OPEN CHANNEL FLOW. One-dimensional - neglect vertical and lateral variations in velocity. In other words, Q v = (1) A. Figure 1. One-dimensional Flow

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1 OPEN CHANNEL FLOW Page 1 OPEN CHANNEL FLOW Open Channel Flow (OCF) is flow with one boundary exposed to atmospheric pressure. The flow is not pressurized and occurs because of gravity. Flow Classification In terms of time and space Steady - flow properties do not vary over time at-a-point. Unsteady - flow properties vary over time at-a-point. Uniform - flow properties do not vary in space. Non-uniform - flow properties vary in space. Typically caused by the influence of a control. In terms of the velocity profile One-dimensional - neglect vertical and lateral variations in velocity. In other words, Q v (1) A Using dimensionless parameters Laminar - low Reynolds number. Layered flow. Turbulent - high Reynolds number. Lots of mixing. Subcritical - Froude number <1. Critical - Froude number 1. Supercritical - Froude number >1. Conservation Principles Figure 1. One-dimensional Flow Most open channel flow problems are solved using one or more of the conservation principles for: 1. Mass (Continuity). Momentum. Energy ECIV 6 Introduction to Water Resources Engineering

2 OPEN CHANNEL FLOW Page Continuity When applied to a control volume, continuity requires: Inflow Outflow Rate of Change in Storage () Figure. Continuity in Open Channel Flow For this channel situation, we write continuity as Q (AΔx) Q - (Q + Δx) () x t This simplifies to: Q A + 0 (4) x t The first term describes non-uniform flow. The second term describes unsteady flow. This is the continuity equation for 1-, unsteady, gradually-varied, open channel flow. For steady flow, we write continuity in its simplest form Q va (1) where Q discharge in cfs (cms) v cross-sectional average velocity in fps (m/s) A cross-sectional area in sq ft (m ) Conservation of Momentum According to Newton's second law of motion, the rate of change in momentum of a body of water is equal to the summation of applied external forces acting on it. Consider the free body diagram in Figure. For flow in the x-direction. F x ma x (5) The applied forces are: 1. Pressure. Gravity (weight). Friction (due to shear stress along channel boundary) ECIV 6 Introduction to Water Resources Engineering

3 OPEN CHANNEL FLOW Page Figure. Forces in Open Channel Flow Expressed mathematically Fx γh1a1 - γh A + γaδx sin α - τo PΔx (6) where γ is the unit weight of water h is the depth to the centroid of the flow x-section A is the cross-sectional area of flow τ o is the shear stress or frictional drag P is the wetted perimeter Consider acceleration, the rate of change of velocity dv a (7) dt Since velocity is a function of space and time, vv(x,t), dv v dx v + () dt x dt t And since dx v (9) dt acceleration can be expressed as v v a v + (10) x t The x-derivative is the spatial or convective acceleration. The t-derivative is the local or temporal acceleration. For steady flow, the time derivative vanishes v Steady Flow : 0 (11) t and momentum simplifies to F x ρ Q Δv (1) If the flow is also uniform, then the spatial derivative vanishes and F x 0 (1) Conservation of Energy Water flows in a channel from a region of higher energy to a region of lower energy. Three forms of energy are normally considered: Potential Energy or elevation head (z)- ability to do work because of the elevation of a mass of water. ECIV 6 Introduction to Water Resources Engineering

