in natural and artificial channels at siphons, aqueducts, weirs, falls, bridges ... summary of the empirical hydraulic design of transitions for flumes and siphons.
Tanta University Faculty of Engineering Irrigation and Hydraulics Department
State of the art of
Open channel Transitions Submitted to the scientific committee
(Water Resources and Public Works) Committee no. 51
By Ibrahim Mohamed Hussein rashwan Associate Professor Irrigation and Hydraulics Department Faculty of Engineering Tanta University
2013
Contents List of Figures..…..………….…….…….…….…………………………………3 List of Table.....…..……………….…….…….…………………..……………...4 Abstract…………..……………….…….…….………………………………….5 1. Introduction………………….……..…….………………………………….6 2. Design of Transitions…………………….………………………………….6 3. Governing Equations……………….…….……………….……………...….6 3.1. Specific Energy Equation.…………….……………….…………..…….6 3.2. Momentum Equation…………………..……………….……………….7 4. Types of Transitions…..……….…………..……………….…………….….8 4.1. Sudden Transitions………..….………….……….…………………….8 4.1.1. Sudden Horizontal Transitions..….…….….……….…...….…8 4.1.2. Sudden Vertical Transitions.…...……….....…………………..9 4.1.3. Sudden Combined Transitions……...……….………………….9 4.2. Gradually Transitions……………..………………………………….9 4.2.1. Gradually Combined Transitions……………………………….9 4.3. Constrictions…………………..…………..…………….…..……….10 5. Problems of Open channel Transitions……………………………………10 5.1. Horizontal Problems… …………………………………..….…….…10 5.2. Vertical Problems…..………..……………….……....…..……..……11 5.3. Combined Problems………………..……………………..….…..…..11 6. Solutions of the Problems of Open Channel Transitions …………..….….11 6.1. Analytical Solutions (trial-and-error and direct solutions)…..…..…12 6.2. Graphical Solution…………..……..…………….…………...…..….14 6.2.1. Specific Energy Diagram ………………..….……….…………14 6.2.2. Specific Discharge Diagram ………………………..……....…15 7. Energy Losses in Open Channel Transitions……………...….………….…17 8. Characteristics of Transitions...………………………………………….…21 9. Short Constrictions……………………………..…………….………….…33 9.1. Subcritical Flow through Constrictions………………………….…34 9.2. Chocking in Open Channel Transitions………………………….…35 10.Transitions as Discharge Devices..……………..…………….………….…39 Conclusions ………………………...…..……………..…………….………….…46 Suggestion for Future Research ….…..……………….…………….………….…46 References…….……………….………..……………..…………….………….…47 Appendix………………….…….……....……………..…………….………….…53
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List of Figures Figure 1 Energy in open channels…………………………………………...….…..7 Figure 2 Analysis of horizontal contraction………………………………..……….8 Figure 3 Horizontal transitions……………………………………………...………8 Figure 4 Vertical transitions……………………………………………..….………9 Figure 5 Gradually horizontal and vertical transitions for rectangular section….….9 Figure 6 Combined transitions……………………………………….…………....10 Figure 7 Sudden and gradually constriction for rectangular section constriction…10 Figure 8 Boundary conditions for open-channel transitions……………………..13 Figure 9 Channel contraction……………………………………...…………….13 Figure10 Distribution of water depth along the channel centerline……..…….…14 Figure 11 Specific discharge diagram……………………………………………..15 Figure 12 Specific energy diagram equations……………………………..………15 Figure 13 Variation of Q with y for exponential and trapezoidal channels……..…16 Figure 14 Variation of Q with y for circular channel…………………………....16 Figure 15 Variation of Q with y for exponential and rectangular channels for different values of z …………………………………………………...17 Figure 16 Various designs of sudden transitions for experiments (After G. Formica (1956))……………………………………………………………..18 Figure 17 Comparison of head loss of the models………………………………...20 Figure 18 Hydraulics efficiency of the Transition Models…...…………..…….....20 Figure 19Efficiency of the Transition Models………………………………….....20 Figure 20 Definition sketch: Rectangular expansion transition…………………...21 Figure 21 Definition sketch: abrupt expansion……………………………………22 Figure 22 Designs of straight-wall contractions…………………………………..22 Figure 23 Cross section of contraction…………………………………………….23 Figure 24 Separation of flow in an expansion of rapid divergence……………….23 2 Figure 25 Plot of F1 against y3/y1 using b3/b1 as a parameter for the analysis of horizontal contractions and expansions……………...….……….…….24
Figure 26 Plan view velocity distribution for a rapid expansion: B/b = 3.0………25 Figure 27 Channel expansion types, all the expansions have geometry symmetric about the centerline……………………………………………………26 Figure 28 Velocity distributions across the width and along the length of abrupt expansion……………………………………………………………….29 Figure 29 Velocity distributions in proposed transition…….……………………..30 Figure 30 Configuration with notation for expansion……....……………………..30 Figure 31 Possible shock wave patterns in a linear channel contraction….………31 Figure 32 Three inlets analyzed for the subcritical contraction…………….……..32 Figure 33 Sketch of the gradual channel transition………………………….…….32 3
Figure 34 Physical model of gradual channel transition…………………….…….32 Figure 35 Flow pattern in canal coarse mesh size......………...…………….…….33 Figure 36 Constriction in uniform flow channel (a, b) in subcritical flow; (e, d) in supercritical flow………………………………………………….…….33 Figure 37 Definition sketch of flow through constriction…………………………34 Figure 38 Sketch of choked flows through short horizontal contractions.………...36 Figure 39 Rating curve of critical flow for closed conduit with free surface.…….36 Figure 40 Definition sketches for three cases for different circular channel transitions…………………………………………………………..38 Figure 41 Water Surface Profiles of Choked Flows……………………………....39 Figure 42 Parshall flume…………………………………………………………..40 Figure 43 Cutthroat flume…………………………………………………………40 Figure 44 Mobile cylinders in various channel shapes……………………………41 Figure 45 Circular cones in rectangular channel…………………………..............42 Figure 46 Circular cones in triangular channel…………………………………....42 Figure 47 Schematic plot of mobile device unit…………………………………..43 Figure 48 Cross section of Venturi in circular flume……………………………...44 Figure 49 Mobile prism dimension detail…………………………………………44 Figure 50 Plan view of rectangular flume with side contraction………………….45
List of Table
Table 1Expansion coefficient ( )…………………………...…………………….18
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Open Channel Transitions Abstract In general, channel transitions are defined as changes in cross-sectional area in the direction of open channel flow. Transitions can also include changes in bed level. When a change of channel shapes, variation of bed elevation, transition is provided as a link between the original and the new channel. Such changes are often required in natural and artificial channels at siphons, aqueducts, weirs, falls, bridges and barrages for economic as well as practical reasons. The transitions may be sudden or gradual, horizontal (contracting or expanding) or vertical (rise or falling in bed) or combined and required for subcritical or supercritical flows. Open channel transitions can be solved with the use of specific energy equation or the momentum equation. The phenomenon is usually so complicated that the resulting flow pattern is not readily subject to any analytical solution. The existing methods of solving the problems of transition using specific energy equation are slide-rule operation, trialand-error solution, graphical solution, design tables prepared from the specific energy equation expressed in dimensionless form and direct solution. Previously, many researchers have been studied the problems of transitions. In 1920, the first laboratory investigation in the United States to the flow through a constriction in water way was mad. Some earlier investigators reported that, the purpose was to design a transition with minimum flow separation and hence small energy head losses. The calculation of energy losses and determination of the transition profile to provide a good velocity distribution at transition are two main problems areas that need the attention of hydraulic engineers. Many researchers assumed and discussed the water-surface profile in the transitions. Others gave a summary of the empirical hydraulic design of transitions for flumes and siphons and made specific recommendations for various geometric shapes. Some previous researchers studied designing of transitions and recommended to avoid the conditions that increase the limit value which cause choke. Many researchers used the transitions phenomenon to measure he discharge through open channels. They developed two types of discharge measurement structure, either the structure is mounted permanently in the channel or the structure is temporarily positioned for discharge evaluation. The previous study investigated experimentally the flow velocities, the head losses; the separation zone and the water surface profile at contraction, expansion and constriction with and without hump. Although the reviews for horizontal and vertical open channel transitions were numerous, it is rare for combined transitions The main objective of this study is to collect and show the previous study of the transitions and make a conclusion and give suggestion for future research.
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Open Channel Transitions 1. Introduction When a change of channel shapes, variation of bed elevation, transition is provided as a link between the original and the new channel. Such changes are often required in natural and artificial channels at siphons, aqueducts, weirs, falls, bridges and barrages for economic as well as practical reasons. Design and performance of transitions are critically dependent on subcritical and supercritical flow states. The calculation of energy losses and determination of the transition profile to provide a good velocity distribution at the end of the transition are two problems areas that need the attention of hydraulic engineers. 2. Design of Transitions The transitions may be horizontal (contracting or expanding) or vertical (rise or falling in bed) or combined, sudden or gradual and required for subcritical or supercritical flows. The transition in a channel is a structure designed to change the shape of cross-sectional area of the flow. Under normal design and installation conditions, practically all canals and flumes require some type of transition structure to and from the waterways. The function of such a structure is to avoid excessive energy losses, to eliminate cross waves and other turbulence and to provide safety for the structure and waterway. The form of transition may vary from straight-line headwalls normal to the flow of water to very elaborate streamlined warped structures. Straight-line headwalls are usually found satisfactory for small structures or where head is not valuable. For the sake of economy, the U.S. Department of Agriculture (Chow [1]) has tested a cylinderquadrant transition as a substitute for the expensive warped structure. For the same reason, the trend of practice in the U.S. Bureau of Reclamation has been toward simplification, with the top edges of walls and the intersections between warped walls and floors, designed in straight lines. The common types of transition are inlet and outlet transitions between canal and flume, inlet and outlet transitions between canal and tunnel, and inlet und outlet transitions between canal and inverted siphon.
3. Governing Equations Open channel transition can be solved with the use of specific energy equation or the momentum equation.
3.1. Specific Energy Equation The concept of specific energy was introduced the first time by Bakhmeteff in 1912 as, (Chow [1]), Fig. (1): 6
E d cos
V 2
(1) 2g where E is the specific energy, d is the normal water depth of flow, V is the mean velocity, is the slope angle of channel bed and g is the acceleration due to gravity.
Figure 1 Energy in open channels For a channel of small slope and uniform velocity distribution, =0.0 and =1.0, V2 Ey (2) 2g where y is the vertical water depth. By using the discharge equation Q AV , Eq. (2) can be written as: Q2 Ey (3) 2gA2 where A is area of the water cross section.
