Wave interactions in a screeching jet

Abstract

We use a series of global models to investigate the linear and nonlinear interactions between shock cells, Kelvin-Helmholtz waves, guided jet modes, and other fluctuations in a screeching jet. First, we identify a set of lightly damped global eigenmodes of the Navier-Stokes operator linearized about the mean flow and show that they result from interactions with different shock-cell wavenumbers. Second, we use resolvent analysis to study the linear input-output behavior of the jet and obtain a time-periodic representation of the screech mode, which compares favorably with experimental data. Third, we use harmonic resolvent analysis to study triadic interactions, including inter-frequency energy transfer, between the screech mode determined from resolvent analysis and other fluctuations in the jet. The components of the optimal harmonic resolvent mode at harmonics of the screech frequency match experimental observations that have not been previously predicted by global models. Fourth, we leverage a novel bilinear formulation of harmonic resolvent analysis to study the impact of the screech mode's nonlinear self-interaction on other fluctuations in the jet. We show that the forcing provided by this nonlinear self-interaction of the screech mode, along with its triadic interactions with other frequencies embedded within the harmonic resolvent operator, is sufficient to explain the redistribution of energy to other frequencies and the associated experimental observations. In aggregate, these findings underscore the critical role of triadic and nonlinear interactions in shaping screech dynamics and offer a promising workflow for studying similar interactions in other flows dominated by periodic motions.

Figures (14)

• Figure 1: Analysis of the shock-cell structure in the mean flow: (a) Mean streamwise velocity showing weak shock cells, with dashed lines indicating the shock cell spacing. (b) Wavenumber spectrum of the mean pressure, with the yellow line representing the normalized wavenumber along the centerline at $r = 0$, and cyan dashed lines mark the local maxima of the spectrum along the centerline.
• Figure 2: Eigenspectrum of the linearized operator around the mean flow. The red eigenvalue is the least-stable mode, which closely matches the observed screech frequency. We show that the two green eigenvalues are additional candidate screech modes created by interactions with different shock-cell wavenumbers. The blue eigenvalue is caused by resonance between upstream- and downstream-traveling acoustic duct modes.
• Figure 3: The eigenmodes associated with isolated eigenvalues of the screeching jet. Panels (a, c, e, g) show the streamwise velocity, and panels (b, d, f, h) show the pressure at frequencies of 0.31 (cyan), 0.66 (green), 0.72 (red), and 0.77 (green), respectively. The color coding in parentheses corresponds to the spectrum shown in figure \ref{['fig:screechEigSpectrum']}.
• Figure 4: Wavenumber spectra of the pressure component of the eigenmodes at (a-d) $St = 0.31$, $0.66$, $0.72$, and $0.77$, respectively. White dashed lines indicate the acoustic wavenumbers $k_x = \pm \omega$.
• Figure 5: The resolvent gain $\sigma^2$ of the screeching jet at $m=0$, obtained via linearization about the PIV mean flow: (blue) optimal gain $\sigma_1^2$; (orange) suboptimal gains $\sigma_2^2$ and $\sigma_3^2$.