Hydraulic Jump Calculator - Fr1 to y2, head loss, length, and efficiency

A hydraulic jump calculator turns four open-channel inputs (g, B, Q, y1) into the full set of jump properties: y2, upstream and downstream Froude numbers, head loss, jump length, jump height, efficiency, and jump type. A hydraulic jump is the abrupt transition from supercritical to subcritical flow in a rectangular open channel and the standard energy dissipator on spillways.

• • Stilling-basin design check: Set basin length to L and tailwater to y2 so the jump stays inside the basin.
• • Spillway and chute toe sizing: Use y2 and Delta E to size the apron and stilling wall.
• • Grade-control structure verification: Confirm a planned drop produces Fr1 in the steady range so it dissipates energy without oscillating waves reaching downstream property.
• • Classroom and homework problems: Plug in g, B, Q, and y1 from a textbook problem and read off the conjugate depth, Froude numbers, head loss, and jump type.

Hydraulic jumps are governed by conservation of mass and momentum across a short control volume, so the upstream and downstream conditions are coupled through the Belanger momentum equation rather than friction alone. The same momentum balance gives the calculator its most useful result, y2.

Like the Hydraulic Gradient Calculator, this calculator takes two physical quantities and returns a derived ratio that drives the rest of the open-channel design.

The calculator chains the upstream Froude number from continuity and the wave celerity, then applies the Belanger momentum equation to get the conjugate depth ratio, then derives everything else from those two numbers.

• g: Gravitational acceleration in m/s^2; default 9.81 for sea-level Earth.
• B: Width of the rectangular channel in metres; same B applies upstream and downstream.
• Q: Volumetric discharge in m^3/s, from the spillway or chute design value.
• y1: Upstream supercritical depth in metres, taken at the spillway toe or chute end.
• Fr1: Upstream Froude number Fr1 = v1 / sqrt(g * y1); the dimensionless number that controls every other jump property.

After the Belanger equation gives y2 / y1, downstream velocity v2 follows from continuity, Fr2 = v2 / sqrt(g * y2), and Delta E = (y2 - y1)^3 / (4 * y1 * y2). L = 220 * y1 * tanh((Fr1 - 1) / 22), h = y2 - y1, and eta end the chain.

According to the FHWA HDS-4, Introduction to Highway Hydraulics, Fr1 = v1 / sqrt(g * y1) classifies the inflow regime, the Belanger momentum equation sets the conjugate depth ratio y2 / y1 = 0.5 * (sqrt(1 + 8 * Fr1^2) - 1) in a rectangular channel, and the head loss is Delta E = (y2 - y1)^3 / (4 * y1 * y2).

According to the FHWA HEC-14, Hydraulic Design of Energy Dissipators for Culverts and Channels, Fr1 selects the jump form from the USBR hydraulic-jump classification reproduced in Figure 6.2 (citing USBR 1987, Peterka): oscillating between 2.5 and 4.5, stable and well-balanced between 4.5 and 9.0 with 45 to 70 percent of upstream specific energy dissipated, and highly efficient above 9.0, and the stilling-basin length scales to L = 220 * y1 * tanh((Fr1 - 1) / 22).

Once L is in hand, the Broad Crested Weir Calculator applies the same critical-flow vocabulary to an overflow weir crest, where flow also transitions from subcritical to supercritical.

Four ideas explain what the calculator does with the four inputs and why it returns eleven outputs instead of just one number.

Fr1 sets the conjugate depth ratio, the ratio sets y2, and y2 sets the head loss, jump length, jump height, downstream velocity, and efficiency.

When the flow inside the jump is laminar or turbulent, the Reynolds Number & Flow Regime Calculator adds the viscous regime check the Froude number cannot give.

The form walks the four inputs in pairs, and the result panel returns eleven outputs so the user can sanity-check the inputs before reading the answer. Downstream depth comes first, then the Froude numbers, then the derived jump properties.

1. 1 Enter gravitational acceleration g: Default 9.81 m/s^2 for sea-level Earth; use 1.62 for the Moon or 3.71 for Mars.
2. 2 Enter the channel width B: Use the inside-of-the-walls width for the rectangular section at the jump.
3. 3 Enter the design discharge Q: Use the spillway or chute design discharge in cubic metres per second.
4. 4 Enter the upstream depth y1: Take y1 from the spillway toe or chute end. Subcritical incoming flow will be flagged.
5. 5 Read y2, Fr1, Fr2, and the jump type first: The primary output is y2; Fr1, Fr2, and the jump type confirm the inputs describe a real jump.
6. 6 Use Delta E, L, h, and eta to size the basin: Set the basin length to L, the basin depth to y2, and the energy-dissipation credit to (100 - eta) percent.

With g = 9.81, B = 2 m, Q = 10 m^3/s, y1 = 0.4 m, the form returns Fr1 = 6.31, steady jump, y2 = 3.375 m, Delta E = 4.877 m, L = 20.84 m, h = 2.975 m, eta = 41.69 percent.

When the stilling-basin flow is a pipe, the Bernoulli Equation Calculator applies the same specific-energy idea to a pressurised conduit.

Why a hydraulic jump calculator belongs next to the open-channel design spreadsheet.

• • Belanger conjugate depth in one step: Substitutes v1, Fr1, and the Belanger momentum equation into one calculation so the user does not hand-square Fr1 and hand-divide by 2.
• • Eleven outputs from four inputs: Returns y2, v1, Fr1, depthRatio, v2, Fr2, Delta E, L, h, eta, and the jump type, replacing a page of textbook algebra.
• • Jump type flag for designers: Marks undular, weak, oscillating, steady, strong, no-jump, and no-flow regimes so the designer can tell at a glance whether the basin will behave.
• • Energy-dissipation credit ready for the basin design: Returns Delta E in metres and eta as a percent for the spillway or chute energy balance.
• • Useful for coursework, design review, and field checks: Lets a student check a homework problem, a designer sanity-check a stilling-basin plan, and a reviewer verify a third-party calc.

Where the jump still leaves too much specific energy for the downstream channel, the Friction Factor Calculator picks up the residual head loss through pipe friction or channel roughness.

Five physical factors decide whether the output is a design number or a screening value.

• • The calculator assumes a horizontal, prismatic, rectangular channel with negligible friction across the jump. Sloped, rough, or strongly curved channels need a numerical model.
• • The jump length uses L = 220 * y1 * tanh((Fr1 - 1) / 22). Real jump lengths scatter by about plus or minus twenty percent around this fit.

Treat this as a screening tool, then move to a physical model or CFD run once basin length, baffle height, or apron depth depend on the numbers.

According to the FHWA HEC-14, Hydraulic Design of Energy Dissipators for Culverts and Channels, hydraulic jumps are the standard energy dissipator downstream of spillways and chutes, with the conjugate depth ratio set by the Belanger equation and the basin length scaled to L = 220 * y1 * tanh((Fr1 - 1) / 22) before adding baffle or chute blocks.

Where the jump sits over a porous foundation and seepage matters as much as surface dissipation, the Hydraulic Conductivity Calculator extends the same hydraulic vocabulary into the subsurface flow that the jump does not dissipate.