3.2. Momentum Equation Jaeger (1956) (Chow [1]) applied the momentum equation for transitions, Fig. (2): Q 3V3 1V1 P1 P2 P3 F f g (4) 1 2 b1 y1 1 2 (b1 b3 ) y2 1 2b3 y3 F f 2
2
2
where Q is the discharge, is unite weight of water, g is acceleration due to gravity, 1 and 3 are the momentum coefficients at sections 1 and 3, respectively, V1 and V3 are mean velocities sections 1 and 3, respectively, P1 , P2 and P3 are the hydrostatic pressures at sections 1, 2 and 3, respectively, b1 , b2 and b3 are the breadths of channel at sections 1, 2 and 3, respectively, y1 , y 2 and y3 are the water depths at sections 1, 2 and 3, respectively and F f is friction force. 7
Figure 2 Analysis of horizontal contraction 4. Types of Transitions The transitions may be sudden or gradual, horizontal (contracting or expanding) or vertical (rise or falling in bed) or combined and required for subcritical or supercritical flows.
4.1. Sudden Transitions Transitions with the change of cross-sectional dimensions occurring in a relatively short distance will introduce rapidly varied flow. Such transitions include: Horizontal Transitions: contraction or expansion, Vertical Transitions: rising in bed (hump) or falling in bed (sump), Combined Transitions: contacting with hump, contracting with sump, expansion with hump and expansion with sump.
4.1.1. Sudden Horizontal Transitions For the case of contacting or expanding in horizontal dimensions only, the specific energy is assumed constant along the transition. Thus E1=E2, where E1 and E2 are the specific energies before and after the transition, respectively, Fig. (3). Total Energy Line
Total Energy Line
E
Water Surface Critical Depth Line
b1
E2
E1
y2
y1
E1
E
Water Surface
y1
b1
b2
Critical Depth Line
y2
b2 Sudden expansion
Sudden contraction
Figure 3 Horizontal transitions 8
E2
4.1.2.
Sudden Vertical Transitions
For the case of rise in bed or falling only, the specific energies are related as E1=E2 Z , where Z is the rise or falling in bed level, Fig. (4). Total Energy Line
Total Energy Line
E
Critical Depth Line y1 E1
E2
E1
E Critical Depth Line E2
y1
y2
y2 Z
Z
Sudden rise in bed (hump)
Sudden falling in bed (sump)
Figure 4 Vertical transitions
4.1.3.
Sudden Combined Transitions
For the case of combined transition, the specific energies are related as E1=E2 Z , where Z is the rise or falling in bed level with change in the channel width.
4.2. Gradually Transitions Transitions with the change of cross-sectional dimensions occurring in a relatively long distance will introduce gradually varied flow. Such transitions include gradually contraction, expansion, rising in bed and falling in bed, Fig. (5). b1
b2
Z
(a) Gradually contraction b1
© Gradually rise in bed (hump)
b2
Z
(d) Gradually falling in bed (sump) Vertical transitions
(b) Gradually expansion Horizontal transitions
Figure 5 Gradually horizontal and vertical transitions for rectangular section
4.2.1.
Gradually Combined Transitions
Transitions include contraction or expansions with rise in bed or falling at the same site are considered as combined transitions. Such transitions include sudden or 9
gradually contractions or expansions with vertically sudden or gradually hump or sump, Fig. (6). Z
Z
b1
b2
b1
(b) Gradually contraction with rise in bed
© Gradually contraction with falling in bed
Z
b1
b2
Z
b1
b2
b2
(d) Gradually expansion (b) Gradually expansion with falling in bed with rise in bed Figure 6 Combined transitions
4.3. Constriction A constriction in an open channel constitutes a reach of sudden reduction in the channel cross section, Fig. (7). b1
b2
b1
b1
b2
b1
Sudden constriction Gradually constriction Figure 7 Sudden and gradually constriction for rectangular section constriction
5. Problems of Open Channel Transitions
The following are some problems that usually involved in the hydraulic design for the transitions.
5.1. Horizontal Problems The following problems of horizontal transitions for subcritical or supercritical flow: 1. To estimate the size of contraction just required to make the flow critical (which lead to a critical contraction case); 10
2. To predict the flow behavior after the transition when the transition is smaller than that of the critical; 3. To predict the flow behavior after the transition as well as upstream when the size of the contraction is larger than that of the critical size; and 4. To predict the flow behavior after the transition is due to an expansion in channel.
5.2. Vertical Problems The following are problems that usually involved in the hydraulic design for the vertical transitions. 1. To estimate the size of rising just required to make the flow critical (which lead to a maximum rise case); 2. To predict the flow behavior after the transition when the transition is smaller than that of the maximum; 3. To predict the flow behavior after the transition as well as upstream when the size of the rise is larger than that of the maximum size; and 4. To predict the flow behavior after the transition is due to a fall in channel bed.
5.3. Combined Problems The following are some problems that usually involved in the hydraulic design for the vertical transitions. 1. To estimate the size of contraction just required to make the flow critical (which lead to a maximum rise case), with constant vertical change; 2. To estimate the size of rising just required to make the flow critical (which lead to a maximum contraction case), with constant horizontal change; 3. To predict the flow behavior after the transition (vertical and horizontal) when the transition is smaller than that of the maximum; and 4. To predict the flow behavior after the transition as well as upstream when the size of the rise or contraction is larger than that of the maximum size.
6. Solutions of the Problems of Open Channel Transitions The existing methods of solving the problems of transition using specific energy equation are; 1. Slide-rule operation (Henderson 1966 [2]) 2. Trial-and-error solution (analytical solution) (Subramanya 1986 [3]) 3. Graphical solution (Vittal 1978 [4]; Ranga Raju 1993 [5]) 4. Design tables prepared from the specific energy equation expressed in dimensionless form (Subhash 2001 [6]); and 5. Direct solution of rectangular channel (Abdulrahman Abdulrahman 2008 [7]). Wikipedia [8] is existing solutions of open channel transitions, Appendix 1. In real problems, the theoretically impossible flow may become possible because the 11
assumptions made in the theoretical derivation may not be true under actual circumstances.
6.1. Analytical Solutions (trial-and-error and direct solutions) Brater and King (1963) [9] developed for the first time equations and prepared tables for solving constriction problems. Tables were prepared to aid in the construction of curves specific energy and specific discharge. The specific energy for rectangular section mat be written q2 (5) x 2 x3 2gE 3 where x y E The specific energy for Triangular section mat be written Q2 4 5 x x (6) 2 gz 2 E 5 where z is side slope. The specific energy for trapezoidal section mat be written 2
Q2 zE 2 3 (7) 1 b x x x 2 gb 2 E 3 where b is bed width. Henderson (1966) [2] announced that, the problems of horizontal transitions may also be solved by writing the specific energy equation before and after the transition section. The existing method in this time for solving the problems of open channel transitions involves trial-and-error solutions of higher degree equations. Bhallamudi and Chaudhri (1992) [10] solved the flows in channel expansions and contractions, two-dimensional, depth-averaged, unsteady flow equations in a transformed coordinate system numerically by using the MacCormack scheme. The non-rectangular physical domain is converted into a rectangular computational domain. The unsteady flow model is used to obtain steady flow solutions by treating the time variable as an iteration parameter and letting the solution converge to the steady state. The results of the mathematical model are compared with experimental data. The capability of the model for handling mixed supercritical and subcritical flows in a channel transition is demonstrated. Abdulrahman (2008) [7] studied the direct solution to problems of open channel transitions for rectangular channels. He presented an analytical solution for transitions located in rectangular channels to achieve accurate results than those as a result of using existing methods. The direct solution for supercritical and subcritical depths may be derived as: 2 1 Ec 3 2 y(sup er) 1 1 (8) y(sup er ) cos2 sin cos cos E 3 3 E 12
where y(super) is supercritical water depth, E is specific energy and Ec is critical specific energy. 2 1 Ec 3 2 y( sub ) 1 1 (9) y( sub ) cos2 sin cos cos 120 0 E 3 3 E where y(sub ) is subcritical water depth. Ladopoulos (2010) [11] studied numerical solution for transitions by using singular integral equation methods for subcritical and supercritical simultaneously. With a known flow rate Q and known velocities upstream and downstream the transition under study, then the remaining velocities on the boundary of the transition and in internal points can be calculated. Beyond the above, the free surface elevations can be determined in every boundary or internal point of the transition. The boundary conditions which corresponds to the flow for open channel transitions as shown in Fig. (8):
Figure 8 Boundary conditions for open-channel transitions The free surface profile will be determined in a channel contraction, with inlet conditions of velocity u0 = 2.1879 m/s, water depth h0 = 0.0305 m, which corresponds to a Froude number F0 = 4.0. Furthermore, for the outlet conditions of the channel contraction the water depth is h = 0.061 m. The width of the inlet channel is 0.61 m, the width of the outlet channel 0.305 m, and the length of the contraction L = 1.45 m. Also, a steady flow of constant flow discharge Q = 0.0407 m3/s is assumed, Fig. (9).
Figure 9Channel contraction 13
An application is finally given to the determination of the free-surface profile in an open-channel contraction (Fig. (10)) and comparing the numerical results with corresponding results by finite differences.
Figure10 Distribution of water depth along the channel centerline Das (2011) [12] described a methodology for determining the channel dimensions leading to the critical state of flow to maintain given specific energy or specific force conditions. The developed methodology works on extracting two equal roots/factors of the specific energy and specific force expressions, where the two equal roots/factors yield the critical flow depth. The methodology is illustrated for trapezoidal, rectangular, triangular, and parabolic channels.
6.2. Graphical Solution Brater and King (1963) [9] developed the specific energy diagram and the specific discharge curve to solve the transitions in open channel. For a given discharge, the specific energy E varies with the depth of flow y. Figure 11 shows that for a discharge Q, there are two possible values of y for a given value of E. These are known as alternative depths, namely subcritical depth on the upper limb and supercritical depth on the lower limb of the curve. The alternate depths coincide at the point of minimum specific energy Ec. The depth at this point is critical depth yc.
6.2.1.
Specific Energy Diagram
For a given specific energy E, the unite discharge q (for rectangular section) varies with the depth of flow y as: q2 Ey (10) 2gy 2 where q is unite discharge (Q/b).
14
y
y2
Q1 Q2
yc y1
Q2
Ec E
E Figure 11 Specific energy diagram 6.2.2. Specificequations Discharge Diagram Figure 12 shows that for a specific energy E, there are two possible values of y for a given value of q. These are known as alternative depths, namely subcritical depth on the upper limb and supercritical depth on the lower limb of the curve. The alternate depths coincide at the point of maximum unite discharge qmax. The depth at this point is critical depth yc. y2 yc y y1
q
qmax
Figure 12 Specific discharge diagram Vittal (1978) [4] proposed dimensionless discharge-depth relationships for exponential, trapezoidal and circular channels to give direct solution to all the problems of open channel transitions. For exponential channels he gave the following particular form: n Q y (1 y ) (11) where Q Q 2l 2 gc 2 E 2 n1 is dimensionless discharge and y is dimensionless
depth (y/E). For trapezoidal channels he gave the following particular form: 15
Q y (1 K1 y ) (1 y )
(12)
where Q Q 2l 2gE 3 is dimensionless discharge and K1 zElb is dimensionless variable. Also he gave an equation for circular channels. He proposed graphical solutions for transitions. The result is unique dimensionless discharge-depth curve covering all discharges for exponential and trapezoidal channels. It includes both supercritical and subcritical limbs, Fig. (13).
