Eulerian–Lagrangian relations in decaying two-dimensional incompressible Navier–Stokes fluids across initial vorticity packing and Reynolds number

Abstract

A recent study [Phys. Fluids 38, 045159 (2026)] showed that, at a fixed high Reynolds number (Re), variations in the initial vorticity packing fraction (VPF) were associated with a general Eulerian–Lagrangian relationship during the inverse-cascade regime and a more direct correspondence between Eulerian statistical organization and Lagrangian transport in the late coherent-dipole regime. Here, we examine the robustness of these connections across a broader (VPF,Re) parameter space, together with random-phase, long-time, and alternative initial-vorticity geometry tests. During the inverse cascade, the broad VPF-associated organization of transport remains comparatively robust, although the quantitative transport exponents vary non-monotonically across Re. In the late coherent-dipole regime, however, Lagrangian transport varies strongly and non-monotonically across Re and exhibits considerably greater sensitivity to the initial random-phase realization and continued coherent-state evolution. High-VPF Eulerian states nevertheless remain consistently KMRS-like across the investigated Re range. Thus, similar Eulerian statistical organization can coexist with substantially different late-time Lagrangian transport, demonstrating that the direct Eulerian–Lagrangian correspondence identified at fixed Re is not universal under variations of Reynolds number, initial random-phase realization, coherent-state evolution, and initial-vorticity geometry.