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This lecture discusses the concepts of hydraulic radius and roughness parameter in surface water flow. It explores measures of roughness such as Manning roughness parameter, Chezy constant, Strickler constant, and friction factor. The lecture also includes calculations and explanations for wetted perimeter, hydraulic radius, and velocity. It further examines uniform flow, varied flow, friction factor, velocity distribution, and gradually and rapidly varied flow. Equations of motion, including continuity and momentum equations, are explained along with their assumptions. The lecture concludes with a discussion on hydraulic jump and various methods for measuring stream flow.
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Hydraulic radius and roughness parameter • Measure of contact area is the hydraulic radius , • Where is cross-sectional area and is the wetted perimeter • Many measures of roughness • Most common are Manning roughness parameter , Chezy constant , Strickler constant and friction factor • Manning roughness parameter most fully dependent on roughness alone (bed properties not channel geometry)
Consider this channel: 2 ft 20 ft What is the wetted perimeter? What is the hydraulic radius? The discharge is 400 ft3/sec, what is the velocity? (show all units)
Uniform flow: flow that does not change with distance downstream, neither in area nor velocity Varied or nonuniform flow: flow that changes with distance down stream
Speed of water • Uniform flow easiest - not changing forces are zero • What does it seem like the speed of water in a channel will depend on, intuitively? • Gravity • Gradient certainly • Some measure of resistance • This it turns out is related to the area of the bed, and a measure of the properties, mostly roughness, of the bed materials
Friction and drag Volume of water moved/time, • Imagine bottom of stream with clasts (rocks)
Stream power • Mass moved/time, , where • So since energy (work), • Work/time power, or • Remember this is the power that it takes to move the fluid out of the way of the bed obstacles
Friction Factor • Friction factor, , characterizes the overall effect of the shape of the bed particles in counteracting flow motion • varies with the change from form drag (blunt) to skin friction (smooth) • is dimensionless • What does its value depend on? • So:
Balance of gravity and friction forces • Balance to get velocity for uniform flow • That is: (since
Derivation of Manning’s Equation What is ? • But friction factor is hard to measure, so we use • Chezy constant, or • Manning roughness parameter • The result is the Manning equation for uniform flow: • Or the Chezy equation: • Question: , , . So mean stream velocity, …?
Velocity Distribution • from the Manning equation is the mean speed. How is this related to the speed at the surface, which is easy to measure? • within the fluid: • parabolic velocity profile with depth for streamflow • not quite correct because of turbulence • one result though is that • (the from Manning’s equation) • So through a vertical in uniform flow :
Gradually- and Rapidly-varied Flow • Uniform flow doesn’t occur everywhere • Gradually varied flow occurs where stream depth and speed change smoothly • Rapidly varied flow occurs where stream depth and speed change suddenly • In gradually and rapidly varied flow, it is a rather complicated calculation to find speed need to look at equations of motion a bit, and perhaps use a numerical program • One can map out the reaches of any stream, such as the Niagara River defined by stretches of Uniform Flow and separated by short stretches of Gradually Varied or Rapidly Varied Flow • To analyze motion in these changing stretches, Manning’s Equation and its assumptions do not hold
Equations of Motion Continuity equation conservation of mass which is discharge , and is constant for no inflow or outflow (either by tributaries, through groundwater, or by evapotranspiration) • Assumes no inflow or outflow (no tributaries or distributaries; losses or gains with groundwater; evaporation) Momentum equation expresses Newton’s second law or Bernoulli’s equation which is a constant called the Bernoulli integral or total mechanical energy • Bernoulli’s equation works within a streamtube - tube made up of mean stream lines , which are parallel to channel walls and the free water surface in the case of streams
Hydraulic Jump • In a region of rapidly varied flow, the height above a datum ( in the Bernoulli equation) does not change significantly, thus is a constant. • Either can be big, or can be big, to get the same • What happens on a graph of (-axis) versus (-axis)? • There are two solution branches, defined by the Froude number, = ratio of kinetic energy to potential energy:
Measuring stream flow USGS steam gauges with “real time” data http://waterdata.usgs.gov/ny/nwis/rt
Measuring stream flow Discharge – reported in cfs by the USGS Or, not measure at every “X,” but instead Just measure at 60% of water depth
Measuring stream flow Discharge – reported in cfs by the USGS Stage – height of water in a creek. Measured with respect to some datum
Measuring stream flow Discharge – reported in cfs by the USGS Stage – height of water in a creek. Measured with respect to some datum. Based on the notion that it is easier to measure stream height (dipstick) than discharge. Founded on relationship intrinsic to the specific river of discharge versus stage (rating curve)
