Combined characteristics and finite volume methods for sediment transport and bed morphology in...

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Mathematics and Computers in Simulation 81 (2011) 2073–2086

Combined characteristics and finite volume methods for sedimenttransport and bed morphology in surface water flows

Fayssal Benkhaldoun a,∗, Mohammed Seaïd b

a LAGA, Université Paris 13, 99 Av J.B. Clement, 93430 Villetaneuse, Paris, Franceb School of Engineering and Computing Sciences, University of Durham, South Road, DH1 3LE, UK

Received 30 December 2009; received in revised form 18 June 2010; accepted 8 December 2010Available online 31 December 2010

bstract

We propose a new numerical method for solving the equations of coupled sediment transport and bed morphology by free-surfaceater flows. The mathematical formulation of these models consists of the shallow water equations for the hydraulics, an advection

quation for the transport of sediment species, and an Exner equation for the bedload transport. The coupled problem forms ane-dimensional hyperbolic system of conservation laws with geometric source terms. The proposed numerical method combineshe method of characteristics with a finite volume discretization of the system. The combined method is simple to implement andccurately resolves the governing equations without relying on Riemann problem solvers. Numerical results are presented for severalest examples on sediment transport and bed morphology by free-surface water flows.

2011 IMACS. Published by Elsevier B.V. All rights reserved.

eywords: Shallow water equations; Sediment transport; Bed morphology; Finite volume scheme; Method of characteristics

. Introduction

The main concern of morphodynamics is to determine the evolution of bed levels for hydrodynamic systems suchs rivers, estuaries, bays and other nearshore regions where water flows interact with the bedload geometry. Examplesf applications include among others, beach profile changes due to severe wave climates, seabed response to dredgingrocedures or imposed structures, and harbour siltation. The ability to design numerical methods able to predicthe morphodynamic evolution of the coastal seabed has a clear mathematical and engineering relevances, compare13,15,14,3] among others. In practice, morphodynamic problems involve coupling between a hydrodynamic model,hich provides a description of the flow field leading to a specification of local sediment transport rates, and an equation

or bed level change which expresses the conservative balance of sediment volume and its continual redistribution withime. In the current study, the hydrodynamic model is described by the shallow water equations, the suspended sediments modelled using an advection equation accounting for deposition and erosion effects, and the transport of the bedload
s modelled by the Exner equation. The coupled models form a hyperbolic system of conservation laws with sourceerms. It is well known that the solutions of these systems may present steep fronts and even shock discontinuities,hich need to be resolved accurately in applications and often cause severe numerical difficulties, see for example [5,3].
∗ Corresponding author. Tel.: +33 1 49 40 36 15; fax: +33 1 49 40 35 68.E-mail address: [email protected] (F. Benkhaldoun).

378-4754/$36.00 © 2011 IMACS. Published by Elsevier B.V. All rights reserved.doi:10.1016/j.matcom.2010.12.025

dx.doi.org/10.1016/j.matcom.2010.12.025

mailto:[email protected]

dx.doi.org/10.1016/j.matcom.2010.12.025

2074 F. Benkhaldoun, M. Seaïd / Mathematics and Computers in Simulation 81 (2011) 2073–2086

The object of this study is to devise a numerical approach able to accurately approximate solution to morphodynamicproblems. Our aim is to develop a family of finite volume methods that incorporate techniques from the method ofcharacteristics into the reconstruction of numerical fluxes. Our main goal is to present a class of numerical methodsthat are simple, easy to implement, and accurately solves the sediment transport equations without relying on Riemannproblem solvers. The method has been recently investigated in [2] for numerical solution of shallow water equations onfixed bed and it is adapted in the current work for the numerical solution of morphodynamic problems. The proposedfinite volume scheme belongs to the class of methods that employ only physical fluxes and averaged states in theirformulations. It can be interpreted as a predictor–corrector scheme. In the corrector stage, the considered equations areintegrated over an Eulerian time–space control volume whereas in predictor stage, the sediment transport equationsare rewritten in a non-conservative form and integrated along the characteristics defined by the water velocity. Themain features of such a finite volume scheme are on one hand, the capability to satisfy the conservation propertyresulting in numerical solutions free from spurious oscillations in significant morphodynamic situations, and on theother hand the achievement of strong stability for simulations of slowly varying bedload as well as rapidly varyingflows containing also shocks or discontinuities. These features are verified using several test examples of the sedimenttransport problems. Results presented in this paper show high resolution of the proposed finite volume schemes andpermit the straightforward application of the method to more complex, physically based sediment transport models.

