QUASI-CONSERVATIVE FORMULATION OF THE ONE DIMENSIONAL SAINT VENANT-EXNER MODEL

Transcription

1 QUASI-CONSERVATIVE FORMULATION OF THE ONE DIMENSIONAL SAINT VENANT-EXNER MODEL Annunziato Siviglia 1, Giampiero Nobile 2 and Marco Colombini 3 ABSTRACT Coupling the Saint-Venant equations with the Exner equation creates a morphodynamic model that can be used to describe flow in rivers the shape of which is determined by the flow itself. Such models are described by a system of hyperbolic equations expressed in non-conservative form, thus requiring the use of non-conservative solution approaches that are known to incorrectly compute both the strength and the celerity of shock waves (bores). The present work proposes a quasi-conservative formulation of the governing equations that may reduce these errors. Comparison of model predictions with measurements obtained by means of a physical model are satisfactory and overcome predictions based on a fully primitive formulation. Keywords: Numerical model, movable bed, sediment transport. INTRODUCTION The most widely used hydro-morphodynamic model is based on the Saint-Venant equations for hydrodynamic flow routing, with bed evolution being described by the Exner equation (Holly and Rahuel 199; Cui et al. 1996; Wu et al. 24). The majority of such models are uncoupled (i.e. bed and flow solutions proceed separately) while only few of them solve simultaneously the complete set of governing equations. This choice of the solution strategy ultimately relies on the relative order of magnitude of the celerities associated with the characteristic curves of the hyperbolic system. The behavior of such curves is well known: far 1 Dept. of Civ. and Env. Engrg., Univ. of Trento, via Mesiano 77, 385 Trento. nunzio.siviglia@ing.unitn.it. 2 Dept. of Env. Engrg., Univ. of Genova, via Montallegro 1, Genova. gp@dicat.unige.it. 3 Dept. of Env. Engrg., Univ. of Genova, via Montallegro 1, Genova. col@dicat.unige.it. 1 Siviglia

2 from criticality the celerity of a bed wave is considerably smaller than that of a surface wave. The bed interacts only weakly with the water surface, thus justifying an approach in which the equations that govern the dynamics of the liquid phase are solved separately from those governing the solid phase. However, there exists a transcritical region,.8 F 1.2 (Sieben 1999), where F is the Froude number, in which two of the three characteristic celerities have the same order of magnitude; in this region, surface waves strongly interact with bed waves (Lyn and Altinakar 22). Moreover, as discussed by Lanzoni et al. (26), near critical conditions a decoupled approach predicts morphodynamic influence (i.e. propagation of information on bed variations) either upstream or downstream, while only a coupled approach is able to propagate bed information in both directions. Therefore, adopting a coupled approach for solving the governing equations is a stringent requirement in order to obtain reliable results when flow is close to critical conditions, a situation that characterizes most mountain rivers and, at least locally, is often encountered in natural rivers due to the presence of obstructions or sudden variations in bed slope and river width. Coupled solutions should then be used for the study of long term evolution of natural rivers (Cao et al. 22; Correia et al. 1992; Singh et al. 24), or, more generally, when it is not possible to assume that the flow adapts instantaneously to changes in bed elevation (Colombini and Stocchino 25). When the numerical properties of the finite difference approximation of the differential system are considered, the choice between conservative or non-conservative formulations of the problem arises. Lax and Wendroff (196) proved that conservative methods, if convergent, converge to correct solutions. On the other hand, Hou and LeFloch (1994) have shown that non-conservative schemes do not correctly converge if a shock wave is present in the solution, a situation encountered in morphodynamic problems when discontinuities (hydraulic jumps and sediment bores) of the solution appear. In this sense, the coupled approach reveals an important drawback: a conservative formulation is not available, so that, as in all the aforementioned studies, a primitive (non- 2 Siviglia

