NUMERICAL SIMULATION OF FLOW OVER STEPPED SPILLWAYS

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1 ISSN: RCEE Research n Cvl and Envronmental Engneerng Research n Cvl and Envronmental Engneerng 014 (04) NUMERICAL SIMULATION OF FLOW OVER STEPPED SPILLWAYS Rasoul Daneshfaraz a, Sna Sadeghfam b, Mahsa H. Kashan c* a Department of Cvl Engneerng, Unversty of Maragheh, Maragheh, IRAN. b Department of Cvl Engneerng, Unversty of Tabrz, Tabrz, IRAN. c Department of Water Engneerng, Unversty of Tabrz, Tabrz, IRAN. Keywords energy dsspaton FLUENT software stepped spllway turbulence models water surface profle A B S T R A C T In ths study, flow over four types of stepped spllways wth dfferent step confguratons s smulated usng the FLUENT software. The numercal analyss s carred out by usng three turbulence models: (a) standard k- epslon model, (b) Renormalzed Group Theory k-epslon (RNG) model and (c) k-omega model. These models are verfed by comparng the results obtaned wth the expermental ones n terms of water surface profles usng the Root Mean Square Error (RMSE) crteron. Moreover, the rate of energy dsspaton was determned for the stepped spllways. The results obtaned showed that the models have acceptable error values. However, the performance of the RNG k-epslon model was n close agreement wth the expermental ones for all types of the spllways. Furthermore, the results proved the hgh capablty of the stepped spllway wth bgger steps to dsspate energy sgnfcantly. Ths s due to creatng large vortexes by usng such spllways. 1 INTRODUCTION Study of negatve effects of water energy at supercrtcal condtons and dsspaton of the energy s one of the most mportant ssues n hydraulc engneerng. In order to reduce these effects and damages, the engneers recommend usng energy dsspaton structures. Exstence of obstacles on flow path causes to dsspate the water energy sgnfcantly. Stepped obstacles known as stepped spllways are used to generate energy losses on dams nstead of usng the tradtonal smooth ogee-profle spllways. The stepped spllways are popular due to the hgh effcency and low-cost n constructon. The stepped spllways dsspate the energy over a wde range of flow heads wth the steps actng as roughness elements that reduce the flow veloctes. Hence, there wll be no need to use other * Correspondng author (Phone: + 98 (914) ; mahsakashan003@yahoo.com).

2 Rasoul Daneshfaraz et al - Research n Cvl and Envronmental Engneerng 014 (04) dsspaton structures such as the stllng basns at the end of the stepped spllways; f usng such structures, ther dmensons wll be reduced. Cassdy (1965) was probably the frst one who used the Laplace s Equaton and the Fnte Dfference Method for analyzng the flow over the ogee spllways. He analyzed the water pressure on the spllway and water surface usng the Potental Flow Theory and found a close agreement between the obtaned and expermental values. Raaratnam (1990) presented an approxmate method for determnng the characterstcs of skmmng flow over stepped spllways. A flow equaton has been developed by usng two terms of a dynamc pressure and a flud frcton coeffcent for expressng the turbulent shear stress between the flud trapped between the steps and the skmmng stream. Olsen and Kellesvg (1998) modeled water flow over a spllway n two and three dmensons numercally for varous geometres. They solved the Naver-Stokes equatons wth the k-t turbulence model on a structured non-orthogonal grd. The capacty and the coeffcent of dscharge for the spllway were calculated usng locaton of the water surface. Fnally, the water surface was compared wth that obtaned by a physcal model. Furthermore, coeffcent of dscharge was compared wth emprcal formulas. Chanson and Toombes (00) presented an expermental study and observed a transton flow between nappe and skmmng flows on a large-sze stepped chute. They conducted ar-water flow measurements n both transton and skmmng flows. Fnally, the results showed that the transton flows are hghly aerated and requre the desgn of comparatvely hgh chute sdewalls. Tabbara et al. (005) predcted water flow on stepped spllways wth some step confguratons numercally. They smulated the water surface profle and determned the rate of skmmng flow over corner vortces, and value of the energy dsspaton usng the ADINA fnte element analyss software and the k ε flow model. The results showed that the predcted water surface profle was n agreement wth the expermentally obtaned water surface profle. Furthermore, the predcted energy dsspaton values were comparable to the expermentally obtaned values. Cheng et al. (006) appled the fnte volume module of the FLUENT software to smulate the characterstcs of the flow over a stepped spllway. They used the RNG k-ɛ turbulence model and the mxture flow model wth the PISO arthmetc technque for ar-water two-phase flow. The results showed the exstence of complex flow over a stepped spllway, the nteracton between cavty recrculaton and ar bubbles n the skmmng flow, the pressure profles and velocty dstrbuton on the step surface. In ths research, the flow over the stepped spllways wth dfferent steps confguratons used by Tabbara et al. (005) s numercally smulated usng the FLUENT software. The performance of the turbulence models: (a) the standard k-epslon model, (b) the Renormalzed Group Theory k-epslon (RNG) model and (c) the k-omega model are compared for solvng the governng equatons. The models are verfed usng the expermentally measured water surface profles. Moreover, the energy dsspaton rates are determned and compared wth each other for all the stepped spllways. FLOW REGIMES AND GOVERNING EQUATIONS The flow over a stepped spllway s dvded nto three flow regmes dependng upon the flow rate: skmmng, transton and nappe flow regmes wth decreasng flow rates. The skmmng flow regme s observed at the largest flow rates. Ths flow regme acts as a coherent turbulent flow and skms on the pseudo-bottom whch s formed by the step edges. There are some recrculaton and vertcal structures 191

