advertisement

International Journal of Mechanical Engineering and Technology (IJMET) Volume 10, Issue 01, January 2019, pp. 1849-1861, Article ID: IJMET_10_01_183 Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=10&IType=01 ISSN Print: 0976-6340 and ISSN Online: 0976-6359 © IAEME Publication Scopus Indexed A SIMPLE UNSTRUCTURED FINITE VOLUME SCHEME FOR SOLVING SHALLOW WATER EQUATIONS WITH WET/DRY INTERFACE Elmiloud Chaabelasri*, Mohamed Jeyar and Najim Salhi LME, Faculty of sciences, First Mohammed University, Oujda, Morocco Imad Elmahi ENSAO, First Mohammed University, Oujda, Morocco *corresponding author ABSTRACT A simple unstructured finite volume scheme is used to solve the two-dimensional shallow water equations (SWEs) for simulation of dam break flows over irregular beds involving wet and dry interfaces. In space, we construct a balancing higher order upwind scheme on unstructured triangular mesh to approximate the convective and source term due to bed slop. We improve the scheme to treat the cases of flow over dry beds. For temporal derivative approximation, a second-order explicit Runge-Kutta is used. We apply the scheme for several theoretical two-dimensional numerical experiments involving dam-break flows with moving wet-dry fronts over irregular bed topography. Promising results are obtained. Keywords: Finite volume, upwind scheme, Shallow water equations, Dam break, Irregular bed, Wet/dry bed. Cite this Article: Elmiloud Chaabelasri, Mohamed Jeyar, Najim Salhi and Imad Elmahi, A Simple Unstructured Finite Volume Scheme For Solving Shallow Water Equations with Wet/Dry Interface, International Journal of Mechanical Engineering and Technology, 10(01), 2019, pp.1849–1861 http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=10&Type=01 1. INTRODUCTION In this paper we consider the two-dimensional shallow water equations with source term: ht + ( hu ) x + ( hv ) y = 0 2 2 ( hu )t + ( hu + 0.5 gh ) x + ( huv ) y = − ghZ x 2 2 ( hv )t + ( huv ) x + ( hv + 0.5 gh ) y = − ghZ y http://www.iaeme.com/IJMET/index.asp (1) 1849 [email protected] Elmiloud Chaabelasri, Mohamed Jeyar, Najim Salhi and Imad Elmahi where h is the total depth from the bed to the free surface, u and v are the depth-averaged velocity components in the Cartesian x and y directions, Z is the bed elevation above a fixed horizontal datum and g is the acceleration due to gravity. There are many difficulties to overcome for solving numerically the shallow water equations, one main difficulty is the treatment of the source term, which need to be balanced by the flux gradient at the steady state. The SWEs admit several steady state solutions, this is dependent of the studied case, but the most important steady state is lake at rest state satisfying, u = v = 0 and h + Z = const. (2) Another difficulty arising when numerically solving the SWEs is to simulate the rapidly moving wet/dry interfaces. Due to the calculation of the flow velocity by dividing the unit discharge by the water depth, unphysical high velocities appeasers at the wet/dry interfaces. Thus, the water depth might become negative, and the numerical calculation will break down. A good numerical method for the SWEs should be able to simulate the flow over irregular bed, wellbalanced to be able to exactly preserve the lake at rest steady states, and solve the problems when the wet/dry appears. Recently, several high order well-balanced methods were successfully developed. We refer the reader, for example, to the finite volume [1, 2, 3, 4, 5 and 6] and discontinuous Galerkin methods [7, 8 and 9]. The aim of this paper is to develop a simple second-order balanced positivity preserving upwinding scheme for solving shallow water with irregular bed on unstructured triangular mesh and able to treating the wet/dry interface. Most existing wetting and drying treatments are focused on post-processing reconstruction of the data obtained from the numerical solution at each time level. Following the approaches proposed in [10], we introduce a simple positivitypreserving technique, which preserves the high order accuracy. This paper is organized as follows: Section 2 presents the general formulation of high order upwind scheme on unstructured mesh, Moreover, the extension to second order in space, the wet dry interface moving are described. Section 6 discusses validation and application of the method; several numerical experiments are carried out for previously published benchmark cases in order to confirm the potential of the proposed scheme. Section 7 summarizes the main findings. 