When developing a finite-volume discretization of the 1-D advection equation: $$ \frac{\partial \phi }{\partial t} + u\frac{\partial \phi}{\partial x} = 0$$
Where $u$ is the advection velocity in m/s and $\phi$ is a scalar.
It is required to write this in conservation form i.e. $$ \frac{\partial \phi }{\partial t} + \frac{\partial F}{\partial x} = 0$$
Where F($\phi$) represents some flux function. It is of course necessary to define a flux function F($\phi$) such that $\frac{\partial F}{\partial x} = u\frac{\partial \phi}{\partial x}$.
My question is... how exactly does integrating the 2nd term with respect to $x$ over a single cell (with the result being $u\phi$ evaluated between cell faces) yield a net flux? The physical significance of this is difficult for me to grasp... units of $\phi*[L/T]$ doesn't seem very intuitive and I'm having a very hard time conceptualizing this step and its significance.
Its sounds fine to say that, per the first form of the equation, the time rate of change of $\phi$ at a point in the flow field (in this case a 1-D grid) is balanced by the rate of advection at that point. It is much harder to visualize or understand how the time rate of change of $\phi$ is balanced by a flux gradient, or, like I mentioned earlier, understand the seemingly odd units of the net flux. Does anybody have any tips to help me visualize\understand this?
