Improvement of the stability and accuracy of solid-wall immersed boundary schemes for the linearized Euler equations using boundary constraints

Abstract

A Cartesian cut-cell method with explicit high-order finite-difference stencils for solid walls at internodal wall positions has been developed to solve the linearized Euler equations. For this, additional boundary constraints besides the physical impermeability condition have been incorporated into Hermite-based finite-difference stencils near the boundary. We demonstrate that our methodology and the ghost point formulations of Tam & Dong (1994) and Gloerfelt (2001) are equivalent when a wall intersects with a grid point. Other than Tam & Kurbatskii's (1997) extrapolation-based cut-cell treatment, our formulation can be regarded as a straightforward generalization of the above-mentioned ghost point formulations. A numerical reflection problem demonstrates improved accuracy in 1D for high-order schemes of sixth- and 12th-order. The scattering of a plane wave with 8 PPW for a 7-point-, and 5 PPW for an 11-point boundary scheme at a 2D-cylinder in fluid at rest demonstrates the extension of the proposed method to higher dimensions. For the 1D case, stability analysis has been carried out to show the significant impact of using boundary constraints to improve the numerical stability of a boundary scheme. In the multidimensional case, weak instabilities have to be suppressed by selective filtering. Our new formalism for boundary methods enables the assessment of the propagation characteristics of modified 1D-boundary stencils for first derivatives by Fourier analysis. It is shown that the incorporation of boundary constraints improves the modified wavenumber signature of the boundary stencils significantly.