Self-similar multi-layer exchange flow through a contraction

Abstract

The multi-layer exchange equations for gravitationally driven flows between two basins with stable Boussinesq type of stratification in discrete layers, specified far upstream on either side of a connecting strait, result in a hydraulic control condition that must be satisfied at the narrowest part of the contraction, the control point. If one stagnant layer is present at the control point, the control condition that applies to all layers collectively may be separated into two such conditions that apply independently to two groups of layers going in opposite directions separated by the stagnant layer. Such bidirectional flow regimes exist if the structure of the prespecified density profiles permits each of the opposing groups to vertically reduce their thickness by the ratio 2/3 relative to their upstream thicknesses, leaving space for the stagnant layer to protrude through the contraction. Under these restrictions, the bidirectional flow is controlled by the fastest propagating wave mode and the stationary solution then relies on the superposition of two previously known unidirectional self-similar flow regimes that are completely decoupled. Techniques for their numerical computation are presented. The transition into loosely coupled and fully coupled flow is discussed. The decoupling principle also applies when several non-adjacent stagnant layers are simultaneously present at control in which case multiple groups of decoupled layers flow in alternating directions.