Nonlinear weighting process in ghost-cell immersed boundary methods for compressible flow

Computational challenges arise for the immersed boundary method (IBM) when dealing with compressible flow, where discontinuous and smoothly varying flow regions appear near the immersed boundary. The conventional ghost-cell IBM provides inaccurate results for smoothly varying regions when a low-order interpolation is used, or it suffers from a numerical instability for the discontinuous flow when a high-order interpolation is used. In this study, a new ghost-cell approach, nonlinear-weighted IBM (NWIBM), is developed to address the abovementioned issues. Inspired by a variety of weighted nonoscillatory interpolation methods, this work combines the high- and low-order polynomials during ghost-cell value estimation to enforce proper boundary conditions in the immersed boundary. A multidimensional smoothness indicator is designed to evaluate flow discontinuities. The nonlinear weighting obtained from the smoothness indicator makes the high-order polynomial dominant and the low-order polynomial negligible in the smoothly varying region, and vice versa in the discontinuous region. The enhanced performance and applicability of the proposed method were validated through various numerical tests in compressible flow. It was demonstrated that the NWIBM provides more stable and more accurate numerical solutions compared with conventional ghost-cell approaches.