In this paper, first the governing equations for the sediment transport problems are formulated. Thereafter, thecombined finite volume characteristics method employed to solve the sediment transport problems is presented. Afterexperiments with the proposed approach for a variety of morphodynamic examples, accuracy and efficiency of thecombined characteristics and finite volume schemes are discussed. Concluding remarks end the paper.

2. Mathematical formulation

We assume that the flow is almost horizontal, the vertical component of acceleration is vanishingly small, thepressure is taken to be hydrostatic, the free-surface gravity waves are long with respect to the mean flow depth andwave amplitude, and the water-species mixture is vertically homogeneous and non-reactive. The governing equationsare obtained by balancing the net inflow of mass, momentum and species through boundaries of a control volumeduring an infinitesimal time interval while accounting for the accumulation of mass, resultant forces and species withinthe control volume, compare for example [1]. Thus, the equations for mass conservation and momentum flux balanceare given by

∂h

∂t+ ∂(hu)

∂x= 0,

∂(hu)

∂t+ ∂

∂x

(hu2 + 1

2gh2)

= −gh∂B

∂x− τb

ρw

+ τω

ρw

,

(1)

where u is the depth-averaged water velocity, h is the water depth, B is the bottom topography, g is the gravitationalacceleration, ρw is the water density, τb and τω are respectively, the bed shear stress and the shear of the blowing winddefined by the water and wind velocities as

τb = ρwCbu|u|, τω = ρwCωω|ω|, (2)

where ω is the velocity of wind at 10 m above water surface, Cb and Cw are respectively, the bed friction coefficientand the coefficient of wind friction which may be either constant or estimated [7],

Cb = n2b

h4/3 , Cw = ρa (0.75 + 0.067|ω|) × 10−3,

with nb being the Manning roughness coefficient at the bed and ρa is the air density. The equation for mass conservation
of species is modelled by
∂(hc)

∂t+ ∂ (huc)

∂x= E − D, (3)

wta

w

w[

w

wd

wo

w

w

wtcA

wetmpud

F. Benkhaldoun, M. Seaïd / Mathematics and Computers in Simulation 81 (2011) 2073–2086 2075

here c is the depth-averaged concentration of suspended sediment, E and D represent the erosion and depositionerms in upward and downward directions, respectively. In the current study, the erosion–deposition term is calculateds

E − D = ξωs (c∗ − c) , (4)

here c∗ is the sediment concentration close to the bed defined by Van Rijn [19],

c∗ = 0.015d50τ

δbd0.3∗, (5)

ith d50 is the median diameter of the sediment and δb is a reference level fixed in our computations to 0.05 h, compare13]. The excess bed shear stress τ is defined as

τ = u2∗ − u2c

u2c

, (6)

here the shear velocity u∗ and the critical bed shear velocity uc for the sediment are given by

u2∗ = |τb|

ρw

, uc =√

a

(d50

10

)b

,

ith a and b are constants depending on the median diameter of the sediment particles. In (5), the particle-size diameter∗ is defined as

d∗ = d50

(gρs − ρw

ρw

ν2)1/3

, (7)

here ρs is sediment density. In (4), ωs is the settling velocity of sediment particles and ξ is the recovery coefficientf the suspended sediment defined following [8] as

