FUNDAMENTALS OF FLUID MECHANICS Chapter 10 Flow in Open Channels

FUNDAMENTALS OF FLUID MECHANICS Chapter 10 Flow in Open Channels Jyh-Cherng Shieh Department of Bio-Industrial Mechatronics Engineering National Taiwan University 1

MAIN TOPICS General Characteristics of Open-Channel Flow Surface Waves Energy Considerations Uniform Depth Channel Flow Gradually Varies Flow Rapidly Varies Flow

Introduction Open channel flow involves the flows of a liquid in a channel or conduit that is not completely filled. There exists a free surface between the flowing fluid (usually water) w and fluid above it (usually the atmosphere). The main deriving force is the fluid weight-gravity gravity forces the fluid to flow downhill. Under steady, fully developed flow conditions, the component if the weight force in the direction of flow is balanced by the equal and a opposite shear force between the fluid and the channel surface. 3

Open Channel Flow vs. Pipe Flow There can be no pressure force driving the fluid through the channel or conduit. For steady, fully developed channel flow, the pressure distribution ion within the fluid is merely hydrostatic. 4

Examples of Open Channel Flow The natural drainage of water through the numerous creek and river systems. The flow of rainwater in the gutters of our houses. The flow in canals, drainage ditches, sewers, and gutters along roads. The flow of small rivulets, and sheets of water across fields or parking lots. The flow in the chutes of water rides. 5

Variables in Open-Channel Flow Cross-sectional sectional shape. Bends. Bottom slope variation. Character of its bounding surface. Most open-channel flow results are based on correlation obtained from model and full-scale experiments. Additional information can be gained from various analytical and numerical efforts. 6

General Characteristics of Open-Channel Flow 7

Classification of Open-Channel Flow For open-channel flow, the existence of a free surface allows additional types of flow. The extra freedom that allows the fluid to select its free-surface location and configuration allows important phenomena in open- channel flow that cannot occur in pipe flow. The fluid depth, y, varies with time, t, and distance along the channel, x, are used to classify open-channel flow: 8

Classification - Type I Uniform flow (UF): The depth of flow does not vary along the channel (dy/dx( dy/dx0). Nonuniform flows: Rapidly varying flows (RVF): The flow depth changes considerably over a relatively short distance dy/dx~1. Gradually varying flows (GVF): The flow depth changes slowly with distance dy/dx <<1. 9

Classification - Type II R e ρvr / h μ V is the average velocity of the fluid. R h is the hydraulic radius of the channel. Laminar flow: Re < 500. Transitional flow: Turbulent flow: Re > 1,500. Most open-channel flows involve water (which has a fairly small viscosity) and have relatively large characteristic lengths, it is uncommon to have laminar open-channel flows. 10

Classification - Type III F r V / gl Critical Flow: Froude number Fr 1. Subcritical Flow: Froude number Fr <1. Supercritical Flow: Froude number Fr >1. 11

Surface Wave 1

Surface Wave The distinguishing feature of flows involve a free surface (as in i open-channel flows) is the opportunity for the free surface to distort into various shapes. The surface of a lake or the ocean is usually distorted into ever- changing patterns associated with surface waves. 13

Kinds of Surface Wave Some of the surface waves are very high, some barely ripple the surface; some waves are very long, some are short; some are breaking wave that form white caps, others are quite smooth. small amplitude Finite-sized solitary Continuous sinusoidal shape 14

small amplitude Wave Speed 1/5 Consider a single elementary wave of small height, by δy,, is produced on the surface of a channel by suddenly moving the initially stationary end wall with speed δv. 15

small amplitude Wave Speed /5 The water in the channel was stationary at the initial time, t0. A stationary observer will observe a single wave move down the channel with a wave speed c, with no fluid motion ahead of the wave and a fluid velocity of δv behind the wave. The motion is unsteady. For a observer moving along the channel with speed c, the flow will w appear steady. v v To this observer, v the fluid v velocity will be V -c i on the observer s right and V (-c + δv) i to the left of the observer. Momentum Equation + Continuity Equation 16

small amplitude Wave Speed 3/5 With the assumption of uniform one-dimensional flow, the continuity equation becomes cyb ( c + δv)(y + δy)b (y + δy) δv δv c y (1) δy δy δy << y Similarly, the momentum equation F F γy F 1 c A 1 γy 1 γy b 1 b F 1 γ(y + δy) γy c1 A 1 b ρbcy 1 γ(y + δy) [(c δv) c] b 17

