Vorticity Packing Effects on Long Time Turbulent Transport in Decaying Two-Dimensional Incompressible Navier-Stokes Fluids

TL;DR

This study addresses how the initial circulation, quantified by the vortex packing fraction $VPF$, shapes long-time transport in decaying 2D incompressible NS turbulence. Using a coupled GPU-accelerated solver (GHD2D-TP) to track passive tracers, it connects Lagrangian transport metrics such as mean-square displacement and PDFs to the Eulerian evolution governed by either point-vortex or KMRS equilibria. The results reveal a clear $VPF$-controlled progression from delayed, anisotropic, sub-diffusive transport in loosely packed cases to rapid, isotropic, super-diffusive transport in tightly packed configurations, with late-time dynamics governed by either coherent dipole trapping or linear dipole motion. This work establishes a strong link between late-time Eulerian statistics and underlying Lagrangian transport, providing a unified view of how initial vorticity packing influences mixing and dispersion in decaying 2D turbulence with potential implications for geophysical and plasma contexts.

Abstract

Recent high-resolution, high-Reynolds-number simulations have shown that the initial total circulation, quantified by the vorticity packing fraction (VPF), strongly influences the late-time Eulerian statistical equilibria of decaying incom- pressible two-dimensional Navier-Stokes turbulence (Biswas et al., 2022, Physics of Fluids 34, 065101), revealing a transition from point-vortex--dominated to finite-size (patch-vortex) equilibria with increasing vortex packing, and emphasizing the role of of the classical exclusion principle (i.e., incompressibility) and total circulation in determining the final statistical states. The present study examines how the associated Lagrangian tracer transport evolves with VPF across the early (linear-nonlinear turbulence onset), intermediate (turbulence development), and late (coherent dipole evolution) stages, and how it correlates with the corresponding Eulerian states. Turbulence, triggered by the Kelvin-Helmholtz instability and sustained by inverse energy cascades, forms large-scale coherent vortices that govern long-time transport. Tracer dynamics, analyzed via mean-square displacement and position-velocity probability distri- bution functions (PDFs), reveal that increasing VPF accelerates turbulence onset, drives a transition from sub- to super- diffusive transport with decreasing anisotropy in the intermediate stage, and determines late-time behavior dominated by either orbital coherent vortex trapping (sub-diffusive) or linear translational dipole motion (super-diffusive). These dis- tinct long-time transport characteristics, evolving from sub- to super-diffusive behavior with increasing vorticity pack- ing, demonstrate a strong correspondence between the transition from point-vortex- to finite-size-vortex-dominated Eulerian equilibria and the underlying Lagrangian transport in decaying incompressible 2D Navier-Stokes turbulence.

Figures (16)

• Figure 1: (a) Initial condition: Two finite strips of fluid with alternating vorticity, flowing in opposite directions, forming discontinuous shear layers (commonly referred to as “oppositely directed broken jets”) to study Kelvin–Helmholtz instability. The Reynolds number of the flow is $R_E = 122$, and the peak vorticity is $\omega_0 = \pm 25$. (b) The growth rates of the Kelvin–Helmholtz instability in the oppositely directed broken jets problem, computed using the GHD2D solver for various mode numbers, are in excellent agreement with the analytical results of Drazin et al. in the two-strip configuration. Simulations were performed at grid resolutions of $256^2$ and $2048^2$, and the results are found to be invariant with respect to resolution.l
• Figure 2: (a) Comparison of vorticity profiles for the two-strip configuration in the oppositely directed broken-jet problem, implemented using sharp and tapered initial conditions. (b) Comparison of the growth rates of the Kelvin–Helmholtz instability for sharp and tapered initial conditions in a fluid with $R_E = 228{,}576$ and $\omega_0 = \pm 1$, showing overall similar behavior.
• Figure 3: Variation of the Kelvin–Helmholtz instability growth rate with mode number for different combinations of Reynolds number ($R_E = 122$ and its multiples) and initial vorticity strip numbers (2, 4, 8, 16, 20), using a tapered profile with vorticity $\omega_0 = \pm 25$. Each color represents a distinct Reynolds number, and within each color group, different lines correspond to increasing strip numbers (2 to 20). For a given $R_E$, the growth rate increases with strip number. Each bell-shaped curve highlights the effect of both Reynolds number and strip count on the peak growth rate and the dominant mode. (b) Variation of the Kelvin–Helmholtz instability growth rate with mode number for a sharp profile of the initial vortex strips at a $R_E$ = 228576 and $\omega_0=\pm 1$.
• Figure 4: (a) Initial conditions of the 2D kinematic flow described by the velocity functions in Eqs. \ref{['eq:u']}--\ref{['eq:v']}. The flow exhibits a lattice of non-steady square cells, as shown by the contour map of the total kinetic energy $E_k$. Streamlines (red lines) indicate the instantaneous flow directions, while quiver arrows represent the velocity vectors at discrete points, illustrating both the magnitude and orientation of the flow. (b) Transport results (single-particle mean-square displacement (MSD) or absolute dispersion) from our tracer particle solver, showing 1,000 advected particles in the kinematic chaotic flow (see Fig. \ref{['fig:kinv']}) compared with the results of Folgia et al. FL2022, which has been reproduced with the corresponding author’s permission. This figure shows results obtained from both velocity ($\mathrm{(uv)_x}, \mathrm{(uv)_y}$) and stream function ($\mathrm{(\psi)_x}, \mathrm{(\psi)_y}$) interpolation and illustrates a similar behavior to Folgia et al. FL2022($\mathrm{(x\text{-}data)_{ref}}, \mathrm{(y\text{-}data)_{ref}}$) , with an initial ballistic regime ($\propto \mathrm{t}^2$) transitioning into a normal diffusive regime ($\propto \mathrm{t}$), and an eddy turnover time of approximately 1 hr.
• Figure 5: Initial vorticity, stream function and associated tracer particles distribution at $T$ = 0 for various cases of turbulence in our numerical simulations. (a) 2 strip configuration with lowest vorticity packing fraction (VPF 6.25%) (b) 4 strip configuration with low VPF (12.5%) (c) 8 strip configuration with moderate VPF (25%) (d) 16 strip configuration with high VPF (50%). The vorticity value for blue, red and green regions are -1, 1 and 0 respectively.