Question 1

What is the conjugate depth equation?

Accepted Answer

The conjugate (or sequent) depth y₂ downstream of a hydraulic jump is given by the Bélanger equation: y₂/y₁ = ½·(√(1 + 8·Fr₁²) − 1), derived from momentum conservation across the jump on a horizontal floor. The ratio depends only on the upstream Froude number Fr₁ = V₁/√(g·y₁).

Question 2

What's a USBR Type II vs Type III stilling basin?

Accepted Answer

USBR Type II is for high-velocity, high-discharge spillways (V₁ > 60 ft/s or q > 200 cfs/ft) and uses chute blocks plus a dentated end sill, with basin length L = 4.3·y₂. Type III is for lower velocities (V₁ < 60 ft/s and q < 200 cfs/ft) and adds baffle blocks in the floor, allowing a shorter basin of L = 2.7·y₂. Type IV is for the oscillating-jump range (Fr₁ = 2.5–4.5) and uses very large chute blocks plus an end sill.

Question 3

Why does the jump 'sweep out' if tailwater is too low?

Accepted Answer

The hydraulic jump location is set by where supercritical flow at depth y₁ meets subcritical flow at the conjugate depth y₂. If actual tailwater is below y₂, the jump moves downstream looking for the right depth, eventually leaving the apron entirely — the supercritical flow continues unattenuated into the natural channel and erodes it. The fix is either lowering the basin floor (depressing the jump) or adding baffle blocks to artificially raise the conjugate depth requirement.

Question 4

What jump regime corresponds to each Froude number?

Accepted Answer

Fr₁ < 1: subcritical, no jump. 1 < Fr₁ < 1.7: undulating jump, smooth surface, no real basin needed. 1.7–2.5: weak jump, low energy loss. 2.5–4.5: oscillating jump, surface waves propagate downstream — this is the hardest regime to manage. 4.5–9.0: steady jump, well-defined roller, ~70% energy loss. > 9: strong jump, ~85% energy loss but rough surface and high splash.

Question 5

How is jump energy loss computed?

Accepted Answer

ΔE = E₁ − E₂ = (y₂ − y₁)³ / (4·y₁·y₂), in units of head. Energy loss as a fraction of incoming specific energy ranges from a few percent at Fr₁ = 1.7 to about 85% at Fr₁ = 20. Higher Fr₁ means a stronger jump and more energy dissipated within the basin.

Question 6

Is the conjugate-depth equation valid on a sloped floor?

Accepted Answer

The Bélanger equation assumes a horizontal floor. For sloped floors (chute spillways), the weight component of the water in the jump enters the momentum balance — y₂/y₁ depends on slope as well as Fr₁. USBR Engineering Monograph 25 (Peterka 1958) gives charts for sloped-floor jumps. For preliminary sizing, treat the floor as horizontal at the toe and apply this calculator.

Fr₁	Regime	Description	Basin
< 1.0	Subcritical	No jump possible	—
1.0–1.7	Undular	Standing waves, smooth surface, ~5% loss	None required
1.7–2.5	Weak	Small surface roller, ~5–15% loss	None / simple apron
2.5–4.5	Oscillating	Wave train propagates downstream, scour-prone	USBR Type IV
4.5–9.0	Steady	Stable roller, ~45–70% loss, best regime	USBR Type III (or II)
> 9.0	Strong	~70–85% loss but rough, splashy	USBR Type II

Hydraulic Jump & Stilling Basin Calculator

Jump regimes vs Froude number

USBR basin selection rules

Worked example

Example — toe of a chute spillway

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