Rver Flow 1 - Dttrch, Koll, Aberle & Gesenhaner (eds) - 1 Bundesanstalt für Wasserbau ISBN 978-3-9393--7 wo Dmensonal Modelng of Dam-Break Flows Ahmad ahershams & Mahnaz Ghaen Hessaroeyeh Department of Cvl Engneerng, Amrkabr Unversty of echnology, ehran, Iran Masoud Montazer Namn Department of Cvl Engneerng, Unversty of ehran, ehran, Iran ABSRAC: A two-dmensonal flow model based on shallow water equatons s developed. he spatal dscretsaton s obtaned by the FVM cell centered type method. he numercal system s solved n explct way. he flux modelng has been deployed by VD WAF scheme wth a second order accuracy both n space and tme. he local Remann problem s solved by the HLLC method. Verfcaton of the model was carred out by comparson of model results and analytcal and numercal solutons. Comparson of these two set of results present a reasonable degree of smlarty. Fnally, the results of numercal model were compared wth avalable expermental data of dam- break test case. Keywords: Numercal models, Dam falure, Shallow water equatons, VD WAF method 1 INRODUCION In recent years free-surface flow models have been ncreasngly developed usng explct schemes. Because the result of numercal models behave better when used to smulate flows wth sharp gradent free surfaces, such as dam-break flows (Namn et al. 4). Several technques have been publshed n the lterature concernng the use of the fnte volume method to solve the shallow water equatons to model free surface flows. Shock capturng technques n the framework of fnte volume dscretsaton, especally Godunov type methods, have recently drawn more attentons. At least fve approxmate Remann solvers.e. Roe, FVS, Osher, HLL and HLLC can be found n the lterature, all of whch are based on the characterstcs theory (oro, 1). Zoppou and Roberts (3) examned explct conservatve schemes for the soluton of the one dmensonal homogeneous shallow water equatons. In the concluson, for the ease of mplementaton, effcency and robustness, the HLLC and Osher schemes have been recommended for frst-order schemes. Also n comprehensve study of Erduran et al. (), the performance of the approxmate Remann solvers has been evaluated accordng to fve crtera ncludng ease of mplementaton, accuracy, applcablty, smulaton tme and stablty. Fnally, HLL and HLLC method were determned as the hgh-ranked schemes n terms of ease of mplementaton. It was hghlghted that a frst-order accurate soluton algorthm usng ether Osher or HLLC schemes can be recommended for the smulaton of all knds of applcatons. here have been a number of studes, amed at developng numercal models to predct dambreak flows. Fraccarollo and oro (1995) utlzed a numercal model usng a shock-capturng method of Godunov type. hey used HLL method for modelng fluxes at the cell nterface and VD WAF method for achevng the second order accuracy. hey compared the numercal model wth ther experments of dam-break flow. Soares- Frazão and Zech () used numercal smulatons of the flow wth three Roe-type fnte volume schemes, ncludng 1D, D and the hybrd approaches. hey compared the results of numercal models wth ther expermental data for the dam break waves n a channel wth a 9 bend. Valan et al. () smulated the flood wave usng a Godunov-type method. In the model, the local Remann problem s solved by the HLL method. hey used the VD MUSCL technque n order to acheve second order accuracy n space and also tme. hey utlzed the model for smulatng Malpasset dam-break. Yu-chuan and Dong (7) smulated dam-break flows n curved boundares by the fnte-dfference method, n a 555
