(Why) Jets quantise trapped acoustic modes

Abstract

High-subsonic turbulent jets trap acoustic modes in their potential core. The modes lie in narrow frequency bands that are $O(0.1)$-wide in Strouhal number, $St$. By studying a Mach 0.9, Reynolds number $10^6$ jet, this work confirms an earlier hypothesis: each band is made up of $O(10)$ discrete frequencies, $ΔSt \sim O(0.01)$ apart. To resolve them, we collect $O(100\,000)$ snapshots from dedicated schlieren experiments and large-eddy simulations (LES), and apply spectral proper orthogonal decomposition (SPOD). Our SPOD algorithm uses sine tapers and enforces the jet's rotational and reflectional symmetries. For both the schlieren and the LES, the trapped SPOD modes are bounded in the $St$-$x$ and $St$-$k$ planes, where $x$ and $k$ are respectively the axial coordinate and wavenumber. The bounds quantise the modes in $St$, $x$, and $k$, such that the $n_x$th mode in $St$ has $n_x$ antinodes along $x$ and $k$. We then predict trapped modes with local, global, modal, and non-modal stability theories. We find evidence of a competition between modal and non-modal stability. From only the theoretical bounds, we build a simple in-phase standing-wave resonance model that predicts the quantised global-mode frequencies.