Fluid-Structure Interaction and Immersed Boundary Method for the Compressible Navier–Stokes Equations Using High Order Methods

Abstract

There has been a growing interest in the study of biological flows in biomechanical systems in recent years. Such flows mostly appear in domains with complex geometries, flexible moving boundaries and usually involve fluidstructure interaction (FSI). The simulation of such problems is computationally challenging. The demand for simulating more complex flow problems more efficiently has motivated the development of the more sophisticated and more accurate numerical models. This dissertation presents a framework for simulating the fluid dynamical behaviour of viscous compressible flows around moving bodies and complex geometries as well as FSI. The first part of the dissertation considers fluid-structure interaction in a simplified 2D model of the upper airways in order to investigate the flowinduced oscillation of the soft palate in the pharynx. This study is related to disorders in the human upper airways, in particular those associated with snoring and obstructive sleep apnoea syndrome (OSAS). A simplified 2D model has been developed to simulate the interaction between the soft palate and compressible viscous flow. The Arbitrary Lagrangian–Eulerian (ALE) formulation is employed to handle the fluid flow in Eulerian description using moving fluid grids and the plate structure in a Lagrangian formulation using stationary structure grids. The coupling between the fluid and the structure is handled by a partitioned approach where forces and deformations are exchanged between the flow and the deformable structure in each time step. To enable the solver to be applicable to larger simulations and to accommodate geometric flexibility with high order summation-by-parts (SBP) difference operators, a multi-block approach is employed to decompose the computational domain and to parallelize the solver. The idealized soft palate is first modeled by the Euler–Bernoulli thin beam theory and then by an inextensible thin beam model. Effects of kinematic as well as structural properties are examined. It has been illustrated that the structure oscillation induces sound generation. The second part of the dissertation is devoted to devise an efficient and versatile immersed boundary method (IBM) for simulating compressible viscous flows with complex and moving boundaries employing high order summation-by-parts difference operators. The proposed Cartesian grid based immersed boundary method builds on the ghost point approach in which the solid wall boundary conditions are applied as sharp interface conditions. The interpolation of the flow variables at image points and the solid wall boundary conditions are used to determine the flow variables on three layers of ghost points within the solid body in order to introduce the presence of the body interface in the flow computation and to maintain the overall high order of accuracy of the flow solver. Two different reconstruction procedures, bilinear interpolation and weighted least squares method, are implemented to obtain the values at the ghost points. A robust high order immersed boundary method is achieved by using a hybrid approach where the layers of ghost points are treated differently. The first layer of ghost points is treated by using a third order polynomial combined with the weighted least squares method and the second and third layers of ghost points are treated by finding the image points of the corresponding ghost points and using bilinear interpolation to find the values at the image points. After demonstrating the accuracy of the present IBM for low Mach number flow around a circular cylinder, it is applied to simulate flow in the upper airways with the cross-section of the complex geometry of a specific OSAS patient. The IBM solver has been further verified and validated for moving boundaries by applying it to a transversely oscillating cylinder in free-stream flow and an in-line oscillating cylinder in an initially quiescent fluid. Sound waves generated by the in-line oscillation of the cylinder exhibit both quadrupole and monopole types.