Liquid vibrations in cylindrical tanks with flexible membranes

Abstract

The objectives of this paper are in studying the liquid vibrations in rigid circular cylindrical shells with internal flexible membranes or covered by membranes. The liquid in the container is supposed to be an ideal and incompressible one, and the fluid motion is irrotational. In above formulated suppositions the velocity potential is introduced; it satisfies the Laplace equation. The boundary value problem is formulated for the velocity potential. To obtain boundary conditions on the liquid free surface, the membrane deflection is considered, and the equality of normal components of liquid and membrane velocities is satisfied. The incompressible and inviscid liquid is supposed to perform irrotational motion in the fluid domain divided into two sub-domains by internal flexible membrane that is installed at the given height. For solutions of the problems both numerical and analytical methods are in use. The analytical solutions of two boundary value problems are obtained for unknown velocity potential and membrane deflection as the Fourier–Bessel series with coefficients depending on unknown frequency. Satisfying boundary conditions, we obtain the system of homogeneous algebraic equations. The condition of a non-trivial solution of this system gives the non-linear equation for evaluating the frequencies. The coupled membrane and liquid vibrations in cylindrical tanks are studied also by FEM and BEM methods. The comparison of results obtained using analytical approach with ones received with boundary and finite element method is provided. The main results are as follows. As follows from numerical simulations, if the membrane is installed inside the cylinder, then the most important parameter affecting the result, is the height of the membrane installation. If the membrane is installed at a considerable distance from the free surface, then the sloshing frequency practically does not change, and more precisely, it slightly increases. The dependencies of frequencies via the filling level are identified. The novelty of proposed approach consists in possibility to study the influence of elastic baffles and roofs in the liquid-filled tanks.