ωs = ν

d50

(√2.5 + 1.2d2∗ − 5

)1.5

, ξ = min

(2,

1 − p

c

),

ith ν is the water kinematic viscosity. To update the bedload, we used the Exner equation given by

∂B

∂t+ ∂Q

∂x= −E − D

1 − p, (8)

here the sediment discharge Q can be evaluated by the simple formula introduced in [10]

Q = As

1 − pu|u|m, (9)

ith p is the sediment porosity assumed to be constant, m and As are coefficients usually obtained from experimentsaking into account the grain diameter and the kinematic viscosity of the sediment. In practice, the values of theoefficient As are between 0 and 1 depending on the interaction between the sediment transport and the water flow.nother formula frequently used for the sediment discharge Q is given in [12]

Q =8√

g(s − 1)d350

1 − p

(n2

bu2

(s − 1)d50h1/3 − 0.047

)3/2

, (10)

here the grain specific gravity s = ρs/ρw. Note that most of existing formulations for sediment transport models arempirical to differing extents and have been derived from experiments and measured data. It should be stressed thathe method described in this paper can be applied to other forms of sediment transport fluxes without major conceptual

odifications. For instance, the bedload sediment transport functions proposed in [14,15] can also be handled by theroposed finite volume method. Notice that, the parameters nb, d50, ξ, ρs, a, and b appeared in above equations areser-defined constants in the sediment transport model. In practice, the selection of these coefficients are problemependent and their discussion is postponed for Section 4 where numerical examples will be presented.


3. Combined characteristics and finite volume methods

In a wide spectrum of practical applications, the sediment transport of the bed occurs on a transport time scale muchlonger than the flow time scale, see for example [9,5,3]. It is therefore desirable to construct numerical schemes thatpreserve stability for all time scales. In the present work, we adapt the quasi-steady approach in which the coupledEqs. (1) and (3) are solved first followed by Exner Eq. (8) to update the bedload. Let us rewrite Eqs. (1) and (3) in acompact form as

∂W∂t

+ ∂F(W)

∂x= S(W), (11)

where W is the vector of conserved variables, F is the flux function and S is the source term defined as

W =

⎛⎜⎝

h

hu

hc

⎞⎟⎠ , F(W) =

⎛⎜⎜⎝

hu

hu2 + 1

2gh2

huc

⎞⎟⎟⎠ , S(W) =

⎛⎜⎜⎝

0

−gh∂B

∂x− τb

ρw

+ τω

ρw

E − D

⎞⎟⎟⎠ .

Once the solution of the system (11) is calculated, the bedload is obtained by solving Eq. (8). This procedure isappropriate for morphodynamical models where the sediment flux does not depend on the bedload and it is widelyused in engineering applications, compare [9,3] among others.

Integrating the system (11) with respect to time and space over the time–space control domain [tn, tn+1] × [xi−1/2,xi+1/2], one obtains the following discrete problem

Wn+1i = Wn

i − t

x

(Fn

i+1/2 − Fni−1/2

)+ tSn

i , (12)

where Wni is the space average of the solution W in the control volume [xi−1/2, xi+1/2] at time tn, Fn

i±1/2 = F(Wni±1/2)

are the numerical fluxes at x = xi±1/2 and time tn. In (12), Sni is a difference notation for the discretized source terms

in (11). The spatial discretization of Eq. (12) is complete when a numerical construction of the intermediate statesWn

i±1/2 and source terms Sni is chosen. In general, the construction of numerical fluxes requires a solution of Riemann

problems at the interfaces xi±1/2. From a computational viewpoint, this procedure is very demanding and may restrictthe application of the method for which Riemann solutions are not available.

Our objective in the present work is to present a class of finite volume characteristics (FVC) methods that are simple,easy to implement, and accurately solves Eqs. (11) and (8) without relying on a Riemann problem solver. This objectiveis reached by integrating twice the system (11) in time and space. In the first integration, Eq. (11) is integrated overan Eulerian time–space control volume. We term this step by corrector stage applied to the conservation equations. Inthe second integration, the equations are rewritten in a non-conservative form and integrated along the characteristicsdefined by the water velocity. This step is called predictor stage and used to calculate the numerical fluxes required inthe corrector stage. In what follows we present the formulations for the two stages.

