Natural vibration frequencies of horizontal tubes partially filled with liquid

This work presents an experimental and numerical study on the flexural vibration of horizontal circular tubes partially filled with liquid. The tube is configured as a free-free beam with attention being directed to the case of small amplitudes of transverse oscillation whereas the axial movements of the tube and liquid are disregarded. At first vertical and horizontal polarizations of the flexural tube are investigated experimentally for different amounts of filling liquid. In contrast with the empty and fully-filled tubes, it is observed that natural frequencies of the vertical and horizontal polarizations are different due to asymmetry induced by the liquid layer, which acts like an added mass. Less mass of liquid is added to the tube when oscillating horizontally; as a consequence, eigenfrequencies for the horizontal polarization are found to be greater than the case of the vertically polarized tube. A simple method to calculate the natural vibration frequencies using coefficients of added mass of liquid is proposed. It is shown that the added mass coefficient increases with the liquid's level and viscosity. At last a numerical investigation of the interaction between the liquid and the tube is carried out by solving in two-dimensions the full Navier-Stokes equations via a finite volume method, with the free-surface flow being modeled with a homogeneous multiphase Eulerian-Eulerian fluid approach. Vertical and horizontal polarizations are imposed to the tube with pressure and shear stresses being determined numerically to assess the liquid's forcing onto the tube's wall. The coefficient of added mass of liquid is then estimated by the ratio between the resulting force and the acceleration imposed to the wall. A good agreement is found between experimental and numerical results, especially for the horizontally oscillating tube. It is also shown that viscosity can noticeably affect the added mass coefficients, particularly at low filling levels.