Rayleigh–Bénard convection in the presence of a radial ramp of the Rayleigh number (original) (raw)
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Physical Review Letters, 1999
We present experimental results for wave numbers q s selected in a thin horizontal fluid layer heated from below. The cylindrical sample had an interior section of uniform spacing d d 0 for radii r , r 0 (G 0 ϵ r 0 ͞d 0 43) and a ramp d͑r͒ for r. r 0. For Rayleigh numbers R 0. R c 1708 in the interior, straight or slightly curved rolls with an average ͗q s ͘ q c 1 ae 0 ͑e 0 ϵ R 0 ͞R c 2 1͒ and q c , q c 3.117 were selected, and q s varied on two length scales approximately equal to G 0 and to four roll wavelengths. For e & 0.03 and e * 0.18 the pattern repeatedly formed defects.
Physica D: Nonlinear Phenomena, 2003
Rayleigh-Bénard convection is studied and quantitative comparisons are made, where possible, between theory and experiment by performing numerical simulations of the Boussinesq equations for a variety of experimentally realistic situations. Rectangular and cylindrical geometries of varying aspect ratios for experimental boundary conditions, including fins and spatial ramps in plate separation, are examined with particular attention paid to the role of the mean flow. A small cylindrical convection layer bounded laterally either by a rigid wall, fin, or a ramp is investigated and our results suggest that the mean flow plays an important role in the observed wavenumber. Analytical results are developed quantifying the mean flow sources, generated by amplitude gradients, and its effect on the pattern wavenumber for a large-aspect-ratio cylinder with a ramped boundary. Numerical results are found to agree well with these analytical predictions. We gain further insight into the role of mean flow in pattern dynamics by employing a novel method of quenching the mean flow numerically. Simulations of a spiral defect chaos state where the mean flow is suddenly quenched is found to remove the time dependence, increase the wavenumber and make the pattern more angular in nature. Published by Elsevier B.V.
Journal of Fluid Mechanics, 2005
Nonlinear solutions in the form of squares and rolls are investigated for Rayleigh-Bénard convection in an infinite-Prandtl-number fluid enclosed between two symmetric slabs. It is found that the heat transfer depends strongly on the thickness and thermal conductivity of the slabs, but hardly on the planform of convection. Examples of stability regions of rolls are calculated, showing that for certain slab selections, rolls remain stable at even larger Rayleigh numbers than with fixed temperatures at the boundaries. The region of stable squares is restricted by a zigzag and a longwavelength cross-roll instability in addition to a new three-dimensional instability. As the slab conductivity is increased, the stability region of the squares shrinks onto a point located well above the critical point for the onset of convection. For a small range of slab conductivities, stability regions for squares and rolls both exist for the same set-up. In the present calculations, the regions never overlap. An example, where both patterns are stable at the same Rayleigh number, provides an explanation for the co-existence of rolls and squares where transparent slabs with a low thermal conductivity were applied.
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.
Square Patterns in Rayleigh-Bénard Convection with Rotation about a Vertical Axis
Physical Review Letters, 1998
We present experimental results for Rayleigh-Bénard convection with rotation about a vertical axis at dimensionless rotation rates 0 # V # 250 and e ϵ DT ͞DT c 2 1 & 0.2. Critical Rayleigh numbers and wave numbers agree with predictions of linear-stability analysis. For V * 70 and small e the patterns are cellular with local fourfold coordination and differ from the theoretically expected Küppers-Lortz unstable state. Stable as well as intermittent defect-free square lattices exist over certain parameter ranges. Over other ranges defects dynamically disrupt the lattice but cellular flow and local fourfold coordination is maintained. [S0031-9007(98)06696-4]
Dynamics of defects in Rayleigh-Bénard convection
Physical Review A, 1981
The behavior of an extra roll extending into an otherwise regular convection pattern is studied as a function of Rayleigh number, Prandtl number, P, and wavelength, by means of a fully resolved numerical simulation of the Boussinesq equations with free-slip boundary conditions. For .reduced Rayleigh numbers of order one or less and P &40, numerical simulations of the lowest-order amplitude equations reproduce the Boussinesq results semiquantitatively. In particular, we find that when this class of defects is stable, they move with constant velocity v, parallel to the roll axis and give rise to a slow modulation of the roll pattern of the form f(x,yvt). Both f and v have been calculated analytically within a linearized theory. The envelope function f depends in an essential way on v such that the limit v~0 cannot be sensibly taken.
