Hexagons and spiral defect chaos in non-Boussinesq convection at low Prandtl numbers

Madruga, Santiago; Riecke, Hermann

doi:10.1103/physreve.75.026210

Cited by 13 publications

(14 citation statements)

References 52 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Similar results exist for other spatially periodic patterns such as hexagons, see e.g. [33]. We have seen that in the present problem the coefficients a 1 , a 2 may render the Eckhaus-stable region highly asymmetrical with respect to k → −k and may reduce dramatically its extent, perhaps eliminating it altogether (see e.g.…”

Section: Discussionsupporting

confidence: 83%

Weakly subcritical stationary patterns: Eckhaus instability and homoclinic snaking

Kao

Knobloch

2012

Phys. Rev. E

View full text Add to dashboard Cite

The transition from subcritical to supercritical stationary periodic patterns is described by the one-dimensional cubic-quintic Ginzburg-Landau equationwhere A(x, t) represents the pattern amplitude and the coefficients µ, a 1 , a 2 and b are real. The conditions for Eckhaus instability of periodic solutions are determined, and the resulting spatially modulated states are computed. Some of these evolve into spatially localized structures in the vicinity of a Maxwell point, while others resemble defect states. The results are used to shed light on the behavior of localized structures in systems exhibiting homoclinic snaking during the transition from subcriticality to supercriticality.

show abstract

Section: Discussionsupporting

confidence: 83%

Weakly subcritical stationary patterns: Eckhaus instability and homoclinic snaking

Kao

Knobloch

2012

Phys. Rev. E

View full text Add to dashboard Cite

show abstract

“…For small Prandtl numbers the signature of the long filaments and of the target patterns disappears and instead a much broader spectrum of smaller, more compact components arises. In a separate investigation 45 we find that the number of small compact components strongly increases when the fluid properties depend significantly on the temperature, i.e., when non-Boussinesq effects become important. They introduce a resonant triad interaction between the stripe modes that enhances the tendency towards hexagonal ͑cellular͒ patterns.…”

Section: Discussionmentioning

confidence: 99%

Geometric diagnostics of complex patterns: Spiral defect chaos

Riecke

Madruga

2006

Chaos: An Interdisciplinary Journal of Nonlinear Science

Self Cite

View full text Add to dashboard Cite

Motivated by the observation of spiral patterns in a wide range of physical, chemical, and biological systems, we present an automated approach that aims at characterizing quantitatively spiral-like elements in complex stripelike patterns. The approach provides the location of the spiral tip and the size of the spiral arms in terms of their arc length and their winding number. In addition, it yields the number of pattern components ͑Betti number of order 1͒, as well as their size and certain aspects of their shape. We apply the method to spiral defect chaos in thermally driven RayleighBénard convection and find that the arc length of spirals decreases monotonically with decreasing Prandtl number of the fluid and increasing heating. By contrast, the winding number of the spirals is nonmonotonic in the heating. The distribution function for the number of spirals is significantly narrower than a Poisson distribution. The distribution function for the winding number shows approximately an exponential decay. It depends only weakly on the heating, but strongly on the Prandtl number. Large spirals arise only for larger Prandtl numbers ͑Prտ 1͒. In this regime the joint distribution for the spiral length and the winding number exhibits a three-peak structure, indicating the dominance of Archimedean spirals of opposite sign and relatively straight sections. Many systems in nature exhibit complex patterns that may be stationary or time dependent, possibly in a chaotic fashion. To understand these patterns and any transition they might undergo, it is important to have quantitative measures that characterize the relevant properties of the patterns and their time dependence. Due to the multitude of different types of patterns, it is not to be expected that a single measure would be sufficient to capture the qualitatively different aspects of the various patterns. Thus, while quite a few different measures have been developed over the years, so far no convincing approach is available that extracts the characteristic features of patterns dominated by spiral-like pattern components. This paper presents an automated method that allows one to determine quantitatively features like the length of a spiral and its winding number. The approach is, however, not limited to proper spirals and yields additional insightful measures.

show abstract

“…The dependence of NOB effects, especially in numerical simulations, is usually characterized by the coefficients (Madruga et al 2006;Madruga & Riecke 2007a) is equal to 0.015, 0.040, 0.010 and 0.009 for E-I, E-II, E-III and E-IV, respectively, for 6 3.0. The temperature difference T is increased in each experiment from onset at a constantT to reach values for which SDC is fully developed.…”

Section: Methodsmentioning

confidence: 99%

Measuring the departures from the Boussinesq approximation in Rayleigh–Bénard convection experiments

2011

View full text Add to dashboard Cite

Algebraic topology (homology) is used to characterize quantitatively non-OberbeckBoussinesq (NOB) effects in chaotic Rayleigh-Bénard convection patterns from laboratory experiments. For fixed parameter values, homology analysis yields a set of Betti numbers that can be assigned to hot upflow and, separately, to cold downflow in a convection pattern. An analysis of data acquired under a range of experimental conditions where NOB effects are systematically varied indicates that the difference between time-averaged Betti numbers for hot and cold flows can be used as an order parameter to measure the strength of NOB-induced pattern asymmetries. This homology-based measure not only reveals NOB effects that Fourier methods and measurements of pattern curvature fail to detect, but also permits distinguishing pattern changes caused by modified lateral boundary conditions from NOB pattern changes. These results suggest a new approach to characterizing data from either experiments or simulations where NOB effects are expected to play an important role.

show abstract

Hexagons and spiral defect chaos in non-Boussinesq convection at low Prandtl numbers

Cited by 13 publications

References 52 publications

Weakly subcritical stationary patterns: Eckhaus instability and homoclinic snaking

Weakly subcritical stationary patterns: Eckhaus instability and homoclinic snaking

Geometric diagnostics of complex patterns: Spiral defect chaos

Measuring the departures from the Boussinesq approximation in Rayleigh–Bénard convection experiments

Contact Info

Product

Resources

About