Chapter 9. Steady Flow in Open channels

Transcription

1 Chapter 9 Steady Flow in Open channels

2 Objectives Be able to define uniform open channel flow Solve uniform open channel flow using the Manning Equation

3 9.1 Uniform Flow in Open Channel Open-channel flows are characterized by the presence of a liquid-gas interface called the free surface. Natural flows: rivers, creeks, floods, etc. Human-made systems: freshwater aqueducts, irrigation, sewers, drainage ditches, etc.

4 In an open channel, Velocity is zero on bottom and sides of channel due to noslip condition Velocity is maximum at the midplane of the free surface In most cases, velocity also varies in the streamwise direction Therefore, the flow is 3D Nevertheless, 1D approximation is made with good success for many practical problems.

5 differences between pipe flow and open channel flow The flow of water in a conduit may be either open channel flow or pipe flow. The two kinds of flow are similar in many ways but differ in one important respect. Open-channel flow must have a free surface, whereas pipe flow has none. A free surface is subject to atmospheric pressure. In Pipe flow there exist no direct atmospheric flow but hydraulic pressure only.

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7 Open channel flow is driven by gravity rather than by pressure work as in pipes.

8 9.2 Classification of Open-Channel Flows The most common classification method is by rate of change of free-surface depth. The classes are summarized as 1. Uniform flow (constant depth and slope) 2. Varied flow a. Gradually varied (one-dimensional) b. Rapidly varied (multidimensional)

9 Classification of Open-Channel Flows Flow in open channels is also classified as being uniform or nonuniform, depending upon the depth y. Uniform flow (UF) encountered in long straight sections where head loss due to friction is balanced by elevation drop. Depth in UF is called normal depth y n

10 Classification of Open-Channel Flows Obstructions cause the flow depth to vary. Rapidly varied flow (RVF) occurs over a short distance near the obstacle. Gradually varied flow (GVF) occurs over larger distances and usually connects UF and RVF.

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12 9.3 Properties of open channels Artificial channels These are channels made by man. They include irrigation canals, navigation canals, spillways, sewers, culverts and drainage ditches. They are usually constructed in a regular cross-section shape throughout and are thus prismatic channels (they don t widen or get narrower along the channel. Natural channels Natural channels can be very different. They are not regular nor prismatic and their materials of construction can vary widely (although they are mainly of earth this can possess many different properties.) The surface roughness will often change with time distance and even elevation.

13 Properties of open channels Geometric properties necessary for analysis Depth (y) the vertical distance from the lowest point of the channel section to the free surface. Stage (z) the vertical distance from the free surface to an arbitrary datum Area (A) the cross-sectional area of flow, normal to the direction of flow Wetted perimeter (P) the length of the wetted surface measured normal to the direction of flow. Surface width (B) width of the channel section at the free surface Hydraulic radius (R) the ratio of area to wetted perimeter (A/P) Hydraulic mean depth (Dm) the ratio of area to surface width (A/B)

14 The wetted perimeter does not include the free surface.

15 Multiple Choice Consider an open rectangular channel 3m wide laid on a 1 slope. If the water depth is 2m, the hydraulic radius is: (a) 0.43m (b) 0.60m (c) 0.86m (d) 1.00m

16 9.4 Continuity and Energy Equations 1D steady continuity equation can be expressed as 1D steady energy equation between two stations Head loss h L is expressed as 2 L V h L = f 4R 2g

17 9.5 Uniform Flow in Channels Occurs in long straight runs of constant slope The velocity is constant with V = V o Water depth is constant with y = y n Slope is constant with S o = tanα

18 Head loss in a Channel Recall: Darcy-Weisbach Equation For circular pipes: L V h = f f D 2g 2 For non-circular conduits: 2 L V h = f R = f 4R 2g A χ

19 Head loss in a Channel 2 L V h f = S L = f 0 4R 2g Solve for V: V = 4S 0 R2g f V = 8g f R 1/ 2 S 1/ 2 0

20 Chezy Formula For a given channel shape and roughness: 8g = Constant f This leads to the Chezy Formula = C Named after Antoine Chezy, who did experiments in the River Seine in 1760s V C RS = = 0 C Ri C Chezy Coefficient i the bed slope of the channel

21 Manning s Analysis Robert Manning (Irish engineer, 1880s) Found C increases with channel size: C = 1 R 1/ 6 n Manning s formula Where n = a roughness coefficient

22 Values of Manning n Lined Canals n Cement plaster Untreated gunite Wood, planed Wood, unplaned Concrete, trowled Concrete, wood forms, unfinished Rubble in cement Asphalt, smooth Asphalt, rough Natural Channels Gravel beds, straight Gravel beds plus large boulders Earth, straight, with some grass Earth, winding, no vegetation Earth, winding with vegetation 0.050

23 How to get n? Field studies expensive! References Estimate based on roughness height

24 9.6 Computations in uniform flow We can use Manning's formula for discharge to calculate steady uniform flow. Two calculations are usually performed to solve uniform flow problems. 1. Discharge from a given depth 2. Depth for a given discharge In steady uniform flow the flow depth is know as normal depth.

25 example 1 Discharge from depth in a trapezoidal channel A concrete lined trapezoidal channel with uniform flow has a normal depth is 2m.The base width is 5m and the side slopes are equal at 1:2 Manning's n can be taken as And the bed slope S0 = What are: a) Discharge (Q) b) Mean velocity (V) c) Reynolds number (Re)

26 Solution: Calculate the section properties the mean velocity And the Reynolds number This is very large - i.e. well into the turbulent zone - the application of the Manning's equation was therefore valid.

27 Example: Manning Formula What is the flow capacity of a finished concrete channel that drops 1.2 m in 3 km? m 3 m

28 Example solution 1 Q = AR n 2/3 S 1/ 2 0 A = ( 3m)(1.5m) + (3m)(1.5m) = 9m P = = ( 3m) + 2 (3m) + (1.5m) 9.71m n = S 1 Q = (9m Q = 14.3m / s m = = m )(0.927m) 2/3 2 (0.0004) 1/ 2 A R = = m P

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31 9.7 Best Hydraulic Cross Sections Best hydraulic cross section for an open channel is the one with the minimum wetted perimeter for a specified cross section (or maximum hydraulic radius R h ) Also reflects economy of building structure with smallest perimeter

32 Best Hydraulic Cross Sections Example: Rectangular Channel Cross section area, A c = yb Perimeter, p = b + 2y Solve A c for b and substitute Taking derivative with respect to To find minimum, set derivative to zero Best rectangular channel has a depth 1/2 of the width

33 Best Hydraulic Cross Sections Same analysis can be performed for a trapezoidal channel Similarly, taking the derivative of p with respect to q, shows that the optimum angle is For this angle, the best flow depth is

34 Multiple Choice Consider an open rectangular channel 3m wide laid on a 1 slope. The most efficient water depth (best depth for a given flow and resistance ) is: (a) 1.0m (b) 1.5m (c) 2.0m (d) 2.5m