small amplitude Wave Speed 4/5 ( δy) << yδy δv δy g c () (1)+() c gy (3) Energy Equation + Continuity Equation 18

small amplitude Wave Speed 5/5 The single wave on the surface is seen by an observer moving with the wave speed, c. Since the pressure is constant at any point on the free surface, the Bernoulli equation for this frictionless flow is V VδV + y cons tan t + δy g g The continuity equation 0 Vy cons tan t yδv + Vδy 0 Combining these two equations and using the fact Vc c gy 19

finite-sized solitary Wave Speed More advanced analysis and experiments show that the wave speed for finite-sized solitary wave c 1/ δy δy gy 1+ c gy 1+ > gy y y 1/ The larger the amplitude, the faster the wave travel. 0

Continuous sinusoidal shape Wave Speed 1/ A more general description of wave motion can be obtained by considering continuous (not solitary) wave of sinusoidal shape. By combining waves of various wavelengths, λ,, and amplitudes, δy. The wave speed varies with both the wavelength and fluid depth as c gλ πy tanh π λ 1/ & (4) 1

Continuous sinusoidal shape Wave Speed / c & gλ π tanh πy λ 1/ y λ c tanh gλ π πy λ 1 Deep layer y λ 0 c tanh gy πy λ πy λ Shallow layer Wave speed as a function of wavelength.

Froude Number Effects 1/3 Consider an elementary wave travelling on the surface of a fluid. If the fluid layer is stationary, the wave moves to the right with speed c relative to the fluid and stationary observer. When the fluid is flowing to the left with velocity V. If V<c, the wave will travel to the right with a speed of c-v. c If Vc, the wave will remain stationary. If V>c, the wave will be washed to the left with a speed of V-c. V Froude Number F r V / gl V / c Is the ratio of the fluid velocity to the wave speed. 3

Froude Number Effects /3 When a wave is produced on the surface of a moving stream, as happens when a rock is thrown into a river. If V0, the wave speeds equally in all directions. If V<c, the wave can move upstream. Upstream locations are said to be in hydraulic communication with the downstream locations. Such flow conditions, V<c, or Fr<1, are termed subcritical. If V>c, no upstream communication with downstream locations. Any disturbance on the surface downstream from the observer will be washed farther downstream. Such conditions, V>c, or Fr>1, are termed supercritical. 4

Froude Number Effects 3/3 If Vc or Fr1, the upstream propagating wave remains stationary and the flow is termed critical. 5

Energy Considerations 6

Energy Considerations 1/3 S 0 z1 z The slope of the channel bottom or bottom l slope is constant over the segment Very small for most open-channel flows. x and y are taken as the distance along the channel bottom and the depth normal to the bottom. 7

Energy Considerations /3 With the assumption of a uniform velocity profile across any section of the channel, the one-dimensional energy equation become p 1 V1 p V + + z 1 + + z + h L γ g γ g h L is the head loss due to viscous effects between sections (1) and (). (5) (5) z p 1 1 / γ z y 1 V1 V 1 + So y + h L y + l + (6) g g S o l p / γ y 8

Energy Considerations 3/3 (6) S f h L y / l V V1 1 y + (Sf So )l (7) g (7) y 1 y g For a horizontal channel bottom (S 0 0) and negligible head loss (S f 0) V V 1 9

Specific Energy 1/4 Define specific energy, E E y + V g (8) E1 E + (Sf S o )l (9) (9) E 1 + z1 E + z The sum of the specific energy and the elevation of the channel bottom Head losses are negligible, S f 0 remains constant. S This a statement of the Bernoulli ol z z1 equation. 30

Specific Energy /4 If the cross-sectional sectional shape is a rectangular of width b q (10) E y + gy Where q is the flowrate per unit width, qq/bvyb/b Vyb/bVy For a given channel b constant q constant E E (y) Specific energy diagram 31

Specific Energy 3/4 E y + q gy (10) For a given q and E, equation (10) is a cubic equation with three solutions, y sup y sub, and y neg. If E >E> min, two solutions are positive and y neg is negative (has no physical meaning and can be ignored). These two depths are term alternative depths. sup, 3

