Development of drop number performance for estimate hydraulic jump on vertical and sloped drop structure

International Journal of the Physical Sciences Vol. 5(), pp. 678-687, 8 September, 00 Available online at http://www.academicjournals.org/ijps ISSN 99-950 00 Academic Journals Full Length Research Paper Development of drop number performance for estimate hydraulic jump on vertical and sloped drop structure Mohammad Sholichin and Shatirah Akib * Department of Water Resources, Faculty of Engineering, University of Brawijaya, Indonesia. Department of Civil Engineering, University Malaya, Malaysia. Accepted 3 August, 00 Hydraulic jumps primarily serve to dissipate the excess energy of flowing water downstream of hydraulic structures, such as spillways, weir, sluice gates etc. This type of jump is of practical importance when tail water depth is inadequate to give a good jump. The research conducted at the hydraulics laboratory aimed to obtain suitability of drop number (D) and an estimated hydraulic jump using the drop number. The objective was to obtain the affectivities of application of D on vertical drop and sloped drop structures and to develop the model equation for the sloped drop structure. The hydraulic laboratory conducted research on a hydraulic rectangular flume. Results showed that the D equation could predict drop length (L d ), and hydraulic jump length (L j ) for vertical drop structures; however, the equation could not estimate the hydraulic jump length for sloped drop structure. Therefore, development of a D equation that could be used to estimate hydraulic jump for sloped drop structures, with analysis of non dimensional relationships based on mathematical equations, was needed. These equations have correlation regression with a range between 60-87% and relative error with a range between 5-3%. Key word: Drop number, hydraulic jump and drop structure. INTRODUCTION The hydraulic jump is a practical subject in hydraulic engineering because the hydraulic jump is the best way to dissipate energy present in a moving fluid. The hydraulic jump causes a reduction in the total energy of a moving fluid, which, in turn, prevents the fluid from scouring the channel banks. The hydraulic jump also enables conversion of a portion of the fluid's kinetic energy, which can stabilie downstream flow conditions. The theory of the hydraulic jump was developed by Safrane in 99, in Germany, and a series of twodimensional experiments were verified (Elevatorski, 959). However, the theory of the hydraulic jump is not yet fully developed (Cheng, 995). Drop structures are commonly used for flow control and energy dissipation. Changing the channel slope from steep to mild, by placing drop structures at intervals along the channel reach, changes a continuous steep slope into *Corresponding author. E-mail: shatirahakib@yahoo.co.uk. a series of gentle slopes and vertical drops. Instead of slowing down and transferring high erosion producing velocities into low non- erosive velocities, drop structures control the slope of the channel in such a way that the high, erosive velocities never develop. The kinetic energy or velocity gained by the water as it drops over the crest of each structure is dissipated by a specially designed apron or stilling basin. Design of irrigation systems has followed the book of irrigation design standards with small modifications situated on site. In irrigation systems, the three important parts are the main structure, the irrigation channel and the drainage channel. The drop structure is a support structure that has function in dropping water elevation (Anonymous, 986). Henderson (966) and French (986) explained the drop number for the design of drop structures in irrigation systems. Kindsvater (944) observed the effects of a sloping channel on the hydraulic jump. Wilson (97) investigated the boundary layer effects in the hydraulic jump locations. Rhone (977) studied the effects of channel bottom roughness on the

