Sharp-Crested Weir Discharge Coefficient

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1 2011, Scencelne Publcaton Journal of Cvl Engneerng and Urbansm Volume 3, Issue 3: (2013) (Receved: December 13, 2012; Accepted: May 7, 2013; Publshed: May 30, 2013) ISSN Sharp-Crested Wer Dscharge Coeffcent Had Arvanagh 1, Navd Naseh Oskue 2 1 Assstant Professor, Water Engneerng Department, Faculty of Agrculture, Unversty of Tabrz, Tabrz, Iran 2 Former M.Sc. Student, Hydraulc Structures Engneerng, Faculty of Agrculture, Unversty of Tabrz, Tabrz, Iran *Correspondng Author s E-mal address: [email protected] ABSTRACT: Wers are useful and common devces n flow measurements. The essental parameter of each wer s to determne the flow coeffcent. In ths study, laboratory measurements of the water surface profle, approach flow velocty and flow rate carred out over three dfferent rectangular sharp-crested wers mounted n a flume. The flow was then numercally smulated by the CFD commercal software Fluent v. 6.2 n several stages. Results show that Fluent can model the flow over ths type of wers precously. Then the effect of three dmensonless parameters of wer heght (H/W), Froude number and Reynolds number on wer dscharge coeffcent s nvestgated usng expermental and numercal data. For a specfc value of these parameters, dscharge coeffcent s tendng to the fxed value of 0.7. Keywords: Sharp-Crested Wer, CFD, Dscharge Coeffcent, Wer Heght, Froude Number, Reynolds Number INTRODUCTION Wers have been wdely used for the flow measurement, flow dverson and ts control n the open channels (Kumar et al., 2011). Wers are categorzed n two man types: sharp-crested and broad-crested. Sharpcrested wers normally consst of a vertcal plate mounted at rght angles to the flow and havng a sharpedged crest, as shown n Fgure 1. (Henderson, 1966). In Fgure 1 W s wer heght, H s water statc head and V 0 s approach flow velocty. The smplest form of these wers conssts of a plate set perpendcular to the flow n a rectangular channel, ts horzontal upper edge runnng the full wdth of the channel. Ths last feature means that the flow s essentally two-dmensonal, wthout lateral contracton effects. Also, so called contracted wers exsts whch are contracted from the sdes as well as n the vertcal plane. Ths last type wll nvolve three-dmensonal flow (Henderson, 1966). Wers have varous shapes such as rectangular, trangular, trapezodal, crcular etc. and especal applcatons and governng equatons. The Typcal form of the governng equaton of these wers s as followng (Bos, 1989): n Q kh where Q = flow rate, k = coeffcent dependng on the sze and shape of the wer and n = dmensonless number dependng on the shape of the wer. For rectangular and trangular wers n equals to 1.5 and 2.5, respectvely. Dscharge equaton for a rectangular sharpcrested wer of the same wdth of the channel can be smplfed as (Henderson, 1966): Q C 2gLH (2) 3 d where L = crest length of the wer, C d = dscharge coeffcent, g = acceleraton due to gravty and H = statc head over the crest. The C d depends on flow characterstcs and geometry of the channel and wer (Kumar et al., 2011). For smplcty we can consder that C d s ust dependents on the raton H/W, for example we can use so called Rehbock equaton for H/W <= 5 (Henderson, 1966): H C (3) W Nowadays, we can smulate turbulent flow usng advanced numercal methods. These methods are useful to determne velocty dstrbuton, water surface profle, flow rate and some other coeffcents. Computatonal Flud Dynamcs (CFD) commercal software such as Fluent and Flow 3D are applcable and strength tools to evaluate mentoned parameters. Sarker and Rhodes (1999) frst carred out laboratory measurements of free surface profle over a rectangular broad-crested wer. Then they numercally smulated the free surface flow by the Fluent n several stages. They also appled standard k-ε turbulence model to solve Nover-Stocks equatons. They results have good conformty n comparson wth expermental data. (1) d Lu et al. (2002) smulated water surface profle on semcrcular wers usng k-ε turbulence model. Haun et al. (2011) calculated water flow over a trapezodal broad-crested wer by two dfferent CFD codes of Flow 3D and SSIM 2, where frst one uses Volume of Flud (VOF) method wth a fxed grd, whle last one uses an algorthm based on the contnuty equaton and the Marker-and-Cell method. They compared the results wth measurements from a physcal model study usng dfferent dscharges and they state that the devaton between the computed and measured upstream water level was between 1.0 and 3.5 %. Kumar et al. (2011) To cte ths paper: Arvanagh, H., Naseh Oskue, N Sharp-Crested Wer Dscharge Coeffcent. J. Cvl Eng. Urban, 3(3): Journal homepage: 87

