A calculation of transient solutions describing roll and hexagon formation in the convection instability (original) (raw)
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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
Hexagonal Patterns in a Model for Rotating Convection
International Journal of Bifurcation and Chaos, 2004
We study a model equation that mimics convection under rotation in a fluid with temperature-dependent properties (non-Boussinesq (NB)), high Prandtl number and idealized boundary conditions. It is based on a model equation proposed by Segel [1965] by adding rotation terms that lead to a Küppers–Lortz instability [Küppers & Lortz, 1969] and can develop into oscillating hexagons. We perform a weakly nonlinear analysis to find out explicitly the coefficients in the amplitude equation as functions of the rotation rate. These equations describe hexagons and oscillating hexagons quite well, and include the Busse–Heikes (BH) model [Busse & Heikes, 1980] as a particular case. The sideband instabilities as well as short wavelength instabilities of such hexagonal patterns are discussed and the threshold for oscillating hexagons is determined.
Analysis of the approach to the convection instability point
A spectral analysis is presented of the fluctuations in a horizontal fluid layer subject to a downward directed temperature gradient, which, for a critical value, drives the system in a convective instability state. It is found that the external force resulting from the combination of the temperature gradient and the gravitation force gives rise to a coupling between the heat diffusion mode and a shear mode. As a result of this mode coupling the damping constant of the heat diffusion mode goes to zero when the temperature gradient increases towards its critical value, i.e. the heat diffusion mode behaves like a "soft mode". The implications of the mode coupling and of the ensuing softening of the heat diffusion mode on the light scattering spectrum are discussed.
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.
Oscillatory instabilities of convection rolls at intermediate Prandtl numbers
Journal of Fluid Mechanics, 1986
The analysis of the instabilities of convection rolls in a fluid layer heated from below with no-slip boundaries exhibits a close competition between various oscillatory modes in the range 2 [lsim ] P [lsim ] 12 of the Prandtl number P. In addition to the even-oscillatory instability known from earlier work two new instabilities have been found, each of which is responsible for a small section of the stability boundary of steady rolls. The most interesting property of the new instabilities is their close relationship to the hot-blob oscillations known from experimental studies of convection. In the lower half of the Prandtl-number range considered the B02-mode dominates, which is characterized by two blobs each of slightly hotter and colder fluid circulating around in the convection roll in a spatially and time-periodic fashion. At higher Prandtl numbers the BE 1-mode dominates, which possesses one hot blob (and one cold blob) circulating with the convection velocity. Just outside t...
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.
We consider surface-tension driven convection in a rotating fluid layer. For nearly insulating boundary conditions we derive a long-wave equation for the convection planform. Using a Galerkin method and direct numerical simulations we study the stability of the steady hexagonal patterns with respect to general side-band instabilities. In the presence of rotation steady and oscillatory instabilities are identified. One of them leads to stable, homogeneously oscillating hexagons. For sufficiently large rotation rates the stability balloon closes, rendering all steady hexagons unstable and leading to spatio-temporal chaos.
Dynamics of zero-Prandtl number convection near the onset
2010
In this paper we present various convective states of zero-Prandtl number Rayleigh-Bénard convection using direct numerical simulations (DNS) and a 27-mode lowdimensional model containing the energetic modes of DNS. The origin of these convective states have been explained using bifurcation analysis. The system is chaotic at the onset itself with three coexisting chaotic attractors that are born at two codimension-2 bifurcation points. One of the bifurcation points with a single zero eigenvalue and a complex pair (0, ±iω) generates chaotic attractors and associated periodic, quasiperiodic, and phase-locked states that are related to the wavy rolls observed in experiments and simulations. The frequency of the wavy rolls are in general agreement with ω of the above eigenvalue of the stability matrix. The other bifurcation point with a double zero eigenvalue produces the other set of chaotic attractors and ordered states such as squares, asymmetric squares, oscillating asymmetric squares, relaxation oscillations with intermediate squares, some of which are common to the 13-mode model of Pal et al.
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.