2. BALANCING HIGHER ORDER UPWIND SCHEME ON UNSTRUCTURED TRIANGULAR MESH For the case of presentation, we denote the shallow water equations (1) by: Wt + F (W ) x + G (W ) y = S ( h, Z ) (3) Where W is the vector of dependent variables, F and G are the inviscid flux vectors, S is the vector of source terms. In full, the vectors are W = (h, hu , hv )T ; F = (hu , hu 2 + 0.5 gh 2 , huv )T ; G = ( hv, huv, hv 2 + 0.5 gh 2 )T ; S = (0, − ghZ x , − ghZ y )T . (4) The flux Jacobian matrix in the normal-pointing direction is given by, 0 2 J = ( gh − u )nx − uvn y −uvn + ( gh − v 2 ) n x y nx 2unx + vn y vnx http://www.iaeme.com/IJMET/index.asp un y unx + 2vn y ny 1850 (5) [email protected] A Simple Unstructured Finite Volume Scheme For Solving Shallow Water Equations with Wet/Dry Interface Which has real and distinct eigenvalues, confirming hyperbolicity, and eigenvectors given by: λ = u.n + v.n + gh x y 1 λ2 = u.nx + v.ny λ3 = u.nx + v.n y − gh (6) e1 = (1, u + gh . nx , v + gh . n y )T T e2 = (0, − gh . nx , gh . n y ) T e3 = (1, u − gh . nx , v − gh . ny ) (7) Matrix J can be diagonal zed as J = P Λ P −1 or Λ = P −1 JP (8) Where P is the matrix of the right eigenvectors of J and Λ is the diagonal matrix with the diagonal elements being the eigenvalues of J. The flow domain is partitioned into a set of triangular cells or finite volumes. Let ij be the common edge of two neighbouring cells i and j, with Γ ij its length, N (i) is the set of neighbouring triangles of cell i, and n ij is the unit vector normal to the edge ij and points toward the cell j (Figure 1) Figure 1: Unstructured mesh, triangular control volume and notations. Integrating the system (2) in a considering triangular control volume Ω, and invoking the Gauss theorem: ∂ ∂ W d Ω + ∫ ∆F d Ω = ∫ Sd Ω ⇒ W d Ω + ∫ ( F .n ) ∂Ω = ∫ S d Ω ∫ ∂t Ω ∂t Ω∫ Ω Ω ∂Ω Ω (9) Assuming a piecewise constant representation, the integral around an element is approximated as the sum of the numerical flux contributions from each edge, such that Win +1 -Win V + i Δt ∑ ∈ j N(i) δ Fij Γ ij = ∫ S (W n i )dV Vi http://www.iaeme.com/IJMET/index.asp (10) 1851 [email protected] Elmiloud Chaabelasri, Mohamed Jeyar, Najim Salhi and Imad Elmahi where Wi n = W ( xi , tn ) is the vector of conserved variables evaluated at time level tn = n∆t , n is the number of time steps, ∆t is the time step, Vi is the area of cell i, F = Fnx + Gny is normal flux, δ Fij is the numerical flux through the adge ij that has a length Γij . 2.1. Convective flux approximation Herein, the following upwind scheme based on Roe’s approximate Riemann solver is employed to determine the numerical flux on the control volume surfaces. At each cell edge [10, 20], δ Fij = 1 1 F(W i , nij ) + F(W j , nij ) − J (W , nij ) (W j −W i ) 2 2 ( ) (11) In which % , nˆ ) Λ(W , n ) P −1 (W , n ) J (W , nij ) = P ( W ij ij ij (12) Where J (W , nij ) is the flux Jacobian evaluated using Roe’s average state W : h + h j u i hi + u j h j vi hi + v j h j W i , h, h 2 hi + h j hi + h j T (13) The normal flux can then be written as 1 3 k =1 δ Fij = F(W i , nij ) + F(W j , nij ) − ∑ϕk λk ek 2 (14) Where they ϕ are the wave strengths given by 1 2 ϕ1,3 = ∆h ± 1 h∆u and ϕ 2 = h∆v 2c (15) 2.2. Balancing source term The source terms are balanced by means of a two-dimensional implementation of the upwind scheme proposed by Vazquez et al. [9] [12] for treating the homogeneous part of Saint-Venant equations, and which satisfies the exact conservation