New analytical formulation of the De Marchi model for zero height side. weir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 New analytical formulation of the De Marchi model for zero height side weir Giovanni Michelazzo 1 Abstract One-dimensional modeling is often used to simulate the hydrodynamics of free-surface flows, including spatially-varied flows as encountered along the side weirs. This paper deals with the particular case of a side weir with zero height crest acting on fixed bed and subcritical flow, for which a new analytical model is formulated starting from the De Marchi hypothesis. The proposed model provides an original interpretation of the side weir problem for which the solution, formulated in dimensionless form in terms of average flow and geometrical variables of the side weir and main channel, is the explicit result of the imposed boundary conditions at the upstream and downstream cross-sections. The proposed model appears to be able to analyze a wide range of hydraulic problems similar to the side weir flow, such as lateral diversions, and to solve design and verification problems as first approximation. The analytical model is used to predict experimental data coming from literature as well as from a new set of laboratory data in a very direct way without any need of numerical techniques. 18 19 20 Subject headings: Channel flow; Hydraulic models; Laboratory tests; One dimensional flow, Weirs. 21 22 23 Key-words: 1D analytical model; De Marchi hypothesis; Laboratory investigations; Side weir; Zero height crest. 1 PhD, Postdoctoral researcher Department of Civil and Environmental Engineering, University of Florence Via S. Marta 3, 50139 Firenze, Italy Phone: 0039 055 4796458 Email: giovanni.michelazzo@dicea.unifi.it Corresponding author. 1

24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 Introduction to the side weir modeling Side weir flows are frequently modeled by means of numerical techniques relying on onedimensional (1D) approaches. Actually, the flow field in the main channel close to the side weir is more complex, since the lateral outflow induces changes in the flow pattern, velocity distribution and mass transport (Neary and Odgaard 1993; Lee and Holley 2002; May et al. 2003; Rosier 2007). Though several contributions have been developed in order to simulate the flow field into details by means of two-dimensional (2D) and three-dimensional (3D) numerical models (Neary et al. 1999; Vasquez 2005; Li and Zeng 2009; Zhou and Zeng 2009), a strong interest still exists on predicting the main flow features in side weir problems by means of simpler 1D approaches (May et al. 2003), which can be applied straightforwardly to several practical situations frequently found in environmental and river engineering. In fact, the hydraulic design of a side weir is often addressed by means of simplified approaches that are based on the equations of the steady, spatially-varied, one-dimensional flow (Subramanya and Awasthy 1972; Hager 1987; Borghei et al. 1999; Oliveto et al. 2001; Durga Rao and Pillai 2008; Emiroglu et al. 2011). Such scheme can predict the flow variables as averages on the cross-section along the main channel, whereas the variations in the vertical and transverse directions are not reproduced. One-dimensional approaches are theoretically more appropriate for those situations in which the flow develops mainly along a longitudinal direction and the secondary flow structures do not get high importance. The longitudinal dimension should be larger than transverse and vertical ones in order to have a good reproduction by 1D-models and, even if it is not possible to define a precise and generally valid range of the geometrical dimensions for which a 1D model may be acceptable since the answer depends on the case and on the goal of the study, it 2

49 is usually accepted the ratio between the flow depth and the channel width to approximately 50 get values around Y/B 10-1. If the prediction of the flow only in its longitudinal development 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 is considered to provide a sufficiently detailed description of the hydrodynamics, then the 1Dmodels may be applied, with the restriction that transverse effects are neglected. The choice of a 1D-scheme provides benefits in terms of easy implementation and possibility to formulate general solutions of the side weir problem. One-dimensional models can be formulated in order to achieve analytical solutions, that offer expeditious tools to predict the longitudinal rate of change of the flow depth and the water discharge in the main channel along a side weir. Within this context, an important contribution to the analytical modeling was given by De Marchi (1934), who developed an equation to predict the longitudinal profile of flow depth under the assumption that the energy per unit weight of the flow remains constant along the side weir. Several past and more recent studies about side weirs rely on the De Marchi assumption (Singh et al. 1994; Borghei et al. 1999; Paris et al. 2012; Namaee et al. 2013), which can reasonably hold in the case of subcritical flows and mild slope of the channel. Recently, some authors developed the De Marchi solution in order to take into account additional effects, such as the variations along the side weir of the specific energy, discharge coefficient and Coriolis coefficient (Venutelli 2008), or different shapes of the main channel, such as trapezoidal (Cheong 1991), triangular (Vatankhah 2012a), circular (Vatankhah 2012b) and U-shaped (Vatankhah 2013) cross-sections. All these studies highlight the scientific interest in proposing analytical or semi-analytical approaches to the side weir problem as useful and handy tools to predict the diverted discharge through existing or designed side weirs. The particular case of a side weir with zero height crest (i.e., with the weir crest at the same elevation of the main channel bottom) has been less addressed, despite it has interesting 3

