Uniform Open Channel Flow – Ch 7 Uniform flow - Manning’s Eqn in a prismatic channel - Q, V, y, A, P, B, S and roughness are all constant Critical flow - Specific Energy Eqn (Froude No.) Non-uniform flow - gradually varied flow (steady flow) - determination of floodplains Unsteady and Non-uniform flow - flood waves

Normal depth implies that flow rate, velocity, depth, bottom slope, area, top width, and roughness remain constant within a prismatic channel as shown below UNIFORM FLOW b Q = C V = C y = C S0 = C A = C b = C n = C Wetted perimeter P remains constant A P

Uniform Flow Force Balance

Uniform Open Channel Flow – Chezy and Manning’s Eqn. Hydrostatic, Friction, and Weight all act together. Derivation of these eqns requires a force balance (x) Actual forces (F = hydrostatic) are summed across C.V.

Chezy and Manning’s Eqn. Since hydrostatic forces are equal, and Slopes are very mild 2. Define R = A/P, the hydraulic radius

Chezy and Manning’s Eqn. Finally, we can equate the two eqns for shear stress C = Chezy Coefficient (1768) in Paris Manning was an Irish Eng and 1889 developed his EQN.

Manning’s OCF Equation Manning’s Eqn for velocity or peak flow rate where n = Manning’s roughness coefficient R = hydraulic radius = A/P S = channel slope

Uniform Open Channel Flow – Brays B. Brays Bayou Concrete Channel

Normal depth is function of flow rate, and geometry and slope. One usually solves for normal depth or width given flow rate and slope information B b

Optimal Channels - Max R and Min P

Water surface slope = Bed slope = dy/dz = dz/dx Uniform Flow Energy slope = Bed slope or dH/dx = dz/dx Water surface slope = Bed slope = dy/dz = dz/dx Velocity and depth remain constant with x H

Bernoulli Flow Eqn H

Critical Depth and Flow My son Eric

Critical depth is used to characterize channel flows -- based on addressing specific energy E = y + v2/2g : E = y + Q2/2gA2 where Q/A = q/y and q = Q/b Take dE/dy = (1 – q2/gy3) and set = 0. q = const E = y + q2/2gy2 y Min E Condition, q = C E

For a rectangular channel bottom width b, Solving dE/dy = (1 – q2/gy3) and set = 0. For a rectangular channel bottom width b, 1. Emin = 3/2Yc for critical depth y = yc yc/2 = Vc2/2g yc = (Q2/gb2)1/3 Froude No. = v/(gy)1/2 We use the Froude No. to characterize critical flows

Y vs E E = y + q2/2gy2 q = const

Critical Flow in Open Channels In general for any channel shape, B = top width (Q2/g) = (A3/B) at y = yc Finally Fr = v/(gy)1/2 = Froude No. Fr = 1 for critical flow Fr < 1 for subcritical flow Fr > 1 for supercritical flow Critical Flow in Open Channels