Figure 13 Variation of Q with y for exponential and trapezoidal channels The variation of Q with y for circular channel is shown in Fig. (14).
Figure 14 Variation of Q with y for circular channel Sobeih and Rashwan (2002) [13] proposed new dimensionless discharge-depth relationships for rectangular section that facilitates a direct solution for all vertical transitional problems, Fig. (15): (13) Q y2 1 y2 z 16
In which Q is the dimensionless discharge, y is the dimensionless water depth and z is the dimensionless rising or falling in channel bed.
Figure 15 Variation of Q with y for exponential and rectangular channels for different values of z
7. Energy Losses in Open Channel Transitions The head loss produced by the transition is most important as it is reflected as increased up stream stages. The calculation of energy losses and determination of the transition profile to provide a good velocity distribution at the end of the transition, are two problems areas that need the attention of hydraulic engineers. Jaeger (Chow [1]) calculated the head loss E by applying specific energy equation before and after the transition as follow: 2 2 V3 V1 E ( y1 ) ( y3 ) (14) 2g 2g where E is the head loss. 2 2 F1 y3 F1 E 1 or (15) 2 2 y1 2 y1 2( y3 / y1 ) (b3 / b1 ) Formica (1955) (Chow [1]) applied the equations of energy loss in closed conduits in open channel flow. The energy loss in a sudden contraction may be expressed as: 2 V3 E K (16) 2g where K is coefficient for contraction. And in a sudden expansion by
17
(v1 v3 ) 2 (17) E 2g where is coefficient for expansion. For subcritical flow through sudden transitions, experiments on various designs were made by Formica as shown in Fig. (16). According to the experimental data obtained by Formica, the values of K for sudden contractions seem to vary in a wide range, generally increasing with the discharge. The approximate median value of K for design I is 0.10 and, for designs II to IV, 0.06.
Figure 16 Various designs of sudden transitions for experiments (After G. Formica (1956)) Also He obtained the following average values of for sudden expansions, Table (1): Table 1Expansion coefficient ( ) Type of designs
1 0.82
2 0.87
3 0.68
4 0.41
5 0.27
6 0.29
7 0.45
8 0.44
Simmons (1964) [14] studied energy losses produced by transitions from pipes to canals and from canals to pipes. Applying the momentum and energy equations for an abrupt expansion, for a bed width b1 to a bed width b2 in a rectangular channel, Henderson (1966) gave the following equation for head loss: 2 2 2 3 V1 b1 2 F1 b1 b2 b1 E (18) 1 4 2 g b2 b 2 Smith and Yu (1966) [15] calculated the energy loss, hL, as the difference between the upstream and downstream specific energy. For their proposed shortened outlet structure with three square baffles to rapidly spread the flow on expansion outlet. Skogerboe et al. (1971) [16] studied the energy losses produced by channel expansions. They used empirical equation describing energy losses in an open channel constriction. Morris and Wiggert (1972) [17] Applied the loss due to 18
surface resistance is neglected, as it is small. The form loss, hL , is assumed to vary uniformly along the transition length and is expressed as: Vo 2 VL 2 hL K H (19) 2 g where, Vo and VL = the velocity at the inlet and outlet of expansion respectively, K H = the loss coefficient lying between 0.3 and 0.75 (Morris and Wiggert, 1972), and g = the acceleration due to gravity. Bos and Reinink (1981) [18] developed an equation to calculate head loss over long throated flume as: h f c L (vc v 2 ) 2 1/ 4 (20) E2 / E1 Cd E1 2 gE1 where, C d is the discharge coefficient, C L is the energy loss coefficient, h f is the energy loss due to friction, vc is the critical flow velocity and v 2 is the mean velocity in the section in which y2 and E2 are measured. Nashta and Garde (1988) [19] did obtain head loss curves by fitting experimental data, but the data were from less common sudden expansion experiments. The head loss was expressed by the Borda-Carnot relationship, involving the velocity head, the depth of flow and the expansion ratio. It is interesting to note that the analysis of Nashta and Garde included the energy loss due to friction in the total energy loss. Alauddin and Basak (2006) [20] studied the velocity distributions of flow through the sudden as well as gradual expansion models. They showed a transition profile for expansion of flow with minimum separation has been evolved by streamlining the boundary shape of the transition, and the performance of the transition is evaluated to compare with the existing profiles. Their Model V is tested for performance and compared with the other transition profiles used commonly in the field, designed by Hartley et al. (1940), Chaturvedi (1963), Nashta and Garde (1988), Swamee and Basak (1993); Model I, II, III and IV, respectively. The minimum loss was found in case of profile suggested by Swamee and Basak and their profile (Fig. 17), using the efficiency of transition from the expression as: Qg ( y 2 y1 ) Qg ( y 2 y1 ) (21) 1 1 V12 V2 2 2 2 QV1 1 QV2 2 1 2 g 2 2 g 2 2 The values of 1 and 2 were calculated from the following expressions: v 3 dA v 3 dA 1 for inlet and 2 for outlet section. 3 3 A1V1 A2V2 The comparison of head loss of the transition models is made and minimum loss is found in case of profile IV and their profile as shown in Fig. (18). 19
Figure 17 Comparison of head loss of the models
Figure 18Hydraulics efficiency of the Transition Models Basak and Alauddin(2010) [21] based on the difference of kinetic energy at inlet and outlet, and head recovery, the percentage efficiency, can be examined. Figure 19 shows the overall hydraulic efficiency of the various transition profiles, where dominance of the Profile IV is observed over others.
Figure 19Efficiency of the Transition Models 20
The overall hydraulic efficiency of the transition models decreases from Model I to III, and these are 75.8%, 74.7 %, and 71.4 % respectively. Efficiency of the Mod el IV is the highest among the models, and it is 78.7%. Azita (2011) [22] studied channel expansions in open channels. They connect a relatively narrow upstream section of channel with a large downstream section of channel. His study aims to quantify the energy losses in a lateral expansion and to further investigate how effective a hump fitted on the channel-bed of the expansion is at reducing energy losses. Estimates of the energy loss coefficients ranged from 0.46 to 0.62. These results would be useful for the design of channel expansions, and for calibrating and validating numerical hydrodynamics models. The presence of the hump has been shown to accelerate the flow, convert adverse to favorable pressure gradient, and lower the energy loss coefficients by more than 50% when compared with the corresponding values without the hump.
8. Characteristics of Transitions Lane (1920) (Chow [1] made the first laboratory investigation in the United States to the flow through a constriction in a water way. This investigation dealt with simple constrictions of flows having values of Froude’s number slightly higher than those usually found in natural channels. Hinds (1927)[23] assumed the watersurface profile in the transition to be composed of two reverse parabolas of equal length connected at the center of the transition, and found the bed width profile corresponding to the assumed water- surface profile, Fig. (20). He gave a summary of the empirical hydraulic design of transitions for flumes and siphons and made specific recommendations for various geometric shapes. The basic design objective is to achieve flow transition in a short distance with a minimum amount of flow disturbance.
Figure 20 Definition sketch: Rectangular expansion transition 21
Hartley et al. (1940) [24] devised a simple design for an expansion transition based on the assumptions of a constant depth and a constant rate of change of velocity with distance. The expression for the expansion is hyperbolic in nature as, Fig. (21):
1
1
1
1
(22) b bo bL bo where b is the bed width, bo is bed width at start of expansion, bL is bed width at end of expansion the transition and x L is dimensionless parameter.
Figure 21 Definition sketch: abrupt expansion Ippen and Dawson (1951) [25] found that the straight contractions are better than curved one of equal length in hydraulic performance and costs. Accordingly, they have proposed a procedure of design for straight contractions. For the design of curved contractions, the cross-wave pattern may be determined experimentally by model test or analytically by the method of characteristics. In supercritical flow through a straight contraction (Fig. 22), symmetrical shock waves are developed at points A and A at the entrance.
Figure 22 Designs of straight-wall contractions 22
From the geometry of this situation, the length of the contraction can be shown to be, Fig. (23): b b (23) L 1 3 2 tan I II II I
b 1 I
b 2
b 3
L II II I of contraction Figure 23 Cross section
Channel expansion in supercritical flow occurs frequently at places where flow, emerges at high velocity from a closed conduit, sluice gate, spillway, or steep chute. Studies by Hom-ma and Shima [26] indicate that separation of flow in channel expansion in supercritical flow like that shown in Fig. (24). The separation surfaces shown by the dashed lines act as solid boundaries within, which the flow has the characteristics to those in a channel of decreasing width.
Figure 24 Separation of flow in an expansion of rapid divergence From both experimental and analytical studies, Rouse et al. [27] have obtained the many results, which be found useful in the preliminary design of channel expansions in supercritical flow. They concluded that the generalization of experimental data for channel expansions may be expressed by the following relationship: x z y (24) f , y1 b1 F1 b1 where y is the depth of flow, y1 is the depth of the approach flow F1 is the Froude number of the approach flow, b1is the channel width, x is the longitudinal coordinate measured from the outlet section and z is the lateral coordinate measured from the center line of the channel. 23
They developed a dimensionless diagram for abrupt expansions. From the experiments, the most satisfactory boundary form for an efficient expansion was found to be: 32
z 1 x 1 2 (25) 2 bF b1 1 1 For expansions designed for this form, the surface contours for a mean value of b1/y1 and for various values of Fl are given. Blasdell [28] made tests on supercritical flow through expansions in conjunction with the SAF stilling basin. The tests were made on transitions with straight flaring side walls and 1% channel slope. Kindsvater and Carter (1955) [29] achieved a new development in the study of constriction in subcritical flow. They produced more exact results for different given conditions. Jaeger (1956) (Chow [1]) applied the momentum equation for transitions, Fig. (2): He obtained the following equation for a sudden contraction, ( y3 / y1 )(( y3 / y1 ) 1) 2 (26) F1 2(( y3 / y1 ) 1/(b3 / b1 )) 2
where F1 is the Froude number at section 1. These equations can be plotted as shown in Fig. (25), using b3/b1 as a parameter.
2
Figure 25 Plot of F1 against y3/y1 using b3/b1 as a parameter for the analysis of horizontal contractions and expansions For a sudden expansion, the equation can be written as follows: 24
(b3 / b1 )(( y3 / y1 )(1 ( y3 / y1 ) 2 ) (27) F1 2(1/(b3 / b1 ) ( y3 / y1 )) Chaturvedi (1963) [30] derived an equation for the expansion transition curve as: 2
n
n
n
1 n
(28) b bo bL bo Which is eventually reduces to Eq. (24) when n=1. On the basis of his experimental results, he found that a transition designed with above equation performs better than Mitra’s hyperbolic transition when the value of n=3/2. Smith and Yu (1966) [15] considered expansions as a rapid expansion when a total central angle, , between sidewalls reached 28°10' or a 1:4 (Lateral: Longitudinal) rate of flare. A gradual expansion may be considered as one with values smaller than 28°10'. Flow separation is expected to occur when increases to 19°10', corresponding to a 1:5.98 rate of flare, unless the width ratio B/b < 2 (Smith and Yu 1966). An attempt to avoid flow separation in an expansion by reducing may not be practical under many instances, because the length required for the expansion will be excessively long and the cost to build such an expansion will be too high. However, They found that the S-curved warped wall expansion recommended by Hinds (1927) was one of the least effective designs among the types of expansions that they tested. Also, they suggested that a straight walled diverging expansion (straight line type) was more efficient than a curved wall expansion of the same length, Fig. (26).