74 75 76 77 78 79 80 81 82 similarities with other free-surface problems, such as branch channels in river bifurcation, lateral intakes and levee breaches. A new analytical formulation of the De Marchi model is developed in this study for steady flow and fixed bed conditions which allows to evaluate flow and geometrical features of a zero height side weir related to the main channel characteristics as functions of the flow conditions imposed at the upstream and downstream sections. The proposed formulation is used to simulate the main flow features of experimental data from the existing literature and from the new set of laboratory investigations performed in the framework of this study. 83 84 85 86 New analytical formulation of the De Marchi model A side weir of zero height crest with rectangular shape is considered along a free-surface channel with a rectangular cross-section of width B, fixed bottom and mild slope (see Fig. 1). 87 88 Fig. 1. Side weir problem: plan view (a); longitudinal section (b); cross section (c). 89 90 91 92 93 94 95 The overflow through the side weir length L s is considered not to be influenced by backwater effects from the tailwater and the De Marchi is assumed, i.e. the total head of the flow remains constant along the side weir. Assuming a negligible variation of the bed elevation between the cross-sections 1 and 2, also the specific energy head can be considered to be constant along the side weir, so that the mass and energy conservation provides the following system: 96 dq(x) dx q C 2g Y(x) s de(x) dx d 0 3/ 2 (1) 4

97 98 99 100 101 102 where E [m] is the specific energy head in the main channel; Y [m] is the flow depth; Q [m 3 /s] is the water discharge in the main channel; q s [m 2 /s] is the discharge diverted through the side weir per unit length; x [m] is the longitudinal axis starting at the beginning of the side weir and directed as the main flow direction (see Fig. 1); g is the gravity acceleration [m/s 2 ]; C d [-] is a discharge coefficient assumed to be constant along the side weir. The traditional solution provided by De Marchi (1934) states: 103 Cd x B E Y 1 E Y 2 3sin Y E (2) 104 where is a dimensionless flow function and C is an integration constant determined from 105 106 107 108 109 110 111 112 113 114 115 the boundary conditions. The new formulation of the De Marchi model is proposed by means of a rearrangement of the equation of conservation of mass, the equation of balance of energy and the De Marchi equation (2) in order to express the main variables of the system as characteristic dimensionless ratios between the side weir and the main channel features depicted in Fig. 1: - Flow depth ratio r Y = Y d / Y u, where Y u and Y d are the flow depths at the upstream and downstream section of the side weir, respectively - Discharge ratio r Q = Q s / Q u, where Q s and Q u are the water discharges diverted through the side weir and incoming from upstream, respectively - Length ratio r L = L s / B 116 117 118 119 These dimensionless ratios are here expressed as functions of the flow conditions achieved at the upstream and downstream cross-sections of the side weir (sections 1 and 2, respectively, in Fig. 1) in terms of the Froude number Fr. 5

120 In particular, the balance of the specific energy head E, together with the assumption that 121 the Coriolis coefficients at the upstream and downstream cross-sections can be taken equal 122 to unity as a first approximation, gives the flow depth ratio r Y : 123 E E r u d Y Y Y d u 2 2 Fr Fr 2 u 2 d (3) 124 125 The continuity principle is used to get the discharge ratio r Q : Q Fr Q Q Q r 1 r s d 3/ 2 s u d Q Y Qu Fru (4) 126 The length ratio r L is finally determined from the system 2: 127 r L L B s d u C d (5) 128 where the dimensionless flow function is expressed in terms of Fr: 129 2 3sin 1 1 2 Fr 2 (6) 130 131 Moreover, the ratio between the average flow velocities at the downstream and upstream sections is directly deduced: 132 r U U U d u 1 rq r Y (7) 133 134 The characteristic ratios are solved once the flow conditions at the upstream and downstream sections are provided. The side weir problem is then described as a combination 135 of the state variables Fr u and Fr d that define a plane, here called, in which the 136 137 138 139 behavior of the flow along a generic side weir is represented as referred to different conditions (see Fig. 2): a) The channel has a mild slope, the flow is subcritical everywhere, the flow depth profile increases downstream and Fr decreases downstream; 6

140 141 142 143 144 145 b) The channel has a steep slope, the flow is supercritical everywhere, the flow depth profile decreases downstream and Fr increases downstream; c) The incoming flow is supercritical, the downstream flow is subcritical and a hydraulic jump occurs within the side weir length. In this case, it is not possible to state a priori whether the downstream flow depth is greater or lower than the upstream one. 146 147 Fig. 2. Side weir situation in the Froude graph: a) subcritical; b) supercritical; c) hydraulic jump. 148 149 150 151 152 153 The dashed areas in Fig. 2 identify flow combinations which are not physically possible: a subcritical approaching flow remains subcritical along the side weir, whereas a supercritical approaching flow gets greater Fr downstream, except when a downstream control requires a depth larger than the critical one Y cr. An analytical demonstration of such behavior is provided by Montes (1998) for a main channel with a rectangular section: 154 2 dfr(x) Fr(x) dq(x) 2 Fr (x) 2 dx 2Q(x) dx 1 Fr (x) (8) 155 156 157 158 159 160 161 162 163 The dimensionless solution of the side weir problem expressed by Eqs. (3) to (7) is computed for every admissible combination of the Froude numbers in the considered case of a subcritical flow (area a in Fig. 2). Any considered couple (Fr u ; Fr d ) can be obtained as a combination of different values of the imposed independent variables, which are: main channel geometry, bed roughness, bed slope, incoming discharge, side weir length, and boundary conditions, e.g. downstream rating curve. Model results are shown by Fig. 3 in terms of contours of the dimensionless ratios in the Froude graph. 7

