A06_anisotropy.tex

anisotropy measures

velocity anisotropy tensor (symmetric, traceless, no elements in isotropic flow):
\begin{equation}
	b_{i,j} = \frac{<u_i'u_j'>}{<u_k'u_k'>} - \frac{\delta_ij}{3}
\end{equation}
can be expressed similarly for the scalar gradient field (if you have the fluctuations along x,y,z axes)
and the vorticity field where $\omega = \grad x u$

\begin{equation}
b_{11} = \frac{<u_1'u_1'>}{<u_2'u_2'>} 
+ \frac{<u_1'u_1'>}{<u_3'u_3'>} 
\end{equation}
\begin{equation}
	b_{12} = \frac{<u_1'u_2'>}{<u_1'u_1'>} 
	              + \frac{<u_1'u_2'>}{<u_2'u_2'>} 
	              + \frac{<u_1'u_2'>}{<u_3'u_3'>}
\end{equation}
\begin{equation}
b_{22} =  \frac{<u_1'u_1'>}{<u_2'u_2'>} 
+ \frac{<u_1'u_1'>}{<u_3'u_3'>}
\end{equation}
\begin{equation}
b_{23} = \frac{<u_2'u_3'>}{<u_1'u_1'>} 
+ \frac{<u_2'u_3'>}{<u_2'u_2'>} 
+ \frac{<u_2'u_3'>}{<u_3'u_3'>}
\end{equation}
\begin{equation}
b_{13} = \frac{<u_1'u_3'>}{<u_1'u_1'>} 
+ \frac{<u_1'u_3'>}{<u_2'u_2'>} 
+ \frac{<u_1'u_3'>}{<u_3'u_3'>}
\end{equation}
\begin{equation}
b_{33} = \frac{<u_3'u_3'>}{<u_1'u_1'>} 
+ \frac{<u_3'u_3'>}{<u_2'u_2'>} 
\end{equation}
where angle brackets are a volume average

code up the anisotropy tensor:
b11 = u*2/v*2 + u*2/w*2
b22 = v*2/u*2 + v*2/w*2
b33 = w*2/u*2 + w*2/v*2
b12 = u*v*/u*2 + u*v*/v*2 + u*v*/w*2
b13 = u*w*/u*2 + u*w*/v*2 + u*w*/w*2
b22 = v*w*/u*2 + v*w*/v*2 + v*w*/w*2

look at the evolution of these through time
and look at the depth distribution