Specific Energy 4/4 Approach ye Very deep and very slowly y V sup sup < > y V sub sub E > E min Two possible depths of flow, one subcritical and the other supercritical Approach y0 Very shallow and very high speed 33

Determine Emin To determine the value of E min 1/ 3 de dy 0 de dy 1 Sub. (11) into (10) q gy 3 0 y c q g (11) 3yc q Emin Vc gyc Frc 1 y 1. The critical conditions (Fr1) occur at the location of E min. Flows for the upper part of the specific energy diagram are subcritical (Fr<1) 3. Flows for the lower part of the specific energy diagram are supercritical (Fr>1) c min. 34

Example 10.1 Specific Energy diagram - Qualitative Water flows under the sluice gate in a constant width rectangular channel as shown in Fig. E10.1a. Describe this flow in terms of the specific energy diagram. Assume inviscid flow. 35

1/ Example 10.1 Solution 1/ Inviscid flow S f 0 Channel bottom is horizontal z 1 z (or S o 0) E 1 E q 1 q The specific energy diagram for this flow is as shown in Fig. E10.1b. The flowrate can remain the same for this channel even if the upstream depth is increased. This is indicated by depths y 1 and y in Fig E10.1c. To remain same flowrate,, the distance between the bottom of the gate and the channel bottom must be decreased to give a smaller flow area (y < y ), and the upstream depth must be increased to give a bigger head (y 1 > y 1 ). 36

/ Example 10.1 Solution / On the other hand, if the gate remains fixed so that the downstream depth remain fixed (y y ), the flowrate will increase as the upstream depth increases to y 1 > y 1. q >q 0 37

Example 10. Specific Energy diagram Quantitative Water flows up a 0.5-ft ft-tall tall ramp in a constant width rectangular channel at a rate of q 5.75 ft /s as shown in Fig. E10.a. (For now disregard the bump )) If the upstream depth is.3 ft, determine the elevation of the water surface downstream of the ramp, y + z. Neglect viscous effects. 38

1/4 Example 10. Solution 1/4 With S 0 l z 1 -z and h L 0, conservation of energy requires that p 1 V 1 g 1.90 y γ + V y V y + y 1 z V 1 1 + p V 5.75ft γ 64.4 + The continuity equation /s V g + z (10.-1) 1) (10.-) ) y 3 1.90y + 0.513 y y 1.7ft 0.638ft 0.466ft 0 39

/4 Example 10. Solution /4 The corresponding elevations of the free surface are either y y + + z z 1.7ft + 0.638ft 0.50ft + 0.50ft.ft 1.14ft Which of these flows is to be expected? This can be answered by use of the specific energy diagram obtained from Eq.(10), which for this problem is E y + 0.513 y The diagram is shown in Fig.E10.(b). 40

3/4 Example 10. Solution 3/4 41

4/4 Example 10. Solution 4/4 The upstream condition corresponds to subcritical flow; the downstream condition is either subcritical or supercritical, corresponding to points or. Note that since E 1 E +(z -z 1 )E +0.5 ft, it follows that the downstream conditions are located to 0.5 ft to the left of the upstream conditions on the diagram... The surface elevation is y + z 1.7ft + 0.50ft.ft 4

Channel Depth Variations 1/3 Consider gradually varying flows. For such flows, dy/dx<<1, and it is reasonable to impose the one- dimensional velocity assumption. At an section the total head V H + y + z g and the energy equation The slop of the energy line dh dx dz dx dh L S f S dx o H + 1 H hl dh dx d dx V g + y + z V g dv dx + dy dx + dz dx 43

Channel Depth Variations /3 dh dx V g L dv dx V g + dv dx dy dx + S dy dx f S + S o 0 (1) For a given flowrate per unit width, q, in a rectangular channel of constant width b, we have V q y dv dx y q dy dx V y dy dx V dv V dy dy (1) F r (13) g dx gy dx dx 44

Channel Depth Variations 3/3 dy dx S S f o Sub. (13) into (1) (14) (1 Fr ) Depends on the local slope of the channel bottom, the slope of the energy line, and the Froude number. 45