164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 Fig. 3. Dimensionless ratios from the analytical model: a) r Y contours; b) r Q contours; c) r L contours; d) r U contours. The 45 line defines the condition Fr u = Fr d, which occurs when no lateral outflow takes place in the channel (Q s = 0). In such situation the flow depth and the discharge remain unchanged from upstream to downstream (r Y = 1, r Q = 0, r L = 0, r U = 1), since the side weir flow does not exist. On the other hand, the presence of a side weir makes an outflow possible in the region where Fr u > Fr d : the main channel flow is spatially varied in terms of flow depth increasing downstream (r Y > 1), flow discharge and velocity decreasing (r Q > 0, r U < 1) along the side weir length (r L > 0). This trend develops for increasing Fr u and decreasing Fr d and the influence of the downstream condition on the side weir flow is particularly evident from the contours of r Q in Fig. 3b, in which a significant increase of the diverted discharge is obtained by imposing a boundary condition associated with lower Fr d rather than greater Fr u. An increase of Fr u, which may be related to a greater acceleration of the upstream flow due to a longer side weir, does not make a so efficient increase of the diverted discharge as a decrease of Fr d does, at least if the flow is required to be subcritical. A limiting case is recognized when the critical state occurs in the upstream section (Fr u = 1) and no flow takes place downstream (r U = 0 and Fr d = 0), which is depicted in the upper left corner of the graphs of Fig. 3. In that situation, the characteristic ratios regarding flow depth, water discharge and side weir length achieve their maximum values: r Ymax = 1.5 is a constant for a rectangular cross-section, r Qmax = 1 and r Lmax tends to a value which is on the order of the channel width and which depends on the discharge coefficient C d. The curves of Fig. 3c were calculated by setting the classical value for broad-crested weirs C d = 0.385 and the maximum length ratio resulted r Lmax 1.12. 8

188 The length ratio r L is directly determined by the De Marchi model and it depends on the 189 -function in addition to the coefficient C d. The variation of with Fr is shown in Fig. 4: the 190 function gets a minimum min 0.43 at the critical state and two conjugated roots of are 191 detected for subcritical (Fr = 0) and supercritical (Fr 2.05) flow. The maximum variation of 192 the -function in the subcritical range is therefore given for a critical state at the upstream 193 section and no flow at the downstream section, as already discussed with reference to Fig. 3c. 194 195 196 Fig. 4. Trend of -function in a subcritical and supercritical flow. 197 198 199 200 In order to analyze the influence of the discharge coefficient on the length ratio, some of the existing formulations for C d are compared in Fig. 5. The discharge coefficient is then calculated for each grid point of the Froude graph and the effect of each considered formulation of C d on r L is shown by graphs of Fig. 6. 201 202 Fig. 5. Variation of C d with Fr u according to different formulations. 203 204 205 206 207 208 209 210 211 The choice of C d affects the values of r L, but it does not influence the general behavior of the solution, as r L is still increasing for greater Fr u and smaller Fr d and its maximum value is always in the range [1 4]. In particular, formulations proposed by Hager (1987), in Fig. 6d, and by Borghei et al. (1999), in Fig. 6f, show similar results of the solution, which are also obtained by imposing a constant C d, as in Fig. 6a. On the other hand, the formulation by Ranga Raju et al. (1979), in Fig. 6c, gives values comparable with Singh et al. (1994), in Fig. 6e, whereas the formulation by Subramanya and Awasthy (1972), in Fig. 6b, provides the most different results in terms of values and trend of the r L -function. 212 9

213 214 215 Fig. 6. Comparison of r L contours according to different formulations of C d : a) C d = 0.385; b) Subramanya and Awasthy (1972); c) Ranga Raju et al. (1979); d) Hager (1987); e) Singh et al. (1994); f) Borghei et al. (1999). 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 Laboratory investigations on a zero height side weir New laboratory experiments were carried out at the hydraulic laboratory of Civil and Environmental Engineering of the University of Florence in a 5.10 m long, 0.43 m wide and 0.30 m deep, glass-walled, water-recirculating, tilting flume with a rectangular cross-section (see the experimental set-up in Fig. 7). The flume was subdivided longitudinally into two channels separated by a vertical 0.30 m high glass wall and it was tilted with an inclination S 0 = 0.1 %. The main channel, where the incoming water discharge Q u was put in, was 0.30 m wide and 0.16 m deep, being its bottom raised in order to avoid backwater effects from the tailwater toward the main flow. The lateral channel, 0.12 m wide and 0.30 m deep, allowed to evacuate the discharge Q s diverted by a side weir placed at about 2 m from the flume entrance section on the left side of the main channel. The side weir crest was at the same elevation of the main channel bottom and its length L s could be set up to about 0.5 m: the main and the lateral channels were separated, so that the water could flow in the lateral channel only through the side weir. At the downstream end of the main channel a vertical sluice gate was set to regulate the downstream boundary condition in terms of the flow depth and rate, while the lateral channel ended over the storage basin. A fixed roughness composed by granular gravel with a median diameter D 50 = 6.8 mm was stuck down to the bottom of the main channel in order to provide a realistic resistance to the flow (dimensionless Chézy coefficient in the order of C h 10). 10

