A hydrometer

Here, I construct a simple, elegant hydrometer model measuring the fluid exit from an outlet in the triangular reservoir or a mountain lake. This two-parameter hydro-mechanical system is based on some dynamically justified physical principles for energy dissipations through the discharge coefficient and the complementary friction. It is expressed as a simple differential equation whose exact-analytical solutions have been presented. The new model is an extension to the classical Torricelli-Bernoulli-law, however, it differs fundamentally from the classical law. As the fluid surface area evolves with the hydraulic head, this becomes the game-changer characterizing the hydrometer. For technical use, the model provides the hydrometer and the hydrometer-velocity measuring the evolving depth of the frictional fluid and its exit velocity. Analytical solutions are constructed for the hydrometer for contrasting scenarios: without and with narrow conduit approximation, and dam collapse, resulting in completely different dynamics. I showed that a hypergeometric function analytically solves the full hydrometer. The pleasing compact dam collapse model possesses an exact analytical solution. Although structurally drastically simplified, the dam collapse solution appears plausible. As a whole, the hypergeometric solution for narrow exit without approximation and the dam collapse solution behave similarly with strong non-linearity for both the hydrometer and the hydrometer-velocity. The results for conduit approximation are fundamentally different than those for the dam collapse and the hypergeometric solution. I proved that the model approximation loses the fundamental physical essence of the hydrometer resulting in paradoxically invalid solution. Contrary to the prevailing perception, the hydrometer is critical to the reservoir free-surface kinetic energy. The crucial revelation is that hydro-mechanically, this kinetic energy provides a vital ingredient to the system that cannot just be ignored; if done so, it destroys the physical system. This is a paradigm-shift in hydro-mechanics. The new hydrometer can be applied to different engineering and environmental problems, including the draining and emergency evacuation of the glacial or mountain lakes, hydropower reservoirs, and the fluid vessels; and their engineered discharges. For mountain flash flood simulation, my method provides cost-effective, efficient and probably the best analytical solutions with the initial hydrographs.