Figure 26 Plan view velocity distribution for a rapid expansion: B/b = The 3.0 length of transition governed by side splay of 7: 1claimed to be the optimal value by Mazumder (1967) [31]. The idea of using a triangular sill in a channel expansion to suppress flow separation and eddy formation was first reported in Seetharamiah and Ramamurthy (1968) [32]. They simplified the problem by dropping an unknown amount of energy losses that would take place between the inlet and outlet of the expansion. Ramamurthy et al. (1970) [33] suggested that the use of a simple hump in an expanding transition accelerates the flow and hence reduces flow separation and limits the area in which the reversal of flow occurs. No extensive experimental study was conducted earlier about the performance of the humps. The study aims at verifying the effectiveness of hump in larger expansion angles, to investigate the possibility of using splitter vanes, and finally to conduct a few numerical simulations by Computational Fluid Dynamics (CFD) analysis in 325
dimensional perspective. Mehta (1979) [34] investigated the flow characteristics in two-dimensional expansion. Mehta (1981) [35] investigated the effects of sudden channel expansion on turbulence characteristics over smooth surfaces. He found that the high intensity occurs either close to the surface or near the bed because of the Prandalt’s second kind secondary flows developed at the channel transitions. According to Graber (1982) [36], flows are symmetric in symmetric, twodimensional rectangular channels with the expansion ratio of less than 1.5. He presented the cause of the asymmetric behavior of the flow is a static instability of the flow system. The results of the stability analysis of Graber (1982) show that channels have the limitation that the maximum Froude number is less than 0.2. The result also predicts instability for expansion ratios greater than 1.5 that is in good agreement with experimental observations. Vittal and Chiranjeevi (1983) [37] developed the following equation of average separating streamline by considering an abrupt expansion from a rectangular channel to a trapezoidal channel: 0.80.2 m (29) b bo bL bo 1 1 U.S. Department of Transportation (1983) [38] classified channel expansions into five different types: a cylindrical quadrant expansion, a straight line expansion, a square end expansion, a warped expansion, and a wedge expansion. These different expansions are illustrated in Fig. (27).
L
Figure 27 Channel expansion types, all the expansions have geometry symmetric about the centerline El-Sheiwy (1985) [39] studied flow characteristics through vertical transitions. He established interrelationships between all hydraulic parameters derived from the theoretical analysis. Rashwan (1987) [40] studied horizontal transitions to develop a computer program to solve three cases which arise in the hydraulic design and develop a graphical solution for horizontal contraction. He established interrelationships between all hydraulic parameters derived from the theoretical 26
analysis. He studied an effect of bed roughness and wall flaring system on the flow characteristics specially the chocking phenomenon and an influence of the live bed on the horizontal transitions. Nashta and Garde (1988) [19] based on minimization of the form loss and friction loss recommended the following equation for the transition: 0.55 (30) b bo bL bo 1 1 They presented the results of analytical and experimental investigation in channels with a sudden expansion with expansion ratios of 1.5 to 3 for subcritical flow. Babarrutsi et al. (1989) [41] investigated experimentally the influence on recirculating flows in a shallow open-channel expansion, and compared their experimental results with the field measurements of island wakes made by Ingram and Chu (1987). Their velocity measurements showed that the bed friction caused both the length of the re-circulating zone and re-circulating flow rate to decrease. Zidan et al. (1990) [42] studied an open channel transition (sudden contraction and limiting constriction) in subcritical flow. They presented relationships between parameters in dimensionless forms; effects of contraction ratio, flared entrance on the flow pattern and chocking phenomenon are given. A set of graphical relationships has been deduced which could be used for design purposes to provide an accurate solution to the theoretical one. They found that the experimental value for the critical width of a channel is bigger than the corresponding theoretical value given by the equation: 2 2 (31) (bc / b) 2 F1 (1.5 /(1 F1 / 2) 3 Babartutsi et al. (1991) [43] mad experiments with a sudden expansion but used dye technique in order to improve wake flow pattern visualization. The implication is that an experimental setup to investigate the effect of channel expansion on flow separation should be built using materials with insignificant roughness height. In his study, smooth-surfaced Plexiglas is used to the channel sections and expansions. Energy losses due to surface friction are not expected to be significant. Swamee and Basak (1991) [44] presented a design method for subcritical expansions for rectangular channels, in order to achieve a minimum head loss. By analyzing a large number of profiles, they obtained an equation for the design of rectangular expansion, producing the optimal bed-width profile. They applied the optimal-control theory to improving the design of rectangular open-channel expansions. Swamee and Basak (1992) [45] extended the application of the optimal-control theory to further improve the geometric shape of the trapezoidal expansion reported in Vittal and Chiranjeevi (1983). He discussed an analytical method for the design of expansions that connect a rectangular channel section with a trapezoidal channel section for subcritical flow. They suggested that flow separation in the expansion and the associated energy losses were considerably reduced through the optimal design of bed-width as well as side-slope profiles, on the basis of the momentum and energy equations. They claimed that the optimal
27
profiles represent an improvement from the design of Vittal and Chiranjeevi (1983) in terms of reducing flow head losses. Swamee and Basak (1993) [46] combined the design ideas presented in Swamee and Basak (1991, 1992). A common limitation of all the above-mentioned investigations is that the energy loss coefficient is assumed. They used the optimal control theory for the design of rectangular-to-trapezoidal expansions for gradually varied subcritical flow. They obtained equations for bed-width, side-slope and bed profiles based on the minimization of the transition head losses as: 0.775
1.35 L (32) b bo bL bo 2.52 1 1 x Mohapatra and Bhalamudi (1994) [47] considered the case where the bed level varied temporally in gradual channel expansions with an erodible bed. The authors used analytical and numerical models to obtain equilibrium solutions of the bed level variation in the expansions. Their work was limited to small expansion angles under steady conditions. The analytical and numerical models were based on the continuity and momentum principles, assuming that the flow depth and velocity did not vary across the channel width. Foumeny et al. (1996) [48] carried out experiments and showed that the asymmetric behavior of the flow is related to the Reynolds number. Chaudhry (1997) [49] studied super critical flow in a channel expansion and contraction to obtain suitable numerical scheme that can efficiently and accurately simulate the flow with all characteristics features. He found that the solution grid adaptive technique is very effective to produce a better solution and resolution of the changes of the flow variables both for subcritical and supercritical flows. Since the physical equations and the grid equations are decoupled, they can be solved independent of each other. This allows the application of any existing finite difference or finite element code without major modifications. Negm (2001) [50] studied the flow characteristics due to asymmetric sudden contraction based on experimental investigation when the flow through the contracted section passed from subcritical to critical flow. He found that Lower contraction ratios produce more backed up depth ratio with higher values corresponding to bigger values of the relative contraction length at constant contraction ratio. Papanicolaou and Hilldale (2002) [51] studied the effects of a channel transition on turbulence characteristics. They determined the distributions of the mean local velocities in the transverse and vertical directions, the distributions of the turbulent intensities and the distributions of the shear stresses in the stream wise and transverse directions. They found that a gradual channel expansion creates an unbalanced turbulent stress distribution of the intensities. Escudier et al. (2002) [52] conducted an experimental study of turbulent flow in a sudden expansion with an expansion ratio of 4 and an aspect ratio of 5.33. A laser Doppler anemometer was used to measure mean flow velocity fluctuations and the Reynolds shear stress. They reported that the flow downstream of the expansion is
28
asymmetric. They concluded that the effect of inlet of expansion is the reason for asymmetrical behavior of flow. Negm et al. (2003) [53] made an experimental study to investigate the effect of both lateral and vertical contraction on energy loss through the constricted length, in sloped open channels. The dimensional analysis was used to correlate the important variables affecting on the energy loss within the transition zone. Different discharges were used for each bed slope. Total energy loss through the transition length were computed and correlated to the relevant parameters. The effect of all these parameters on the energy loss through the contraction is analyzed and discussed. It has been found that the energy loss increases with the increase of the bed slope, decrease of the lateral contraction ratio, increase of the Froude number of approach and increase of the incoming passing discharge. Negm et al. (2003) [54] made an experimental investigation to study the effect of horizontal and vertical transitions on the relative protection length of sudden expansion in sloping open channels. Alauddin and (Alauddin and Basak) (2006) [55, 56] developed an expansion transition in open channel subcritical flow. To develop an expansive transition profile, investigation was made on an abrupt expansion in a laboratory flume. The velocity distributions of flow through the sudden as well as gradual expansion models are made, Fig. (28) and Fig. (29). Thus a transition profile for expansion of flow with minimum separation has been evolved by streamlining the boundary shape of the transition, and the performance of the transition is evaluated to compare with the existing profiles.
Figure 28 Velocity distributions across the width and along the length of abrupt expansion A methodology for design of expansive transition in open channel subcritical flow has been presented; an empirical equation for the expansion transition is evolved as: 29
L n b bo bL bo a 1 1 x where a , n and m are constants.
m
(33)
Figure 29 Velocity distributions in the proposed transition model A computer programmer in FORTRAN has been developed adopting grid-search algorithm to find the optimal value of a , n and m. Having known the constants a , n and m, the final form of the transition equation is obtained as: 0.8 1.35 L (34) b bo bL bo 2.52 1 1 x Akers and Bokhove (2008) [57] recognized that when the incoming flow is supercritical then the flow pattern is distinctly two-dimensional. However they used the one-dimensional approach to study channel flow through a linear contraction, and approximate the shock configuration in Fig. 30, which is actually established in their experiments as a normal jump in the contraction.
Figure 30 Configuration with notation for expansion 30
Solid lines in Figure 30 denote shock wave fronts, M is the Mach stem; shaded area denotes subcritical flow. Basak and Alauddin(2010) [58] studied a development of an expansion transition in open channel subcritical flow. The velocity profiles of Model IV are close to flat and near ideal; those of Model I, II, and III depict central deformations indicating one sidedness of maximum velocity thread. Defina and Viero (2010) [59] studied the open channel flow through a linear contraction within the framework of shallow 2D flows. They were focused on the stable reservoir state with a Mach stem discussed by Akers and Bokhove. They proposed and discussed a theoretical model for the flow configuration pattern with an irregular reflection (IR) off the converging wall (Fig. 31). They presented new experimental results which complement those of Akers and Bokhove. A series of possible flow configurations and shock wave patterns is shown in Fig. 33.
