Physical Meaning of Divergence of Convective Velocity Term 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?
 A: 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.
A: We may be able to find the meaning of this term by considering an easier problem, an incompressible fluid. Taking the divergence of the Navier-Stokes equation in this case yields:
$$\nabla^2p=-\rho\left(\nabla\mathbf u\right)\cdot\left(\nabla\mathbf u\right)^T$$
We can see how the term in question is directly related to the Laplacian of the pressure field. Since this term exists in an incompressible flow, we can say it has physical meaning beyond or even different than dilation.
If we think of a Stokes flow, where this term is considered negligible due to the dominance of viscous terms, the nullity of this term serves to highlight the fact that there is no source or sink of convective acceleration due to pressure. 
In a case where this is non-zero like in a turbulent flow (high Reynolds number), first: this tells us about the non-locality property of the pressure field (which you could see if you think of integrating the equality shown above). Second: It tells us about how pressure acts as a source or sink of of a fluid not by dilating or contracting it but by the purely non-linear nature of turbulence. The reason behind that is the fact that the term in question comes in by taking the divergence of the convective term of acceleration in the NS equation. The divergence of the acceleration field at a point, if non-zero, indicated the presence of a source or sink of acceleration.
A: I will start off by apologizing for being unfamiliar with the specifics of this problem, and the notation commonly used. I will therefore use notation and terminology that I am used to; we can decompose the gradient of the velocity field as
$$
\partial_ju_i = \omega_{ij} + \sigma_{ij} + \frac{1}{3}\delta_{ij}\theta,
$$
where $\omega_{ij} = \partial_{[j}u_{i]}$ is the anti-symmetric part (vorticity), $\sigma_{ij} = \partial_{(j}u_{i)} - \delta_{ij}\theta/3$ is the symmetric and trace-free part (shear), and $\theta = \partial_iu_i$ is the trace/divergence (expansion parameter, $q$ in your notation). Then the term
$$
\partial_iu_j\partial_ju_i = \sigma^2 + \frac{1}{3}\theta^2 - \omega^2,
$$
where $\sigma^2 = \sigma_{ij}\sigma_{ij}$, and $\omega^2 = \omega_{ij}\omega_{ij}$. 
Since the other term gives the change of the expansion parameter along the fluid flow, we can make the interpretation (by moving the squared terms to the other side of the equation) that non-zero shear and expansion serves to reduce expansion along the fluid flow, while non-zero vorticity serves to increase expansion along the fluid flow. 
