Eulerian–Lagrangian bridge for the energy and dissipation spectra in isotropic turbulence

Abstract

We study, numerically and analytically, the relationship between the Eulerian spectrum of kinetic energy, E E(k, t), in isotropic turbulence and the corresponding Lagrangian frequency energy spectrum, E L(ω, t), for which we derive an evolution equation. Our DNS results show that not only E L(ω, t) but also the Lagrangian frequency spectrum of the dissipation rate \({\varepsilon_{\rm L} (\omega, t)}\) has its maximum at low frequencies (about the turnover frequency of energy-containing eddies) and decays exponentially at large frequencies ω (about a half of the Kolmogorov microscale frequency) for both stationary and decaying isotropic turbulence. Our main analytical result is the derivation of equations that bridge the Eulerian and Lagrangian spectra and allow the determination of the Lagrangian spectrum, E L (ω) for a given Eulerian spectrum, E E (k), as well as the Lagrangian dissipation, \({\varepsilon_{\rm L}(\omega)}\), for a given Eulerian counterpart, \({\varepsilon_{\rm E} (k)=2\nu k^2 E_{\rm E}(k)}\). These equations were derived from the Navier–Stokes equations in the sweeping-free coordinate system (intermediate between the Eulerian and Lagrangian frameworks) which eliminates the effect of the kinematic sweeping of the small eddies by the larger eddies. We show that both analytical relationships between E L (ω) and E E (k) and between \({\varepsilon_{\rm L} (\omega)}\) and \({\varepsilon_{\rm E} (k)}\) are in very good quantitative agreement with our DNS results and explain how \({\varepsilon_{\rm L} (\omega, t)}\) has its maximum at low frequencies and decays exponentially at large frequencies.