Numerical Simulation of a Dam-Break Flood Wave

Transcription

1 European Water 33: 45-53, E.W. Publcatons Numercal Smulaton of a Dam-Brea Flood Wave V. Bellos 1 and V. Hrssanthou 1 Natonal Techncal Unversty of Athens, Postgraduate Course "Water Resources: Scence and Technology", 9 Iroon Polytechnou, 1578, Athens Greece, e-mal: ps119@mal.ntua.gr Department of Cvl Engneerng, Democrtus Unversty of Thrace, Vas. Sophas 1, 671 Xanth Greece, e-mal: vhrssan@cvl.duth.gr Abstract: Key words: In ths paper, the propagaton of a flood wave after the brea of a renforced concrete dam s smulated numercally, assumng one-dmensonal (1D) unsteady flow. Two numercal models are developed based on the 1D Shallow Water Equatons (SWE) or Sant-Venant's Equatons, usng the numercal schemes Lax-Wendroff and McCormac, respectvely. In order to valdate the smulaton results, a comparson wth expermental data s made. The expermental set-up conssts of a water tan that smulates the reservor of a dam, followed downstream by a horzontal dry bottom secton, a trangular bottom sll wth hgh slopes and a small pool of water at rest that ends up n a vertcal daphragm. The algorthms can smulate successfully the flow over the trangular bottom sll wth a hgh negatve value of slope, wthout any complcate consderatons. The comparson of expermental and numercal data shows a hgh degree of convergence. The use of an artfcal dffuson factor s favorable for the mplementaton of numercal schemes that are unable to descrbe abrupt changes of hydraulc flow propertes (velocty, water depth, bottom bent etc.). There s a weaness n descrbng the phenomenon of flood wave reflectons on vertcal daphragms due to the nduced supercrtcal flow condtons, as well as the smultaneous formaton of a hydraulc jump. dam-brea, expermental data, numercal schemes, numercal smulaton. 1. INTRODUCTION Durng a dam falure, a flood wave wth destructve consequences for the downstream area s created. The dam-brea and the flood wave whch s created by a dam falure, s wdely studed by many researchers both expermentally and theoretcally (especally numercally). The two basc areas of research nclude: a) defnton of flow characterstcs at the tme the dam breas such as flood hydrograph or breach type, whch depends on dam's materal, b) study of flood wave propagaton whch conssts n choosng governng equatons and decdng n how many dmensons they are solved (one-dmensonal [1D], two-dmensonal [D] or three-dmensonal [3D]). The frst approach by Rtter (189) was theoretcal, but wth many smplfcatons, such as gnorng hydraulc resstance. Many researchers studed ths phenomenon expermentally, for example Schlotsch (1917) studed the small wave propagaton, Eguazaroff (1935) performed wave propagaton measurements, Dressler (195, 1954) found that hydraulc resstance s the controllng factor of the propagaton of the dam-brea wave, Montuor (1965), Yevjevch and Barnes (197) and more recently Lauber and Hager (1998), Stansby et al. (1998) and Spnewne and Zech (7) valdated ther proposed models wth expermental data. The development of computng machnes led researchers to numercal smulatons. Chanson (6) notes that there s no theoretcal development of ths phenomenon snce the basc wor of Dressler (195) and Whtham (1955) for horzontal slope and Hunt (198, 1984) for slopng channel. Numercal modelng methods n lterature can be classfed, n general, nto three categores: Fnte-Dfference (FDM), Fnte-Element (FEM) and Fnte-Volume methods (FVM). Concernng FDM, Terzds and Streloff (197) studed surges and shocs n open channel flows, Xanthopoulos

