Three-Dimensional Convective Planforms for Inclined Darcy-Bénard Convection

D Andrew S Rees,Antonio Barletta

doi:10.3390/fluids5020083

Abstract

We investigate the onset of convection in an inclined Darcy-Bénard layer. When such a layer is unbounded in the spanwise direction it is generally known that longitudinal rolls comprise the most unstable planform. On the other hand, when a layer has a sufficiently small spanwise width, then transverse rolls form the most unstable planform. However, the layer remains stable to transverse roll disturbances when the inclination is above roughly 31 degrees from the horizontal. This paper considers the transition between these two extreme cases where the spanwise width takes moderate values and where rectangular cells are considered. It is found that the most unstable planform is quite strongly sensitive to the magnitude of the spanwise width and that there are large regions of parameter space within which three-dimensional convection patterns have the smallest critical Darcy-Rayleigh number.

Highlights

The Rayleigh-Bénard instability in a porous layer saturated by a fluid and heated from below has been the subject of many investigations over the last decades
In this paper we have studied the effect of having a spanwise constraint on the onset of
When the layer is unconstrained in this way, it is well-known that longitudinal rolls always form the most unstable convection pattern and that the critical Darcy-Rayleigh number is 4π 2 / cos α

Summary

Introduction

The Rayleigh-Bénard instability in a porous layer saturated by a fluid and heated from below has been the subject of many investigations over the last decades. The alternative name used for this instability when a porous medium is involved is Darcy-Bénard as the momentum transfer for seepage flows is modelled through Darcy’s law. The porous layer where the thermoconvective instability arises in the form of cellular patterns is assumed to be horizontal and bounded by impermeable walls kept at different uniform temperatures. This situation has a stationary basic state where no flow occurs and a vertical temperature gradient directed downward causes the instability. There is a different scenario when the porous layer is inclined at an angle α to the horizontal

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