C-property. Integration of the source term on the control volume Vi is written, ∫S n ( W )d V = ∑ ∫S n ( W , n ij ) dΓ j∈N ( i ) Γij Vi (16) Following Bermudez [13, 20], this approximation is upwinded and the source term S n replaced by a numerical source vector ψ n , such that ∫ S ( W, n n ij )dΓ = δ S n ( W i , W j , nij ) Γij Γij (17) At each cell interface Γ ij , the contribution of the source term is defined as the projection of the source term vector in the basis of eigenvectors of the Jacobian matrix. Thus the function source term is, ) % , n ) A(W % , n )] ⋅ S n ( X , X , W , W , n ) δ S n ( W i , W j , n ij ) = [I − A ( W ij ij i j i j ij (18) http://www.iaeme.com/IJMET/index.asp 1852 [email protected] A Simple Unstructured Finite Volume Scheme For Solving Shallow Water Equations with Wet/Dry Interface ) % , n ) is the Roe flux Jacobian, and S n ( X , X , W , W , n ) Where I is the identity matrix, A( W ij i j i j ij represents an approximation of the source term on the cell interface Γ ij . Its choice is crucial to ) obtain accurate results. Using states Wi and Wj , the approximation S n is defined by [13,20] as: 0 ) (h − h j ) 1 S n ( Xi , X j , Wi , W j , nij ) = g hi h j i nij d ij g h h (hi − h j ) n 2 i j ij dij (19) 2.3. Improved higher order upwind scheme To obtain higher-order spatial accuracy, the fluxes at each edge are calculated using a piecewise linear function of the state variable W inside the control volume. For the current work, the MUSCL approach is adopted, whereby the left and right values of the state’s variable are evaluated from: 1 L Wij = Wi + 2 ∇Wi ⋅ Nij ur WR = W − 1 ∇W ⋅ N ij j j ij 2 (20) in which Nij is the vector distance between the barycentre coordinates of cells Vi and Vj. The MUSCL approach gives a second-order spatial approximation. However, numerical oscillations can occur when evaluating the normal gradients of the state variables, and so a slope limiter is usually applied. Here we consider the van Albada limiter, limVal ( x, y ) = x 2 + xy x2 + y 2 (21) The resulting discretization is therefore, second-order accurate in space and monotone. We incorporate slope limiters to the reconstruction 1 L Wij = Wi + 2 limVal (∇Wi ⋅ Nij , Wj − Wi ) ur WR = W − 1 lim (∇W ⋅ N , W − W ) ij j Val j ij j i 2 (22) 2.4. Non-negative water depth reconstruction The wet/dry treatment developed in this study is based on the concept presented in [10], in which the non-negative water depth is maintained by enforcing the negative ones at interfaces to zero. The water level and bed elevation are then modified accordingly so as to preserve the wellbalanced condition and mass conservation at wet–dry interfaces. This concept can be read in many recent works, see for example [11, 12 and 13]. At a wet/dry interface, the bed elevation is modified temporarily when computing the numerical flux. In fact, as illustrated in figure 2, if the water surface of the wet cell is lower than the bed elevation of its adjacent dry cell, the bed elevation of the dry cell is set to be the water http://www.iaeme.com/IJMET/index.asp 1853 [email protected] Elmiloud Chaabelasri, Mohamed Jeyar, Najim Salhi and Imad Elmahi surface level of the wet cell. In the other hand, if the water surface of the wet well is higher than the bed elevation of its adjacent dry one, the bed elevation in this case is modified and set to be the bed level of the wet cell. Figure 2: Illustration of wet/dry interface bed reconstruction. Generally the above method treats the dry/wet interfaces satisfactorily. However, the modified bed elevation Z can be formulated as Z Mj = Zi − ∆Z (23) Where if Z j ≤ Z i Z j − Z i ∆Z = Z j − (hi + Z i ) if Z j ≥ Z i + hi (24) −5 Note that a threshold flow depth hc = 10 m is used to determine whether the cell is wet or dry. A dry cell is defined when its flow depth h < hc. 2.4. Time integration and stability condition To date, the forward Euler method has been mainly used as the preferred time stepping scheme for many numerical methods. However, the forward