Eigenvalue solution for the convected wave equation in a circular soft wall duct

Abstract

A numerical approach to find the eigenvalues of the wave equation applied in a circular duct with convective flow and soft wall boundary conditions is proposed. The characteristic equation is solved in the frequency domain as a function of a locally reacting acoustic impedance and the flow Mach number. In addition, the presence of the convective flow couples the solution of the eigenvalues with the axial propagation constants. The unknown eigenvalues are also related to these propagation constants by a quadratic expression that leads to two solutions. These two solutions replaced into the characteristic equation generate two separate eigenvalue problems depending on the direction of propagation. Given that the resulting nonlinear complex-valued equations do not provide the solution explicitly, a numerical technique must be used. The proposed approach is based on the minimization of the absolute value of the characteristic equation by the Nelder–Mead simplex method. The main advantage of this method is that it only uses function evaluations, rather than derivatives, and geometric reasoning. The minimization is performed starting from very low frequencies and increasing by small steps to the particular frequency of interest. The initial guess for the first frequency of calculation is provided as the hard wall eigenvalue solution. Then, the solution from the previous step is used as the initial value for the next calculation. This approach was specifically developed for applications with resonator-type liners commonly used in the commercial aviation industry, where the low-frequency behavior resembles that of a hard wall and agrees with the first initial guess for the first frequency of calculation. The numerical technique was found to be very robust in terms of convergence and stability. Also, the method provides a physical meaning for each eigenvalue since the variation as a function of frequency can be clearly followed with respect to the values that are originally linked to the hard wall modes.