Uniform Depth Channel Flow 46

Uniform Depth Channel Flow 1/3 Many channels are designed to carry fluid at a uniform depth all along their length. Irrigation canals. Nature channels such as rivers and creeks. Uniform depth flow (dy/dx( dy/dx0) can be accomplished by adjusting the bottom slope, S 0, so that it precisely equal the slope of the energy line, S f. A balance between the potential energy lost by the fluid as it coasts c downhill and the energy that is dissipated by viscous effects (head loss) associated with shear stress throughout the fluid. 47

Uniform Depth Channel Flow /3 Uniform flow in an open channel. 48

Uniform Depth Channel Flow 3/3 Typical velocity and shear stress distributions in an open channel: (a)) velocity distribution throughout the cross section. (b)( ) shear stress distribution on the wetted perimeter. 49

The Chezy & Manning Equation 1/6 Control volume for uniform flow in an open channel. 50

The Chezy & Manning Equation /6 Under the assumption of steady uniform flow, the x component of the momentum equation F 1 x ρq(v V ) 0 F F τ Pl + Wsin θ 0 (15) 1 w where F 1 and F are the hydrostatic pressure forces across either end of the control volume. P is wetted perimeter. 51

The Chezy & Manning Equation 3/6 y 1 y F 1 F (15) τ w Pl + Wsin θ 0 τ w Wsinθ Pl W γal R h A P τ w γals Pl o rr h S o (16) Wall shear stress is proportional to the dynamic pressure (Chapter 8) τ w Kρ V τ w ρ K is a constant dependent upon the roughness of the pipe V 5

The Chezy & Manning Equation 4/6 V (16) K ρ rr (17) hso V C R h So C is termed the Chezy coefficient Chezy equation Was developed in 1768 by A. Chezy (1718-1798), 1798), a French engineer who designed a canal for the Paris water supply. (17) 1/ V S o Reasonable V R h / 3 V R h Manning Equation 53

The Chezy & Manning Equation 5/6 In 1889, R. manning (1816-1897), 1897), an Irish engineer, developed the following somewhat modified equation for open-channel flow to more accurately describe the R h dependence: V R / 3 h n S 1/ o (18) Manning equation n is the Manning resistance coefficient. Its value is dependent on the surface material of the channel s s wetted perimeter and is obtained from experiments. It has the units of s/m 1/3 or s./ft 1/3 54

The Chezy & Manning Equation 6/6 / 3 1/ (18) V κ R h S o (19) Where κ1 if SI units are n used, κ1.49 if BG units / 3 1/ Q κ AR h S o (0) are used. n The best hydraulic cross section is defined as the section of minimum area for a given flowrate Q, slope, S o, and the roughness coefficient, n. Q R h κ A n A P A P / 3 S 1/ o κ n A 5/ 3 P S / 3 1/ o A nq ks 1/ o 3/5 P /5 constant A channel with minimum A is one with a minimum P. 55

Value of the Manning Coefficient, n 56

Uniform Depth Examples 57

Example 10.3 Uniform Flow, Determine Flow Rate Water flows in the canal of trapezoidal cross section shown in Fig. F E10.3a. The bottom drops 1.4 ft per 1000 ft of length. Determine the flowrate if the canal is lined with new smooth concrete. Determine the Froude number for this flow. 58

1/ Example 10.3 Solution 1/ / 3 1/ (0) Q κ AR h S o κ1.49 if BG units are used. n 5 A 1ft(5ft) + 5ft ft 89.8ft tan 40 P 1ft + (5/ sin 40 ft) 7.6ft R h A / P 3.5ft From Table 10.1, n0.01 Q 1.49 n (89.8ft )(3.5ft) / 3 (0.0014) 1/ 10.98 n 915cfs V V Q / A 10.ft / s Fr... gy 0.804 59

/ Example 10.3 Solution / 60

Example 10.4 Uniform Flow, Determine Flow Depth Water flows in the channel shown in Fig. E10.3 at a rate o Q 10.01 m 3 /s. If the canal lining is weedy, determine the depth of the flow. 61

Example 10.4 Solution A 1.19y P R h 3.66 A P + + 3.66y y sin 40 1.19y 3.11y + + 3.66y 3.66 3.11y + 3.66 From Table 10.1, n0.030 Q 10 κ n / 3 1/ AR h S o 1.0 0.03... (1.19y + 3.66y) 5 515(3.11y + 3.66) 0 y1.50 m 6

