1265. A well-balanced unsplit finite volume model with geometric flexibility

Transcription

1 1265. A well-balance unsplit finite volume moel with geometric flexibility Byunghyun Kim 1, Taehyung Kim 2, Jinho Kim 3, Kunyeon Han 4 1 UC Center for Hyrologic Moeling, Department of Civil an Environmental Engineering University of California, Irvine, CA 92697, Unite States 2, 4 School of Architecture an Civil Engineering, Kyungpook National University, Daegu, , Korea 3 Department of Mechanical Engineering, Yeungnam University, Gyeongsan, , Korea 3 Corresponing author 1 bh.kim@uci.eu, 2 sunz3515@hotmail.com, 3 jinho@ynu.ac.kr, 4 kshanj@knu.ac.kr (Receive 24 February 2014; receive in revise form 5 April 2014; accepte 11 April 2014) Abstract. A two-imensional finite volume moel is evelope for the unsteay, an shallow water equations on arbitrary topography. The equations are iscretize on quarilateral control volumes in an unstructure arrangement. The HLLC Riemann approximate solver is use to compute the interface fluxes an the MUSCL-Hancock scheme with the surface graient metho is employe for secon-orer accuracy. This stuy presents a new metho for translation of iscretization technique from a structure gri escription base on the traitional (, ) uplet to an unstructure gri arrangement base on a single inex, an efficiency of propose technique for unsplit finite volume metho. In aition, a simple but robust well-balance technique between fluxes an source terms is suggeste. The moel is valiate by comparing the preictions with analytical solutions, experimental ata an fiel ata incluing the following cases: steay transcritical flow over a bump, am-break flow in an averse slope channel an the Malpasset am-break in France. Keywors: shallow water equations, unsplit scheme, finite volume metho, well-balance, unstructure gri, Riemann solver, am-break. 1. Introuction A suen increase in ischarge from am or levee breaks can generate shock waves of iscontinuous flow properties. Discharge iscontinuity ultimately leas to water surface iscontinuity. It is ifficult to fin exact solutions to these iscontinuous flows through the linearization of non-linear equations. With the introuction of the total variation iminishing (TVD) metho an an approximate Riemann solver, there has been an innovative avance in secon or higher orer accurate schemes that can be applie to analyzing iscontinuous shock waves such as am-break waves. However, the high resolution schemes face challenges such as an imbalance between fluxes an source terms in steay flows over an irregular be. The imbalance is cause by the inaccurate iscretization of source terms, such as the be slope in shallow water equations. Accoringly, various numerical techniques for treating source terms in an effective way, so-calle well-balance schemes, which woul make it possible to maintain the balance between fluxes an source terms uner steay-state conitions, have been propose. Bermuez an Vázquez [1] introuce the C-property, which shows valiity for conservation laws in still water by iviing the vector an scalar terms in the shallow water equations an iscretizing the source terms by upwin schemes. LeVeque [2] propose a quasi-steay wave algorithm that recomposes the fluxes by introucing an artificial iscontinuity within each computational cell. However, this algorithm cannot provie accurate solutions for a steay transcritical flow with a shock. Zhou et al. [3] propose a surface graient metho (SGM) for accurately treating source terms such as the be slope. The metho was extene to the SGMS [4], which is capable of treating be topographies incluing a vertical step, an was valiate by laboratory ata with various be slopes. Brafor an Saners [5] introuce a new technique for hanling numerical truncation errors inuce by the pressure an be slope terms in an arbitrary topography with moving lateral bounaries. Burfau et al. [6] solve the 1D an 2D unsteay 1574 JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAY VOLUME 16, ISSUE 3. ISSN

