Hydrostatic relaxation scheme for the 1D shallow water - Exner equations in bedload transport

Transcription

1 Hydrostatic relaxation scheme for the D shallow water - Exner equations in bedload transport Putu Harry Gunawan a,b,, Xavier Lhébrard b a Industrial and Financial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jalan Ganesha, Bandung 432, Indonesia. b Universitée Paris-Est, LAMA UMR85, F-77454, Marne-la-Vallée, France. Abstract This paper describes the numerical scheme of the one-dimensional Exner - shallow water equations for sediment modelling. Here, our novel numerical scheme is called hydrostatic relaxation scheme which is a robust and straightforward scheme. This scheme is given as the hydrostatic reconstruction of the relaxation solver. Some numerical tests are presented such as analytical solution, transcritical flow over a granular bump, and erodible bottom due to dam-break. The comparisons of numerical and data experiments are also given in dam-break over granular bed simulation. The results of our simulation have a good agreement with analytical solution and the data experiments. Moreover, satisfactory convergence rate in L -norm is obtained. Keywords: Shallow water equations, Exner equation, Bedload sediment, Corresponding author. Tel addresses: putu-harry.gunawan@u-pem.fr; harry.gunawan.putu@gmail.com (Putu Harry Gunawan), xavier.lhebrard@u-pem.fr (Xavier Lhébrard) The first author acknowledge the support of the Ensemble Estimation of Flood Risk in a Changing Climate project funded by The British Council through their Global Innovation Initiative, the scholarship support from DIKTI, and wish to thanks to François Bouchut for the valuable advise and for the great discussions. Preprint submitted to Journal of Elsevier August 26, 25

2 Hydrostatic reconstruction, Relaxation solver, Dam-break.. Introduction Sediment transport is an important phenomena in nature but least understood processes and complex, e.g. the morphological change under the actions of flooding, rain runoff, wave currents, and tidal flow. The flood can cause the biggest disasters to our environment such as erosion and deposition process. Hence the study of interaction between the flow and the transport dynamics and how this relation influence the morphology surface change are needed. The numerical simulation of sediment process becomes important. However, the reliability of numerical result relies on how the mathematical model described the sediment process. Therefore, the accuracy of the governing equations and discretization of numerical scheme are required. Moreover, the numerical results still need to be verified and validated using analytical solution or data experiment to see the applicability of the scheme. Some models and numerical approaches for sediment transport have been demonstrated Wu & Wang (27); Capart & Young (998); Simpson & Castelltort (26); Cao et al. (24); Cordier et al. (2); Audusse et al. (22). Some of them are quite complex because they consider physical characteristics of sediment in detail. In addition, their models include some type of sediment such as bedload and suspended load. For example, one of the simple model is called shallow water-exner equations to study the bedload sediment transport and is introduced in Cordier et al. (2), and Audusse et al. (22). 2

3 The shallow water - Exner equations consists of three equations: mass conservation, momentum conservation and Exner equation which is a mass conservation of sediment in interaction with the fluid flow. In order to approximate this model, some methods are already studied such as splitting and coupled method. In splitting method, shallow water equations and Exner equation are computed separately. First, the shallow water equations can be approximated by the finite volume method using collocated approach (see Audusse et al. (24); Greenberg & Leroux (996); LeVeque (998); Tang et al. (24); Tao & Xu (999)) or staggered approach (Doyen & Gunawan (24); Stelling & Duinmeijer (23); Herbin et al. (24)) and afterward the Exner equation can be computed by a simple finite difference method. However, splitting method using finite volume collocated approach seems not a robust scheme for shallow water - Exner equations (see Cordier et al. (2)). In Cordier et al. (2), they show that the splitting method fails to approximate shallow water - Exner equations because of the instability issue. Hence, they proposed coupled Roe scheme which is shown as a stable and robust scheme. Moreover, another coupled scheme is also introduced by Audusse et al. (22), where the relaxation flux and simple solver are used to obtain some robust results as their expected. Here we introduce a novel method, which is the combination of hydrostatic reconstruction and relaxation flux solver. The idea of this method is follow the paper of Audusse et al. (24) and the book of Bouchut (24). We apply the local hydrostatic reconstruction to determine a well-balanced scheme from the numerical flux of Audusse et al. (22). According Audusse et al. (24), the hydrostatic reconstruction method is designed to be applied 3

4 with a numerical flux for the homogeneous problem. However, here the numerical flux we used is for the non-homogeneous problem (i.e non-flat topography and friction). Moreover, we applied the apparent topography method for treatment the friction term, which is introduced in Bouchut et al. (24). In order to validate the numerical results, we compared our results with the analytical solution and data experiments obtained in Berthon et al. (22) and Capart & Young (998) respectively. The comparison of data experiments and numerical computations is given in dam-break flow over granular bed simulation. The outline of this paper is following: in Section 2, the model of coupled system shallow water and Exner equations are introduced. The discretization of numerical method such as hydrostatic reconstruction and relaxation flux solver described in Section 3. Some numerical tests and comparisons with analytical solution and data experiments are given in the Section 4. Last in Section 5, the conclusions are constructed. 2. Governing equations The shallow water - Exner model The one-dimensional coupled model of shallow water - Exner equations is given by t h + x (hu) = () t (hu) + x ( hu gh2 ) + gh x Z = ghs f (2) Φ t Z + x Q s = (3) 4

5 7 where h is the total water depth, u is the velocity in the x direction, Z = (z H) is the bed elevation and H the depth of fixed bedrock layer to the reference level (see Fig. ). The friction term is defined by S f, Φ = ( φ) with φ is the bed sediment porosity, Q s is the bed load and g is the acceleration due to gravity. Reference level h H z Fixed bedrock Figure : The one-dimensional sketch of shallow water with a sediment layer. 75 Some estimations for the sediment discharge Q s have been obtained by empirical formulas. Some of most important empirical formulas of the bed load exist, such as Grass formula, Meyer-peter and Müller s formula or Van Rijn s formula (see Cordier et al. (2); Van Rijn (984); Rijn (984) for more detail). The Grass formula for the solid transport discharge are given as, Q s = A g (u) u m u (4) where A g is a constant depending on the sediment properties (such as the grain diameter, cinematic viscosity, etc) and the experimental data, u is the 5

6 8 horizontal velocity of fluid, and the parameter m 4 is a given number depend on sediment. The value m = 3 is the usual value for the exponent m. The interval value A g is such that A g. If the value closed to, it means the interaction between the fluid and the sediment is weak. Meanwhile, the strong interaction between fluid and sediment is described with the value of A g closed to. For the friction terms, we consider the friction slope which depends on flow conditions (Simpson & Castelltort (26); Kadlec (99); Julien & Simons (985)). Here we can use either Manning s friction law or Darcy-Weisbach s friction law S f = µ2 u u h 4/3 (5) S f = C fu u 8gh (6) 85 where µ is Manning s roughness coefficient, and C f is the Darcy-Weisbach friction factor which depends on the Reynolds number. In this work, the coefficients µ and C f are set to be constant. Note that, the friction law (5) is often used for the case when flow is laminar, and the later friction slope (6) is often used for the case when flow is turbulent in overland flow. The empirical formula for the sediment discharge by Meyer-peter and Müller s is given by Q s = sgn(u)8(τ τ c ) 3/2 (φ r )gd 3, (7) where g is gravitational force, d is the diameter size of sediment grain, φ r = ρ s /ρ w is a relative density of fluids, ρ w and ρ s are the density of fluid and sediment respectively, τ is the non-dimensional shear stress and τ c is the 6

7 non-dimensional critical shear stress (see Cao et al. (26) for more detailed). The non-dimensional shear stress τ is defined by τ = τ (γ s γ)d, (8) where τ = γr h S f is the shear stress, γ = gρ w denotes the specific weight 9 of fluid, γ s = gρ s denotes the specific weight of the sediment, R h h is the hydraulic ratio and S f is the friction slope defined as (5) or (6) depends on type of water flow. 3. Numerical method 3.. The D numerical scheme We consider a grid of points x i+, i Z, < x 2 2 define the cells and their lengths < x 2 < and we Ω i = [x i, x 2 i+ ] x i = x 2 i+ x 2 i 2 >. (9) We shall define the discrete times by t n = n t, n N with a constant time step t >. Here we would like to approximate the solution of Eqs. () - (3), by discrete values U n i = (h n i, h n i u n i, Z n i ) T, i Z, n N. The discrete value U n i is computed by considering the averages of the exact solution over the cells, Then we can write the scheme as Ui n U(t n, x) dx. () x i Ω i U n+ i = U n i ξ t x i ( F i+ 2 L F i 2 R ), () 95 where ξ = [,, /Φ] T. 7

