Autocorrelations of velocity and acceleration components

The following are autocorrelations defined as follows:

\[ \rho(\tau)= \frac{\langle v(t) v(t+\tau)\rangle}{\sqrt{\langle v ^2(t) \rangle  \langle  v ^2(t+\tau) \rangle  } } \quad \quad \text{Where} \quad v(t) = V(t) - \langle V(t) \rangle \quad \text{i.e. the velocity less the Lagrangian mean velocity at time = t} \]

Attention: the wall-bins are not all the same size! They have a width of $y^+= 25$ near the wall, and a width of $y^+ = 75$ away from the wall.

The data used to make these plots can be found here and the script used to plot these results can be found here

Velocity

source:trunk/Figures/Correlations/Normalized_correlations/v_x(t)v_x(t+\tau).png

source:trunk/Figures/Correlations/Normalized_correlations/v_y(t)v_y(t+\tau).png

source:trunk/Figures/Correlations/Normalized_correlations/v_z(t)v_z(t+\tau).png

Acceleration

source:trunk/Figures/Correlations/Normalized_correlations/a_x(t)a_x(t+\tau).png

source:trunk/Figures/Correlations/Normalized_correlations/a_y(t)a_y(t+\tau).png

source:trunk/Figures/Correlations/Normalized_correlations/a_z(t)a_z(t+\tau).png