Acoustic Nonlinearities in a Quasi 1-D Duct with Arbitrary Mean Properties and Mean Flow

Nonlinear temporal dynamics of acoustic oscillations in a quasi one-dimensional (1-D) duct are investigated using both numerical and analytical methods. The spatiotemporal nonlinear wave equation is derived for pressure oscillations in a quasi 1-D duct with axially varying cross-section and spatially inhomogeneous mean properties such as the velocity, temperature, density and pressure. Using the finite element method with quadratic interpolation functions, the linear Helmholtz equation is solved for the modal shapes and frequencies. With the modal shape as the weighting function, the standard Galerkin method is applied to transform the spatiotemporal wave equation into a second-order nonlinear ordinary differential equation (ODE) governing the time evolution of modal amplitudes. The limit-cycle amplitude and frequency of pressure oscillations are quantified analytically using the Lindstedt--Poincar\'{e} perturbation method. Furthermore, to capture the transient evolution to the limit cycle, the method of averaging is applied to the nonlinear temporal ODE for the modal amplitude. From the Lindstedt--Poincar\'{e} method, it is seen that a limit cycle exists when the linear damping coefficient $\mu$ in the nonlinear ODE is of the opposite sign as the quantity $S$ that is a function of the coefficients of the quadratic and cubic terms in the ODE.