8 The fluxes are defined by hydrostatic reconstruction scheme as F i+ 2 L = F(U n i+ 2 L, U n i+ 2 R) + F i 2 R = F(U n i 2 L, U n i 2 R) + g(h n i )2 g(hn i+ 2 L)2 2 2 g(h n i )2 g(hn i 2 R)2 2 2, (2) (3) with F is a numerical flux which is defined by relaxation solver of shallow water - Exner equations introduced by Audusse et al. (22) and will be recalled in next Subsection 3.3. Moreover, the reconstructed states U i+ 2 L and U i+ R are defined by 2 where U i+ 2 L = (hn i+ 2 L, hn i+ 2 Lun i, Z n i ) T, (4) U i+ 2 R = (hn i+ 2 R, hn i+ 2 Run i+, Z n i+) T, (5) h i+ 2 L = (hn i ( Z) + ) +, (6) h i+ 2 R = (hn i+ ( Z) + ) +, (7) Z = Z n i+ Z n i, (8) with the notation a + = max(a, ) and Z n i+ = (z n i+ H i+ ) Apparent topography The source term (i.e friction term) in momentum conservation (2) is treated by apparent topography. This method was introduced by Bouchut 8

9 (24) and some applications can be found in Delestre et al. (23) and Delestre (2). To discretize the friction terms Eqs. (5) and (6), we define F i+ 2 = f n i+ 2 x i, (9) where f n i+ 2 for Manning s is given as f n i+ 2 = if h i = h i+ =, otherwise. (h n ) 4/3 i+ 2 µ 2 u n u n i+ i+ 2 2, (2) and for Darcy-Weisbach is written as if h i = h i+ =, f n = C i+ f u n u n i+ 2 i+ 2 2 otherwise. 8gh n i+ 2. (2) Meanwhile, the intermediate values u n i+ 2 and h n i+ 2 are defined by and u n i+ 2 h n i+ 2 = hn i+u n i+ + h n i u n i h n i+ + hn i, (22) = 2 (hn i+ + h n i ), (23) respectively. Eventually, we can modify the reconstruction of Eqs. (6) - (8) become where Z is defined by Eq. (8). h i+ 2 L = (hn i ( Z app ) + ) +, (24) h i+ 2 R = (hn i+ ( Z app ) + ) + (25) Z app = Z + F i+ 2 9 (26)

10 Remark 3.. Second-order. To improve the order of accuracy into higher 5 order in the numerical scheme, we can use for instance the MUSCL reconstruction for space and Heun method for time. The detail about this method can be found in Bouchut (24) and Delestre (2) Numerical flux As we already mentioned in previous section, we need to find the value of numerical flux F. A general way to construct numerical flux F is the notion of approximate Riemann solver which come from the Godunov approach (Bouchut (24)). Now we consider the initial data U n (x) which is piecewise constant, and we define an approximation solution for t n t < t n+ and x R by U appx (t, x) = R( x x i+/2 t t n, U n i, U n i+) (27) for x i < x < x i+ with x i = (x i+/2)+xi /2 )/2. This approximation is possible until time t n+ under a half CFL condition, in sense that x/t < x i 2 t R(x/t, U i, U i+ ) = U i, (28) x/t > x i+ 2 t Then, the new values at time t n+ are given as R(x/t, U i, U i+ ) = U i+. (29) U n+ i = U appx (t n+, x) dx. (3) x i Ω i Note that our system is not conservative, however we can follow the same computations in Bouchut (24), Chapter 2 to obtain U n+ i = U n i ξ t x i ( F(U n i, U n i+) F(U n i, U n i ) ). (3)

