The effect of distant sidewalls on the transition to finite amplitude Benard convection

1978 ◽  
Vol 358 (1693) ◽  
pp. 173-197 ◽  
Keyword(s):  
Heat Transfer ◽  
Theoretical Model ◽  
Rayleigh Number ◽  
Fluid Layer ◽  
Critical Value ◽  
Finite Amplitude ◽  
Smooth Transition ◽  
Two Dimensional ◽  
Bénard Convection ◽  
Benard Convection

The theory of the nonlinear development of Benard convection in an infinite fluid layer confined between horizontal boundaries predicts that the amplitude of the motion undergoes a bifurcation as the Rayleigh number passes through the critical value for instability predicted by linear theory. Segel (1969) has shown that this is also the case if the flow is confined laterally by rigid perfectly insulating sidewalls. In the present paper it is shown that if there is a small heat transfer through these walls (so that the boundary conditions there are inconsistent with a state of no motion) the bifurcation is in general replaced by a smooth transition to finite amplitude convection. This effect also ensures that the motion is stable. Although an idealized theoretical model is assumed in which the flow is two-dimensional and stress-free at the horizontal boundaries, the results apply qualitatively to more realistic models.

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).

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 ).

Large amplitude convection in porous media

1974 ◽  
Vol 64 (1) ◽  
pp. 51-63 ◽  
Author(s):  
Joe M. Straus
Keyword(s):  
Rayleigh Number ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Bénard Convection ◽  
Benard Convection ◽  
Two Dimensional Flow ◽  
The Stability ◽  
Finite Band ◽  
Rayleigh Numbers ◽  
Convection In Porous Media

The properties of convective flow driven by an adverse temperature gradient in a fluid-filled porous medium are investigated. The Galerkin technique is used to treat the steady-state two-dimensional problem for Rayleigh numbers as large as ten times the critical value. The flow is found to look very much like ordinary Bénard convection, but the Nusselt number depends much more strongly on the Rayleigh number than in Bénard convection. The stability of the finite amplitude two-dimensional solutions is treated. At a given value of the Rayleigh number, stable two-dimensional flow is possible for a finite band of horizontal wavenumbers as long as the Rayleigh number is small enough. For Rayleigh numbers larger than about 380, however, no two-dimensional solutions are stable. Comparisons with previous theoretical and experimental work are given.

On the occurrence of cellular motion in Bénard convection

1967 ◽  
Vol 30 (4) ◽  
pp. 651-661 ◽  
Author(s):  
E. Palm ◽  
T. Ellingsen ◽  
B. Gjevik
Keyword(s):  
Kinematic Viscosity ◽  
Silicone Oil ◽  
Fluid Layer ◽  
Critical Value ◽  
Free Boundaries ◽  
Bénard Convection ◽  
Benard Convection ◽  
Rayleigh Numbers ◽  
Effect Of Surface ◽  
Cellular Motion

The interval of Rayleigh numbers in Bénard convection corresponding to cellular motion is determined in the case of free-free boundaries, rigid-free boundaries and rigid-rigid boundaries, taking into account the variation of the kinematic viscosity with temperature. Neglecting the effect of surface tension, it is shown that this interval is largest for the rigid-rigid case. The most important feature from the obtained formula (6.1) is, however, that the interval is extremely dependent on the depth of the fluid layer. To obtain a cellular pattern it is therefore necessary to have very small fluid depths. For example, with Silicone oil and a fluid depth of 7 mm, cellular motion will, according to the theory, be observed for Rayleigh numbers larger than the critical value and less than 1·07 times the critical value. For a fluid depth of 5 mm, however, the formula (6.1) gives that cellular motion will be observed for Rayleigh numbers up to 1·54 times the critical value.

On finite amplitude Bénard convection in a cylindrical container

1978 ◽  
Vol 360 (1703) ◽  
pp. 455-469 ◽  
Keyword(s):  
Rayleigh Number ◽  
Linear Theory ◽  
Large Cell ◽  
Finite Amplitude ◽  
Cylindrical Container ◽  
Amplitude Function ◽  
Bénard Convection ◽  
Benard Convection

A study analogous to that of Daniels (1977) is undertaken for finite amplitude Bénard convection in a shallow cylindrical container with an imperfectly insulated sidewall. A novel feature of the investigation is the singularity which develops in the amplitude function as the centre of the cylinder is approached. This singularity results in an unexpectedly large cell amplitude at the origin, and a slight increase in the value of the Rayleigh number at which the convection cells spread throughout the fluid from that predicted by linear theory.

scholarly journals Turbulent transition in Rayleigh-Bénard convection with fluorocarbon (a)

2021 ◽  
Vol 136 (1) ◽  
pp. 10003
Author(s):  
Lucas Méthivier ◽  
Romane Braun ◽  
Francesca Chillà ◽  
Julien Salort
Keyword(s):  
Heat Transfer ◽  
Velocity Field ◽  
Rayleigh Number ◽  
Working Fluid ◽  
Turbulent Transition ◽  
Bénard Convection ◽  
Benard Convection ◽  
Image Velocimetry ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

Abstract We present measurements of the global heat transfer and the velocity field in two Rayleigh-Bénard cells (aspect ratios 1 and 2). We use Fluorinert FC770 as the working fluid, up to a Rayleigh number . The velocity field is inferred from sequences of shadowgraph pattern using a Correlation Image Velocimetry (CIV) algorithm. Indeed the large number of plumes, and their small characteristic scale, make it possible to use the shadowgraph pattern produced by the thermal plumes in the same manner as particles in Particle Image Velocimetry (PIV). The method is validated in water against PIV, and yields identical wind velocity estimates. The joint heat transfer and velocity measurements allow to compute the scaling of the kinetic dissipation rate which features a transition from a laminar scaling to a turbulent Re 3 scaling. We propose that the turbulent transition in Rayleigh-Bénard convection is controlled by a threshold Péclet number rather than a threshold Rayleigh number, which may explain the apparent discrepancy in the literature regarding the “ultimate” regime of convection.

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