viscous means flows in nearly inviscid faraday waves
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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
Nearly inviscid Faraday waves
This coupling has effect in the dynamics beyond threshold
Weakly nonlinear dynamics of nearly inviscid Faraday waves is coupled to the associated viscous mean flow (streaming flow)
small parametric forcing
(wforcing ~ 2w0)
Drift Instabilities
Douady, Fauve & Thual (Europhys. Lett. 10, 309, 1989)
Reflection symmetry breaking of the mean flow
Drift instabilities of spatially constant and spatially modulated drift waves in annular containers
.
Drift modes Compresion modes
Usual amplitude equations
Weakly damped + spatially uniform + monochromatic Faraday wave
SW
Simplified model = 2D + x-Periodic, no wave modulation
• Nondimensional model
free surface:
Formulation (Martin, Martel & Vega 2002, JFM 467, 57-79)
x-periodic functions, period L
Boundary layers and bulk regions
Matching with the
bulk region
Limit Singular perturbation problem
Linear analysis (Martel & Knobloch 1997)
Slow non-oscillatory mean flow
Inviscid modes
water
Viscous modes
Infinite non-oscillatory modes exist for each k, whose damping grow with the wave number k
)
Weakly nonlinear analysis: bulk expansions
Amplitude equations
Weakly damped + spatially uniform + monochromatic Faraday wave
Drifted
SW
usual Navier Stokes equations
+
Mean flow equations
const.
Coupled spatial phase-mean flow equations
Mean flow stream functionMean flow vorticity
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
Re = 260, k = 2.37, L = 2.65 (kL=2
Numerical Results: Primary Instability of SW
• Hopf bifurcacion
k = 4
SW(L/2)
Numerical results: bifurcation diagrams (depend strongly on k, L)
k = 4, L =
Numerical Results: Primary Instability of SW
• Hopf bifurcacion
SW(L/2)
k = 2.37
Numerical results: bifurcation diagrams (depend strongly on k, L)
k = 2.37, L = 2.65
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=2
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=2
Numerical results: TW
´
Surface waves: Drifted standing waves, constant drift
Symmetries: None, Stability:
k = 2.37, L = 2.65 (kL=2
Numerical results: SW
Surface waves: Standing waves
Symmetries: x-reflexion Stability:
k = 2.37, L = 2.65 (kL=2
Numerical results: 2L SW
Surface waves: Standing waves
Symmetries: x-reflexion, periodicity (L/2)
k = 2.37, L = 5.3 (kL=2
Numerical results: 2L SW
Surface waves: Standing waves
Symmetries: x-reflexion
k = 2.37, L = 5.3 (kL=2
Numeric results: 2L TW
Surface waves: Drifted standing waves, constant drift
Symmetries: None
k = 2.37, L = 5.3 (kL=2
Numerical results: chaotic solutions
k = 2.37, L = 5.3 (kL=2
Formulation with surface contamination (Marangoni elasticity+surface viscosity), Martin & Vega 2006, JFM 546, 203-225
Surface contamination: upper boundary layer changes
Matching with the
bulk region
Coupled spatial phase-mean flow equations
Surface contamination parameter
Standing wave solutions SW(L/2)
Surface contamination
k = 2.37, L
k =2.37, L
Primary instability of SW(L/2)
k =2.37, L
Primary instability of SW(L/2)
Bifurcation diagram k L
Complex attractors
Re =274
Re =276.4
k =2.37, L
More complex attractors
Re =780
Re =1440
k =2.37, L
Mean flow effects in the Faraday internal resonance
Forcing 2(k) 33k)
excites nonlinear interaction
Forcing 6(3k) k)
Faraday internal resonance 1:3 (Martin, Proctor & Dawes)
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)
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