Fully conservative leak-proof treatment of thin solid structures immersed in compressible fluids [electronic resource]

In this dissertation we develop and present a leakproof approach to strongly coupled fluid-structure interactions in the presence of compressible fluids. First we present a novel semi-implicit formulation of the Euler equations that separates the flux terms into an advection component and an acoustic component. The advection terms are treated explicitly using a standard flux-based scheme, and an implicit system of equations are derived for the pressure of the flow field. The implicit system of equations for pressure closely mirrors Poisson's equation that arises for {\em in}compressible flow, and indeed one obtains the Poisson equation for pressure in the limit as the sound speed goes to $\infty$. By treating the pressure implicitly we can alleviate the often-stringent acoustic component of the CFL restriction, and the resulting well-conditioned method depends only on the bulk velocity of the flow field for its time step restriction. Next the implicit system of equations for pressure are integrated into a monolithic coupled system that robustly and stabily captures two-way coupled fluid-structure interactions. This formulation is quite general, and works with arbitrary fluid equations of state as well as both rigid and deforming structures without any special treatment. This tightly-coupled system captures the entire feedback loop that arises as fluid pressures and structure velocities interact, and so the method is suitable for capturing the behavior of flow near infinitesimally light structures (unlike partitioned methods) as well as extremely heavy structures. We exactly conserve momentum and kinetic energy within the coupled system, and hence naturally handle highly non-linear phenomena such as shocks, contacts and rarefactions near the fluid-structure interface. Note that this the method is {\em not} conservative near the interface during the advection step. The advection stage is addressed in the third chapter, and a conservative semi-Lagrangian advection scheme is developed that works by supplementing a standard semi-Lagrangian advection with a conservative limiter and a forward-advection step. The conservative limiter clamps material motion from the first semi-Lagrangian step, guaranteeing that no new material is created, while the second forward-advection step is used to push forward any material that was left behind by the first step. We consider this advection scheme in its original habitat (incompressible flows), and show that this method can be used to exactly conserve mass and momentum in such a flow. More interestingly, as the method works by tracing characteristic curves and interpolating values it is unconditionally stable. With this in mind we demonstrate that this {\em unconditionally stable} conservative advection scheme can be used to remove any and all remaining time step restrictions from the flux-split compressible flow previously introduced. Finally, we introduce cut cells and partial volumes into the fluid-structure solver from Chapter 3 and modify the conservative semi-Lagrangian advection scheme to capture these small, irregular cell volumes without introducing any of the time step restrictions typically associated with cut cells. The semi-Lagrangian advection is limited to first order accuracy both in time and space, and so it is hybridized with a flux-based scheme and total variation-diminishing Runge-Kutta time integration, yielding a method that maintains high resolution accuracy in the bulk of the flow. The semi-Lagrangian method works by tracing characteristics, and so we modify it to enforce non-penetration through the structure interface by incorporating a temporal visibility map into the advection and clamping stages of the conservative semi-Lagrangian advection solver. Unlike previous methods, we do not require any complex geometric time evolution of volumes of material, nor do we require any special treatment for swept or uncovered cells. The resulting method can handle thin, moving solid structures in a fully conservative manner without any material leaking across the interface.

Advisor: Fedkiw Ronald P.
Darve Eric advisor. Thesis advisor
Farhat Charbel advisor. Thesis advisor