Hydraulic Jump — Physica

Equivalent forms

Fr_1 = \frac{v_1}{\sqrt{g\,y_1}}

y_2 = \tfrac{y_1}{2}\left(\sqrt{1+8Fr_1^2}-1\right)

\frac{\Delta E}{E_1} = \frac{(y_2-y_1)^3}{4 y_1 y_2}

Conserve momentum across a wall of water and the depth ratio falls out of a single square root.

Symbol	Name	SI unit	Dimension	Range
$y_1$	Upstream Depth Supercritical flow depth before the jump	m	L	0.001 – 2
$v_1$	Upstream Velocity Fast supercritical flow speed	m/s	L·T⁻¹	0 – 15
$Fr_1$	Upstream Froude Numberoutput Ratio of flow speed to shallow-water wave speed	dimensionless	1	0 – 12
$y_2$	Downstream Depthoutput Subcritical depth after the jump	m	L	0 – 5

Unit systems

Symbol	SI	CGS	Imperial
$y_1$	m	cm	ft
$v_1$	m/s	cm/s	ft/s
$Fr_1$	—	—	—
$y_2$	m	cm	ft

Where it holds

One-dimensional, rectangular-channel momentum balance assuming hydrostatic pressure on each side and negligible bed friction over the short jump. Real jumps add air entrainment and 3D structure but follow the depth ratio closely.

Dimensional analysis

$Fr_1 = v$ /\ $\sqrt{g y} = (L\cdot T^{-1})$ /\ $\sqrt{(L\cdot T^{-2})(L)} = (L\cdot T^{-1})/(L\cdot T^{-1}) = 1 \checkmark$ (dimensionless); y_2/y_1 is a pure ratio $\checkmark$

Discovery

Jean-Baptiste Bélanger / Giorgio Bidone · 1828

Bidone first documented the abrupt rise of water in open channels in the 1820s; Bélanger derived the momentum balance giving the conjugate-depth ratio. It is now the design backbone of spillway stilling basins worldwide.

Try this

The shimmering ring where tap water hits your sink is a shock wave — what sets its radius?

Fast, shallow water (depth 0.01 m) races along at 2 m/s before abruptly thickening. With upstream Froude number Fr₁ = v/√(g·y₁), find the downstream depth from the conjugate-depth relation.

Research status: stable

Real-world applications

Spillway and stilling-basin energy dissipators below dams
Tidal bores in rivers
Mixing and aeration in wastewater channels
The circular jump in a kitchen sink

Common misconceptions

Energy is conserved across the jump — momentum is, but a hydraulic jump deliberately destroys mechanical energy as turbulence and heat
Jumps can occur in subcritical flow — they require $Fr_{1}$ > 1 upstream
The kitchen-sink ring is unrelated — it is exactly a circular hydraulic jump

Experimental verification

Laboratory flumes reproduce the Bélanger ratio to within a few percent across

Fr_{1} = 1.5

–10; spillway stilling-basin models confirm the predicted energy dissipation used to protect dam foundations.

Derivation

Apply conservation of mass

(v_{1}y_{1} = v_{2}y_{2})

and momentum

(\tfrac{1}{2}\rho g y_{1}^{2} + \rho y_{1} v_{1}^{2} = \tfrac{1}{2}\rho g y_{2}^{2} + \rho y_{2} v_{2}^{2})

across the jump.

Eliminating

v_{2}

yields a quadratic in

y_{2}/y_{1}

whose positive root

is \tfrac{1}{2}(\sqrt{1+8Fr_{1}^{2}} - 1)

.

Limiting cases

Fr_1 \to 1⟶ y_2 \to y_1At critical flow there is no jump — the surface is smooth and depths are equal.

Fr_1 < 1⟶ No jumpSubcritical flow already 'feels' downstream conditions; a hydraulic jump only forms from supercritical flow.

Fr_1 \gg 1⟶ y_2/y_1 \approx \sqrt{2}\,Fr_1Strong jumps grow nearly linearly with Froude number and dissipate most of the incoming energy.

What if…

What if the upstream flow is subcritical

(Fr_{1}

< 1)?

The square-root term gives $y_{2}/y_{1}$ < 1, which is unphysical — no jump forms; the surface stays smooth.

What if you open the tap wider?

Higher $Fr_{1}$ pushes the circular jump outward and deepens the downstream pool — watch the bright ring expand.

1

Jump downstream of a fast sheet

Given ·

y 1:: 0.01
v 1:: 2

Find · y_2

Steps

Froude number: $Fr_{1} = v_{1}/\sqrt{g y_{1}} = 2/\sqrt{9.8\times 0.01} \approx 6.39$
Depth ratio: $y_{2}/y_{1} = \tfrac{1}{2}(\surd (1+8(6.39)^{2}) - 1) \approx 8.55$
$y_{2} \approx 8.55 \times 0.01 \approx 0.086\,\mathrm{m}$ — the sheet jumps $to \sim 8.5\times its$ depth

Result ·

Fr_{1} = 2/\sqrt{9.8\times 0.01} = 2/0.313 \approx 6.39

;

y_{2}/y_{1} = \tfrac{1}{2}(\sqrt{1+8\times 40.8}-1) \approx 8.55

, so

y_{2} \approx 0.0855\,\mathrm{m}

.

Froude Number→Bernoulli's Equation→Torricelli's Law→