Two-step method for evolving nonlinear acoustic systems to a steady state

Abstract

A two-step method for evolving two-dimensional nonlinear acoustic systems with flow to a periodic steady state is presented. In the first step of the method, the full nonlinear system governing acoustic disturbances is integrated numerically starting with arbitrary initial conditions using the explicit predictor-corrector method developed by MacCormack. In the second step, the Fourier Time Transform of the computed field is calculated to determine its frequency components. The transient wave field is then filtered from the spectrum and the inverse transform taken to provide an approximation to the steady-state wave field. This approximate field provides a new initial condition for subsequent iterations on the method. The method is tested on a benchmark acoustic problem for which exact steady-state solutions are known and on a nonlinear problem for which a steady-state solution has not been given before. Excellent agreement with the benchmark acoustic solution was obtained within four iterations for planar and nonplanar sources. Convergence to a steady state for the nonlinear problem occurred in six iterations. The two-step method eliminates the need to develop nonreflecting boundary conditions in order to obtain periodic steady-state solutions of two-dimension al acoustic problems and is easily extended to any number of spatial dimensions and to other hyperbolic systems. The procedure is shown to be numerically stable and may provide the only alternative for obtaining steady-state solutions to problems for which nonreflecting boundary conditions are not known or are physically incorrect.