Sholichin and Akib 679 jump. According to the results of the experiments by Moore (943; Backheteff (93); Rand (965); Locher and Hsu (984), drop number (D) function can explain the geometric flow in drop structures. Hager (99) examined the hydraulic jump on free surface profiles, and developed an empirical equation to determine the length of the jump. The value of the drop number can determine length (L d ) in the stilling basin. There are two classifications of hydraulic conditions in the drop structure: low flow and high flow (Iwao, 99). Much of the literature does not give detailed explanations of drop numbers, especially the ratio of critical depth (y c ) and water fall (). The purpose of the current study was to obtain the boundary conditions for application of Drop Number on vertical and sloped drop structures and to develop the model equation for the sloped drop structure. Theoretical expression Similarities between the two different sied phenomena provided the basis for the hydraulic model s development. In all branches of engineering, mechanical, electrical, civil and others, the laws of similarity provide the foundation for the testing of models. These laws state that the equations used to express the behaviour of the machine or other structure must be dimensionally homogeneous. For complete similarity between the model and the fullsied structure (prototype), this similarity must be geometric, kinematics and dynamic. (i) Form geometric similarity, all the relevant linear dimensions of the model must be directly proportional to the corresponding full-sie dimension, (ii) For kinematics similarity, there must be a linear relationship between the two sets of corresponding velocities, (iii) For dynamic similarity, there must be a linear relationship between the corresponding sets of forces. The basic principle used to analye steady incompressible flow through a channel is the law of conservation of energy expressed by the Bernoulli (energy) equation. The energy equation, generalied to apply to the entire cross section of flow, expresses the energy at any point on the cross section in feet of water by equation, H P V Z + γ +α. g = () Where, H = total energy head in meter of water above the datum plane (m), Z = height above a datum plane (m), P = pressure at the point, pounds per square meter, V = average flow velocity, meters per second (m/sec), g = acceleration due to gravity, m/sec, = contante. The mean presure, at any location along a chute, is determined using the principal of conservation of energy, expresed by the energy equation. Conservation of energy requires that the energy at one location on the spillway will be equal to the energy at any downstream location plus all intervening energy losses. Expressed in equation form and in units of meter, P V P V Z + + α = Z + + α + γ. g γ. g H L. () In order for a hydraulic jump to occur, the flow must be supercritical. The jump becomes more turbulent and more energy dissipates, as Froude s number increases. A jump can occur only when the Froude s number is greater than.0. Froude s number (Fr) is a ratio relating inertia and gravity forces. V Fr = (3) g.y This number, representing the ratio of inertial and gravity forces, is expressed by the average flow velocity V and the celerity of gravity wave in shallow water g. y. Using the Froude number one can distinguish: (i) Critical flow when Fr =, (ii) Supercritical flow when Fr >, (iii)subcritical flow when Fr <. According to Chow (959), jumps in a horiontal channel can be further classified into several types (Table ). Drop number functions that have two variables, water fall () and specific discharge (q), can explained the geometric flow in drop structures (Moore, 943). An equation can determine critical depth (y c ) on a rectangular channel. q g y 3 c =, and with substitution, we can obtain y c q = 3. (4) 3 Z g.z Therefore, 3 y c = Z q g.z 3, or q D = (5) 3 g.z

680 Int. J. Phys. Sci. Table. Types of hydraulic jumps. Name Froude s number Energy dissipation % Characteristics Undular jump.0 -.7 < 5 Standing waves Weak jump.7 -.5 5-5 Smooth rise Oscillating jump.5-4.5 5-45 Unstable; avoid Steady jump 4.5-9.0 45-70 Best design range Strong jump > 9.0 70-85 Choppy, intermittent Figure. Common vertical drop structure (Bos, 976). Figure. Sloping drop structure (Anonymous, 986). Where, D = drop number q = specific discharge (m 3 /sec/m) = water fall (m) g = gravitational constant = 9.8 (m /sec) The common vertical and sloped drop structure with aerated free-falling napped hits the downstream basin floor with turbulent circulation in the pool beneath the napped contributing to the energy dissipation are shown in Figure and. The value of the vertical drop number can predict drop length (L d ), hydraulic jump length (L j ), water depth in upstream (y ) and water depth in down stream (y ) using the following formulae (Moore, 943; Rand, 965): L d = 0.7 (6) L j 4.30D ( y ) = 6.9 y (7) y = 0.45 (8) 0.54D y = 0.7 (9).66D

