Fluid mechanics

A fountain is a coupled flow system: the pump adds head, the piping dissipates head, the nozzle converts pressure and elevation into velocity, and the free jet converts velocity back into height. A useful model keeps those conversions in one balance.

Continuity fixes the speed scale

For a steady incompressible stream, mass entering a control volume equals mass leaving it:

ρA₁v₁ = ρA₂v₂.

The density cancels, leaving Q = Av. A 40 mm pipe carrying water at 2 m/s has Q = π(0.04²)/4 · 2 = 0.00251 m³/s. A 20 mm nozzle has one quarter the area, so its mean speed is 8 m/s. The smaller opening does not create flow; it converts the same flow into a larger velocity and usually adds loss.

Energy identifies what the pump must supply

Start with the ideal parcel energy balance. Pressure work per volume is p, kinetic energy per volume is ½ρv², and gravitational energy per volume is ρgz. Along a streamline,

p₁ + ½ρv₁² + ρgz₁ = p₂ + ½ρv₂² + ρgz₂.

A real pump and real pipe require the extended form

p₁/ρg + v₁²/2g + z₁ + h_p = p₂/ρg + v₂²/2g + z₂ + h_L.

Here h_p is pump head added and h_L is head lost to friction, fittings, filters, and the nozzle. Every term is a length in metres of fluid.

A jet-height calculation

Suppose the reservoir surface and nozzle are both at atmospheric pressure, the reservoir speed is negligible, elevation rise is 3 m, and measured losses are 0.8 m. The available velocity head is H = 3 − 0.8 = 2.2 m, so

v = √(2gH) = √(2 · 9.81 · 2.2) = 6.57 m/s.

If the jet rises freely and air drag is small, its ideal additional height is v²/(2g) = 2.2 m. A measured height much lower than that points to a wrong flow estimate, extra loss, wind, or a nozzle coefficient below one.

Momentum locates structural load

Energy predicts speed; momentum predicts reaction. For a 90° bend, use R + F_pressure + W = ṁ(v_out − v_in). Do not infer pipe load from pump watts: equal power can mean high flow at low speed or low flow at high speed, while reaction depends on ṁΔv.

Calculation

Record basin level, Q, nozzle diameter, pressure immediately upstream of the nozzle, elevation, water temperature, and electrical real power. Compute continuity speed, Reynolds number Re = ρvD/μ, extended-Bernoulli head, and the bend reaction. Mark each term as measured, estimated, or neglected; that classification is part of the physics.