3 Rasoul Daneshfaraz et al - Research n Cvl and Envronmental Engneerng 014 (04) under the pseudo-bottom, whch fll the cavtes. The transton flows occur at ntermedate flow rates. In ths flow regme, there are some strong fluctuatons, splashng near the free surface. The nappe flows occur at small dscharge condtons. In the nappe flows, free-fallng nappes are observed at each step edge, followed by ther mpacts on the followng step (Raaratnam, 1990). Fg. 1 shows the three flow regmes over the stepped spllways. Fg. 1 Schematc vew of dfferent flow regmes ((a): skmmng flow regme, (b): transton flow regme and (c): nappe flow regme) n stepped spllways The governng equatons of ncompressble vscous flud at turbulence condton are expressed by the Reynolds Averaged Naver-Stokes equatons. These equatons are known as the contnuty and momentum equatons: u x t u 0 uu x 1 P g x x S (1) () horzontal component of acceleraton of gravty and S s the stress tensor whch s defned for the turbulent flow as follows: S x u u u v v k v t x 3 t x (3) The turbulence shear stress s the sum of two terms: a lamnar term whch s smlar to Newton's Law of Vscosty, but ncludes the tme averaged velocty, and a turbulent term (or Reynold's Stress term) that uses the varaton n the veloctes and s gven by S u x u ' ' uu vt k (4) x 3 19

4 Rasoul Daneshfaraz et al - Research n Cvl and Envronmental Engneerng 014 (04) In Eq. (3) and (4), vt s the vortex vscosty that depends on flow characterstcs and turbulence. The knetc energy per unt mass of the flud s defned by Eq. (5) as follows: 1 k u ' u ' u ' k (5) δ 1 0 (6) In order to solve the turbulent flow feld usng the contnuty and Reynolds equatons, the Reynols Stress terms n the equatons should be calculated. In ths case, there wll be four unknown parameters (three velocty normal components and a pressure component) for a three-dmensonal flow wth four equatons (contnuty equaton and momentum equatons n three dmensons). For governng these equatons the turbulence models, the RNG k-epslon model, the standard k-omega model and the standard k-epslon model, are employed. 3 EXPERIMENTAL SETUP For desgnng the stepped spllway profle, we used the USACE-WES hydraulc desgn charts 111-/1. The upstream porton of the spllway before the crest s conssted of a vertcal face followed by two crcular arcs wth rad 0.H d and 0.5H d, where H d s the desgn head. The downstream porton of the profle after the crest s conssted of three segments. The frst segment s defned by the followng equaton x n KH 1n d y where the orgn of the x and y coordnates s located at the crest. The desgn head, H d, n and K parameters are taken as 5.08 cm, 1.85 and, respectvely; and the followng equaton s obtaned: 1.85 y 0.156x (7) The second segment s a straght lne wth a 60 nclnaton (slope of 1.73V:1H). The thrd segment s a crcular arc whch provdes a transton between the straght-lne segment and the horzontal stllng basn. A total of four stepped spllway confguratons are consdered (Table 1). The desgn head of all these spllways s taken as 1.5H d. Each spllway s bult by assemblng two plexglass parts: an upper part that ncluded the crest regon and whose heght s 1/3H spll and a bottom part that ncluded the toe regon and whose heght s /3H spll. The total heght of the model spllway, H spll, s equal to 380 mm. In the upper part as well as n the bottom part steps were ntroduced along the chute such that the envelope of ther tps followed the smooth spllway chute profle. In each part the steps are ether all large steps of heght 1/H spll (19 mm), or all small steps of heght 1/4H spll (9.5 mm) (Tabbara et al., 005). 193

