Shallow Water Tuned Liquid Dampers: Modeling, simulation and experiments

The use of sloshing liquid as a passive means of suppressing the rolling motion of ships was proposed already in the late 19th century. Some hundred years later the use of liquid sloshing devices, often termed Tuned Liquid Dampers (TLD), began to find use in the civil engineering community. The TLDs studied in this thesis essentially consist of a rectangular container partially filled with liquid in the form of plain tap water. The frequency of the liquid sloshing motion, which is adjusted by varying the length of the tank and the depth of the wa- ter, is tuned to the structural frequency of interest. When, due to various disturbances, the structure starts vibrating, the liquid motion or sloshing is initiated due to the fre- quency tuning and sloshing forces are imposed on the structure. The main focus of the present work has been the development of a mathematic model capable of predicting the interaction between a structure and fluid sloshing forces. A mathematical model describing liquid sloshing in shallow water is formulated by simplifying the full Navier-Stokes equations expressed in a moving frame of reference. The resulting set of equations are known as the Nonlinear Shallow Water (NSW) equations, or the St. Venant equations, named after the originator who derived the set of equations in 1871. The set of equations are developed with the purpose of describing sloshing in tanks with relatively large base amplitudes which result in the formation of moving hydraulic jumps or bores, by some researchers on TLDs termed wave breaking. A large part of the energy dissipation in the fluid is anticipated to stem from the turbulence in the vicinity of the moving hydraulic jump, and in order to verify this supposition the effect of bottom friction is included in the mathematical model. Studies reveal that for realistic roughness parameters the bottom friction has very limited effect on the liquid sloshing behavior and can be neglected. Herby the postulate is verified. Based on the mathematical model three dimensionless parameters are derived showing that the response of the damper depends solely on ratio of the base amplitude and tank length, A/L, on the frequency ratio of the forcing frequency and sloshing frequency, Ω/ωw, and finally on a friction parameter γ. These dimensionless parameters have been postulated by several researches in the field of TLDs but has not been derived rigorously as in the present thesis. In the derivation of the dimensionless parameters it is assumed that the proposed mathematical model captures the relevant physical processes in the problem. The model is based on a shallow water assumption and an extensive measurement campaign is carried out to establish an appropriate upper limit for the filling or depth defined by the ratio of the water depth and tank length, h/L, for which the mathematical model is valid. Moreover, the experiments are used to determine the effects of the forcing ratio A/L and frequency ratio Ω/ωw on the sloshing behavior. The current study is novel in its rigorous approach and brings valuable information on the range of application for the proposed mathematical model. In order to solve the mathematical model an extensive amount of work has been invested in computational fluid dynamics. Other similar reported studies of the NSW equations in connection with sloshing has used cumbersome, computationally expensive and somewhat outdated numerical solution schemes. We compare a state of the art, high order, shock capturing method with a simpler low order scheme and find that the simple scheme is adequate for simulating shallow water sloshing. The interaction between a shallow water TLD and a structure, the main focus of the work, is analyzed experimentally and by simulation. The mathematical sloshing model is coupled to a simple Single-Degree-Of-Freedom (SDOF) system in a general framework and a time integration scheme is proposed. A number of interaction experiments are performed where TLDs are coupled to an elastic structure. The elastic structure is given an initial horizontal displacement and then released. The mathematical interaction model captures the transient free surface behavior of liquid sloshing as well as the position of the hydraulic jump very well. By coupling the shallow water TLD to the structure, the total structural damping is increased and the increased damping is estimated precisely by the model. The mathematical model is further verified using experimental results from the literature on the interaction of shallow water TLDs and elastic structures. The interaction model predicts the structural amplitude satisfactorily but introduces a small error in the frequency location of the maximum structural amplitude.