Coupled nonlinear dynamics of acoustic modes in a quasi 1-D duct with inhomogeneous mean properties and mean flow

Abstract

This study investigates the coupled nonlinear evolution of acoustic modes in a quasi 1-D duct with axially inhomogeneous mean velocity and mean thermodynamic properties. A novel modification of the Krylov–Bogoliubov method of averaging (KBMA) is developed to analytically solve the modal-amplitude equations that are linearly and nonlinearly coupled with both quadratic and cubic nonlinearities. Approximate analytical solutions based on the method of multiple scales (MMS) are also derived for the coupled modal-amplitude equations with cubic nonlinearities. The KBMA and MMS solutions are presented for the cases with and without internal resonance among the acoustic modes. Furthermore, novel initial conditions are derived for the KBMA and MMS modal amplitude and phase equations that are essential to validate the analytical solutions using numerical solutions. While both the analytical solutions are seen to be in good to excellent agreement with the numerical solutions, only KBMA captures the low-frequency oscillations in the peak amplitude of the limit cycle oscillations. The role of the order of nonlinear terms in the evolution to limiting oscillations is elucidated by successively considering coupled modal equations with both quadratic and cubic nonlinearities, only cubic nonlinearities and only quadratic nonlinearities. It is seen that while the cubic nonlinearities determine the asymptotic temporal behavior, the quadratic nonlinearities give rise to complex oscillation patterns characterized by a range of non-integral frequencies. For coupled oscillators with only quadratic nonlinearities, it is observed that for mean temperature gradients below a threshold, the pressure fluctuations are characterized by intermittent bursts of high-amplitude fluctuations, as well as by a frequency spectrum that is essentially continuous. The effects of linear coupling of the modal-amplitude equations on the limit-cycle behavior of the solutions are also illustrated.