Rayleigh–Bénard convection in the presence of spatial temperature modulations

2011 ◽  
Vol 673 ◽  
pp. 318-348 ◽  
Author(s):  
G. FREUND ◽  
W. PESCH ◽  
W. ZIMMERMANN
Keyword(s):  
Rayleigh Number ◽  
Numerical Solutions ◽  
Critical Rayleigh Number ◽  
Lower Boundary ◽  
Spatial Modulation ◽  
Bénard Convection ◽  
Benard Convection ◽  
Wide Range ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

Motivated by recent experiments, we study a rich variation of the familiar Rayleigh–Bénard convection (RBC), where the temperature at the lower boundary varies sinusoidally about a mean value. As usual the Rayleigh number R measures the average temperature gradient, while the additional spatial modulation is characterized by a (small) amplitude δm and a wavevector qm. Our analysis relies on precise numerical solutions of suitably adapted Oberbeck–Boussinesq equations (OBE). In the absence of forcing (δm = 0), convection rolls with wavenumber qc bifurcate only for R above the critical Rayleigh number Rc. In contrast, for δm≠0, convection is unavoidable for any finite R; in the most simple case in the form of ‘forced rolls’ with wavevector qm. According to our first comprehensive stability diagram of these forced rolls in the qm – R plane, they develop instabilities against resonant oblique modes at R ≲ Rc in a wide range of qm/qc. Only for qm in the vicinity of qc, the forced rolls remain stable up to fairly large R > Rc. Direct numerical simulations of the OBE support and extend the findings of the stability analysis. Moreover, we are in line with the experimental results and also with some earlier theoretical results on this problem, based on asymptotic expansions in the limit δm → 0 and R → Rc. It is satisfying that in many cases the numerical results can be directly interpreted in terms of suitably constructed amplitude and generalized Swift–Hohenberg equations.

scholarly journals Rayleigh–Bénard convection in a creeping solid with melting and freezing at either or both its horizontal boundaries

2018 ◽  
Vol 846 ◽  
pp. 5-36 ◽  
Author(s):  
Stéphane Labrosse ◽  
Adrien Morison ◽  
Renaud Deguen ◽  
Thierry Alboussière
Keyword(s):  
Boundary Condition ◽  
Rayleigh Number ◽  
Phase Change ◽  
Time Scale ◽  
Critical Rayleigh Number ◽  
Classical Case ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

Solid-state convection can take place in the rocky or icy mantles of planetary objects, and these mantles can be surrounded above or below or both by molten layers of similar composition. A flow towards the interface can proceed through it by changing phase. This behaviour is modelled by a boundary condition taking into account the competition between viscous stress in the solid, which builds topography of the interface with a time scale $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$, and convective transfer of the latent heat in the liquid from places of the boundary where freezing occurs to places of melting, which acts to erase topography, with a time scale $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D719}}$. The ratio $\unicode[STIX]{x1D6F7}=\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D719}}/\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D702}}$ controls whether the boundary condition is the classical non-penetrative one ($\unicode[STIX]{x1D6F7}\rightarrow \infty$) or allows for a finite flow through the boundary (small $\unicode[STIX]{x1D6F7}$). We study Rayleigh–Bénard convection in a plane layer subject to this boundary condition at either or both its boundaries using linear and weakly nonlinear analyses. When both boundaries are phase-change interfaces with equal values of $\unicode[STIX]{x1D6F7}$, a non-deforming translation mode is possible with a critical Rayleigh number equal to $24\unicode[STIX]{x1D6F7}$. At small values of $\unicode[STIX]{x1D6F7}$, this mode competes with a weakly deforming mode having a slightly lower critical Rayleigh number and a very long wavelength, $\unicode[STIX]{x1D706}_{c}\sim 8\sqrt{2}\unicode[STIX]{x03C0}/3\sqrt{\unicode[STIX]{x1D6F7}}$. Both modes lead to very efficient heat transfer, as expressed by the relationship between the Nusselt and Rayleigh numbers. When only one boundary is subject to a phase-change condition, the critical Rayleigh number is $\mathit{Ra}_{c}=153$ and the critical wavelength is $\unicode[STIX]{x1D706}_{c}=5$. The Nusselt number increases approximately two times faster with the Rayleigh number than in the classical case with non-penetrative conditions, and the average temperature diverges from $1/2$ when the Rayleigh number is increased, towards larger values when the bottom boundary is a phase-change interface.