237 238 239 D-. Fig. 7. Experimental set-up in the tilting flume: a) plan view; b) longitudinal section A- 240 241 242 243 244 245 The same constant inflow discharge Q u of about 10 l/s was set during the investigations, whereas L s was changed according to ten different values ranging from 3 to 47 cm. Moreover, two different downstream boundary conditions were imposed by choosing two different configurations for the sluice gate. The two configurations,, were set by choosing the opening extent of the vertical sluice gate: in particular, 246 247 248 249 250 251 252 253 254 255 discharge, a larger flow depth took place in the downstream reach of the flume. Finally, no sediment was put into the flume. Table 1 provides a synthesis of the experimental conditions of the runs. The goal of the experimental activity was not to scale a singular case of study, but it was to investigate a process. The experimental set-up allowed to reproduce a flow that was characterized by typical values of the Reynolds number of about Re = 10 5, that, for a relative roughness R = 10-1 (where R is the ratio between the absolute roughness of the channel bed and the hydraulic radius), is referred to a fully turbulent flow (rough flow regime). Therefore, it was considered that the main turbulent flow structures could be reasonably reproduced. 256 257 258 Measurements of the water surface and the water discharge were taken. The water surface was measured along the centerline (x-axis in Fig. 7) of the main channel every 1 cm of 259 longitudinal distance by means of ultrasonic sensors having a precision of 1 mm and the 260 261 local flow depth Y referred to the fixed bottom of the main channel was then calculated. The data regarding the flow depth at the upstream and downstream sections of the side weir are 11

262 263 264 reported in this paper, whereas further experimental results regarding the observed flow field will be presented in a forthcoming paper. The inflow discharge Q u was recirculated from the storage basin and it was measured by an 265 electromagnetic flow meter installed at the inflow pipe with an accuracy of 10-4 m 3 /s. 266 267 268 269 The diverted discharge Q s through the side weir was calculated by using a rating curve previously calibrated at the downstream zone of the lateral channel and with the water level measured at the same location by an ultrasonic sensor. The water discharge flowing in the main channel downstream the side weir Q d was calculated as the difference between the 270 inflow and the diverted discharge (Q d = Q u Q s ) and, as a further control, by using another 271 272 273 calibrated rating curve at the downstream zone of the main channel. The main experimental results in terms of the flow depth and the water discharge at the upstream and downstream sections of the side weir are reported in Table 1. 274 275 276 Table 1. Summary of conditions and results for the performed tests (for each test: main channel width B = 0.3 m, side weir with zero height crest). 277 278 279 280 281 282 283 Performances of the proposed model The proposed model is now tested with the experimental data that have been found in the analyzed literature and that have been collected within the present work. Moreover, a comparison with the classical models is provided, in order to highlight the novelty of the proposed approach. 284 285 Comparison between the experimental data and the proposed model 12

286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 The proposed analytical model is applied to the new laboratory investigations regarding side weirs with a zero height crest presented in the previous Section. Only the runs related to the subcritical flow conditions are taken into account in order to satisfy the model assumptions and such verification is made by checking the flow depth profile to be increasing along the entire length side weir: as a consequence, tests B7 to B13 and C17 to C19 are selected, for which the flow conditions ranged within Fr u = [0.18 0.85] and Fr d = [0.12 0.51]. Other tests, as B14 to B16 and C20, are not selected, since they showed a decreasing flow depth profile at the beginning of the side weir, that might be related to the occurrence of a modest submerged hydraulic jump similar to the condition in Fig. 2c, with the particular case of subcritical approaching flow and critical state at the upstream section. Moreover, additional data sets were searched in the literature regarding side weirs. Unfortunately, very few experimental data regarding the case of side weir with a zero height crest and conditions of a subcritical flow have been found: data from Awasthy (1970) and Subramanya and Awasthy (1972) were the only data that considered the aforementioned conditions, so that they could be used in the present work, whereas other data did not satisfy the conditions (as data by Hager, 1981). Therefore, data from Awasthy (1970), that are included in Subramanya and Awasthy (1972), are here used to test the reliability of the proposed model for a zero height side weir and subcritical flow conditions, which ranged within Fr u = [0.23 0.84], Fr d = [0.03 0.63] 305 and Q u = [0.0032 0.0782] m 3 /s. 306 307 308 309 310 Finally, a third set of data was analyzed regarding the case of a lateral diversion, since it has some analogies compared to the side weir case. A lateral diversion represents an outflow which affects the main channel flow in a way similar to a side weir, even if the resulting flow field depends also on the conditions acting in the branch channel. Despite the analytical model does not take into account backwater effects from the outflow, which can occur for the 13

