부산대학교 도서관

Sill-controlled flows involving the acceleration across a crest or control section with significant flow curvature are characterized by the Euler equations of an inviscid fluid. The problem is of both theoretical and practical interest in fluid mechanics research, given the role of such flows in water discharge measurements structures, underwater oceanic currents, river flows and shallow bar-built estuary inlets, among others. While three-dimensional computations are feasible, vertically averaged solutions are sought to gain efficiency. The vertically averaged representation of sill overflows based on the Serre–Green–Naghdi (SGN) theory is limited to very shallow flows, and a good averaged approach for large overflows is currently not available. High-level Green–Naghdi (GN) theory forms a hierarchy of theories of increasing accuracy based on expanding the kinematic field and vertically-averaging in a weighted-residual sense. These theories have been successfully applied to ocean research but so far, they have not been applied to flow in open channels and structures. In this work, the high-level Green–Naghdi theories, of which SGN equations are the lowest level possible (GN level I theory), are formulated and newly applied to the sill-controlled flow problem. High-level GN theory is compared with detailed experiments from the literature and new ones conducted, and with a new fully non-linear vertically resolved potential flow solver which uses a x–ψ mapping. It was found that the GN expansions are convergent in the sill problem, in contrast to former perturbation solutions, which are asymptotic. The GN level V theory was found to be in good agreement with experiments for the sill-controlled flow problem, which excellently reproduced the free surface, bottom pressure, and vertical distributions of velocity and pressure. The GN level V theory was found to be applicable for relatively wide range of overflows up to E/R = 3, where E is the minimum specific energy and R the sill crest radius of curvature. From a practical viewpoint, large improvements were observed when using GN level II theory, instead of GN level I (e.g. the SGN theory), producing good results up to E/R = 1.75, which is the minimum level thus recommended in practice.