Standing and travelling waves in cylindrical Rayleigh–Bénard convection

Boronska, Katarzyna; Tuckerman, Laurette S.

doi:10.1017/s0022112006000309

Cited by 40 publications

(30 citation statements)

References 43 publications

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“…The latter would correspond to those studied in the half-zone model 46 and discussed in detail in the Rayleigh-Bénard configuration. 47 In both studies the initial standing waves, which result from superposition of the rotating waves traveling in opposite directions, eventually become unstable in favor of one rotating wave in the non-linear regime. With no azimuthal velocity in the base flow (Re ¼ 0), the matrices in the eigenvalue problem have real coefficients.…”

Section: B Linear Stability Analysismentioning

confidence: 99%

Thermocapillary instabilities in a laterally heated liquid bridge with end wall rotation

2011

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The effect of rotation on the stability of thermocapillary driven flow in a laterally heated liquid bridge is studied numerically using the full-zone model of the floating-zone crystal growth technique. A small Prandtl number (0.02) fluid, relevant for semiconductor melts, is studied with an aspect ratio (height to diameter of the melt) equal to one. Buoyancy is neglected. A linear stability analysis of three-dimensional perturbations is performed and shows that for any ratio of angular velocities, a weak rotation rate has the surprising effect of destabilizing the base flow. By systematically varying the rotation rate and ratio of angular velocities, the critical threshold and azimuthal wave number of the most unstable mode is found over a wide range of this two parameter space. Depending on these parameters, the leading eigenmode is a wave propagating either in the positive or negative azimuthal direction, with kinetic energy typically localized close to one of the end walls. These results are of practical interest for industrial crystal growth applications, where rotation is often used to obtain higher quality crystals.

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Section: B Linear Stability Analysismentioning

confidence: 99%

Thermocapillary instabilities in a laterally heated liquid bridge with end wall rotation

2011

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“…(20) In this expression m is related to the number of half wavelengths of the unstable structure (sine or cosine as appropriate) growing in the rescaled domain. Restrictions on k given by condition (20) are replaced in the dispersion relation R = R(k) to provide a dispersion relation R = R(m, Γ ) for each integer m as a function of the aspect ratio Γ . These critical curves for different m are displayed in Fig.…”

Section: Stationary Equations and Linear Stabilitymentioning

confidence: 99%

“…However, recent studies like those of [20,21] suggest that a complete description of the solutions to a convection system may require further analysis. These works address convection problems at constant viscosity with different geometries and have proposed complete nonlinear solutions by combining direct numerical simulations and bifurcation studies based on branch continuation techniques.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation phenomena in a convection problem with temperature dependent viscosity at low aspect ratio

Plá

Mancho

Herrero

2009

Physica D: Nonlinear Phenomena

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a b s t r a c tIn this paper we analyse convective solutions of a two dimensional fluid layer in which viscosity depends exponentially on temperature. This problem takes in features of mantle convection, since large viscosity variations are to be expected in the Earth's interior. These solutions are compared with solutions obtained at constant viscosity. Special attention is paid to the influence of the aspect ratio in the solutions presented. The analysis is assisted by bifurcation techniques such as branch continuation, which has proven to be a useful, systematic method for gaining insight into the possible stationary solutions satisfied by the basic equations. One feature presented by the fluid with non constant viscosity is the presence of pitchfork and saddle-node subcritical bifurcations and the presence of convective solutions below the linear critical threshold. The analysis also provides limits of existence of stationary solutions and draws the boundaries for time dependent convection.

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“…One of these two effects can prevail or they can balance each other. 11 The crossing point tends to be at lower P r for increasing Ra. This means that the P r interval, in which the viscous boundary layer would influence the thermal one, increases with Ra.…”

Section: Vertical Profiles and Boundary Layers As Functions Of Ra Andmentioning

confidence: 95%

“…Almost all the pattern-development studies concern convective systems with large aspect ratio (Γ ≫ 1) [9]. For cells of moderate aspect ratio (1 Γ 10) a short review can been found in [11] concerning the work on the first convective states. Through experiments, numerical simulations and theoretical calculations the work just cited mainly provides a stability analysis of convective states.…”

Section: Introductionmentioning

confidence: 99%

Numerical Simulations of Thermal Convection at High Prandtl Numbers

Silano

Sreenivasan

Verzicco

2010

Direct and Large-Eddy Simulation VII

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In this thesis we present the results of an extensive campaign of direct numerical simulations of Rayleigh-Bénard convection at high Prandtl numbers (10 −1 ≤ P r ≤ 10 4 ) and moderate Rayleigh numbers (10 5 ≤ P r ≤ 10 9 ). The computational domain is a cylindrical cell of aspect-ratio (diameter over cell height) Γ = 1/2, with the no-slip condition imposed to the boundaries.By scaling the results, we find a 1/ √ P r correction to apply to the free-fall velocity, obtaining a more appropriate representation of the large scale velocity at high P r. We investigate the Nusselt and the Reynolds number dependence on Ra and P r, comparing the results to previous numerical and experimental work. At high P r the scaling behavior of the Nusselt number with respect to Ra is generally consistent with the power-law exponent 0.309.The Nusselt number is independent of P r, even at the highest Ra simulated. The Reynolds number scales as Re ∼ √ Ra/P r, neglecting logarithmic corrections. We analyze the global and local features of viscous and thermal boundary layers and their scaling behavior with respect to Rayleigh and Prandtl numbers, and with respect to Reynolds and Peclet numbers.We find that the flow approaches a saturation regime when Reynolds number decreases below the critical value Re s ≃ 40. The thermal boundary layer thickness turns out to increase slightly even when the Peclet number increases. We explain this behavior as a combined effect of the Peclet number and the viscous boundary layer influences.The range of Ra and P r simulated contains steady, periodic and turbulent solutions.A rough estimate of the transition from steady to unsteady flow is obtained by monitoring the time-evolution of the system until it reaches stationary solutions (Ra U ≃ 7.5 × 10 6 at P r = 10 3 ). We find multiple solutions as long-term phenomena at Ra = 10 8 and P r = 10 3 which, however, do not result in significantly different Nusselt number. One of these multiple solutions, even if stable for a long time interval, shows a break in the mid-plane symmetry of the temperature profile. The result is similar to that of some non-Boussinesq effects. We analyze the flow structures through the transitional phases by direct visualizations of the temperature and velocity fields. We also describe how the behavior of the flow structures changes for increasing P r. A wide variety of large-scale circulations and plumes structures are found. The single-roll circulation is characteristic only of the steady and periodic solutions. For other solutions, at lower P r, the mean flow generally consists of two opposite toroidal structures; at higher P r, the flow is organized in multi-cell structures extending mostly in the vertical direction. At high P r, plumes detach from sheet-like structures. The different large-scale-structure signatures are generally reflected in the data trends with rei spect to Ra, but not in those with respect to P r. In particular, the Nusselt number is independent of P r, even when the flow structures appear strongly diffe...

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Standing and travelling waves in cylindrical Rayleigh–Bénard convection

Cited by 40 publications

References 43 publications

Thermocapillary instabilities in a laterally heated liquid bridge with end wall rotation

Thermocapillary instabilities in a laterally heated liquid bridge with end wall rotation

Bifurcation phenomena in a convection problem with temperature dependent viscosity at low aspect ratio

Numerical Simulations of Thermal Convection at High Prandtl Numbers

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