2 shallow water equations using the cell-centere Roe finite volume scheme an an upwin iscretization of the be slope. Rogers et al. [7] propose an algebraic technique using Roe s approximate Riemann solver for keeping the balance between flux graients an source terms. Valiani an Begnuelli [8] introuce the ivergence form for be slope source term (DFB) as a possible basis of any numerical integration metho for the 1D or 2D SWE. Caleffi an Valiani [9] presente a new metho calle a finite volume well-balance weighte essentially nonoscillatory (WENO) scheme to manage be iscontinuities. Pu et al. [10] propose a surface graient upwin metho (SGUM) to combine the merit of SGM an epth graient metho (DGM) by incluing the source terms into the invisci flux iscretization. This stuy evelops a well-balance finite volume moel applicable to natural rivers using the unsplit metho. The unsplit metho consiers the conserve variables in all irections of space simultaneously. Hence, this approach is very effective an flexible in ealing with unstructure meshes which are robust in generating computational omains with highly complex bounary shapes. To calculate the flux at the interfaces between the cells through the use of the unsplit metho, information for both computational an neighboring cells shoul be known. To achieve this, this stuy introuces an efficient algorithm to automatically fin neighboring cells of computational cells an to etermine the orer of the neighboring cells even in complex calculation omains compose of unstructure meshes. Also, this work proposes a simple technique to preserve a well-balance solution in a ry be an a new bounary-hanling technique that can enable the effective application of an unsplit metho. The propose moel is valiate through the comparison of numerical solutions an analytical solutions to steay transcritical flow over a bump. It is also applie to the am-break flow in an averse slope channel using laboratory ata. Finally, the accuracy an reliability of the numerical moel base on a Gounov-type scheme are verifie with a natural topography through the simulation of the Malpasset am-break by comparing simulate results, fiel surveye ata an experimental measurements. 2. Governing equations The numerical moel solves the two-imensional shallow water equations as governing equations. The two-imensional shallow water equations can be written in matrix form as: +() +() =(), (1) h h h =h, ()=h h h 2 gh, ()= h h + 1, ()=gh! "#, (2) 2 gh gh! " where is the conserve variable vector, () an () are the fluxes in the $ an % irections, respectively, () represents a source term, an are the velocity in the $ an % irections, respectively, h is the water epth an an! are the be slope an friction slope terms, respectively. 3. Numerical moel 3.1. Finite volume metho The governing equations are iscretize base on the cell centere finite volume metho. By integrating Eq. (1) over an arbitrary computational cell, the basic equation of finite volume metho can be obtaine as: JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAY VOLUME 16, ISSUE 3. ISSN

3 & ' '( )* + +,- / )Ω 1 = &)*, (3) + where Ω is the bounary of the cell, / is the outwar unit vector that is normal to the bounary Ω, an - / is the normal flux vector an can be etermine by using the rotational invariance property [11]: - /=()cos5+()sin5=8(5) 9: -8()", (4) where 5 is the angle between the outwar unit vector / an the $-axis, ;=8() represent the transforme conserve variables, -;"=-8()" is the transforme normal flux, an 8(5) an 8(5) 9: are the transformation matrix an its reverse, respectively: (5)= 0 cos5 sin5, 8(5) 9: = 0 cos5 sin5, (5) 0 sin5 cos5 0 sin5 cos5 therefore, Eq. (3) can be iscretize as * ( + =8(5)9: -;">? = &)*, (6) B?A: where * is the area of an integration cell, C is the number of cell interfaces, D is the inex of the cell interface, an >? is the with of each interface HLLC approximate Riemann solver The HLL approximate Riemann solver [13] cannot take into account intermeiate waves, such as shear waves or contact iscontinuities [11]. To correct the efects of the HLL approach, Toro et al. [14] propose a moifie metho calle the HLLC scheme, in which C stans for Contact. The HLLC scheme provies results that satisfy the entropy conition [15]. If wave spees are properly calculate an applie, the HLLC scheme can easily hanle ry bes as well [16]. It also provies successful approximate solutions to practical applications [17]. Fig. 1. Wave structure of the HLLC approximate Riemann problem Fig. 1 shows the structure of waves assume in the HLLC approximate Riemann solver. The conserve variable is ivie into four parts accoring to wave spees S F, S G an S. If the conserve variables in each region have been etermine, the HLLC numerical fluxes are evaluate from [11]: 1576 JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAY VOLUME 16, ISSUE 3. ISSN

4 F, 0 F, IJJK =L F = F + F ( F F ), F 0, G = G + G ( G G ), 0 G, G, 0 G, (7) where F =( F ), G =( G ), an F an G are the left an right Riemann states at each interface, respectively. F an G are the numerical fluxes in the left an right parts of the star region of the Riemann solution separate by a contact wave, respectively, an F, an G are the spees of the left, contact an right waves, respectively. To properly apply the HLLC approach to shallow water equations, it is important to calculate correct wave spees F, an G [18]. In this stuy, a computational cell an a neighboring cell are set as the left an right han sies of the Riemann interface for correctly calculating wave spees using the unsplit metho (Fig. 2(b)). At the same time, the wave spees F, an G are estimate by taking into account the wet an ry be conitions. The combinations of existing relations among wave spees F an G are shown in Table 1. Table 1. Wave spees for wet an ry bes Wet be Dry be Rarefaction wave Shock wave Computational cell (>) Neighboring cell (O) S F = F PQh F S F = F PQh F R F S F = G 2PQh G S F = F PQh F S G = G +PQh G S G = G +PQh G R G S G = G +PQh G S G = F +2PQh F In Table 1, F, G, h F an h G are the left an right initial values for a local Riemann problem. The contact wave spee, water epth h an R S (T=>,O) are calculate from [11]: = Fh G ( G G ) G h F ( F F ) h G ( G G ) h F ( F F ) h = 1 Q U1 2 PQh F+PQh G "+ 1 4 ( F G )W, (8), (9) R S =X 1 2 Y(h +h S )h h Z. (10) 3.3. Linear reconstruction of unsplit finite volume scheme The MUSCL-Hancock scheme is a secon-orer extension of the Gounov upwin metho. This scheme reconstructs conserve variables at the interface between cells prior to the calculation of fluxes for secon-orer accuracy in space an preicts fluxes at (/2 for secon-orer accuracy in time. To achieve secon-orer accuracy in space an time, the two-step MUSCL-Hancock metho consisting of preiction an correction proceures is employe. In the preiction step, the reconstruction of conserve variables is implemente at the interface between cells by using the ata of both computational cells an neighboring cells. The simultaneous reconstruction of conserve variables at each interface is require to preict fluxes using the unsplit metho. Thus, information about a computational cell an its neighboring cells, incluing cell numbers an primitive variables, must be known, an the orer of the neighboring cells nees to be etermine for two-imensional calculations. To obtain the information necessary for calculation, the following algorithm is introuce in this stuy: 1) The angles prouce by the centroi coorinate of a computational cell an the coorinates of noes consisting of a computational cell are calculate using Eq. (11) for all cells in the computational omain: JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAY VOLUME 16, ISSUE 3. ISSN