Figure 31 Possible shock wave patterns in a linear channel contraction Howes et al. (2010) [60] used a three-dimensional computational fluid dynamics (CFD) model to design a subcritical rapidly varied flow contraction that provides a consistent linear relationship between the upward-looking ADVM sample velocity and the cross-sectional average velocity in order to improve ADVM accuracy without the need for in situ calibration. CFD simulations validated the subcritical contraction in a rectangular and trapezoidal cross section by showing errors within +1.8 and 2.2%. They were tested three contraction inlet designs shown in Fig. (32). Inlet A is a simple inlet condition with a 0.2 m radius rounded entrance. Inlet B is a combination of straight and rounded entrances described by Smith (1967). Inlet C is an elliptical entrance shown by Montes (1998). 31
Figure 32 Three inlets analyzed for the subcritical contraction Najmeddin (2012) [61] extended earlier investigations about fitting a hump in the vertical to eliminate flow separation. This study uses the Computational Fluid Dynamics (CFD) modeling approach. The model results are validated using existent analytical solutions under simplified conditions and available experimental data for a limited number of cases. Yan and Wai (2012) [62] investigated gradual channel transition in open channel designs by using physical modeling tests and simple 2-D numerical simulations, Fig. (33).
Figure 33 Sketch of the gradual channel transition A commercial numerical model was employed to simulate the 2-dimensional flow structure in the gradual channel transition. The computed results of the longitudinal mean flow velocity and flow depth have a good agreement with the experimental results, Fig. (34).
Figure 34 Physical model of gradual channel transition 32
Mamizadeh and Ayyoubzadeh (2012) [63] investigated three-dimensional flows in sudden expansions of rectangular channels. The CFD simulation model (Fig. (35)) is used. The k-ε turbulence model is used to simulate turbulence. The results in canal with sudden expansion showed that the vortex formed by coarse mesh size and number of iteration more than 400 is asymmetric and for small and medium mesh size is symmetric. Results showed that, asymmetrical flow pattern can be viewed with some minor changes such as roughness coefficient in any part of flume.
Figure 35 Flow pattern in canal coarse mesh size
9. Short Constrictions
Chow [1] showed a constriction in an open channel constitutes a reach of sudden reduction in the channel cross section. The phenomenon is usually so complicated that the resulting flow pattern is not readily subject to any analytical solution. The flow through a constriction may be subcritical or supercritical, Fig. (36).
Figure 36 Constriction in uniform flow channel 33
When the flow is subcritical the constriction will induce a pronounced backwater effect that extends a long distance upstream (Fig. 36a and b). When the flow is supercritical, the constriction will disturb only the water surface that is adjacent to the upstream side of the constriction and will not extend the effect farther upstream (Fig. 36c). If the upstream water surface is dammed up to a depth greater than the critical depth, the surface will form of an S1profile, extending upstream only for a short distance and then ending with a hydraulic jump (Fig. 36d).
9.1. Subcritical Flow through Constrictions When an area constriction is introduced to an otherwise uniform, friction controlled flow in a prismatic channel of mild slope (Fig. 37), a backwater of M1type profile is first developed upstream from the constriction. The upstream end point of the backwater curve is assumed to be at section 0. Near the constriction at section 1 the central body of water begins to accelerate, deceleration occurs along the outer boundaries, and separation zones are created in the corners adjacent to the constriction. An adequate approximation for the location of section 1 may be taken at a point one opening width b from the center of the opening. Between sections 0 and 1, the flow is gradually varied.
Figure 37 Definition sketch of flow through constriction 34
At the constriction, the flow is rapidly varied, characterized by marked acceleration in directions both normal and parallel to the streamlines. As the water passes through the contraction, the contracted stream reaches a minimum width at section 2, which corresponds to the vena contracta in an orifice flow. After the vena contracta, the live stream begins to expand until it reaches downstream section 4, where the uniform flow is reestablished in the full width channel. Between sections 3 and 4, the flow is gradually varied. The discharge through section 3 is h (35) Q 8.02CA3 2 2 2 1 1C A3 A1 2 gC 2 A3 K 3 L La K 3 K1 where h y1 y3 , K1 and K3 are the conveyance of the sections 1 and 3, respectively, A1 and A3 are the water area of the sections 1 and 3, respectively, K1 and L is the contracted reach length, La is the approach length from sections 1 to the upstream side of the contracted opening, 1 is energy coefficient at section 1and C is an over-all coefficient of discharge, equal to Cc (36) C 3 ke k p where 3 is energy coefficient at section 3, Ke is a coefficient and Kp is a coefficient responsible for the non-hydrostatic pressure distribution. For purposes of practical application, the value of C may be expressed as (37) C C K f K r KW K K y K x K e K t K j where C' is the standard value of the coefficient of discharge corresponding to a standard condition of all effects mentioned above; and where the k' are coefficients that can be used to adjust the value of C' to a given nonstandard condition of the Froude number, entrance rounding, chamfer, angularity, side depths, side slope, bridge submergence, and bridge piles and piers, respectively. From the laboratory investigation by the (Kindsvater et al. (U.S. Geological Survey [64], a set of curves were developed for the determination of these coefficients for four different types of constriction.
9.2. Chocking in Open Channel Transitions For designing transitions, it is recommended to avoid the conditions that increase the limit value which cause choke. Choked flows may significantly increase the upstream backwater elevation and may flood the upstream reach. It may also increase the velocity and bed shear stress severely within the contracted reach, causing local scour and endangering the safety of the hydraulic structure. Therefore, it is of practical importance to investigate the potential for the occurrence of choking phenomenon due to channel contractions and to compute the resulting hydraulic conditions in irrigation canals and natural rivers. For flows subject to choking, major concerns to engineers include the determination of threshold conditions for choking, discharge computations, and computation of 35
maximum back-water elevations and energy losses. Yarnell (1934 a,b) [66, 67] developed two theoretical equations to determine the limit opening ratios at which the flow chokes. The first of these equations was derived from energy conservation by assuming no change in specific energy between the upstream and contracted sections (sections 1 and 2 in Fig. 38), whereas the second equation was based on the momentum principle between the contracted and downstream sections (sections 2 and 4 in Fig. 38).
Figure 38 Sketch of choked flows through short horizontal contractions Steven (1936) (Chow [1]) proposed rating curves Fig. 39 of a critical flow proposed for a closed conduit with free surface flow.
Figure 39 Rating curve of critical flow for closed conduit with free surface 36
Bradley (1973) [68] based on data with limited opening ratios (σ = 0 .25–0.59), presented the coefficient CL as a function of the opening ratio σ and the encroachment shape. The primary objectives of this study are to investigate the threshold conditions for choking and the variations of energy loss coefficients with dominant parameters such as the opening ratio, σ , encroachment structure shape, inlet angle, α , and contraction length, L. In the following analysis, an equation to predict the limit opening ratios for choking that appropriately accounts for the energy losses is to be derived. Also a general equation for energy loss coefficient is developed based on a wide range of experimental data derived from all past studies of choking. The analysis is limited to short, lateral contractions in subcritical flow. Allen (1980) [69] was proposed the solution of choke-free flow with a maximum rise in bed elevation. Liong (1984) [70] analyzed the case of a rectangular channel, using the energy principle. He used graphic solutions for one dimensional, steady, uniform flow without choke flow. He modified the Allen’s work and gave the solutions of choke-free flow due to change in bed elevation and the change of the widths of rectangular channels. Hager and Dupraz (1985) [71] examined the effects of opening ratio σ (ratio of contracted channel width to unobstructed channel width as shown in Fig. 1), inlet angle, α, and contraction length, L, on the discharge characteristics. However, in Hager and Dupraz’s analysis, the energy losses were neglected. Hager (1986) [72] investigated local head losses in different zones along the contraction and their effects on the discharge characteristics. Dey et al. (1990, 1994, 1998) [73, 74, 75] reported the generalized solutions of choke-free flow for other shapes (triangular; parabola; and trapezoidal). Dey (1998) [76] obtained physical solutions for the flow through circular channel due to increase in bed elevation. He considered three cases of channel transitions commonly found in practice. As most of the sewers or conduits in practical cases are of circular section, the main three cases of channel transitions are considered here as following, Fig. (40): (1) The channel bed is raised due to change in position of the centerline. (2) The channel bed is raised due to decrease in channel diameter. (3) The channel bed downstream the transition zone is raised with the formation of a flat base. Molinas and Marcus (1998) [77] examined the effects of energy losses on the threshold choking conditions for short, abrupt contractions. They found that when energy losses are taken into account choking takes place under much less severe contraction ratios than those computed by Yarnell’s equation. The results of Molinas and Marcus indicated that, for short vertical wall contractions, the energy loss corresponding to threshold choking conditions is about 25% of the upstream specific energy. Wu and Molinas (2001) [78] extended the Molinas and Marcus (1998) study to include effects of different encroachment structure shapes, inlet angles, and relative contraction lengths on discharge coefficients. 37
Sec. 2
Sec. 1
E1 . d o1
y1
T.E.L .
Z
y 2 E2 . do2
Case I: Change in position of the centerline Sec. 2 Sec. 1 T.E.L .
E1 .
d o1
E2 . y1
Z
y2
do2
Case II: Change in channel diameter Sec. 2 Sec. 1 T.E.L .
yc 2
E1 . y1
Z max .
do
E2 .
yc 2
do
Z max .
Case III: Change bed elevation with the formation of a flat base. Figure 40 Definition sketches for three cases for different circular channel transitions Wu and Molinas concluded a “short contraction” is defined as a reach that has a length less than 1.5 times the approach channel width. Characteristics of Choked flows in subcritical channels result in severe backwater effects generate high downstream velocities and consume the excessive energy by accompanying hydraulic jumps. As shown in Fig. 41, under choking conditions the water surface profile along the centerline dramatically deviates from the normal flow profile. All choked flows (cases 1–5) shown in Fig. 41(a) are subjected to the same analysis since in all cases the flow passes through the critical depth in contraction. Figure 41(b) shows examples of choked-flow water surface profiles measured by Molinas and Kheireldin (1994) where the depth upstream from the contraction is unaffected by the downstream conditions. Rashwan (2004) [79] presented solve to the problems of transitions through circular sections of open channel flow for the three different cases of circular crosssectional channel transitions are considered, Fig. (40). The maximum rise in bed yields critical flow at downstream section without affecting upstream flow, critical size, is predict for the three cases analytically, graphically and using the relevance table. 38
Figure 41 Water surface profiles of choked flows The maximum rising (limit) in bed elevation at downstream gives critical flow at the downstream section without affecting upstream flow and considering the allowable value for design condition of a choke-free flow.
1 0.25 yc / d o 2 0.16 yc / d o 2 2 Z max . K yc d o 2 1 2 do2 3 1.25 yc / d o 2 1.12 yc / d o 2
(38)
where y c = critical water depth at section 2, and d o 2 = internal diameter of circular channel at section 2, where K = dimensionless parameter ( E1 / d o ). Wu and Molinas (2005) [80] studied the energy losses and threshold conditions for choking in short, lateral contractions in subcritical open channel flows. They derived a theoretical equation to predict the limiting opening ratio for choking from the conservation of energy and continuity principles. This equation accounts for the critical flow conditions in the contraction and the local energy losses. For the computations of energy losses, an expression for the energy loss coefficient is developed based on a total of 186 sets of choking experiments conducted in the past by various researchers.
