Mixed convection in a periodically heated channel

Abstract

Mixed convection in a channel with flow driven by a pressure gradient and subject to spatially periodic heating along one of the walls has been studied. The pattern of the heating is characterized by the wavenumber${\it\alpha}$and its intensity is expressed in terms of the Rayleigh number$\mathit{Ra}_{p}$. The primary convection has the form of counter-rotating rolls with the wavevector parallel to the wavevector of the heating. The resulting net heat flow between the walls increases proportionally to$\mathit{Ra}_{p}$but the growth saturates when$\mathit{Ra}_{p}=O(10^{3})$. The most effective heating pattern corresponds to${\it\alpha}\approx 1$, as this leads to the most intense transverse motion. The primary convection is subject to transition to secondary states with the onset conditions depending on${\it\alpha}$. The conditions leading to transition between different forms of secondary motion have been determined using the linear stability theory. Three patterns of secondary motion may occur at small Reynolds numbers$\mathit{Re}$, i.e. longitudinal rolls, transverse rolls and oblique rolls, with the critical conditions varying significantly as a function of${\it\alpha}$. An increase of${\it\alpha}$leads to the elimination of the longitudinal rolls and, eventually, to the elimination of the oblique rolls, with the transverse rolls assuming the dominant role. For large${\it\alpha}$, the transition is driven by the Rayleigh–Bénard mechanism; while for${\it\alpha}=O(1)$, the spatial parametric resonance dominates. The global flow characteristics are identical regardless of whether the heating is applied at the lower or the upper wall.