Stability and accuracy of aeroacoustic time-reversal using the pseudo-characteristic formulation

Abstract

This paper investigates the stability and accuracy of the aeroacoustic Time-Reversal (TR) simulation using the Pseudo-Characteristic Formulation (PCF). To this end, the forward simulation of acoustic wave propagation in 1-D and 2-D computational domain with a uniform mean flow was implemented using the PCF of the Linearised Euler Equations (LEE). The spatial derivatives in the opposite propagating fluxes of the PCF were computed using an overall upwind-biased Finite-Difference (FD) scheme and a Runge-Kutta scheme was used for time-integration. The anechoic boundary condition (ABC) was implemented for eliminating spurious numerical reflections at the computational boundaries, thereby modelling a free-space. The stability of 1-D forward and TR (with only time- reversed acoustic pressure as the input at the boundary nodes) simulations were analysed by means of an eigenvalue decomposition, wherein it was shown that opposite upwinding directions must be considered while using the overall upwind-biased FD scheme. Furthermore, the implementation of ABC was found to be crucial for ensuring the stability of the forward simulation over a large time duration and the 2-D TR simulations. The overall central Dispersion-Relation Preserving (DRP) FD schemes were however, found to be unstable and unsuitable for TR simulation. The accuracy of both the forward and the TR simulations using the PCF was assessed by comparing the simulation results against the corresponding analytical solutions of a spatially and temporally evolving Gaussian pulse. It was shown that numerically reversing the mean flow direction during TR (using the PCF) and only the time-reversed acoustic pressure as input at the boundaries is sufficient to accurately back-propagate the waves and localise the initial emission point of the pulse in 1-D or 2-D computational domain.