DUET Journal Vol., Issue, June of an Expansive Transition in an Open Channel Subcritical Flow B. C. Basak and M. Alauddin Department of Civil Engineering Dhaka University of Engineering & Technology, Gazipur, Bangladesh E-mail: mauddin_duet@yahoo.com ABSTRACT Open channel transitions involving an expansion of width are a coon feature of canals and flumes. Subcritical flow through an expansion transition can result in significant head loss due to separation of flow and subsequent eddy formation. The body of the hydraulic structure is subjected to lateral vibrations due to intermittent shedding of eddies, which are dangerous and hence, undesirable. Moreover, uneven distribution of velocity may cause asyetry of flow and thus develop scour at places of highly concentrated velocities. This paper presents the results of experimental investigations on subcritical flow through gradual expansion in rectangular rigid-bed channels. The velocity distributions of flow through the transition models are made, thus, the efficiencies of the transitions evolved by different investigators are evaluated.. INTRODUCTION Open channel expansions for subcritical flow are encountered in the design of hydraulic structures such as aqueducts, siphons, barrages, and so on. In these structures the flow tends to separate while subjected to the positive pressure gradient associated with flow deceleration, thus resulting in a considerable loss of energy. In an expanding flow, the distribution of velocity in the cross section can be extremely uneven, and uneven distribution of velocity may cause asyetry of flow and thus develop scour at places of highly concentrated velocities. This study involves the performance-evaluation of transitions evolved by different investigators in an open channel subcritical flow. To evaluate the transition profiles, efficiencies of the transition models are determined in a laboratory setup flow, defining this as the ratio of gain of potential energy to loss of kinetic energy.. AVAIABE METHODS Because of the importance of knowledge concerning expansions in rigid-bed channels, several investigators studied with different aspects of flow in expansion. The methods available for the design of expansion transitions were contributed by Hinds [], Hartley et al [], Chaturvedi [3], Nashta and Garde [4], and Swamee and Basak [5]. A brief outline of each method is given as follows (Referred to Fig. ). Hinds [] assumed the water-surface profile in the transition to be composed of two reverse parabolas of equal length connected at the centre of the transition, and found the bed-width profile corresponding to the assumed water- b y Flow x S b x PAN y x EEVATION Fig. : Definition sketch: Rectangular expansion transition surface profile. For this purpose Eq. () is taken as a form loss equation. The loss due to surface resistance is neglected, as it is small. The form loss, h, is assumed to vary uniformly along the transition length and is expressed as: V V h = K () H g where, V and V = the velocity at the inlet and outlet of expansion respectively, K H = the loss coefficient lying between.3 and.75 [6], and g = the acceleration due to gravity. The equations of the two reverse parabolas representing the water depth y at a distance x from the inlet are given by y = y + ( y y ) ξ ; ξ.5 (a) and y = y ( y y )( ξ) ;.5 ξ (b) y b Dhaka University of Engineering & Technology, Gazipur 7
