I'm currently studying the derivation of the RANS (Reynolds Averaged Navier Stokes) equations, used in the study of turbulence, and I've stumbled upon a step wich I don't understand very well.
The problem is on the following identity (one of the steps of the formal derivation of the RANS equations, present in many online texts):
$$ u \bullet \nabla u = \nabla \bullet (u \otimes u) $$
, where $ u = \{ u \; v \; w \} $ is a 3D vector and $ u \otimes u $ is the tensor dot product / outer product of vector $u$ by itself:
$$ u \otimes u = \begin{pmatrix} u^2 & uv & uw \\ vu & v^2 & vw \\ wu & wv & w^2 \\ \end{pmatrix} $$
My question is: where can I find a proof of the previous equality?
When I try to derive it by hand I don't end up with the result above (I'm probably commiting an error somewhere).
Can someone derive it (in a simple way) here by hand?
I've searched online, but I haven't found anything this specific.