Implicit-Explicit Time stepping for a Two-Dimensional Inviscid Fluid-Structure Interaction Solver.

摘要

This thesis describes the development of a two-dimensional, high-order, fluid-structure interaction (FSI) solver. The well-established spectral difference (SD) method is used for spatial discretization of the Euler equations over deforming, unstructured quadrilateral grids. The Geometric Conservation Law (GCL) is incorporated into the conservative Euler equations, before discretization. After simplification, the equations reduce to a form, in the computational domain, identical to the equations in the physical domain. In this form, the equations can be integrated implicitly in time, without the requirement of any additional source term, to guarantee free-stream preservation. The fluid and structure sub-systems are individually integrated in time using the explicit first stage, single diagonal, diagonally implicit Runge-Kutta (ESDIRK) method. As the first step to solving the coupled, non-linear Euler equations, implicit in time, we linearize the governing equations. The resulting linearized simultaneous equations are then solved sequentially using lower-upper symmetric Gauss-Seidel (LU-SGS) relaxation sweeps. The fluid and structure sub-systems are loosely coupled and the coupling term is integrated in time using an explicit RK method, resulting in an implicit-explicit (IMEX) RK coupling. The spatial accuracy and the free-stream preserving ability of the solver are demonstrated by testing a supersonic, isentropic vortex in a curved channel. Next, the temporal accuracy of the solver is established using an Euler vortex propagation test case. It is also demonstrated that the four-stage ESDIRK is capable of handling time-steps 50 times larger than the four-stage explicit RK. In each of these cases, third- and fourth-order SD for spatial discretization and second-order backward difference (BDF2) and third-order, four stage ESDIRK for time integration were tested. Since the loose (explicit) FSI coupling restricts permissible structural deformation, we limit ourselves to small harmonic oscillations resulting from linearized perturbed Euler equations. The interaction between a linear piston and an inviscid, compressible fluid is simulated to demonstrate that the IMEX coupling does not contaminate the spatial or temporal accuracy of the implemented high-order methods. Through rigorous testing, this development is expected to lay a foundation for a powerful computational framework for various fluid-structure interaction problems.