3 conservative) formulation is usually adopted. Coupling and conservative formulation are therefore mutually exclusive while both strategies should be adopted in the morphodynamic modelling of natural rivers. To overcome this problem, a quasi-conservative formulation of the fully coupled Saint- Venant Exner model, is proposed. Appropriate quasi-conservative predictor corrector numerical schemes are described in the following sections and tested making use of a model system of equations for which analytical solutions are available. The proposed model is applied to a schematic mobile bed case and to a natural environment making use of the measurements obtained on a physical model of a flood defense system to be realized on the Vara River (Northern Italy). PROBLEM FORMULATION The riverbed is assumed to be composed of well-graded sediments with the same particle size d s as those transported by the water flow. The solid transport is characterized as bed-load, assumed to be in local and instantaneous equilibrium with the flow. The governing equations impose mass conservation for the fluid and solid phases and the momentum principle on an open channel flow over a movable bed consisting of cohesionless sediment. In a coordinate system (x, y, z) with the x-axis longitudinal, y-axis transversal and the z axis vertical upward (see figure 1), they are: Q t + x A t + Q x = (1) ) (β Q2 A + ga h x + gas f = (2) (1 λ p ) A s t + Q s x = (3) where A is the flow cross sectional area, Q and Q s are the flow rate and the sediment 3 Siviglia

4 discharge, A s is the bed material area defined as follows: A s = z b (y)dy ; (4) B f where B f is the width of the active portion of the bed, z b is the bottom elevation λ p is the porosity, h is the water level, β is the momentum correction coefficient, g is the gravitational acceleration and S f is the friction term. The continuity equations for both the liquid and the sediment phases are already written in conservative form. The momentum equation can be rewritten in the following conservative form (Cunge et al. 1994): Q t + ) (β Q2 x A + gi 1 = gi 2 + ga(s S f ) (5) where S is the local bottom slope, I 1 represents a hydrostatic pressure force term and I 2 accounts for the pressure forces in a volume of constant depth due to longitudinal variations. Finally, the friction resistance law and the sediment-transport formula required to evaluate S f and Q s are obtained assuming that, locally and instantaneously, relationships available for steady and uniform flow conditions still hold. To evaluate S f we adopt the Gauckler- Strickler closure relationship, according to which S f = Q 2 Q2. = (6) Ks 2 A 2 R4/3 I 2 where K s is the Strickler coefficient and R is the hydraulic radius. In order to properly account for the irregular geometry of the cross-sections the term Ks 2 A 2 R 4/3 is evaluated as integrals over the cross-section, following the approach originally proposed by (Engelund 1964). In the present work we are concerned with gravel-bed rivers, we thus neglect suspended load and we only account for bedload transport. In the case of irregularly shaped 4 Siviglia

5 cross-sections the sediment discharge Q s is computed as Q s = (s 1)gd 3 s B f Φ [ϑ(y)] dy, ϑ(y) = [h z b(y)]. Q2 (s 1)d s5 I 2 (7a,b) where s is the relative sediment density, d s5 the median sediment diameter and ϑ(y) the local Shields stress which, using equation (6), assumes the expression (7b). Finally, the dimensionless transport function Φ is evaluated in the following through the relationship proposed by Parker (199). CONSERVATIVE, PRIMITIVE AND QUASI-CONSERVATIVE FORMULATIONS Two different solution strategies can be adopted for the solution of the hyperbolic system given by (1), (5) and (3): decoupled or fully-coupled. Different roles are played in the two approaches by the term S = η x in (3), the local slope of the bottom, where η is the thalweg bottom elevation. This term is treated as a source in the decoupled framework since in the geometry of the river bed is updated only at the end of the time step. In contrast, in the coupled formulation, S is related to the first derivative of one of the three independent variables of the complete problem ( η B f x = As x A ) s x η (8) and can no longer be considered as a source term. Instead, it belongs to the flux term, thus influencing the eigenstructure of the whole system of governing equations. In the decoupled approach, equations (1) and (5) can be solved separately from equation (3) and the complete set of equations can be expressed in conservative form: U t + F x = S, (9) 5 Siviglia

6 where the unknowns U, the fluxes F and the sources S are given by U = A Q A s (1 λ p ), F = Q β Q2 A + gi 1 Q s, S = gi 2 ga(s f S ). (1) The above conservative formulation is made possible by the inclusion of the bed slope term in the sources, which is formally correct only in the decoupled framework. In the fully coupled approach, all the governing equations are solved simultaneously at each time step. Incorporating gas in the fluxes is still an open issue, leading to adoption of primitive versions of governing equations: U t + J U x = S. (11) where the Jacobian matrix J and the source term vector S are given by J = ga b 1 βq2 A 2 Q s A Q 2 βq A Q s Q ga B f A, S = gas f + g A B f A s x η. (12) The system of governing equations may be written usefully as (Toro 26): U t + F x + A V x = S, (13) where V is any choice of variables related to the unknowns U. Setting A to zero in (13) produces the conservative form while setting F to zero leads to the primitive form. The rationale for a quasi-conservative formulation is that errors are reduced considerably if all the terms that can be written in a conservative fashion are collected in the fluxes, isolating as much as possible terms that cannot be expressed in conservative form. For the 6 Siviglia