2 nvestgated the capacty of the trangular and rectangular sharp-crested wers and they stated that the effcency of trangular wers s better than the rectangular one and also hgh for low vertex angle. They also examned the senstvty of the wer.e., change of dscharge due to unt change n head whch ndcates that the wer s more senstve at the low head and low vertex angle. In ths paper, we examned the effect of the heght, Froude number and Reynolds number on dscharge coeffcent of rectangular sharp-crested wer. Tests are carred out both expermentally and numercally. Ths study s amed to specfy a range for mentoned parameters n whch the wer dscharge coeffcent s constant. Fgure 1. Scheme of a sharp-crested wer. MATERIALS AND METHODS Experments have been done n a laboratory flume wth glass made walls. The flume length, wdth and heght are 10, 0.25 and 0.5 m, respectvely. Water enters the flume through a stllng system. Then flow passes over the wer and fnally flow dscharges to the settlng basn of the flume. There s a V-notch wer downstream the settlng basn whch s calbrated to measure the flow rate. Water surface elevaton n the flume s measured by a needle type levelmeter whch accuracy s about 0.1 mm. Sharp-crested wers of the same wdth of the flume (L = b = 0.25 m) are made from PVC plates n three dfferent heghts of 10, 15 and 20 cm. The wers are mounted 6 m downstream the flume entrance (see Fgure 2). Tests are terated for each wer at least 10 tmes. Dscharge measurements have been done after that the water head s balanced. Each wer s tested wth dfferent heads from 1 cm up to the flume heght. Fgure 2 The poston and dmensons of wer n the flume. In ths study, numercal modellng s carred out by Fluent v Fluent s one of the powerful and common CFD commercal software. It frst transforms the governng equatons to the algebrac equatons by fnte volume method then solves them. Fluent has the ablty to solve 2D and 3D problems of open channel flow, confned condut flow and sedment transport by advanced turbulence models. Also t s possble to solve contnuum and momentum equatons so called Nover- Stocks equatons around sharp-crested wers. So, relable CFD codes such as Fluent could be successfully used to pre-optmze the aforementoned ssues, savng a lot of money, tme and effort. It s relatvely easy to modfy the geometry n a numercal model; changes n physcal models are usually hard and costly to be mplemented. Naturally, the CFD codes have to be effcent and accurate, and t has to be able not only to deal successfully wht the nstabltes of the non-lnear equatons of moton, but also to compute satsfactorly the turbulence propertes and to fnd the free surface locaton. The governng equatons of our nterest are unsteady ncompressble 2-dmensonal contnuty and Reynolds-averaged Nover-Stockes equatons (RANS) for lqud and ar (Lu et al., 2002). u 0 (4) t x p u u u t x x u u 2 u ' ' uu x x x 3 x x (5) where, ρ = flud densty, u = velocty components, x = space dmensons, t = tme, p = hydrostatc pressure, µ = dynamc vscosty, ' ' uu = Reynolds stress tensor and δ = crooner delta. There are dfferent methods to solve RANS. In ths study, k-ε RNG turbulence model s used whch s defned as (Papageorgaks and Assans, 1999): k ku t x (6) k G G Y S k eff k b M k x x u eff t x x x (7) 2 C G C G C R S 1 k 3 b 2 k k where k = knetc energy, ε = energy dsspaton rate, G k = turbulence knetc energy generaton due to mean velocty gradent, G b = knetc energy due to floataton, Y M = turbulence Mach number and the other parameters are model coeffcents. To start Fluent we frst need to specfy channel geometry and then generate a mesh for t. Gambt software s used to do ths. Because three dfferent To cte ths paper: Arvanagh, H., Naseh Oskue, N Sharp-Crested Wer Dscharge Coeffcent. J. Cvl Eng. Urban, 3(3): Journal homepage: 88

