Instabilities of convection rolls in a fluid of moderate Prandtl number

1979 ◽  
Vol 91 (2) ◽  
pp. 319-335 ◽  
Author(s):  
F. H. Busse ◽  
R. M. Clever
Keyword(s):  
Prandtl Number ◽  
Fluid Layer ◽  
Experimental Methods ◽  
Two Dimensional ◽  
Theoretical Predictions ◽  
Convection Rolls ◽  
Rayleigh Numbers ◽  
Horizontal Fluid Layer

The instabilities of two-dimensional convection rolls in a horizontal fluid layer heated from below are investigated in the case when the Prandtl number is seven or lower. Two new mechanisms of instability are described theoretically as well as experimentally. The knot instability causes the transition to spoke-pattern convection at higher Rayleigh numbers while the skewed varicose instability accomplishes a change to larger horizontal wavelengths of the convection rolls. Both instabilities disappear in the limits of small and large Prandtl number. Although the experimental methods fail in realizing closely the infinitely conducting boundaries assumed in the theory, the observations agree in all qualitative aspects with the theoretical predictions.

Finite-amplitude convection in the presence of finitely conducting boundaries

2001 ◽  
Vol 432 ◽  
pp. 351-367 ◽  
Author(s):  
M. WESTERBURG ◽  
F. H. BUSSE
Keyword(s):  
Thermal Conductivity ◽  
Prandtl Number ◽  
Fluid Layer ◽  
Finite Amplitude ◽  
Onset Of Convection ◽  
Theoretical Predictions ◽  
Square Cells ◽  
Convection Rolls ◽  
Rayleigh Numbers ◽  
Horizontal Fluid Layer

Finite-amplitude convection in the form of rolls and their stability with respect to infinitesimal disturbances is investigated in the case of boundaries of the horizontal fluid layer which exhibit a thermal conductivity comparable to that of the fluid. It is found that even when rolls represent the preferred mode at the onset of convection a transition to square cells may occur at slightly supercritical Rayleigh numbers. The phenomenon of inertial convection in low Prandtl number fluids appears to become more pronounced as the conductivity of the boundaries is reduced. Modulated convection rolls have also been found as solutions of the problem. But they appear to be unstable in general. Experimental observations have been made and are found in general agreement with the theoretical predictions.

Nonlinear interaction of magnetic field and convection

1975 ◽  
Vol 71 (1) ◽  
pp. 193-206 ◽  
Author(s):  
F. H. Busse
Keyword(s):  
Magnetic Field ◽  
Magnetic Energy ◽  
Fluid Layer ◽  
Boussinesq Equations ◽  
Finite Amplitude ◽  
Magnetic Reynolds Number ◽  
Convection Rolls ◽  
Rayleigh Numbers ◽  
Horizontal Fluid Layer ◽  
Chandrasekhar Number

The interaction between convection in a horizontal fluid layer heated from below and an ambient vertical magnetic field is considered. The analysis is based on the Boussinesq equations for two-dimensional convection rolls and the assumption that the amplitude A of the convection and the Chandrasekhar number Q are small. It is found that the magnetic energy is amplified by a factor of order R½m, where Rm is the magnetic Reynolds number. The ratio between the magnetic and kinetic energies can reach values much larger than unity. Although the magnetic field always inhibits convection, this influence decreases with increasing amplitude of convection. Thus finite amplitude onset of steady convection becomes possible at Rayleigh numbers considerably below the values predicted by linear theory.

Instabilities of convection rolls in a high Prandtl number fluid

1971 ◽  
Vol 47 (2) ◽  
pp. 305-320 ◽  
Author(s):  
F. H. Busse ◽  
J. A. Whitehead
Keyword(s):  
Prandtl Number ◽  
Three Dimensional ◽  
Wave Number ◽  
Stationary Convection ◽  
Wave Numbers ◽  
Theoretical Predictions ◽  
High Prandtl Number ◽  
The Stability ◽  
Convection Rolls ◽  
Rayleigh Numbers

An experiment on the stability of convection rolls with varying wave-number is described in extension of the earlier work by Chen & Whitehead (1968). The results agree with the theoretical predictions by Busse (1967a) and show two distinct types of instability in the form of non-oscillatory disturbances. The ‘zigzag instability’ corresponds to a bending of the original rolls; in the ‘cross-roll instability’ rolls emerge at right angles to the original rolls. At Rayleigh numbers above 23,000 rolls are unstable for all wave-numbers and are replaced by a three-dimensional form of stationary convection for which the name ‘bimodal convection’ is proposed.

The oscillatory instability of convection rolls in a low Prandtl number fluid

1972 ◽  
Vol 52 (1) ◽  
pp. 97-112 ◽  
Author(s):  
F. H. Busse
Keyword(s):  
Prandtl Number ◽  
Reasonable Agreement ◽  
Fluid Layer ◽  
Three Dimensional ◽  
Convective Motion ◽  
Critical Value ◽  
Two Dimensional ◽  
Free Boundaries ◽  
Vertical Vorticity ◽  
Convection Rolls

The instability of convection rolls in a fluid layer heated from below is investigated for stress-free boundaries in the limit of small Prandtl number. It is shown that the two-dimensional rolls become unstable to oscillatory three-dimensional disturbances when the amplitude of the convective motion exceeds a finite critical value. The instability corresponds to the generation of vertical vorticity, a mechanism which is likely to operate in the case of a variety of roll-like motions. In all aspects in which the theory can be related to experiments, reasonable agreement with the observations is found.