DUET Journal Vol., Issue, June in which y and y = the depth of flow at inlet and outlet of x channels respectively; and ξ =. Equating total energies at the inlet and at a section ξ.5, and using Eq. () and (a), the bed width profile is obtained as Q K H Q ( K H ) b = + g S ξ y + ( y y ) ξ b y 4g( y y ) ξ 4g( y y ) ξ. 5 ] (3a) in which Q = the discharge, and S = the channel-bed slope. Similarly, applying energy equation at the outlet and at a section.5 ξ and using Eq. () and (b), the corresponding bed-width profile is Q K H Q ( K H ) b = y ( y y )( ξ) b y. 5 g S ( ξ) + 4g( y y )( ξ) ] (3b) Hartley et al [] assumed the following linear variation of the velocity: V = V + V V ) ξ (4) ( in which V = the velocity of flow at a distance x and further assumed constant depth throughout the transition, b V = bv = bv (5) Combining Eqs. (4) and (5), the bed-width profile was obtained as: [ ( ) ] = b + b b ξ b (6) Chaturvedi [3] generalized Eq. (6) in designing the rectangular expansion transition in the following manner: n n n n [ b + ( b b ) ] b = ξ (7) where from experimental investigation the best value of n was claimed to be.5. Nashta and Garde [4], based on minimization of the form loss and friction loss recoended the following equation for the transition:.55 b = b + ( b b ) ξ[ ( ξ) ] (8) Based on optimal control theory, a methodology has been presented for optimal design of a rectangular subcritical expansive transition by Swamee and Basak [5]. Analyzing a large number of optimal profiles, an equation for the design of rectangular transitions was presented as.775.35 b = b + ( b b ).5 + (9) x However, no experimental evidence is available for the latest development. 3. EXPERIMENTA SETUP AND PROCEDURE The experiments were carried out in a flume of Water Resources Engineering aboratory, DUET, Gazipur, which consisted of steel frame and bed, side walls of Perspex sheet. To test the transition models a contracted reach of width 9.55 cm and length.5 m was constructed and placed at upstream portion of the flume, the walls of which were of varnished wood. Then the gradual expansion of length 55.56 cm was provided in the reach to have the normal channel width 5.4 cm and then continued for remaining.5 m length. The transition models were made of wooden bed and the side walls of Perspex sheet. The length of transition governed by side splay of 7: has been used in the present study, which is claimed to be the optimal value [7], and all the experiments were conducted in a rectangular channel with an expansion ratio of.67. A tail gate was provided at the downstream end of the flume for depth regulation. Water was circulated through the channel by one electrically driven centrifugal pump with constant speed closed impellers. The rails were provided along the entire working length of the flume, which supports a moving carriage, and a continuous scale calibrated in millimeters is provided along the length of one of the rails. The carriage with pointer gauge and Pitot tube is used for depth of flow and velocity measurements respectively (Fig. ). Fig. : Water surface profile and setup for velocity measurements To know the velocity distribution of flow through gradual expansion and hence to evaluate the efficiency of transition, the (i) velocity at various points, and (ii) depth of flow at various sections were to be known. Pitot tube with its setup was to be held at various points to know the velocities for those points. The velocity was evaluated from the differential head (known as velocity head, ( V / g ) in the Pitot tube over the water surface in the channel. Water level along the centre line of the channel was measured with a pointer gauge. Velocities were measured at near surface,.y,.4y,.6y,.8y and near bottom in the vertical. These measurements were made at a number of sections across the width at /6, ½, and 5/6 times the width, (Fig. 3) and along the length of transition at inlet, mid-length, and outlet sections. Each transition was tested for five different discharges, Q as.5,.3,.45,.6, and.75 cumec. For each discharge, experiments were conducted at five different depths so that Froude number F, at entry were.5,.3,.4,.47, and.55 respectively. A typical table (Table ) is given below which reveals the data organizing. Dhaka University of Engineering & Technology, Gazipur 8
