How to visualize divergence correctly?

Question

I have posted two figures. 1st is for $F=xi+ yj$ and 2nd is for $F=yi+ xj$

For the first $div F=2$ and for second $div F=0$.

I have the idea that if the bigger vectors come out of the circle, the divergence is positive and if the same size and number of vectors go in as out of the circle then divergence is zero.

But here no such differences are shown. For both cases, it seems to me that bigger vectors come out of the circle. But divergence is zero for the second figure. What to do? How to visualize correctly?

Edit: Please give me a proper visualization technique of divergence from the plot of vector field...

Answer 1

To see the flux through the circle you have to take the projection of the vector field ($\boldsymbol{\color{blue}{blue}}$) to the outward normal vectors at the circle (black). This gives you the $\boldsymbol{\color{red}{red}}$ vectors. In the case of $$ \begin{pmatrix}x\\y\end{pmatrix} $$ which has divergence two we see clearly that there is net flux out of the circle:

In the case of $$ \begin{pmatrix}y\\x\end{pmatrix} $$ the net flux is zero: