Coherent fundamental-harmonic interactions in a compressible shear layer via integral nonlinear instability approach

Abstract

A physics-based integral approach is adopted here to develop a theory for studying bi-modal excitation of the shear layer of a Mach 1.5 planar jet. The transverse shapes of the mean flow quantities are given by analytical functions modeling the corresponding jet experiment and Reynolds-averaged Navier–Stokes solution. The transverse shapes of the frequency modes of the coherent large-scale structure are obtained as the eigenfunctions of the locally parallel linear stability theory. The Navier–Stokes equations are then reduced to a set of ordinary differential equations. The solution of this set, subject to the initial conditions, describes the nonlinear interaction among the excited frequency modes, as well as their interaction with the mean flow and the background turbulence. Our analysis shows that the time-averaged interaction among the modes is non-zero only if the two frequency modes are related to each other by the fundamental-harmonic frequency. We label the fundamental mode here, “f,” as the most amplified mode developing nonlinearly along the jet in the absence of other imposed modes. We then use the resulting theory to study the effect of bimodal excitation by harmonically related pairs (f, f/2) or (f, 2f) to see under which conditions this bimodal excitation can suppress the fundamental. We found that the combination of fundamental and harmonic (f, 2f) can effectively reduce the fundamental at an optimized phase lag. By viewing the fundamental as the most dominant sound source in the jet, it is, thus, possible to reduce the jet noise via bi-modal excitation.