Wavepacket instability in a rectangular porous channel uniformly heated from below

Abstract

This paper is aimed to investigate the transition to absolute instability in a porous layer with horizontal throughflow. The importance of this analysis is due to the possible experimental failure to detect growing perturbations which are localised in space and which may be convected away by the throughflow. The instability of the uniform flow in a horizontal rectangular channel subject to uniform heating from below and cooled from above is studied. While the lower wall is modelled as an impermeable isoflux plane, the upper wall is assumed to be impermeable and imperfectly conducting, so that a Robin temperature condition with a given Biot number is prescribed. The sidewalls are assumed to be adiabatic and impermeable. The basic state considered here is a stationary parallel flow with a vertical uniform temperature gradient, namely the typical configuration describing the Darcy–Bénard instability with throughflow. The linear instability of localised wavepackets is analysed, thus detecting the parametric conditions for the transition to absolute instability. The absolute instability is formulated through an eigenvalue problem based on an eighth–order system of ordinary differential equations. The solution is sought numerically by utilising the shooting method. The threshold to absolute instability is detected versus the Péclet number associated with the basic flow rate along the channel.