Open Channel Hydraulics Environmental Hydrology Lecture 12

Winooski Falls, Photo by Jim Westphalen

Conditions of flow In space In time Uniform flow – no change in velocity, width, depth with distance Non-uniform flow – velocity, width, depth can change (gradually varying, rapidly varying) In time Steady flow – no change in flow with time Unsteady flow – flow changes with time

Forces operating on open channels Driving force: w sinq q w Resisting force: friction

Metrics of flow conditions Reynolds Number (Re) – ratio of inertial forces to viscous forces Re = v R u Re < 500 laminar flow 500 < Re < 2000 transition Re > 2000 turbulent flow where: v = average velocity R = “characteristic depth” (i.e. hydraulic radius) u = kinematic viscosity

Metrics of flow conditions Froude Number (Fr) – ratio of inertial forces to gravity forces Fr = v √g y Fr < 1 subcritical flow Fr = 1 critical flow Fr > supercritical flow where: V = average velocity g = acceleration due to gravity (9.81m/sec2, 32.2 ft/sec2) y = flow depth

Nash Stream at Whitcomb Peak. Image Source: Cohostrail.org

Uniform Open Channel Flow Continuity equation Resistance equations Energy & momentum equations Connecticut River at East Haddam. Image Source: Franklin Academy

Continuity Inflow 3 A 3 Outflow 1 A’ 2 Section AA’ Image source: Andy Ward Inflow – Outflow = Change in Storage

Continuity Flow or Discharge (Q) = V x A 3 where: V = average flow velocity at cross section (ft/sec, m/sec) A = cross sectional area (ft2, m2) 3 Section AA’ Image source: Andy Ward

velocity profile in a river Depth-averaged velocity is above the bed at about 0.4 times the depth

“resistance” in the channel Particle size distribution Ward & Trimble, Fig 7.3

Julius Ludwig Weisbach Antoine Chezy (1718-1798) Henri Darcy (1803-1858) Henri Emilie Bazin (1829-1917) Robert Manning (1816-1897) Julius Ludwig Weisbach (1806-1871)

Resistance Manning’s equation where: v = velocity (ft/sec*) R = wetted cross-sectional area/perimeter (ft*) S = slope (ft/ft*) n = Manning’s roughness coefficient * 1.49 is conversion factor for English units, use 1 if v, R, and S are in SI units

Ward & Trimble, Table 7.1

Resistance Darcy Weisbach equation where: v = velocity (m/s) g = gravitational constant (9.81m/s2) R = wetted cross-sectional area/perimeter (m) S = slope (m/m) f = Darcy-Weisbach friction factor

Application of resistance equations Roughness characterization Discharge estimation Flood reconstruction 1995 flood in Madison Co., Va. Image: Craig Kochel

Potential and Kinetic Energy in Open Channels Velocity head Pressure of fluid column (P = rgy = gy) y1 y2 Elevation above datum Total head (H) = Z + P/g + v2/2g Bernoulli equation

Specific Energy Specific energy (E) = y + v2/2g See also Ward & Trimble, Fig 8.1 Specific energy (E) = y + v2/2g

Hydraulic Jump “Meatgrinder” – So. Fork American River Image source: Greg Pasternack, UC Davis