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Vietnam J ournal of Mechanics, VAST, Vol. 29, No. 4 (2007), pp. 539 - 550 CONSTRUCTION , VERIFICATION OF A SOFTWARE FOR THE 2D DAM-BREAK FLOW AND SOME ITS APPLICATIONS HOANG VAN LAI , NGUYEN THANH DON Institute of Mechanics Abstract . In this paper the numerical method for the shallow water equations is studied. The paper consists of 3 sections. In t he section 1 t he t heoretical basis and software IMECI-L2DBREAK for simulation of the 2D dam-break or dyke- break flows is o utlined. In the section 2 some results in verification of the IMECH_2DBREAK by the test cases proposed in the big European Hydraulics Laboratories are shown . In the last section some appli cations of IMECH_2DBREAK for the inundation problem in t he Red river delta in the Northern of Vietnam are presented. 1. INT R ODUCTI ON Analyses of the dam-break or t he dyke-break flows play an essential role when considering reservoir and dyke safeties for developing emergency plans. The rapid and continuing development of computing power a . . 1d techniques during the last years has allowed significant advances in the numerical modelling techniques in that difficult and important problem. The shallow water equations (SWE) are accepted for many practical applications as properly modelling the unsteady flow of water in general, modelling the dam-break and the dyke-break flows in particular. Many computational methods have been reported successful for SvVE. In the last few years a lot of effort has been devoted to the development of the finite volume method (FVM) in modelling the dam-break and the dyke-break flows and many papers have been published (see, for instance, [1] - [6]). · In this paper we concentrate on the Roe technique in FVM for SWE, especially in the case of the flow with shock waves . The paper is organized as follows. After introduction in the section 1 the main formulas of the FVM for SvVE are outlined, the boundary conditions for FVM are described. Applying those formulas and condit ions, a software IMECH _2DBREAK for simulation of the 2D dam-break or dyke-break flows has been constructed. In the section 2 the results in verification of IMECH_2DBREAK for the well known test cases are presented. In section 3 some applications of IMECH_2DBREAK software for studying the inundation problem in the Red River Delta are demonstrated. 2. THEORETICAL BASIS AND SOFTWARE CONSTRUCTION. The two -dim ensional shallow-water equations: The 2D SWE in a conservative form are written as follows (see /3]) : au aE ac _ H at + ax+ ay - ' (2.1) Hoang Van Lai, Nguyen Thanh Don 540 with H = H( 1 ) + H(2) ' qx U= H(l) U:)' = ( ~h So ,x ) 2 , H( ) gh2 q; h qxqy h + - 2- - E = I = ( gh So ,y G= ' qy qxqy h q2 y h + -gh2 - J 2 (2.2) ~(qx) ) F(qy) where h , qx, qy, are unknown functions, h = h(x, y, t) is the flow depth, qx = qx(x , y , t) and qy = qy( x, y, t) are the unit -width discharge components (qx = uh and qy = vh with u, v are the depth-averaged velocit ies) in x and y directions , respectively, g is the gravity acceleration, So ,x, So ,x are the bed slopes 8 zb So ,x = - ax ; 8 zb So,y = - ay Zb is bottom elevation and F( qx ), F( qy) are the bed shear stresses in x and y directions, respectively: F( qx) = - F( qx) = - gqx Ju2+v2 . El ; 2 F( qy) = - gqy h,x Ju2+v2 , 02h,y (2 .3) Ch,x = Ks ,x h 213 , Ch ,y = K s,y h 213 , where Ch ,x, Ch,y and Ks ,x, K s,y are Chezy and Strickler coefficients. The Finite Volume Method: Given a computational (for example t ri angle) mesh (see Fig. 1). This mesh divide the domain into small cells (triangles) Vi, Vi, .... The integrated form of equation (2 .1) for a fixed area Vi is: Jau atdV Vi + j ac] By [aE Bx+ dV Vi ~~ dV + f (E, G).iidc c Vi j HdV. (2.4) Vi The application of the Green t heorem to (2.4) for j = = Vi , yields: J H dV. (2.5) Vi The contour integral in (2.5) can be approximated as follows: fc 3 (E , G).iidC = L (E , G) kiikdCk, (2.6 ) k where k represents the index of the edge k of the volume Vi, nk = ( nx , ny) is the unit outward normal, dC k is the length of the edge k. The evaluation of t he numerical flux for the normal flux (E, G)k.nk = Fk+ 1 ; 2 in (2.6) used in t his paper is based on the Riemann problem defined by t he· conditions on the left Construction, verification of a software for the 2D dam-break fiow and some its app lications 541 Fig. 1. Unstructured mesh (UL) and right (UR) sides of the edges . For this purpose, the system (2.1) is rewritten in the following form: au + aE au + ac au = H at au ax where aE au [ o_(cu2v_ ~u u2) au ay ac au Ouo ] v = (2.7) ' [ ~uv (c2 - v2) ~ ~l2v ] 0 The J acobian matrix An of the normal flux in (2. 7) is evaluated as (see [3]) : An = 0 [ ( c2 - u 2 ) nx - uvny 2 -uvnx c - v 2 ) ny +( nx 2unx vnx ny + vny uny unx ] + 2vny The eigenvalues of An are .\1 = u nx .\2 = u nx ,\3 = u nx where c = vf9Fi, The eigenvectors of An are f 1 ,3 = [ ~v ±cny ± cnx +v +v +v ny + c, ny, ny - c, l; Main computational formulas: In solving of that Riemann problem, as suggested by Roe (see [7]), instead of An, we consider the matrix An, where An has the same shape as An but is evaluated at an average state given by the following quantities and c, u, v v= VhRvR+ VhLvL v1hR + ./hi_, and and the normal flux (E, G)k.nk = Fk+i ; 2 in (2.6) is evaluated as (see [1]): Fk+1 ; 2 = ~ [!(UL)+ f (UR) - L :=l l.\kl(Jh - ak)fk J , (2 .8) Hoang Van Lai, Nguyen Thanh Don 542 f31,3 - n1,3 = hk - hL ? ,_, ± - 1 [(qx,R - qx, L)nx 2T + ( qy,R ~ - qy,L)ny - ('llnx ~ + vny)(hR - hL)] 1 f32 - CX2 = -; [((qy ,R - qy,L)v( hk - hL))nx - ((qx, R - qx ,L) - u(hk - hL)ny] Finally, we get the following computational formula: U n+l i - 6.t un i v; i +L N (E, G )k·nk. _, dCk -- H i·· (2.9) k= l In the formula (2.9): ut+ 1 is unknown numerical solution at time tn+l = tn + 6.t , Ut is known numerical solution at time tn, (E, G)k.nk = Fk+i ; 2 are the normal fluxes computed by the formula (2.8), Hi is an approximation of the integral J H dS for source Si terms. Boundary conditions. For calculation of Uin+l in (2 .9) for the volume Vi we need the value Ut of the volume Vi and the values U'f: of all surrounding volumes Vi . So, we can use (2.9) only for the case when Vi is the inside volume. In the case, when Vi is a boundary volume, we do not know the value U for some edge k and we need to use boundary conditions . It is well known that (see, for example, [3], [4]) , depending on both the value of the normal velocity through the boundary ( u · n) = unx + vny and the local Froude number Fr = (u · n) /c, (c = sqrt(gh)), there are four possibilities: (i) Supercritical inflow: (u·n) :S - c: all the variables must be imposed. (ii) Subcritical inflow: -c < (u · n) :S 0: two v'ariables must be imposed . (iii) Supercritical outflow: (u · n) > c: none of the variables must be imposed. (iv) Subcritical outflow: 0 < (u · n) :S c: one variable must be imposed. Using now these boundary conditions and computational formulas for boundary volumes in [1] we can compute the flux (E, G)k.nk in (2.9) for the boundary volumes . Software description. Software name: IMECH _2DBREAK Software purpose: Simulation of 2D complicated water flow (supercritical flow , flow with shock waves , flows on dry beds , ... ) in the case of the instant of dam or dyke failures. Software