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


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

\[ av = a_xv_x + a_yv_y + a_zv_z\]


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