Autocorrelations of acceleration magnitude, acceleration spherical angles, and A.V

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

source:trunk/Figures/Correlations/Normalized_correlations/a_{mag}(t)a_{mag}(t+\tau).png

The following plot traces the autocorrelation of the inner product of acceleration and velocity, defined as

\[ av = a_xv_x + a_yv_y + a_zv_z\]

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

The two spherical angles of the acceleration vector are defined as follows:

\[ \phi = atan2 \left( \frac{a_y}{a_z} \right) \quad \quad \theta = acos \left( \frac{a_z}{|a|} \right) \]

Where $atan2$ is a function that avoids the singularity, and has a range of $-\pi/2, \pi/2$, more  here

source:trunk/Figures/Correlations/Normalized_correlations/a_{\phi}(t)a_{\phi}(t+\tau).png

source:trunk/Figures/Correlations/Normalized_correlations/a_{\theta}(t)a_{\theta}(t+\tau).png

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