Resonant mode interactions in Rayleigh-Bénard convection

Prat, Josep; Mercader, Isabel; Knobloch, Edgar

doi:10.1103/physreve.58.3145

Cited by 32 publications

(39 citation statements)

References 24 publications

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“…1, as well as in all stability diagrams corresponding to other 2D and 3D cases. It should be mentioned that the increase of the number of two-dimensional Rayleigh-Bénard convective rolls with the increase of the aspect ratio is well known [13][14][15].…”

Section: Results For the Two-dimensional Casementioning

confidence: 99%

Different Modes of Rayleigh–Bénard Instability in Two- and Three-Dimensional Rectangular Enclosures

Gelfgat

1999

Journal of Computational Physics

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The article describes a complete numerical solution of a recently formulated benchmark problem devoted to the parametric study of Rayleigh-Bénard instability in rectangular two-and three-dimensional boxes. The solution is carried out by the spectral Galerkin method with globally defined, three-dimensional, divergent-free basis functions, which satisfy all boundary conditions. The general description of these three-dimensional basis functions, which can be used for a rather wide spectrum of problems, is presented. The results of the parametric calculations are presented as neutral curves showing the dependence of the critical Rayleigh number on the aspect ratio of the cavity. The neutral curves consist of several continuous branches, which belong to different modes of the most dangerous perturbation. The patterns of different perturbations are also reported. The results obtained lead to some new conclusions about the patterns of the most dangerous perturbations and about the similarities between two-and three-dimensional models. Some extensions of the considered benchmark problem are discussed.

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Section: Results For the Two-dimensional Casementioning

confidence: 99%

Different Modes of Rayleigh–Bénard Instability in Two- and Three-Dimensional Rectangular Enclosures

Gelfgat

1999

Journal of Computational Physics

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“…In principle, k can take any positive real value in the interval [0, 1]. For k/k 0 = m/n ∈ Q (n ∈ N, m = 1, ..., n), the instability of a travelling wave of wavenumber k 0 can be understood as the superharmonic instability of n replicas of the travelling wave filling a periodic domain of length Λ = nλ 0 to periodic perturbations that fit m times in the domain (Prat et al 1998;Melnikov et al 2014). Floquet theory analysis shows that modes with m = j and m = n − j are related by conjugation, so that only half the possible wavelengths need be explored.…”

Section: Numerical Approachmentioning

confidence: 99%

A mechanism for streamwise localisation of nonlinear waves in shear flows

Mellibovsky

Meseguer

2015

J. Fluid Mech.

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We present the complete unfolding of streamwise localisation in a paradigm of extended shear flows, namely, two-dimensional plane-Poiseuille flow. Exact solutions of the Navier-Stokes equations are computed numerically and tracked in the streamwise wavenumber -Reynolds number parameter space to identify and describe the fundamental mechanism behind streamwise localisation, a ubiquitous feature of shear flow turbulence. Unlike shear flow spanwise localisation, streamwise localisation does not follow the snaking mechanism demonstrated for plane-Couette flow.

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“…The resulting system has Boussinesq symmetry if the Biot numbers at the top and bottom are likewise identical. In earlier work [Prat et al, 1998] we showed, following Armbruster [1987], that even two-dimensional convection is strongly affected by the presence of Boussinesq symmetry. This is because Boussinesq symmetry changes the relative importance of certain spatial resonances.…”

Section: Introductionmentioning

confidence: 78%

“…Since the dynamics near the 1 : 2 and 1 : 3 resonances are quite different [Porter & Knobloch, 2000, 2001 it is of interest to explore the crossover from one type of behavior to the other, particularly in low Prandtl number fluids since low Prandtl numbers favor dynamics. In previous papers we have used numerical branch following techniques to explore the behavior near the 1 : 3 resonance in systems with Boussinesq symmetry [Prat et al, 1998], and to investigate in detail how the 1 : 2 resonance comes to dominate the dynamics as the Boussinesq symmetry is progressively broken Prat et al, 2002]. We accomplished this by homotopically continuing the velocity boundary conditions at the top from no-slip to free-slip, while keeping the lower boundary no-slip, and discovered that the progressive loss of symmetry opens up intervals in Rayleigh number in which none of the simple solutions known to be present near the 1 : 2 mode interaction (the n = 1 and n = 2 steady states, and traveling waves) is stable.…”

Section: Introductionmentioning

confidence: 99%

Robust Heteroclinic Cycles in Two-Dimensional Rayleigh–bénard Convection Without Boussinesq Symmetry

Mercader

Prat

Knobloch

2002

Int. J. Bifurcation Chaos

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The onset of convection in systems that are heated via current dissipation in the lower boundary or that lose heat from the top boundary via Newton's law of cooling is formulated as a bifurcation problem. The Rayleigh number as usually defined is shown to be inappropriate as a bifurcation parameter since the temperature difference across the layer depends on the amplitude of convection and hence changes as convection evolves at fixed external parameter values. A modified Rayleigh number is introduced that does remain constant even when the system is evolving, and solutions obtained with the standard formulation are compared with those obtained via the new one. Near the 1 : 2 spatial resonance in low Prandtl number fluids these effects open up intervals of Rayleigh number with no stable solutions in the form of steady convection or steadily traveling waves. Direct numerical simulations in two dimensions show that in such intervals the dynamics typically take the form of a nearly heteroclinic modulated traveling wave. This wave may be quasiperiodic or chaotic.

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Resonant mode interactions in Rayleigh-Bénard convection

Cited by 32 publications

References 24 publications

Different Modes of Rayleigh–Bénard Instability in Two- and Three-Dimensional Rectangular Enclosures

Different Modes of Rayleigh–Bénard Instability in Two- and Three-Dimensional Rectangular Enclosures

A mechanism for streamwise localisation of nonlinear waves in shear flows

Robust Heteroclinic Cycles in Two-Dimensional Rayleigh–bénard Convection Without Boussinesq Symmetry

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