4 OPEN CHANNEL FLOW Page 4 Flow-work Energy (Stored Energy) or pressure head (P/γ)- ability to do work because of pressure in excess of atmospheric pressure. Kinetic Energy or velocity head (v /g)- exists because of the velocity of flowing water. Energy conservation is evaluated with the following form of the Bernoulli equation. v 1 v y 1 + z 1 + y + z + + h f (14) g g where y is the pressure head term and h f is the head (energy) loss due to friction. Uniform Flow Figure 4. Energy Terms for Open Channel Flow Uniform flow occurs in prismatic channels when the flow depth is constant throughout the reach of interest. As such, all streamlines are parallel to the channel bed, including the free water surface. When we speak of uniform flow, steady, uniform flow is generally what is considered. If the flow is both steady and uniform, it is not accelerating and the momentum equation simplifies to F x 0 (1) Since the flow depth is the same at both ends of the channel segment, there is no pressure imbalance, meaning the gravity and friction forces are equal and opposite (i.e., off-set each other). Fx γ A Δx sin α - τo P Δx 0 (15) Solving for shear stress A τo γ sin α (16) P For small sloped channels (θ<5 o ), sinθ closely equals the slope of the channel bed, S o, expressed in the units of ft/ft (m/m). A τ o γ S o (17) P The term A/P represents the hydraulic radius, R, defined as the flow area divided by the wetted perimeter. τ o γ RS o (1) which is known as the generalized resistance law. Experimental studies on turbulent flow have shown shear stress is proportional to the arcy-weisbach friction factor and the square of the flow velocity. That is v τ fρ (19) Combining the two preceding equations to eliminate shear stress, and solving for velocity gives γ v RS o (0) fρ By letting ECIV 6 Introduction to Water Resources Engineering

5 OPEN CHANNEL FLOW Page 5 C γ fρ (1) Chezy's equation for open channel flow is obtained as V C RS o () where C is a factor related to the roughness of the channel. Chezy's equation for discharge is Q VA CA RS o () Values for C are found in textbooks on open channel flow. Manning's Equation In 19 an Irish engineer named Robert Manning presented a formula, which was later modified to its present form v R S (4) n where v is the cross-sectional mean velocity in fps, R is the hydraulic radius in feet, S is the slope of the energy line (EGL), and n is the coefficient of roughness, specifically known as Manning's n. This formula was developed from seven different formulas, based on published experimental data, and further verified by 170 observations. For uniform flow, the slope of the EGL is the same as slope of the channel bed, S o, since EGL, HGL and channel bed are all parallel lines. For uniform flow calculations, S o replaces S. Accordingly, Manning's equation for discharge, Q, in the English system of units is 1.49 Q AR S o (5) n where flow rate is in cfs, and for the SI system of units, 1 Q AR S o (6) n where flow rate is in cms. Manning's n-value is viewed as a roughness coefficient, but is actually affected by many factors: Surface roughness Vegetation Channel irregularity and alignment Silting and scouring Obstructions Manning n-values are found in most hydraulics textbooks and drainage design manuals. Normal epth The normal depth of flow, designated y n, is the depth when the flow is uniform. We determine normal depth using Manning's equation. Normal depth and critical depth are two parameters that we use to classify flows and channels. ECIV 6 Introduction to Water Resources Engineering

6 OPEN CHANNEL FLOW Page 6 Manning n-values for storm sewers and culverts Example 1 Application of Manning's Equation to a Trapezoidal Channel Given the following channel geometry. Longitudinal slope is 0.% and the Manning's n-value is (a) etermine the flow rate when the normal depth of flow is.0 feet. (b) etermine the normal depth of flow, y n, when the flow rate is 0 cfs. z.0 w10 ft Both solutions utilize Manning's equation. Since the data are given in U.S. Customary units, use Eq Q AR S o n (5) For a trapezoidal geometry A wy + zy (7) P w + y 1+ z () where A R P (9) Part (a): A 10() + ( ) ft ( ) ft P 10 + R 1.41 ft.65 ECIV 6 Introduction to Water Resources Engineering

7 OPEN CHANNEL FLOW Page ( )( 1.41) cfs Q Answer Part (b): Rearrange Manning's equation as follows nq fn ( y n ) AR 1.49 S (0) By T&E, find 10y + y ( ) ( 0) 10y + y o y y n.47 ft Answer Specific Energy Specific Energy (E) - total energy at a point (cross-section) in open channel flow with respect to the channel bed. For a given discharge, Q (va) This curve has two asymptotes: As y 0, As y, v E y + g Q y + g A E E E y (Abscissa) A plot of the specific energy equation is called the specific energy diagram, as shown in Figure 5. Convention is to plot specific energy, E, on the abscissa (x-axis) and depth of flow, y, on the ordinate. This curve is plotted for a given Q. To obtain this curve, simply evaluate E for a range of y (and corresponding A) values while holding Q constant. Plot the resulting (y,e) pairs (45 o line) (1) () Figure 5. Specific Energy iagram For given discharge, at each value of specific energy there are two possible flow depths, one on the upper limb and one on the lower limb. These are known as alternate depths of flow. Critical epth ECIV 6 Introduction to Water Resources Engineering

