Transient motion of a confined stratified fluid induced simultaneously by sidewall thermal loading and vertical throughflow

Abstract

An analytical study is made of the transient adjustment process of an initially stationary, stably stratified fluid in a square container. The boundary walls are highly conducting. The overall Rayleigh number Ra is large. Flow is initiated by the simultaneous switch-on of a temperature increase (δT) at the vertical wall and a forced vertical throughflow (Ra^−1/4δw) at the horizontal walls. The principal characteristics are found by employing the matched asymptotic expansion method. The flow field is divided into the inviscid interior, vertical boundary layers and horizontal boundary layers and analyses are conducted for each region. The horizontal boundary layers are shown to be of double-layered structure, and explicit solutions are secured for these layers. Evolutionary patterns of velocity and temperature in the whole flow domain are illustrated. Both opposing (δw/δT > 0) and cooperating (δw/δT < 0) configurations are considered. The flow character in the opposing configuration can be classified into (a) a forced-convection dominant mode (δw/δT > 1/ √2), (b) a buoyancy-convection-dominant mode (0 < δw/δT < 1/√2), and (c) a static mode (δw/δT ≈ 1/√2). Global evolutionary processes are depicted, and physical rationalizations are provided.