programming languag e: FORTRAN 90 Software organization: IMECH _2DBREAK consists of a computational program and a data directory SL2D. The main part of the computational program is the MODULE: MH2D_HMD.FOR, in which the water depth h and the unit-width discharge components qx, qy are computed by the formula (10). The input data of the IMECH_2DBREAK store in the directory SL2D and consist of 4 FILES: - FVUNMESH.DAT describes the unstructured mesh for the computational region. - FVCODATA.TXT defines common use parameters (computational precision, depths for definition of wet/dry front, ... ) - INIVHQ.TXT defines the initial values in the volumes. - FVBOU3.TXT describes the boundary condition in the case of subcri t ical outflow (the boundary condition of the type (iv)). Construction, verification of a software fo r the 2D dam-break fiow and some its applications 543 The information for the other boundary conditions gives in the main program. Namely, no any information is needed for the boundary condition of the type (iii). In the case of inflow boundaries the discharge and water level at the boundaries will be given. IMECH_2DBREAK will compute qx, qy for the boundary condition of the type (ii) and all variables for the boundary condition of the type (i) . The output data of the IMECH_2DBREAK also are stored in the directory SL2D . The output data consist of 2 FILES: - KTRHUV.TXT consists of the values h, qx, qy in the all volumes at the end of the computational t ime. - RESOl.DAT store the values of the water levels, the velocities, the wet/dry front at the nodes of the mesh in the TECPLOT format. With t his FILE it is easy to get different presentations of the computational results 3. SOFTWARE VERIFICATION The developed IMECH _2DBREAK software has been verified by the test cases proposed in the big European Hydraulics Laboratories (see [14]) . 3.1. Case 1: Total dam-break problem (see [12)) The computational domain is a rectangular basin of 16 m x 0.1 m. Initially, the free surface elevation presents a discontinuity at abscissa x= 0. m, where a dam is located. The left side of the basin is covered with water at rest, the water depth being 6 m, whereas the right side is dry (Fig. 2). At the start of the simulation the dam is totally and instantly broken and the dam location becomes a critical flow section where the velocity and the depth are constant in time (see (13]). The flow is subcritical to the left of the dam location (upstream part), and supercritical to the right (downstream part). Fig. 4 and 5 present some results of the IMECH_2DBREAK. "( Fig. 2. (Case 1) Water surface at the start of the simulation 3.2. Case 2: Partial dam-break problem (see [2]): The geometry of the problem consists of a 200x200 m 2 basin with a non-symmetrical breach. The initial water level of the dam is 10 m and the tail water is 5 m high (Fig. 6). At the instant of dam failure, water is released into the downstream side through a breach 75 m wide. Fig. 8 presents the computed water surface at 7 .3 sec. Fig. 9 shows the water surface obtained in [2) at the same time. Hoang Van Lai, Nguyen Thanh Don 544 ~ 1~ I '- ."- 8 4 ··... ~ i~ : : l ~24 l ~32 '··""16 12 "' 20 ·,, 28 " '<·· ··----->. :.- -------------><::·-------------:·::-::k~---- ---------~\ ~7 ~15 3 11 '""-, 23 ·~ '.~ · ~ ~j I 3 l \ 2 I\ i l 1 ___ ·---- -·- - , -1 \ j ~- 545 - - h tt I ·-------· h =:_ ex ~ I L - -- _ __ - -~ ~ · --~----------- ·~---------< -2 :bo -1 oo o 1 oo 200 300 40o .... ...... ... ....... .............. ... _·1- ····'. .. ... .. .... ... ... ........... ...... .................. ..... ...... .. ... .... ..... ............... : Fig . 