Hydraulics of stratified sill flows within varying channel geometries: investigating energy loss and mixing of maximal two-layer exchange

The paper presents internal-flow hydraulics for stratified flows generated in laboratory-scale channels with different geometries, forcing conditions and two-layer flow regimes. Analytical model formulations are presented for the internal-flow head function of quadratic-type channels. While the study focuses on maximal two-layer exchange, where two hydraulic transitions of critical flow are required for the bi-directional stratified flow to be fully controlled, for the channel geometries where only one critical-flow hydraulic transition is present, the two-layer exchange is considered to be sub-maximal, i.e. partially controlled, or frictionally determined without this hydraulic transition. In particular, for the latter case, it is shown that dominant frictional shear and interfacial mixing processes, driving buoyancy flux between the counter-flowing layers, may be a reason that maximal two-layer exchange conditions are not achieved for these specific channel-sill geometries. This interfacial mixing is largely associated with internal dynamics of stably stratified, two-layer, bi-directional flows along the channel by externally imposed barotropic net-exchange flow components, which can restrict or wholly block one of the counter-flowing layers. A novel buoyancy-flux transfer model is therefore incorporated into the internal-flow hydraulic model to provide partially-controlled or frictionally-determined conditions for the uni- and bi-directional stratified flows generated. Previous and new experimental investigations are used to justify the extended internal-flow hydraulic model solutions proposed in the present study, including (1) buoyancy-driven exchange across a descending barrier within a rectangular channel-basin configuration, (2) dense gravity flows along an upsloping and constricted triangular channel, and (3) externally forced two-layer exchange (i) across a submerged sill obstruction with a rectangular cross-section and (ii) through an elongated sill-channel with a trapezoidal cross-section. As these previous laboratory studies were often motivated by the observations for stratified flows in incompletely blocked river estuaries, across fjordic sills and through sea straits, the hydraulic modelling results are discussed in the context of applicability to these natural stratified-flow environments.