About Hydraulic Jump Calculator
The hydraulic jump calculator analyses the abrupt transition from supercritical to subcritical flow in a horizontal, rectangular open channel. Enter the upstream (supercritical) depth y1 together with either the upstream velocity v1 or the flow rate Q and channel width b, and the tool returns the upstream Froude number, the sequent (conjugate) depth y2, the downstream velocity and Froude number, the energy dissipated, and an approximate jump length.
The sequent depth follows from the momentum (Belanger) equation y2/y1 = 0.5*(sqrt(1 + 8*Fr1^2) - 1), and the specific-energy loss from the algebraic identity dE = (y2 - y1)^3 / (4*y1*y2). The jump is classified by its upstream Froude number into undular, weak, oscillating, steady, or strong regimes, and the upstream and downstream states are plotted on the specific-energy diagram.
How It Works
- Choose how to supply the upstream flow: either an upstream depth y1 and velocity v1, or a flow rate Q with the channel width b (the calculator then derives v1 = Q / (b*y1)).
- It computes the upstream Froude number Fr1 = v1 / sqrt(g*y1). A jump only forms when the upstream flow is supercritical (Fr1 > 1); subcritical input is flagged with no jump.
- The sequent depth is found from the momentum equation y2/y1 = 0.5*(sqrt(1 + 8*Fr1^2) - 1), and the downstream velocity from continuity v2 = v1*y1/y2.
- It evaluates the specific energy E = y + v^2/(2g) upstream and downstream, the energy loss dE = (y2-y1)^3/(4*y1*y2), the efficiency E2/E1, the approximate length L ~ 6*y2, and the USBR jump type, then plots the specific-energy curve.
Worked Example
Supercritical flow enters with depth y1 = 0.5 m and velocity v1 = 6.644 m/s. The upstream Froude number is Fr1 = 6.644 / sqrt(9.81*0.5) = 3.0. The sequent depth ratio is y2/y1 = 0.5*(sqrt(1 + 8*3^2) - 1) = 0.5*(sqrt(73) - 1) = 0.5*(8.544 - 1) = 3.772, so y2 = 3.772*0.5 = 1.886 m. By continuity v2 = v1*y1/y2 = 6.644*0.5/1.886 = 1.761 m/s, and Fr2 = 1.761/sqrt(9.81*1.886) = 0.41 (subcritical). The energy loss is dE = (1.886 - 0.5)^3 / (4*0.5*1.886) = 2.663/3.772 = 0.706 m. With E1 = 0.5 + 6.644^2/(2*9.81) = 2.750 m, the jump dissipates 0.706/2.750 = 25.7% of the upstream energy. The approximate jump length is L = 6*1.886 = 11.3 m, and with Fr1 = 3.0 the jump is classified as oscillating.
Formulas
- Upstream Froude number
Fr1 = v1 / sqrt(g * y1)- Sequent (conjugate) depth ratio
y2 / y1 = 0.5 * (sqrt(1 + 8*Fr1^2) - 1)- Continuity (per unit width)
v2 = v1 * y1 / y2- Specific energy
E = y + v^2 / (2 g)- Energy loss across the jump
dE = (y2 - y1)^3 / (4 * y1 * y2)- Jump length and classification
L ~ 6*y2; Fr1: 1-1.7 undular, 1.7-2.5 weak, 2.5-4.5 oscillating, 4.5-9 steady, >9 strong
Standards & References
- Ven Te Chow, Open-Channel Hydraulics (1959)
- Belanger / momentum equation for the sequent depth
- USBR Engineering Monograph 25 (jump classification & stilling basins)
Frequently Asked Questions
When does a hydraulic jump form?
A hydraulic jump forms only when the approaching flow is supercritical, i.e. the upstream Froude number Fr1 is greater than 1. The flow then transitions abruptly to a subcritical (deeper, slower) state. If Fr1 is less than or equal to 1 the flow is already subcritical or critical and no jump occurs, so the calculator flags this case.
What is the sequent (conjugate) depth?
The sequent depth y2 is the downstream subcritical depth that has the same momentum function as the upstream supercritical depth y1. For a rectangular channel it follows from the momentum equation y2/y1 = 0.5*(sqrt(1 + 8*Fr1^2) - 1). It is not the same as the alternate depth, which shares the same specific energy rather than momentum.
Why does a hydraulic jump dissipate energy?
The jump involves intense turbulence, air entrainment, and surface rollers that convert mechanical energy into heat. Because momentum (not energy) is conserved across the jump, there is a net specific-energy loss dE = (y2 - y1)^3 / (4*y1*y2). This energy dissipation is exactly why jumps are used in stilling basins below spillways and gates to protect the downstream channel from erosion.
How accurate is the jump length L = 6*y2?
L = 6*y2 is a widely used engineering approximation for the length of a well-developed jump. The true length depends on the Froude number; USBR experimental curves give L/y2 ratios from about 4 to 6 over the common range Fr1 = 4.5 to 9 (steady jumps). Use L = 6*y2 for preliminary stilling-basin sizing and refine with USBR design charts for final design.