scholarly journals Boundary conditions and linear analysis of finite-cell Rayleigh–Bénard convection

1992 ◽  
Vol 241 ◽  
pp. 549-585 ◽  
Author(s):  
Yih-Yuh Chen
Keyword(s):  
Boundary Conditions ◽  
Critical Rayleigh Number ◽  
System Size ◽  
Perturbative Approach ◽  
Bénard Convection ◽  
Reaction Diffusion Model ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
Finite Cell

The linear stability of finite-cell pure-fluid Rayleigh–Bénard convection subject to any homogeneous viscous and/or thermal boundary conditions is investigated via a variational formalism and a perturbative approach. Some general properties of the critical Rayleigh number with respect to change of boundary conditions or system size are derived. It is shown that the chemical reaction–diffusion model of spatial-pattern-forming systems in developmental biology can be thought of as a special case of the convection problem. We also prove that, as a result of the imposed realistic boundary conditions, the nodal surfaces of the temperature of a nonlinear stationary state have a tendency to be parallel or orthogonal to the sidewalls, because the full fluid equations become linear close to the boundary, thus suggesting similar trend for the experimentally observed convective rolls.

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.

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.

scholarly journals Mixed insulating and conducting thermal boundary conditions in Rayleigh–Bénard convection

2017 ◽  
Vol 835 ◽  
pp. 491-511 ◽  
Author(s):  
Dennis Bakhuis ◽  
Rodolfo Ostilla-Mónico ◽  
Erwin P. van der Poel ◽  
Roberto Verzicco ◽  
Detlef Lohse
Keyword(s):  
Heat Transfer ◽  
Boundary Conditions ◽  
Heat Transport ◽  
Bottom Plate ◽  
Bulk Temperature ◽  
Stripe Pattern ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

A series of direct numerical simulations of Rayleigh–Bénard convection, the flow in a fluid layer heated from below and cooled from above, were conducted to investigate the effect of mixed insulating and conducting boundary conditions on convective flows. Rayleigh numbers between $Ra=10^{7}$ and $Ra=10^{9}$ were considered, for Prandtl numbers $\mathit{Pr}=1$ and $\mathit{Pr}=10$. The bottom plate was divided into patterns of conducting and insulating stripes. The size ratio between these stripes was fixed to unity and the total number of stripes was varied. Global quantities, such as the heat transport and average bulk temperature, and local quantities, such as the temperature just below the insulating boundary wall, were investigated. For the case with the top boundary divided into two halves, one conducting and one insulating, the heat transfer was found to be approximately two-thirds of that for the fully conducting case. Increasing the pattern frequency increased the heat transfer, which asymptotically approached the fully conducting case, even if only half of the surface is conducting. Fourier analysis of the temperature field revealed that the imprinted pattern of the plates is diffused in the thermal boundary layers, and cannot be detected in the bulk. With conducting–insulating patterns on both plates, the trends previously described were similar; however, the half-and-half division led to a heat transfer of about a half of that for the fully conducting case instead of two-thirds. The effect of the ratio of conducting and insulating areas was also analysed, and it was found that, even for systems with a top plate with only 25 % conducting surface, heat transport of 60 % of the fully conducting case can be seen. Changing the one-dimensional stripe pattern to a two-dimensional chequerboard tessellation does not result in a significantly different response of the system.

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.

scholarly journals Mixed thermal conditions in convection: how do continents affect the mantle’s circulation?

2017 ◽  
Vol 822 ◽  
pp. 1-4 ◽  
Author(s):  
R. Ostilla-Mónico
Keyword(s):  
Boundary Conditions ◽  
Plate Tectonics ◽  
Large Scale ◽  
Thermal Conditions ◽  
Experimental Proof ◽  
Bénard Convection ◽  
Benard Convection ◽  
Large Scale Flow ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

Natural convection is omnipresent on Earth. A basic and well-studied model for it is Rayleigh–Bénard convection, the fluid flow in a layer heated from below and cooled from above. Most explorations of Rayleigh–Bénard convection focus on spatially uniform, perfectly conducting thermal boundary conditions, but many important geophysical phenomena are characterized by boundary conditions which are a mixture of conducting and adiabatic materials. For example, the differences in thermal conductivity between continental and oceanic lithospheres are believed to play an important role in plate tectonics. To study this, Wang et al. (J. Fluid Mech., vol. 817, 2017, R1), measure the effect of mixed adiabatic–conducting boundary conditions on turbulent Rayleigh–Bénard convection, finding experimental proof that even if the total heat transfer is primarily affected by the adiabatic fraction, the arrangement of adiabatic and conducting plates is crucial in determining the large-scale flow dynamics.

scholarly journals Robust feedback control of Rayleigh–Bénard convection

2001 ◽  
Vol 437 ◽  
pp. 175-202 ◽  
Author(s):  
A. C. OR ◽  
L. CORTELEZZI ◽  
J. L. SPEYER
Keyword(s):  
Feedback Control ◽  
Critical Rayleigh Number ◽  
Linear Quadratic ◽  
Parameter Uncertainties ◽  
Sensor Model ◽  
Motion State ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

We investigate the application of linear-quadratic-Gaussian (LQG) feedback control, or, in modern terms, [Hscr ]2 control, to the stabilization of the no-motion state against the onset of Rayleigh–Bénard convection in an infinite layer of Boussinesq fluid. We use two sensing and actuating methods: the planar sensor model (Tang & Bau 1993, 1994), and the shadowgraph model (Howle 1997a). By extending the planar sensor model to the multi-sensor case, it is shown that a LQG controller is capable of stabilizing the no-motion state up to 14.5 times the critical Rayleigh number. We characterize the robustness of the controller with respect to parameter uncertainties, unmodelled dynamics. Results indicate that the LQG controller provides robust performances even at high Rayleigh numbers.

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