Cell Velocity

The cell velocities can be computed with the weighted least-squared method as

1)	$\begin{array}{l}\displaystyle \displaystyle u_c = \frac{1}{\Delta} \left( a_{2,2} b_1- a_{1,2} b_2 \right ), v_c = \frac{1}{\Delta} \left( a_{1,1} b_2- a_{1,2} b_1 \right )\end{array}$

in which the coefficients are:

$\begin{array}{l}\displaystyle a_{1,1} = \sum_k w_k n_{k,1}^2\end{array}$ , $\begin{array}{l}\displaystyle a_{2,2} = \sum_k w_k n_{k,2}^2\end{array}$ , $\begin{array}{l}\displaystyle a_{1,2} = a_{2,1} = \sum_k w_k n_{k,1} n_{k,2}\end{array}$ , $\begin{array}{l}\Delta = a_{1,1} a_{2,2} - a_{1,2} a_{2,1}\end{array}$ , $\begin{array}{l}\displaystyle b_1 = \sum_k w_k n_{k,1} u_{N,k}\end{array}$ , $\begin{array}{l}\displaystyle b_1 = \sum_k w_k n_{k,2} u_{N,k}\end{array}$

The weights $\begin{array}{l}w_k\end{array}$ are assigned to the face areas, $\begin{array}{l}(n_{k,1}, n_{k,2})\end{array}$ is the face normal unit vector, and $\begin{array}{l}u_{N,k}\end{array}$ is the face normal velocity.