Exploring the severely confined regime in Rayleigh–Bénard convection

2016 ◽  
Vol 805 ◽  
Author(s):  
Kai Leong Chong ◽  
Ke-Qing Xia
Keyword(s):  
Rayleigh Number ◽  
Flow Topology ◽  
Thermal Plumes ◽  
Bénard Convection ◽  
Benard Convection ◽  
Wide Range ◽  
Classical Boundary ◽  
Global And Local ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

We study the effect of severe geometrical confinement in Rayleigh–Bénard convection with a wide range of width-to-height aspect ratio $\unicode[STIX]{x1D6E4}$, $1/128\leqslant \unicode[STIX]{x1D6E4}\leqslant 1$, and Rayleigh number $Ra$, $3\times 10^{4}\leqslant Ra\leqslant 1\times 10^{11}$, at a fixed Prandtl number of $Pr=4.38$ by means of direct numerical simulations in Cartesian geometry with no-slip walls. For convection under geometrical confinement (decreasing $\unicode[STIX]{x1D6E4}$ from 1), three regimes can be recognized (Chong et al., Phys. Rev. Lett., vol. 115, 2015, 264503) based on the global and local properties in terms of heat transport, plume morphology and flow structures. These are Regime I: classical boundary-layer-controlled regime; Regime II: plume-controlled regime; and Regime III: severely confined regime. The study reveals that the transition into Regime III leads to totally different heat and momentum transport scalings and flow topology from the classical regime. The convective heat transfer scaling, in terms of the Nusselt number $Nu$, exhibits the scaling $Nu-1\sim Ra^{0.61}$ over three decades of $Ra$ at $\unicode[STIX]{x1D6E4}=1/128$, which contrasts sharply with the classical scaling $Nu-1\sim Ra^{0.31}$ found at $\unicode[STIX]{x1D6E4}=1$. The flow in Regime III is found to be dominated by finger-like, long-lived plume columns, again in sharp contrast with the mushroom-like, fragmented thermal plumes typically observed in the classical regime. Moreover, we identify a Rayleigh number for regime transition, $Ra^{\ast }=(29.37/\unicode[STIX]{x1D6E4})^{3.23}$, such that the scaling transition in $Nu$ and $Re$ can be clearly demonstrated when plotted against $Ra/Ra^{\ast }$.

Effects of Slip Boundary Conditions on Rayleigh-Bénard Convection

2009 ◽  
Vol 25 (2) ◽  
pp. 205-212 ◽  
Author(s):  
L.-S. Kuo ◽  
P.-H. Chen
Keyword(s):  
Rayleigh Number ◽  
Boundary Conditions ◽  
Knudsen Number ◽  
Critical Rayleigh Number ◽  
Slip Boundary Conditions ◽  
Slip Boundary ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

AbstractThis work studied the Rayleigh-Bénard convection under the first-order slip boundary conditions in both hydrodynamic and thermal fields. The variation principle was applied to find the critical Rayleigh number of instability. The exteneded relations of the critical Rayleigh number (Rc) and the wavenumber (ac) under partially slip boundary conditions were derived. The numerical results showed that both Rc and ac are decreasing with increasing the Knudsen number. The dependence of Rc on the Knudsen number (K) shows that when K≤10−3, the boundary can be considered as nonslip, while K≥10, it can be considered as free boundaries. The maximum change rate occurs when the Knudsen number is around 0.1, indicating that the system would be affected significantly in that range.

ON THE ONSET OF RAYLEIGH–BÉNARD CONVECTION IN A LAYER OF NANOFLUID IN HYDROMAGNETICS

2013 ◽  
Vol 12 (06) ◽  
pp. 1350038 ◽  
Author(s):  
RAMESH CHAND
Keyword(s):  
Magnetic Field ◽  
Rayleigh Number ◽  
Critical Rayleigh Number ◽  
Mode Analysis ◽  
Free Boundaries ◽  
Weighted Residuals ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

Rayleigh–Bénard convection in a horizontal layer of nanofluid in the presence of uniform vertical magnetic field is investigated by using Galerkin weighted residuals method. The model used for the nanofluid describes the effects of Brownian motion and thermophoresis. Linear stability theory based upon normal mode analysis is employed to find expressions for Rayleigh number and critical Rayleigh number. The boundaries are considered to be free–free, rigid–rigid and rigid–free. The influence of magnetic field on the stability is investigated and it is found that magnetic field stabilizes the fluid layer. It is also observed that the system is more stable in the case of rigid–rigid boundaries and least stable in case of free–free boundaries. The expression for Rayleigh number for oscillatory convection has also been derived for free–free boundaries.

Roll-pattern evolution in finite-amplitude Rayleigh–Beńard convection in a two-dimensional fluid layer bounded by distant sidewalls

1984 ◽  
Vol 143 ◽  
pp. 125-152 ◽  
Author(s):  
P. G. Daniels
Keyword(s):  
Rayleigh Number ◽  
Temporal Evolution ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

