A high-order ghost-point immersed boundary method coupled with auxiliary differential equations for time-domain impedance walls

Abstract

A ghost-point immersed boundary method is coupled with auxiliary differential equations to impose a locally-reacting impedance condition on a wall that does not conform to a Cartesian grid, in the framework of the linearized Euler equations. The ghost-point reconstruction follows the normal-stencil approach: an elliptical least squares cloud of fluid nodes is used to build a high-order polynomial along the local wall normal, which is then extrapolated onto the ghost points by one-dimensional La grange interpolation. The impedance, modeled as a massspringdamper oscillator, is advanced in time through two auxiliary state variables per ghost point, coupled to the reconstruction at every RungeKutta stage. The method is validated against exact and semi-analytical references on a flat immersed wall, a cylindrical wall reflecting an acoustic pulse, and a cylinder scattering a harmonic monopole. Convergence orders ranging from about three on the curved wall to about four and a half on the flat wall are measured, and a cross-check against an exact harmonic reference shows that a persistent offset observed against a semi-analytical reference is a property of that ref erence rather than of the coupled solver. A spectral stability analysis of the linearized semi-discrete operator further shows that the normal-stencil reconstruction reduces, without eliminating, a known instability that occurs when the immersed wall becomes tangent to the grid.