Example 10.5 Uniform Flow, Maximum Flow Rate Water flows in a round pipe of diameter D at a depth of 0 y D, as shown in Fig. E10.5a. The pipe is laid on a constant slope of S 0, and the Manning coefficient is n. At what depth does the maximum flowrate occur? Show that for certain flowrate there are two depths possible with the same flowrate.. Explain this behavior. 63

1/ Example 10.5 Solution 1/ / 3 1/ (0) Q κ AR h S o κ1.49 if BG units are used. n D A ( θ sin θ) 8 Dθ A D( θ sin θ) P R h P 4θ 8 / 3 5/ 3 κ 1/ D ( θ sin θ) Q S o / 3 / 3 n 8(4) θ This can be written in terms of the flow depth by using y D [1 cos( θ / )] 64

/ Example 10.5 Solution / A graph of flowrate versus flow depth, Q Q(y), has the characteristic indicated in Fig. E10.5(b). The maximum flowrate occurs when y0.938d, or θ303º Q Qmax when y 0.938D 65

Example 10.6 Uniform Flow, Effect of Bottom Slope Water flows in a rectangular channel of width b 10 m that has a Manning coefficient of n 0.05. Plot a graph of flowrate,, Q, as a function of slope S 0, indicating lines of constant depth and lines of constant Froude number. 66

1/ Example 10.6 Solution 1/ A (19) by 10y (gy) S o V 1/ Fr κ n R R h A P by (b + y) / 3 / 3 1/ 1.0 10y 1/ h S o S o 1.0 0.05 0.00613Fr 0.05 10 + 10y 10 y + 5 + y y 5y / 3 4 / 3 S y 1/ o (10.6-) (10.6-1) 1) 67

/ Example 10.6 Solution / For given value of Fr, we pick various value of y, determine the corresponding value of S o from Eq (10.6-) ),, and then calculate QVA, with V from either Eq (10.6-1) 1) or V(gy) 1/ Fr. The results are indicated in Fig. E10.6 68

Example 10.7 Uniform Flow, Variable Roughness Water flows along the drainage canal having the properties shown in Fig. E10.7a. If the bottom slope is S 0 1 ft/500 ft0.00, estimate the flowrate when the depth is y 0.8 ft + 0.6 ft 1.4 ft. 69

Example 10.7 Solution Q Q + 1 + Q Q3 Q i 1.49 n i A i R / 3 h i S 1/ o Q... 16.8ft 3 / s 70

Example 10.8 Uniform Flow, Best Hydraulic Cross Section Water flows uniformly in a rectangular channel of width b and depth y. Determine the aspect ratio, b/y,, for the best hydraulic cross section. 71

1/3 Example 10.8 Solution 1/3 (0) / 3 1/ Q κ AR h S o n κ1.49 if BG units are used. A by P b + y A by A Ay R h P (b + y) (b + y) (y + A) Q κ n A A 5/ Ay (y + y A) K(y / 3 + S A) 1/ o K constant nq κs 1/ o 3/ 7

/3 Example 10.8 Solution /3 The best hydraulic section is the one that gives the minimum A for all y. That is, da/dy 0. da dy A 0 5/ 5 A 4ky 3/ da dy y + A 5/ K 4y + da dy y by The rectangular with the best hydraulic cross section twice as wide as it is deep, or b / y 73

3/3 Example 10.8 Solution 3/3 The best hydraulic cross section for other shapes 74

Gradually Varied Flow dy << dx 1 75

Gradually Varied Flow 1/ Open channel flows are classified as uniform depth, gradually varying or rapidly varying. If the channel bottom slope is equal to the slope of the energy line, S o S f, the flow depth is constant, dy/dx0. The loss in potential energy of the fluid as it flows downhill is i exactly balanced by the dissipation of energy through viscous effects. If the bottom slope and the energy line slope are not equal, the flow depth will vary along the channel. 76

Gradually Varied Flow / dy dx S f (1 F S o r (14) ) The sign of dy/dx,, that is, whether the flow depth increase or decrease with distance along the channel depend on S f -S o ad 1-Fr1 77

Classification of Surface Shapes 1/3 The character of a gradually varying flow is often classified in terms of the actual channel slope, S o, compared with the slope required to produce uniform critical flow, S oc. The character of a gradually varying flow depends on whether the fluid depth is less than or greater than the uniform normal depth, y n. 1 possible surface configurations 78

