Application of compact upwind biased finite difference schemes for 2-D time reversal simulations

Abstract

The 2-D Linearized Euler equations (LEE) with a superimposed uniform mean flow subject to anechoic boundary conditions are numerically solved using the Pseudo-Characteristic Formulation (PCF) and the compact upwind biased Finite Difference (FD) schemes (developed in a companion paper). The third-order total variation diminishing (TVD) Runge-Kutta scheme is used for time integration. The anechoic boundary conditions in a 2-D computa-tional domain are modeled by setting the incoming fluxes (in the PCF) to zero at the boundaries of the rectangular (square) domain and also by simultaneously using the sponge layer to damp the incoming fluxes. This ensures a reasonable suppression of the contaminant reflected waves from the boundaries. An initial monopole condition of the acoustic pressure field modeled by a Gaussian pulse is considered as a test case which initiates an acoustic wave that propagates and expands in time. The numerical simulations are carried for sufficient time duration to ensure that the wave completely propagates out of the computational domain, wherein only the acoustic pressure at the boundaries is stored after each step of time-integration. The results are presented as forward time-evolution of acoustic pressure field at different time instants. These stored acoustic pressure time-histories are reversed in time and used as input data during the implementation of Time-Reversal (TR) simulations of the 2-D LEE to compute the backward time-evolution of the acoustic field variables, whereby the same numerical solver is used. The acoustic pressure field predicted by the TR procedure at final time-instant of simulation is found to be in excellent agreement with the initial conditions considered during the forward problem. This suggests the potential use of proposed numerical scheme (based on the PCF) with only the time-reversed acoustic pressure history as the input inflow boundary conditions during TR simulations, in localizing an acoustic source in a 2-D domain.