Open Channel Flow. M. Siavashi. School of Mechanical Engineering Iran University of Science and Technology

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1 M. Siavashi School of Mechanical Engineering Iran University of Science and Technology W ebpage: webpages.iust.ac.ir/msiavashi Landline: Fall 2013 Introduction to Fluid Mechanics II 1

2 Objectives Understand how flow in open channels differs from flow in pipes Learn the different flow regimes in open channels and their characteristics Predict if hydraulic jumps are to occur during flow, and calculate the fraction of energy dissipated during hydraulic jumps Learn how flow rates in open channels are measured using sluice gates and weirs Introduction to Fluid Mechanics II 2

3 Classification of Open-Channel Flows 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: fresh-water aqueducts, irrigation, sewers, drainage ditches, etc. Introduction to Fluid Mechanics II 3

4 Classification of Open-Channel Flows In an open channel, Velocity is zero on bottom and sides of channel due to no-slip 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. Introduction to Fluid Mechanics II 4

5 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 Introduction to Fluid Mechanics II 5

6 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. Introduction to Fluid Mechanics II 6

7 Classification of Open-Channel Flows Like pipe flow, OC flow can be laminar, transitional, or turbulent depending upon the value of the Reynolds number Where r = density, m = dynamic viscosity, n = kinematic viscosity V = average velocity R h = Hydraulic Radius = 1/4D h = A c /p A c = cross-section area P = wetted perimeter Note that Hydraulic Diameter was defined in pipe flows as D h = 4A c /p = 4R h (D h is not 2R h, BE Careful!) Introduction to Fluid Mechanics II 7

8 Classification of Open-Channel Flows The wetted perimeter does not include the free surface. Examples of R h for common geometries shown in Figure at the left. Introduction to Fluid Mechanics II 8

9 Froude Number and Wave Speed OC flow is also classified by the Froude number Resembles classification of compressible flow with respect to Mach number Introduction to Fluid Mechanics II 9

10 Froude Number and Wave Speed Critical depth y c occurs at Fr = 1 y c At low flow velocities (Fr < 1) Disturbance travels upstream y > y c At high flow velocities (Fr > 1) Disturbance travels downstream y < y c 2 V = = g Q ga 2 c Introduction to Fluid Mechanics II 10

11 Froude Number and Wave Speed Important parameter in study of OC flow is the wave speed c 0, which is the speed at which a surface disturbance travels through the liquid. Derivation of c 0 for shallowwater Generate wave with plunger Consider control volume (CV) which moves with wave at c 0 Introduction to Fluid Mechanics II 11

12 Froude Number and Wave Speed Continuity equation (b = width) Momentum equation is a balance between net hydrostatic pressure and momentum 1 - gb y y y c by c V c 2 r é + d - = r - d - ë û 2 2 ( ) ù ( ) Introduction to Fluid Mechanics II 12

13 Froude Number and Wave Speed Combining the momentum and continuity relations and rearranging gives For shallow water, where dy << y, Wave speed c 0 is only a function of depth Introduction to Fluid Mechanics II 13

14 Froude Number and Wave Speed why we pay so much attention to flow being subcritical or supercritical? the liquid level drops gradually in the flow direction in subcritical flow, but a sudden rise in liquid level, called a hydraulic jump, may occur in supercritical flow (Fr>1) as the flow decelerates to subcritical (Fr<1) velocities Introduction to Fluid Mechanics II 14

15 Specific Energy Total mechanical energy of the liquid in a channel in terms of heads z is the elevation head y is the gage pressure head V 2 /2g is the dynamic head Taking the datum z=0 as the bottom of the channel, the specific energy E s is Introduction to Fluid Mechanics II 15

16 Specific Energy Q =constant For a channel with constant width b, Q = AcV = ybv E s = y + Q 2 2b y 2 2 Plot of E s vs. y for constant Q and b Introduction to Fluid Mechanics II 16

