OK, I can't stand it any more so here are some suggestions.
gbam wrote:
"OK -- An invert is the lowest point of a conveyance. This can be a channel or pipe."
This is, I believe, what most engineers understand by the word invert

. The definition offered by PELS is correct if you are using invert as a verb.
gbam also wrote:
"A flow line is similar. When used together to define a culvert, the flowline is the top of dirt and the invert is the bottom of culvert."
This simply confuses the issue, I believe. The word flowline is never, in my experience, used to refer to the dirt over the top of a culvert or any other conveyance structure.
dicksewerat wrote:
"I thought the flow line was the elevation of the water surface as the water moved down the pipe. "
This is also confusing. I have never heard the word flowline used this way. In a pipe or culvert with a free water surface the highest elevation to which the water may rise may be called any of the following: " water surface elevation" sometimes the "hydraulic grade line" or just "water surface".
Flowline may be used to refer to the low point in a pipe, culvert, ditch, stream or other structure that conveys water, WHEN VIEWED IN PROFILE. That term may sometimes be used to refer to the thread of a stream WHEN VIEWED IN PLAN. This use is synonymous to the German word "thalweg".
When words are our only tools it seems important to agree on their meaning. Fortunately, engineers have other tools, such as pictures, drawings and numbers and symbols.