# Interactions between fluids and elastic solids

Project leader: Professor Aric Wheeler
Project manager: Kevin Dembski
Team members: Zachary Robers, Michael Thomas, Olly Yang

We aim to study a simpler version of the iceberg calving problem. In our model, we simulate waves approaching an iceberg using the shallow water equations and deploy the linear wave equation to model the iceberg's compression. This setup allows us to study how waves interact with the iceberg through a series of equations and conditions designed to preserve energy and mass. To analyze this system, we use two main approaches. First, we make a linear approximation of the system, which helps us understand the conditions under which the system is stable. Stability here means that small perturbations to the system don’t evolve into large changes which could lead to behavior such as iceberg calving. Second, we program a simulation to study the problem in more detail. By using classical numerical methods, we can track the fluid and solid velocities, the height of the waves, the compression of the iceberg, and the position of the boundary between the water and the iceberg. Together, these approaches give us valuable insights into how the system evolves under different conditions, helping us better understand the processes involved in fluid-structure interaction. While this work remains a simplified means to represent and analyze iceberg calving, extending this work to incorporate more realistic physical conditions could lead to insights on how and why icebergs calve.

We study a simplified model of a one-dimensional fluid-structure interaction problem consisting of nonlinear shallow water waves coupled to a linearized elasticity model with the boundary between the fluid and solid domains allowed to freely move. In order to couple the fluid and solid problems together and describe the motion of the interface, we turn to physical principles such as mass and energy conservation in order to provide the necessary coupling conditions. Our analysis of the system consists of a two-pronged approach of linear stability analysis and numerics. In our linear stability analysis, we follow the approach of Lax and Majda’s work on the stability of shocks to analyze the stability of the associated linearized problem. MacCormack’s Method for finite difference serves as the basis of our numerical model. This classical numerical method achieves quadratic convergence for either the shallow water equation or linear wave equation in isolation. We then combine these equations along the moving boundary using a scheme rooted in energy conservation, to study the behavior of the full coupled system with a moving boundary. Our analysis provides a first step towards studying important physical applications such as the iceberg calving problem. In our simplified setting, we are able to simulate the dynamics of an incoming fluid swell by using suitably chosen boundary conditions on the left side of the computational window.