Energy Balance Between Two Sections with Losses

Water flowing down a stream carries energy as elevation, depth of water and speed, and this page balances that energy between two cross sections, following the Natural Resources Conservation Service's National Engineering Handbook (Part 630, chapter 14, Figure 14-1). At a section the energy, as a head, is the channel bottom elevation $Z$ plus the depth $d$ plus the velocity head $\alpha V^{2}/2g$. Because $Z + d$ is the water surface elevation $W$, the page takes $W$ directly. The upstream section is numbered 2 and the downstream section 1, and the page keeps that order.

The coefficient $\alpha$ corrects the velocity head for the way the velocity varies across the section. It is 1.0 if the velocity is the same everywhere in the section and greater than 1.0 if it varies, as in a section with a channel and a floodplain. The loss has two parts. Friction is the friction slope $S_f$ times the reach length, with $S_f = (Q/K)^{2}$ and $K$ the mean of the conveyances of the two sections. Contraction or expansion is a coefficient times the change in velocity head, 0.1 in the handbook's example.

The handbook's Example 14-1 balances the energy of the Concho River between two sections 680 ft apart to estimate a flood discharge. At the trial discharge of 230,000 ft³/s that balances, the upstream water surface is 56.50 ft with $\alpha = 1.56$ and a velocity of 6.62 ft/s, and the downstream one is 55.98 ft with $\alpha = 1.47$ and 7.02 ft/s. The handbook's loss is 0.46 ft (friction 0.45 and contraction 0.01). The page gives 0.45666 ft: it uses standard gravity, 9.80665 m/s² (32.174 ft/s²), where the handbook uses 64.4 for $2g$, and the table prints the loss to 0.01 ft.

For a check by hand, take nearly still water, 10.5 m upstream and 10.0 m downstream: the velocity heads are negligible and the whole 0.5 m is loss (the page needs a non-zero velocity, so 0.001 m/s stands in for none). Velocity changes the split. With $\alpha = 1$ at both sections, a loss of 0.4 m and a downstream velocity of 3 m/s, the surface falls 0.5 m, 0.4 m of it is lost and the remaining 0.1 m appears as extra velocity head downstream, so the upstream velocity is 2.65304 m/s.