5 5 S =tan 9:`%S $ S a. (11) 2) As shown in Fig. 2(a), the inex of noes is etermine in ascening orer base on the angles calculate in the previous step. 3) Two noes from the lowest number are rea counter-clock wise for a computational cell. At the same time, the noe numbers for the other cells, except the computational cell, are rea in the same manner. The process is repeate to fin the noes matche with the two previously rea noes. 4) The cells with two share noes become the computational cell an one of its neighboring cells. The orer of neighboring cells is etermine in the orer of angles prouce between the centroi an the noes of the computational cell. The propose algorithm can be easily applie without any limits to the configuration of cells, even in complex flow omains such as natural rivers. θ k Fig. 2. Schematic iagram of topological relationships between cells a) The orer of neighboring cells an b) Linear reconstruction at interface for unsplit scheme The SGM [3] is couple with the MUSCL-Hancock scheme so as to keep the balance between flux graients an source terms. The SGM uses the water level b instea of the water epth h of epth graient metho (DGM) for reconstruction of conserve variables at interfaces in the preiction proceure. Accoringly, c, which is the variation of =[ b, h, h] g, instea of the variation of =[ h, h, h] g, is estimate as: c h,: = SA: h, c h, = h SA, c h,i = h SAi, c h,j = SAj h, (12a) (12b) where k is a computational cell an T= 1, 2, 3 an 4 represent the inex of neighboring cells composing cell k (Fig. 2(a)). To prevent numerical oscillations that may occur near iscontinuities with the use of the secon-orer-accurate MUSCL-Hancock metho, the TVD scheme, which limits the slopes of conserve variables to be reconstructe at the interface, is applie in this stuy. The Superbee limiter [11] is extene to the $ an % irections for unsplit scheme application as Eq. (13): c h, c h,t =l maxo0,min2 c h,,c h,j ",minc h,,2 c h,j "p, mino0,max2 c h,,c h,j ",maxc h,,2 c h,j "p, =l maxo0,min2 c h,i,c h,: ",minc h,i,2 c h,: "p, mino0,max2 c h,i,c h,: ",maxc h,i,2 c h,: "p, if c h,j >0, if c h,j <0, if c h,: >0, if c h,: <0. (13a) (13b) 1578 JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAY VOLUME 16, ISSUE 3. ISSN

6 For linear reconstruction of flow variables at interface, the metho which uses the istance between the center of a computational cell an mipoint of eges is employe (Fig. 2(b)). This technique is efficient to treat uniform or nor-uniform mesh since it can capture the correct behavior of linear function within a cell [13]. Therefore, =[ b, h, h] g for each interface between cells is extrapolate as: h,: h,i > u: c h, = h +, > u: +> h, = h u: > ui = h (> ui +> ui ) c h,, h,j = h + > u c > u +> h, u > uj c > uj +> h, uj, (14a), (14b) where > us (T= 1, 2, 3 an 4) is the istance from the center of cell k to each mipoint of interface (Fig. 2(b)). To estimate the conserve variable =[ h, h, h] g neee to calculate the fluxes at the interfaces, h an h, both of which are conserve variables of momentum equation calculate in Eq. (14), an h, which is the conserve variable of the continuity equation calculate in Eq. (15), are use [13]: h h,s = b h,s v h,s, (15) where h h,s, b h,s an v h,s refer to the water epth, water level an average be elevation at interfaces between a computational cell (k) an neighboring cells (T). Auusse et al. [20] suggeste h h,s =max0, b h,s v h,s " instea of Eq. (15) to preserve nonnegative water epth. However, this suggestion can cause imbalance between fluxes an sources terms in ry be [21]. In this stuy, a simple technique propose to preserve nonnegative water epth an the well-balance solutions in a ry be application as Eq. (16): w h h,s =0.0, h h,s =h h,s +(b h,s v h,s ), if h h,s <0.0, (16) where T is the counterpart number of T (e.g., T= 1, T = 3). The preiction fluxes y an y are obtaine by substituting the conserve variables at interfaces into Eq. (2). For secon-orer accuracy in time, a conserve variable in Δ(/2 is compute as: { : h = { h Δ( }= y 2* h,s / h,s + y h,s / { h,s "~ h,s h h SA:, (17) where * h is the area of the cell k, Δ( is the computational time step, h is the number of the neighboring cells of cell k, / h,s an / h,s are the $ an % components of the normal outwar unit vector of the interface (k,t) an ~ h,s is the with of the interface. In the correction step, a conserve value at the time level ƒ+1 is achieve by solving a series of Riemann problems at the interface base on ata from the preiction step [3, 19]: { : h = { h Δ( }= u * h,s / h,s + u h,s / { : h,s "~ h,s h h SA:, (18) where the correction fluxes u an u in the $ an % irections are compute using the HLLC approximate Riemann solver. JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAY VOLUME 16, ISSUE 3. ISSN