Euler method is only first-order accurate in time and so may introduce excessive numerical dissipation in the computed solutions. To achieve a higher order of accuracy, we use the explicit Runge-Kutta method [14]. The procedure to advance the solution from the time tn to the next time tn+1 is carried out as http://www.iaeme.com/IJMET/index.asp 1854 [email protected] A Simple Unstructured Finite Volume Scheme For Solving Shallow Water Equations with Wet/Dry Interface W (1) = L (W n ) W (2) = L (W n + ∆tW (1) ) 1 W n+1 = W n + ∆t (W (1) + W (2) ) 2 (25) Where n represents the time level, and ∆ t is the time step. To achieve stability, for this explicit scheme, the time step must meet the following criterion: ∆t = CFL Ai + A j 2Γ ij max(λ1 , λ 2 , λ 3 )ij (26) Where the CFL is the Courant number, such that 0 < CFL < 1, Гij is the edge between two triangles i and j, 3. NUMERICAL RESULTS 3.1. Test for balanced property The following test case was studied to verify the balanced property of the proposed scheme, on a still water steady state problem with complex bottom configuration. The bottom configuration is given by the depth function, describing three bumps in the bottom: 1 3 (x − 10) 2 + (y − 11) 2 , 1 − (x − 10) 2 + (y − 31) 2 , 8 10 4 1− (x − 27) 2 + (y − 20) 2 10 Z(x, y) = max[0, 1 − (27) In the domain [0, 40] × [0,40], and the still water steady state as the initial data: h(x, y) + z(x, y) = 4m, hu = hv = 0m / s (28) The computed surface water level and bottom configuration at t=200s are plotted in figure 3. A set of rounding errors are presented in figure 5 to demonstrate that the still water steady state solution is maintained. Here, we analyzed the effect of mesh number, structured and unstructured mesh (see figure 4) and the scheme order on error minimization. We can clearly see that the errors are at the level of rounding errors for these precisions, which verify the desired balanced property. Figure 3: Test for balanced property: Surface water level at t=200s. http://www.iaeme.com/IJMET/index.asp 1855 [email protected] Elmiloud Chaabelasri, Mohamed Jeyar, Najim Salhi and Imad Elmahi Figure 4: Test for balanced property: Different mesh types, Coarse(left), refined (middle) unstructured and (right) structured meshes. Figure 5: Test for balanced property: Errors norms of water depth h as time. 3.2. Two dimensional partial dam break simulation with a wet river bed For this first test case, we consider a hypothetical two dimensional dam break problem with nonsymmetric breach that is a typical validation made in many presented papers, see for example [15, 16, and 17]. An illustration of this problem is shown in Figure 6, in wish the domain is defined by 200mⅹ500m channel with horizontal bed. A dam is located in the middle of the domain, and the non-symmetric breach is 75 m wide with negligible thickness and is located 95m from the left side of the domain. The initial water discharge and depth are given by: hu (t = 0, x, y ) = hv(t = 0, x, y) = 0 10 if x ≤ 95 h(t = 0, x, y ) = 5 otherwise http://www.iaeme.com/IJMET/index.asp (29) 1856 [email protected] A Simple Unstructured Finite Volume Scheme For Solving Shallow Water Equations with Wet/Dry Interface Figure 6: Two dimensional partial dam break with a wet river bed: Computational domain and unstructured mesh. As boundary conditions, at the left and right ends of the channel, a zero discharge and a free boundary are considered, and a solid walls for the others. Figure 7 shows the numerical results of flow after the dam fails, at three different times in terms of water depth and velocity. As we can show, the upstream water is released into the downstream side through the breach, creating surge wave propagating to downstream and neglective wave to upstream. No analytical solution is available for this case, but the results can be compared with those of other numerical schemes. In figure 8, we present a comparison with the results obtained by Wen et al. [18]. Generally, the results are in agreement references