Temporal correlation function in 3-D Turbulence

We observe oscillatory decay in the two-point, non-equal time, velocity correlation function of homogeneous, isotropic turbulence. We found this through a direct numerical simulation (DNS) of the three dimensional Navier-Stokes ($3-D$ NS) equation. We give an approximate analytic theory which explains this oscillatory behaviour. The wave-number and frequency dependent effective viscosity turns out to be complex; the imaginary part gives rise to the temporal oscillation. We find that, at least for the decay at short times, data collapse occur among the inertial range velocity wave-vector modes with the long time dynamic exponent $z=2/3$, but the time period of the temporal oscillation is not universal.