311 312 313 314 315 316 317 318 319 320 lateral diversion, the effect of the branch channel flow on the upstream and downstream flow is implicitly considered by the model, since it uses the boundary flow features Fr u and Fr d to close the balance of the mass and the energy within the control volume given by the portion of the main channel in front of the lateral outflow. Therefore, the application of the proposed model to the lateral diversion case may be performed in terms of the prediction of flow depth and flow discharge, with the restriction that the width of the branch channel cannot be specifically developed for side weirs. Data from Hsu et al. (2002) are used as a case of a subcritical, right-angled, lateral diversion, for which the flow conditions ranged within Fr u = [0.32 0.77], Fr d = [0.14 0.55] and Q u = [0.0030 0.0054] m 3 /s. 321 322 The comparison between the experime the 323 324 325 326 327 328 329 330 of the flow depth (Fig. 8), flow discharge (Fig. 9) and weir length (Fig. 10). The discharge coefficient C d for the calculation of r L regarding the data of the present study is chosen according to the formulation proposed by Borghei et al. (1999) which gives the best fit, whereas the data from Subramanya and Awasthy (1972) are predicted considering the formulation proposed by the same authors. The r L ratio is not computed for the lateral diversion case, since the side weir law cannot be used to model a branch channel. The results presented in Figs. 8, 9 and 10 show that the proposed analytical model gives a good prediction of the flow conditions in terms of the flow depth ratio and flow discharge 331 ratio and most of data are predicted within an error of 10 %. The validity of the De Marchi 332 333 334 hypothesis is confirmed for the zero height side weir experiments as well as for the lateral diversion tests and most of the collected data are well predicted (Fig. 8). The comparison between the model prediction and the experimental data suggests 14

335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 assumption can hold for the cases of subcritical flows along the side weir of zero height and fixed bed, and in particular the following ranges of validity have been verified: - Fr u = [0.18 0.85], Fr d = [0.03 0.63], r Q = [0.17 0.89], r L = [0.1 1.1] for the side weirs - Fr u = [0.32 0.77], Fr d = [0.14 0.55], r Q = [0.12 0.59], r L = [1] for the lateral diversions. The mass balance is verified for all tests regarding the lateral diversion and for most of the present study experiments, whereas a greater scatter is observed for data coming from Subramanya and Awasthy (1972) resulting in a general underestimation of the diverted discharge for tests having a discharge ratio smaller than 0.5 (Fig. 9). The tests showing the underestimation are associated with large Froude numbers at the upstream and downstream sections (about Fr u > 0.7 and Fr d > 0.4). The errors for such flow conditions may arise from assuming the Coriolis coefficients equal to 1, that may lead to errors in the prediction of dimensionless ratios. By considering that larger Coriolis coefficients are expected at the downstream section than at the upstream one for a subcritical flow (i.e., d > u according to May et al. 2003) and that they should be taken into account in equation 3, the predicted flow depth ratio r Y without considering Coriolis coefficients may be overestimated. As a consequence, equation 4 suggests that the predicted discharge ratio r Q without considering the Coriolis coefficients may be underestimated. A preliminary application of the model by taking into account Coriolis coefficients showed that the percentage of Subramanya and Awasthy data reproduced within the error range 10 % regarding r Q changes from about 48% (in the case u = d = 1) to 72% (by assuming u = 1.15 and d = 1.75, as suggested by May et al. 2003). 358 The prediction of the length ratio is affected by higher errors (about 20 %), which may 359 be due to the approximation introduced by means of a constant value for the discharge 15

360 361 362 363 coefficient along the side weir and to the uncertainty related to the formulation chosen for such coefficient. Data regarding the present work are all predicted within the error range, whereas some data from Subramanya and Awasthy (1972), corresponding to about 25 % of their dataset, are again underestimated. 364 365 366 Fig. 8. Comparison between the experimental and predicted flow depth ratios for collected data. Fig. 9. Comparison between the experimental and predicted discharge ratios for collected data. 367 368 Fig. 10. Comparison between the experimental and predicted length ratios for collected data. 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 Comparison between the proposed model and the classical approaches The proposed model is herein compared with other models coming from the classical approaches in order to test its predictive capability and to highlight the original approach that has been proposed. The classical models usually aim to solve the system of differential equations (1) that govern the side weir flow in order to get the flow depth profile along the side weir length and, as a consequence, the diverted discharge Q s is obtained. In such procedure, a numerical scheme may be used to solve for the flow depth once the upstream or the downstream condition is known for, respectively, a supercritical or subcritical flow. The law for the diverted discharge is taken in a form similar to the weir overflow, and the discharge coefficient C d gets high importance in order to solve the entire system. Several studies have been focused on the determination of C d as a function of manifold parameters (as the upstream Froude number) and some of them have been reported in the present work. The problem of determining a generally valid and reliable formula for C d is still open. Subramanya and Awasthy (1972) and Hager (1987) proposed formulations for C d that 16