results and as with the results obtained by Wen et al. [18] and other numerical methods in aforementioned literatures. Figure 7: Two dimensional partial dam break with a wet river bed: (top) The water depth contours and (bottom) the surface of the water level at time t = 7,2s . http://www.iaeme.com/IJMET/index.asp 1857 [email protected] Elmiloud Chaabelasri, Mohamed Jeyar, Najim Salhi and Imad Elmahi Figure 8: Two dimensional partial dam break with a wet river bed: Comparison of the contour lines of the water surface level, (left) present method and (right) Wen et al. results [ref] 3.3. Dam break over three humps The current shallow water model is applied to simulate a dam-break over an initially dry floodplain with three humps, recommended by Kawahara and Umetsu [19] for this challenging problem that involves complex flow hydrodynamics, bottom topography and wetting and drying. The simulated setup is sketched in Figure 9, where the dam break occurs in a 75ⅹ30m rectangular domain with the dam located 16 m away from upstream end. A reservoir with a still water surface elevation of 1.875m is assumed upstream of the dam. The bed topography of the domain is defined as: 1 1 (x − 30) 2 + (y − 6) 2 , 1 − (x − 30) 2 + (y − 24) 2 , 8 8 3 1− (x − 47.5) 2 + (y − 15) 2 10 z(x, y) = max[0, 1 − (30) Figure 9 : Dam break over three humps : Computational domain, bed profile, and unstructured mesh. http://www.iaeme.com/IJMET/index.asp 1858 [email protected] A Simple Unstructured Finite Volume Scheme For Solving Shallow Water Equations with Wet/Dry Interface The floodplain is initially dry and a constant Manning coefficient of 0.018 s/m1/3 is used throughout the domain. The domain walls are assumed to be solid. The dam collapses instantly at t = 0s and simulation is run for 300s on a 5779 unstructured triangular mesh. After the dam fails, the initial still water in the reservoir rushes onto the downstream floodplain. In Figure 10, we present the profile of water depth and contours at different output times t = 2, 6, 12, 30 and 300s. As can be observed from these results, After about 2s, the wet– dry front reaches the two small humps and begun to climb over them. At t = 6.0 s, the two small humps are entirely submerged and the wave front has hit the large hump. At t=12 s, the wave front passes through both sides of the large hump and begins to submerge the lee of the hump. Finally the flow becomes steady due to the dissipation caused by bed friction, as shown at t=300 s when the flow is nearly motionless and the peaks of the humps are no longer submerged. The overall flow pattern for this example is preserved with no spurious oscillations appearing in the results. Obviously, the obtained results verify the stability and the wet/dry interface properties of the proposed method. Figure 10: Dam break over three humps: Profile of water depth and depth contours of the dam break over three humps at different output times: t = 2 s; t = 6 s; t = 12 s and t = 300 s. 4. CONCLUSIONS http://www.iaeme.com/IJMET/index.asp 1859 [email protected] Elmiloud Chaabelasri, Mohamed Jeyar, Najim Salhi and Imad Elmahi This paper has investigated a simple unstructured finite volume method for numerical simulation of dam break flows by solving the shallow water equations including non-uniform bed elevation terms. The space derivative has been approximated by an upwinding scheme for numerical fluxes due to convective and bed slope behavior of flow. An explicit second order Runge-Kutta scheme was used for time integration. The method has been found to be flexible and straightforward to implement in unstructured triangular mesh. The method was tested against several theoretical dam break problems, includes dam break with a non-symmetric breach, circular dam and a dam break on an irregular beds involving wet/dry interface. The numerical model gives promising predictions by comparison with previously references and published results. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] Gallardo J, Pares C, Castro M. On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas. J Comput Phys 2007; 227:574–601. Noelle S, Xing Y, Shu C-W. High-order well-balanced finite volume WENO schemes for shallow water equation with moving water. J Comput Phys 2007; 226:29–58. J. Fernandez-Pato, M. Morales-Hernandez, P. Garcia-Navarro, Implicit finite volume simulation of 2D shallow water flows in flexible meshes, Comput. Methods Appl. Mech. Engrg. 328 (2018) 1–25 Fayssal Benkhaldoun, Saida Sari, Mohammed Seaid, Projection finite volume method for shallow water flows, Mathematics and Computers in Simulation, 118( 2015) 87-101. Benkhaldoun F., Elmahi I., Sari S. ,Seaid M., An unstructured finite‐volume method for coupled models of suspended sediment and bed load transport in shallow‐water flows, International Journal for Numerical Methods in Fluids Chaabelasri, E., N. Salhi, I. Elmahi. F. Benkhaldoun, Second order well balanced scheme for treatment of transcritical flow with topography on adaptive triangular mesh, International Journal of Physical and Chemical. News 53 (2010) 119-128. Yulong Xing, Xiangxiong Zhang, Chi-Wang Shu, Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations, Advances in Water Resources 33 (2010) 1476–1493. Yulong Xing, Exactly well-balanced discontinuous Galerkin methods for the shallow water equations with moving water equilibrium, Journal of Computational Physics 257 (2014) 536–553 Gang Li, Lina Song, Jinmei Gao, High order well-balanced discontinuous Galerkin methods based on hydrostatic reconstruction for shallow water equations, Journal of Computational and Applied Mathematics 340 ( 2018) 546-560 Liang Q. flood simulation using a well-balanced shallow flow model. J Hydraul Eng 2010; 136:669– 75. Zhiguo He, Ting Wu, Haoxuan Weng, Peng Hu, Gangfeng Wu, Numerical simulation of dam-break flow and bed change considering the vegetation effects, International Journal of Sediment Research, (2015)http://dx.doi.org/10.1016/j.ijsrc.2015.04.004 Chunchen Xia, Zhixian Cao, Gareth Pender, Alistair Borthwick, "Numerical algorithms for solving shallow water hydro sediment-morphodynamic equations", Engineering Computations (2017), https://doi.org/10.1108/EC-01-2016-0026 ingming Hou, Run Wang, Qiuhua Liang, Zhanbin Li, Mian Song Huang, Reihnard Hinkelmann, Efficient surface water flow simulation on static Cartesian grid with local refinement according to key topographic features, Computers and Fluids (2018), doi: 10.1016/j.compfluid.2018.03.024 Chaabelasri E., Borthwick A., Salhi N., Elmahi I., Balanced adaptive simulation of pollutant transport in Bay of Tangier. (Morocco), World Journal of Modelling and Simulation Vol. 10 ( 2014 ) No. 1, pp. 3-19. Liang, D., Lin, B. and Falconer, R.A. ‘‘Simulation of rapidly varying flow using an efficient TVDMacCormack scheme’ (2007), Internat. J. Numer. Methods Fluids, 53, pp. 811–826. Nikolos IK, Delis AI. An unstructured node-centered finite volume scheme for shallow water flows with wet/dry fronts over complex topography. Comput Methods Appl Mech Eng 2009;198(4748):3723–50. http://www.iaeme.com/IJMET/index.asp 1860 [email protected] A Simple Unstructured Finite Volume Scheme For Solving Shallow Water Equations with Wet/Dry Interface [17] Chou C.K., C.P. Sun, D.L. Young, J. Sladek, V. Sladek, Extrapolated local radial basis function collocation method for shallow water problems, Engineering Analysis with Boundary Elements 50(2015)275–290 [18] Huda.Altaie, Performance of Two -Way Nesting Techniques for Shallow Water Models. International Journal of Mechanical Engineering and Technology, 7(6), 2016, pp. 425–434. [19] Xiao Wen, Zhen Gao, Wai Sun Don, Yulong Xing, Peng Li, Application of Positivity-Preserving WellBalanced Discontinuous Galerkin Method in Computational Hydrology, Computers and Fluids (2016), doi: 10.1016/j.compfluid.2016.04.020 Kawahara M, Umetsu T. Finite element method for moving boundary problems in river flow (1986). Int J Numer Meth Fluids (6)365-86. [20] http://www.iaeme.com/IJMET/index.asp 1861 [email protected]