On the solution of the Bénard problem with boundaries of finite conductivity

1967 ◽  
Vol 296 (1447) ◽  
pp. 469-475 ◽  
Keyword(s):  
Thermal Diffusivity ◽  
Rayleigh Number ◽  
Hydrodynamic Stability ◽  
Critical Rayleigh Number ◽  
Finite Conductivity ◽  
Stationary Convection ◽  
Bénard Problem ◽  
Benard Problem ◽  
The Media

The Bénard problem in hydrodynamic stability is formulated under conditions where the media bounding the fluid have finite thermal diffusivity. It is shown that the principle of the exchange of stabilities remains valid in this case so that instability in the fluid first sets in as stationary convection. Solutions are obtained for various values of the ratio of the thermal diffusivity of the fluid to that of the bounding media; the critical Rayleigh number at which the instability occurs is markedly reduced when this ratio is large.

Stabilization of the no-motion state in the Rayleigh–Bénard problem

1994 ◽  
Vol 447 (1931) ◽  
pp. 587-607 ◽  
Keyword(s):  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Control Flow ◽  
Feedback Controller ◽  
Linear Stability Theory ◽  
Motion State ◽  
Conduction State ◽  
Bénard Problem ◽  
Benard Problem ◽  
Rayleigh Benard

Using linear stability theory and numerical simulations, we demonstrate that the critical Rayleigh number for bifurcation from the no-motion (conduction) state to the motion state in the Rayleigh–Bénard problem of an infinite fluid layer heated from below and cooled from above can be significantly increased through the use of a feedback controller effectuating small perturbations in the boundary data. The controller consists of sensors which detect deviations in the fluid’s temperature from the motionless, conductive values and then direct actuators to respond to these deviations in such a way as to suppress the naturally occurring flow instabilities. Actuators which modify the boundary’s temperature or velocity are considered. The feedback controller can also be used to control flow patterns and generate complex dynamic behaviour at relatively low Rayleigh numbers.

Thermal convection during the directional solidification of a pure liquid with variable viscosity

1988 ◽  
Vol 188 ◽  
pp. 547-570 ◽  
Author(s):  
Marc K. Smith
Keyword(s):  
Rayleigh Number ◽  
Directional Solidification ◽  
Biot Number ◽  
Critical Rayleigh Number ◽  
Density Ratio ◽  
Bulk Liquid ◽  
Pure Liquid ◽  
Solidification Velocity ◽  
Finite Conductivity ◽  
Two Phases

The onset of a buoyancy-driven instability during the directional solidification of a pure liquid with a strongly temperature-dependent viscosity and an arbitrary Prandtl number is investigated using linear stability theory. The Rayleigh number for this system contains the lengthscale Ls defined as the ratio of the thermal diffusivity of the liquid and the solidification velocity times the density ratio of the two phases. It is independent of the actual depth of the liquid and it reflects the fact that increasing the solidification velocity stabilizes the system. The theory also shows that the difference in material properties between the two phases and the properties of the solidifying interface itself cause the interface to look like a boundary of finite conductivity measured by a wavenumber-dependent Biot number. For large viscosity variations, convection occurs below a stagnant layer which forms just beneath the interface where the liquid is immobilized by its very large viscosity. The thickness of this layer is measured by the natural logarithm of the viscosity contrast in the liquid times the lengthscale Ls. In this limit, the influence of the solidifying boundary is shielded from the bulk liquid by the stagnant layer and so the effect of the Biot number on the critical Rayleigh number is small. However, inertial effects, being associated with the bulk liquid, are very important for small Prandtl numbers of the fluid far from the interface. The model has applications to the solidification of magma chambers or lava lakes and to the material processing of polymeric liquids.

On the two-component Benard problem a numerical solution

1975 ◽  
Vol 80 (1) ◽  
pp. 76-88 ◽  
Author(s):  
J.C. Legros ◽  
D. Longree ◽  
G. Chavepeyer ◽  
J.K. Platten
Keyword(s):  
Numerical Solution ◽  
Two Component ◽  
Bénard Problem ◽  
Benard Problem

Linear stability of rotating convection in an imposed shear flow

1997 ◽  
Vol 350 ◽  
pp. 271-293 ◽  
Author(s):  
PAUL MATTHEWS ◽  
STEPHEN COX
Keyword(s):  
Rayleigh Number ◽  
Shear Flow ◽  
Eigenvalue Problem ◽  
Linear Stability ◽  
Thermal Convection ◽  
Critical Rayleigh Number ◽  
Rotation Vector ◽  
Linear Stability Theory ◽  
Fixed Temperature ◽  
Differential Motion

In many geophysical and astrophysical contexts, thermal convection is influenced by both rotation and an underlying shear flow. The linear theory for thermal convection is presented, with attention restricted to a layer of fluid rotating about a horizontal axis, and plane Couette flow driven by differential motion of the horizontal boundaries.The eigenvalue problem to determine the critical Rayleigh number is solved numerically assuming rigid, fixed-temperature boundaries. The preferred orientation of the convection rolls is found, for different orientations of the rotation vector with respect to the shear flow. For moderate rates of shear and rotation, the preferred roll orientation depends only on their ratio, the Rossby number.It is well known that rotation alone acts to favour rolls aligned with the rotation vector, and to suppress rolls of other orientations. Similarly, in a shear flow, rolls parallel to the shear flow are preferred. However, it is found that when the rotation vector and shear flow are parallel, the two effects lead counter-intuitively (as in other, analogous convection problems) to a preference for oblique rolls, and a critical Rayleigh number below that for Rayleigh–Bénard convection.When the boundaries are poorly conducting, the eigenvalue problem is solved analytically by means of an asymptotic expansion in the aspect ratio of the rolls. The behaviour of the stability problem is found to be qualitatively similar to that for fixed-temperature boundaries.Fully nonlinear numerical simulations of the convection are also carried out. These are generally consistent with the linear stability theory, showing convection in the form of rolls near the onset of motion, with the appropriate orientation. More complicated states are found further from critical.

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