7 coupled morpho-hydrodynamic problem, the governing system can be written in the form (13), in which U and F are identical to (1), but A is a 3 3 matrix where the only non-vanishing coefficient is A 23 = ga and V = η, S = gi 2 gas f. (14) NUMERICAL SOLUTION Numerical solution of equation 13 requires the adoption of non-conservative schemes. Deriving effective non-conservative schemes from well known conservative schemes is not straightforward (Toro and Siviglia 23); choices related to the discretisation of the nonconservative terms are crucial. In this section the quasi-conservative version of the MacCormack scheme, is applied to a benchmark test. The scheme incorporates with total variation diminishing (TVD) dissipation capable of making the solution oscillation free while retaining second-order accuracy both in space and time. It is worth noting that the TVD theory applies to scalar, homogeneous one-dimensional equations and the construction of TVD methods for systems of non conservative methods is done on an empirical basis (Toro 1999). Quasi-Conservative MacCormack The conservative version of the MacCormack scheme is still widely used for solving openchannel problems of real cases (Tseng 23; Garcia-Navarro et al. 1999), because of its easy implementation with variable spatial steps and its ability to capture shock waves with second order accuracy both in time and in space. The predictor (p) and corrector (c) steps of a quasi-conservative MacCormack scheme are written as: U p i = Un i t [ ] F n i+1 F n t i x x A [ ] i+ 1 V n 2 i+1 Vi n + ts (U n i, x) (15) U c i = U n i t [ F p i x ] t [ Fp i 1 x Ap i V p 1 i ] Vp i 1 + ts (U p i, x) (16) 2 7 Siviglia

8 where the source terms are evaluated using pointwise discretisation (more sophisticated treatment of source terms can be found in Tseng (23)). By setting A = or F =, the conservative or primitive formulation can be obtained, where A represents the Jacobian matrix and V U. In the current work a spatial average (.5 (A i + A i+1 )) is used for A i The solution at the new time level n + 1, is evaluated as an average of the predictor and corrector steps: U n+1 i =.5 (U p i + Uc i) (17) NUMERICAL RESULTS AND DISCUSSION Application to the inviscid shallow water equations The proposed quasi-conservative method is assessed by considering the time-dependent non-linear inviscid shallow water equations. These describe non-linear wave propagation, with smooth and discontinuous solutions and, for sufficiently simple initial conditions (e.g. Riemann problem), analytical solutions are easily obtained in the form of rarefaction and shock waves. Moreover, this system of equations, written in the generic form (13) with vanishing source terms, can be cast in different quasi-conservative formulations, starting from the fully conservative (C) U = Y Y u, F = Y u Y u gy 2, A =, V =. (18) up to the fully primitive (P ), where U V U = Y u, F =, A = u g Y u, V = Y u. (19) 8 Siviglia

9 Three different quasi-conservative formulations are examined, namely: (QC1) U = (QC2) U = (QC3) U = Y Y u Y Y u Y Y u, F =, F =, F = Y u Y u 2 Y u 1 2 gy 2 Y u 1 2 gy 2, A =, A =, A = gy u Y u 2 Y, V =, V = Y Y u, V = Y u 1 2 u2 Y u 2, (2), (21), (22) In all the above equations, Y = Y (x, t) is the water depth, u = u(x, t), is the vertically averaged longitudinal velocity. The test solves the model equations in the domain x 5 with a regular mesh of M = 1 cells and with initial conditions Y (x, ) = Y L = 1 if x 25 Y (x, ) = Y R =.1 if x > 25 u(x, ) = x 5 (23) which physically represent a typical dam-break problem in the inviscid context. The analytical solution of this problem contains a left-facing smooth rarefaction wave and a right-facing shock wave, which is associated with the nonlinear fields u ± gy. In Figure 2 the solution obtained with the QC-MacCormack scheme at time t = 7 s with the C, P, QC1, QC2, QC3 formulations are compared with the analytical solution. The C formulation gives the most accurate result, for both the shock and the rarefaction waves. In contrast, results obtained using the P formulation clearly show that both the intensity and the position of the shock are badly predicted. The solution is sharply resolved and spurious oscillations are absent, due to the presence of the TVD correction. All the quasi-conservative solutions clearly outperform the primitive one. The errors in the computation of the shock speed and strength 9 Siviglia