10. Transitions as Discharge Devices There are two types of discharge measurement structure, either the structure is mounted permanently in the channel or the structure is temporarily positioned for discharge evaluation. The device may be mounted and removed quickly. Such a 39
structure might then be referred to as mobile device. Its major advantages include low cost, reading precision and rapid instillation in running water. The critical-flow flume, also known as the Venturi flume, has been designed in various forms. It is usually operated with an un-submerged or free-flow condition having the critical depth at a contracted section and a hydraulic jump in the exit section. Under certain conditions of flow however, the jump may be submerged. One of the most extensively used critical-flow flumes is the Parshall flume (Fig. 42) which was developed in 1920 by Parshall [81]. The depth-discharge relationships of Parshall flumes of' various sizes are calibrated empirically
Figure 42 Parshall flume Cutthroat flume developed by Skogerboe and Hyatt at 1967 [82]. It was designed to measure flows in flat gradient stream as shown in Fig. (43), for free flow condition the discharge equation is found as (39) Q Cf yn1 u where C f is the free flow coefficient, y u is the upstream flow depth and n1 is the free flow exponent. B=b+L/4.5
B=b+L/4.5
b
1/3 L
2/9 L
5/9 L
2/3 L
Figure 43 Cutthroat flume 40
For submerged flow condition the discharge equation is developed as n C s y u y d Q= (40) logs n where yd is the downstream flow depth, yu is the upstream depth, n1 is the free flow exponent, n2 is the submerged flow exponent, Cs is the submerged flow coefficient and s is the submerged ratio (yd/yu). In 1985, Hager [83] used a cylinder made of high plastic type material to make a contracted section to have critical flow, and that, for contracted rectangle channel as shown in Fig. 44 the discharge equation can be expressed as 1.5 2B - d 2 2 3 (41) Q B - d gy a 1 2 9B 3 where B is the width of channel, d is the diameter of cylinder, ya are the approach water depth and g is the gravity acceleration. 1
2
y
1
y
m
d
d B
(a) Rectangular section
y
B
(b) Trapezoidal section
d D
(c) U-shaped section
Figure 44 Mobile cylinders in various channel shapes For trapezoidal channel shows in Fig.44b the dimensionless discharge equation can be written as follows: 3 y * 1 y * 2 (42) Q* 1 2y * where y * is the dimensionless water depth for approach section mya B d , m is the cotangent of the lateral side wall slope and Q * is the dimensional discharge
m Q
g B d . The U-shaped profile is composed by a semi-circular profile at its base and a rectangular profile as shown in Fig.44c. Dimensionless upstream energy head is found as: Q*2 D E * Ya y< (43) 2 2 2 4 3Ya1.5 1 Ya 3 δYa δ 3 12 Q *2 D E * Ya y (44) 2 2 2 π 4 8 1 δ Yo δ 3 12 where δ is the contraction ratio (d/D), Ya is the dimensionless approach water depth 3
5 12
2
(ya/D), Q* is the dimensional discharge Q 2 gD 5 and E* is the dimensionless specific energy (E/D). 12
41
In 1986, Hager [84] used a circular cone to made contraction section through a rectangular channel as shown in Fig. (45).
m 1
bc B
Figure 45 Circular cones in rectangular channel The presence of cone inside rectangular section made the water section as double trapezoidal section and discharge equation can be described as 1.5 2 2mE Q ca b gE 3 1 0 < E* < 1 (45) 3b 3 2.5
gE 5 5b 4 Q ca m E* ≥ 1 (46) 1 2 4mE 5 where Qca is the calculated discharge, b is the channel width at contraction section (B-bc), E is the specific energy, E* is the dimensionless specific energy head mE b ,
B is the channel width and bc is the cone base diameter. The discharge correction factor can be written as Q me U (47) 1 Q ca 271 U 27 where Qme is the measured discharge, U is the dimensionless curvature E 2 brc , rc is the radius of curvature at critical cross section. For triangular section as shown in Fig. 46, cone used to contract the section by put the half of it on the section wall. P c
p
b) Cross section view a) plan view Figure 46 Circular cones in triangular channel The discharge equation can be written as 0.5
2 2 4 gm c E Q ca (48) 5 2 The relation between experimental and theoretical discharge may be defined as (49) Q me Qca 1+ 4T 75 2.5
42
where mc is the cotangent of the lateral side wall slope at critical (contracted) cross section, z is the elevation of the channel bottom to a reference datum and T is the dimensionless curvature which equaled E 2 m c zr . In 1987, Hager [85, 86] presented a modified Venturi type discharge measurement. The model used was a sharp edge constriction plates placed symmetrically in a rectangular channel. A constriction plate was completed with a short rectangular plates fixed at 90o. The discharge equation of this model can be written as follows 1.5 E *2 2 4 (50) Q b g E 0.828 0.057ψ 2ψ 1 2 3 3 5E * where b is the contraction width, B channel width, E * is the dimensionless specific energy (E/b) and ψ is the contraction ratio (b/B). In 1988, Hager [87] used a mobile device as shown in Fig. 47 to measure the discharge in partially field pipe.
d
D ya
E
D y
Side view
Longitudinal view
Figure 47 Schematic plot of mobile device unit The device is a cylinder of diameter d < D and the base of cylinder is rounded to D/2. Using specific energy equation and Froude number lead to have the following expressions The relation between measured and calculated discharge can be written as Qme Qca 0.985 0.205E* (51) In 1991, Samani et al. [88] replaced graphical approach presented by W. H. Hager, which used to calibrate the water measuring device based on measured value of upstream energy. The replacement was developed using computer model based on the measurement of the water depth at the vertical cylinder pipe and eliminates the need for measuring the upstream energy. The relation between measured and calculated discharge is found as: Qme Qca 1.057 0.2266E* (52) In 1993, Samani and Magallanez [89] used flume consisted of pipe installed axially inside a trapezoidal channel with side slope 1:1. The relation between measured and calculated discharge was related as: Qme Qca 1.121E B 0.0701 (53) In 1997, Kohler and Hager [90] improved the circular mobile flume. The modified device was provided with a series of holes drilled into the cylinder, with an inter distance of roughly 0.3d which water can enter. This model removes 43
practically effect of surface fluctuation and pipe slope. The depth reading can be made either visually from downstream through the Plexiglas or with pressure cell mounted at the bottom of the device and connected to data logger. The effects of device distortion, device position, bottom slope of pipe and submergence have been evaluated. In 1997, Oliveto and Hager [91] used a mobile device as sector Venturi which is characterized by a ring of variable diameter in which two sharp-crested sectors are attached as showing Fig. (48). When mounting the sector Venturi the ring is expanded to match with the pipe section, the sector elements are sharp crested and have a V-notch at the invert. The dimensionless discharge can be written as 1 (54) Q*2 1 2.4So E *2.5 3 Where, So is the pipe slope and E* is the dimensionless specific energy (E/D).
E
D
Figure 48 Cross section of Venturi in circular flume In 1997, Peruginelli and Bonacci [92] used mobile pier-shaped prism device to measure the discharge in a rectangular channel as, Fig. (49). Q Fra y a μB b gy a (55) where Fra is the Froude number at approach section (0.566+0.0745Ya), ya is the approach water depth, B is the channel width, b is the prism width and μ is the contraction coefficient ( 0.832b -0.13 ). b B
γ
L
Plan Figure 49 Mobile prism dimension detail In 2000, Samani and Magallanez [93] attached a two semi cylinder of polyvinyl chloride (PVC) to the side wall of rectangular channel. Assuming uniform velocity distribution and neglecting energy loss between upstream and contracted section, the actual discharge equation was derived as follows 44
(56) Q Cd B d g2 E 3 Where Cd is the discharge coefficient 1.33 - 0.44 d B sin0.21E B d , B is the channel width, d is the cylinder diameter and E is the specific energy. WU et al. (2001) [75] presented a new discharge equation, which developed based on the conservation of energy and experimental data with a wide range of opening ratios (ratio of contracted width-to-total width). The discharge coefficients Cd* expressedas: 1.04 2.08(sin( / 2))1.2 1 * (57) Cd * 2 0.1 (1 sin( / 2)) (2.77 K s K L / ) 3 2 where Cd* is the coefficient of discharge, KL* is the coefficient to account for relative contraction length effects, Ks is the coefficient to account for shape effects, s is the opening ratio defined as width ratios of contracted and non-contracted sections (b/B) and a is the inlet angle. In 2006, Gole [94] estimated a discharge equation for free and submerged flow condition. The estimation is for contracted rectangular section with two prisms fixed on side wall as shown in Fig. (50). 3
b/2
b
B
b
b
b/2
Figure 50 Plan view of rectangular flume with side contraction The actual discharge equation was expressed as 1.5 0.5 Q 2 3 g C c C s bE (58) where Cc is the characteristic discharge coefficient, Cs is the submerged flow coefficient and b is the contraction zone width. A dimensionless curve was developed to determine C and Cdr. Rashwan and Idress (2012) [95] studied the evaluation efficiency for mobile flume as discharge measurement device for partially filled circular channel. To evaluate the efficiency of this type of device mathematical and experimental studies are presented. Specific energy, discharge and Froude number equations are used to develop mathematical model. The experimental data is used to evaluate the mobile flume as a device to measure discharge. A general equation was developed for discharge coefficient (Cd) as: Cd 7.3839δ 1.9348 4.4912δ - 0.2883E*me (59) where E *me is the dimensionless specific energy E D ,δ is contraction ratio(d/D).
45
Conclusions In this study previous work for open channel transitioned are collected as possible as can. The previous study can be classified as: 1. Theoretical formulation of the flow through transitions has been derived from the momentum and energy principles. 2. Laboratory experiments have been conducted to investigate the behavior of open-channel transitions. 3. The flow in the transitions is subcritical or supercritical. 4. The existing methods of solving the problems of transition using specific energy equation are; Slide-rule operation (Henderson 1966 [2]) Trial-and-error solution (analytical solution) (Subramanya 1986 [3]) Graphical solution (Vittal 1978 [4]; Ranga 1993 [5]) Design tables prepared from the specific energy equation expressed in dimensionless form (Subhash 2001 [6]); and Direct solution of rectangular channel (Abdulrahman 2008 [7]). 5. The phenomenon is usually so complicated that the resulting flow pattern is not readily subject to any analytical solution. 6. For designing transitions, it is recommended to avoid the conditions that increase the limit value which cause choke. 7. There are two types of discharge measurement structure, either the structure is mounted permanently in the channel or the structure is temporarily positioned for discharge evaluation.
