When taking the divergence of the convective velocity term, I get the following:
\begin{align}
\nabla\cdot\left[\mathbf u\cdot\nabla\mathbf u\right]&=\frac{\partial}{\partial x_i}\left[u_j\frac{\partial u_i}{\partial x_j}\right]\\
&=\frac{\partial u_j}{\partial x_i}\frac{\partial u_i}{\partial x_i}+u_j\frac{\partial^2u_i}{\partial x_j\partial x_i} \\
&=\mathbf u\cdot\nabla q+\left(\nabla\mathbf u\right)\cdot\left(\nabla\mathbf u\right)^T
\end{align}
where $q=\nabla\cdot\mathbf u$.
I know the first term on the right hand side represents the convective term for the dilatation component of the velocity field (from Helmholtz decomposition), but I can't quite get the physical meaning of the second term. The gradient of velocity is a 2nd order tensor, but what is the physical meaning of the product of a second order tensor with its transpose? Is there a way to manipulate it to get a better physical meaning out of it?
Best Answer
The term in equation is:
$$\frac{\partial u_i}{\partial x_j}\frac{\partial u_j}{\partial x_i}$$
So let's take a step back and think about what kinds of terms can appear in conservation equations. There can be a production term, a transport term, and a dissipation term. The transport term is the $\vec{u}\cdot\nabla q$ term that you noted. When you look at the full coupled set of equations (vorticity and dilatation conservation equations), there are some production and dissipation terms that transfer dilational velocity into vorticity and vice-versa.
Now, I'm unfamiliar with the decomposition here specifically. However, looking at some other equations which I am familiar with (turbulent kinetic energy), I will go out on a limb and say that that term is a dissipation term. In all the conservation laws I have seen, terms that look like the term in question are dissipation terms -- this goes to answer your question about how to think about terms like this in general.
This hypothesis seems to be backed up by a few papers I've found and scanned quickly, and this thesis in Eq 2.14d which lumps the term in question into a viscous dissipation term.
My vote -- it's a dissipation of dilatation.