On the Lagrangian Power Spectrum of Turbulence Energy in Isotropic Turbulence

We present, for the first time, a derivation of the transport equation of the Lagrangian frequency power spectrum, ${ E_{L}(t,\omega)}$, of turbulence energy in isotropic turbulence starting from the autocorrelation of the Lagrangian velocity. The new equation is: ${\partial E_L (t,\omega)} \big / {\partial t} = $ $ {\mathcal T}_L (t,\omega) -$ $ \varepsilon_L (t,\omega) +$ $ \Psi_L (t,\omega) $, where $ {\mathcal T}_{L} (t,\omega)$ is the transfer rate of ${ E_{L}(t,\omega)}$ across the frequency spectrum, $\varepsilon_{L} (t,\omega)$ is the viscous dissipation rate of ${ E_{L}(t,\omega)}$, and $ \Psi_L (t,\omega)$ is the external forcing rate. Our DNS shows that $\varepsilon_{L} (\omega)$ is maximum at low frequencies and vanishes at high frequencies. We also performed an analytical study which confirms the DNS result and shows that $\varepsilon_{L}(\omega) \sim (\omega_{\eta} - \omega)$, i.e. there is non-locality for {$\varepsilon_{L}(\omega)$} in the $\omega$ domain, whereas $E_{L}(\omega) \sim (1/\omega^2 - 1/\omega_{\eta}^2)$, i.e. the locality is valid for $E_{L}(\omega)$, where $\omega_{\eta}$ is the Kolmogorov scale frequency.