Sammendrag
This research is motivated by the aim to simulate fluid-structure interaction (FSI) in the human upper airways, in particular FSI associated with snoring and obstructive sleep apnea (OSA). The Arbitrary Lagrangian–Eulerian (ALE) formulation is employed to describe the fluid flow using the compressible Navier-Stokes equations and the structure. The coupling between the fluid and the structure is handled by a partitioned approach exchanging the positions, velocities and forces at the fluid-structure interface in each time step. The compressible Navier-Stokes equations are discretized by globally fourth order summation-by-parts (SBP) difference operators with in-built stability properties and the classical fourth order explicit Runge-Kutta method. For FSI of viscous compressible flow with an elastically mounted circular cylinder, the second order ordinary differential equations (ODEs) governing the cylinder motion are expressed as a first order ODE system and solved synchronously with the fluid equations using the same Runge-Kutta method.
An efficient and versatile immersed boundary method (IBM) for simulating compressible viscous flows with complex and moving convex boundaries employing globally fourth order SBP operators has been developed. The proposed Cartesian grid based IBM 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 at 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 in which 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 using bilinear interpolation to find the values at the image points of the corresponding ghost 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 cross-section of the upper airways of a specific OSA patient. The IBM solver has been further verified and validated for moving boundaries by applying it to an in-line oscillating circular cylinder in an initially quiescent fluid. Sound waves generated by the in-line oscillation of the cylinder exhibit both quadrupole and monopole types. The present IBM is also verified for FSI of an elastically mounted circular cylinder in freestream flow at Reynolds number 200, and the rate of energy transferred between the fluid and the structure is investigated.
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