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)