Hydrodynamic Turbulence: Sweeping Effect and Taylor’s Hypothesis via Correlation Function
Tóm tắt
We demonstrate the sweeping effect in turbulence using numerical simulations of hydrodynamic turbulence without a mean velocity. The velocity correlation function,
$$C(\mathbf{k},\tau )$$
, decays with time due to the eddy viscosity. In addition,
$$C(\mathbf{k},\tau )$$
shows oscillations due to the sweeping effect by “random mean velocity field”
$${ \tilde{\mathbf{U}}}_0$$
. We also perform numerical simulation with mean velocity
$$\mathbf{U}_0= 10\hat{z}$$
(10 times the rms speed) for which
$$C(\mathbf{k},\tau )$$
exhibits damped oscillations with the frequency of
$$|\mathbf{U}_0| k$$
and decay time scale corresponding to the
$$\mathbf{U}_0=0$$
case. For
$$\mathbf{U}_0=10\hat{z}$$
, the phase of
$$C(\mathbf{k},\tau )$$
shows the sweeping effect, but it is overshadowed by oscillations caused by
$$\mathbf{U}_0$$
. We also demonstrate that for
$$\mathbf{U}_0=0$$
and
$$10\hat{z}$$
, the frequency spectra of the velocity fields measured by real-space probes are respectively
$$f^{-2}$$
and
$$f^{-5/3}$$
; these spectra are related to the Lagrangian and Eulerian space-time correlations respectively.
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