385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 were specifically obtained for zero height crest of the side weir and they will be considered in the next analysis as the ones that are theoretically most appropriated to the present case. On the other hand, the proposed approach solves the problem in an integral formulation by setting the control volume around the side weir. In such a way, the proposed model does not solve the longitudinal water depth profile along the side weir, as the classical methods do, but the global variables, like the total diverted discharge Q s through the side weir, are obtained directly from the knowledge of the upstream and downstream flow conditions. According to Eqs. 3-4-5, the proposed approach gives Q s without any need to know the formulation for the discharge coefficient C d, that is necessary only to get the length ratio r L. Finally, another difference with the classical approaches is that the proposed model is analytically formulated and it does not need any numerical schemes for the solution of the problem, since the system of equations (1) is not solved in the differential form. The classical approaches and the proposed one may be compared on the basis of their predictive capabilities in terms of the dimensionless ratios. Data presented in the previous Section are here used again and the proposed model is applied as well as the classical approach with the two formulations of C d given respectively by Subramanya and Awasthy (1972) and by Hager (1987). The classical models are implemented by means of an explicit numerical scheme based on the finite differences, that solves for the governing differential equations (1) by imposing the upstream and the downstream conditions, in the same framework of the proposed model. Since subcritical flow data are used, the algorithm is started by providing a downstream condition in terms of Fr d and Y d and the flow depth profile is then numerically solved toward upstream until the flow condition Fr u is found. A discretization step of L s /100 is chosen and the following formulations for C d, that depend on Fr u, are used: 17

409 C dhager 2 2 Fr 0.485 3 2 3 Fr 2 u 2 u (9) 410 C dsubramanya & Awasthy 2 3 0.611 1 3 2 Fr Fr 2 u 2 u (10) 411 412 413 The comparison between the predictive capability of the proposed and the classical approaches are reported in Table 2 in terms of the following mean errors on the prediction of the dimensionless ratios: 414 ry N Ntest 1 r r Y expi Y modi r test i 1 Y expi (11) 415 rq N Ntest 1 r r Qexpi Q mod i r test i 1 Qexpi (12) 416 rl N Ntest 1 r r Lexpi Lmodi r test i 1 Lexpi (13) 417 With N test = number of analyzed tests for each data set. 418 419 420 Table 2. Predictive capabilities of the proposed model and the classical models (Hager 1987 and Subramanya and Awasthy 1972) on the analyzed data sets. 421 422 423 424 425 426 427 428 429 The results summarized in Table 2 show that the predictions of the proposed model are usually better than the two other models. The only case in which the proposed approach provides larger errors is on the r Y -prediction for data coming from Subramanya and Awasthy (1972) and from the present study: however, a slight difference with the prediction given by classical models is obtained for these cases. For all the other cases, the proposed model gives smaller errors than the classical models in terms of r Q - and r L -prediction, and the improvement is significant regarding the prediction of the discharge ratio for every data sets. Finally, it is worth to note that a very good 18

430 431 432 433 434 reproduction is achieved for r Q of data regarding lateral diversions by Hsu et al. (2002), whereas classical models predict larger errors: the resolution of the outflow process in the integral form given by the proposed model seems to be a better choice for lateral diversion flows. As it was already mentioned, the r L ratio is not computed for the lateral diversion case, since the side weir law cannot be used to model a branch channel. 435 436 437 438 439 440 441 442 443 444 445 Practical applications The proposed analytical model can be used for approximate solutions in the design practice and for verification purposes. The analytical results of the model highlight the governing effect of the downstream boundary condition on the side weir flow and a way to compute the downstream Froude number in subcritical flow condition is proposed. For applications to the field cases in which the side weirs are encountered along quite regular channels, one may consider a uniform flow regime as downstream boundary condition. In such a case, Fr d is directly related to the channel features in terms of flow roughness and bed slope S 0, whereas it is little influenced by the variation of the flow depth due to the diminishing of water discharge. As a matter of fact: 446 U Frd Ch S gy 0 (14) 447 Where C h 7.66 (Y/ r ) 1/6 is the Chézy coefficient, r [m] is the bed roughness and a wide 448 449 450 451 452 453 rectangular section of the main channel is considered as a first approximation. According to eq. (14), the variation of Fr with the flow depth for a uniform flow regime is small, since the uniform flow slope is constant and the resistance coefficient C h changes only slightly with the flow depth. As a consequence, the downstream flow condition may be predicted a priori and the efficiency of the diversion may be planned by analyzing not only the side weir features (such as the length), but also the main channel characteristics. 19