17 Specific Energy Q = constant A small change in Es near the critical point causes a large difference between alternate depths and may cause violent fluctuations in flow level. Operation near this point should be avoided. This plot is very useful Easy to see breakdown of E s into pressure (y) and dynamic (V 2 /2g) head E s as y 0 E s y for large y E s reaches a minimum called the critical point. There is a minimum Es required to support the given flow rate. 2 2 de s d æ Q ö Q = ç y = - = 2 3 dy dy è 2gb y ø gb y 2 1/3 æ Q ö y c = ç 2 è gb ø Noting that V c = gy c For a given E s > E s,min, there are two different depths, or alternating depths, which can occur for a fixed value of E s Introduction to Fluid Mechanics II 17

18 Example: Character of Flow and Alternate Depth Find: 1- Flow velocity 2- subcritical or supercritical? 3- Alternate flow depth? Q Q Q Q Ü Q Introduction to Fluid Mechanics II 18

19 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 in pipe flow, using the friction factor, and either the hydraulic diameter or radius Introduction to Fluid Mechanics II 19

20 Continuity and Energy Equations The change in elevation head can be written in terms of the bed slope a Introducing the friction slope S f The energy equation can be written as Introduction to Fluid Mechanics II 20

21 Uniform Flow in Channels Uniform depth occurs when the flow depth (and thus the average flow velocity) remains constant Common in long straight runs Flow depth is called normal depth y n Average flow velocity is called uniform-flow velocity V 0 Introduction to Fluid Mechanics II 21

22 Uniform Flow in Channels Uniform depth is maintained as long as the slope, cross-section, and surface roughness of the channel remain unchanged. During uniform flow, the terminal velocity reached, and the head loss equals the elevation drop We can solve for the velocity (or flow rate) Where C is the Chezy coefficient. f is the friction factor determined from the Moody chart or the Colebrook equation Introduction to Fluid Mechanics II 22

23 Uniform Flow in Channels Manning found that the Chezy coefficient can be expressed by the following relation: Experimental values of n (and the corresponding roughness height) are listed in Table 10.1 (white s book) for various channel surfaces Introduction to Fluid Mechanics II 23

24 Uniform Flow in Channels Introduction to Fluid Mechanics II 24

25 Example: Uniform Flow in Channels The asphalt-lined trapezoidal channel carries 300 ft 3 /s of water under uniform-flow conditions when S= What is the normal depth y n? From Table 10.1 n Introduction to Fluid Mechanics II 25

26 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 R h =A c /P Introduction to Fluid Mechanics II 26

27 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 y To find minimum, set derivative to zero Best rectangular channel has a depth 1/2 of the width Introduction to Fluid Mechanics II 27

28 Best Hydraulic Cross Sections For any angle q,the most efficient cross section for uniform flow occurs when the hydraulic radius is half the depth To find the correct depth y, these relations must be solved in conjunction with Manning s flowrate formula To find the best angle q, set dp/da=0, for constant A and y: To minimize P, evaluate dp/dy for constant A and a and set equal to zero For this angle, the best flow depth is Introduction to Fluid Mechanics II 28

29 Gradually Varied Flow In GVF, y and V vary slowly, and the free surface is stable In contrast to uniform flow, S f ¹ S 0. Now, flow depth reflects the dynamic balance between gravity, shear force, and inertial effects To derive how the depth varies with x, consider the total head Introduction to Fluid Mechanics II 29

30 Gradually Varied Flow Take the derivative of H Slope dh/dx of the energy line is equal to negative of the friction slope Bed slope has been defined Inserting both S 0 and S f gives Introduction to Fluid Mechanics II 30

31 Gradually Varied Flow Introducing continuity equation, which can be written as Differentiating with respect to x gives Substitute dv/dx back into equation from previous slide, and using definition of the Froude number gives a relationship for the rate of change of depth Introduction to Fluid Mechanics II 31