Sholichin and Akib 68 Table. Matrix formulation for dimensional. Variables Depended variable Free variable Other k k k 3 k 4 k 5 k 6 k 7 k 8 K 9 k 0 Q v L d y c y y L j h Z g M 0 0 0 0 0 0 0 0 0 0 L T - - 0 0 0 0 0 0 0 - DIMENSIONAL ANALYSIS Dimensional analysis, one approach to identify useful parameter combinations, requires dimensional consistency in the equation governing a process of interest. Although, the requirement for dimensional consistency applies to equations that have dimensions in each term, it is invariably applied in ways that convert all the terms to dimensionless groups. Main physical quantity is some combination of length, mass and time (denoted L, M, and T, respectively). A matrix method formulates a dimensional equation is shown in Table. The Buckingham i theorem is a tool to provide the relationships between N quantities with M dimensions. This theorem arranges the quantities as N-M independent dimensionless parameters. Therefore, the functional relation must exist as (Hanche-Olsen, 004). k k k 3 kn π = ρ, ρ, ρ,..., ρ, (0) 3 n These parameters contain three main variables, such as M, L and T. If has dimension M i, L i, T i, then the dimension of is as follows: = (M L T ) k, (M L T i ) k.(m n L n T n ) kn () It can be () non-dimensional if k + k +. n k n = 0 () k + k + n k n = 0 (3) k + k + k = 0 (4) The Buckingham i method analyed hydraulic parameters involved in a drop structure and hydraulic jump as follows: q = the flow discharge per unit width v = the mean velocity of upstream section. L d = drop length y c = critical depth y = the initial depth of the jump upstream y = the sequent depth of the jump downstream L j = the length of the hydraulic jump h = water depth in upper drop structure = water fall g = the acceleration due to gravity Statistical analysis Data consisting of two variables or more require the variables to correlate with each other. A mathematical equation that expresses the functional relationship between any variables generally expresses the correlation. To determine the equation that expresses any variables requires transformation of the collected data, showing the free variable value perceived. Therefore, a scatter diagram can express the results of the measurements. This results in a depiction of a regression curve that expresses many variables. This correlation test (R ) indicates the degree of correlation between any variables (Henriksen et al., 003): [ ( x x)( y y) ] ( x x) ( y y) R (5) = Where, R = coefficient of correlation x = free of variable y = depended of variable x = average value of free variable y = average value of depended variable The range of values for R is.0 (best) to 0.0. The R coefficient measures the fraction of the variation in the measured data that is replicated in simulated model results. A value of 0.0 for R means that none of the variance in the measured data is replicated by the model predictions. On the other hand, a value of.0 indicates that all of the variance in the measured data is replicated by the model predictions. Henriksen et al. (003) suggest that a R value > 0.85 is excellent for a model, a value between 0.65-0.85 is very good, values between 0.5-0.65 are good, those between 00-0.50 are poor and any < 0.0 are very poor. In order to know the relative