This paper considers the temporal evolution of two-dimensional Rayleigh–Bénard convection in a shallow fluid layer of aspect ratio 2L ([Gt ] 1) confined laterally by rigid sidewalls. Recent studies by Cross et al. (1980, 1983) have shown that for Rayleigh numbers in the range R = R0 + O(L−1) (where R0 is the critical Rayleigh number for the corresponding infinite layer) there exists a class of finite-amplitude steady-state ‘phase-winding’ solutions which correspond physically to the possibility of an adjustment in the number of rolls in the container as the local value of the Rayleigh number is varied. It has been shown (Daniels 1981) that in the temporal evolution of the system the final lateral positioning of the rolls occurs on the long timescale t = O(L2) when the phase function which determines the number of rolls in the system satisfies a one-dimensional diffusion equation but with novel boundary conditions that represent the effect of the sidewalls. In the present paper this system is solved numerically in order to determine the precise way in which the roll pattern adjusts after a change in the Rayleigh number of the system. There is an interesting balance between, on the one hand, a tendency for the number of rolls to change by the least number possible and, on the other, a tendency for the even or odd nature of the initial configuration to be preserved during the transition. In some cases this second property renders the natural evolution susceptible to arbitrarily small external disturbances, which then dictate the form of the final roll pattern.The complete transition involves an analysis of the motion on three timescales, a conductive scale t = O(1), a convective growth scale t = O(L) and a convective diffusion scale t = O(L2).

Direct Numerical Simulation of a Low Prandtl Number Rayleigh–Bénard Convection in a Square Box

2019 ◽  
Vol 11 (6) ◽  
Author(s):  
Ojas Satbhai ◽  
Subhransu Roy ◽  
Sudipto Ghosh
Keyword(s):  
Nusselt Number ◽  
Steady State ◽  
Rayleigh Number ◽  
Prandtl Number ◽  
Dimensionless Temperature ◽  
Bénard Convection ◽  
Benard Convection ◽  
Wide Range ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

Direct numerical simulations for low Prandtl number fluid (Pr = 0.0216) are used to study the steady-state Rayleigh–Bénard convection (RB) in a two-dimensional unit aspect ratio box. The steady-state RB convection is characterized by analyzing the time-averaged temperature-field, and flow field for a wide range of Rayleigh number (2.1 × 105 ⩽ Ra ⩽ 2.1 × 108). It is seen that the time-averaged and space-averaged Nusselt number (Nuh¯) at the hot-wall monotonically increases with the increase in Rayleigh number (Ra) and the results show a power law scaling Nuh¯∝Ra0.2593. The current Nusselt number results are compared with the results available in the literature. The complex flow is analyzed by studying the frequency power spectra of the steady-state signal of the vertical velocity at the midpoint of the box for different Ra and probability density function of dimensionless temperature at various locations along the midline of the box.

scholarly journals Rayleigh-Bénard Convection for Nanofluids for More Realistic Boundary Conditions (Rigid-Free and Rigid-Rigid) Using Darcy Model

2019 ◽  
Vol 4 (1) ◽  
pp. 139-156 ◽  
Author(s):  
Jyoti Ahuja ◽  
Urvashi Gupta
Keyword(s):  
Magnetic Field ◽  
Rayleigh Number ◽  
Boundary Conditions ◽  
Critical Rayleigh Number ◽  
Galerkin Approximation ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
Realistic Boundary Conditions

In this article, Rayleigh-Bénard convection for nanofluids for more realistic boundary conditions (rigid-free and rigid-rigid) under the influence of the magnetic field is investigated. Presence of nanoparticles in base fluid has introduced one additional conservation equation of nanoparticles that incorporates the effect of thermophoretic forces and Brownian motion and the inclusion of magnetic field has introduced Lorentz’s force term in the momentum equation along with Maxwell’s equations. The solution of the Eigen value problem is found in terms of Rayleigh number by implementing the technique of normal modes and weighted residual Galerkin approximation. It is found that the stationary as well as oscillatory motions come into existence and heat transfer takes place through oscillatory motions. The critical Rayleigh number for alumina water nanofluid has an appreciable increase in its value with the rise in Chandrasekhar number and it increases moderately as we move from rigid-free to both rigid boundaries. The effect of different nanofluid parameters on the onset of thermal convection for two types of boundaries is investigated.

Interpretation of shadowgraph patterns in Rayleigh-Bénard convection

1988 ◽  
Vol 190 ◽  
pp. 451-469 ◽  
Author(s):  
D. R. Jenkins
Keyword(s):  
Rayleigh Number ◽  
Linear Theory ◽  
Vertical Component ◽  
Fluid Flows ◽  
Bénard Convection ◽  
Benard Convection ◽  
Higher Order Terms ◽  
The Relationship ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

The relationship between observations of cellular Rayleigh-Bénard convection using shadowgraphs and theoretical expressions for convection planforms is considered. We determine the shadowgraphs that ought to be observed if the convection is as given by theoretical expressions for roll, square or hexagonal planforms and compare them with actual experiments. Expressions for the planforms derived from linear theory, valid for low supercritical Rayleigh number, produce unambiguous shadowgraphs consisting of cells bounded by bright lines, which correspond to surfaces through which no fluid flows and on which the vertical component of velocity is directed downwards. Dark spots at the centre of cells, indicating regions of hot, rising fluid, are not accounted for by linear theory, but can be produced by adding higher-order terms, predominantly due to the temperature dependence of a material property of the fluid, such as its viscosity.

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