2 46 V. Bellos & V. Hrssanthou and Kouttas (1976) smulated numercally a D flow after a dam-brea, Katopodes and Streloff (1978) developed a numercal D model based on the characterstcs method, Bellos and Saas (1987) developed a 1D numercal model of flood wave propagaton n a dry bed. Mller and Chaudhry (1989) developed an equvalent 1D numercal model for curved channels, Rahman and Chaudhry (1998) notced the mportance of local grd adaptaton n dam-brea flow. The well nown DAMBRK software and ts replacement FLDWAV should also be mentoned. These numercal models were developed by Fread (1984a, 1984b), Fread and Lews (1988, 1993) and are based on the Pressman mplct numercal scheme. Concernng D or 3D numercal models of FEM, Gee and McArthur (1978) developed a D numercal model based on the Galern technque, Katopodes (198) developed a D numercal model for small Froude numbers and later (1984) developed a numercal model based on the Petrov-Galern technque whch has a good shoc capturng ablty. FVM s smlar to FEM but n comparson easer and needs less computatonal wor. Concernng FVM, Bermudez and Vazquez (1994) proposed a dscretzaton method of bed slope for Sant- Venant s equatons, Hubbard and Garca-Navarro () proposed a numercal scheme based on Bermudez and Vasquez method, Zoppou and Roberts () developed a numercal method for D dam-brea flows. The man goal of the present paper s the study of flood wave propagaton after an mmedate and abrupt brea of a renforced concrete dam. When a renforced concrete dam breas, t s destroyed mmedately and completely, unle the brea of flexble dams, where water slps through falure breaches. For ths purpose, two numercal models are developed, based on the 1D Shallow Water Equatons (SWE). In cases where flood waves propagate wthn a rver, the vertcal and traverse velocty components are much smaller than the streamwse velocty component, hence they can be consdered neglgble. Due to ths smplfcaton, the problem of propagaton of a flood wave can be smulated assumng 1D flow (streamwse drecton). The numercal method used n the models s FDM, whch s the most approprate and less complcated n 1D approach n comparson wth FEM or FVM.. GOVERNING EQUATIONS The governng equatons that descrbe 1D, unsteady, open channel flow are wdely nown as Shallow Water Equatons (SWE) or Sant-Venant s Equatons. These partal dfferental equatons are based on mass and momentum conservaton and are vald under a number of assumptons. Wth sutable modfcatons, these equatons can be transformed to the matrx form shown below: W F D t x (1) where W= A F= Q V Q A gay D= l S f qlvl ga S q where A s the area of wetted cross-secton, Q s the dscharge, V s the water velocty, q l s the lateral nflow per channel unt length, S s the slope of channel bottom, S f s the slope of energy lne, V l s the velocty component of the nflow n man flow drecton and y s the vertcal dstance of the centre of gravty of the cross-secton from the water surface.

3 European Water 33 (11) 47 Equatons (1) consttute a hyperbolc type system of dfferental equatons n a form approprate to be numercally solved. The man advantage of Equatons (1) s that they are n a conservaton law form, proved to be approprate for descrbng a dscontnuous flow (e.g. hydraulc jump). 3. NUMERICAL MODELS The only way to solve the governng partal dfferental equatons (1) s an approxmate numercal smulaton usng a numercal scheme, for example based on FDM. The frst numercal model of ths study contans the well nown explct numercal scheme Lax- Wendroff (Lax and Wendroff, 1964), whch s usng forward tme dfferences and central space dfferences n two steps and has an accuracy of second order. These two steps, as well as the soluton process of the numercal scheme are descrbed below (Equatons () and (3), Fgure 1): Frst Step W 1 1 ( W 1 W 1 ) ( F x 1 F 1 ) D () Second Step W W ( F x 1 1 F 1 1 ) D (3) Fgure 1. Lax-Wendroff numercal scheme The second numercal model of ths study s an mproved verson of Lax-Wendroff numercal scheme, called McCormac numercal scheme, whch s more sutable for dscontnuous flows. Ths scheme s an explct numercal scheme whch ncludes two steps (predcton and correcton), wth an accuracy of second order, but forward and bacward, nstead of central, space dfferences are used. These two steps as well as the soluton process of the numercal scheme are descrbed below (Equatons (4) and (5), Fgure ): Predctor Step W * W ( F 1 F ) D x (4)

