λ₂ criterion of visualisations for flow past a circular cylinder for Re =4000.

Описание: The λ₂ criterion identifies vortices in fluid flow by analysing the pressure Hessian eigenvalues. Mathematically, for the velocity gradient tensor ∇u, it computes the symmetric tensor S² + Ω² (where S is the strain rate, Ω is the rotation rate). Vortex regions are where the second eigenvalue λ₂ less than 0, indicating local pressure minima in the plane perpendicular to the vortex axis. The λ₂ criterion provides a more physically rigorous identification of vortical structures than the Q-criterion, particularly in transitional and turbulent regimes like Re=4000. While both methods capture the large-scale Kármán vortex shedding, λ₂ offers superior discrimination by detecting local pressure minima in planes perpendicular to vortex axes, ensuring that identified structures correspond to true rotational cores rather than shear-dominated regions.

At this Reynolds number 4000, λ₂ reveals critical insights missed by Q-criterion: finer vortical filaments connecting primary vortices, internal sub-structure within breaking vortex cores, and a more precise distinction between attached shear layers and separated vortical regions. Unlike Q, which requires positive thresholds, λ₂’s natural threshold (negative λ₂) directly links vortex cores to pressure minima, offering better correlation with surface pressure measurements and aerodynamic forces. The visualisation highlights earlier detection of three-dimensional instabilities and a more precise vortex skeleton, enabling detailed tracking of vortex reconnection and energy transfer mechanisms in the turbulent wake.