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Frictional exchange flow through a wide channel with application to the Burlington Ship Canal Gu, Li

Abstract

The maximal frictional two-layer exchange flows in general and the exchange of water through the Burlington Ship Canal in particular were studied through both theoretical analysis and laboratory experiments. Gravitational two-way exchange flows are often driven by a slight density difference due to temperature, salinity and/or sediment concentration variations across a constriction connecting two water bodies. The exchange of two fluids of differing density through the constriction such as a channel or strait is of importance in a wide range of geophysical, oceanographic, and engineering contexts. One such example, which has largely motivated this study, is the exchange of water through the Burlington Ship Canal connecting the heavily polluted Hamilton Harbour with Lake Ontario. Previously analytical solutions to exchange flow problems have often ignored frictional and/or non-linear inertial effects. As a result, the applicability of such solutions is fairly limited, especially for the Burlington Ship Canal, where both frictional and inertial effects are important. To this end, the fully non-linear one-dimensional shallow-water equation must be used to describe general frictional exchange flow problems. So far, solutions to such problems have been exclusively obtained through numerical integration. The maximal, steady, frictional exchange flow through a horizontal channel of constant width was studied analytically in the context of two-layer internal hydraulics. The fully non-linear hydraulic or shallow water equation for a two-layered flow system was solved through direct integration as a Boundary Value Problem (BVP). The resulting analytical exchange flow solutions predict the maximal exchange flow rate and interface profile throughout the channel for given frictional parameters. A laboratory facility was also designed specifically for the purpose of modelling the two-layer exchange flow. The laboratory experiments aim to validate the theoretical predictions and provide additional in-depth understanding of the dynamics of exchange flows. The exchange flows established in the laboratory experiments were studied using conductivity profiling, flow visualization, particle image velocimetry, and image processing techniques. The solution shows that friction significantly increases the overall interface slope and reduces the exchange rate. Despite the pronounced non-linear nature of exchange flow problems, the linear density interface profile has been widely used as the first approximation in the previous theoretical formulations of analytical solutions on frictional exchange flows. The linear density interface profile assumes that the density interface follows a straight line linking two hydraulic controls. The resulting non-linear exchange flow solution indicates that the interface profile is not only non-linear, but also non-symmetric in nature. The theoretical predictions compared well with both laboratory experiments and field measurements in the Burlington Ship Canal as well as several famous sea straits.

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