11 Hence, by a relaxation approach of the coupled system of shallow water - Exner equations in Audusse et al. (22), the numerical flux F is defined as F(U l, U r ) = F (U l ), if σ F (Ûl), if σ σ 2 F (Ul ), if σ 2 σ 3 F (Ur ), if σ 3 σ 4 (32) where F (Ûr), if σ 4 σ 5 F (U r ), if σ 5 F (U l ) = (h l u l, h l u 2 l + Π l, Q s (u l )) T, (33) F (Ûl) = (ĥlû l, ĥlû 2 l + ˆΠ l, Q s (û l )) T, (34) F (U l ) = (h l u l, h l u 2 l + Π l, Q s (u l )) T, (35) F (U r ) = (h ru r, h ru 2 r + Π r, Q s (u r)) T, (36) F (Ûr) = (ĥrû r, ĥrû 2 r + ˆΠ r, Q s (û r )) T, (37) F (U r ) = (h r u r, h r u 2 r + Π r, Q s (u r )) T, (38) with Π = 2 gh2 and the waves speeds (see Fig. 2) are given by σ < σ 2 < σ 3 < σ 4 < σ 5, (39) with σ = u l β h l, σ 2 = û l α ĥ l, σ 3 = u, σ 4 = û r + α ĥ r, σ 5 = u r + β h r. (4)

12 The superscripts (ˆ) and ( ) distinguish the intermediate and internal states respectively. In Fig. 2, we can see clearly that each wave separates the constant states: U l, Ûl, U l, U r, Ûr, and U r from the left to the right. Moreover, according Audusse et al. (22), to ensure the stability of the scheme, the parameter α and β are chosen such that α < β. Hence, they are defined by ( α = max i β = max gh 3 i ( max i ), (4) ) ) ( (h i u i ) 2 + gh 2i u(q s ) i, κα. (42) where i Z, and κ >. numerical simulations. Further, in this paper we use κ =.5 for all σ 2 σ 3 σ 4 U l U r σ Û l Û r σ5 U l U r Figure 2: The Riemann solver for shallow water with a sediment layer. 5 According Audusse et al. (22), the sediment elevation Z and sediment load Q s are continuous through the intermediate and internal waves, because of the structure of the eigenvectors of the relaxation scheme. Therefore we define Ẑl = Z l = Z r = Ẑr and ˆQ sl = Q sl = Q sr = ˆQ sr as Z and Q s respectively. The definition of the intermediate sediment elevation Z and 2

13 2 sediment load Q s can be seen in Audusse et al. (22)( Eqs. (5) and (6) respectively). Finally the Courant-Friedrichs-Levy (CFL) condition of this scheme is given as x t = ν (43) max i σ i where ν is a Courrant number. In the end, we can summarize the previous information into an algorithm. It can be seen in Algorithm. Algorithm The hydrostatic relaxation scheme. Step. Give the initial conditions at n =. Step 2. Compute (4) and (42) to get α and β respectively. Step 3. Compute t by (43) to preserve the stability. Step 4. Solve (24) - (26) to apply the apparent topography for the friction term. Step 5. Define the reconstructed states by (4) and (5). Step 6. Compute the fluxes (2) and (3) with numerical flux F defined by (32) at each interface of cells. Step 7. Finally, update the discrete value U n i into U n+ i by solving () Numerical simulations In this section we elaborate some numerical simulations such as the comparison between the analytical solution and the numerical scheme, transcritical flow over a granular bump, and the comparison water surface of dambreak over a granular bed with the data experiments. For all simulations, 3

14 3 the depth of bedrock to the reference level H is set to be constant, such that we have x H =. 4.. Analytical solution 35 Here we consider the numerical simulation where the analytical solution is available. This simulation follow the paper of Berthon et al. (22) with no friction term (S f = ). Z,h+Z (m) (h+z) Initial Z Initial (h+z) Analytic Z Analytic (h+z) HR Z HR x (m) Figure 3: Comparison of hydrostatic relaxation (HR) scheme with the analytical solution at final time t = 7 s. The space domain is Ω = [, 7] and the initial conditions of this simulation are given by (hu) ini (x) =, (44) [ ax + b ] /3 u ini (x) =, A g (45) h ini (x) = (hu) ini(x), u ini (x) (46) Z ini (x) = u3 ini(x) + 2g(hu) 2 ini(x), 2gu ini (x) (47) 4