5. Exact and computational water levels Fig. 6. (Case 2) Water surface at the start of the simulation Fig. 8. Water surface at 7.3 sec. after dambreak (computed by IMECH_2DBREAK) Fig. 1. Computational mesh Fig . 9. \i\Tater surface at 5 sec. after dambreak (obtained in [2]) The "flood plain" is 2.3 m wide and 6. 75 m long (Fig. 12) .The variation of the water-depth with time were obtained with four capacitance probes. Fig. 13 presents the measured and computed water levels in the prober N2. Hoang Van Lai, Nguyen Thanh Don 546 ~ ! lo~:~ d 1.00 .. , __ ., Upstream reS91Voir > · Gate r--'t~'M%\%\j 1 =/ constriction I o.~ 7~0 & 10 19 .JO Fig. 10. Channel configuration 0.25 ,, ____,,. ,. 1 ____ "' +--' i \,r'.il,"' .) __- - ---------------1-fv__:_ - --: - - ----- -- - ----- :.: ;-_":"= [ ) ~-=r~;;J 4 6r - -- - h _ lt ···· h_ cx -- 8 10 12 Fig. 11. The measured and computed water levels in the prober N2 Upstream reservoir r---e·i-o--·c,f~-----B:4s--1 Floodplain ~ ~: 19.30 Fig . 12. Channel with floodplain ·-·-···· · ························ ··· 002s t I = o02 ·- . """ 0.01 -----~---------/~~~~------------: ~ ' I i\ ;,;'•11,1,,.f. 1 !':~' ~\Mi n¥v " J!1~ J_it:~~- -- - !~ ~\l]v, __ --~''"i,111,,'lw - --- -- -:\\[~-I, I ·--· :\ u ~j_ ··-· I-!~/ -.t\:y\1ii ~ .: ~~ , """' '__:::=:_ : _ ==--~-J U I! " - - -- J l .. - h _ lt ······ h_c x - ~. , 10 12 Fig. 13. The measured and computed vvater levels in the prober N2 ! Construction, verification of a software for the 2D dam-break flow and some its applications 547 4. SOFTWARE APPLICATION 4 .1. Simulation of dyke break flows in Thanh-Ha region Fig. 14. The Thanh Ha region Fig. 15. Inundation in Tha nh Ha region at 10 hours after the dyke breach The geometry of t he problem consists of a cell N240 in the Thai Binh river basin (Fig. 14) . The area of that region is about 3100 ha . The region is relatively fl at (the elevation of the land: 0.9 - 1. 2 m) , except some roads (the elevation of roads: 1. 5 - 1. 8 m .). In t he 1996 year flood 2 places (near the cross-sections STBINH _32 a nd STBINH_34) of t he Thai Binh river dyke were broken. Fig. 15. presents the computational result of the inundation at 10 hours aft er the dyke breach. This result is agreed wit h the real inundation process in this region in t he 1996 year flood. 4.2. Simulation of flow caused by an operation of the emergency spillway in Dong-Anh region For the flood control in the Red and Thai Binh river system some emergency spillways are studied . One of them is the emergency spillway in Dong Anh region. This region 548 Hoang Van Lai, Nguyen Thanh Don consists of 3 cells (N136 , N137 and N154) in the Red river basin (Fig. 16). When extreme Fig. 16. The Dong Anh region floods occur , emergency spillways should be operated to avoid passive responses to dyke breakings and severe disasters. It means that the dyke elevation in spillway places will be suddenly decreased and flood water in the river system will overflow temporary dykes in riverbanks. So , operating emergency spillways in the extreme flood will accept certain damage levels but avoiding catastrophe. The software IMECH _2DBREAK can be used in simulation of a n operation of the emergency spillway. Fig. 17. presents the computational result of the inundation at 5 hours after the operation of the emergency spillway in Dong Anh region . • • ... ( .......... ~ &I ~ . . . . .• ............ .... .............. ' ''·---·-- ... ,,____,_\._____ \',,___ /') (, \\ / / i / ,,// ,/' .··'" i ,· / 1'> I •• I 5000 500000 505000 v 510000 I / I I C/ I I 515000 Fig. 17. Inundation in Dong Anh region at 5 hours after the operation of the emergency spillway