An improved seventh-order WENO scheme with a novel smoothness indicator for flows containing discontinuity-disturbance interaction

Abstract

This article presents a modified version of the WENO numerical method with an increased order of accuracy over critical points and higher resolution in detecting shock-turbulence interactions. The proposed method is an improved version of the WENO-η-Z scheme. The optimization is based on a new Global Smoothness Indicator definition that produces less numerical error at relative extremum points as an indicator of fluctuations in the flow field. Both 1-D and 2-D benchmark problems are implemented to verify the proposed scheme’s accuracy. The convergence of the presented scheme is compared with that of a standard and optimal WENO-η-Z, in the linear wave transfer problem, which shows better convergence for the proposed method. The modified method’s capability to detect discontinuity and shocks in the flow-field has been evaluated by solving two shock-tube problems, namely the Lax shock tube problem and Sod’s problem. The proposed method’s ability to detect fluctuations and disturbances in the flow-field in the presence of shocks has also been assessed in two problems, including the 1-D Shu-Osher shock-disturbance interaction and the 2-D shock-turbulence interaction. Improvements is observed in convergence and reduction in numerical errors in the proposed method compared to the standard WENO and WENO-η-Z method, whilst the capability to detect shocks has not reduced in the modified version.