Classification of Surface Shapes /3 Fr<1 : y>y c Fr>1 : y<y c 79

Examples of Gradually Varies Flows 1/5 Backwater curve Drop-down profile Typical surface configurations for nonuniform depth flow with a mild slope. S 0 < S 0c. 80

Examples of Gradually Varies Flows /5 Typical surface configurations for nonuniform depth flow with a critical slope. S 0 S 0c. 81

Examples of Gradually Varies Flows 3/5 Typical surface configurations for nonuniform depth flow with a steep slope. S 0 > S 0c. 8

Examples of Gradually Varies Flows 4/5 Typical surface configurations for nonuniform depth flow with a horizontal slope. S 0 0. 83

Examples of Gradually Varies Flows 5/5 Typical surface configurations for nonuniform depth flow with a adverse slope. S 0 <0. 84

Classification of Surface Shapes 3/3 The free surface is relatively free to conform to the shape that satisfies the governing mass, momentum, and energy equations. The actual shape of the surface is often very important in the design d of open-channel devices or in the prediction of flood levels in natural channels. The surface shape, yy(x y(x), can be calculated by solving the governing differential equation obtained from a combination of the Manning equation (0) and the energy equation (14). Numerical techniques have been developed and used to predict open-channel surface shapes. 85

Rapidly Varied Flow dy dx ~ 1 86

Rapidly Varied Flow Rapidly varied flow: flow depth changes occur over a relatively short distance. Quite complex and difficult to analyze in a precise fashion. Many approximate results can be obtained by using a simple one-dimensional model along with appropriate experimentally determined coefficients when necessary. 87

Occurrence of Rapidly Varied Flow 1/ Flow depth changes significantly un a short distance: The flow changes from a relatively shallow, high speed condition into a relatively deep, low speed condition within a horizontal distance e of just a few channel depths. Hydraulic Jump 88

Occurrence of Rapidly Varied Flow / Sudden change in the channel geometry such as the flow in an expansion or contraction section of a channel. Rapidly varied flow may occur in a channel transition section. 89

Example of Rapidly Varied Flow 1/ The scouring of a river bottom in the neighborhood of a bridge pier. p Responsible for the erosion near the foot of the bridge pier. The complex three-dimensional flow structure around a bridge pier. 90

Example of Rapidly Varied Flow / flow-measuring devices are based on Many open-channel flow principles associated with rapidly varied flows. Broad-crested weirs. Sharp-crested weirs. Critical flow flumes. Sluice gates. 91

Hydraulic Jump 9

The Hydraulic Jump 1/6 Under certain conditions it is possible that the fluid depth will change very rapidly over a short length of the channel without any a change in the channel configuration. Such changes in depth can be approximated as a discontinuity in the free surface elevation (dy/dx( dy/dx ). This near discontinuity is called a hydraulic jump.. 93

The Hydraulic Jump /6 A simplest type of hydraulic jump in a horizontal, rectangular channel. c Assume that the flow at sections (1) and () is nearly uniform, steady, and one-dimensional. 94

The Hydraulic Jump 3/6 The x component of the momentum equation F1 F ρq(v V) ρv1 y1b(v V 1) F1 pc1a1 γy F pca γy 1 b / b / y y V y 1 1 1 (V g V) (1) y bv1 ybv () The conservation of mass equation () 1 Q The energy equation V1 V 1 y + hl y + + (3) g g The head loss is due to the violent turbulent mixing and dissipation. 95

The Hydraulic Jump 4/6 (1)+()+(3) Nonlinear equations Other solutions? (1)+() Solutions One solution is y 1 y, V 1 V, h L 0 y1 y V1 y1 V1 y1 V y V 1 1 1 (y1 g y gy y y F r1 y + 1 y 1 y 1 1± 1+ 8F y 1 r1 0 Fr 1 V 1 gy y 1 1+ 1+ 8F y 1 y ) 1 r1 (4) 96

The Hydraulic Jump 5/6 (3) h y L 1 1 y y 1 + r1 F y 1 y 1 (5) (4)+(5) Depth ratio and dimensionless head loss across a hydraulic jump as a function of upstream Froude number. 97

The Hydraulic Jump 6/6 The head loss is negative if Fr 1 <1. Violate the second law of thermodynamics Not possible to produce a hydraulic jump with Fr 1 <1. 98