32 Gradually Varied Flow Subcritical flow (Fr<1): (1-Fr 2 >0) Supercritical flow (Fr>1): (1-Fr 2 <1) In uniform flow (y=y n ): S 0 =S f If y>y n, velocity decreases and cause a decrease in h L & S f S < S ÞS - S > f 0 0 f 0 If y<y n, velocity increases and cause an increase in h L & S f S > S Þ S - S < f 0 0 f 0 for horizontal (S 0 =0) and upward (S 0 <0) channels S0 - S f < 0 Introduction to Fluid Mechanics II 32

33 Gradually Varied Flow gradually varied flow involves acceleration and deceleration of liquid, and the surface profile reflects the dynamic balance between liquid weight, shear force, and inertial effects. This result is important. It permits classification of liquid surface profiles as a function of Fr, S 0, S f, and initial conditions. Bed slope S 0 is classified as Steep : y n < y c Critical : y n = y c Mild : y n > y c Horizontal : S 0 = 0 Adverse : S 0 < 0 Initial depth is given a number 1 : y > y n 2 : y n < y < y c 3 : y < y c Introduction to Fluid Mechanics II 33

34 Gradually Varied Flow 12 distinct configurations for surface profiles in GVF. S > S 0 c Introduction to Fluid Mechanics II 34

35 Gradually Varied Flow 12 distinct configurations for surface profiles in GVF. Introduction to Fluid Mechanics II 35

36 Gradually Varied Flow Typical OC system involves several sections of different slopes, with transitions Overall surface profile is made up of individual profiles described on previous slides Introduction to Fluid Mechanics II 36

37 Rapidly Varied Flow and Hydraulic Jump Flow is called rapidly varied flow (RVF) if the flow depth has a large change over a short distance Sluice gates Weirs Waterfalls Abrupt changes in cross section Often characterized by significant 3D and transient effects Backflows Separations Introduction to Fluid Mechanics II 37

38 Rapidly Varied Flow and Hydraulic Jump Consider the CV surrounding the hydraulic jump Assumptions 1. V is constant at sections (1) and (2), and b 1 and b 2» 1 2. P = rgy 3. t w is negligible relative to the losses that occur during the hydraulic jump 4. Channel is wide and horizontal 5. No external body forces other than gravity Introduction to Fluid Mechanics II 38

39 Rapidly Varied Flow and Hydraulic Jump Continuity equation X momentum equation Substituting and simplifying Quadratic equation for y 2 /y 1 Introduction to Fluid Mechanics II 39

40 Rapidly Varied Flow and Hydraulic Jump Solving the quadratic equation and keeping only the positive root leads to the depth ratio Energy equation for this section can be written as Head loss associated with hydraulic jump Introduction to Fluid Mechanics II 40

41 Rapidly Varied Flow and Hydraulic Jump Often, hydraulic jumps are avoided because they dissipate valuable energy However, in some cases, the energy must be dissipated so that it doesn t cause damage A measure of performance of a hydraulic jump is its fraction of energy dissipation, or energy dissipation ratio Introduction to Fluid Mechanics II 41

42 Rapidly Varied Flow and Hydraulic Jump Experimental studies indicate that hydraulic jumps can be classified into 5 categories, depending upon the upstream Fr Introduction to Fluid Mechanics II 42

43 Flow Control and Measurement Flow rate in pipes and ducts is controlled by various kinds of valves In OC flows, flow rate is controlled by partially blocking the channel. Weir : liquid flows over device Underflow gate : liquid flows under device These devices can be used to control the flow rate, and to measure it. Introduction to Fluid Mechanics II 43

44 Flow Control and Measurement Underflow Gate Free outflow Drowned outflow Underflow gates are located at the bottom of a wall, dam, or open channel Outflow can be either free or drowned In free outflow, downstream flow is supercritical In the drowned outflow, the liquid jet undergoes a hydraulic jump. Downstream flow is subcritical. Introduction to Fluid Mechanics II 44

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