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Rapidly Varying Flow The Hydraulic Jump

Key Concepts: When the upstream flow is supercritical, F_r > 1, (Froude number is greater than 1)

the flow is unstable. Local nonuniform flow causes a steep upward slope of the surface profile with violent turbulence transitioning to subcritical, downstream uniform flow. See the figure below.

Key assumptions in the analysis of the hydraulic jump for a rectangular channel of width w are: Steady, uniform, incompressible, frictional, and one fluid type (typically water). Also, neglect the wall shear stress since distance between (1) and (2) is relatively small. The

principles of conservation of mass, linear momentum, and energy apply to hydraulic jumps.

Conservation of Mass: ρ₁A₁V₁ = ρ₂A₂V₂ , A₁V₁ = A₂V₂ ,

Or wy₁V₁ = wy₂V₂ = Q , ρ₁ = ρ₂ = ρ = γg and finally y₁V₁ = y₂V₂

Conservation of Linear Momentum: ρ₂A₂V₂² - ρ₁A₁V₁² = Σ F_x = F₁– F₂

or ρQV₂ - ρQV₁ = ρQ(V₂ - V₁) = F₁– F₂

where F₁= P₁A₁ = γ y₁² w/2 and F₂ = P₂A₂ = γ y₂² w/2 , (hydrostatic pressure distribution)

ρQ(V₂ - V₁) = ρwy₂V₂² - ρwy₁V₁² = ρg y₁² w/2 - ρg y₂² w/2 divide by ρgw

y₂V₂² /g - y₁V₁² /g = y₁² /2 - y₂² /2 and finally reorganize

y₁V₁² /g + y₁² /2 = y₂V₂² /g + y₂² /2

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