Effect of a finite external heat transfer coefficient on the Darcy-Bénard instability in a vertical porous cylinder

Abstract

The onset of thermal convection in a vertical porous cylinder is studied by considering the heating from below and the cooling from above as caused by external forced convection processes. These processes are parametrised through a finite Biot number, and hence through third-kind, or Robin, temperature conditions imposed on the lower and upper boundaries of the cylinder. Both the horizontal plane boundaries and the cylindrical sidewall are assumed to be impermeable; the sidewall is modelled as a thermally insulated boundary. The linear stability analysis is carried out by studying separable normal modes, and the principle of exchange of stabilities is proved. It is shown that the Biot number does not affect the ordering of the instability modes that, when the radius-to-height aspect ratio increases, are displayed in sequence at the onset of convection. On the other hand, the Biot number plays a central role in determining the transition aspect ratios from one mode to its follower. In the limit of a vanishingly small Biot number, just the first (non-axisymmetric) mode is displayed at the onset of convection, for every value of the aspect ratio. C 2013 American Institute of Physics. [http://dx

The impact of imperfect heat transfer on the convective instability of a thermal boundary layer in a porous media

“…They calculated the magnitude of this effect and proposed a Biot-modified Rayleigh number defined by Rp Bi/(1 + Bi) to account for this change. Their Biot-modified Rayleigh number tends to a constant value in both the high and low Biot number limits, with a monotonic decrease as the Biot number decreases, in agreement with the observation of Barletta et al (2015) that insulating boundaries are less restrictive. Robin boundary conditions have also been considered in a variety of different geometries, including a horizontal cylindrical pipe (Barletta & Storesletten 2011) and a vertical cylinder (Barletta & Storesletten 2013).…”

“…Slim & Ramakrishnan 2010;Tilton et al 2013). The net deviation causes a weak increase in stability, and is consistent with a competition between reduction of available potential energy by vertical confinement of the background profile, and the tendency of the modified perturbation boundary condition to reduce stability by allowing greater variation in surface temperature (see Barletta et al 2015). For smaller Biot numbers, the reduction in available potential energy strongly dominates and leads to a substantial increase in the critical Rayleigh number, which scales asRp c ∼ 1.9/Bi forBi 1.…”

“…It is also consistent with a tendency for stronger and deeper-penetrating convection, leading to larger convective cells with a longer wavelength and smaller wavenumber, k b c < 0. Barletta et al (2015) noted a similar effect for steady-state convection in a finite horizontal domain. As the Biot number decreases they observed that the critical Rayleigh number and critical wavenumber also decrease as the boundary conditions become less restrictive on the allowable perturbations (note that their parameters a i and b i (i = 1, 2) are equivalent to 1/Bi).…”

“…Barletta & Storesletten 2012). Barletta, Tyvand & Nygård (2015) took this further and applied a Robin boundary condition to both the thermal and vertical velocity perturbations on the upper and lower boundaries. They found that Dirichlet boundary conditions (constant temperature or no normal flow) are the most restrictive of convective instability, while Neumann boundary conditions (no heat flux or no horizontal flow) are the most liberating.…”

Adiabatic eigenflows in a vertical porous channel

“…When they are put in increasing order, the correspondence between each pair (p, q) and the index n of the eigenvalue, employed in the general equation (4.5), comes out naturally. A discussion on this ordering can be found, for instance, in Beattie (1958), Barletta & Storesletten (2013) and Barletta (2014).…”