channel-ftted orthogonal curvlnear coordnate system. In ths paper, the HLLC approxmate Remann solver s selected for computng the fluxes n the nterface of the cells. In order to acheve secondorder accuracy, the Weghted Average Flux (WAF) method whch was ntroduced by oro (1989) s used. he followng part conssts of how to develop a two dmensonal fnte volume model. hen the model s verfed by dam-break wth a left dry bed and partal dam-break through a sluce gate. hen the model s appled for the dam-break n channel wth 9 bend (CADAM test). whch yelds by applcaton of the dvergence theorem: NUMERICAL MODEL.1 Governng equatons Under the assumpton of hydrostatc pressure, and neglectng the dffuson terms and Corols and wnd effects, the obtaned two-dmensonal model of depth averaged shallow water equatons appears as: U t F( U ) H ( U ) + + = S( U ) x y U = ( h, hu, hv) F ( U ) = ( hu, hu + gh /, huv) + H ( U ) = ( hv, huv, hv gh / ) S ( U ) = (, gh( S S ), gh( S S )) ox fx oy fy (1) where h = water depth, u, v = depth-averaged velocty components n the x and y drectons, g = acceleraton due to gravty, S ox, S oy = bed slopes n the x and y drectons, S f x, S f y = frcton terms n the x and y drectons, respectvely (Valan et al., ). S oy z, y S ox z () x n v u v n u u, 4/3 4/ 3 + + v S fy S fx = (3) h h where n = Mannng s roughness coeffcent.. Dscretsaton he computatonal doman s dvded nto a set of quadrlateral fnte volumes. hen the governng equatons are ntegrated over each control volume, Fgure 1. Generc control volume and notatons U d S( U )dv (4) t V + G( U ) nds = v s v where G ( U ) = ( F, H ), n =the unt normal vector outward from control volume that s shown n Fgure 1. Equaton (4) can be rewrtten as: du ΔV dt m j= 1 G( U ) n j ds j + S ( U ) ΔV (5) where ΔV= the area of each control volume, m= the number of cell sdes, ds j =the length of sde j. he Rotatonal Invarance Property s utlzed between F and H over each sde (oro, 199): du ΔV dt m j= 1 1 nj F( nj U, nj U ) ds j j + S ( U ) ΔV (6) Consequently, n order to compute the fluxes n the cell nterface, the local one-dmensonal Remann problems are utlzed n the normal drectons of the cell sdes. Whch F( nj U, nj U j ) s resolved va an HLLC method. he numercal flux of HLLC, F +1/, evaluated as follows (oro et al., 1994): FL f SL F* L f SL S* F + 1/ = (7) F* R f S* SR FR f SR where S L, S * and S R are the speeds of the left, contact and rght waves, respectvely that s shown n Fgure, and F L =F(U L ), F R =F(U R ), U L and U R are the left and rght Remann states of a local cell nterface, respectvely, F *L and F *R are the numercal fluxes n the left and rght sdes of 556
the star regon of the Remann soluton whch s dvded by a contact wave (oro et al., 1994). Fgure Wave structure of HLLC Remann solver In order to acheve second-order accuracy, the WAF method was selected. he WAF scheme, as beng second-order accurate n space and tme, produces spurous oscllatons near steep gradents, and so needs VD stablzaton. Accordng to VD WAF scheme the flux at the cell nterface s calculated by the relaton (oro, 199): VD WAF 3 ( k) ( k ) + 1 / = 1/ ( F + F + 1) 1/ = sgn( C ) + Δ k 1 k A 1/ F + 1/ F (8) ( k ) ( k+ 1) ( k ) = F( U ), ΔF + 1/ = F + 1/ F + 1/, Ck = ( Δt / Δx Sk (9) ( k ) A + 1/ s a WAF flux lmter functon. F ) where here are varous choces for computng the lmter functon n the present model such as Superbee, Van Leer, Van Albada and Mnbee lmter functons (oro, 1). In modelng the bottom slope source terms, the numercal pontwse treatment of S o s not so dffcult f the bottom slope n the x and y drectons can be easly determned. However, generally the four vertces of each cell do not le on the same plane; therefore the slope of the cell s not trvally computable. o avod ths problem, the technque has been developed by Valan et al. () s used. he test has been desgned by oro (1). he computatonal doman s a channel 5 m long, unt wdth, frctonless and horzontal bed. he downstream secton of the dam s dry n ths test case. Let h L =, h R =1 m, u L = and u R =, depth and velocty n the left and rght sde of x =3 m whch s the poston of ntal dscontnuty. In