5 Rasoul Daneshfaraz et al - Research n Cvl and Envronmental Engneerng 014 (04) Type Table 1 Stepped spllway confguratons. Number and heght of steps at upper part 19 mm Number and heght of steps at bottom part 19 mm 9.5 mm 9.5 mm 19 mm 9.5 mm 9.5 mm 19 mm 4 NUMERICAL ANALYSIS The fnte element mesh conssted of rectangular four-node element was used for all the stepped spllways. Fgure ndcates the mesh for the frst type of the spllways. The mesh resoluton s hghlghted at three dfferent locatons labeled 1, and 3. A mesh wth hgh resoluton (small-szed elements) was used for the steps n order to capture the formed vortces. Smlar meshes were employed n the other three types. Fgure 3 llustrates the boundary condtons and ntal condtons for the frst type of stepped spllway. As t can be seen from ths fgure, a fxed wall boundary condton was consdered on the bottom edges and along the spllway (all lnes labeled A). The boundary condton s used when the flud s surrounded by a sold phase. In vscous flows, the condton of no-slp flow near the wall has been consdered n the FLUENT software. However, one can estmate the velocty of flow near the wall by determnng the shear stress value. The ntal water surface was modeled by the one straght lne labeled B. At the nlet (lne labeled C) a unform water velocty of m/s was predefned. Ths value was estmated by dvdng the flow dscharge by the upstream water depth. Moreover, at the nlet (lne labeled D) a unform ar velocty of m/s was prescrbed. The E and F boundares are consdered as the nlet and outlet boundares of pressure. The precedng descrpton of the computatonal doman s applcable to all the other spllway types. For smulatng flow usng the FLUENT software, frst the ntal condtons of water surface should be specfed (Fg. 3), then the equatons are solved teratvely wth tme step of The procedure s repeated untl the convergence s acheved. Here, the convergence crteron s that the relatve error between two consecutve teratons s less than or equal to The flow feld s assumed to be conssted of separate control volumes. The flow governng equatons are ntegrated n each control volume; and the algebrac equatons are become dscontnued usng dfferent dscontnuty schemes. The GAMBIT software was appled n order to generate the geometry of flow feld and mesh. Moreover, the PRESTO scheme was used for nterpolatng the pressure; the Quck scheme was mplemented for dscretzng the terms of the momentum equaton; the frst order upwnd scheme was employed for dscretzng the turbulence equatons and the PISO algorthm was used for couplng the pressure and velocty. Usng the under-relaxaton factors less than one for the pressure, momentum and Reynolds stress avods soluton dvergence. 194

6 Rasoul Daneshfaraz et al - Research n Cvl and Envronmental Engneerng 014 (04) Fg. The fnte element mesh for the frst type of the stepped spllway Fg. 3 Boundary condtons and ntal condtons for the frst type of stepped spllway 5 RESULTS AND DISCUSSION The derved water surface profles usng the three turbulence models are ndcated n fgure 4 (a-d) for the four types of the stepped spllways. In ths fgure, the derved water profles are compared wth the expermental results obtaned by Tabbara et al. (005). As t can be seen from Fg. 4, the RNG k-epslon model predcts the water surface profle more accurately. For comparng the turbulence models performance statstcally, the root mean square error (RMSE) crteron was estmated usng Eq. (8). RMSE N 1 y m y N p (8) where, N s the number of data, ym and yp s the modeled and expermentally measured water depths (mm), respectvely. The lower the RMSE value s, the more accurate the model s. Table presents the RMSE values of the models for the four spllway types. Accordng to Table, the RNG k-epslon model wth less RMSE s selected as the best model for stepped spllways modelng. Type Table RMSE values of the three turbulence models for all four types of the spllways Standard k-omega model RNG k-epslon model Standard k-epslon model