8 OPEN CHANNEL FLOW Page In Figure 5, note there is a value for the depth of flow where the specific energy is a minimum. Mathematically, this is known as a critical point. So let us call this depth the critical depth and designate it with the variable name, y c (Figure 6). Figure 6. Location of Critical epth de d Q At yy c y 0 dy dy + ga () performing the indicated differentiation As seen in the Figure 7 da T dy where T is the top width of flow. Substituting, we obtain or de Q T 1-0 dy g A de Q 1 + dy g A (5) (6) da dy Q T 1 g A when y y c (4) (7) To determine y c for any channel, one can solve the following relationship A Q fn(yc ) constant () T g ECIV 6 Introduction to Water Resources Engineering

9 OPEN CHANNEL FLOW Page 9 Figure 7. Incremental Area in an Irregular Geometry Channel In the case of a rectangular channel, Awy, where w is the bottom width, and since the top width is the same as the channel bottom width Q T Q w q (9) g A g w w y g y where q is flow per unit width (qq/w). Solving for yy c q y c (40) Note: this equation applies only to rectangular channels. For non-rectangular channels, use, Eq. 7. g Froude Number For a rectangular channel, we have seen that The left hand side simplifies to taking the square root Q T q g A g y q g y 1 v y g y when y y v gy v F 1 gy where F is the Froude number, one of the dimensionless parameters. In other words, When y yc, F 1 We see that Q T F g A ( 1when y y ) c c (9) (41) (4) (4) ECIV 6 Introduction to Water Resources Engineering

10 OPEN CHANNEL FLOW Page 10 Figure. Specific Energy and Froude Number Relationship Classification of Flows F<1 subcritical flow F1 critical flow F>1 supercritical flow Relatively speaking, subcritical flow is deep, slow moving flow, and supercritical flow is shallow, fast moving flow. Example - Computing Normal and Critical epths in Circular Sewers Given a flow of 40 cfs in a 6-inch RCP sewer on 0.5% grade. etermine the normal (uniform) and critical depths of flow. Classify the flow. Use n Solution for normal depth Use Manning's equation and the imensionless Geometric Elements table. Rearrange Manning's equation as Form it dimensionless as where AR 1.49 AR n Q fn( y ) AR n AR AR nq 1.49 S o S o nq 1.49 S y fn( ) nq 1.49 S o 0.015(40) () o 0.04 ECIV 6 Introduction to Water Resources Engineering

11 OPEN CHANNEL FLOW Page 11 Locate this value in the last column of the Geometric Elements table and choose the closest value of dimensionless depth in the first column. The normal depth of flow is y 0.0 y n 0.0 ().40 ft Added note: If you compute a dimensionless quantity larger than the highest value given in the table, i.e., AR > 0.5 this means the pipe will flow full for the stated discharge and the sewer likely will surcharge. Solution for critical depth Set Froude number equal to 1 and rearrange it as A Q fn(y c ) T g Form it dimensionless as A y fn( ) 5 T where A Q (40) T g.() Use the Geometric Elements table to find y/. y 0.69ft The critical depth of flow is y c 0.69 ().07 ft Since y n > y c, the flow is subcritical. ECIV 6 Introduction to Water Resources Engineering

12 OPEN CHANNEL FLOW Page 1 Nonuniform Flow Nonuniform flow varies in depth along the channel reach. Figure 9. Nonuniform Flow Most flows are nonuniform because Most channels are non-prismatic. Flow may be nonuniform in a prismatic channel due the influence of a control, e.g., backwater created by a high tailwater depth or drawdown at a free overfall. Control A control is a channel feature, usually structural, where there is a unique (one-to-one) relationship between depth and discharge. A control regulates (controls) the state of flow. Examples: Free overfall at the end of a mild channel Weirs, Flumes (critical controls) Infinitely long prismatic channel (control reach) Subcritical flow is controlled by downstream conditions. Supercritical flow is controlled by upstream conditions. Channel Classification Channel bed slopes are classified hydraulically as mild, steep, critical, horizontal, and adverse. A channel bed slope may classify as mild for one flow rate and steep for another. The classification is based on the relationship between the normal depth of flow, y n, and the critical depth of flow, y c, for a given flow rate. The bed slope is called: Mild if y n >y c Steep if y c >y n Critical if y n y c For these three cases, the channel slope is sustaining, meaning the bed slopes downward in the direction of flow. Channels with horizontal or adverse bed slopes are termed non-sustaining because the channel bed does not slope downward in the direction of flow. Channels with horizontal bed slopes have a flat (zero) slope. As such, y n does not exist; only y c exists. Channels with adverse bed slopes slope upward in the direction of flow. Similar to the horizontal bed slope, y n does not exist, only y c. ECIV 6 Introduction to Water Resources Engineering