Fgure, the model results and analytcal water surface profles are shown at t= 4 s. he soluton conssts of a sngle rght rarefacton wave, wth the wet/dry front attached to the tal of t. he propagaton of wet/dry front at the correct speed s one major dffculty of numercal methods. In a real applcaton n whch such fronts are to be propagated by several klometers, the propagaton speed and thus the predcted wave arrval tme wll suffer from consderable errors (oro, 1). he model result s shown n Fgure by the VD WAF scheme wth Superbee lmter functon. he model has a good agreement wth analytcal soluton especally n the begnnng of the rarefacton part that s ndcated by a crcle n Fgure. Most of the models wth the second order accuracy are more dffusve n ths part. h (x ) 1. 1.8.6.4. Exact 1 3 4 5 x (m) Fgure 3. Dam-break problem wth left dry bed 3. Partal Dam-break through a sluce-gate he am of ths test case s to study the capacty of the present model to smulate the front wave propagaton over a dry bed. he spatal doman s a m flat regon (Fgure 4). 3 MODEL ESING 3.1 Dam break problem wth left dry bed Fgure 4. Partal dam-break layout (Loukl and Soulaïman, 7) he bottom s frctonless. he computatonal doman s a 4 4 square mesh. he upstream dscharge s zero. he ntal water levels are 1 m upstream and m n downstream. he asymmetrc breach s 75 m wde. here s no analytcal reference soluton for ths test case, but n the lterature numercal results of varous authors are avalable e.g. Mngham and Causon (1998), Loukl and Soulaïman (7). Fgures 5 and 6 show the re- 557
sults of present model for water surface and water depth contours. he duraton of the smulaton s 7. s. Fgure 7 shows the water depth contour of Loukl and Soulaïman (7). he present model result shows a good agreement wth the publshed results. the calculaton s taken equal to.11 s.m -1/3 (Prokof ev, ). Fgure 5. Results of present model for water depth contour Fgure 6. he results of present model for water surface 3.3 Dam-break n channel wth 9 bend hs test problem s the expermental case desgned by Soares-Frazão et al. (1998) for verfyng the capablty of numercal methods to smulate dam-break flows. he flow doman conssts of a square reservor and L-shaped channel as shown n Fgure 8. he bottom level of the channel s.33 m hgher than the bottom level of the reservor, whch means that there s a step at the entrance of the channel. Intally, the water depth was.53 m n the reservor whch s separated by a gate from the channel and then the gate s suddenly opened to produce a dam-break stuaton. he water depth n the channel was set to.1 m. akng nto account the effect of the bottom and the chute walls the average frcton coeffcent n Fgure 7. Results of Loukl and Soulaïman for water depth contour (Loukl and Soulaïman, 7) he varaton of water surface elevaton wth tme s compared wth the expermental data at the dfferent gauge postons as shown n Fgure 9. he comparson of results for gauge P1 s satsfactory as t shown n Fgure 9 (a). It shows that the numercal model computes the rght dscharge comng n to the channel. Fgure 9(b) shows the arrval of the frst shock travelng downstream of the reservor and the arrval of the second shock reflected from the channel bend. he agreement for the second shock s very good. However, there are dfferences at P3 and P4 for the arrval of second shock. At most gauge postons, there s good agreement between the present model results and the expermental data. he devaton between the model results and expermental data may be due to the local head loss caused by sudden change n flow geometry at the entrance to the channel and due to eddy losses, whch are not taken nto account n the numercal computatons. Fgure 8 Plane vew of Channel wth 9 bend (dmensons n cm) (Prokof ev, ) 558