7 3.4. Treatment of source terms In eveloping a numerical moel to compute shallow water flows with an irregular be topography, the numerical treatment of source terms largely affects the accuracy of the calculate results [17, 22-23]. The SGM is accurate an effective in treating source terms an can be easily couple with the MUSCL-Hancock scheme [3]. Accoringly, this stuy improve the accuracy an stability of the evelope moel by combining the two schemes. The source terms consist of two parts, incluing the be slope an the friction slope. For the iscretization of the be slope terms, the application of the ivergence theorem yiels the rate of change of bottom elevation v in the $ an % irections of cell k as follows [22]: = v $ = (v j v )(% : % i ) ($ j $ )(% : % i ) (v : v i )(% j % ) ($ : $ i )(% j % ), (19a) = 'v '% =(v j v )($ : $ i ) (% j % )($ : $ i ) (v : v i )($ j $ ) (% : % i )($ j $ ), (19b) where subscripts 1, 2, 3 an 4 represent the inex of noes composing the computational cell, as shown in Fig. 2(b). The friction slopes in the $ an % irections are estimate using Manning s formula with the wie channel assumption [24]:! = ƒ + h j/i,! = ƒ + h j/i, (20) where ƒ is Manning coefficient, an the primitive variables h, an are estimate using the average values over the computational cell k Bounary conitions Fig. 3. Assignment of ghost cell numbers for bounary conition treatment The finite volume metho requires ghost cells for the treatment of the bounary conition. Fig. 3 epicts the computational omain an ghost cells. In this stuy, the numbers to be assigne to each ghost cell are etermine using bounary cell numbers as shown in Fig. 3. This technique elivers an efficient treatment of complex topographies at the bounary. For general shallow water problems, two types of bounary conitions incluing the transmissive bounary conition an the reflective bounary conition are taken into account without consiering the flow regime [11]. The Eq. (21) an (22) are use for a transmissive, an a reflective bounary conition, respectively: 1580 JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAY VOLUME 16, ISSUE 3. ISSN

8 h =h F, = F, = F, h =h F, = F, = F, (21) (22) where the subscripts > an refer to the left han sie an right han sie of the Riemann states at the interface between bounary cells an ghost cells, respectively. 4. Applications The evelope moel is applie to various test cases to assess the efficiency an accuracy of the algorithm propose in this stuy. A steay flow over a bump an a am-break flow in an averse slope channel with flow conitions incluing steay transcritical flow, hyraulic jump, wet an ry fronts an reverse flow are consiere, an the calculate results are compare with analytical solutions an experimental ata. Aitionally, the numerical moel is applie to the Malpasset am-break in France, an the application results are compare with fiel surveye ata an experimental ata D Steay flow over a bump If bounary conitions remain constant in a computational omain in which stationary flow is taken as the initial state, the flow will become steay-state espite the presence of a be slope. However, nonphysical oscillations occur aroun the iscontinuity of the be slope if the source terms are not treate appropriately. Simulations investigating the convergence to a steay state are generally use to verify the balance between the flux graients an source terms of a numerical moel. In orer to emonstrate convergence to a steay state, this stuy takes into account a channel propose by Goutal an Manual [25]. The channel is frictionless an rectangular, 25 m in length an 1 m in with. The be elevation is efine as: 0, $ 8 m, v($)= ($ 10), 8 m<$<12 m, (23) 0, $ 12 m. Accoring to the initial an bounary conitions, the steay flow over a bump can be subcritical or transcritical flow with or without a shock [25]. Of the three cases, subcritical flow an steay transcritical flow with a shock are selecte for moel verification, because the two cases are consiere most challenging for a numerical moel. We refer to Goutal an Maurel [25] for analytical solutions Subcritical flow Water Level (m) Numerical Solution Analytical Solution Be Elevation Unit Dischagre (m 2 /s) Numerical Solution Analytical Solution Distance (m) Distance (m) Fig. 4. Steay subcritical flow over a bump a) Water level an b) Discharge per unit with JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAY VOLUME 16, ISSUE 3. ISSN