15 with coefficients a = b both equal to 5 3. The analytical solutions for h, u and Z can be found in the paper of Berthon et al. (22). In this simulation we consider the Grass formula with Q s = A g u u 2, where A g = 5 3 and we used grid points. In our simulation result, 4 the scheme has a good behavior which is the convergence of our scheme is close to the analytical solution (see Fig. 3 and 4) even in coarse grids. Fig. 3 shows the free surface and sediment profile, and Fig. 4 shows the velocity profile at the final time t = 7 s. We can see clearly in velocity profile (Fig. 4), the hydrostatic relaxation (HR) scheme is less diffusive than the simple 45 solver (SR) scheme obtained in Berthon et al. (22). 2.8 u Analytic u HR u SR u (m/s) x (m) Figure 4: Comparison the velocity profile of hydrostatic relaxation (HR) scheme and simple solver (SR) scheme with the analytical solution at final time t = 7 s. Tabel collects the discrete L -norm errors for h, u, and Z at the final time with difference mesh sizes. Moreover, Tabel also shows the rate of convergence for first-order computations, computed by comparison between two discrete L -norm errors with different mesh sizes. Here, we can confirm 5

16 Table : L -norm error of relaxation scheme at t = 7 s. Cells h h analytic L τ h u u analytic L τ u Z Z analytic L τ Z E- / 2.266E- /.34E-2 / E E E E E E E E E E E E E E E that our scheme has a good convergence rate τ where it tends to since computed in first-order for each unknown variables (h, u and Z). The CFL number used is in this simulation Transcritical flow over a granular bump Here, we consider the numerical simulation of subcritical and supercritical flow over a granular bump. Before we get the steady state solution, first we start with the initial conditions: Z ini (x) =. +.e (x 5)2, (48) (hu) ini (x) =.6, (49) h ini (x) + Z ini (x) =.4, (5) A g =. (5) where the boundary conditions are (hu)(, t) =.6 and h(, t) =.4 with 55 the space domain Ω = [, ]. The points of grids are used and the CFL condition is set to be in this simulation. 6

17 Water surface Sediment t= Z, h+z (m) x (m) Figure 5: The initial condition of dam-break simulation over a granular bed 6 65 Once the steady state is reached (see Fig. 5), we set immediately the coefficient of Grass bedload flux with A g = 5 4 hence the sediment could evolve in time. As noted in Cordier et al. (2), the important information is that in this simulation one can obtains a negative eigenvalue in supercritical regions. For instance, the splitting method does not take into account about this information, hence the instability occurs in the simulation (see Fig. 5 in paper of Cordier et al. (2)). However, with our scheme (even in coarse grid and CFL =) the simulation remains stable. The results in various final times t = 5 and t = 3 are shown in Fig. 6 and 7 respectively Dam-break over a granular bed 7 This simulation, we introduce the numerical experiment of dam-break flow over a granular bed. In Capart & Young (998), the laboratory experiment has been done by mobile bed measurement.2 m long with 6 cm deep 7

18 Z, h+z (m) h+z initial Z initial (h+z) HR Z HR t= x (m) Figure 6: The solution of transcritical flow over a granular bed with A g = 5 4 at final time t = 5 Z, h+z (m) h+z initial Z initial (h+z) HR Z HR t= x (m) Figure 7: The solution of transcritical flow over a granular bed with A g = 5 4 at final time t = 3 of particle layer. The dam is located at the left of domain with height and width are set to be. m and.4 m respectively (see Fig. 8). In this simulation we used points of gird and the parameters are A g =., g = 9.8 m/s 2 and CFL condition is. The sequence comparisons 8

19 Z, h+z (m) Water surface Sediment x (m) Figure 8: The initial condition of dam-break simulation over a granular bed of our numerical scheme and data experiment in several final times are shown in Fig. 9. With hydrostatic relaxation scheme, we can see that our scheme has a good agreement with the rarefaction wave of dam-break. In the shock area, our numerical results are far away from the experiment data. However it is reasonably well since in the real experiment the sediment and water mixed in that area. Overall we can conclude that, the comparisons of numerical scheme and experiment observation are equitable well. In the previous simulations, we consider the physical parameters of sediment such as, the diameter of sediment, porosity, friction etc into one constant value A g by Grass formula. When we want to consider about the characteristics of sediment directly into simulation, the Grass formula is not suitable. Hence the Meyer-peter and Müller s formula is acceptable for that problem. In Wu & Wang (27); Capart & Young (998), two experiments on dam-break flow over movable beds which are performed in Taipei (University of Taiwan) and Louvain-la-Neuve (Uinversité Catholique de Louvain) have reported. In their report, the information about the characteristics of sediment are written in detail. Therefore, it will also be interesting if we compare our numerical simulation using the Meyer-peter and Müller s formula 9