DUET Journal Vol., Issue, June Inlet tank Perspex sheet sidewall Transition model Control weir 9.55 cm 5.4 cm Outflow Inlet transition Wooden channel 65 cm 35 cm 5 cm 55.56 cm 5 cm a) Top view of flume Flow depth, y b/6 b/ 5b/6 Bed width, b Near surface. y.4 y.6 y.8 y Near bottom b) Cross-section at inlet of transition model with various points of measurements Fig. 3: Top view of flume with the section showing the various points of velocity measurements Table : ocal velocity (V, Q =.45 cumec) Q m 3 /s F inlet y at Froude No., F.45.4 Section y, 4. BACKGROUND OF EFFICIENCY For finding the efficiency, it is necessary to consider the velocity variation across the channel. Referring to section - at the inlet, Fig. 3, let v be the velocity of flow through an infinitesimal area da. Assuming that the flow is essentially in the direction of the axis of the transition, the volume of fluid per unit time passing through the elementary area is vda, and then the mass rate of flow through this area is vda. The kinetic energy of this fluid y at outlet mass per unit time is ( ) y, y 3 at /4 y 4 at / y 5 at 3 /4.5 53. 3.6 55.3 3. 53.8 54.3 54.8.3 9. 5.8 3.5 6.3 3. 3.8 3.4.4.. 4..8.3.3 3..47.. 5.4..8 4.3 5..55 89. 7.8 94.8 9. 9. 9.8 94. At Depth s Reading, h in At b/6 At b/ At 5b/6 Velocity, Velocity, Reading, v in v in h in m/sec m/sec V. at mid-length Velocity, Reading, h Reading, v in at.6y4, h in m/sec Avg. Vel, v in m/sec. y..443.5.454..443. y.5.454..465.5.454 Inlet.4 y..443.5.454..443.6 y 9..4 9.5.43 9..4 3..43.8 y 8..396 8.5.48 7.5.384 5.5.38.9 y 6..343 7..37 6..343 5.5.38. y 5..33 6..343.5. 4.5.97. y 6.5.357 7.5.384 3..43.5..4 y 6..343 6.5.357.5. at b/3, and b/3 at I/O Outlet.6 y 3..43 4..8.5.7 4.5.97.8 y..4..98..4 4..8.9 y...... vda v, or v 3 da. The total A kinetic energy per unit time passing over the entire section -, is obtained by integrating the term v 3 da, i.e., 3 v da, where A indicates integration over the cross- section -. In a similar manner, the kinetic energy per unit time for section -, at the outlet of transition is 3 v da. As the fluid passes through the transition, the A actual reduction in kinetic energy per unit time, or the power available for transformation in the transition, is the 3 3 difference of the above two, i.e., v da v da. A A All of this energy is not transformed into useful work. It may be helpful to imagine the transition as a pump. The pump raises the pressure of the fluid entering. The term 3 3 v da v da A might be regarded as the power A supplied or input to the pump. The head gained by the fluid flowing through the transition is(, where y y y ) and y refer to the depths of flow at sections - and - respectively. If Q be the rate of flow per sec, potential energy gained by the fluid per sec will be Q g( y y ) which gives the actual power, the pump adds to the fluid. The purpose of transition is to convert kinetic energy into useful pressure energy. Hence, the efficiency of the transition is defined by Qg y y ( ) ε = () 3 3 v da A v da A Dhaka University of Engineering & Technology, Gazipur 9
DUET Journal Vol., Issue, June Suppose, V be the average velocity at section - and V be the average velocity at section -. Then one can write 3 α v da = QV () A and 3 α v da = QV () A where, α and α are numerical constants, known as energy coefficients for non-uniformity of velocity distribution. Now Eq. () takes the form Q g( y y ) ( y y ) (3) ε = = V V QV α QV α α α g g The energy coefficients, α and α are obtained numerically from Eqs. () and (). 