The Open-Channel Toolbox TM Peter Wilcock Conservation Relations Conservation of Mass (Continuity) Conservation of Energy Conservation of Momentum Constitutive Relations Flow Resistance Sediment Transport The open channel toolbox consists of 5 physical relations to describe flow and transport in river channels. Additional relations are needed including conservation of sediment mass and at-a-station hyd. geom. Also energy min. used for regime channels and L&L 62.

On this slide we show the three-dimensional unsteady form of the Navier-Stokes Equations. These equations describe how the velocity, pressure, temperature, and density of a moving fluid are related. The equations were derived independently by G.G. Stokes, in England, and M. Navier, in France, in the early 1800's. The equations are extensions of the Euler Equations and include the effects of viscosity on the flow. These equations are very complex, yet undergraduate engineering students are taught how to derive them in a process very similar to the derivation that we present on the conservation of momentum web page.

Conservation of Mass (Continuity) Mass is neither created nor destroyed Inputs = outputs Inputs and outputs for fluid flow are discharge Vel x Flow Area All this means is that discharge is constant, flow rate remains the same from 1 xs to the next. Unless you are in a losing/gaining reach, where inputs do not equal outputs. Water is incompressible, so the same amt of water must go through a smaller space as a bigger one. (Continuity applies to all fluid flows, even air. Continuity is not true in aerodynamics, where air flow becomes compressible when approaching Mach 1) U1A1 = U2A2

NCED and “beige” slides by Conservation of Momentum (Force-balance) NCED and “beige” slides by Peter Wilcock/ Johns Hopkins Univ A

Unsteady, nonuniform flow Flow accelerates in space and time 1-d St. Venant eqn. Rearranged 1-d St. Venant eqn.

Potential Energy and Kinetic Energy Bernoulli energy equation H = d + Z + V2/2g + losses d = depth Z = elevation above datum, e.g. sea level V = velocity of flow g = gravity H1 H1

Conservation of Energy Energy is neither created nor destroyed Two components kinetic ( ) potential (z+h) Energy is also converted to heat, hf H1 =H2 + hf Energy has units of length (head). Potential energy is bed elevation z plus flow depth h. As flow goes downstream, mech. energy is converted to heat through friction. We call this head loss.

http://ga.water.usgs.gov/edu/hyhowworks.html

Multiply by 1.49 for English units Flow Resistance (Metric) Multiply by 1.49 for English units Relation between velocity, flow depth, basal shear stress, and hydraulic roughness A variety of relations exist including Manning’s Chezy Empirical The big unknown: n Using continuity, He picked DuBuat(1786), Eytelwein(1814), Weisbach(1845), St. Venant(1851), Neville(1860), Darcy/Bazin(1865), and Ganguillet/Kutter(1869). He calculated the vel. from each formula for a given slope and for a range of R (.25-.3 m.). Took the average of the 7 velocities, and generated a best fit line. V = 32 RS(1+R^1/3) or V=CS^.5R^.66

LWD covering less than 2% of the streambed can provide half the total roughness or flow resistance. This results in a finer streambed substrate. Buffington and Montgomery 1999, WRR 36, 3507-3521 Manga and Kirchner, 2000, WRR 36, 2373-2379.

Sediment Load Sources: Chemical weathering (dissolved) Human activity Mass wasting Slopewash Rill and gully formation Channel scour Bed Cutbanks

What does transport depend on? The strength of the flow, the fluid, and the sediment Strength of the flow = shear stress The sediment = grain size and density The fluid = water density and water viscosity (its resistance to deformation)

Sediment transport Emmett and Wolman (2001) Directly expressed in terms of sediment supply and water supply Shear stress is a descriptor of transport rate Directly and simulataneously measuring Q and Qs we develop a sediment rating curve, but most of the action occurs when it’s too dangerous to be measuring! Alternatives: use shear stress rather than discharge to describe transport rates. Requires that shear stress and critical shear be estimated. Empirically derived. q* is dimensionless shear stress. Meyer-Peter and Muller: General Form:

How to measure sediment transport?

Sediment Transport and Incipient Motion They are not the same sed trans = mass flow rate per unit time incipient motion = moves or not moves (binary: 0 or 1) What they share f(shear stress) transport depends on the fluid force applied to the bed Where would you use Incipient motion? Sizing riprap for rd protection. Why would calculate sed trans? Reservoir sizing/life

Tractive Force or Incipient Motion Shield’s equation Grain motion is driven by shear stress, t Units of force/unit area: psf, psi, Pa Critical shear stress, tc Shear stress needed to get a grain of a given size moving Shield’s number or dimensionless shear stress t* We can calculate shear as a function of grain size and flow condition from the Shields eqn. Tau* is the ratio of hydrodynamic force (flow force) to submerged weight. If you sum the forces acting on a particle c.g., you will get this result from the ratio of the drag force to the submerged wt. We can calculate the shear from the depth-slope product, but we need to know if that shear exceeds the critical shear to entrain the particle size of interest. So we need to calculate critical shear for that particle. First we need dimensionless critical shear. (s-1)rhogD is the grain wt per area. s = relative density of grain to density of water.

Shield’s diagram motion no motion From the experimentally determined Shields diagram, we can get tau*. The eqn of the curve is given. S* is the dimensionless viscosity. For S*>10,000, tau*c is roughly a constant value of 0.045. S* is something like a particle Reynolds number that describes the flow condition from laminar, transitional, to fully turbulent. If tau varies with S*, then viscosity influences the dimensionless crit shear for smaller grains. Now the great simplification, for grains slightly greater than sand size (2 mm), we can express tauc as a function of grain size. Now apply a safety factor. This is the simplest case. There are a lot of other considerations including variation in shear on side slopes, flow around bends.

http://www.uwsp.edu/gEo/faculty/lemke/geomorphology/lecture_outlines/03_stream_sediment.html

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Problem Site 9 on the White River

Try it For a rectangular channel 5 m. wide by 2 m. deep (running full), with a slope of 2% What is the basal shear stress? 5 m 2m Remember, hydraulic radius R = A/WP

Try It A basketball has a diameter of ~ 10 inches. What is the critical shear stress required to just move a basketball-sized rock in a river? For the channel in the last problem is the rock mobile?