Oscillations of Water Levels in Vertical Semi-Immersed Tubes: Analytical Solutions and Experimental Verification

Abstract

Experimental results for water level oscillations in vertical tubes, together with a theoretical solution for the flow in such tubes considering local and distributed energy losses, are presented and compared. The experimental data were obtained in small scale experiments, allowing adequately controlling the oscillations. The governing equation for the oscillations was obtained by applying the conservation laws of mass, momentum and energy for fluids. It is a second order nonlinear differential equation which was reduced to a first order differential Bernoulli equation. The obtained solution is composed by two different equations, one for the upwards motion and the other for the downwards motion, which together reproduce the oscillatory damped behavior of such flows. Numerical solutions of the differential equation were also checked. The experimental data and the theoretical and numerical results showed a good agreement between measured and calculated values of velocity and surface level for the first periods of oscillation.