9 For the case of a subcritical flow, a constant ischarge per unit with of R= 4.42 m 2 /s is specifie at the upstream bounary an a fixe water level of b= 2 m is impose at the ownstream bounary. b= 2 m is also use as an initial value for the entire omain. For the computational omain, 100 cells with $= 0.25 m are compose. The Courant number is 0.9, an the simulation time is 360 secons. Fig. 4 inicates that the moel preictions are in goo agreement with the analytical solutions for water level an ischarge per unit with. Deviations between the simulation results an the analytical solutions are similar to other researchers results [3, 26-27] Transcritical flow with a shock For the transcritical flow with a shock, a ischarge per unit with of R= 0.18 m 2 /s is impose at the upstream bounary an a water level of b= 0.33 m is fixe at the ownstream bounary. The initial water level is b= 0.33 m throughout the entire computational omain, in which case the hyraulic jump occurs at the en of the bump. To investigate accuracy epenence on cell size, simulations for 100 an 200 cells were carrie out, respectively. Fig. 5 shows a comparison between the numerical preictions an analytical solutions for the water level an ischarge per unit with estimate after 360 secons. In both cases, numerical errors occur at the en of the bump. When the number of cells is 100, ischarge near the point where the hyraulic jump occurs cannot meet steay-state conitions. On the other han, when the number of cells is increase to 200, the numerical preiction is shown to have goo agreement with analytical solutions. The maximum eviation between the ischarge preictions an the analytical solutions is 15 % an 2 % for 100 an 200 cells, respectively. It is clearly shown that the error between the numerical solution an analytical solutions ecreases as the number of cells increases cells 200 cells Analytical Be Elev cells 200 cells Analytical Water Level (m) Unit Dischagre (m 2 /s) Distance (m) Distance (m) Fig. 5. Steay transcritical flow over a bump with a shock a) Water level an b) Discharge per unit with The simulate results of the steay flow over a bump show close agreement with analytical solutions without spurious oscillations. This suggests that the unsplit metho applie to the propose algorithm in this stuy an the source-term treatment technique combine with the SGM maintains a numerical balance between fluxes an source terms. In orer to assess the efficiency an accuracy of the source term treatment with the SGM in the evelope moel, a case with the DGM is employe for comparison. The SGM is the same as the DGM in proceure, except that it uses the water level instea of the water epth for the reconstruction of conserve variables at the interfaces between cells. The SGM an DGM are applie to the previous two test cases of steay flow over a bump, respectively. In those applications, 200 cells are use. Fig. 6 an 7 show a comparison of the unit ischarge preictions of the SGM an DGM to subcritical flow an transcritical flow with a shock, respectively. In the comparison results, the numerical treatment of source terms using the DGM resulte in far larger 1582 JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAY VOLUME 16, ISSUE 3. ISSN

10 numerical errors than that of source terms using SGM. This occurs because the DGM oes not reflect the real variation of water epth [3]. As a result, the DGM cannot accurately reconstruct water epth at the interface. Thus, the incorrect reconstruction of water epth affecte the entire calculation by causing an inaccurate estimation of fluxes at the interface. This emonstrates that the application of the SGM for the treatment of source terms can improve the accuracy an efficiency of the evelope moel DGM Analytical SGM Analytical Unit Dischagre (m 2 /s) Unit Dischagre (m 2 /s) Distance (m) Distance (m) Fig. 6. Comparison of the DGM an SGM using the propose moel for steay subcritical flow over a bump a) DGM an b) SGM DGM Analytical SGM Analytical Unit Dischagre (m 2 /s) Unit Dischagre (m 2 /s) Distance (m) Fig. 7. Comparison of the DGM an SGM using the propose moe for steay transcritical flow over a bump with a shock a) DGM an b) SGM 4.2. Dam-break flow in an averse slope channel Distance (m) In this section, the accuracy of the propose algorithm is verifie through simulating ambreak experiments esigne by Aureli et al. [28]. Laboratory experiments set up with ifferent upstream epths an averse slopes were intene to inuce shock origination an propagation, a reverse flow an wetting an rying conitions. As shown in Fig. 8, the experimental omain was compose of a rectangular reservoir an a channel with an averse slope connecte to it. The length an with of this channel were set at 7 m an 1 m, respectively. To apply the numerical moel, a computational omain consisting of 140 cells with $= 0.05 m an %= 1.0 m was compose. The Manning roughness coefficient is set to m -1/3 s, an the reservoir has an initial water epth of 0.25 m. The initial water epth ownstream of the am is zero for the ry be test, an the total simulation time is set to 40 secons. The sie walls an upstream region of the channel have close bounary conitions, an the ownstream en has transmissive bounary conitions. Uner these same conitions, Aureli et al. [28] an Tseng [29] valiate a numerical moel using the TVD-Maccormack scheme an the flux-limite TVD scheme, respectively. Fig. 8 JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAY VOLUME 16, ISSUE 3. ISSN