68 Int. J. Phys. Sci. Figure 3. Flume channel in hydraulic laboratory. error, the relative error equation () was used. x y = 00% x (6) Where, = relative error x = teoritical results y = model measurement results EXPERIMENTAL SETUP The experiments used a rectangular laboratory flume at the University of Brawijaya. The flume was 7.5 m long, 0.078 m wide, and 0.35 m high; a hydraulic circuit, connected to the flume, allowed for a recirculation of discharge. The walls and the bed of the flume were made of Plexiglas sheets (Figure 3). The experiments included both vertical drop and sloping drop structures. A pump supplied water from an underground storage tank in the laboratory. A point gauge measured the flow depth h. The runs were carried out for selected values of the flow depth y, y and the Froude number Fr ; the jump was set within the experimental measuring reach, using the downstream gate. For each run, the discharge Q, the flow depths y, y and the jump length L j were measured. A weight method measured the discharges, which varied from 75-950 cm 3 /s (Bos, 976; Novak et al., 98). The model vertical drop structure and sloped drop structure were constructed by using wooden material with a high quality performance. Figure 4 (a) and (b) show the model before and after installation in the flume. The vertical drop structure had several heights of drop. The heights of the drops were 4.0, 6.0, 8.0, 0.0, and.0 cm, respectively. The discharge flows were 75, 0, 350, 500, 700 and 940 cm 3 / sec, respectively. Then, the same methods were repeated for sloping drop structure. Table 3 shows the description of scenarios of the hydraulic test. In part I, measurements of water jump length (L j) were measured at the vertical drop structure by the following discharges, (Q) 75, 0, 350, 500, 700 and 940 cm 3 /sec, respectively, and with condition of water fall height () of 4, 6, 8 0 and cm, respectively. In part II, III and IV similar experiments were conducted with discharge and water fall height variations as in part I but with different drop structures. RESULTS AND DISCUSSION The jump length, the distance between the beginning and the end of a hydraulic jump, usually moves up and down in the channel and is difficult to measure. Basically, the section of water surface starting to rise and having rollers coming out was regarded as the beginning of the jump, and the section of the water surface having the most bubbles coming out from the water surface was considered the end of the jump. The average of the distances from the beginnings to the ends of the jumps constitutes an average of jump length. The initial depth and sequent depth of the jump are the depths of the beginning and the end of a hydraulic jump, respectively. In order to know the suitability of the drop number used, a comparison between theoretical graphs based on the Moore equation is required, with measurement

Sholichin and Akib 683 Figure 4. Wood material for vertical and sloped drop structure with ratio (m = :.0; m = :.5; m = :.5; m = :.75; m=:.0); where (a) before install in flume, (b) after install in flume. Table 3. Design experimental. Part Drop structure High drop (Z) (cm) Discharge (cm 3 /sec) Vertical drop 4, 6, 8, 0, and 75, 0, 350, 500, 700 and 940 Sloping with ratio :.0 4, 6, 8, 0, and 75, 0, 350, 500, 700 and 940 3 Sloping with ratio :.5 4, 6, 8, 0, and 75, 0, 350, 500, 700 and 940 4 Sloping with ratio :.0 4, 6, 8, 0, and 75, 0, 350, 500, 700 and 940 00 00 Ratio (y/) and (Lj/ 0 0. 0.0 Fr Lj/ Ld/ y/ y/ Lj/ Ld/ Fr 0 0. 0.0 Froud number 0.00 0.0000 0.000 0.00 0.0 0. Drop number [q /(g. 3) ] Figure 5. Theoretical curve of relationship between L j/, L d/, y / and Fr with D based on Rand (955); Moore (943) equation. 0.00 records on the rectangular flume at vertical drop structure and sloped drop structure. Statistical analysis is used to determinate relative error. Figure 5 shows the 684 Int. J. Phys. Sci. theoretical curve of the relationship between L j /, L d /, y / and Fr with D based on the Rand (955) and Moore (943) equation. The increase of drop number resulted to

00 00 Ratio (y/) and (Lj/ ratio (y /) and (L j /) 0 0. 0.0 0.00 Lj/ Ld/ y/ Fr 0.000 0.00 0.0 0. Drop number [q /(g. 3) ] y/ Lj/ Ld/ Fr Figure 6. Curve of relationship between L j/, L d/, y / and Fr with D based on measurement in hydraulic flume. 0 0. 