454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 The Froude graphs presented in Fig. 3 may be used to analyze the flow features in case of a side weir with zero height crest. For example, given L s of an existing or planned side weir and the features of the main channel (as B, C h, S 0 ), the Fr d is assessed by means of the eq. (14). Then, the knowledge of Fr d and the ratio r L allows to predict the upstream flow features in terms of Fr u through the use of the graph of Fig. 3c. Therefore, all the other characteristic ratios are calculated from Eqs. (3), (4) and (7), or, equivalently, from Figs. 3a-b-d, and the diverted discharge Q s is evaluated if the inflow Q u is given. Moreover, the flow variables Y, Q and U at the upstream and downstream sections are obtained and additional considerations may be deduced: for instance, the flow acceleration due to the lateral outflow may induce sediment transport mechanisms in case of a movable bed and the erosion of the bed in the upstream zone is then possible, whereas the aggradation can take place at the downstream section due to the decreasing of the flow velocity along the side weir (see Fig. 3d). A way to induce modifications the lateral outflow process may be studied by acting on the downstream condition if all the other conditions have to be the same: it is evident that the diverted discharge can be modified by changing the flow depth profile at the downstream reach, since, for instance, a backwater profile increases Q s in comparison with the case when a drawdown profile occurs. Finally, eq. 4 allows to investigate the discharge ratio as function of the Froude number and the trend of the maximum r Q that can be achieved under imposed conditions is analyzed by Fig. 11, in which r Q is plotted as a function of Fr d for given Fr u (Fig. 11a) and as a function of Fr u for given Fr d (Fig. 11b). In particular, there exists a limiting curve describing the maximum r Q, that is consistent with the assumption of subcritical flow, which is represented by the continuous line of Fig. 11a and by eq. 15: 20

478 r 1 Fr 3 Q max d 2 2 Frd 3/ 2 (15) 479 480 481 482 483 484 On the other hand, the influence of the upstream Froude number on the discharge ratio is shown by Fig. 11b, in which r Q increases with Fr u toward an asymptotic value that depends on the imposed Fr d. The graphs of Fig. 11 represent another result of the analytical model that can be used (i) to predict the discharge to be diverted under the imposed flow conditions and (ii) to analyze the influence that a variation of the boundary conditions in terms of the Froude number has on the discharge ratio. 485 486 Fig. 11. Trend of r Q as function of Fr d (a) and Fr u (b). 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 Summary and concluding remarks A new analytical solution of the one-dimensional flow problem for side weirs is presented in this paper. The proposed model is based on the De Marchi hypothesis and it is developed for the specific case of side weirs with a zero height crest acting on fixed bed and subcritical flow conditions. The governing equations of the conservation of mass and energy and the De Marchi equation are re-arranged in order to formulate the solution in terms of dimensionless ratios of flow and geometrical variables at the upstream and downstream cross-sections expressed as functions of the Froude number of the upstream (Fr u ) and the downstream (Fr d ) flow. The plane defined by Fr u and Fr d provides an original way to represent the side weir problem according to the flow features and the significant effect on the lateral outflow given by the downstream boundary condition, which actually governs subcritical flows, is focused in an explicit way. The analytical model provides a set of five equations (Eqs. 3-4-5-6-7) that, once the flow conditions at the upstream and downstream sections are given in terms of Fr, express the 21

502 503 504 505 506 solution as flow depth ratio, discharge ratio, length ratio and flow velocity ratio. The results are discussed with reference to the assumed case of subcritical flow and the sensitivity of the model with respect to the discharge coefficient is analyzed by considering six different formulations of C d, for which the maximum length ratio ranges within [1 4]. The proposed model is tested by using a new set of laboratory data as well as existing data 507 from the literature and most of data are predicted within an error range of 10 %, regarding 508 r Y and r Q, and of 20 % regarding r L. Since the zero height side weir model may represent 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 other free-surface situations, a first promising application to the case of lateral diversion is performed and discussed by processing data from Hsu et al. (2002). Moreover, the proposed approach is compared with the classical ones to highlight its originality and to analyze its predictive capability. The classical approach is taken into account by means of the governing system of differential equations, that is numerically solved, and with the formulations for C d proposed by Hager (1987) and by Subramanya and Awasthy (1972). The predictive capabilities of the proposed and classical models are tested on the collected experimental data and it results that the performances of the proposed model in terms of the prediction of the dimensionless ratios are usually better than by using the classical methods. Finally, a practical procedure to calculate the geometrical and flow features of planned or existing side weirs is presented, starting from the uniform flow as downstream condition for which the eq. 14 takes implicitly into account the influence of the bottom roughness and the channel slope. Moreover, the maximum discharge ratio that can be achieved under the subcritical assumption is analyzed in terms of Fr u and Fr d and eq. 15 relates the limiting r Qmax with the given Fr d. The proposed approach provides an original interpretation of the lateral outflow phenomenon, that may be further investigated. For instance, additional studies could be 22