11 shows the experimental configurations an simulation conitions. The be topography is efine as: v($)=w 0, 0 $ $ u, 0.1($ $ u ), $ u $ 7 m. (24) Fig. 8. Schematic view of the am-break experiment in an averse slope channel Fig. 9 shows a comparison of the calculate water epth an measure water epth at four ifferent points. The simulate results for preicting the propagation time of the forwar an reverse waves, wet/ry fronts an water epth are in close agreement with the observe values. From these simulate results, it can be emonstrate that a new metho for a linear reconstruction of conserve variables in effectively applying an unsplit scheme an a simple technique to preserve balance between fluxe an source terms in a ry be can hanle reverse flow an wetting an rying bes very well Water Depth (m) Water Depth (m) Measure This Stuy 0.05 Measure This Stuy Time (sec) Measure This Stuy Time (sec) Measure This Stuy Water Depth (m) Water Depth (m) Time (sec) c) ) Fig. 9. Dam-break flow in an averse slope channel: comparison of water epths at a) G1, b) G2, c) G3 an ) G Malpasset am-break flow Time (sec) To verify the applicability of the evelope moel to a natural river with an irregular be 1584 JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAY VOLUME 16, ISSUE 3. ISSN

12 elevation an ry terrain, it is applie to the Malpasset am-break simulation. The am was locate in the valley of the Reyran River about 12 km upstream of the Frejus city in France an was constructe to provie irrigation an rinking water for that city in It was an arch am with a maximum height of 66.5 m, a crest length of 223 m an a storage capacity of m 3. It collapse ue to a suen rise of water level in the reservoir resulting from exceptional rainfall on December 2, 1959, an the resulting floo waves were propagate along the Reyran valley to Frejus, leaving 421 casualties [30]. This am-break case was simulate by other researchers [30-33] to verify their numerical moel. The high water marks of floo waves resulting from the am-break were surveye by the police along the left an right banks of the Reyran River. Fig. 10 shows the surveye points, incluing P1-P17. In 1964, a laboratory experiment on the Malpasset am-break was conucte using a physical moel scale at 1:400, mae by LNH-EDF. In the experiment, the arrival time of the floo wave an the maximum free surface elevations were measure in S6-S14 gauges as shown in Fig. 10 [32]. These ata are use to valiate the numerical moel in this stuy. The bounary conition for the upstream region is an impose reflective bounary conition without any inflow into the reservoir because it was very small in comparison to the ischarge resulting from the am-break [30]. The ownstream bounary conitions are set as a free flow which floo waves run into the sea. For the initial conitions, the water level of the reservoir is impose at 100 m, an the ownstream region of the am is consiere as a ry be. The 13,550 quarilateral meshes composing the computational omain are use an Manning coefficient is specifie at m -1/3 s for the entire area. In Fig. 11, the preicte arrival time of the floo front an the maximum water surface elevation are compare with the measurements of the physical moel. Fig. 12 shows a comparison of the calculate maximum water surface elevation with fiel ata surveye in the left an right banks. The preictions in the propose moel emonstrate relatively goo agreement with the fiel an experimental ata. However, there are small iscrepancies between the simulation results an high water marks at surveye points, P4, P5, P7, P8, P13 an P16. It shoul be note that, in the simulation, bottom elevation ata igitize from maps ating back to 1931 are use [30]. However, the fiel ata were collecte after the am-break, which was accompanie by abrupt changes in topography. This may be the reason for ifferences between the numerical results an the high water marks. Fig. 10. Topography an locations of fiel surveye points (P1-P17) an gauges at physical moel (S6-S14) JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAY VOLUME 16, ISSUE 3. ISSN

