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Numerical Hydraulics Open channel flow 1 Wolfgang Kinzelbach with Marc Wolf and Cornel Beffa 1 Saint Venant equations in 1D b(h) lb A(h) z h • continuity • momentum equation 2 Saint Venant equations in 1D • continuity (for section without inflow) Q A 0 x t • Momentum equation from integration of Navier-Stokes/Reynolds equations over the channel cross-section: 0 hP v v v g t x x Rhy 3 Saint Venant equations in 1D • In the following we use: 1 • and 0 8 v 2 (Re, k / R ) hy 4 Saint Venant equations in 1D The friction can be expressed as energy loss per flow distance: 0 E / V gI R Rhy x Using friction slope and channel slope IR 1 v2 8 Rhy g z IS x Alternative: Strickler/Manning equation for IR 5 Saint Venant equations in 1D we finally obtain v v h v g IS IR g t x x (vA(h)) A(h) 0 x t For a rectangular channel : A bh h h v v h 0 t x x 6 Approximations and solutions • • • • Steady state solution Kinematic wave Diffusive wave Full equations 7 Steady state solution (rectangular channel) dv dh v g IS IR g dx dx dh dv v h 0 dx dx Solution: 1) approximately, 2) full 8 Steady state solution (rectangular channel) Approximation: Neglect advective acceleration IS IR 0 Normal flow Full solution (insert second equation into first): 2 I I dh v 2 S R 0 with Fr 2 dx 1 Fr gh yields water surface profiles 9 Classification of profiles hgr = water depth at critical flow hN = water depth at uniform flow Is = slope of channel bottom Igr = critical slope Horizontal channel bottom Is = 0 H2: h > hgr H3: h < hgr 10 Classification of profiles Mild slope: hN > hgr Is < Igr M1: hN <h > hgr M2: hN > h > hgr M3: hN > h < hgr Steep slope: hN < hgr Is > Igr S1: hN <h > hgr S2: hN < h < hgr S3: hN > h < hgr 11 Classification of profiles Critical slope hN = hgr IS = Igr C1: hN < h C3: hN > h Negative slope IS < 0 N2: h > hgr N3: h < hgr 12 Numerical solution (explicit FD method) Subcritical flow: Computation in upstream direction h( x x) h( x) I S I E ( x x) 2 x 1 Fr ( x x) Solve for h(x) Supercritical flow: Computation in downstream direction h( x x) h( x) I S I E ( x) 2 x 1 Fr ( x) Solve for h(x+x) 13