Open Channel Hydraulics

Environmental Hydrology

Lecture 12

Winooski Falls, Photo by Jim Westphalen

Conditions of flow

• In space– 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

w

Forces operating on open channels

Driving force: w sin

Resisting force: friction

Metrics of flow conditions

• Reynolds Number (Re) – ratio of inertial forces to viscous forces

Re = v R

where:v = average velocityR = “characteristic depth” (i.e. hydraulic radius) = kinematic viscosity

Re < 500 laminar flow500 < Re < 2000 transitionRe > 2000 turbulent flow

Metrics of flow conditions

• Froude Number (Fr) – ratio of inertial forces to gravity forces

Fr = v √g y

where:V = average velocityg = acceleration due to gravity (9.81m/sec2, 32.2 ft/sec2)y = flow depth

Fr < 1 subcritical flowFr = 1 critical flowFr > supercritical flow

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 – Outflow = Change in Storage

Inflow

1 2

A

A’

3

Section AA’

Outflow

3

Image source: Andy Ward

Continuity

Flow or Discharge (Q) = V x A

where: V = average flow velocity at cross

section (ft/sec, m/sec)A = cross sectional area (ft2, m2)

Section AA’

3

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

Henri Emilie Bazin (1829-1917)

Henri Darcy (1803-1858)

Julius Ludwig Weisbach (1806-1871) Robert Manning (1816-1897)

Antoine Chezy(1718-1798)

Resistance

Manning’s equation

2132 SRn

49.1v

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

f

gRSv

82

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

Total head (H) = Z + P/ + v2/2gBernoulli equation

Elevation above datum

Pressure of fluid column(P = gy = y)

Velocity head

y1y2

Specific Energy

See also Ward & Trimble, Fig 8.1

Specific energy (E) = y + v2/2g

“Meatgrinder” – So. Fork American RiverImage source: Greg Pasternack, UC Davis

Hydraulic Jump