13 Propagation Time (s) Exp. Data Compute Maximum Water Level (m) Exp. Data Compute S6 S7 S8 S9 S10 S11 S12 S13 S14 Gauge Points S6 S7 S8 S9 S10 S11 S12 S13 S14 Gauge Points Fig. 11. Comparison between preicte results an experimental ata a) Propagation time at gauges, b) Maximum water level at gauges Maximum Water Level (m) Fiel Data Compute Maximum Water Level (m) Fiel Data Compute P1 P3 P5 P7 P9 P12 P13 P15 P17 Surveye Points P2 P4 P6 P8 P10 P11 P14 P16 Surveye Points Fig. 12. Comparison between preicte results an fiel ata a) Maximum water level at right banks, b) Maximum water level at left banks Fig. 13 illustrates the three-imensional floo inunation extent an water epth istributions at 0, 3, 12, 21, 30 an 35 minutes after the am-break. As shown in Fig. 13, the main channel in the center of the river valley is inclue in the topography an floo waves ue to am-failure are rapily propagate along the steep an narrow valley, starting at the am. However, as the floo waves travel ownstream, where the flooplain becomes wier, they are propagate more slowly an the water epth ecrease. Aitionally, the wave front is also propagate over a wier area. The water level, which is initially set to 100 m in the reservoir, ramatically rops for 35 minutes after the am-break. Fig. 13 shows that the main characteristics of the floo resulting from the Malpasset am-break can be well reprouce by the propose moel. It also reveals that the techniques presente in this stuy can be simply an efficiently applie to irregular topographies such as natural channels JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAY VOLUME 16, ISSUE 3. ISSN

14 c) ) e) f) Fig D inunation water epth an extent at a) 0, b) 3, c) 12, ) 21, e) 30 an f) 35 min 5. Conclusion This stuy evelope a well-balance finite volume moel that can be applie to irregular bounaries an complex be topographies. The propose moel is base on the conservative form of the two-imensional shallow water equations, an the HLLC approximate Riemann solver with JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAY VOLUME 16, ISSUE 3. ISSN

15 MUSCL-Hancock scheme solves the equations by applying the unsplit metho to achieve geometrical flexibility. An algorithm for fining neighboring cells to a computational cell for the efficient application of the unsplit metho an a new scheme for hanling ghost cells for the treatment of complex bounaries are propose. In aition, a simple an robust technique to preserve nonnegative water epth an well-balance solutions in a ry be is suggeste. The propose moel is valiate through the comparison of preictions with analytical solutions, experimental ata an fiel ata. In this stuy, three test cases are consiere. One is a steay flow over a bump, another is am-break flow with an averse slope an the last one is the Malpasset am-break in France. Simulation results for the first two cases show that a balance between the flux graient an source terms can be reache with the evelope moel. In aition, the propose moel gives accurate solutions to complex flows, incluing the interactions of multiple flows, reverse flow, ry/wet shock fronts an hyraulic jumps. In the Malpasset am-break simulation, the evelope moel is capable of proviing accurate preictions for the arrival time an the maximum water surface elevation of floos occurring in natural rivers. From the reasonable results of several test cases chosen for a numerical verification, it is emonstrate that the propose moel can provie high accuracy, flexibility an efficiency in hanling various types of shallow water flows with arbitrary topography. Acknowlegements This research was supporte by Yeungnam University research grant in 2014 an a Development of Flooing Disaster Mitigation Stanar Moel an Avance Management Technology [NEMA-NH ] from the Natural Hazar Mitigation Research Group, National Emergency Management Agency of Korea. References [1] Bermuez A., Vázquez M. E. Upwin metho for hyperbolic conservation laws with source terms. Computers & Fluis, Vol. 23, Issue 8, 1994, p [2] LeVeque R. J. Balancing source terms an flux graients in high-resolution Gounov methos: the quasi-steay wave-propogation algorithm. Journal of Computational Physics, Vol. 146, Issue 1, 1998, p [3] Zhou J. G., Causon D. M., Mingham C. G., Ingram D. M. The surface graient metho for the treatment of source terms in the shallow-water equations. Journal of Computational Physics, Vol. 168, Issue 1, 2001, p [4] Zhou J. G., Causon D. M., Ingram D. M., Mingham C. G. Numerical solutions of the shallow water equations with iscontinuous be topography. International Journal for Numerical Methos in Fluis, Vol. 38, Issue 8, 2002, p [5] Brafor S. F., Saners B. F. Finite-volume moel for shallow-water flooing of arbitrary topography. Journal of Hyraulic Engineering, Vol. 128, Issue 3, 2002, p [6] Brufau P., Vázquez-Cenón M. E., Garcia-Navarro P. A. Numerical moel for the flooing an rying of irregular omains. International Journal for Numerical Methos in Fluis, Vol. 39, Issue 3, 2002, p [7] Rogers B. D., Borthwick A. G. L., Taylor P. H. Mathematical balancing of flux graient an source terms prior to using Roe s approximate Riemann solver. Journal of Computational Physics, Vol. 192, Issue 2, 2003, p [8] Valiani A., Begnuelli L. Divergence form for be slope source term in shallow water equations. Journal of Hyraulic Engineering, Vol. 132, Issue 7, 2006, p [9] Caleffi V., Valiani A. Well-balance bottom iscontinuities treatment for high-orer shallow water equations WENO scheme. Journal of Engineering Mechanics, Vol. 135, Issue 7, 2009, p [10] Pu J. H., Cheng N. S., Tan S. K., Shao S. Source term treatment of SWEs using surface graient upwin metho. Journal of Hyraulic Research, Vol. 50, Issue 2, 2012, p [11] Toro E. F. Shock-capturing methos for free-surface shallow flows. John Wiley & Sons, New York, JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAY VOLUME 16, ISSUE 3. ISSN

