I have a basic question regarding the definition of a curvature.
Most of my searches revealed the following:
κ = -T⋅$dN/ds$,
where T is the tangent vector, N is the normal vector, s is the arc length, k is the curvature.
I use a CFD code where the normal vector is computed and for the curvature the following equation is used:
$k = -divergence.N$, This equation also makes it way into Wiki Wikipedia curvature calculation unfortunately without proof.
Can anyone help me understand how the 1st and 2nd equations are related.
You seem to be totally mixing things.
On the one hand, $\kappa = - \textrm T \cdot \dfrac {\Bbb d \textrm N} {\Bbb d s}$ is obtained from the second Frenet formula by taking the scalar product with $\textrm T$. It is about the curvature of a curve.
On the other hand, $H = -\frac 1 2 \text{div } N$ is just another way of computing the mean curvature of a hypersurface.
There is no connection between these two types of curvatures, let alone the fact that there exist other types of curvature as well.