The velocity gradient invariants in turbulent channel flow: An analysis based on proper orthogonal decomposition

The dynamical evolution of multi-scale topological characteristics in channel turbulence is examined based on direct numerical simulation. In the original flow field, the joint probability density function (PDF) constructed from the velocity-gradient tensor invariants Q and R exhibits a classic teardrop shape, which reflects the dominance of vortex stretching and sheet-like structures in small-scale motions. The fluctuating velocity field extracted from turbulent channel flow is broken down into orthogonal modes via proper orthogonal decomposition (POD), with each mode arranged according to the magnitude of turbulent kinetic energy (TKE), in highest-to-lowest order. Flow field reconstructed with the first four high-energy modes (10% TKE contribution) is explored in this work. Quasi-periodic temporal energy oscillations are observed in this reconstructed field, with its spatial patterns linked to large-scale coherent structures. The joint PDFs of Q and R in the four-mode reconstruction display a Gaussian behavior, with approximately balanced signatures of vortex stretching and vortex compression. In addition, regarding the fully-developed channel turbulence investigated herein, behavior consistent with inverse energy cascade is observed in the near-wall region of the four-mode reconstruction. For comparison, flow fields reconstructed with modes that cumulatively contribute 50% and 90% of the TKE are also examined. In these reconstructions, the disappearance of quasi-periodic behavior is accompanied by a recovery of the canonical teardrop pattern in the joint PDFs of Q and R , indicating that when the reconstruction covers a sufficiently broad range of scales, topological features under different truncations remain highly similar.