Bélanger’s hydraulic jump (1828)

Water shooting fast and shallow from under a sluice gate cannot slow down smoothly: it rises abruptly, in a turbulent roller, to a deeper, slower stream. Jean-Baptiste Bélanger studied this hydraulic jump in his 1828 essay on steady flow in open channels. The rule used today, that the pressure force plus the momentum flux is the same on both sides, gives the sequent depth \(y_2/y_1 = (\sqrt{1 + 8\mathrm{Fr}_1^2} - 1)/2\) (sequent_depth()). The energy balance cannot be used across the jump, because the roller dissipates energy (hydraulic_jump_head_loss()). This example finds the jump below a sluice gate and how much of the stream’s energy it destroys.

import matplotlib.pyplot as plt
import numpy as np

from engineeringkit.fluid_engineering import G, froude_number, hydraulic_jump_head_loss, sequent_depth, specific_energy

The jump below a sluice gate

A gate releases 4 m²/s per metre of width at a depth of 0.4 m.

q, y1 = 4.0, 0.4
v1 = q / y1
y2 = sequent_depth(y1, v1)
loss = hydraulic_jump_head_loss(y1, y2)
E1 = specific_energy(y1, q)
print(f"Fr1 = {froude_number(v1, y1):.2f}: depth rises from {y1} m to {y2:.2f} m (Fr2 = {froude_number(q / y2, y2):.2f})")
print(f"head lost in the jump {loss:.2f} m, {100 * loss / E1:.0f}% of the upstream specific energy")

Fr1 = 5.05: depth rises from 0.4 m to 2.66 m (Fr2 = 0.29)
head lost in the jump 2.72 m, 49% of the upstream specific energy

Momentum, not energy

The momentum function \(M = y^2/2 + q^2/(gy)\) is equal at the two depths, while the specific energy drops.

M1, M2 = y1**2 / 2 + q**2 / (G * y1), y2**2 / 2 + q**2 / (G * y2)
print(f"M1 = {M1:.3f} m², M2 = {M2:.3f} m²; E1 = {E1:.2f} m, E2 = {specific_energy(y2, q):.2f} m")

M1 = 4.159 m², M2 = 4.159 m²; E1 = 5.50 m, E2 = 2.78 m

Stronger jumps

Stilling basins below spillways are designed around the upstream Froude number: a weak undular jump near 1 loses little energy, a steady jump at 4.5 to 9 loses most of it.

Fr1 = np.linspace(1, 10, 100)
ratio = sequent_depth(1.0, Fr1 * np.sqrt(G))
loss_fraction = hydraulic_jump_head_loss(1.0, ratio) / specific_energy(1.0, Fr1 * np.sqrt(G))
fig, ax = plt.subplots(figsize=(6, 4))
ax.plot(Fr1, ratio, label="depth ratio $y_2/y_1$")
ax.plot(Fr1, 10 * loss_fraction, label="energy lost × 10")
ax.axvspan(4.5, 9, color="0.9", zorder=0, label="steady jump")
ax.set(xlabel="upstream Froude number $Fr_1$", title="Bélanger's hydraulic jump")
ax.legend()
fig.tight_layout()

assert np.isclose(M1, M2)
assert np.isclose(loss, E1 - specific_energy(y2, q))
assert froude_number(q / y2, y2) < 1 < froude_number(v1, y1)
assert np.isclose(ratio[0], 1.0) and loss_fraction[0] == 0

plt.show()