Numerical methods for fluid-structure interaction and free surface flows

Serino, Daniel A.
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Henshaw, William D.
Banks, Jeffrey W.
Kapila, Ashwani K.
Sahni, Onkar
Schwendeman, Donald W.
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In the final part of this thesis, an algorithm is developed for free surface flows driven by gravity and surface tension. Deforming composite grids are used to handle the motion of the free surface and the governing equations are solved in an Arbitrary Eulerian-Lagrangian (ALE) frame using an IMEX scheme. The free boundary is represented as a parametric surface, which allows for an efficient computation of surface tension forces using a differential geometry approach involving the first and second fundamental forms. Depending on the numerical scheme used to evolve the free surface, different coupling options are possible. The free surface uncoupled (FSU) scheme splits the evolution of the free surface position, velocity, and pressure into separate stages, while the free surface coupled (FSC) scheme is a variation of the FSU scheme which couples the solution for the pressure and free surface position. A model problem analysis is performed to derive time-step restrictions for the FSU and FSC schemes, and numerical results are presented to verify stability and second-order accuracy. Asymptotic expressions for curvature are derived for small surface perturbations to a plane, cylindrical shell, and sphere. For these geometries, new traveling wave solutions which model capillary waves, cylindrical stream flow, and oscillating bubbles, respectively, are derived.
A number of useful benchmark problems are developed including elastic piston problems in Cartesian and polar geometries. These solutions are exact and include finite interface deformations either normal or tangent to the interface. Traveling wave solutions are also derived and numerically verified for a fluid disk surrounded by a solid annulus. Fluid flow in a channel past a deformable solid annulus is computed and errors are estimated from a self-convergence grid refinement study. The AMP scheme is found to be stable and second-order accurate for a wide range of solid densities.
In this thesis, numerical methods for simulations involving fluid-structure interaction (FSI) and free surface flows are developed, analyzed, and tested. In the first part of this thesis, a stable added-mass partitioned (AMP) algorithm is developed for FSI problems involving viscous incompressible flow and compressible elastic solids. The AMP scheme is stable and second-order accurate even when added-mass and added-damping effects are large. Deforming composite grids are used to effectively handle the evolving geometry and large deformations. An explicit upwind scheme is used to advance the solid and an implicit-explicit (IMEX) fractional-step scheme is used to update the fluid. In the IMEX scheme, the velocity is advanced in one step, treating the viscous terms implicitly, and the pressure is computed in a second step. The AMP interface conditions for the fluid arise from the outgoing characteristic variables in the solid and are partitioned into a Robin (mixed) interface condition for the pressure, and interface conditions for the velocity. The latter conditions include an impedance-weighted average between fluid and solid velocities using a fluid impedance of a special form. A similar impedance-weighted average is used to define interface values for the solid. While the impedance for the solid is well defined, the fluid impedance is not, and a semi-discrete local analysis is used to inform this choice. The properties of the new scheme are analyzed and numerical results are presented to confirm the stability and accuracy of the scheme.
May 2019
School of Science
Dept. of Mathematical Sciences
Rensselaer Polytechnic Institute, Troy, NY
Rensselaer Theses and Dissertations Online Collection
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