20 Z, h+z (m) Experiment (h+z) HR Z HR x (m) (a) Z, h+z (m) Experiment (h+z) HR Z HR x (m) (b) Z, h+z (m) Experiment (h+z) HR Z HR x (m) (c) Z, h+z (m) Experiment (h+z) HR Z HR x (m) (d) Figure 9: Comparison of water surface using hydrostatic relaxation scheme and data experiment by Capart & Young (998): (a) dimensionless time t = t ; (b) t = 2t ; (c) t = 3t ; and (d) t = 4t, where t = g/h, h =. m. with the data experiments given in Fraccarollo & Capart (22). In Fig., we show the series of comparison of our numerical computation and data experiment by Louvain-la-Neuve experiment only. The characteristics of sediment are given similar as in Wu & Wang (27). The sediment porosity φ =.3, the diameter of sediment d = 3.2 mm, the density of sediment, 54 kg/m 3, and the Manning s coefficient. and.25 are given for water and sediment respectively. The domain of this simulation is Ω = [.25 :.25], the initial conditions are h ini (x) = h x<, Z ini (x) =, u ini (x) =, 95 where the initial water depth h =. m. Fig. shows the nice comparison of our numerical simulation and data 2

21 Z, h+z (m) (h+z) HR Z HR (h+z) Experiment Z Experiment (a) Z, h+z (m) (h+z) HR Z HR (h+z) Experiment Z Experiment (b) x (m) x (m) Z, h+z (m) (h+z) HR Z HR (h+z) Experiment Z Experiment x (m) (c) Figure : Comparison of water surface using hydrostatic relaxation scheme and data experiment in Louvain case: (a) dimensionless time t = 3t ; (b) t = 7.5t ; and (c) t = t, where t = g/h, h =. m. 2

22 experiment both in water height and sediment profile. Thus our scheme is capable to handle simulation when considering the physical characteristic of sediment. 5. Conclusion In the present study, we introduced the hydrostatic relaxation scheme to approximate the shallow water-exner equations. This scheme is based on the hydrostatic reconstruction and relaxation solver. The relaxation solver was introduced in Audusse et al. (22), and is stable and well-posed for shallow water-exner equations. Further we modify their scheme by using hydrostatic reconstruction on their solver, which enabled us to get better numerical results. Indeed, some numerical experiments show the robustness and accurateness of the scheme. The comparison of numerical results with the analytical solution and data experiments are obtained nicely comparable. Additionally, the convergence properties to an analytical solution is satisfied (the convergence rate tends to in discrete L -norm errors since the first order scheme is applied). The scheme also shows efficient and as well as stable even in stiff conditions. Moreover, this scheme is simple and robust, therefore it is flexible for many applications. Finally, the comparison between this paper and the papers of Cordier et al. (2) and Audusse et al. (22) are drawn up in Table 2. References Audusse, E., Berthon, C., Chalons, C., Delestre, O., Goutal, N., Jodeau, M., Sainte-Marie, J., Giesselmann, J., & Sadaka, G. (22). Sediment trans- 22

23 Table 2: The comparison between this paper and the papers of Cordier et al. (2) and Audusse et al. (22). This paper Cordier et al. (2) Audusse et al. (22) Numerical method: Hydrostatic Coupled Roe Relaxation reconstruc- scheme simple solver tion + Relaxation solver Comparison of numerical Done Undone Done results and analytical solution by Berthon et al. (22) Table L -norm error Done Done Undone Dam-break over a granular bed simulation Done Undone Done Transcritical flow over a Done Done Undone granular bump simulation Comparison between dambreak over a granular bed simulation and data experiments Done Undone Undone 23

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