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End of Chapter 4 Movement of a Flood Wave and begin Chapter 7 Open Channel Flow, Manning’s Eqn. Overland Flow

End of Chapter 4 Movement of a Flood Wave and begin Chapter 7 Open Channel Flow, Manning’s Eqn. Overland Flow . Monoclinal Wave Velocity: Celerity.

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End of Chapter 4 Movement of a Flood Wave and begin Chapter 7 Open Channel Flow, Manning’s Eqn. Overland Flow

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  1. End of Chapter 4 Movement of a Flood Waveand begin Chapter 7 Open Channel Flow, Manning’s Eqn. Overland Flow

  2. Monoclinal Wave Velocity: Celerity • Suppose there is a big storm upstream, or a catastrophic dam breach, or an emergency lowering of reservoir level*, and a tall wave of much deeper water rolls downstream. How fast does it go? *Remember Issaquah, HW1?

  3. 2 1 • Divide the cross-section in two. The flow rate of the advancing wave front • Qwave = uAwave = Q2 – Q1 = A2V2 – A1V1 • Awave = A2 –A1 • So u(A2 –A1)= A2V2 – A1V1 = Q2 – Q1 • where the speed of the wave is u, the speed of the water is V. • Divide by A2 – A1 • Then the wave speed u is approximately • u = A2V2 – A1V1 = Q2 – Q1 rise over run • (A2 –A1) (A2 –A1) • and precisely c = dQ/dA, where c is the precise version of wave velocity u, the celerity. Recall last time Q and A functions of y=H • If we express the Area A as the product of a constant channel width B and variable depth y, Area A = B . y, then • c = (1/B) . dQ / dy

  4. Manning’s Equation • The average velocity V in an open channel is given by Manning’s equation: • where R (the hydraulic radius) = A/P [length], and • A = cross sectional Area [length2] • P = wetted Perimeter [length] • S = the energy slope [length/length] • n = Manning’s Roughness [unitless] • k = 1.49 [ft 1/3 /sec] or 1.0 [m1/3 /sec] for SI

  5. Manning’s for a wide rectangular channel • Here wetted perimeter P is mostly B, • the cross-sectional area is A = B . y • so the hydraulic radius R = y approximately • Multiplying Manning’s by the area = By gives the flow rate Q • where Q and y are the only variables.

  6. Celerity • If we differentiate the flow form of Manning’s we can evaluate the celerity. Since R ~ y everything to the right of B is a Manning velocity so Monoclinal waves move 5/3 faster than the stream water.

  7. Steady Flow • Now, lets jump to Chapter 7 and continue our use of Manning’s Equation. Steady-state flow refers to the condition where the fluid properties at a point in the system do not change over time. Uniform flow is prismatic flow (flow in prismatic channels: constant cross section and slope) for which the slope of total energy (EGL slope) equals bottom slope.

  8. Normal Depth • Uniform flow problems use Manning’s to compute Normal depth yn, the only depth where flow is uniform. • Normal depth depends on channel geometry and the roughness coefficient, n

  9. Overland Flow Manning’s n Roughness coefficients for overland flow onto the floodplain vary greatly.

  10. Two Examples • As usual, we will have some examples and similar class work / homework. • Example 7.1 calculates normal depth in a rectangular channel given n and the flow rate. • Example 7.3 calculates normal flow rate given normal depth in a trapezoidal channel

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