Sweeping effect and Taylor's hypothesis via correlation function

We performed high-resolution numerical simulations of hydrodynamic turbulence with and without mean velocity ($U_0=0,10$), and demonstrate the sweeping effect. For $U_0=0$, the velocity correlation function, $C(\mathbf{k},\tau )$ decays with time due to eddy viscosity, but it also shows fluctuations due to the sweeping effect. For $U_0=10$, $C(\mathbf{k},\tau )$ exhibits damped oscillations with the frequency of $U_{0}k$ and decay time scale corresponding to the $U_0=0$ case. A closer examination of $\mathrm{\Im }[C(\mathbf{k},\tau )]$ also demonstrates sweeping effect for $U_0=10$. We also demonstrate that the frequency spectra of the velocity fields measured by real-space probes are respectively $f^{-2}$ and $f^{-5\mathrm{/}3}$ for $U_0=0$ and 10; these spectra are related to the Lagrangian and Eulerian space-time correlations.