Classification of Hydraulic Jump 1/ The actual structure of a hydraulic jump is a complex function of Fr 1, even though the depth ratio and head loss are given quite accurately by a simple one-dimensional flow analysis. A detailed investigation of the flow indicates that there are essentially five type of surface and jump conditions. 99

Classification of Hydraulic Jump / 100

Hydraulic Jump Variations 1/ Hydraulic jumps can occur in a variety o channel flow configurations, not just in horizontal, rectangular channels as discussed above. Other common types of hydraulic jumps include those that occur in i sloping channels and the submerged hydraulic jumps that can occur just downstream of a sluice gate. 101

Hydraulic Jump Variations / Hydraulic jump variations: (a)) jump caused by a change in channel slope, (b)) submerged jump 10

Example 10.9 Hydraulic Jump Water on the horizontal apron of the 100-ft ft-wide spillway shown in Fig. E10.9a has a depth o 0.60 ft and a velocity of 18 ft/s. Determine ermine the depth, y, after the jump, the Froude numbers before and after the jump, Fr 1 and Fr, and the power dissipated, P d, with the jump. 103

1/3 Example 10.9 Solution 1/3 Conditions across the jump are determined by the upstream Froude number Fr (4) V 18ft /s 1 1 gy 1 (3.ft /s )(0.60ft) y 1 1 1 1 8F + + r... y 1 y 5.3(0.60ft) 4.10 3.19ft Since Q 1 Q, or V (y 1 V 1 )/y 3.39ft/s 5.3 104

/3 Example 10.9 Solution /3 Fr V 3.39ft /s gy 1 (3.ft /s )(3.19ft) 0.334 The poser dissipated, Pd, by viscous effects within the jump can be determined from the head loss P d γqh L γby 1 V 1 h L (3) V V h y 1 y L 1 + +....6ft g g Pd γqhl γby1v1 hl...1.5 10 ft lb /s 5 77hp 105

3/3 Example 10.9 Solution 3/3 q q 1 E y + q q gy Q b y 1 V 1 1.81 y + y 10.8ft /s Various upstream depth 106

Weirs and Gate 107

Weir A weir is an obstruction on a channel bottom over which the fluid must flow. Weir provides a convenient method of determining the flowrate in an open channel in terms of a single depth measurement. 108

Sharp-Crested Weir 1/4 A sharp-crested weir is essentially a vertical-edged edged flat plate placed across the channel. The fluid must flow across the sharp edge and drop into the pool downstream of the weir plate. 109

Sharp-Crested Weir - Geometry /4 Sharp-crested weir plate geometry: (a) rectangular, (b) triangular, (c) trapezoidal. 110

Sharp-Crested Weir Flowrate 3/4 Assume that the velocity profile upstream of the weir plate is uniform and that the pressure within the nappe is atmosphere. Assume that the fluid flows horizontally over the weir plate with a nonuniform velocity profile. 111

Sharp-Crested Weir Flowrate 4/4 With P B 0, the Bernoulli equation for flow along the arbitrary streamline A-B A B indicated can be written as p A V1 + + za (H + Pw h) γ g + p V z A 1 A + + H + pw + γ g V1 g u g (6) Since the total head for any particle along the vertical section (1) is the same V1 (6) u g h + u da g h H u h 0 Q l dh (7) 11

Rectangular Weir Flowrate 1/ For a rectangular weir, lb Q gb 0 H h + 1 V g gb H 3/ (8) Q gbh (9) 3 1/ dh 3 + 1 V g 3/ 1 V g 3/ (8) V 1 << g H Because of the numerous approximations made to obtain Eq.. (9) Q C wr 3 gbh 3/ (30) 113

Rectangular Weir Flowrate / C wr C wr wr is the rectangular weir coefficient. wr is function of Reynolds number (viscous effects), Weber number (surface tension effects), H/P w (geometry effects). In most practical situations, the Reynolds and Weber number effects are negligible, and the following correction can be used. C wr 0.611+ 0.075 H P w (31) 114

Triangular Weir Flowrate 1/ For a triangular weir V 1 << g H Q 8 15 θ tan 5/ gh l (H h) tan θ An experimentally determined triangular weir coefficient, C wt, is used to account for the real world effects neglected in the analysis so that Q C wt 8 15 tan θ gh 5/ (3) 115

