Standing-Wave Dynamics in Low-Frequency Breathing of a Turbulent Separation Bubble

Abstract

This study investigates the low-frequency dynamics of a turbulent separation bubble (TSB) 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. Spectral proper orthogonal decomposition (SPOD) of the streamwise velocity field reveals a dominant low-rank mode at low Strouhal numbers ($St < 0.05$), 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-traveling three-dimensional modes. The model accurately reproduces the measured SPOD modes, demonstrating that sidewall reflections lead to the formation of standing wave-like patterns. To gain further insight into the driving mechanisms of the low-frequency dynamics, a global stability analysis is performed, revealing a zero-frequency eigenmode whose growth rate depends on the spanwise wavenumber. This eigenmode originates from a centrifugal instability. 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, particularly those relying on spanwise-periodic boundary conditions, and offers a low-order framework for predicting sidewall-induced modal dynamics in separated flows.