Interaction of Nearly-Inviscid, Multi-mode Faraday Waves and Mean Flows

Higuera, Marı́a; Vega, José M.; Knobloch, Edgar

doi:10.1007/3-540-44698-2_21

Cited by 6 publications

(8 citation statements)

References 22 publications

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“…In fact, almost all of the behavior described here can also be found in a simpler threedimensional system derived from Eqs. (23)-(25) that permits additional analysis, as reported elsewhere [44].…”

Section: Fast-slow Systems: Faraday Oscillationsmentioning

confidence: 95%

A Finite-Dimensional Mechanism Responsible for Bursts in Fluid Mechanics

Knobloch¹

2004

World Scientific Series on Nonlinear Science Series B

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A simple finite-dimensional mechanism is described that appears to be responsible for the presence of bursts near onset of primary instability in a number of fluid systems. The manifestation of the mechanism differs dramatically from system to system, but the mechanism is common to all of them. The geometrical ideas behind the mechanism are discussed, and the results applied to three systems, (a) binary fluid convection, (b) natural doubly diffusive convection, and (c) Faraday oscillations in an almost circular container. Where possible the results are confronted with the results of direct numerical simulations in two or three dimensions.

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Section: Fast-slow Systems: Faraday Oscillationsmentioning

confidence: 95%

A Finite-Dimensional Mechanism Responsible for Bursts in Fluid Mechanics

Knobloch¹

2004

World Scientific Series on Nonlinear Science Series B

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“…In the simplest possible case, the coefficient of the nonlinear term is approximated by an imaginary constant which has already been rescaled to i in Eq. (17). Second, it is also known that linear damping is not sufficient to produce wave saturation in this system [9].…”

Section: (K Lt ) = ^(K0 + Y^-(k T ) (U)mentioning

confidence: 95%

“…The mean flow can be decomposed into its inviscid and viscous parts, as has been done in a number of studies [13,17,18]. Alternatively, a decomposition of the mean flow variables into a short wave component (oscillatory in the horizontal direction) and a long wave component (slowly varying in the horizontal direction) is a convenient choice.…”

Section: Mean Flow Equationsmentioning

confidence: 99%

Mean Flow Effects in Model Equations for Faraday Waves

Rüdiger

Prada

2004

World Scientific Series on Nonlinear Science Series B

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We review model equations for parametric surface waves (Faraday waves) in the limit of small viscous dissipation. The equations account for two effects of viscosity, namely damping of the waves and slowly varying streaming and large scale flows (mean flow). Equations for the mean flow can be derived by a múltiple scale analysis and are coupled to an order parameter equation describing the evolution of the surface waves. In addition, the equations incorpórate a phenomenological damping term due to viscous dissipation. The nonlinear terms, which are undetermined by the derivation of the equation for the surface waves, are chosen so that the primary bifurcation is to a set of standing waves in the form of stripes. Results for the secondary instabilities of the primary waves are presented, including a weak amplification of both Eckhaus and Transverse Amplitude Modulation instabilities due to the mean flow, and a new longitudinal oscillatory instability which is absent without mean flow. Generation of mean flow due to dislocation defects in regular patterns is studied by numerical simulations.

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“…We remind the reader that the limit of vanishing viscosity is a singular limit which does commute with the limit ε → 0. We could decompose the mean flow into its inviscid and viscous parts, as is done in [13,14], but for convenience this is not done here.…”

Section: Mean Flowmentioning

confidence: 99%

“…However, the contribution from mean flows to the equations governing the slow evolution of the surface waves can be of the same order as the standard cubic nonlinear and conservative terms which are always retained. Thus weakly nonlinear corrections to surface waves and mean flows must be considered simultaneously, which has not been done in three dimensions in the limit considered below (see [12] for the analysis of this limit in two dimensions and [13,14,15] for the analysis of related limits). The effects considered here are not unlike other known phenomenology that includes the streaming flow produced by fixed surface waves, [16,17,18,19] and references therein, and the evolution of surface waves in the presence of a fixed vortical flow [20,21].…”

Section: Introductionmentioning

confidence: 99%

Phenomenological model of weakly damped Faraday waves and the associated mean flow

2004

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A phenomenological model of parametric surface waves (Faraday waves) is introduced in the limit of small viscous dissipation that accounts for the coupling between surface motion and slowly varying streaming and large scale flows (mean flow). The primary bifurcation of the model is to a set of standing waves (stripes, given the functional form of the model nonlinearities chosen here). Our results for the secondary instabilities of the primary wave show that the mean flow leads to a weak destabilization of the base state against Eckhaus and Transverse Amplitude Modulation instabilities, and introduces a new longitudinal oscillatory instability which is absent without the coupling. We compare our results with recent one dimensional amplitude equations for this system systematically derived from the governing hydrodynamic equations.

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Interaction of Nearly-Inviscid, Multi-mode Faraday Waves and Mean Flows

Cited by 6 publications

References 22 publications

A Finite-Dimensional Mechanism Responsible for Bursts in Fluid Mechanics

A Finite-Dimensional Mechanism Responsible for Bursts in Fluid Mechanics

Mean Flow Effects in Model Equations for Faraday Waves

Phenomenological model of weakly damped Faraday waves and the associated mean flow

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