On the Nature of Symmetry Breaking Seeds in Boundary Layer Transition: A Data-Driven Stability Analysis

We elucidate the exact nature of temporal and spatial symmetry breaking in canonical K-type boundary layer transition through modal decomposition, dynamical systems theory, and data-driven stability analysis. Specifically, we use space-time proper orthogonal decomposition (STPOD), spectral POD (SPOD), and D1 symmetry decomposition to extract coherent structures with unique optimality properties. SPOD of the early transitional regime identifies a prototypical harmonic resonant response on a periodic limit cycle, captured by a few modes at the fundamental frequency and its harmonics. In the late transitional regimes beyond the skin friction maximum, STPOD uncovers periodic as well as non-periodic, symmetric and anti-symmetric modes linked to limit cycle instability, with phase space dynamics driving the flow toward chaos and broadband turbulence. To reveal the origin of these symmetry-breaking seeds, we derive stability equations from the symmetric and anti-symmetric Navier-Stokes equations, linearized around the harmonic base flow. A data-driven residual method confirms that this base flow satisfies the governing equations to high accuracy, ensuring that it is a valid reference flow, such that small fluctuations represent linear perturbations. We then use a multi-trajectory Koopman inference to learn linear operators that predict seed evolution for both symmetry-breaking types, with eigenspectra exhibiting dominant unstable modes that match observed growth rates. Our findings suggest that leading STPOD modes capture Floquet instabilities of the fundamental harmonic response, offering a dynamical systems view of the organized, linear origins of symmetry breaking in transition.