First of all, make sure you understand the three rules for water and the three rules for pumps on Kanddak's
Hydrodynamics Education post. I refer to the rules for non-pumped water as W-1, W-2, and W-3; the rules for pumped water are similarly P-1, P-2, and P-3.
The designs I've come up with (and
Grimlocke's design, I think) take advantage of the restriction on diagonal movement in rules W-2 and P-2 as well as the fact that water moving according to rule W-3 seems to create flow in the source tile almost all the time. Here is my hypothesis regarding the process used in my designs:
- On manual startup, the pump moves the 7/7 water in its input tile on the lower level to its output tile on the upper level. (Rule P-1)
- The water in the output tile cannot move under pressure because there are no orthogonally-connected empty tiles on the lower level (Rule W-2), so it stays in place. There is currently no water in the pump's input tile, so the pump does nothing.
- At some point, the water in one of the tiles under the waterwheel spreads out diagonally into the pump's input tile (Rule W-3). This creates flow in the source tile, which appears to persist for an unknown number of ticks.
- The pump now collects the water from the input tile and moves it according to Rule P-2. The output tile is full, so the water is teleported through the 7/7 tiles until it reaches the source tile from step 3. This refills the source tile back to 7/7.
- The process repeats from step 3. There is now 7/7 water with flow beneath the waterwheel, so it powers the pump from this point on.
I believe that steps 3 and 4 happen in that order during a single tick, which is why no water appears in the input tile. A pressure plate in the input tile never stops detecting water, which supports this hypothesis.
Furthermore, I suspect that using a single diagonally-connected tile per waterwheel fails because during the tick when the water in that tile spreads out, the water in the tile under the waterwheel is at 3/7 when the waterwheel performs its checks.