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

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 ( w forcing ~ 2 w 0 )

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 modesCompresion 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 LimitSingular 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,

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) 3  3k) 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   



 PTW CPTW Chaotic

Bifurcation diagram: forcing frequency 2    with mean flow without mean flow

Results: forcing frequency 6    with mean flowwithout 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 flowwithout mean flow Competition between the resonant basic state |A|=|B|, |C|=|D| obtained for 2   frequency and the non resonant solution |A|=|B|=0, |C|=|D| obtained for   frequency For the case m 1 =m 3, 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, Vega, J.M., Knobloch, E. & Martel, C. 2001, “Nearly inviscid Faraday waves in annular containers of moderately large aspect ratio”, Physica D 154, Martín, E. & Vega, J.M. 2006, “The effect of surface contamination on the drift instability of standing Faraday waves”, J. Fluid Mech. 546, Martel, E. & Knobloch, E. 1997, “Damping of nearly inviscid Faraday waves”, Phys. Rev. E 56, Nicolas, J.A. & Vega, J.M. 2000, “A note on the effect of surface contamination in water wave damping”, J. Fluid Mech. 410, Martín, E., Martel, C. & Vega, J.M. 2003, “Mean flow effects in the Faraday instability”, J. Modern Phy.B 17, nº 22, 23 & 24, Lapuerta, V., Martel, C. & Vega, J.M “Interaction of nearly-inviscid Faraday waves and mean flows in 2-D containers of quite large aspect ratio, Physica D, Higuera, M., Vega, J.M. & Knobloch, E “coupled amplitude-mean flow equations for nearly-inviscid Faraday waves in moderate aspect ratio containers” J. Nonlinear Sci. 12,

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)