10 are sharply reduced by quasi-conservative formulations but are not all equivalent. For the QC MacCormack scheme the solutions QC1 and QC3 are identical to C, i.e. the Mac- Cormack scheme gives rise to the same finite difference approximation, while QC2 solution displays larger errors. Computations with different quasi conservative schemes (e.g. Toro (26)) were carried with similar though not identical results. We can conclude that smaller error solutions are obtained using the QC formulations. The choice among the available QC formulations should be made empirically, identifying the numerical method that minimizes the errors related to the non-conservative term. Application to a schematic mobile bed case The test is based on the flume experiments of Bellal et al. (23) conducted in a steepsloped, rectangular channel. The initial condition is given by a uniform supercritical flow over an equilibrium bottom. At reference time t =, a subcritical condition is imposed at the outlet by suddenly raising the water level, while the water and sediment discharges at the inlet are kept constant giving rise to a hydraulic jump and a sediment bore. In the experimental work of Bellal et al. (23), the flume is 6.9 m long,.5 m wide and the slope is equal to.32. Uniform coarse sand with a mean diameter of 1.65 mm, and porosity of.42 are considered. Strickler bed roughness is K s = 6.6 m 1/3 s 1. Finally, the water discharge is equal to 12 l/s and the imposed downstream water surface elevation is h = 2.93 cm. The degree of solid mass conservation is used to measure the performance of a method. The total volume of sediment V mob (t) mobilized, is calculated at each time t as V mob (t) = (1 λ p ) [A s (x, ) A s (x, t)] dx. (24) L while the volume V in (t) entering the upstream (x = ) section and the volume V out (t) transported through the downstream boundary (x = L) are, respectively, V in (t) = t Q s (x =, τ) dτ V out (t) = t Q s (x = L, τ) dτ. (25) 1 Siviglia

11 V in and V out are defined by the imposed boundary conditions but V mob crucially depends on the numerical method adopted because it is obtained by integrating the solid area which results from the numerical simulation over the entire computational reach. Therefore, the percentage of mass error E mass (t) = 1 V in V out V mob V in V out (26) is taken as the performance indicator. In table 1 numerical results obtained using the Mac- Cormack scheme from two different numerical simulations adopting the QC and P formulations are given. In the QC case E mass is very small (say O(1 8 ) after 3 s) compared to the corresponding error for the P formulation (O(1 2 )). The error increases in time, as expected, but it remains about four order of magnitude smaller for the QC than for the P formulation. Sediment front propagation is also affected by the formulation of the numerical model. In figure 3, the location of the sediment front x f predicted by the QC and P formulations is plotted as a function of t and compared with the experimental data. The celerity of the front is given by the inverse of the slope of the above curves. The QC solution compares well with the experimental data, apart for an almost constant shift that may be due to the uncertainties associated to the initial condition, while, a faster downstream propagation of the front is found for the P solution. Application to a natural mobile bed environment The quasi conservative model was also applied to a reach of the Vara River that is about 1 km long and has an average width and slope of about 3 m and.1, respectively. Along the right side of the main channel, a flood plain is present, which is planned to be used as a storage area. The proposed flood defense system consists of a transverse dam with a bottom gate and a top spillway and a longitudinal embankment, which has a lateral spillway near the dam. The dimension of the gates are m, the top spillway of the dam is placed 11 Siviglia

