A UNIFIED FLUID-STRUCTURE-INTERACTION FORMULATION FOR DIRECT NUMERICAL SIMULATION OF DEFORMABLE HYPERELASTIC SOLIDS IMMERSED IN VISCOUS INCOMPRESSIBLE FLOW

Abstract

It is of great interest for the metal casting industry to understand the physics associated with the pouring of highly oxidative liquid metals, such as steel or aluminum alloys, to reduce dross formation for improved production quality. The most common approach for the prediction and control of dross formation currently practiced in the industry is trial and error, which usually results in excessive waste of material and energy, as well as degradation in product quality. Therefore, it is important to develop more reliable and systematic design tools to predict and control the formation and transport of dross. The ultimate objective is to develop high-fidelity direct numerical simulation tools, which can model the complete behavior of the pouring process, including the formation of dross and its interactions with the surrounding fluid. As a first step towards that end, a multidimensional direct numerical simulation methodology is developed to model the dynamics of deformable elastic solids immersed in an incompressible viscous fluid. In this approach, the solid domain is represented by a signed distance function that is defined on a fixed and uniform Cartesian grid. The strain field associated with the solid domain is tracked using a set of three reference level set functions that form a dynamic grid which allows the solid deformation to be determined with reference to the fixed underlying Cartesian grid. The initial state of the dynamic grid coincides with the Cartesian grid, representing the unstressed equilibrium state of the solid. The deformation gradient tensor is formed by comparing the reference level sets with the underlying Cartesian grid. A finite-strain constitutive model for hyper-elastic, Neo-Hookean materials is used to relate the deformation gradient tensor to the solid stress tensor. Due to the modular nature of the formulation, other material models can be incorporated in a straightforward manner. The divergence of the stress tensor is added to the momentum equation to represent the solid stress as a source term only in the solid region. The jump in normal stress across the solid-fluid boundary is captured implicitly by creating a finite width solid-fluid boundary across which the respective material stresses are smoothly blended together using a continuous Heaviside function. In order to ensure a smooth continuation of the stress fields in the blended boundary region, an improved extrapolation methodology is developed to propagate the dynamic grid from the solid to the fluid domain using Hamilton-Jacobi and topology projection strategies for reinitializing the solid level set indicator field to a signed distance function. In contrast to Lagrangian-Eulerian methods for immersed elastic solids, our methodology results in a unified formulation, where a single Poisson equation needs to be solved to obtain pressures and velocities within both the solid and the fluid domains. The implementation has been embedded within the FLASH 4 solver framework to enable scalable parallel computations using HYPRE with an algebraic multigrid solver. We demonstrate the accuracy of the numerical implementation in comparison with Lagrangian-Eulerian methods using canonical test cases. We also demonstrate that the methodology is robust and easy to extend to the axisymmetric and 3-D domains.