Suggestion for Future Research
Future studies should consider the influence of distributed flow velocities at the transitions, and remove the assumption that the energy coefficient is unity. For this purpose distributed flow velocities at selective cross sections in transitions need to be measured. Future studies should also investigate the effects of transitions of different dimensions and configurations on flow behavior. For this purpose, the dual approach of combining laboratory experiments with numerical modeling would be effective. Previous study has limited to the case of transitions of rectangular cross section. Future research should consider transitions of other shapes (trapezoidal shape and circular shape). Future studies should be extended to include more cases where the Froude number is high value than 0.5. Future studies should be extended to include more cases for the combined transitions. 46
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[35] Mehta, P. R., Separated flow through large sudden expansions, Journal of the Hydraulics Division – ASCE 107(HY4), 1981, 451–460. [36] Graber, S. D., Asymmetric flow in symmetric expansions, Journal of the Hydraulics Division – ASCE 108(HY10), 1982, 1082–1101. [37] Vittal, N. and Chiranjeevi, V. V., Open channel transitions: Rational method of design, Journal of Hydraulics Engineering, ASCE, 109 (1), 1983, 99-105. [38] U.S. Department of Transportation, Hydraulic design of energy dissipaters for culverts and channels, Hydraulic Engineering Circular Number 14, Third Edition, 1983. [39] El-Sheiwy, M. I. A., “Flow Characteristics through Vertical Transitions”. Master thesis, presented to Mansoura University, Faculty of Engineering, 1985. [40] Rashwan, I. M. H., “Flow Characteristics through Horizontal Transitions”, Master thesis, presented to Mansoura University, Faculty of Engineering, 1987. [41] Babarrutsi, S., Ganoulis, J., and Chu, V. H., Experimental investigation of shallow recirculating flows, Journal of Hydraulic Engineering 115(7), 1989, 906–924. [42] Zidan, A.R.A., Owais, T.M., and Rashwan, I.M.H., “Horizontal Transitions in Subcritical Flow”, Mansoura Engineering Journal, )MEJ(, Faculty of Engineering, Mansoura University, Vol. 15, No. 1, June 1990. [43] Babarrutsi, S., and Chu, V. H., Dye-concentration distribution in shallow recirculating flows, Journal of Hydraulic Engineering 117(5), 1991, 643– 659. [44] Swamee, P. K., and Basak, B. C., Design of rectangular open-channel expansion transitions, Journal of Irrigation and Drainage Engineering – ASCE 117(6), 1991, 827–838. [45] Swamee, P. K., and Basak, B. C., Design of trapezoidal expansive transitions, Journal of Irrigation and Drainage Engineering, ASCE 118(1), 1992, 61–73. [46] Swamee, P. K. and Basak, B. C., A comprehensive open-channel expansion transition design, Journal of Irrigation and Drainage Division, ASCE, 119(1), 1993, 1-17. [47] Mohapatra, P. K., and Bhalamudi, S. M., Bed-level variation in channel expansions with movable beds, Journal of Irrigation and Drainage Engineering – ASCE 120(6), 1994, 1114–1121. [48] Foumeny, E. A, Ingham, D. B, and Walker, A. J., Bifurcations of incompressible flow through plane symmetric channel expansions, Journal of Computers and Fluids 25(3), 1996, 335–351. [49] Chaudhry MH. Computation of flow in open-channel transitions. J Hydr Res IAHR 1997; 35:243–256. 49
[50] Negm, A.M., Flow Characteristics at Asymmetric Sudden Contraction, Proc. of IWTC-VI 2001, Alex., Egypt, March 23-25, 2001, pp.5-19. [51] Papanicolaou, A., and Hilldale, R., Turbulence characteristics in a gradual channel transition, Journal of Engineering Mechanics 128(9), 2002, 948-960. [52] Escudier, M. P., Oliveira, P. J. and Poole, R. J., Turbulent flow through a plane sudden expansion of modest aspect ratio, Journal of Physics of Fluids – AIP 14(10), 2002, 3641–3654. [53] Negm, A.M., Elfiky, M. M, Attia, M. I. and Ezzeldin, M.M., Energy loss due to sudden contraction through transition length, Proc. of 7th Alazhar Engineering International Conference, AEIC’2003, April 7-10, Faculty of Engineering, Alazhar University, Cairo, Egypt. [54] Negm, A. M., Elfiky M. M., Attia M. I. and Ezzeldin M. M., “Protection length downstream of sudden transition for incoming subcritical flow”, 1st International Conference of Civil Engineering Science, ICCES1, Vol. 1, 2003. [55] Alauddin, M., Development of an expansion transition in open channel subcritical flow, Journal of Civil Engineering (IEB), 34 (2) (2006) 91-101. [56] Alauddin, M. and B. C. Basak, Development of an expansion transition in open channel subcritical flow, World journal of engineering 3 (3) (2006) 8795. [57] Akers B. and Bokhove O., “Hydraulic flow through a channel contraction: Multiple steady states,” Phys. Fluids 20, 056601 (2008). [58] Basak B. C. and Alauddin, M., Efficiency of an expansive transition in open channel subcritical flow, DUET Journal, Vol. 1, Issue 1, June 2010. [59] Defina A. and Viero D. P., “Open channel flow through a linear contraction,” PHYSICS OF FLUIDS 22, 036602 (2010). [60] Howes, D. J., Burt, C. M. and Sanders, B. F., “Subcritical Contraction for Improved Open-Channel Flow Measurement Accuracy with an UpwardLooking ADVM” Journal of Irrigation and Drainage Engineering, ASCE, September, 2010. [61] Najmeddin, S., “CFD Modeling of Turbulent Flow in Open-Channel Expansions”, M Sc., Civil Engineering, Concordia University, Montreal, Quebec, Canada, 2012. [62] YAN, Xu-F., and WAI, W. H. “Study on Flow Structure and Local Losses of Gradual Transition within a Lotus-shape Open Channel”, IAHR-HK Student Research Forum, November 17, 2012. [63] Mamizadeh J. and Ayyoubzadeh, S. A., “Simulation of Flow Pattern in Open Channels with Sudden Expansions”, Journal of Applied Sciences, Engineering and Technology 4(19): 3852-3857, 2012. © Maxwell Scientific Organization. [64] Kindsvater, C. E., Carter, R. W. and Tracy, H. J., Computation of peak discharge at contractions, U.S. Geological survey, Circular No. 284, 1953. 50
[65] Hubert J. Tracy and Rolland W. Carter: Backwater effects of open-channel constrictions, Transactions, American Society of Civil Engineers, vol. 120, 1955, pp. 993-1006. [66] Yarnell, D.L., Pile Trestles as Channel Obstructions. Technical Bulletin No. 429, U.S. Department of Agriculture, Washington, DC, 1934a. [67] Yarnell, D.L., Bridge Piers as Channel Obstructions. Technical Bulletin No. 442, U.S. Department of Agriculture, Washington, DC, 1934b. [68] Bradley, J.N., Hydraulics of Bridge Waterways. Hydraulic Design Series No. 1, U.S. Department of Transportation, Federal Highway Administration, Washington, DC, 1973. [69] Allen, R. F., Steady solution for river flow.” Journal of Hydraulic Division, ASCE, Vol. 106, No. 4, 1980, pp. 608-611. [70] Liong, S. Y., Channel design and flow operation without choke, Journal of Irrigation and Drainage Engineering, ASCE, July/September, Vol. 110, No. 4, 1984, pp. 403-407. [71] Hager, W.H. and Dupraz, P.A., Discharge Characteristics of Local, Discontinuous Contractions, Journal of Hydraulics Research, IAHR 23(5), 1985, 421–433. [72] Hager, W.H., Modified Trapezoidal Venturi Channel, Journal of Irrigation and Drainage Engineering, ASCE 112(3), 1986, 225–241. [73] Dey, S., Dey, A., Bajpai, S., and Singh, A. K. “Design and flow operation of triangular and parabolic channels without choke.” Indian J. Power and River Valley Development, 40(5-6), 1940, pp. 71-74. [74] Dey, S. “No-choke flow in trapezoidal channels” Journal of Engineering Mechanics, ASCE, Vol. 120, No. 10, 1994, pp. 2224-2231. [75] Dey, S., and Sil, S. N., Choke-free flow in trapezoidal channels with change in bed elevation, Sadhana; Proc., Indian Acad. of Science, 1998, 23(June). [76] Dey, S., Choke-free flow in circular channel with increase in bed elevations, Journal of Irrigation and Drainage Engineering, ASCE, Vol. 124, No. 6, November/December, 1998, pp. 317-320. [77] Molinas, A. and Marcus, K.B., Choking in Water Supply Structures and Open Channels, Journal of Hydraulics Research, IAHR 36(4), 1998, 675– 694. [78] Wu, B. and Molinas, A., Choked Flows through Short Contractions, Journal of Hydraulics Engineering, ASCE 127(8), 2001, pp. 657–662. [79] Rashwan, I.M.H., No Chock Flow in Circular Open Channel with Rise in Bed Elevation, Scientific Bulletin, Ain Shams University, Faculty of Engineering, Vol. 39, No. 1, March 31, 2004. [80] Wu, B. and Molinas, A., Energy losses and threshold conditions for choking in channel contractions, Journal of Hydraulic Research, Volume 43, Issue 2, 2005, pages 139-148. 51
[81] Parshall, R.L., the Improved Venturi Flumes, Trans. ASCE, 89, 1926, pp.841–880. [82] Skogerboe, G.V., Bennett, R.S. and Walker, W.R., Generalized Discharge Relations for Cutthroat Flumes, Journal of Irrigation and Drainage Engineering, ASCE 98(4), 1972, pp.569–583. [83] Hager, W. W., Modified Venturi channel, Journal of Irrigation and Drainage Engineering, ASCE, Vol. 111, No. 1, March, 1985. [84] Hager, W. W., Modified Trapezoidal Venturi channel, Journal of Irrigation and Drainage Engineering, ASCE, Vol. 112, No. 3, August, 1986, pp. 225241. [85] Hager, W.H., Discharge Characteristics of Local, Discontinuous Contractions (Part II), Journal of Hydraulics Research, IAHR, 25(2), 1987, pp. 197–214. [86] Hager, W. W., Venturi Flumes of Minimum Space Requirements, Journal of Irrigation and Drainage Engineering, ASCE, Vol. 114, No. 2, May, 1988, pp. 226–243. [87] Hager, W. W., Mobile Flume for Circular Channel, Journal of Irrigation and Drainage Engineering, ASCE, Vol. 114, No. 3, August, 1988. [88] Samani, Z., Jorat, S. and Yousef, M., Hydraulic Characteristics of Circular Flume, Journal of Irrigation and Drainage Engineering, ASCE 117(4), 1991, pp. 558–566. [89] Samani, Z. and Magallanez, H., Measurement Water in Trapezoidal Canals, Journal of Irrigation and Drainage Engineering, ASCE, Vol. 119, No. 1, January, 1993. [90] Kohler, A and Hager, W. W., Mobile Flume for Pipe Flow, Journal of Irrigation and Drainage Engineering, ASCE, Vol. 123, No. 1, January, 1997. [91] Olveto, G. and Hager, W. W., Discharge Measurement in Circular Sewer, Journal of Irrigation and Drainage Engineering, ASCE, Vol. 123, No. 2, March, 1997. [92] Peruginelli A. and Bonacci F., Mobile Prism for Flow Measurement in Rectangular Channel, Journal of Irrigation and Drainage Engineering, ASCE, Vol. 123, No. 3, May, 1997. [93] Samani, Z. and Magallanez, H., Simple Flume for Flow Measurement in Open Channel, Journal of Irrigation and Drainage Engineering, ASCE, Vol. 126, No. 2, March, 127-129, 2000. [94] Gole, A., Flow meter for discharge measurement in irrigation channel, Flow measurement and instrumentation, Science Direct, Vol. 17, (2006). [95] Rashwan, I.M.H. and Idress M.I., Evaluation efficiency for mobile flume as discharge measurement device for partially filled circular channel, Ain Shams Engineering journal, Elsiver, November 2, 2012.
