## 2 Modeling of TSDs
### Choice of an Adequate Numerical Model  
An appropriate numerical model needs to be identified and set up properly. There are various numerical approaches that can, in principle, simulate the movement of sloshing liquids. Traditionally, models based on the shallow water equations were the choice as the computational requirements remained fairly low. These continue to provide a good initial approximate for certain scenarios of wave motions. Respective theories perform particularly well when the horizontal length scale is considerable larger than the vertical  
Figure 1: Scheme of a TMD attached to a structure (**left**) and the main parameters for defining a TSD (**right**). scale, thus, the naming of _shallow_. Corresponding equations result from depth-integrating the Navier-Stokes relations, which are the generally accepted governing formulas for flow problems. Shallow water equations cannot represent quantities over height, as the resulting variables are the wave height and horizontal travelling velocity of the wave. Consequently, the outcomes solely describe the surface level of the sloshing and its speed. This poses strong limitations for nonlinear dynamic motions, where they are unable to tackle wave breaking or to provide a realistic insight into the inner dynamics, which is linked to the internal energy considerations of the sloshing tank. Another type of numerical model is represented by formulations using a particle analogy. These imply considering the sloshing liquid as the collection of movement of all particles in the container. Two main categories can be identified: one uses an explicit numerical mesh, which deforms according to the sloshing movement; the other type uses an implicit numerical mesh, meaning the existence of a background grid on which the particles are tracked. The former type suffers when large mesh distortions are present (the case of highly dynamic movement), whereas both models can have issues with the conservation of mass. Further investigations would be needed for more insight into the accuracy, robustness and scalability. Nonetheless, this type of approach still remains promising for the design of TSDs. The main result is represented by the positions and velocities of the particles. Internal forces and reactions on the tank can be derived mainly based on collision and cohesion. Specific implementations used for testing are part of the Kratos Multiphysics [14] project. From here, we explored the use of the _ShallowWaterApplication_ (with various considerations outlined in [18]) for the depth-integrated theory, the _PfemFluidDynamicsApplication_ (theory and applications reviewed in [19]) and the _ParticleMechanicsApplication_ (with important principles focusing on material points described in [20]) as the approaches based on the particle analogy (implying a _Lagrangian_ view of the governing physics), as well as the two-fluid formulation (with a _Eulerian_ approach to the flow) of the Navier-Stokes equations contained in the _FluidDynamicsApplication_. A comparison is detailed in the work of [21]. The very last model of the flow equations can consider two phases in the flow: for our purposes, water and air. When used to model a TSD, the expected velocities are moderate, resulting in a Mach number \(Ma<0.3\), such that the assumption of incompressibility is appropriate [22]. The detailed derivation of the formulas below is presented in [23]. Corresponding numerical results describe the flow field being represented by the velocity and pressure in the domain. Under the assumption of the two phases being immiscible, this model enables the identification of the interface between water and air. The entire container domain \(\Gamma\) is divided into two complementary subdomains, seen in Equation (5),  
\[\Gamma=\Gamma_{a(ir)}\cup\Gamma_{w(ater)}. \tag{5}\]  
\[ \begin{cases}\rho_{i}\Big{(}\frac{\partial\mathbf{u_{i}}}{\partial t}+( \mathbf{u_{i}}\
abla)\mathbf{u_{i}}\Big{)}-\mu_{i}\Delta\mathbf{u_{i}}+\
abla p _{i}=\mathbf{f_{i}}&\text{in $\Gamma_{i}$ for $i=\text{air, water}$}.\end{cases}