1. End of Chapter 4 Movement of a Flood Wave and 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. Divide the cross-section in two. The flow rate of the advancing wave front Q wave = uA wave = Q 2 – Q 1 = A 2 V 2 – A 1 V 1 A wave = A 2 –A 1 So u(A 2 –A 1 ) = A 2 V 2 – A 1 V 1 = Q 2 – Q 1 where the speed of the wave is u, the speed of the water is V. Divide by A 2 – A 1 Then the wave speed u is approximately u = A 2 V 2 – A 1 V 1 = Q 2 – Q 1 rise over run (A 2 –A 1 ) (A 2 –A 1 ) 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 21

4. Manning’s Equation open channel 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 [length 2 ] 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 [m 1/3 / 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 y n, the only depth where flow is uniform. Normal depth depends on channel geometry and the roughness coefficient, n

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

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