5. ANAYSIS OF DATA The velocities measured at different depths in the vertical indicated the usual turbulent boundary layer profile (mentioned in data table); as such, only the average velocity over a vertical was used in further analysis. Using the data for depth of flow and local velocity, the flow area and average velocity at the sections were known, and thus energy correction factors were calculated. After then these data were used to evaluate the efficiency of transition from the expression shown above. The table (Table ) given below shows this in brief. Four transition profiles used coonly in the field [, 3, 4 and 5], are tested for performance and compared the efficiency of the models, i.e., II, III, and IV respectively. Table : Average velocity, energy correction factor, and efficiency (V, Q =.45 cumec) Section Avg. Vel., Avg. V / Avg. V / y v y v strip Section v 3 y.448....448... Strip I.43..9.4.8.48..9.5.37..8..459....459... Inlet Strip II.443...434.45.9.4..9.6.389..9.3.448....448... Strip III.43..9.49.8.4..9.4.363..8..335.8.8.9.35.8.8. Strip I.93.8.7.48.6.9.8.4..7.8...363.8.8..37.8.8. Outlet Strip II.39.8.7.78.34.7.39.8.5.3.99.8...3.8.5.3.3.8.5.3 Strip III.97.8.4.77..56.8.4..7.8.. α / Strip....49.36.3 α / Section..39 76. The percentage efficiencies for different discharges and Froude numbers, and also overall efficiencies of the transition models are presented in Table 3a. The efficiencies of the transition models for average of discharges and average of Froude numbers are also suarized in Table 3b, where the comparative feature of performance of the models is observed. The efficiency Table 3a: hydraulic efficiencies of the transition models Discharge, Q, cumec..3 Froude No., F I.5 87.8 84. 8 89..3 8.8 7.3 83.8 89.4 78.4 75.5 8.9 8.9.47 73.4 79. 68.6 76.7.55 76.4 7.7 7.9 79.5 88. 77.8 78. 87..3 83 8 7.8 77.7.4 79.3 76.5 7.8 78.7.47 7. 76.6 74 76.4.55 7.4 68.9 7.4 7.7.5 85.6 84.7 8.7 9.7.3 86.6 8.3 75.9 78..5.4 77.6 75.8 74.4 74.7 68.9 7.4 76..47 77.3 69.8 65.7 7.4.55 68. 65.3 6. 7.9.5 8 8. 76.6 86.7.3 76 75.3 74.3 8.4.6.4 7. 7. 67 77.6.47 64.6 7. 59 7..55 59. 6.3 59.8 69.3.5 78.8 8.7 77.8 88..3 74.8 8.4 7.7 85.5.8.4 68.3 7.7 66. 73.8.47 69.5 7.9 67. 67.9.55 68.7 63.8 6.5 68.9 Table 3b: Hydraulic efficiencies of the transition models (for avg. of Q and avg. of F ) curves for average of discharges and for average of Froude numbers for the four models have been shown in Figs. 4 and 5, respectively. 6. EXPERIMENTA OBSERVATION AND RESUTS V I V Average Average Average Average for for Avg. for for Avg. for Avg. for for Avg. for Avg. Avg. Q F Avg. Q F Q Avg. Q F.5 84. 8.9 78.6 88.8.3 8.6 77.9 75.7 8..4 74.8 73.8 7.8 77.4.47 7. 73.7 66.9 7.9.55 68.8 66.4 65. 7.4. 79.8 76.4 77. 83. Froude No., F Discharge, Q, cumec.3 78. 4 76. 73. 78.3.5 79. 75. 7.5 78.5.6 7. 7.4 67.3 77..8 7. 73.9 69. 76.8 The surface configuration in the transitions I, II, and III exhibited local disturbances leading to diagonal waves, starting in the inlet section and persisting towards the entire length of the transition. The separation of flow took place at the boundaries near the exit of the transition. The points of separation were not syetrical on either side; this was also observed by several investigators earlier. The separation points were found to move forward and 78.7 Dhaka University of Engineering & Technology, Gazipur 3
DUET Journal Vol., Issue, June 95. e, % 9. 85. 8. 75. 7. 65. I V Velocity ine Q =.5 cumec F =.4 6. 55. 5...5.3.35.4.45.5.55.6 Velocity ine Q =.5 cumec F =.4 Froude Number, F Fig. 4: (average of discharges) of transition 9. 85. I e, % 8. 75. 7. I V Velocity ine Q =.5 cumec F =.4 65. 