Energy Equation Lab

Follow the energy balance from pressurized flow in a pipe, through an atmospheric free jet, into a receiving pool and onward through an open flume.

About this model

Water often passes between pressurized pipes, free jets and open channels at outlets, culverts, spillways, hydropower structures and bypass systems. The same energy balance connects these regimes even though pressure, water depth and velocity are represented differently along the way.

Water level alone does not show how much mechanical energy the flow carries. Part of the energy may be stored as pressure, while fast flow can carry a large kinetic contribution v²/2g. The HGL therefore lies one velocity head below the EGL. A contraction can lower pressure while leaving total head almost unchanged, whereas friction, jet impact and mixing actually remove mechanical energy and make the EGL fall.

The elevation term z is always measured from a fixed, arbitrary datum. The reference point used for z₂ depends on the regime:

the pipe centreline in pressurized flow;
the jet centreline in the free jet;
the channel bed in free-surface flow.

The atmospheric jet is especially revealing because its gauge-pressure term disappears while its kinetic energy remains. When the jet enters the pool, much of that kinetic energy is dissipated by turbulence and mixing rather than simply reappearing as water depth. The mean velocity head in the broad pool is negligible; farther downstream, the water accelerates again along the sloping flume.

Scientific basis

Section 1 is fixed in the upstream pipe and section 2 is movable. Discharge is solved from the prescribed total head H₁, pipe geometry and distributed pipe loss. The jet follows a ballistic trajectory at atmospheric pressure with negligible air resistance. At impact, the difference between the jet EGL and the downstream free-surface EGL is treated as mixing loss. A downstream control section closes the free-surface boundary condition and the bed then slopes away. The friction control is deliberately exaggerated for comparison: zero gives an idealized lossless reference, while larger values make distributed losses visually clear rather than representing a calibrated roughness or Darcy friction factor.

\[ H_1 = z_2 + \text{(depth or pressure term)} + \frac{v_2^2}{2g} + h_L \]

White, F. M. (2016). Fluid Mechanics, 8th ed. McGraw-Hill.