.6.5.5. Experment (a) P1.4.3..1 (b) P (c) P3 (d) P4.5..15.1.5.5..15.1.5.5..15.1.5 1 3 4 Experment WAF (superbee) Experment 1 3 4 Experment 1 3 4 Experment 1 3 4.15.1.5 1 3 4 (f) P6 Fgure 9. Dam-break flows on wet bed at dfferent gauges It s reported some dscrepancy between the results of numercal models and the expermental data at the gauge statons P and P5 by prevous researchers such as Prokof ev (). However, the results of the present model at the mentoned statons are good agreement wth expermental data whch can be caused by good accuracy of VD WAF method as mentoned n secton 3.1. 4 CONCLUSIONS A two dmensonal fnte volume model has been developed based on shallow water equatons. In the current study the VD WAF method wth second order accuracy both n space and tme has been used for modelng fluxes at the nterface of the cells. Model verfcaton has been made by comparson of the model results wth analytcal and numercal solutons. Comparson of the two set of results represent a reasonable degree of smlarty. Fnally, the model was appled to one of CADAM test and the results of numercal model compared wth measurements. he results of present model ndcate that: - he model s applcable when the bed s dry. - he model has a good agreement n the begnnng of the rarefacton wave n 1-D dam break problem. - Comparson of the model results and expermental data n dfferent gauge statons n channel wth 9 bend shows that the model operates well n predctng the wave caused by dam-break..5. Experment REFERENCES h(m) (e) P5.15.1.5 1 3 4 t(s) Mngham, C.G., Causon, D.M. 1998. Hgh-Resoluton Fnte Volume Method for Shallow Water Flows. Journal of Hydraulc Engneerng, 14(6), 65-613. Erduran, K.S., Kutja, V., and Hewett, C. J. M.. Performance of Fnte Volume Solutons to the Shallow Water Equatons wth Shock-Capturng Schemes. Internatonal Journal for Numercal Methods n Fluds, 4, 137 173. 559
Fraccarollo, L., oro, E.F. 1995. Expermental and Numercal Assessment of the Shallow Water Model for wo Dmensonal Dam Break ype Problems. Journal of Hydraulc Research, 33(6), 843-863. Loukl, Y., Soulaïman, A. 7. Numercal rackng of Shallow Water Waves by the Unstructured Fnte Volume WAF Approxmaton. Internatonal Journal for Computatonal Methods n Engneerng Scence and Mechancs, 8, 1-14. Namn, M., Ln, B., Falconer, R.A. 4. Modelng Estuarne and Coastal Flows Usng an Unstructured rangular Fnte Volume Algorthm. Advances n Water Resources, 7, 1179-1197. Prokof ev, V.A.. State-of-the-Art Numercal Schemes Based on the Control Volume Method for Modelng urbulent Flows and Dam-Break Waves. Power echnology and Engneerng, 36(4), 35-4. Soares-Frazão, S., Sllen, X., Zech, Y. 1998. Dam-break Flow through Sharp Bends Physcal Model and D Boltzmann Model Valdaton. Proceedng of the CA- DAM meetng, HR Wallngford, U.K. Soares-Frazão, S., Zech, Y.. Dam-Break n Channels wth 9 Bend. Journal of Hydraulc Engneerng, 18(11), 956-968. oro, E. F. 1989. A Weghted Average Flux Method for Hyperbolc Conservaton Laws. Proceedngs of the Royal Socety, London A43, 41 418. oro, E. F. 199. Remann Problems and the WAF Method for Solvng the wo-dmensonal Shallow Water Equatons. Phlosophcal ransactons of the Royal Socety, London A338, 43 68. oro, E.F., Spruce, M., Speares W. 1994. Restoraton of the Contact Surface n the HLL Remann Solver. Shock Waves, 4, 5-34. oro, E. F. 1997. Remann Solvers and Numercal Methods for Flud Dynamcs. Sprnger-Verlag, Berln. oro, E.F. 1. Shock-Capturng Method for Free Surface Shallow Water Flows. J. Wley and Sons. Valan, A., Caleff, V., Zann, A.. Case study: Malpasset Dam-Break Smulaton Usng a wo-dmensonal Fnte Volume Method. Journal of Hydraulc Engneerng, 18(5), 46 47. Yu-chuan, B., Dong, X. 7. Numercal Smulaton of wo-dmensonal Dam-Break Flows n Curved Channels. Journal of Hydrodynamcs, Ser. B, 19(6), 76-735. Zoppou, C., Roberts, S. 3. Explct Schemes for Dam- Break Smulatons. Journal of Hydraulc Engneerng, 19(1), 11 34. 56