527 528 529 conducted to extend the proposed model for the case of supercritical flows. In such a situation, the model should take into account for the increasing trend of the Froude number along the side weir and the energy dissipation, that cannot be neglected anymore. 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 Notation The following symbols are used in this paper: B = width of the cross-section in the main channel; C d = discharge coefficient of the side weir; C h = dimensionless Chézy coefficient; D 50 = median diameter of bed material; E = specific energy head of the flow; E d = specific energy head of the flow at the downstream section of the side weir; E u = specific energy head of the flow at the upstream section of the side weir; Fr = Froude number; Fr d = Froude number of the flow at the downstream section of the side weir; Fr u = Froude number of the flow at the upstream section of the side weir; g = gravity acceleration; L s = side weir length; Q = water discharge flowing through a cross-section; Q d = water discharge downstream in the main channel; Q s = water discharge diverted over the side weir; Q u = water discharge upstream in the main channel; q s = discharge diverted through the side weir per unit length; R = relative roughness; Re = Reynolds number; 23

552 553 554 555 556 557 558 559 560 561 562 563 r L = L s / B = length ratio; r Q = Q s / Q u = flow discharge ratio; r U = U d / U u = flow velocity ratio; r Y = Y d / Y u = flow depth ratio; S 0 = bed slope of the main channel; U = average flow velocity at a cross-section; x = longitudinal axis; Y = flow depth at a cross-section; Y d = flow depth at the downstream section of the side weir; Y u = flow depth at the upstream section of the side weir; = Coriolis coefficient; = mean error between prediction models and analyzed data sets; 564 r Y = mean error on prediction of r Y ratio; 565 r Q = mean error on prediction of r Q ratio; 566 r L = mean error on prediction of r L ratio; 567 568 r = bed roughness; = dimensionless flow function of De Marchi solution. 569 570 571 572 573 574 Acknowledgments The results presented in this paper are part of the PhD thesis of the author (Michelazzo 2014), which was jointly supervised by Prof. E. Paris (University of Florence) and by Prof. H. Oumeraci (University of Braunschweig) within the International Course of Research Doctorate Mitigation of Risk due to Natural Hazards on Structures and Infrastructures 575 24

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Table 1. Summary of conditions and results for the performed tests (for each test: main channel width B = 0.3 m, Test side weir with zero height crest). Side weir Inflow Downstream Diverted Upstream Downstream length Sluice gate discharge discharge discharge flow depth flow depth configuration L s [m] Q u [m 3 /s] Q d [m 3 /s] Q s [m 3 /s] Y u [m] Y d [m] B7 0.03 Setting 1 0.0095 0.0065 0.0030 0.146 0.146 B8 0.08 Setting 1 0.0101 0.0058 0.0043 0.104 0.106 B9 0.13 Setting 1 0.0100 0.0052 0.0048 0.084 0.087 B10 0.18 Setting 1 0.0101 0.0050 0.0051 0.072 0.079 B11 0.23 Setting 1 0.0100 0.0047 0.0053 0.063 0.072 B12 0.28 Setting 1 0.0101 0.0048 0.0053 0.057 0.069 B13 0.33 Setting 1 0.0101 0.0047 0.0054 0.054 0.064 B14 0.38 Setting 1 0.0101 0.0046 0.0055 0.053 0.060 B15 0.43 Setting 1 0.0101 0.0045 0.0056 0.053 0.058 B16 0.47 Setting 1 0.0101 0.0044 0.0057 0.052 0.058 C17 0.03 Setting 2 0.0097 0.0080 0.0017 0.087 0.087 C18 0.08 Setting 2 0.0100 0.0073 0.0027 0.074 0.075 C19 0.13 Setting 2 0.0101 0.0074 0.0027 0.057 0.062 C20 0.18 Setting 2 0.0100 0.0070 0.0030 0.054 0.057 C21 0.23 Setting 2 0.0100 0.0067 0.0033 0.053 0.052 C22 0.28 Setting 2 0.0100 0.0063 0.0037 0.053 0.048 C23 0.33 Setting 2 0.0100 0.0058 0.0042 0.052 0.043 C24 0.38 Setting 2 0.0100 0.0054 0.0046 0.053 0.041 C25 0.43 Setting 2 0.0100 0.0051 0.0049 0.052 0.040 C26 0.47 Setting 2 0.0100 0.0049 0.0051 0.052 0.038

Table 2. Predictive capabilities of the proposed model and the classical models (Hager 1987 and Subramanya and Awasthy 1972) on the analyzed data sets. Subramanya and Awasthy (1972) DATA SETS Present Study Hsu et al. (2002) Proposed model 6.2 3.9 0.5 ry [%] rq [%] rl [%] Hager (1987) model 4.6 2.5 0.6 Subramanya and Awasthy (1972) model 4.3 2.5 0.5 Proposed model 16.7 9.3 2.4 Hager (1987) model 20.9 15.4 9.3 Subramanya and Awasthy (1972) model 20.9 15.4 9.1 Proposed model 12.6 9.4 Hager (1987) model 21.8 10.9 Subramanya and Awasthy (1972) model 15.7 12.6