Three-dimensional global instability modes associated with a localized hot spot in Rayleigh–Bénard–Poiseuille convection

Abstract

Global modes of the thermal convection type in the Rayleigh–Bénard–Poiseuille (RBP) system are analysed for the case of non-uniform heating of the lower wall. Specifically, a single two-dimensional ‘hot spot’ or ‘temperature bump’ giving rise to a finite region of local instability is considered. For the case of the lower wall temperature varying slowly on the scale of the RBP cell height, i.e. for a gentle temperature bump, WKBJ asymptotics are used to construct an analytical approximation of the linear global mode. At the same time, an analytical selection criterion for the critical global mode is derived from the breakdown of the WKBJ expansion at a double turning point located at the top of the temperature bump. The analytical construction and the underlying assumptions are supported by comparison with direct numerical simulations for Gaussian temperature bumps of elliptical planform not necessarily aligned with the mean flow. From these comparisons, it is concluded that the proposed analytical construction indeed yields the most amplified global mode, which is characterized by an essentially transverse orientation (normal to the mean flow direction) of the convection rolls, independent of the planform of the temperature bump. The paper concludes with preliminary DNS results on the saturated global mode shape and a discussion of possible connections to the ‘steep’ fully nonlinear global modes found for one-dimensional inhomogeneities of the basic state.