Mean flow in hexagonal convection: stability and nonlinear dynamics (original) (raw)
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Whirling hexagons and defect chaos in hexagonal non-Boussinesq convection
New Journal of Physics, 2003
We study hexagon patterns in non-Boussinesq convection of a thin rotating layer of water. For realistic parameters and boundary conditions we identify various linear instabilities of the pattern. We focus on the dynamics arising from an oscillatory side-band instability that leads to a spatially disordered chaotic state characterized by oscillating (whirling) hexagons. Using triangulation we obtain the distribution functions for the number of pentagonal and heptagonal convection cells. In contrast to the results found for defect chaos in the complex Ginzburg-Landau equation and in inclined-layer convection, the distribution functions can show deviations from a squared Poisson distribution that suggest non-trivial correlations between the defects.
Stability of hexagonal patterns in Bénard-Marangoni convection
Physical Review E, 2001
Hexagonal patterns in Bénard-Marangoni ͑BM͒ convection are studied within the framework of amplitude equations. Near threshold they can be described with Ginzburg-Landau equations that include spatial quadratic terms. The planform selection problem between hexagons and rolls is investigated by explicitly calculating the coefficients of the Ginzburg-Landau equations in terms of the parameters of the fluid. The results are compared with previous studies and with recent experiments. In particular, steady hexagons that arise near onset can become unstable as a result of long-wave instabilities. Within weakly nonlinear theory, a two-dimensional phase equation for long-wave perturbations is derived. This equation allows us to find stability regions for hexagon patterns in BM convection.
Penta-Hepta Defect Chaos in a Model for Rotating Hexagonal Convection
Physical Review Letters, 2003
In a model for rotating non-Boussinesq convection with mean flow we identify a regime of spatiotemporal chaos that is based on a hexagonal planform and is sustained by the induced nucleation of dislocations by penta-hepta defects. The probability distribution function for the number of defects deviates substantially from the usually observed Poisson-type distribution. It implies strong correlations between the defects in the form of density-dependent creation and annihilation rates of defects. We extract these rates from the distribution function and also directly from the defect dynamics.
Non-Boussinesq Convection at Low Prandtl Numbers: Hexagons and Spiral Defect Chaos
arXiv (Cornell University), 2008
We study the stability and dynamics of non-Boussinesq convection in pure gases (CO$_2$ and SF$_6$) with Prandtl numbers near Prsimeq1Pr\simeq 1Prsimeq1 and in a H$_2$-Xe mixture with Pr=0.17Pr=0.17Pr=0.17. Focusing on the strongly nonlinear regime we employ Galerkin stability analyses and direct numerical simulations of the Navier-Stokes equations. For Prsimeq1Pr \simeq 1Prsimeq1 and intermediate non-Boussinesq effects we find reentrance of stable hexagons as the Rayleigh number is increased. For stronger non-Boussinesq effects the hexagons do not exhibit any amplitude instability to rolls. Seemingly, this result contradicts the experimentally observed transition from hexagons to rolls. We resolve this discrepancy by including the effect of the lateral walls. Non-Boussinesq effects modify the spiral defect chaos observed for larger Rayleigh numbers. For convection in SF$_6$ we find that non-Boussinesq effects strongly increase the number of small, compact convection cells and with it enhance the cellular character of the patterns. In H$_2$-Xe, closer to threshold, we find instead an enhanced tendency toward roll-like structures. In both cases the number of spirals and of target-like components is reduced. We quantify these effects using recently developed diagnostics of the geometric properties of the patterns.
Defect Chaos of Oscillating Hexagons in Rotating Convection
Physical Review Letters, 2000
Using coupled Ginzburg-Landau equations, the dynamics of hexagonal patterns with broken chiral symmetry are investigated, as they appear in rotating non-Boussinesq or surface-tension-driven convection. We find that close to the secondary Hopf bifurcation to oscillating hexagons the dynamics are well described by a single complex Ginzburg-Landau equation (CGLE) coupled to the phases of the hexagonal pattern. At the bandcenter these equations reduce to the usual CGLE and the system exhibits defect chaos. Away from the bandcenter a transition to a frozen vortex state is found.