12 at a height of about 7.5 m over the river bed. The lateral spillway is about 15 m long. During a flood event, the obstruction created by the dam reduces the velocity of the near critical flow and increases its level, inducing the formation of an hydraulic jump which moves upstream as the flow discharge increases. Once the height of the lateral spillway is overtopped the storage area is progressively flooded. The net result in term of peak reduction of the flood water volume is of the order of 2% but, as expected, the dynamics of sediment transport is shown to be strongly affected by the presence of the flood defense system. The backwater effect induced by the dam implies a decrease of the sediment transport capacity of the flow that ultimately leads to the formation of a significant deposit upstream of the dam. The grain size of the uniform sediment used in the simulations corresponds to an actual diameter d s of about 9 cm. The upstream boundary conditions consist of: i) two consecutive hydrographs with a return period of 3 years (Q max = 419m 3 /s); ii) constant bed level. At the end of the reach, the rating curve describing the effect of the dam has been experimentally determined on the physical model. In order to present the results synthetically, in each cross section the initial bed elevation has been subtracted from the bed elevation at the end of each flood event ( η), obtaining the volume deposited per unit length. The averaged thickness of the deposit has then been obtained dividing the latter by the actual width of the deposit itself. Results of the above procedure are plotted in figure 4, where the results of the numerical simulations are compared with the experimental measurements. The analysis of figure 4 shows that the deposit is organized into three distinct deposition areas, due to the geometrical characteristics of the natural sections considered. Comparing the top and bottom panels it can be seen that the aggradation process continues and no equilibrium conditions is reached after one event, even if the total amount of sediment deposited is quite high. The qualitative and quantitative agreement between the numerical predictions and the experimental data is more than satisfactory. 12 Siviglia

13 CONCLUSIONS The role of different non-conservative formulations for the coupled Saint-Venant-Exner model has been investigated. A quasi-conservative formulation of the morphodynamic model was developed and assessed making use of the MacCormack scheme. Numerical tests for the inviscid shallow water equations reveal that different QC formulations gives rise to different results. Particular choices for the evaluation of the non conservative terms are able to reduce considerably the mass conservation error. Comparison between numerical results and the exact solution in this simplified test case show that QC formulations outperform the fullyprimitive one. A quasi-conservative formulation of the coupled Saint-Venant-Exner model is proposed. Numerical simulations show that such formulation leads to an impressive improvement in terms of mass conservation with respect to the fully primitive formulation. Finally, applications of the model to the case of the Vara River, which involves a complex natural geometry and severe flood conditions, reveal its robustness. A good agreement between the numerical results and the experimental observations made in laboratory making use of a physical mobile-bed model has been found. 13 Siviglia

14 APPENDIX I. REFERENCES Bellal, M., Spinewine, C., and Zech, Y. (23). Morphological evolution of steep-sloped river beds in the presence of a hydraulic jump.. Experimental study, paper presented at XXX IAHR congress Cao, Z., Day, R., and Egashira, A. (22). Coupled and decoupled numerical modeling of flow and morphological evolution in alluvial rivers. Journal of Hydraulic Engineering, ASCE, 128(3), Colombini, M. and Stocchino, A. (25). Coupling or decoupling bed and flow dynamics: fast and slow sediment waves at high froude numbers. Physics of Fluids, 17(3). Correia, L., Krishnappan, B., and Graf, W. (1992). Fully coupled unsteady mobile boundary flow model. Journal of Hydraulic Engineering, ASCE, 118(3), Cui, Y., Parker, G., and Paola, C. (1996). Numerical simulation of aggradation and downstream fining. Journal of Hydraulic Research, 34(2), Cunge, J. A., Holly, F. M., and Verwey, A. (1994). Practical Aspects of Computational River Hydraulics. Pitman Publishing Ltd. Reprinted by the University of Iowa. Engelund, F. (1964). Book of abstracts. Report No. 6, University of Denmark. Basic Research. Garcia-Navarro, P., Fras, A., and Villanueva, I. (1999). Dam-break flow simulation: some results for one-dimensional models of real cases. Journal of Hydrology, 216, Holly, F. and Rahuel, J. (199). New numerical/physical framework for mobile bed modelling, part 1: Numerical and physical principles. Journal of Hydraulic Research, 28(4), Hou, T. and LeFloch, P. (1994). Why Non Conservative Schemes Converge to the Wrong Solutions: Error Analysis. Math. of Comput., 62, Lanzoni, S., Siviglia, A., and Seminara, G. (26). Long waves in erodible channels and morphodynamic influence. Water Resources Research, 42. W6D17, doi:1.129/26wr Siviglia

