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Hydraulic Jump Conjugate Depth & Dissipation Calculator

A hydraulic jump is a rapid transition in an open channel from high-velocity supercritical flow (Fr > 1) to tranquil subcritical flow (Fr < 1).

Water depth upstream of the jump.

Flow velocity upstream of the jump.

Width of the rectangular stilling basin.

Calculated Result
2.86 m

Sequent Subcritical Depth (y₂)

Upstream Froude Number (Fr₁)

3.71

Hydraulic Head Loss (ΔE)

1.69 m

Energy Dissipation Efficiency

35.6%

Power Dissipated

446.4 kW

Estimated Jump Stilling Length

17.2 m

Downstream Velocity (V₂)

1.89 m/s

Calculation Breakdown

  1. Inflow Froude NumberFr₁ = V₁ / √(g·y₁) = 9 / √(9.81 · 0.6) = 3.71 (Supercritical)
  2. Bélanger Conjugate Depth Equationy₂ = (y₁/2) · [√(1 + 8·Fr₁²) - 1] = (0.6/2) · [√(1 + 8·3.71²) - 1] = 2.86 m
  3. Energy Dissipation EvaluationΔE = (y₂ - y₁)³ / (4·y₁·y₂) = 1.69 m (35.6% of total energy dissipated)

What Is the Hydraulic Jump Conjugate Depth & Dissipation Calculator?

A hydraulic jump occurs when open channel flow abruptly changes from supercritical (Fr > 1) to subcritical (Fr < 1).

The momentum equation balances the change in hydrostatic pressure against momentum flux across the roller.

How Does the Hydraulic Jump Conjugate Depth & Dissipation Calculator Work?

The Bélanger equation relates initial depth y1 and Froude number Fr1 to downstream sequent depth y2.

Head loss ΔE converts enormous kinetic energy into thermal dissipation and acoustic noise.

Hydraulic Jump Conjugate Depth & Dissipation Calculator Formula & Variables

The core mathematical equation utilized by this calculator is expressed as:

y_2 = \frac{y_1}{2} \left[ \sqrt{1 + 8 Fr_1^2} - 1 \right], \quad \Delta E = \frac{(y_2 - y_1)^3}{4 y_1 y_2}, \quad Fr_1 = \frac{V_1}{\sqrt{g y_1}}

Classic Bélanger conjugate depth and energy dissipation formulas for rectangular channels.

How to Use the Hydraulic Jump Conjugate Depth & Dissipation Calculator

  1. Input upstream depth, velocity, and channel width.
  2. Review sequent depth, energy dissipation percentage, and required stilling basin length.

Step-by-Step Example Calculation

Spillway Chute Stilling Basin

Input Values:

initialDepthMeters:0.5
upstreamVelocityMPerS:10
channelWidthMeters:6
Worked Steps: Calculates sequent depth y2 ~ 2.8m and over 60% energy dissipation.

Understanding Your Result

Froude numbers between 4.5 and 9.0 produce a well-established steady jump with 45–70% energy dissipation.

Factors That Affect the Result

  • Higher inflow Froude numbers result in higher sequent depths and greater relative energy dissipation.

When Should You Use This Calculator?

  • Dam spillway design, stilling basin sizing, irrigation drop structures, and floodway channels.

Assumptions & Limitations

  • Assumes a horizontal, prismatic rectangular channel with negligible boundary friction over the jump length.

Frequently Asked Questions

Calculation Accuracy & Reference Note

Standard USBR (United States Bureau of Reclamation) hydraulic design formulation.

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