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Summary A high‐order immersed boundary method is devised for the compressible Navier‐Stokes equations by employing high‐order summation‐by‐parts difference operators. The immersed boundaries are treated as sharp interfaces by enforcing the solid wall boundary conditions via flow variables at ghost points. Two different interpolation schemes are tested to compute values at the ghost points and a hybrid treatment is used. The first method provides the bilinearly interpolated flow variables at the image points of the corresponding ghost points and the second method applies the boundary condition at the immersed boundary by using the weighted least squares method with high‐order polynomials. The approach is verified and validated for compressible flow past a circular cylinder at moderate Reynolds numbers. The tonal sound generated by vortex shedding from a circular cylinder is also investigated. In order to demonstrate the capability of the solver to handle complex geometries in practical cases, flow in a cross‐section of a human upper airway is simulated.

Fluid-structure interaction in a simplified 2D model of the upper airways is simulated to study flow-induced oscillation of the soft palate in the pharynx. The goal of our research has been a better understanding of the mechanisms of the Obstructive Sleep Apnea Syndrome and snoring by taking into account compressible viscous flow. The inspiratory airflow is described by the 2D compressible Navier-Stokes equations, and the soft palate is modeled as a flexible plate by the linearized EulerBernoulli thin beam theory. Fluid-structure interaction is handled by the arbitrary LagrangianEulerian formulation. The fluid flow is computed by utilizing 4 th order accurate summation by parts difference operators and the 4 th order accurate classical Runge-Kutta method which lead to very accurate simulation results. The motion of the cantilevered plate is solved numerically by employing the Newmark time integration method. The numerical schemes for the structure are verified by comparing the computed frequencies of plate oscillation with the associated second mode eigenfrequency in vacuum. Vortex dynamics is assessed for the coupled fluid-structure system when both airways are open and when one airway is closed. The effect of mass ratio, rigidity and damping coefficient of the plate on the oscillatory behaviour is investigated. An acoustic analysis is carried out to characterize the acoustic wave propagation induced by the plate oscillation. It is observed that the acoustic wave corresponding to the quarter wave mode along the length of the duct is the dominant frequency. However, the frequency of the plate oscillation is recognizable in the acoustic pressure when reducing the amplitude of the quarter wave mode.

In this article the problem of two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability is presented and Homotopy Perturbation Method (HPM) is employed to compute an approximation to the solution of the system of nonlinear differential equations governing the problem. Comparisons are made between the Numerical solution (NM) and the results of the He's Homotopy Perturbation Method (HPM).