Time-scales from autocorrelations of velocity and acceleration:

Time-scales were found from the integration of autocorrelation.

Acceleration time-scales

The following time scales were obtained by integrating the autocorrelation of each component of the acceleration until the value of the autocorrelation reached 0.05, i.e.

\[ T_{i} =\int_0^t \rho_i(\tau) d\tau \quad \text{Where} \quad \rho_i(t) = 0.05 \]

These time scales are plotted below, along with plots of the autocorrelations of accelerations showing the limits of integration.

Data from the following figure can be found here





Velocity time-scales

The autocorrelation of velocity has a decorrelation time which is longer than the duration of our tracks (except very close to the wall). Here we assume that the autocorrelation of velocity has an exponential form, fit an exponetial decay to our data, then calculate the time-scale based on the fitted exponential.

For example:

\[ \rho(\tau)= exp(\tau/L + A), \quad \text{ Where $L$ is the time-constant and $A$ is an offset to account for non-exponential behavior at short times} \]

If we fit an exponential function to the data, we can then evaluate the time-scale in a way consistant with the acceleration time-scale, i. e.,

\[ T_{i} =\int_0^t \rho_i(\tau) d\tau \quad \text{Where} \quad \rho_i(t) = 0.05 \quad \implies \rho(0.05) = \exp(-t/L+A) \implies t= AL-L(log(0.05)) \]

And so

\[ T_{i} =\int_0^{AL-L(log(0.05))} \exp(-\tau/L+A) d\tau \implies T_{i} = L(\exp(A)+log(0.05) ) \]

The figures below show the velocity time-scales obtained by this method, as well as the autocorrelations of velocity, the fitted exponentials, and the part of the curve used for the fitting (between the two vertical red lines). Note that this method fails very close to the wall for the wall-normal component of velocity, and so the first two data points close to the wall were obtained by the method used with the autocorrelations of acceleration.





Ratio of time-scales

The following figure plots the ratio of velocity and acceleration time scales:


 Yeung and Pope, 1989 Reports various Lagrangian time scales in DNS of homogenous isotropic turbulence for several Reynolds numbers ($R_\lambda = 38-93$). Below we plot the ratio of Lagrangian integral scales and Kolmogorov scales ($T_L/\tau_\eta$) as a function of Re\lambda from their paper, as well as the experimentally measured value of this ratio for each component. Here we rescale the experimental results to the $Re_\lambda$ profile determined from the channel flow DNS.