Wave-packet modulation in shock-containing jets

We propose an approach to predict the modulation of wave packets in shock-containing jets. With a modeled ideally expanded mean flow as input, an approximation of the shock-cell structure is obtained from the parabolized stability equations (PSE) at zero frequency. This solution is then used to define a new shock-containing mean flow, which is a function of the shock-cell wave number at each streamwise station. Linearization of the Navier-Stokes equations around this quasiperiodic mean flow allows us to postulate a solution based on the Floquet ansatz, and further manipulation of the equations leads to a system called the parabolized Floquet equations [PFE; Ran, Phys. Rev. Fluids 4, 023901 (2019)2469-990X10.1103/PhysRevFluids.4.023901] that bears several similarities to PSE. The modulation wave numbers are marched spatially together with the central Kelvin-Helmholtz wave number, leading to a modulated wave packet as the final solution. The limitations of PFE are highlighted, and the method is applied to two sample cases: a canonical slowly diverging jet at low supersonic Mach number and a heated overexpanded jet, for which large-eddy simulation (LES) data are available. Good agreement is observed between the wave packets predicted by PFE and the leading spectral proper orthogonal decomposition (SPOD) modes from the LES, suggesting that the method is able to capture the underlying physical mechanism associated with wave-packet modulation: the extraction of energy from the mean flow by the Kelvin-Helmholtz mode and a redistribution of energy to modulation wave numbers due to the interaction of this mode with the shock-cell structure.