Induced defect nucleation and side-band instabilities in hexagons with rotation and mean flow
Physica D: Nonlinear Phenomena, 2003
The combined effect of mean flow and rotation on hexagonal patterns is investigated using Ginzburg-Landau equations that include nonlinear gradient terms as well as the nonlocal coupling provided by the mean flow. Long-wave and shortwave side-band instabilities are determined. Due to the nonlinear gradient terms and enhanced by the mean flow, the penta-hepta defects can become unstable to the induced nucleation of dislocations in the defect-free amplitude, which can lead to the proliferation of penta-hepta defects and persistent spatio-temporal chaos. For individual penta-hepta defects the nonlinear gradient terms enhance climbing or gliding motion, depending on whether they break the chiral symmetry or not.
Stability of oscillating hexagons in rotating convection
Physica D: Nonlinear Phenomena, 2000
Breaking the chiral symmetry, rotation induces a secondary Hopf bifurcation in weakly nonlinear hexagon patterns which gives rise to oscillating hexagons. We study the stability of the oscillating hexagons using three coupled Ginzburg-Landau equations. Close to the bifurcation point we derive reduced equations for the amplitude of the oscillation, coupled to the phase of the underlying hexagons. Within these equation we identify two types of long-wave instabilities and study the ensuing dynamics using numerical simulations of the three coupled Ginzburg-Landau equations.
Journal of Fluid Mechanics, 2020
Stability of hexagonal patterns in Rayleigh-Bénard convection for shear-thinning fluids with temperature dependent viscosity is studied in the framework of amplitude equations. The rheological behavior of the fluid is described by the Carreau model and the relationship between the viscosity and the temperature is of exponential type. Ginzburg-Landau equations including nonvariational quadratic spatial terms are derived explicitly from the basic hydrodynamic equations using a multiple scale expansion. The stability of hexagonal patterns towards spatially uniform disturbances (amplitude instabilities) and to long wavelength perturbations (phase instabilities) is analyzed for different values of the shear-thinning degree α of the fluid (defined in equation 2.12) and the ratio r of the viscosities between the top and bottom walls. It is shown that the amplitude stability domain shrinks with increasing shear-thinning effects and increases with increasing the viscosity ratio r. Concerning the phase stability domain which confines the range of stable wavenumbers, it is shown that it is closed for low values of r and becomes open and asymmetric for moderate values of r. With increasing shear-thinning effects, the phase stability domain becomes more decentered towards higher values of the wavenumber. 2 T. Varé et al. Beyond the stability limits, two different modes go unstable: longitudinal and transverse modes. For the parameters considered here, the longitudinal mode is relevant only in a small region close to the onset. The nonlinear evolution of the transverse phase instability is investigated by numerical integration of amplitude equations. The hexagon-roll transition triggered by the transverse phase instability for sufficiently large reduced Rayleigh number ǫ is illustrated.
Defect Chaos and Bursts: Hexagonal Rotating Convection and the Complex Ginzburg-Landau Equation
Physical Review Letters, 2006
We employ numerical computations of the full Navier-Stokes equations to investigate non-Boussinesq convection in a rotating system using water as the working fluid. We identify two regimes. For weak non-Boussinesq effects the Hopf bifurcation from steady to oscillating (whirling) hexagons is supercritical and typical states exhibit defect chaos that is systematically described by the cubic complex Ginzburg-Landau equation. For stronger non-Boussinesq effects the Hopf bifurcation becomes subcritical and the oscillations exhibit localized chaotic bursting, which is modeled by a quintic complex Ginzburg-Landau equation.
Re-entrant hexagons in non-Boussinesq convection
Journal of Fluid Mechanics, 2006
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. Contents 1 Introduction 1 2 Basic Equations 3 3 Linear Stability of Hexagons 8 3.1 Amplitude Instabilities 8 3.2 Side-Band Instabilities 10 4 Origin of Reentrant Hexagons 12 5 Numerical Simulations 16 6 Conclusions 18