7 Rasoul Daneshfaraz et al - Research n Cvl and Envronmental Engneerng 014 (04) (a) (b) (c) (d) Fg. 4 Comparson of computed and expermentally measured water surface profle for all types of the stepped spllways ((a): type 1, (b): type, (c): type 3 and (d): type 4) In order to compare the rate of energy dsspaton of the four types of spllways, the energy rate at the spllway nlet (secton A) and outlet (secton B) was calculated usng Eq. (9) and (10). E A VA ya (9) g E B VB yb (10) g where, EA and EB are the rate of energy, ya and yb are the water depth, and VA and VB are the flow velocty at sectons A and B, respectvely; and g s the acceleraton of gravty. Note that the channel bed s assumed to be the base level n those equatons. Fgure 5 llustrates the rate of water energy and ts varatons calculated by the RNG k-epslon model at sectons A and B. 196

8 Rasoul Daneshfaraz et al - Research n Cvl and Envronmental Engneerng 014 (04) Fg. 5 Energy dsspaton rate at sectons A and B As t can be seen from ths fgure, the frst and second types of the stepped spllways dsspate the most and least rate of energy, respectvely. Vortex formed on the steps of the spllways leads to a dsspaton of the water energy. Snce the frst type of the stepped spllway has bgger steps than others t can create large vortexes and dsspate hgh energy rates. Table 3 ndcates the relatve loss of energy obtaned by the Eq. (11) for the four spllway types. E EA E E A B (11) proves ts hgh ablty to sgnfcantly dsspate the energy. A low value of the energy loss (0.0938) was obtaned for type. Therefore, t can be concluded that the effect of varyng the step confguraton on the energy loss s mnmal. Ths fndng s n agreement wth those obtaned by Tabbara et al. (005). Type Table 3 The loss of energy for stepped spllway wth dfferent step confguraton CONCLUSION In ths study, flow over the stepped spllway was smulated numercally usng the FLUENT software. To ths end, three turbulence models: (a) the standard k-epslon model, (b) RNG k-epslon model and (c) the k- omega model were employed. The water surface profles over the stepped spllways wth four dfferent step confguratons were predcted usng the turbulence models and compared wth the expermental results obtaned by Tabbara et al. (005). Based on the obtaned results, t was found that that the RNG k- epslon model was more successful than the others. The other two models led to dvergent solutons. Furthermore, the rate of energy dsspaton was calculated usng the energy equatons. It was found that 197

9 Rasoul Daneshfaraz et al - Research n Cvl and Envronmental Engneerng 014 (04) the frst type of the stepped spllways havng bgger steps led to the most rate of energy dsspaton. Therefore, consderng the lmted resstance of the spllways materals, t s suggested to use the stepped spllways wth bgger steps. Acknowledgment We thank the anonymous revewers and edtor for ther constructve and useful comments that helped us mprove the qualty of the paper. References Cassdy, J.J. (1965). Irrotatonal flow over spllways of fnte heght. Journal of Engneerng Mechancs ASCE, 91, Chanson, H., & Toombes, L. (00). Expermental nvestgatons of ar entranment n transton and skmmng flows down a stepped chute. Canadan Journal of Cvl Engneerng, 9, Cheng, X., Chen, Y., & Luo, L. (006). Numercal smulaton of ar-water two-phase flow over stepped spllways. Scence n Chna Seres. E: Technologcal Scences, 49, Olsen, N.B.R., & Kellesvg, H.M. (1998). Three-dmensonal numercal flow modelng for estmaton of spllway capacty. Journal of Hydraulc Research IAHR, 36, Raaratnam, N. (1990). Skmmng flow n stepped spllway. Journal of Hydraulc Engneerng ASCE, 117, Tabbara, M., Chatla, J., & Awwad, R. (005). Computatonal smulaton of flow over stepped spllways. Computers & Structures, 83,