Numerical study on the Eulerian-Lagrangian analysis of Navier-Stokes turbulence

Detailed analyses of the Navier-Stokes equations on the basis of the Euler-Lagrangian formalism are presented with the use of numerical simulations. A singular perturbation property arising in the limit of vanishing viscosity is one issue of this paper. By using the connection matrix, which is related to the geometry of particle paths, we introduce "connection anomaly" for the characterization of the property and confirm it numerically. As a characterization in physical space, we show how regions with small values of a determinant of a derivative of diffusive labels are spatially correlated with vortex structures. Two kinds of initial conditions are examined: (i) Decaying isotropic turbulence developing from a random initial condition and (ii) the orthogonally offset two vortex tubes. For (i), it is found that when turbulence is fully developed, the resetting process occurs very frequently, which defines a short time scale associated with small-scale motion. For (ii), we confirm our previous finding that resetting of diffusive label captures successfully reconnection of vortex at higher Reynolds numbers. Even for this special initial condition, turbulence is developed after the phase of prominent reconnection and frequent resettings are associated with it.