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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Altura conjugada do ressalto hidráulico
O ressalto hidráulico é uma das transições mais espetaculares da hidráulica: o escoamento que desce um vertedouro chega rápido e raso (regime supercrítico, Fr > 1) e, ao encontrar a água mais lenta do rio, sofre uma elevação brusca e turbulenta para um regime subcrítico (lento e profundo, Fr < 1). A profundidade a jusante do ressalto, chamada altura conjugada ou profundidade sequente, é y₂ = (y₁/2)·(√(1 + 8·Fr₁²) − 1), deduzida da conservação da quantidade de movimento (equação de Bélanger), a partir da profundidade de montante y₁ e do número de Froude incidente Fr₁. Essa equação é a base do projeto de bacias de dissipação a jusante de vertedouros e comportas. A água que desce o vertedouro tem energia altíssima — capaz de escavar o leito do rio e descalçar a fundação da própria barragem. A bacia de dissipação força o ressalto a ocorrer dentro de uma estrutura de concreto reforçado, e a altura conjugada y₂ define a profundidade necessária dessa bacia (e a submergência) para que o ressalto fique estável e na posição correta. Se y₂ não for atendida, o ressalto é 'varrido' para jusante, levando a turbulência erosiva para o leito desprotegido. Informe a profundidade de montante e o número de Froude.
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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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