Line-Based High-Order Hybrid Scheme on Unstructured Grids for Compressible Flows

This study explores the use of line-based finite difference techniques on unstructured grids employing high-order stencils for practical aerodynamic applications. The primary focus is on finite-difference explicit and compact spatial discretization up to sixth order. Problems studied include compressible flows containing discontinuities, which necessitated the use of a hybrid finite difference–finite volume approach, where the former is shown to be effective and accurate for smooth regions, while the latter is robust in capturing discontinuities and shocks. To differentiate between discontinuous and smooth zones, two distinct shock indicators are used, one based on pressure gradients and the other on the divergence of velocity. To achieve faster convergence, spatial discretization is applied with line-based alternating direction implicit (ADI) time integration. A key focus of this work is to demonstrate the proposed methodology in flow scenarios relevant to aircraft applications that also test various aspects of the methodology. Consequently, the cases studied are isentropic vortex convection, standing normal shock, transonic NACA0012 airfoil, supersonic cylinder, unsteady flow past tandem circular cylinders, and supersonic flow past a forward-facing step. Comparisons were also done against an existing line-based finite-volume technique for unstructured grids that employed Monotonic Upstream-Centered Scheme for Conservation Laws (MUSCL) and fifth-order Weighted Essentially NonOscillatory (WENO5) spatial reconstruction schemes. It is observed that the hybrid finite difference–finite volume approach performed well with excellent agreement against available results, sometimes with far fewer degrees of freedom. Furthermore, the implicit ADI scheme is effective over unstructured grids and provides a three-time speed-up over the traditional Runge–Kutta fourth-order scheme.