13 OPEN CHANNEL FLOW Page 1 Flow Profiles epending on the relationship between y n and y c, there can be three zones where y(x) occurs. Figure 10. Flow Profile Zones on a Hydraulically Mild Channel 1. If y(x)>y n, then the flow profile exists in Zone 1. In other words, the actual depth of flow, y(x), is greater than y n and occurs in Zone 1.. If y n >y(x)>y c, then the flow profile exists in Zone.. If y c >y(x), then the flow profile exists in Zone. Flow profiles are identified using a two-character label. See Figures 11 and 1. The first is an alpha character that identifies the type of channel slope (e.g., M for mild, S for steep). The second is a numeric character that identifies the zone (e.g., 1, or ). For a hydraulically mild channel. If y(x)>y n, the profile exists in Zone 1, and is an M1 profile. If y n >y(x)>y c, the profile exists in Zone, and is an M profile. If y c >y(x), the profile exists in Zone, and is an M profile. Zone 1 flow profiles are known as backwater profiles because the water "backs-up" due to a downstream control that restricts the outflow and forces the water to pond to a greater depth to "push" the flow through/past the control. Zone profiles are known as drawdown or frontwater curves. Flow Profile Analysis Nonuniform flow varies in depth along the channel reach. Stated mathematically, y 0 x The change in depth is known as the flow profile and is determined mathematically by flow profile analysis. Flow profile analysis is performed to determine the variation in depth, y(x), along a given profile. Calculations involve solving the energy equation at different points along the profile. Flow profile analysis is used in storm sewer design when determining the hydraulic grade line. Theory Consider the definition figure for nonuniform flow in an open channel (Figure 1). Applying energy conservation between sections 1 and gives EGL 1 EGL + hf (44) where EGL 1 is the total energy at section 1 EGL is the total energy at section h f is energy lost to friction between 1 and. ECIV 6 Introduction to Water Resources Engineering

14 OPEN CHANNEL FLOW Page 14 Figure 11. Mild Channel Profiles Figure 1. Steep Channel Profiles Figure 1. efinition Figure ECIV 6 Introduction to Water Resources Engineering

15 OPEN CHANNEL FLOW Page 15 Rewrite the energy equation in terms of specific energy E 1 + z1 E + z + hf (45a) Rearrange it as E E z z + (45b) 1 1 hf The difference in elevation heads is product of channel bed slope, S o, and reach length, Δx. z1 z So Δx (46) and h f is the product of friction slope, S f, and reach length. hf Sf Δ x (47) S f is the slope of the energy grade line. After substitution, E - E1 So Δ x - Sf Δx (4) ividing both sides by Δx, we get E E1 S0 Sf Δx (49) This is the general form of the energy equation we solve to determine the depth profile, i.e., y(x). Because specific energy is evaluated at both ends of a channel reach (Δx), convention is to evaluate the friction slope term at both ends of the reach to approximate the average friction slope over the reach. E E1 So Sf (50) Δx in which Sf1 + S S f f (51) where S f1 is the friction slope evaluated at 1 and S f is the friction slope evaluated at. We use Manning's equation to approximate S f. Simply substitute S f for S o and solve for S f. n Q Sf (English) 4.1A R (5) n Q (SI) 4 A R Solution Methods Most flow profile analysis programs use numerical solution techniques. Two methods are commonly used. 1. irect Step Method. Standard Step Method irect Step Method Involves an explicit (direct) numerical solution of the energy equation. Applies only to prismatic channels. Involves solving for the position (x-location) of user specified y-values along the flow profile. Rule-of-thumb: Given y, find x. ECIV 6 Introduction to Water Resources Engineering