15 Lax, P. and Wendroff, B. (196). Systems of Conservation Laws. Comm. Pure Appl. Math., 13, Lyn, D. and Altinakar, M. (22). St.venant exner equations for near-critical and transcritical flows. Journal of Hydraulic Engineering, ASCE, 128(6), Parker, G. (199). Surface-based bedload transport relation for gravel rivers. J. Hudraul. Res., 28(4), Sieben, J. (1999). A theoretical analysis of discontinuous flows with mobile bed. Journal of Hydraulic Research, 37(2), Singh, A., Kothyari, U., and Ranga Raju, K. (24). Rapidly varying transient flows in alluvial rivers. Journal of Hydraulic Research, 42(5), Toro, E. (1999). Riemann Solvers and Numerical Methods for Fluid Dynamics, Second Edition. Springer Verlag. Toro, E. (26). Riemann solvers with evolved initial conditions. International Journal for Numerical Methods in Fluids. DOI: 1.12/fld Toro, E. and Siviglia, A. (23). Price: Primitive centred schemes for hyperbolic system of equations. International Journal for Numerical Methods in Fluids, 42, Tseng, M. (23). The improved surface gradient method for flow simulation in variable bed topography channel using tvd-maccormack scheme. International Journal for Numerical Methods in Fluids, 43, DOI:1.12/fld.65. Wu, W., Vieira, D., and Wang, S. (24). One-dimensional numerical model for nonuniform sediment transport under unsteady flows in channel networks. Journal of Hydraulic Engineering, 13(9), Siviglia

16 APPENDIX II. NOTATION The following symbols are used in this paper: A = cross sectional flow area (m 2 ); A s = bed material area (m 2 ); b = river width (m); B f = width of the active layer (m); C = Chezy coefficient; CF L = Courant number; d s = sediment diameter (m); F = Froude number; F = vector of fluxes; g = acceleration due to gravity (m/s 2 ) ; h = water surface elevation (m); I 1 = first moment of the wetted cross section with respect to the free surface (m 3 ); I 2 = spatial variation of the first moment (m 2 ); J = Jacobian matrix; K s = roughness coefficient (Strickler) (m 1/3 /s); Q, Q s = water and solid discharge (m 3 /s); λ p = porosity ; t = time (s); x, y, z = spatial coordinates ; R = hydraulic radius (m); s = relative density; S = vector of source terms; S f = friction factor ; S = local slope ; u = velocity (m/s); U = vector of unknowns; 16 Siviglia

17 x f = location of the sediment front (m); Y = water depth (m); z b = bottom elevation (m); x, t = spatial and time steps (m), (s); η = thalweg bottom elevation (m); γ = solid discharge/water discharge ratio; Φ = non dimensional sediment discharge function; θ = Shields parameter. 17 Siviglia

18 List of Tables 1 Percentage of mass error produced by the Quasi-Conservative (QC) and Primitive (P) formulations Siviglia

19 time [s] QC E mass % P E mass % TABLE 1. Percentage of mass error produced by the Quasi-Conservative (QC) and Primitive (P) formulations. 19 Siviglia

20 List of Figures 1 Cross sectional geometry Conservative, Quasi-Conservative(1-2-3) and Fully Primitive solutions using the MacCormack scheme (symbol) and exact solution (line). Results at time t = 7. s for mesh M = 1 cells and CFL = Comparison of front position for quasi-conservative and primitive solutions Bottom variations at the end of the first (top) and second (down) hydrograph having return period T = 3 years Siviglia

21 FIG. 1. Cross sectional geometry 21 Siviglia

22 1 C Q-C-1 Q-C-2 Q-C-3 P.75 Depth h [m] x [m] FIG. 2. Conservative, Quasi-Conservative(1-2-3) and Fully Primitive solutions using the MacCormack scheme (symbol) and exact solution (line). Results at time t = 7. s for mesh M = 1 cells and CFL = Siviglia

23 t [s] Q-C P exp x f [m] FIG. 3. Comparison of front position for quasi-conservative and primitive solutions 23 Siviglia

24 exp num x (m) exp num x (m) ηav (m) ηav (m) FIG. 4. Bottom variations at the end of the first (top) and second (down) hydrograph having return period T = 3 years 24 Siviglia