Interaction of elastic structure with non-uniformly aerated fluid.

for 24rd IWWWFB, Zelenogorsk, Russia, 2009 Interaction of elastic structure with non-uniformly aerated fluid Ten I. , Korobkin A. (1) BUREAU VERITAS DR, Paris, France (igor.ten@bureauveritas.com) (2) School of Mathematics, University of East Anglia, Norwich, UK Introduction The problem of fluid/structure interaction with account for aeration of the fluid is relevant to possible violent sloshing in a tank of LNG carriers accompanied with high slamming loads acting on the tank walls. The overall problem of sloshing impacts is extremely complex. It is known that the impact loads are highly localized both in time and in space. This allows us to model such loads by using simplified models of hydrodynamics [1]. Shape of the wave front just before the impact defines the impact type. Four types of the impact can be distinguished: ’Wagner-type impact’, ’Steep wave impact’, ’Bagnold-type impact’ and ’Aerated fluid impact’. Wagner-type impact can be detected when the wave front is inclined from the wall. The second impact type corresponds to the case where the wave front is almost vertical before the impact and can be modeled as a hydraulic jump. In Bagnold-type impact the wave profile is inclined toward the wall, the wave is almost breaking and its upper part contacts the wall before the wave overturning with trapping an air cavity. The fourth case corresponds to spilling breaker when the wave broke before arriving at the wall and the fluid is mixed with air around the impact region. Steep wave impact was studied in [2] for 2D and [3] for 3D cases. In this report we present the analysis of aerated wave impact. Two-dimensional problem of steep wave impact onto elastic wall is considered with account for non-uniform aeration of the fluid in the impact region. A fluid of finite depth H is assumed aerated near the wall. The horizontal dimension D of the aerated region is finite. Distribution of the air fraction α(x, z) in this region is given. In the main flow domain outside the aerated region, the fluid is incompressible. The flow is assumed irrotational. In the aerated region, compressibility effects matter and the flow is described by the velocity potential which satisfies the wave equation with variable sound speed. In the main flow region the corresponding velocity potential satisfies Laplace’s equation. The sound speed in the aerated fluid can be as small as 25 m/sec. The hydrodynamic pressures induced by the impact are expected to be smaller than those predicted by the model of compressible non-aerated fluid. However, since the dimension of the aerated zone is finite, multiple reflections of the pressure pulses from the interface between aerated and pure fluid and from the wall may lead to certain increase of the pressure magnitude within the aerated region with corresponding increase of magnitude of elastic stresses in the wall. The problem is solved with the help of the modal decomposition method. The derived initial boundary value problem is reduced to a linear system of ordinary differential equations with respect to generalized coordinates of the wall deflection and the velocity potential in the aerated region. The system has been solved by utilizing its general solution and by the Runge-Kutta 4 order method. The numerical code was verified with respect to size of the time step of numerical integration and number of modes to be used for accurate predictions of the quantities of practical interest. The code was validated by comparing the results obtained for constant sound speed with those obtained by Steep Wave Impact numerical code. The effects of non-uniform aeration and thickness of the aerated region on hydrodynamic pressure distribution, elastic deflections of the wall and strains are investigated. It is shown that non-uniform concentration of air in the impact region may increase the stresses in flexible structure compared to the case of non-aerated fluid impact.