Hydraulic Jump Sequent Depth
Calculate the sequent (conjugate) depth downstream of a hydraulic jump, y₂ = (y₁/2)·(√(1 + 8·Fr₁²) − 1), from the upstream depth y₁ (supercritical) and the incoming Froude number Fr₁. The hydraulic jump is the abrupt transition from fast, shallow (supercritical) to slow, deep (subcritical) flow, with strong turbulence and energy dissipation. This Bélanger equation, from momentum conservation, is the basis for designing stilling basins downstream of spillways and gates: water descending a spillway arrives at very high (supercritical) velocity and must be decelerated before returning to the river, otherwise it erodes the bed catastrophically. The sequent depth y₂ sets the required basin depth for a stable jump. Enter the upstream depth and the Froude number.
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Sequent depth of a hydraulic jump
The hydraulic jump is one of the most spectacular transitions in hydraulics: the flow coming down a spillway arrives fast and shallow (supercritical regime, Fr > 1) and, on meeting the slower water of the river below, rises abruptly and turbulently into a subcritical regime (slow and deep, Fr < 1). The depth downstream of the jump, called the sequent depth or conjugate depth, is y₂ = (y₁/2)·(√(1 + 8·Fr₁²) − 1), derived from conservation of momentum (the Bélanger equation), starting from the upstream depth y₁ and the incoming Froude number Fr₁. This equation is the basis for designing stilling basins downstream of spillways and gates. Water coming off a spillway carries enormous energy — enough to scour the river bed and undermine the foundation of the dam itself. The stilling basin forces the jump to occur inside a reinforced concrete structure, and the sequent depth y₂ sets the depth that basin needs (and the required submergence) for the jump to sit stable and in the right position. If y₂ is not met, the jump is 'swept out' downstream, carrying erosive turbulence onto the unprotected bed. Enter the upstream depth and the Froude number.
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Hydraulic Jump Length
Estimate a hydraulic jump's length, L ≈ 6.9·(y₂ − y₁), by the classic empirical formula, from the upstream y₁ and downstream y₂ sequent depths. Unlike the sequent depths (from momentum), jump length is empirical, from lab tests, since the jump has no mathematically sharp end — its length is the distance from the upstream face to where the surface stabilizes. Several formulas exist (Smetana ≈ 6(y₂−y₁), USBR vs Fr, Elevatorski ≈ 6.9(y₂−y₁)); all give the order of magnitude. Jump length sets the stilling basin size downstream of a spillway: the basin must be long enough to contain the whole jump so dissipation completes within the concrete-lined structure before water returns to the natural bed. Undersizing throws the still-erosive jump tail onto the unprotected bed. Enter the upstream and downstream sequent depths.
Hydraulic Jump Energy Loss
Calculate the specific energy dissipated in a hydraulic jump, ΔE = (y₂ − y₁)³ ÷ (4·y₁·y₂), from the upstream y₁ (supercritical) and downstream y₂ (subcritical) sequent depths. The hydraulic jump is one of the most efficient energy dissipators in hydraulics: intense turbulence in the transition converts kinetic energy to heat and sound, removing excess flow energy. This head loss ΔE is exactly what is sought downstream of spillways, gates and bottom outlets — water arrives with very high energy (able to scour the riverbed and undermine the structure), and the stilling basin induces the jump to 'burn' that energy in a controlled way. The higher the incoming Froude number, the greater the dissipated fraction — jumps with Fr > 9 dissipate up to 85%. Enter the upstream and downstream sequent depths.
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The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.