Aqueduct and Inverted Siphon

What is modelled

The open channel is rectangular, of width b, with bed slope S and Manning roughness n. For a given water depth y it carries a steady uniform flow. At the valley the channel feeds an inverted siphon, a closed pipe of diameter D and length L_p running down and up a valley of depth H. The head available is the level drop between the two basins; friction and entry losses use it up, and the pressure at the valley floor follows from the hydraulic grade line.

Equations

• Manning: Q = (1/n) A R2/3 S1/2, with A = b y and R = A/(b + 2y).
• Critical depth yc = (Q²/(g b²))1/3; Froude number Fr = v/√(g y).
• Siphon: Δh = (f Lp/D + ΣK) v²/(2g), with the Colebrook equation for f.
• Pressure head at the valley floor is the elevation of the hydraulic grade line above the pipe, p/ρg = HGL − z.

Checks run in the page

• Normal depth found by bisection reproduces the requested flow.
• Critical depth gives a Froude number of 1.
• Swamee-Jain friction factor agrees with the iterated Colebrook value.
• The siphon flow solved from the head balance gives back the available head.

Honest simplifications

Steady uniform flow, a rectangular channel, a single equivalent pipe, fixed minor-loss coefficients and no air entrainment or transients such as water hammer. Roman engineers used trial and error, not these equations; the numbers show how the physics limits what they could build.

• Choose Gentle channel and compare its flow with the Steep channel preset.
• Choose Deep valley with a thin pipe and watch the pressure approach the pipe limit.
• Enlarge the pipe and see the siphon flow rise much faster than the diameter.