52
Appendix http://en.wikipedia.org/wiki/Classic_energy_problem_in_open-channel_flow Constriction in an open-channel flow due to smooth narrowing of the channel width As the word constriction suggests, the channel width in an open-channel flow reduces in size and that leads the flow to change its characteristics from the upstream values. Let’s consider the following question to understand the concept better. An open rectangular channel in the city of Blacksburg carries a constant discharge of 20.0 ft³/s at a depth of 3.0 ft. The width of the channel upstream and downstream is 4.0 ft and 2.0 ft respectively. Considering this transition to be smooth and the flow to be frictionless, 1. Determine the water surface elevation in the constriction. 2. If instead of 2.0 ft, the width of the channel is reduced to 1.0 ft, would upstream flow undergo any changes in its depth? 3. What is the maximum reduction that one can make in the width of the channel for upstream flow to remain same as before? Solution:
Top view of constriction in a channel Step1: Froude number of the upstream flow: q 5 Fr 0.17 (subcritical flow) gy 3 32.2(33 ) Step2: Specific energy of upstream flow (Eu/p): 2 q1 52 Eu / p y1 3 3.04 ft 2 2( 32.2 )( 32 ) 2 gy1 Step3: Corresponding to this upstream specific energy, the maximum discharge (qmax) possible: 3
( Eu )( g ) (3.043 )(32.2) qmax 16.4 ft 2ls 3 3 (1.5 ) (b2 ) Step4: The minimum width (Wmin) allowed downstream so that upstream flow depth remains unchanged: 53
Q 20 1.22 ft qmax 16.4 Step5: In part (i) of the question, the width (Wd/s) reduces to 2 ft which is greater than Wmin. Therefore the flow depth remains unchanged upstream. Also here discharge per unit width in the constriction (q2): Q 20 q2 10.0 ft 2 / s Wd / s 2 Step6: Since flow is frictionless, the specific energy remains conserved both upstream (Eu/p) and on the constriction (Ed/s): Eu / s Ed / s Wmin
10 2 Ed / s y 2 3.04 ft 2 2(32.2) y2 Solving we get depth on constriction, y2 = 2.85 ft and y2 is less than y1 .Therefore for a constriction in a subcritical flow the depth downstream decreases. Step7: In part (2) of the question, new Width (Wd/s) = 1.0 ft which is less than the Wmin. It’s a choke and the upstream flow conditions will change. Step8: New downstream discharge (q3) on constriction per unit width: Q 20 q3 20.0 ft 2ls Wd / s 1 Step9: Specific energy of the downstream flow (Ed/s) will be equal to the critical specific energy corresponding to the new (q3): 1
1 20 2 3 q3 3 3.47 ft Ed / s 1.5( ) (1.5) g 32.2 Step10: To find the new depth upstream (y1), again apply the concept of specific energy conservation upstream and downstream: Eu / p Ed / s 2
q2 y1 3.47 ft 2 2 gy1 52 y1 3.47 ft 2 2(32.2) y1 Solving we get new depth upstream, y1 = 3.44ft. Step11: The initial transient discharge downstream (q trans) when the flow just encounters the choke conditions, will be the "maximum discharge corresponding to the original upstream specific energy qmax = 16.4ft²/s." Correspondingly, the initial transient depth (y2) on constriction downstream will be the critical depth given by: 54
1 3
(16.4) qtrans 2.02 ft ) g 32.2 An algorithm for solving any such similar questions: From the given value of unit discharge and depth, find the Fr number of upstream flow → Find out the upstream flow specific energy → corresponding to this specific energy, find out the maximum discharge possible (q max)→ corresponding to this qmax, find out the minimum width (wmin) applicable for no choke condition = (Q/qmax)→ Compare this (wmin)with the given widths in question → If the given Width (W) of the constriction is greater than the w min, then no choke conditions developed → If the given Width(W) of the constriction is less than wmin, choke conditions certainly developed and the upstream flow will encounter a change in its depth.→ Draw a rough E-y diagram (both for upstream and downstream). Since q upstream is different than q downstream, we will be having different E-y curves for upstream and downstream flow. The corresponding Energy -Depth diagram has been plotted and shown below: 2
yc (
1 3
2
Constriction in an open channel flow Expansion of the channel width in an open-channel flow Consider the same channel as above, however instead of reducing the width, the width is increased to 6.0 ft. 1. Assuming the transition to be smooth and frictionless, What will be the new surface water elevation downstream ? 2. Will the upstream flow conditions ever change for further increase in the width of the channel? Solution: 55
Top view of channel expansion Step1: Finding the Downstream discharge per unit width of the flow (q d/s): (16.4) Q 20 q ( trans ) Wd / s 6 g 32.2 2
qd / s
1 3
2
1 3
3.33 ft 2 / s
Step2: Since the flow is considered to be frictionless, the specific energy remains same both upstream (Eu/p) and downstream (Ed/s): Eu / p Ed / s q2 3.04 y 2 2 2 gy 2 3.332 3.04 y2 2 232.2 y2 Solving, we get depth on constriction, y2 = 3.03 ft. And y2 is greater than y1. Therefore, in a subcritical flow, as the channel width increases the depth also increases downstream. Here, the flow upstream will always have sufficient energy to carry the flow downstream. So we will never encounter choke conditions in expansions. An algorithm for solving any such similar questions: Calculate the unit discharge downstream (qd/s). Since downstream, the channel width is increasing, value of (qd/s) will be less than (qu/p) → Since specific energy remains same in the case of expansion,(Ed/s) = (Eu/p)→ Using the energy depth formula, find out the depth downstream(y2).→Roughly sketch out the E-y diagram for both upstream and downstream and realise that flow will never encounter choke conditions. The Energy-Depth diagram for this question has been plotted and shown below:
56
Expansion of channel width An upward step in an open channel flow (rise in the channel bed) 2 An open rectangular channel carrying a discharge of 10 ft /s per unit width is flowing at a depth of 6.0 ft. After a certain distance the flow encounters a smooth step which makes the channel bed rise by 2.0 ft. 1. What will be the depth downstream of the step? 2. For an upward step of 4.0 ft, will the flow profile remain the same upstream? If not, then what would be the new upstream depth? Solution:
Upward step in an open channel flow Step1: For this question, the discharge per unit width (q = 10 ft2/s) remains same both upstream and downstream. Froude number of upstream flow (Fr): q 10 Fr 0.12 (subcritical flow) 3 3 32 . 2 ( 6 ) gy1 Step2: Specific energy of the flow upstream (Eu/p): q2 10 2 Eu / p y1 6 6.04 ft 2 2( 32.2 )( 6 2 ) 2 gy1 57
Step3: For q of 10 ft²/s, the minimum possible or the critical specific energy (Ec): 1 3
1 3
q 10 2.20 ft Ec 1.5 1.5 g 32.2 Step4: Safe step: Maximum upward step (Zmax) possible for no choke conditions: Z max Eu / p Ec 6.04 2.20 3.84 ft Step5: For part (i) of the question, the upward step (Z) is equal to 2.0 ft which is less than the Zmax. Therefore, it’s not a choke and the upstream flow depth remains unchanged. Step6: Specific energy of the flow on the step downstream (Ed/s): Ed / s Eu / p Z 6.04 2.0 4.04 ft 2
2
q2 ( y2 ) 4.04 ft 2 2 gy2 10 2 ( y2 ) 4.04 ft 2 232.2 y2 Solving, we get the depth on the step downstream (y2) = 3.95 ft. Therefore from above working we conclude that for a subcritical flow, an upward step leads to decrease in the depth downstream of the flow Step7: In part (ii) of the question, the upward step (Z) is equal to 4.0 ft which is greater than the Zmax. Therefore it’s a choke. The flow upstream will now need to change its depth to gather extra energy. Step8: The specific energy downstream will be equal to critical specific energy (Ec) corresponding to discharge per unit width of 10 ft2/s as calculated in step 3: Ed / s Ec 2.20 ft Step9: The specific energy upstream: Eu / p Ec Z 2.20 4 6.20 ft ( y1
q2 ) 6.20 ft 2 2 gy1
10 2 ( y1 ) 6.20 ft 2 2(32.2) y1 Solving, we get the new depth upstream (y1) = 6.16 ft. Step10: The initial transient discharge (qtrans) on the step, corresponding to the Original specific energy of the upstream flow:
qtrans
Eu / p Z 3 g 6.04 43 32.2 9.00 3
3
ft 2 / s
(1.5) (1.5) * An algorithm for solving any such similar questions: Calculate the froude number and find out whether the flow is subcritical or supercritical→ Find out the specific energy upstream→Since q remains the same 58
for upstream & downstream, the q curve will be common to both upstream & downstream →Draw the E-y sketch and approximately show the specific energies and depths upstream & downstream (Safe step)→ find the critical energy corresponding to the given q→ When the step is less than (E u/p − c) , it is not a choke→ When step is greater than (Eu/p − Ec) ,it is a choke. The Energy-Depth diagram for this question has been plotted and shown below:
An upward step in an open channel flow A downward step in an open channel flow (fall in the channel bed) In the same channel as above, if the bed falls smoothly by 2.0 ft instead of a rise, then: 1. Find out the depth downstream. 2. Will the upstream depth ever change for further decrease in the bed level? Solution:
Downward step in an open channel flow Step1: Since it’s a downward step, the specific energy downstream (Ed/s)will be : 59
Ed / s Eu / p Z 6.04 2 8.04 ft q2 10 2 ( y2 ) 8.04 ft ( y2 ) 8.04 ft 2 2 2 gy2 232.2 y2 Solving, we get the depth on the downward step (y2) = 8.04 ft. Step2: Solving for part (ii) of the question, the upstream depth will never change for further decrease of the bed depth. Reason being : the specific energy downstream is always greater than the specific energy upstream. So the downstream conditions will never reach critical conditions. An algorithm for solving any such similar questions: From the given value of q and depth, find out the specific energy of flow upstream(Eu/p)→ Add the value of downward step to Eu/s)to obtain specific energy Ed/s)→ From specific energy-depth relationship, find out the value of depth downstream→Draw a rough E-y diagram showing the value of calculated depths and specific energies both upstream and downtream.→ since q remains same both upstream and downstream, E-y curve will be same for upstream flow as well as for downstream flow. Energy depth diagram for this question has been plotted and shown below
Downward step in an open channel flow
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