NONUNIFORM FLOW AND PROFILES. Nonuniform flow varies in depth along the channel reach. Figure 1 Nonuniform Flow

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1 Nonuniorm Flow and Proiles Page 1 NONUNIFORM FLOW AND PROFILES Nonuniorm Flow Nonuniorm low varies in depth along the channel reach. Figure 1 Nonuniorm Flow Most lows are nonuniorm because Most channels are non-prismatic. Storm sewers are non-prismatic due to the presence o manholes and changes in pipe diameter, slope and direction. Flow may be nonuniorm in a prismatic channel due the inluence o a control, e.g., backwater created by a high tailwater depth or drawdown at a ree overall. Control A control is a channel eature, usually structural, where there is a unique (one-to-one) relationship between depth and discharge. A control regulates (controls) the state o low. Examples: Free overall at the end o a mild channel Weirs, Flumes (critical controls) Ininitely long prismatic channel (control reach) Subcritical low is controlled by downstream conditions. Supercritical low is controlled by upstream conditions. Channel Classiication Channel bed slopes are classiied hydraulically as mild, steep, critical, horizontal, and adverse. A channel bed slope may classiy as mild or one low rate and steep or another. The classiication is based on the relationship between the normal depth o low, y n, and the critical depth o low, y c, or a given lowrate. ECIV 36 Introduction to Water Resources Engineering

2 Nonuniorm Flow and Proiles Page The bed slope is called: Mild i y n >y c Steep i y c >y n Critical i y n y c Flow Proiles Depending on the relationship between y n and y c, there can be three zones where y(x) occurs. Figure Flow Proile Zones on a Mild Channel I y(x)>y n, then the low proile exists in Zone 1. In other words, the actual depth o low, y(x), is greater than y n and occurs in Zone 1. I y n >y(x)>y c, then the low proile exists in Zone. I y c >y(x), then the low proile exists in Zone 3. Flow proiles are identiied using a two-character label. The irst is an alpha character that identiies the type o channel slope (e.g., M or mild, S or steep). The second is a numeric character that identiies the zone (e.g., 1, or 3). For a hydraulically mild channel: I y(x)>y n, the proile exists in Zone 1, and is an M1 proile. I y n >y(x)>y c, the proile exists in Zone, and is an M proile. I y c >y(x), the proile exists in Zone 3, and is an M3 proile. Zone 1 low proiles are known as backwater proiles because the water "backs-up" due to a downstream control that restricts the outlow and orces the water to pond to a greater depth to "push" the low through/past the control. Zone proiles are known as drawdown or rontwater (HMI, et al., 004) curves. ECIV 36 Introduction to Water Resources Engineering

3 Nonuniorm Flow and Proiles Page 3 Figure 3 Mild Channel Proiles Figure 4 Steep Channel Proiles Flow Proile Analysis The change in depth in nonuniorm low is known as the low proile and is determined mathematically by low proile analysis. Calculations involve solving the energy equation at dierent points along the proile. Flow proile analysis is used in storm sewer design when determining the hydraulic grade line (FHWA, 1996). Theory Consider the deinition igure or nonuniorm low in an open channel (Figure 5). Applying energy conservation between sections 1 and gives EGL 1 EGL + h where EGL 1 is the total energy at section 1, EGL is the total energy at section, and h is energy lost to riction between 1 and. Rewrite the energy equation in terms o speciic energy E 1 + z1 E + z + h ECIV 36 Introduction to Water Resources Engineering

4 Nonuniorm Flow and Proiles Page 4 Figure 5 Deinition Figure The dierence in elevation heads is product o channel bed slope, S o, and reach length, Δx. z1 z So Δx and h is the product o riction slope, S, and reach length. h S Δ S is the slope o the energy grade line. Ater substitution, x ( S S ) x E - E1 o Δ Dividing both sides by Δx, we get E E 1 S 0 S Δx Because speciic energy is evaluated at both ends o a channel reach (Δx), convention is to evaluate the riction slope term at both ends o the reach to approximate the average riction slope over the reach. E E 1 S S o Δ x in which S 1 + S S where S 1 is the riction slope evaluated at 1 and S is the riction slope evaluated at. We use Manning's equation to approximate S. Simply substitute S or S o and solve or S. S Q n.1 A R n A Q R (English ) (SI) ECIV 36 Introduction to Water Resources Engineering

