Complex spatiotemporal convection patterns

Pesch, Werner

doi:10.1063/1.166194

Cited by 56 publications

(40 citation statements)

References 45 publications

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“…Previous simulations have shown that this spatial resolution is sufficient to capture the spiral defect chaos at the very least semiquantitatively. 35,36 Since the data analysis requires relatively long runs, going to a significantly higher spatial resolution is beyond our current computational means. To solve for the time dependence the code uses a fully implicit scheme for the linear terms, whereas the nonlinear parts are treated explicitly using a second-order Adams-Bashforth method.…”

Section: Geometric Analysis Of Spiral Defect Chaosmentioning

confidence: 99%

Geometric diagnostics of complex patterns: Spiral defect chaos

Riecke

Madruga

2006

Chaos: An Interdisciplinary Journal of Nonlinear Science

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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.

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Section: Geometric Analysis Of Spiral Defect Chaosmentioning

confidence: 99%

Geometric diagnostics of complex patterns: Spiral defect chaos

Riecke

Madruga

2006

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…Such phenomena has been observed and modelled in a wide range of systems including Dendritic Solidification (Ben-Jacob and Garik, 1990;Halsey, 2000;Vicsek, 1984), fluid convection (Pesch, 1996;Kadanoff, 2001), surfactant structures (Bachmann et al, 1992;Hanczyc and Szostak, 2004;Mayer et al, 1997;Ono, 2005), bacterial colony formation (BenJacob, 1997(BenJacob, , 1993, and reaction-diffusion systems (Lee et al, 1994;Turing, 1952;Pearson, 1993;Scott, 1985, 1994;Mahara et al, 2008;Virgo, 2011), see also Gollub and Langer (1999); Prigogine (1978); Kondepudi and Prigogine (2014). From an artificial life perspective, the dynamics of these (often very simple) systems are particularly poignant since they often exhibit properties analogous to those of living systems.…”

Section: Introductionmentioning

confidence: 99%

Emergence of Competition between Different Dissipative Structures for the Same Free Energy Source

Bartlett¹,

Bullock²

2015

07/20/2015-07/24/2015

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In this paper, we explore the emergence and direct interaction of two different types of dissipative structure in a single system: self-replicating chemical spot patterns and buoyancyinduced convection rolls. A new Lattice Boltzmann Model is developed, capable of simulating fluid flow, heat transport, and thermal chemical reactions, all within a simple, efficient framework. We report on a first set of simulations using this new model, wherein the Gray-Scott reaction diffusion system is embedded within a non isothermal fluid undergoing natural convection due to temperature gradients. The non-linear reaction which characterises the Gray-Scott system is given a temperature-dependent rate constant of the form of the Arrhenius equation. The enthalpy change (exothermic heat release or endothermic heat absorption) of the reaction can also be adjusted, allowing a direct coupling between the dynamics of the reaction and the thermal fluid flow. The simulations show positive feedback effects when the reaction is exothermic, but an intriguing, competitive and unstable behaviour occurs when the reaction is sufficiently endothermic. In fact when convection plumes emerge and grow, the reaction diffusion spots immediately surround them, since they require a source of heat for the reaction to proceed. Then however, the proliferation of spot patterns dampens the local temperature, eventually eliminating the initial convection plume and reducing the ability of the spots to persist. This behaviour appears almost ecological, similar as it is, to competitive interactions between organisms competing for the same nutrient source.

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“…For that purpose we have extended our previously developed spectral code for the OB-equations (Pesch (1996); Bodenschatz et al (2000)) to include the NOBeffects in (2.13,2.14,2.15). It employs the same vertical modes as the Galerkin stability code but places the wave vectors of the Fourier modes on a rectangular rather than a hexagonal grid.…”

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confidence: 99%

Re-entrant hexagons in non-Boussinesq convection

2006

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While non-Boussinesq hexagonal convection patterns are well known to be stable close to threshold (i.e. for Rayleigh numbers R ≈ R c ), it has often been assumed that they are always unstable to rolls already for slightly higher Rayleigh numbers. Using the incompressible Navier-Stokes equations for parameters corresponding to water as a working fluid, we perform full numerical stability analyses of hexagons in the strongly nonlinear regime (ǫ ≡ R − R c /R c = O(1)). We find 'reentrant' behavior of the hexagons, i.e. as ǫ is increased they can lose and regain stability. This can occur for values of ǫ as low as ǫ = 0.2. We identify two factors contributing to the reentrance: i) the hexagons can make contact with a hexagon attractor that has been identified recently in the nonlinear regime even in Boussinesq convection (Assenheimer & Steinberg (1996);Clever & Busse (1996)) and ii) the non-Boussinesq effects increase with ǫ. Using direct simulations for circular containers we show that the reentrant hexagons can prevail even for side-wall conditions that favor convection in the form of the competing stable rolls. For sufficiently strong non-Boussinesq effects hexagons become stable even over the whole ǫ-range considered, 0 ≤ ǫ ≤ 1.5.

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Complex spatiotemporal convection patterns

Cited by 56 publications

References 45 publications

Geometric diagnostics of complex patterns: Spiral defect chaos

Geometric diagnostics of complex patterns: Spiral defect chaos

Emergence of Competition between Different Dissipative Structures for the Same Free Energy Source

Re-entrant hexagons in non-Boussinesq convection

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