Stabilization of the No-Motion State of a Horizontal Fluid Layer Heated From Below With Joule Heating

1995 ◽  
Vol 117 (2) ◽  
pp. 329-333 ◽  
Author(s):  
J. Tang ◽  
H. H. Bau
Keyword(s):  
Joule Heating ◽  
Fluid Layer ◽  
Control Strategies ◽  
Critical Rayleigh Number ◽  
Linear Stability Theory ◽  
Small Perturbations ◽  
Motion State ◽  
Conduction State ◽  
Rayleigh Numbers ◽  
Horizontal Fluid Layer

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–Be´nard problem of an infinite fluid layer heated from below with Joule heating and cooled from above can be significantly increased through the use of feedback control strategies effecting small perturbations in the boundary data. The bottom of the layer is heated by a network of heaters whose power supply is modulated in proportion to the deviations of the temperatures at various locations in the fluid from the conductive, no-motion temperatures. Similar control strategies can also be used to induce complicated, time-dependent flows at relatively low Rayleigh numbers.

The effect of distant side-walls on the evolution and stability of finite-amplitude Rayleigh-Bénard convection

1981 ◽  
Vol 378 (1775) ◽  
pp. 539-566 ◽  
Keyword(s):  
Rayleigh Number ◽  
Prandtl Number ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Bénard Convection ◽  
Benard Convection ◽  
Side Walls ◽  
Rayleigh Benard Convection ◽  
Rayleigh Numbers ◽  
Rayleigh Benard

A recent study by Cross et al . (1980) has described a class of finite-amplitude phase-winding solutions of the problem of two-dimensional Rayleigh-Bénard convection in a shallow fluid layer of aspect ratio 2 L (≫ 1) confined laterally by rigid side-walls. These solutions arise at Rayleigh numbers R = R 0 + O ( L -1 ) where R 0 is the critical Rayleigh number for the corresponding infinite layer. Nonlinear solutions of constant phase exist for Rayleigh numbers R = R 0 + O ( L -2 ) but of these only the two that bifurcate at the lowest value of R are stable to two-dimensional linearized disturbances in this range (Daniels 1978). In the present paper one set of the class of phase-winding solutions is found to be stable to two-dimensional disturbances. For certain values of the Prandtl number of the fluid and for stress-free horizontal boundaries the results predict that to preserve stability there must be a continual readjustment of the roll pattern as the Rayleigh number is raised, with a corresponding increase in wavelength proportional to R - R 0 . These solutions also exhibit hysteresis as the Rayleigh number is raised and lowered. For other values of the Prandtl number the number of rolls remains unchanged as the Rayleigh number is raised, and the wavelength remains close to its critical value. It is proposed that the complete evolution of the flow pattern from a static state must take place on a number of different time scales of which t = O(( R - R 0 ) -1 ) and t = O(( R - R 0 ) -2 ) are the most significant. When t = O(( R - R 0 ) -1 ) the amplitude of convection rises from zero to its steady-state value, but the final lateral positioning of the rolls is only completed on the much longer time scale t = O(( R - R 0 ) -2 ).

scholarly journals Linear and Nonlinear Convection with an Aligned Magnetic Field

1990 ◽  
Vol 142 ◽  
pp. 135-136
Author(s):  
N. Rudraiah ◽  
I S Shivakumara ◽  
P Geetavani
Keyword(s):  
Magnetic Field ◽  
Fluid Layer ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Horizontal Magnetic Field ◽  
Nonlinear Convection ◽  
Linear And Nonlinear ◽  
Horizontal Fluid Layer ◽  
Aligned Magnetic Field

The effect of horizontal magnetic field on the onset of three-dimensional convection in a horizontal fluid layer is studied. It is found that the two-dimensional solutions are unstable to three-dimensional disturbances. A detailed bifurcation study is reported.

Finite Amplitude Longitudal Convection Rolls in an Inclined Layer

1973 ◽  
Vol 95 (3) ◽  
pp. 407-408 ◽  
Author(s):  
R. M. Clever
Keyword(s):  
Fluid Layer ◽  
Horizontal Layer ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Governing Equations ◽  
Inclined Layer ◽  
Experimental Heat ◽  
Inclined Fluid Layer ◽  
Longitudinal Coordinate ◽  
Convection Rolls

For the case of a large Prandtl number, buoyancy driven flow in an inclined fluid layer, it is shown that all longitudinal-coordinate-independent solutions of the governing equations are obtainable from a knowledge of the existing results for two-dimensional convection in a horizontal layer, heated from below. The rescaling here yields results which compare favorably with those of existing experimental heat transport values.

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