Numerical Results: Instability of SW
• Hopf bifurcacion
• Without taking into account the coupling evolution of the spatial phase and the mean flow:
Pitchfork bifurcation (the usual amplitude equations are faulty)