We investigated the asymptotics of two-dimensional steady solutions simulating the energy-conserving flow in a horizontal duct of finite depth in situations where the flow contains a region spanning the depth of the duct, and a region in which the fluid surface detaches from the ceiling of the duct as a free surface. These asymptotics are constructed using the local hydrostatic approximation, which generalizes the classical long-wave approximation. The initial (zero-order) asymptotics leading to the piecewise constant solutions are obtained from the mass, momentum, and energy conservation laws of the first approximation of shallow water theory. The first-order asymptotics for the liquid depth are constructed using the momentum conservation law of the Green-Nagdi model representing the second approximation of shallow water theory. It is shown that the continuous solution obtained from this asymptotics is in good agreement with the Wilkinson laboratory experiment [D. L. Wilkinson, "Motion of air cavities in long horizontal ducts,"J. Fluid Mech. 118, 109 (1982)] on modeling the energy-conserving steady flow predicted by the classical piecewise constant Benjamin solution [T. B. Benjamin, "Gravity currents and related phenomena,"J. Fluid Mech. 31, 209 (1968)].
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