Numerical Modelling of Sloshing Tanks and Their Application in Tuned Liquid Dampers

Abstract

In this study, we focus on sloshing tanks, particularly Tuned Liquid Dampers (TLDs). When a tank is excited, sloshing waves are generated and TLDs utilize these waves to mitigate the vibrations of the underlying structure. Due to their easy installation and cost-effective maintenance, TLDs are promising damping solutions for tall slender buildings subjected to wind or earthquake forces. Although extensive experimental research exists on TLD-coupled systems, further studies are needed to enhance the understanding of their dynamic characteristics. To address this, our research is divided into two main parts. The first section of the first part focuses on the linear wave theory of shallow water equations and investigates how various widely used finite difference and finite volume methods introduce numerical diffusion errors in the simulation of sloshing waves from both initial perturbations and forced excitations. A model with low numerical diffusion, integrating physical diffusion, is then employed and coupled with a single degree of freedom (SDOF) system to develop a numerical model for TLD simulation. This facilitates a parametric study that includes the frequency ratio and damping ratio. The second section shifts to the nonlinear wave theory of shallow water equations. This model incorporates bottom topography and enables the simulation of high-gradient free surfaces. It employs the central upwind method and Minmod slope limiter function for flux approximation, along with the fourth-order Runge-Kutta method for time discretization. The model is verified for dam breaks on both horizontal and sloped beds, as well as for runup and rundown in a parabolic tank. Subsequently, the model integrates dissipation and dispersion terms to create asymmetric and steady waves. After verification with experimental box sloshing tanks, the model is coupled with an SDOF system to represent the TLD model. Finally, the TLD with a sloped bed demonstrates higher sloshing waves and greater suppression of the underlying structure vibrations compared to conventional box TLDs. The last part includes an innovative analytical approach to model the dynamic behavior of a TLD-coupled system, bridging a significant gap in the literature. The separation of variables technique is employed to convert the linear shallow water equations, including the continuity and momentum equations, into a modal coordinate system. These equations are then strongly coupled with the dynamic equation of a single degree of freedom (SDOF) system under free vibration, resulting in a fourth-order characteristic equation. Using this equation, novel and significant concepts such as the stability of the coupled system, natural frequencies of the coupled system, initial boundary conditions, and response of the coupled system are discussed. In the next stage, the model incorporates both structural and fluid dissipations, leading to a different characteristic equation. This allows for the study of new topics, including the influence of dissipations on the natural frequencies and stability of the coupled system. For the first time, an expression for the response of the coupled system, indicating an exponential reduction in vibration, is proposed. Additionally, the analytical model is modified to consider the effect of a sloshing tank equipped with screens on the coupled system response. Finally, contrary to previous cases focusing on free vibration, this stage examines the forced vibration of the coupled system. It presents a characteristic equation and describes the particular, complementary, and total responses in three scenarios: undamped coupled system, structurally damped coupled system, and fluid and structurally damped coupled system. Regarding the particular solution, the model introduces new concepts, including dynamic amplification and phase difference between the force and displacement of the coupled system. For the first time, closed-form analytical solutions for the response of the coupled system in all three scenarios are proposed. Notably, in each scenario, in addition to the general case, two special cases are analytically investigated: resonant excitation of the coupled system and when the excitation frequency approaches the water tank frequency.