0.0 Froud number 0.00 Figure 7. Vertical drop structure on hydraulic rectangular flume. the increase ratio of L j /, L d / and y /, respectively. In contrast, the Fr value decreased. Figure 6 shows the curve of relationship between L j /, Ld/, y / and Fr with D based on measurement in the hydraulic flume. Most of relationship graphs such as L j /, L d / and y / with D in Figure 6 had similar trend in comparison with graphs in Figure 5. According to statistical analysis and comparison between the resulting calculations by Moore s formula, with reading data from the channel flume for drop length (L d ), there was not much difference. Based on statistical analysis, the maximum error degree was less than 5%. The hydraulic jump length (L j ) value was about 5-8% relative error. The drop number (D) equation predicted drop length (L d ) and hydraulic jump length (L j ) on vertical drop structure. Figure 7 shows the vertical drop structure in the hydraulic flume with discharge 75 cm /sec and height water fall 8 cm. Based on the results of measurements at the flume for the sloped drop structure, such as (m = :, m = :.5 and m = :) and statistic analysis, the relative error between measurement records with the empirical formulation equation was more than > Sholichin and Akib 685

Table 4. Matrix formulation for dimensional. Variables k k k 3 k 4 k 5 k 6 k 7 k 8 k 9 k 0 q v L d y c y y L j h g 0 0 0 0 0 0 0 -.5-0.5 0 0 0 0 0 0 0-0,5-0.5 3 0 0 0 0 0 0 0 -.0 0 4 0 0 0 0 0 0 0 -.0 0 5 0 0 0 0 0 0 0 -.0 0 6 0 0 0 0 0 0 0 -.0 0 7 0 0 0 0 0 0 0 -.0 0 8 0 0 0 0 0 0 0 -.0 0 5%. The drop number could not estimate hydraulic jump in the sloped drop structure. Therefore, in order to develop the drop number (D) to estimate hydraulic jump on the sloped drop structure, the analysis of non dimensional relationships, based on mathematical equations, was needed (Table 4). Formulation based on dimensional analysis: α = 0 β = k + + + + + + + + + k k3 k4 k5 k6 k7 k8 k9 k0 γ = k k k0 By elimination of k 0 + k 9, can be found k = k k k 0 0 =.5k 0. 5 k 0 k ( 0.5 k 0. ) 9 = k k k3 k4 k5 k6 k7 k8 5k 9 =.5 k 0. 5k k3 k4 k5 k6 k7 k8 k In order to finish this equation, Langhaar matrix was used (Table 4). variable had relationship with k, k 9 and k 0 but it had no relationship with k, k 3, k 4, k 5, k 6, k 7, and k 8, respectively. The variable had relationship with k, k 9 and k 0 but it had no relationship with k, k 3, k 4, k 5, k 6, k 7, and k 8, respectively. The 3 variable had relationship with k 3, k 9 and k 0 but it had no relationship with k, k, k 4, k 5, k 6, k 7, and k 8, respectively. Moreover, variable 4, 5, 6, 7, and 7 had relationship with k 9. The final analyses based on Langhaar matrix can be summaried by eight dimensionless relationships as follows: q g. = ; 3 = v g. ; 3 = 686 Int. J. Phys. Sci. L d ; 4 = y c y ; 5 = ; 6 = y ; 7 = L j, and 8 = h The measurements of hydraulic jumps length on sloped drop structure in each condition were conducted 5 times in the range of discharges between 75-940 cm 3 /s and height of waterfall from 4 - cm. The resume of measurement of hydraulic jump is shown in Table 5. Figure 8 shows the development curve of relationship between Lj/ and y/ with D and Figure 9 shows the development curve of relationship between y/, h/ with D. Therefore, this establishes the correlation between with 5, 6, 7 and 8, respectively. Based on statistical analysis, this research has developed a new formula for sloping drop structures in laboratory conditions. 0.68 y = 7.038x with R = 0.867, where y = x = D, therefore L j = 7.038D 0.68 0.40 y = 0.494x with R = 0.809, where y = y = 0. 494 D x = D, therefore 0. 40 0.5 y = 0.73x with R = 0.60, where y = x = D, therefore y = 0. 73D 0.5 L j and y and y and 0.3 y = 0.79x with R = 0.60, where y = and x = D, therefore h = 0.79D 0.3 h

00 0 y = 7.038x 0.68 R = 0.867 Ratio (Lj/) and (y/) 0. 0.0 y = 0.494x 0.400 R = 0.809 =4 cm =6 cm = 8cm = 0 cm = cm 0.00 0.0000 0.000 0.00 0.0 0. Dro p nu mber [q /(g. 3 )] Figure 8. Development curve of relationship between L j/ and y / with D based on measurement data analysis Table 5. Resume of measurement of hydraulic jump Lj (cm) in each sloped drop structure. h Q m = : m = :.5 m = : (cm ) (cm ) (cm ) cm cm 3 /s 4 6 8 0 4 6 8 0 4 6 8 0.0 75 6.3 8.0.5 5.9 9. 8.3 0.0 3.5 7.9..3 3.0 6.5 0.9 4..5 0.4 3.8 9.0.7 7.9 6.4 8.8 4.0 6.7 3.9 0.4.8 8.0 30.7 36.9.0 350 5. 8.7 6.8 8.7 35.9 0. 5.0 3.5 39. 45.0 4. 30.0 40.0 45.0 53.0.5 500 8.8 4.8 35.0 36. 4.9 3.8 3.3 33.8 46.9 5.5 7.8 35.0 45.0 50.5 59.0 3.0 700.0 9.6 4. 43.6 50. 7.0 39.9 40.0 5.0 60.0 3.0 40.0 50.0 56.0 64.0 3.5 940 4.8 34.6 54.3 56.3 59.8 3.8 49.8 50.0 60. 70. 37.8 45.0 55.0 60.0 68.0 00 Ratio (y/) and (hj/ 0 0. y = 0.73x 0.5 R = 0.60 y = 0.79x 0.3 R = 0.60 0.0 0.00 0.0 0. Drop number [q/(g.3)] Figure 9. Development curve of relationship between y /, h/ with D based on measurement data analysis.

