Exact Solutions of Steady Two-Layer Hydraulic Exchange Flow

Abstract

Exchange flows are widely discussed to investigate internal waves in estuaries and straits. These phenomena are typically modeled as two-layer flows through a channel connecting two reservoirs with different densities. This paper proposes a method to calculate the interface profile and flow velocity of each layer in a channel with various contraction and sill configurations. Using the rigid-lid approach, the continuity and momentum balance equations are expressed in terms of the Froude numbers, to obtain both maximal and submaximal exchange solutions. These solutions are satisfied for certain pairs of the Bernoulli constants and the exchange rates. In a channel with a sill, the depth of the reservoir is considered, in contrast to the literature which assumes infinite reservoir depth. Furthermore, this method can be applied even when the contraction and sill are not located at the same position, which has not been discussed in other literature. These exact solutions are valuable for understanding the hydraulics of two-layer flow and can serve as benchmarks for validating numerical models.