Standing-wave dynamics in low-frequency breathing of a turbulent separation bubble

This study investigates the low-frequency dynamics of a turbulent separation bubble (TSB) forming over a backward-facing ramp, with a focus on large-scale coherent structures associated with the so-called `breathing motion’. Using time-resolved particle image velocimetry (PIV) in both streamwise and spanwise planes, we examine the role of sidewall confinement, an aspect largely overlooked in previous research. Spectral proper orthogonal decomposition (SPOD) of the streamwise velocity field reveals a dominant low-rank mode at low Strouhal numbers ( $St \lt 0.05$ ), consistent with prior observations of TSB breathing. Strikingly, the spanwise-oriented PIV data uncover a previously unreported standing-wave pattern, characterised by discrete spanwise wavenumbers and nodal/antinodal structures, suggesting the presence of spanwise resonance. To explain these observations, we construct a resolvent-based model that imposes free-slip conditions at the sidewall locations by superposing left- and right-travelling three-dimensional modes. The model accurately reproduces the spanwise structure and frequency content of the measured SPOD modes, demonstrating that sidewall reflections lead to the formation of standing-wave-like patterns. Global stability analysis reveals a zero-frequency eigenmode originating from a centrifugal instability, giving rise to the observed low-frequency breathing. Downstream, the associated coherent structures are further amplified through non-modal lift-up mechanisms. Our findings highlight the critical influence of spanwise boundary conditions on the selection and structure of low-frequency modes in TSBs. This has direct implications for both experimental and numerical studies relying on spanwise-periodic boundary conditions and offers a low-order framework for predicting sidewall-induced modal dynamics in separated flows.