3 heghts are defned for expermental models of wers, we need to desgn an especal geometry and mesh for each wer (e.g. see Fgure 3). In ths study, more than 40 2D confguratons are generated by gambt. Dmensonless wer heght Fgure 6 to Fgure 8 represent dmensonless heght of wer effect on dscharge coeffcent. As seen, for H/W > 0.4 the numercal results are very close to expermental ones and by ncreasng wer heght ths pont s reached before H/W = 0.4. Fgure 9 shows that after a specfc value of H/W (0.6) the dscharge coeffcent approxmately has the fxed value of 0.7 for dfferent wer heghts. Fgure 3 - Generated mesh by Gambt. RESUTLS AND DISCUSSION Frst, tests are carred out expermentally on three dfferent sharp-crested rectangular wers of the same wdth of the flume wdth. Then all confguratons are smulated by Gambt and fnally are modelled by Fluent. In Table 1 results of expermental and numercal nvestgatons are represented. Usng ths data water surface profle over the wer and the effect of the dmensonless wer heght, Froude number and Reynolds number on dscharge coeffcent s nqured. Water surface profle Durng each test, the water depth upstream and downstream the wer s measured after that the flow balanced. Then the water surface profle s drawn and s compared wth Fluent results. Fgure 4 depcts the water surface profle for one of the confguratons whch has run n the Fluent. Error! Reference source not found. s also a comparson between numercal and expermental profle. The results show the good conformty of numercal result by expermental ones. Fgure 6 - Comparson of numercal and expermental Cd for dfferent H/W (W=10 cm). Fgure 7 - Comparson of numercal and expermental Cd for dfferent H/W (W=15 cm). Fgure 4 - Water surface profle modeled by Fluent. Fgure 8 - Comparson of numercal and expermental Cd for dfferent H/W (W=20 cm). Fgure 5 Comparson of expermental and numercal water surface profle. Fgure 9 - Expermental Cd vs. H/W. To cte ths paper: Arvanagh, H., Naseh Oskue, N Sharp-Crested Wer Dscharge Coeffcent. J. Cvl Eng. Urban, 3(3): Journal homepage: 89

4 No. W (cm) H (cm) y = W+H Table 1 Results of expermental and CFD tests. H/W q (EXP) (lt/s) V (m/s) Re Fr q (CFD) (lt/s) Cd (EXP) Cd (CFD) To cte ths paper: Arvanagh, H., Naseh Oskue, N Sharp-Crested Wer Dscharge Coeffcent. J. Cvl Eng. Urban, 3(3): Journal homepage: 90

5 The effect of froude number on dscharge coeffcent The dscharge coeffcent varaton wth Froude number for dfferent wer heghts s shown n Fgure 10. The dscharge coeffcent for Fr > 0.2 s approxmately the fxed value of 0.7. Fgure 10 - Expermental Cd vs. Fr. The effect of reynolds number on dscharge coeffcent Fgure 11 represents Cd varaton wth Reynolds number for dfferent values of wer heghts. As seen, for Re > Cd has the fxed value of Haun S, Redar NBO, Feurch R. (2011). Numercal modelng of flow over trapezodal broad-crested wer. Engneerng Applcatons of Computatonal Flud Mechancs, 5(3): Henderson FM. (1964). Open-channel flow, Macmllan, New York, Kumar S, Ahmad Z, Mansoor T. (2011). A new approach to mprove the dschargng capacty of sharp-crested trangular plan form wers. Flow Measurement and Instrumentaton, 22(2011): Lu C, Hute A, Wenu MA. (2002). Numercal and expermental nvestgaton of flow over a semcrcular wer. Acta Mechanca Snca, 18: Papageorgaks GC, Assans DN. (1999). Comparson of lnear and nonlnear RNG-based models for ncompressble turbulent flows. Journal of Numercal Heat Transfer, Unversty of Mchgan, 35: Sarker MA, Rhodes DG. (1999). 3D free surface model of laboratory channel wth rectangular broadcrested wer. Proceedng 28 th IAHR Congress, Graz, Austra. CONCLUSION Fgure 11 - Expermental Cd vs. Re. Rectangular sharp-crested wer s one of the flow measurng tools whch are usually used n rrgaton and dranage channels. The most essental parameter of dscharge equaton of ths type of wers s the dscharge coeffcent. In ths study dscharge coeffcent of the rectangular sharp-crested wer s nvestgated expermentally and numercally. The results show that the Cd has the fxed value of 0.7 when the followng condton s mantaned: H/W > 0.6, Fr > 0.2, Re > Beyond these boundares, Cd s not constant and t s not recommended to use a unque Cd for dfferent flow condtons. Furthermore n ths study the water surface profle and dscharge coeffcent s modelled usng Fluent and s compared wth expermental results. A good conformty s yelded between numercal and expermental results. REFERENCES 1. Bos MG. (1989). Dscharge measurement structures. Thrd revsed edton. Internatonal nsttute for land reclamaton and mprovement, To cte ths paper: Arvanagh, H., Naseh Oskue, N Sharp-Crested Wer Dscharge Coeffcent. J. Cvl Eng. Urban, 3(3): Journal homepage: 91