5 Nonuniorm Flow and Proiles Page 5 Solution Methods Most low proile analysis programs use numerical solution techniques. Two methods are commonly used: (1) Direct Step Method and () Standard Step Method. Direct Step Method Involves an explicit (direct) numerical solution o the energy equation. Applies only to prismatic channels. Involves solving or the position (xlocation) o user speciied y-values along the low proile. Rule-o-thumb: Given y, ind x. Standard Step Method Involves an iterative numerical solution o the energy equation. Applies to any channel: natural, prismatic, etc. Involves solving or the depth at user speciied x-locations. Rule-o-thumb: Given x, ind y. Direct Step Procedure: Determine the design lowrate, Q, channel geometry and dimensions, bed slope, and Manning's n-value. Determine normal depth, y n, and critical depth, y c, or the design low. Classiy the channel bed slope. Channel is mild i y n >y c, steep i y c >y n, etc. Determine the location o the control and the depth o low at the control. For example, i you are dealing with a hydraulically mild storm sewer with a ree overall at the outlet, the low will pass through critical depth at the outlet, meaning the control is at the outlet. Classiy the proile type, e.g., M1, M, S1, etc. For the example, the upstream proile is an M or drawdown curve. Knowing the proile type and the depth at the control, establish the range o values or the depth proile, y(x). For the example, y c <y(x)<y n. Create a worktable (Example Problem in Unit on Storm Sewer Design). Starting at the control, speciy several y-values over the range determined in the previous step. Use a small increment between successive y-values near the control. A larger increment can be used arther rom the control. Note: I the low is subcritical, the control is downstream, and the proile is evaluated in the upstream direction, starting at the control. I the low is supercritical, the control is upstream, and the proile is evaluated in the downstream direction, starting at the control. Solve the energy equation between successive y-values to determine the distance, Δx, between them. Sum Δx values to determine the distance o each point rom the control. Plot y(x) versus x. This is the desired low proile. Standard Step Procedure: Determine the design lowrate, Q. Determine the channel x-sectional geometry, slope and Manning's n-value. ECIV 36 Introduction to Water Resources Engineering

6 Nonuniorm Flow and Proiles Page 6 I the channel is not prismatic, determine the cross-sectional geometry, dimensions and Manning's n-value at each cross-section where you desire the depth o low. Also, measure the distance o each x-section rom the control, and determine the bed slope between adjacent x-sections. Determine Δx between adjacent x-sections (known Δx values). Evaluate normal depth, y n, and critical depth, y c. I the channel is not prismatic, do this or each subreach, Δx. Classiy the channel bed slope. I the channel is not prismatic, to have one continuous proile type, all subreaches must have the same bed slope classiication; otherwise, multiple proiles exist. Determine the location and low depth at the control. Classiy the proile. Create a worktable. Starting at the control, choose a trial y value at the irst x-section rom the control. Solve the energy equation between the two x-sections to determine the distance, Δx, between them (predicted Δx). I the predicted Δx is suiciently close to the known Δx, the assumed y-value is good; otherwise, adjust the assumed y-value and repeat the process. Repeat the process or all sub-reaches. Plot y(x) versus x. This is the desired low proile. REFERENCES Federal Highway Administration (1996). Urban Drainage Design Manual, Hydraulic Engineering Circular No., Oice o Technology Applications, th Street SW, Washington, DC. Haestad Engineering Sta, Walski, T.M., Barnard, T.E., Durrans, S.R., and Meadows, M.E. (004). Computer Applications in Hydraulic Engineering, 6 th Edition, Haestad Press, Waterbury, CT, 375 p. ECIV 36 Introduction to Water Resources Engineering