The `u dot del u` term is describing the "self-advection" of the velocity field.
You can interpret `u dot del u` as `(u dot del)u` where `(u dot del)` is an operator which seems to sometimes be called the "advection operator" or the "directional derivative" or the "convection derivative" and probably even more names I don't even know -- which makes it really annoyingly hard to look up information about.
`(u dot del)` expands to `u_x * d(__)/dx + u_y * d(__)/dy` in 2D -- a function taking a scalar (plugged into where I wrote the `__`s) and which evaluates to a scalar. As indicated on the 2nd link above, to operate on a vector just apply it to each component (similar to how you'd normally multiply a scalar and a vector). So that means that `(u dot del)u` becomes:
You can interpret `u dot del u` as `(u dot del)u` where `(u dot del)` is an operator which seems to sometimes be called the "advection operator" or the "directional derivative" or the "convection derivative" and probably even more names I don't even know -- which makes it really annoyingly hard to look up information about.
See https://en.wikipedia.org/wiki/Advection and https://en.wikipedia.org/wiki/Del (section "directional derivative").
`(u dot del)` expands to `u_x * d(__)/dx + u_y * d(__)/dy` in 2D -- a function taking a scalar (plugged into where I wrote the `__`s) and which evaluates to a scalar. As indicated on the 2nd link above, to operate on a vector just apply it to each component (similar to how you'd normally multiply a scalar and a vector). So that means that `(u dot del)u` becomes:
which can be factored into a matrix-vector multiplication (by pulling out the `u_x * ___ + u_y * ___` part into a vector): As you can see, that's what was written in the article. I'm not sure why the author wrote it out with a matrix like this.As DanWaterworth said in the other comment, `del p` is just the gradient of the pressure.