16 [12] Zhao D. H., Shen H. W., Tabios G. Q., Lai J. S, Tan W. Y. Finite-volume two-imensional unsteay-flow moel for river basins. Journal of Hyraulic Engineering, Vol. 120, Issue 7, 1996, p [13] Harten A., Lax P. D., Van Leer B. On upstream ifferencing an Gounov-type schemes for hyperbolic conservation laws. Journal of Computational Physics, Vol. 25, Issue 1, 1983, p [14] Toro E. F., Spruce M., Speares W. Restoration of the contact surface in the HLL Riemann Solver. Shock Waves, Vol. 4, Issue 1, 1994, p [15] LeVeque R. J. Finite volume methos for hyperbolic problems. Cambrige University Press, Cambrige, [16] Fraccarollo L., Toro E. F. Experimental an numerical assessment of the shallow water moel for two-imensional am-break type problems. Journal of Hyraulic Research, Vol. 33, Issue 6, 1995, p [17] Zia A., Banihashemi M. A. Simple efficient algorithm (SEA) for shallow flows with shock wave on ry an irregular bes. International Journal for Numerical Methos in Fluis, Vol. 56, Issue 11, 2008, p [18] Liang Q., Borthwick A. G. L., Stelling G. Simulation of am- an yke-break hyroynamics on ynamically aaptive quatree gris. International Journal for Numerical Methos in Fluis, Vol. 46, Issue 2, 2004, p [19] Soares-Frazão S., Guinot V. An eigenvector-base linear reconstruction scheme for the shallowwater equation on two-imensional unstructure mesh. International Journal for Numerical Methos in Fluis, Vol. 53, Issue 1, 2007, p [20] Auusse E., Bouchut F., Bristeau M. O., Klein R., Perthame B. A fast an stable well-balance scheme with hyrostatic reconstruction for shallow water flows. SIAM Journal on Scientific Computing, Vol. 25, Issue 6, 2004, p [21] Liang Q. Floo simulation using a well-balance shallow flow moel. Journal of Hyraulic Engineering, Vol. 136, Issue 9, 2010, p [22] Guo W. D., Lai J. S., Lin G. F. Finite-volume multi-stage schemes for shallow-water flow simulations. International Journal for Numerical Methos in Fluis, Vol. 57, Issue 2, 2008, p [23] Aliparast M. Two-imensional finite volume metho for am-break flow simulation. International Journal of Seiment Research, Vol. 24, Issue 1, 2009, p [24] Eruran K. S., Kutija V., Hewett C. J. M. Performance of finite volume solutions to the shallow water equations with shock-capturing schemes. International Journal for Numerical Methos in Fluis, Vol. 40, Issue 10, 2002, p [25] Goutal N., Maurel F. Proceeings of the 2n workshop on am-break wave simulation. Technical report HE-43/97/016, EDF-DER, Paris, [26] Vázquez-Cenón M. E. Improve treatment of source terms in upwin schemes for the shallow water equations in channels with irregular geometry. Journal of Computational Physics, Vol. 148, Issue 2, 1999, p [27] Ma D. J., Sun D. J., Yin X. Y. Solution of the 2-D shallow water equations with source terms in surface elevation splitting form. International Journal for Numerical Methos in Fluis, Vol. 55, Issue 5, 2007, p [28] Aureli F., Mignosa P., Tomirotti M. Numerical simulation an experimental verification of Dam-Break flows with shocks. Journal of Hyraulic Research, Vol. 38, Issue 3, 2000, p [29] Tseng M. H. Improve treatment of source terms in TVD scheme for shallow water equations. Avances in Water Resources, Vol. 27, Issue 6, 2004, p [30] Valiani A., Caleffi V., Zanni A. Case stuy: malpasset am-break simulation using a two-imensional finite volume metho. Journal of Hyraulic Engineering, Vol. 128, Issue 5, 2002, p [31] Brufau P., García-Navarro P., Vázquez-Cenón M. E. Zero mass error using unsteay wetting rying conitions in shallow flows over ry irregular topography. International Journal for Numerical Methos in Fluis, Vol. 45, Issue 10, 2004, p [32] Liang D., Lin B., Falconer R. A bounary-fitte numerical moel for floo routing with shock-capturing capability. Journal of Hyrology, Vol. 332, Issue 3-4, 2007, p [33] Kim B., Saners B. F., Schubert J. E., Famiglietti J. S. Mesh type traeoffs in 2D hyroynamic moeling of flooing with a Gounov-base flow solver. Avances in Water Resources, Vol. 68, 2014, p JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. MAY VOLUME 16, ISSUE 3. ISSN