Unsteady shock wave behaviors in supersonic nozzle exit under varying over-expanded conditions

To elucidate the unsteady shock-containing flow features in an over-expanded supersonic jet issuing from an axisymmetric nozzle, a comprehensive study combining schlieren visualization, pressure, and force/moment measurements was conducted at varying nozzle pressure ratios (NPRs). The resulting grayscale images were subsequently analyzed utilizing a multi-modal decomposition framework, including the randomized singular value decomposition-based proper orthogonal decomposition (POD) and its quadruple reconstruction (QPOD), as well as dynamic mode decomposition. Under over-expanded conditions, two dominant unsteady mechanisms were identified: (1) large-scale, low-frequency (approximately Strouhal number (St) of 0.007) shock system migration and reciprocating oscillation; and (2) relatively high-frequency (St of around 0.023) shear layer mixing, corresponding to the Kelvin–Helmholtz instability. POD analysis revealed descending energy ratios distinguishing these behaviors, where the shock migration exhibits a linear variation without a characteristic frequency, while its oscillation manifests as a distinct periodic feature. The QPOD results indicate that the shear layer instability significantly contributes to unsteady energy, influencing the fluctuation of nozzle forces. Despite symmetric exit flow in schlieren images, unsteady circumferential non-uniformities induce off-axis forces. Statistical and modal analysis shows that the flow unsteadiness grossly enhanced with decreasing of NPR, while the characteristic frequencies remain essentially consistent. Specifically, near NPRs transitioning from oblique shocks to Mach disk, the nonlinear coupling results in the “frequency locking” phenomena.