4 48 V. Bellos & V. Hrssanthou Corrector Step W 1 1 * * * W F ) * 1 D ( W ( F x ) (5) In Equatons (), (3), (4) and (5), Δx s the space step and Δt the tme step. Fgure. MacCormac numercal scheme Both above mentoned numercal schemes are stable under Courant-Fredrchs-Lewy (CFL) condton: V c x 1 (6) where c s the celerty. Artfcal dffuson due to truncaton errors s added to the numercal schemes wth a weght factor (OME) at second or corrector step, respectvely (Equatons (3) and (5)), so that problems of abrupt change of flow characterstcs can be solved (Kouttas, 198): 1 (7) 1 OMEW W W OMEW 1 1 The artfcal dffuson s determned as follows (Bellos, 1994): 1 OME x t where OME ranges from to DESCRIPTION OF EXPERIMENTAL DATA 4.1 Expermental Set-Up The expermental set-up was developed by Soares-Frazao (7) and conssts of a channel wth

5 European Water 33 (11) 49 a wdth of b=.5 m and a length of l=5.6 m, wth a horzontal bottom (Fgure 3). A vertcal daphragm s placed at the end of the upstream secton of the channel and another one at a dstance l 1 =.39 m from the frst. The latter daphragm represents the dam, whle the water tan represents the reservor. The water depth n ths secton s y 1 =.111 m. Just downstream of the water tan there s a dry bed secton, whch has a length l =1.61 m. Downstream of ths secton, a symmetrcal trangular bottom sll s placed, whch has a length l 3 =.9 m and a heght y =.65 m. Just downstream of the trangular bottom sll there s a small water pool at rest, whch has a length l 4 =.7 m and a depth y 3 =. m. A vertcal daphragm s also placed at the end of the channel. The purpose of ths daphragm s the preventon of water flow and the creaton of a basn at rest after a dam-brea. The lateral walls of the channel were made of glass and the Mannng s coeffcent has been estmated to be n=.11. Fgure 3. Expermental set-up 4. Experment Implementaton The experment starts wth the abrupt openng of the daphragm whch retans the water of upstream tan and smulates the dam. The abrupt falure of a renforced concrete dam s smulated wth ths process. Water flows n the dry bottom untl t reaches the trangular bottom sll. The abrupt slope change creates a hydraulc jump. Afterwards, the water overcomes the trangular bottom sll and flows n the small water pool at rest, whch s postoned downstream. The flow s reflected when the water front reaches the vertcal daphragm at the end of the channel. The upstream vertcal daphragm reflects the flow too. Pctures of fve water profles have been taen n fve tme steps (t=1.8 s, t=3. s, t=3.7 s, t=8.4 s, t=15.5 s). 5. NUMERICAL MODELS IMPLEMENTATION 5.1 Input Data The two numercal models are mplemented and valdated wth data collected from the above experment. As t s mentoned, the lateral walls of the channel were made of glass. Ths fact allows the assumpton of an nfnte wdth channel whch has many advantages as t allows the smplfcaton of equatons by studyng a band of unt wdth. Hence, the nput data for the wdth of the channel s b=1. m and not b=.5 m as n the experment. The nput data for the water depth at the dry bed secton s h=.1 m. It should be notced that ths depth has no nfluence on the celerty of the wave (Bellos and Saas, 1987), but s needed for computatonal reasons. Water s restrcted by the vertcal daphragms upstream and downstream of the expermental setup, creatng a pool. In order to smulate these condtons, boundary condtons must be specfed. In