6....3.4.5.6.7.8 Velocity ine V Q =.5 cumec Discharge, Q, cumec F =.4 Fig. 5: (average of Froude numbers) of transition backward on either side. The flow in the downstream channel after expansion was found to be unstable and oscillating in nature. At times, flow was found to swing completely from one side to the other, thereby reversing the picture of separation and velocity distribution. Excepting a few cases, flow was never syetrical with respect to the centre line of the channel. The maximum-velocity line coincided with the centre line of the channel, for a short length after entry. Thereafter, it shifted to the side to which the main flow attached. The percentage efficiency, based on the difference of kinetic energy at inlet and outlet, and head recovery, can be examined from Tables 3a and 3b, and Figs. 4 and 5. The superiority of Transition IV is apparent. Provision of smooth outlet in the optimal transition (Profile IV) eliminates the chances of eddy formation and separation to a significant amount. The eddy with reverse flow is observed strong for Transition I to III. The velocity profiles of V are close to flat and near ideal; those of, II, and III depict central deformations indicating one sidedness of maximum velocity thread (Fig. 6). Except Profile IV, all other transitions have abrupt ending at downstream part of transition (Fig. 7), which may be responsible for the separation noticed at the exit section in these transitions. The overall hydraulic efficiency of the transition models decreases from to III, and these are 75.8%, 74.7%, and 7.4% respectively. of the V is the highest among the models, and it is b x- b )/( b - b ) (..8.6.4. All dimensions are in Scale : 5 Velocity- Other dimensions- 5 cm/s cm Fig.6: Velocity distributions across the width and along the length of transition models (Q =.5 cumec, F =.4) Swamee & Basak (993) (Profile IV) Hartley et al (94) (Profile I) Chaturvedi (963) (Profile II)....4.6.8. x / Nashta and Garde (988) (Profile III) Fig.7: Comparison of bed-width profiles Dhaka University of Engineering & Technology, Gazipur 3
DUET Journal Vol., Issue, June 78.7%. Abrupt ending in the transition profiles evolved by Hartley et al [], Chaturvedi [3], and Nashta and Garde [4], shifts the efficiency curves (Figs. 4 and 5) downward, whereas smooth ending in the optimal transition (Profile IV) makes it efficient over others, and keeps the efficiency iii. curve at top. Figure 8 shows the overall hydraulic efficiency of the various transition profiles, where dominance of the Profile IV is observed over others. ε, % 8. 78. 76. 74. 7. 7. 68. 66. 64. 6. 6. 7. CONCUSIONS I V Fig.8: of the Transition Models In the light of the present study the following conclusions could be drawn: i. The Transition Profile IV yields a design that produces the highest efficiency among the existing profiles. So, this can be suggested for field use over others. ii. The velocity distribution after expansion in case of, II, and III becomes highly non-uniform with the result that although the average velocity of flow decreases, the local velocity remains considerably high, which if allowed to persist in erodible channels will scour away bed and sides of channel. REFERENCES [] J. Hinds, The hydraulic design of flume and siphon transitions, Trans., ASCE, 9, pp. 43-459, 98. [] G. E. Hartley, J. P. Jain and A. P. Bhattacharya, Report on the model experiments of fluming of bridges on Purwa branch, Technical Memorandum. 9, United Provinces Irrigation Res. Inst., ucknow (now at Roorkee), India, pp. 94-, 94. [3] R. S. Chaturvedi, Expansive subcritical flow in open channel transitions, J. Inst. of Engrs., India, 43(9), pp. 447-487, 963. [4] C. F. Nashta, and R. J. Garde, Subcritical flow in rigid-bed open channel expansions, J. Hydr. Res., 6(), pp. 49-65, 988. [5] P. K. Swamee and B. C. Basak, A comprehensive open-channel expansion transition design, J. of Irrg. and Drainage Div., ASCE, 9(), pp. -7, 993. [6] H. M. Morris and J. M. Wiggert, Applied Hydraulics in Engineering. Second Ed., The Ronald Press Co., New York, N. Y., 97, pp. 84-88. [7] S. K. Mazumder, Optimum length of transition in open channel expansive subcritical flow, J. Inst. of Engrs., India, 48(3), pp. 463-478, 967. Dhaka University of Engineering & Technology, Gazipur 3