Physics of Fluids, 2013
We present the results of direct numerical simulations of flow patterns in a low-Prandtl-number (P r = 0.1) fluid above the onset of oscillatory convection in a Rayleigh-Bénard system rotating uniformly about a vertical axis. Simulations were carried out in a periodic box with thermally conducting and stress-free top and bottom surfaces. We considered a rectangular box (L x ×L y ×1) and a wide range of Taylor numbers (750 ≤ T a ≤ 5000) for the purpose. The horizontal aspect ratio η = L y /L x of the box was varied from 0.5 to 10. The primary instability appeared in the form of two-dimensional standing waves for shorter boxes (0.5 ≤ η < 1 and 1 < η < 2). The flow patterns observed in boxes with η = 1 and η = 2 were different from those with η < 1 and 1 < η < 2. We observed a competition between two sets of mutually perpendicular rolls at the primary instability in a square cell (η = 1) for T a < 2700, but observed a set of parallel rolls in the form of standing waves for T a ≥ 2700. The three-dimensional convection was quasiperiodic or chaotic for 750 ≤ T a < 2700, and then bifurcated into a two-dimensional periodic flow for T a ≥ 2700. The convective structures consisted of the appearance and disappearance of straight rolls, rhombic patterns, and wavy rolls inclined at an angle φ = π 2 − arctan (η −1) with the straight rolls.
Traveling concentric-roll patterns in Rayleigh-Bénard convection with modulated rotation
Physical Review E, 2002
We present experimental results for pattern formation in Rayleigh-Bénard convection with modulated rotation about a vertical axis. The dimensionless rotation rate ⍀ was varied as ⍀ m ϭ⍀͓1ϩ␦ cos(⍀t)͔ ͑time is scaled by the vertical viscous diffusion time of the cell͒. We used a cylindrical cell of aspect ratio ͑radius/ height͒ ⌫ϭ11.8 and varied ⍀, ␦, , and ⑀ϵR/R c (⍀)Ϫ1 (R is the Rayleigh number͒. The fluid was water with a Prandtl number of 4.5. Sufficiently far above onset even a small ␦տ0.02 stabilized a concentric-roll ͑target͒ pattern. Multiarmed spirals were observed close to onset. The rolls of the target patterns traveled radially inward independent of the sense of rotation. The radial speed v was nearly independent of ⑀ for fixed ⍀, ␦, and. However, v increased with any one of ⍀, ␦, and when all the other parameters were held fixed.
Étude expérimentale de la convection de Rayleigh-Bénard dans le cas d'un fluide à seuil
2013
Les travaux portent sur l'étude expérimentale de la convection de Rayleigh-Bénard dans le cas d'un fluideà seuil (Carbopol 980). Pour différentes puissances de chauffe P, en combinant les mesures, des différences de température entre les deux plans horizontaux parallèles, et la mesure locale du champ de vitesse au sein du fluide, deux régimes distincts ontété observés. Pour des puissances de chauffe inférieuresà une valeur critique P c , un régime purement conductif est observé. Une augmentation progressive de la puissance de chauffe au-delà de ce seuil révèle l'apparition d'un régime convectif qui se manifeste par une dépendance non linéaire, avec la puissance de chauffe, de la différence de température entre les plaques. En parallèle de cette observation, les mesures locales du champ de vitesse montrent une augmentation non linéaire de l'amplitude des cellules. Indépendamment de la concentration en Carbopol, la convection de Rayleigh-Bénard dans le gel de Carbopol apparaît comme une bifurcation imparfaite qui peutêtre modélisée par la théorie de Landau-Ginsburg des transitions de phase. Les résultats obtenus dans cetteétude sont au final comparésà ceux existants dans la littérature.
Rayleigh-Bénard Convection in Large-Aspect-Ratio Domains
Physical Review Letters, 2004
The coarsening and wavenumber selection of striped states growing from random initial conditions are studied in a non-relaxational, spatially extended, and far-from-equilibrium system by performing large-scale numerical simulations of Rayleigh-Bénard convection in a large-aspect-ratio cylindrical domain with experimentally realistic boundaries. We find evidence that various measures of the coarsening dynamics scale in time with different power-law exponents, indicating that multiple length scales are required in describing the time dependent pattern evolution. The translational correlation length scales with time as t 0.12 , the orientational correlation length scales as t 0.54 , and the density of defects scale as t −0.45 . The final pattern evolves toward the wavenumber where isolated dislocations become motionless, suggesting a possible wavenumber selection mechanism for large-aspect-ratio convection.