6 5 V. Bellos & V. Hrssanthou the upper and lower parts of the channel, where the vertcal daphragms are placed, the dscharge s assumed to be zero and the water depth s determned wth the characterstcs method (bacward and forward characterstcs for upstream and downstream boundares, respectvely (Terzdes, 198). Fnally, the artfcal dffuson factor was calbrated to be OME= Expermental and Numercal Results The correspondng water profles at tmes t=1.8 s, t=3. s, t=3.7 s, t=8.4 s, t=15.5 s are llustrated n Fgures 4, 5, 6, 7, 8, respectvely. These profles are derved from expermental data, the numercal smulaton wth the Lax-Wenrdoff (LAX) scheme and the numercal smulaton wth the McCormac (MAC) scheme. The dscontnuty of expermental data s attrbuted to the fact that only the half of the downstream secton was photographed, as well as to the fact that there were beams whch were used to support the lateral walls and hence the optcal feld was not clear enough. A comparson of the expermental and numercal data at tmes t=1.8 s, t=3. s, t=3.7 s, t=8.4 s shows a hgh degree of convergence. At tme t=15.5 s there s a dvergence between the expermental and numercal data, whch s probably due to the mpact of flood wave reflecton on the downstream daphragm. However, the numercal smulaton results can be assumed to be satsfactory. The proposed algorthms can descrbe the overcomng of an obstacle wth hgh enough negatve slopes, wthout any complcated consderatons. Moreover, the algorthms smulate wth hgh convergence the ntal tme steps of the smulated flow, whch are more useful n predctons regardng a real dam-brea and the destructve consequences n the areas downstream. Besdes, the conducted laboratory experment represents just a study case and not a real stuaton, n whch there are no vertcal daphragms downstream of a dam that prevent the propagaton of a flood wave. For future studes, the phenomenon studed n the present paper can be further analyzed tang nto account the effect of wave reflectons on the daphragms. Moreover, the exstence of an obstacle downstream of a dam (such as the trangular bottom sll) and the expected energy reducton due to the hydraulc jump formaton can be nvestgated n order to see f ths obstacle can be assumed as a safety measure whch lmts the destructve consequences of a possble dam-brea ΒΟΤΤΟΜ LAX MAC EXPERIMENTAL DATA Fgure 4. Water profle at tme step t=1.8 s

7 European Water 33 (11) BOTTOM LAX MAC EXPERIMENTAL DATA. Fgure 5. Water profle at tme step t=3. s BOTTOM LAX MAC EXPERIMENTAL DATA. Fgure 6. Water profle at tme step t=3.7 s BOTTOM LAX MAC EXPERIMENTAL DATA Fgure 7. Water profle at tme step t=8.4 s

8 5 V. Bellos & V. Hrssanthou BOTTOM LAX MAC EXPERIMENTAL DATA Fgure 8. Water profle at tme step t=15.5 s 6. DISCUSSION AND CONCLUDING REMARKS Lax-Wendroff and McCormac algorthms descrbe satsfactorly the flood wave propagaton phenomenon n an open channel wth a trangular bottom sll, as a result of an abrupt dam-brea, wthout any complcated consderatons. The governng equatons mentoned above are sutable to descrbe the flood wave propagaton on a dry bed. The use of an artfcal dffuson factor s favorable for the mplementaton of numercal schemes that are unable to descrbe abrupt changes of hydraulc flow propertes (velocty, water depth, bottom bent etc.). There s a weaness n descrbng the phenomenon of flood wave reflectons on vertcal daphragms due to the nduced supercrtcal flow condtons, as well as to the smultaneous formaton of a hydraulc jump. Despte the 1D smplfcaton, the developed numercal models seem to be capable of descrbng the propagaton of flood wave after the falure of a renforced concrete dam wth an open channel downstream. The 1D FDM soluton methods are easer, faster and less complcated when compared wth D and 3D approaches or other numercal methods. The models can be used to estmate the tme t wll tae for a possble flood wave to reach an area downstream of a broen dam as well as the respectve water depth. Ths estmate can ad the desgn of early warnng systems. REFERENCES Bellos, C. V., and Saas, J. G., D dam-brea flood wave propagaton on dry bed. Journal of Hydraulc Engneerng, ASCE, 113(1): Bellos, C., Study of the flood wave propagaton n natural channels. Techna Chrona, Secton A, 14( 3): Bermudez, A., and Vazquez, M. E., Upwnd methods for hyperbolc conservaton laws wth source terms. Comput. Fluds, 3: Bohorquez, P., 8. On the wave-front shape and the advancng of the wettng front of a dam-brea flood over an nclned plane of arbtrary bottom slope. Proceedngs of the Internatonal Worshop on Numercal Modellng of Hydrodynamcs for Water Resources, Zaragoza, Span: Chanson, H., 6. Analytcal solutons of lamnar and turbulent dam brea wave. Proceedngs of the Internatonal Conference on Fluval Hydraulcs, Rver Flow 6, Lsbon, Portugal: Dressler, R. F., 195. Hydraulc resstance effect upon the dam-brea functons. Journal of Research of the Natonal Bureau of Standards, vol. 49, no. 3: 17 5.

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