Triangular Weir Flowrate / Weir coefficient for triangular sharp-crested weirs 116

About Nappe Flowrate over a weir depends on whether the napple is free or submerged. Flowrate will be different for these situations than that give by Eq.. (30) and (3). Flow conditions over a weir without a free nappe: : (a)( ) plunging nappe, (b)) submerged nappe. 117

Broad-Crested Weir 1/3 A broad-crested weir is a structure in an open channel that has a horizontal crest above which the fluid pressure may be considered hydrostatic. 118

Broad-Crested Weir /3 Generally, these weirs are restricted to the range 0.08 < H/L w < 0.50. For long weir block (H/L w < 0.08), head losses across the weir cannot be neglected. For short weir block (H/L w > 0.50), the streamlines of the flow over the weir are not horizontal. Apply the Bernoulli equation V H + P 1 w + yc + Pw + g V c g If the upstream velocity head is negligible H y Vc V1 c g V c g 119

Broad-Crested Weir 3/3 Since H V Vc gyc yc yc yc The flowrate is Q by 3 H 3/ 3/ 3/ V bycvc b gyc b g H Again an empirical broad-crested weir coefficient, C wb, is used to account for the real world effects neglected in the analysis so that 3 Q 3/ 3/ 0.65 Cwbb g H (33) Cwb 1/ 3 (33) (34) H 1 + P w 10

Example 10.10 Sharp-Crested and broad- Crested Weirs Water flows in a rectangular channel of width b m with flowrate between Q min 0.0 m 3 /s and Q max 0.60 m 3 /s. This flowrate is to be measured by using (a) a rectangular sharp-crested weir, (b) a triangular sharp-crested weir with θ90º,, or (c) a broad-crested weir. In all cases the bottom of the flow area over the weir is a distance P w 1 m above the channel bottom. Plot a graph of Q Q(H) for each weir and comment on which weir would be best for this application. 11

1/3 Example 10.10 Solution 1/3 For the rectangular weir with P w 1. (30)+(31) Q Q C wr 3 gbh 3/ 0.611 + 5.91(0.611 + 0.075H)H 3/ 0.075 H P w 3 gbh 3/ For the triangular weir 8 θ 5/ (3) Q Cwt tan gh 15.36C wt H 5/ 1

/3 Example 10.10 Solution /3 For the broad-crested weir (33)+(34) Q C Q wb b (1 + g. H) 3 1/ 3/ H H 3/ 3/ 1 + 0.65 H P w 1/ b g 3 3/ H 3/ 13

3/3 Example 10.10 Solution 3/3 14

Underflow Gates 1/4 A variety of underflow gate structure is available for flowrate control at the crest of an overflow spillway, or at the entrance of an irrigation canal or river from a lake. Three variations of underflow gates: (a)( ) vertical gate, (b)( ) radial gate, (c)( ) drum gate. 15

Underflow Gates /4 The flow under the gate is said to be free outflow when the fluid issues as a jet of supercritical flow with a free surface open to the atmosphere. In such cases it is customary to write this flowrate as q C a (35) d gy 1 Where q is the flowrate per unit width. The discharge coefficient, C d, is a function of contraction coefficient, C c y /a, and the depth ration y 1 /a. 16

Underflow Gates 3/4 Typical discharge coefficients for underflow gates 17

Underflow Gates 4/4 The depth downstream of the gate is controlled by some downstream obstacle and the jet of water issuing from under the gate is overlaid by a mass of water that is quite turbulent. Drowned outflow from a sluice gate. 18

Example 10.11 Sluice gate Water flows under the sluice gate shown in Fig. E10.11. The channel width is b 0 ft, the upstream depth is y 1 6 ft, and the gate is a 1.0 ft off the channel bottom. Plot a graph of flowrate,, Q, as a function of y 3. 19

1/ Example 10.11 Solution 1/ (35) q bq bacd gy1 393Cd cfs (Figure 10.9) Q 393(0.56) cfs Along the vertical line y 1 /a6. For y 3 6 ft, C d 0 The value of C d increases as y 3 /a decreases, reading a maximum of C d 0.56 when y 3 /a3.. Thus with y 3 3.a3.ft 0 cfs For y 3 < 3. ft the flowrate is independent of y 3, and the outflow is a free outflow. 130

/ Example 10.11 Solution / The flowrate for 3.ft y 3 6ft 131