Relationship between the power spectral density of the Lagrangian velocity and the hierarchy of coherent vortices in turbulence

TL;DR

This study uses direct numerical simulations to explore how the hierarchy of coherent vortices influences the power spectral density of Lagrangian velocity in turbulence, revealing universal high-frequency behavior and non-universal low-frequency effects from forcing methods.

Contribution

It introduces a scale-decomposition method for Lagrangian velocity to analyze vortex contributions at different scales, clarifying the impact of large-scale flows on spectral formation.

Findings

Universal high-frequency spectral behavior observed across different forcing methods.

Large-scale flows contaminate the inertial range, affecting the Kolmogorov scaling.

Eulerian velocity spectra are less affected by forcing methods than Lagrangian spectra.

Abstract

We conduct direct numerical simulations of developed turbulence in a periodic cube to investigate the formation mechanism of the power spectral density of the Lagrangian velocity. We compare the power spectral density of the Lagrangian velocity of turbulent flows with different forcing methods and Reynolds numbers. This systematic comparison demonstrates that universal behavior is observed in a narrow high-frequency regime, whereas non-universality originating from the forcing method broadly appears in a low-frequency regime. To reveal the formation mechanism of the spectra in terms of the hierarchy of coherent structures in turbulence, we propose a scale-decomposition method for the Lagrangian velocity, which enables us to evaluate the contribution of vortices at different scales. This scale-decomposition analysis directly demonstrates that the largest-scale flows driven by the external force can contaminate the Kolmogorov scaling of the Lagrangian velocity spectra formed by small-scale vortices in the inertial range, thus leading to the narrow Lagrangian inertial range. Furthermore, we provide evidence that this remarkable effect by the largest-scale flows is specific to the Lagrange velocity by demonstrating that the power spectral density of the Eulerian velocity is less sensitive to the forcing method.