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CH 7 - Open Channel Flow Brays Bayou 1. Uniform & Steady 2. Non-uniform and Steady 3. Non-Uniform and Unsteady Concrete Channel Open Channel Flow 1. Uniform & Steady - Manning’s Eqn in prismatic channel - Q, v, y, A, P, B, S and roughness are all constant 2. Critical flow - Specific Energy Eqn (Froude No.) 3. Non-uniform flow - gradually varied flow (steady flow) - determination of floodplains 4. Unsteady and Non-uniform flow - flood waves Uniform Open Channel Flow Manning’s Eqn for velocity or flow 1 2 /3 v R S S.I. units n 1.49 2 /3 v R S English units n where n = Manning’s roughness coefficient R = hydraulic radius = A/P S = channel slope Q = flow rate (cfs) = v A Normal depth is function of flow rate, and geometry and slope. One usually solves for normal depth or width given flow rate and slope information B b Normal depth implies that flow rate, velocity, depth, bottom slope, area, top width, and roughness remain constant within a prismatic channel as shown below UNIFORM FLOW B Q V y S0 A B n = Const = C = C = C = C = C = C y V So A 1 + Z2 1 a z Common Geometric Properties Cot a = z/1 Optimal Channels - Max R and Min P H = z + y + av2/2g = Total Energy E = y + av2/2g = Specific Energy a often near 1.0 for most channels Energy Coeff. a = S vi2 Qi V2 QT H Uniform Flow Energy slope = Bed slope or dH/dx = dz/dx Water surface slope = Bed slope = dy/dz = dz/dx Velocity and depth remain constant with x My son Eric Critical Depth and Flow Critical depth is used to characterize channel flows -based on addressing specific energy E = y + v2/2g : E = y + Q2/2gA2 where Q/A = q/y and q = Q/b Take dE/dy = (1 – q2/gy3) and set = 0. const q= E = y + q2/2gy2 y Min E Condition, q = C E Solving dE/dy = (1 – q2/gy3) and set = 0. For a rectangular channel bottom width b, 1. Emin = 3/2Yc for critical depth y = yc 2. yc/2 = Vc2/2g 3. yc = (Q2/gb2)1/3 Froude No. = v/(gy)1/2 We use the Froude No. to characterize critical flows Y vs E E = y + q2/2gy2 q = const Critical Flow in Open Channels In general for any channel shape, B = top width (Q2/g) = (A3/B) at y = yc Finally Fr = v/(gy)1/2 = Froude No. Fr = 1 for critical flow Fr < 1 for subcritical flow Fr > 1 for supercritical flow Non-Uniform Open Channel Flow With natural or man-made channels, the shape, size, and slope may vary along the stream length, x. In addition, velocity and flow rate may also vary with x. Non-uniform flow can be best approximated using a numerical method called the Standard Step Method. Non-Uniform Computations Typically start at downstream end with known water level - yo. Proceed upstream with calculations using new water levels as they are computed. The limits of calculation range between normal and critical depths. In the case of mild slopes, calculations start downstream. In the case of steep slopes, calculations start upstream. Calc. Q Mild Slope Non-Uniform Open Channel Flow Let’s evaluate H, total energy, as a function of x. H z y a v / 2g 2 dH dz dy a dv dx dx dx 2 g dx 2 Take derivative, Where H = total energy head z = elevation head, av2/2g = velocity head Replace terms for various values of S and So. Let v = q/y = flow/unit width - solve for dy/dx, the slope of the water surface dy q –S So 1 3 since v = q / y dx gy 2 1 d 2 1 d q q v 2 2g dx 2g dx y g 2 2 1 dy 3 y dx Given the Froude number, we can simplify and solve for dy/dx as a fcn of measurable parameters Fr v / gy 2 2 dy So S So S 2 2 dx 1 v / gy 1 Fr *Note that the eqn blows up when Fr = 1 and goes to zero if So = S, the case of uniform OCF. where S = total energy slope So = bed slope, dy/dx = water surface slope Yn > Yc Uniform Depth Mild Slopes where - Yn > Yc Now apply Energy Eqn. for a reach of length L v1 v 2 y1 = y2 S S o L 2 g 2 g 2 2 v1 v2 y 1 2 g y 2 2g L S S0 2 2 This Eqn is the basis for the Standard Step Method Solve for L = Dx to compute water surface profiles as function of y1 and y2, v1 and v2, and S and S0 Backwater Profiles - Mild Slope Cases Dx Backwater Profiles - Compute Numerically Compute y3 y2 y1 Routine Backwater Calculations 1. Select Y1 (starting depth) 2. Calculate A1 (cross sectional area) 3. Calculate P1 (wetted perimeter) 4. Calculate R1 = A1/P1 5. Calculate V1 = Q1/A1 6. Select Y2 (ending depth) 7. Calculate A2 8. Calculate P2 9. Calculate R2 = A2/P2 10. Calculate V2 = Q2/A2 Backwater Calculations (cont’d) 1. Prepare a table of values 2. Calculate Vm = (V1 + V2) / 2 Energy Slope Approx. 3. Calculate Rm = (R1 + R2) / 2 nV m Manning’s 4. Calculate S 2 3 1.49Rm 2 y1 v12 y 2 v 22 2g 2g 5. Calculate L = ∆X from first equation L S S0 6. X = ∑∆Xi for each stream reach (SEE SPREADSHEETS) 100 Year Floodplain Bridge D Floodplain Tributary C QD QC Main Stream Bridge Section B QB A Cross Sections QA Cross Sections The Floodplain Top Width Floodplain Determination The Woodlands The Woodlands planners wanted to design the community to withstand a 100-year storm. In doing this, they would attempt to minimize any changes to the existing, undeveloped floodplain as development proceeded through time. HEC RAS (River Analysis System, 1995) HEC RAS or (HEC-2)is a computer model designed for natural cross sections in natural rivers. It solves the governing equations for the standard step method, generally in a downstream to upstream direction. It can Also handle the presence of bridges, culverts, and variable roughness, flow rate, depth, and velocity. HEC - 2 Orientation - looking downstream River Multiple Cross Sections HEC RAS (River Analysis System, 1995) HEC RAS Bridge CS HEC RAS Input Window HEC RAS Profile Plots 3-D Floodplain HEC RAS Cross Section Output Table Brays Bayou-Typical Urban System • Bridges cause unique problems in hydraulics Piers, low chords, and top of road is considered Expansion/contraction can cause hydraulic losses Several cross sections are needed for a bridge 288 Bridge causes a 2 ft Backup at TMC and is being replaced by TXDOT 288 Crossing