Computation of the equilibrium position of a liquid with surface tension inside a tank of complex geometry and extension to sloshing dynamic cases

Abstract

To study the vibrations of a tank partially filled with a liquid in low-gravity environment, we first have to find the static position of the liquid. In this paper, we present a three-dimensional finite element approach to find this equilibrium configuration for any tank geometry. Both gravity and capillary effects are taken into account. The nonlinear equations of this problem are derived from the differentiation of the total potential energy of the system, then the problem is transformed into a liquid free surface form-finding. The well-known singularity of this kind of problems is regularized using the updated reference strategy. The equations of the regularized problem are discretized using the finite element method and solved by the Newton-Raphson algorithm. Several examples illustrate the effectiveness of this method, even for complex cases, and two validation tests are presented. The linear sloshing vibrations of the liquid are finally studied near this equilibrium position and two validation cases are proposed for the eigenvalue dynamic problem.

Three‐dimensional modal analysis of sloshing under surface tension

“…Concerning the definiteness of K : Let us first remark that the dynamic stiffness operator is very similar to the tangent stiffness operator established in the previous paper dealing with the static computation [4] (except for the effect of the tank wall curvature in K that has been neglected in the static computation). In that paper, it has been shown that this stiffness operator is singular due to the well-known 'lack of rigidity' in the tangent direction to the free surface plane [13,14].…”

“…Contrary to what has been done for the static meniscus computation [4], we will not use here an energetic approach to establish the variational formulation describing the linear dynamic behavior of an inviscid and incompressible liquid with surface tension. It is easier here to directly write the variational formulation from the linear dynamic local equations of the problem, obtained by linearization of the classical motion equations of an inviscid incompressible fluid with capillarity with respect to the static equilibrium position that has been determined previously.…”

“…In this aim, we use the following relation expressing the first-order variation of the area of deformed by any vector field v [4,12]:…”

“…We denote respectively and L , the liquid/gas and solid/liquid interfaces, and = * the boundary of (see Figure 1). The surface tension phenomenon is characterized by two parameters (supposed constant): the coefficient of capillarity and the contact angle [4].…”

“…In this aim, we propose here a numerical approach to solve the three-dimensional linear sloshing problem of an incompressible inviscid liquid taking into account the effects of surface tension which are predominant in low-gravity environment. Since a procedure to determine the equilibrium position of a liquid in a three-dimensional tank has been proposed in a previous paper [4], it is supposed here that the static position of the liquid inside the tank is known. A variational 88 M. EL-KAMALI, J.-S. SCHOTTÉ AND R. OHAYON formulation is then derived from the linearization of the motion equations of the liquid near this equilibrium state (Sections 2 and 3).…”