Pick a scenario; every control below stays editable.
Typical absolute roughness, not a measurement of any historic conduit.
Gold: energy line (head minus friction and minor losses). Blue: hydraulic grade line (pressure + elevation). Grey: pipe elevation. Where blue dips below grey the pressure is below atmospheric: air and leaks get sucked in.
The science
Energy balance. Between the cistern surface (head H above the nozzle) and the nozzle exit: H = (1+Kn) Vj²/2g + (Kin + K + f L/D) Vp²/2g. Mass conservation fixes Vp D² = n Vj d². Without losses this is Torricelli: Vj = sqrt(2 g H) and a vertical jet could reach H.
Friction. Darcy-Weisbach with Re = Vp D / ν. Laminar f = 64/Re; turbulent flow uses the Colebrook-White equation, iterated; the linear blend between Re 2300 and 4000 is a simplification. Roughness values are typical figures for each material.
The jet. Droplets leave the nozzle at Vj and decelerate under gravity and quadratic air drag, a = -g - k |v| v, k = (3/4)(ρair/ρw) Cd / ddrop with Cd = 0.45. The rise is ln(1 + k v²/g) / (2k). The effective droplet size is an illustrative knob: real columns break up unevenly.
Maximum power. For a long pipe the jet power ½ ρ Q Vj² peaks when the friction loss is about one third of the head; the page finds this numerically.
The cistern. A dh/dt = Qin - Qout(H). If the spring cannot fill it, the level falls to the inlet and then the delivered head sinks until Qout = Qin. Mill power is η ρ g Q hw with η = 0.65 (typical overshot wheel) and a 3 m wheel drop.
Honest simplifications. Steady flow in a pipe of uniform slope from the cistern floor to the nozzle; no air locks, water hammer, valves or pipe corrosion; nozzle coefficient 0.98; the pipe is drawn thicker and its route serpentines on the map only to show its length; hill, castle and town are an invented scene, not a real place. No historical claim is made about any particular water system.