Boundary Layer Transition as Succession of Temporal and Spatial Symmetry Breaking

Abstract

We show that both temporal and spatial symmetry breaking in canonical K-type transition arise as organized hydrodynamic structures rather than stochastic fluctuations. Before the skin-friction maximum, the flow is fully described by a periodic, spanwise symmetric, harmonic response to the Tollmien-Schlichting wave, forming a spatially compact coherent structure that produces hairpin packets. This fundamental harmonic response may visually resemble turbulence, but remains fully periodic and delimits the exact extent of the deterministic regime. A distinct regime change occurs after this point; a hierarchy of new (quasi-)periodic and aperiodic space-time structures emerges, followed shortly by anti-symmetric structures that develop similarly despite no anti-symmetric inputs, marking the onset of aperiodicity and spanwise asymmetry. We identify these structures as symmetry-decomposed spectral and space-time proper orthogonal modes that resolve the full progression from deterministic to broadband dynamics. The key insight is that laminar-turbulent transition can be viewed as a sequence of symmetry breaking events, each driven by energetically dominant, space-time coherent modes that gradually turn an initially harmonic flow into broadband turbulence.

Figures (5)

• Figure 1: Dynamical regimes of the transition process. (a) Instantaneous DNS snapshot. (b) Fundamental harmonic response $\tilde{\mathbf{q}}$ (STPOD mode $\bm{\upphi}_0$). (c) Cyclo-stationary fluctuation $\mathbf{q}"$. (d) Symmetric $\mathbf{q}"^S$ and anti-symmetric $\mathbf{q}"^A$ fluctuation components for time and spatial symmetry breaking. $Q$-criterion isosurfaces ($Q=10^3$ for all); colored in their respective $u$-velocities ($w$-velocity for $\mathbf{q}"^A$). (e) Regimes labeled on the $C_f$ graph.
• Figure 2: SPOD energy spectra for (a) $\mathbf{q}^S$ symmetric and (b) $\mathbf{q}^A$ anti-sym. components; compare to sayadi2013direct. (c) Superposition of $N_n$={1, 2, 4, 8, 16, 32} dominant SPOD modes $\bm{\uppsi}^{(1)}_{n}$ at harmonic peaks $f_n$ (\ref{['eq:SPOD_lincomb']}); $Q$-criterion ($Q=10^3$) colored by $u$.
• Figure 3: Local power spectral densities (PSD) of different quantities as a streamwise function of $Re_x$, colored by the ratio of the dominant local SPOD energy over the full local energy $\uplambda^{(1)}(x)/\sum_{i} \uplambda^{(i)}(x)$. (a) $\mathbf{q}^S(\mathbf{x},t)$ full symmetric data. (b) $\mathbf{q}"^S(\mathbf{x},t)$ fluctuation data. (c) Zoom on amplified region, $r=\{2,8\}$ first two and first eight symmetric STPOD-mode-subtracted fluctuation $\mathbf{q}"^S(\mathbf{x},t)-\sum_{m=1}^{r} \bm{\upphi}_m^{S}(\mathbf{x},\uptau) a_{m,j}^S$.
• Figure 4: (a) $\mathbf{q}"^S$ symmetric and (b) $\mathbf{q}"^A$ anti-symmetric STPOD modal energy spectra; dominant modes with periodic dynamics colored magenta. (c) Streamwise amplitude development of select STPOD modes $\bm{\upphi}_m(\mathbf{x},\tau) a_{m,j}$, enveloped by the total $\mathbf{q}^S$, $\mathbf{q}^{\prime \prime S}$ and $\mathbf{q}^{\prime \prime A}$ components. The TS wave input amplitude level serves as significance threshold.
• Figure 5: Symmetry breaking modes from STPOD. Deterministic mode $\bm{\upphi}_0$ ($\equiv$ FHR $\tilde{\mathbf{q}}$), first three symmetric modes ($\bm{\upphi}^S_{1-3}$) and first two anti-sym. modes ($\bm{\upphi}^A_{1-2}$). (a) $\textbf{a}_m^{\text{POD}}(\uptau)$ trajectory phase-spaces and (b) instantaneous isosurfaces at same phase. Symmetric modes shown by $Q$-criterion ($Q=10^3$) colored by $u$-velocity. Anti-sym. modes shown by $w$-velocity ($w=\pm0.002$). (c) Symmetric and (d) anti-sym. mode relative $\text{L}_2$-norm distances between consecutive phases: Red markers denote distance from last to first phase (periodicity mismatch), error bars show the existing min$/$max consecutive phase distances within the period, all normalized by the average phase distance in a given mode.