16 OPEN CHANNEL FLOW Page 16 Procedure: 1. etermine the design flow rate, Q, channel geometry and dimensions, bed slope, and Manning's n-value.. etermine normal depth, y n, and critical depth, y c, for the design flow.. Classify the channel bed slope. Channel is mild if y n >y c, steep if y c >y n, etc. 4. etermine the location of the control and the depth of flow at the control. For example, if you are dealing with a hydraulically mild storm sewer with a free overfall at the outlet, the flow will pass through critical depth at the outlet, meaning the control is at the outlet. 5. Classify the profile type, e.g., M1, M, S1, etc. For the example, the upstream profile is an M or drawdown curve. 6. Knowing the profile type and the depth at the control, establish the range of values for the depth profile, y(x). For the example, y c <y(x)<y n. 7. Create a worktable (See Example ). Starting at the control, specify 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 farther from the control. Note: If the flow is subcritical, the control is downstream, and the profile is evaluated in the upstream direction, starting at the control. If the flow is supercritical, the control is upstream, and the profile 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 of each point from the control. 9. Plot y(x) versus x. This is the desired flow profile. Standard Step Method Involves an iterative numerical solution of the energy equation. Applies to any channel: natural, prismatic, concrete, etc. Involves solving for the depth at user specified x-locations. Rule-of-thumb: Given x, find y. Procedure: 1. etermine the design flow rate, Q.. etermine the channel x-sectional geometry, slope and Manning's n-value.. If 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 of flow. Also, measure the distance of each x-section from the control, and determine the bed slope between adjacent x-sections. 4. etermine Δx between adjacent x-sections (known Δx values). 5. Evaluate normal depth, y n, and critical depth, y c. If the channel is not prismatic, do this for each sub-reach, Δx. 6. Classify the channel bed slope. If the channel is not prismatic, to have one continuous profile type, all subreaches must have the same bed slope classification; otherwise, multiple profiles exist. 7. etermine the location and flow depth at the control.. Classify the profile. 9. Create a worktable. Starting at the control, choose a trial y value at the first x-section from the control. 10. Solve the energy equation between the two x-sections to determine the distance, Δx, between them (predicted Δx). If the predicted Δx is sufficiently close to the known Δx, the assumed y-value is good; otherwise, adjust the assumed y-value and repeat the process. 11. Repeat the process for all sub-reaches. 1. Plot y(x) versus x. This is the desired flow profile. ECIV 6 Introduction to Water Resources Engineering

17 OPEN CHANNEL FLOW Page 17 Example Flow Profile Analysis Using the irect Step Method Given a 6-inch RCP storm sewer, 500-ft long on 0.5% grade, carrying 0 cfs. If the depth of flow at the outlet is. ft, determine the flow profile in the sewer using the direct step method. Use n etermine y n : Use the dimensionless geometric elements table. y n fn AR 0. y 0. nq 1.49 S ( 0) () Find 65 y n 0.65 o ( ) 1.91 ft etermine y c : Use the dimensionless geometric elements table. fn y c A Q 5 T g 5 Find Classify the channel slope: y n > y c --> Mild channel etermine profile type: TW > y n --> M1 profile Control is at the outlet; upstream flow is subcritical. Specify range of y-values: 1.91 < y(x) <.0 Create a worktable: y y c ( ) ft y A v E ΔE AR / S f S f Δx x ECIV 6 Introduction to Water Resources Engineering

18 OPEN CHANNEL FLOW Page 1 Equations: E Q v A v y + g n Q S f.1 AR ΔE ΔE Δ x S Sf Sf o / / ( ) ( AR ) Interpolation table: y y A AR ECIV 6 Introduction to Water Resources Engineering

19 OPEN CHANNEL FLOW Page 19 Geometric Elements for Circular Pipes Y/ A/ R/ T/ A /T 5 AR / / / Y/ A/ R/ T/ A /T 5 AR / / / ECIV 6 Introduction to Water Resources Engineering

20 OPEN CHANNEL FLOW Page ***** 0.1 ECIV 6 Introduction to Water Resources Engineering