Sholichin and Akib 687 Figure 0. Sloped drop structure with slop; (a) m = :; (b) m = :.5, (c) m = :. Figure 0 shows the sloped drop structure on the rectangular flume with ratio m = :.0, m = :.5 and m = :.0, respectively. Conclusion Drop number use for vertical drop structures was appropriate to conditions from weak to oscillation (transition) hydraulic jump (at the Froude number value range:.70 < Fr < 4.5) and at range 0. < y c / < 0.6 with the specific discharge range: 30 cm 3 /sec/m < q < 0 cm 3 /sec/m. The hydraulic jump length (L j ) value was about 5-8 % relative error. The D equation predicted drop length (L d ) and hydraulic jump length (L j ) on vertical drop structures. Based on the results of the measurements at the flume for sloped drop structure, (m = :, m = :.5 and m = :) and statistic analysis, the relative error between measurement records, with the empirical formulation equation, was more than > 5%. This indicated that drop numbers were not useful in estimating hydraulic jump on sloped drop structures. Development of drop number (D) that can be used to estimate hydraulic jump on sloped drop structures, with analysis of non-dimensional relationships, based on mathematical equations, was needed. These new equations have correlation regression with a range between 60-87% and relative error with a range between 5-3%. Chow VT (959). Open Channel Hydarulics. McGraw, New York, NY. Elevatorski EA (959). Hydraulic Energv Dissigators, McGraw, New York, NY. French RH (986). Open Channel Hydraulics, McGraw-Hill Book Company, New York. Hanche-Olsen H (004). Buckingham's pi-theorem, Retrieved April 9, 007. Hager WH (99). Energy Dissipaters and Hydraulic Jump, Kluwer Academic Publishers, London. Henderson FM (966). Open Channel Flow. MacMillan Company, New York, USA. Henriksen H, Troldborg L, Nyegaard P, Sonnenborg T, Refsgaard J, Madsen B (003). Methodology for construction, calibration and validation of a national hydrological model for Denmark, J. Hydrol., 80: 5-7. Iwao O, Youichi Y (99). Transition from supercritical to sub critical flow at an abrupt drop, J. Hydraulic Res., 9: 3. Kindsvater CE (944). The hydraulic jump in sloping channels. Trans. Am. Soc. Civ. Eng., 09: 07-0. Locher FA, Hsu ST (984). Energy dissipation at high dams-in Development in Hydraulic Engineering. Vol., P. Novak (Ld.), London, Elsevier Applied Science. Moore WL (943). Energy Loss at the Base of a Free Overfall. Transactions, ASCE, 08: 343-360. Discussion: 08: 36-39. Novak P, Cabelka J (98). Models in Hydraulic Engineering- Physical principles and Design Applications. Pitman, London, pp. 60-73, 85-93. Rand W (965). Flow Over A vertical Sill in An Open Channel. J. Hydraulic Div. ASCE, 9(4): 97-. Rhone TJ (977). Baffled Apron as Spillway Energy Dissipater. J. Hydraulic Div., ASCE, 03(): 39-40. Safrane K (99). Untersuchung über den Wechselsprung (Investigation on the hydraulic jump), Der Bauingenieur, 0, Heft 37: 649-65; Heft 38: 668-678. Wilson EH, Turner AA (97). Boundary Layer Effects on Hydraulic Jump Location. J. Hydraulic Div., ASCE, 98(7): 7-4. REFERENCES Anonymous (986). Irrigation Design Standard, Design Criteria, Department of Public Work, Jakarta, Indonesia. Bacheteff BA (93). Hyraulic open Chaneels. McGraw, New York, NY. Bos MG (976). Discharge Measurement Structure, Working Group on Small Hydraulic Structure, Oxford & IBH Publishing CO, New Delhi. Cheng F (995). L-D Etermining The Location of Hydraulic Jump by Model Test and Hec- Flow Routing, thesis, Frit J. and Dolores H. Russ College of Engineering and Technology, Ohio University.