3.1. Discretization of the flux gradients

We reconstruct the numerical fluxes Fni±1/2 using the method of characteristics applied to the system (11) rewritten

in the physical variables as

∂U∂t

+ u∂U∂x

= Q(U), (13)

w

Ttia

Nmts[

ITti

O

wcd

waXa

rsi


here

U =

⎛⎜⎝

h

u

c

⎞⎟⎠ , Q =

⎛⎜⎜⎜⎜⎝

−h∂xu

−g∂

∂x(h + Z) − τb

hρw

+ τω

hρw

E − D

h

⎞⎟⎟⎟⎟⎠ . (14)

he fundamental idea of the method of characteristics is to impose a regular grid at the new time level and to backtrackhe flow trajectories to the previous time level. At the old time level, the quantities that are needed are evaluated bynterpolation from their known values on a regular grid, for more discussions we refer the reader to Refs. [16–18]mong others. Thus, the characteristic curves associated with Eq. (13) are solutions of the initial-value problem

dXi+1/2(t)

dτ= ui+1/2(t, Xi+1/2(t)), t ∈

[tn, tn + t

2

],

Xi+1/2

(tn + t

2

)= xi+1/2. (15)

ote that Xi+1/2(t) is the departure point at time t of a particle that will arrive at point xi+1/2 in time tn + t/2. Theethod of characteristics does not follow the flow particles forward in time, as the Lagrangian schemes do, instead it

races backward the position at time tn of particles that will reach the points of a fixed mesh at time tn + t/2. By doingo, the method avoids the grid distortion difficulties that the conventional Lagrangian schemes have, see for instance18,16]. The solutions of (15) can be expressed as

Xi+1/2(tn) = xi+1/2 −∫ tn+t/2

tn

ui+1/2(Xi+1/2(t))dt,

= xi+1/2 − αi+1/2.

(16)

t is worth remarking that the departure points in (15) are calculated in the interval [tn, tn + t/2] instead of [tn, tn+1].his is motivated by the idea of reconstructing a predictor–corrector scheme where the predictor stage is computed at

he fractional time tn + t/2 completed by a corrector stage computed at the end time tn+1. This fractional time steppings also supported by the analysis reported in [2].

The integral in (16) can be calculated using the simple iteration

α(0)i+1/2 = t

2u(tn, xi+1/2),

α(m+1)i+1/2 = t

2u(tn, xi+1/2 − α

(m)i+1/2), m = 0, 1, . . .

(17)

nce the characteristic curves Xi+1/2(tn) are known, a solution at the cell interface xi+1/2 is reconstructed as

Uni+1/2 = U(tn + t/2, xi+1/2) = U(tn, Xi+1/2(tn)), (18)

here U(tn, Xi+1/2(tn)) is the solution at the characteristic foot computed by interpolation from the gridpoints of theontrol volume where the departure point resides. Thus, applied to Eq. (13), the predictor stage in the FVC method isefined by

Uni+1/2 = Un

i+1/2 − t

2Qn

i+1/2, (19)

here Qni+1/2 is a consistent discretization of the source term Q in (13) and Un

i+1/2 = U(tn, Xi+1/2(tn)) are the solutionst the characteristic foot computed by interpolation from the gridpoints of the control volume where the departure pointi+1/2(tn) belongs. The corrector stage in the FVC method is given by (12) where the intermediate states Wn

i±1/2 areccordingly recovered from the characteristic solutions Un

i±1/2 in (19).
It should be stressed that in general, the method of characteristics fails to conserve mass, compare [16] and further
eferences are therein. However, in our FVC method the mass lost in the predictor step for calculating the intermediatetages Un

i±1/2 will be recovered in the corrector step (12). It is evident from the formulation (12) that the FVC methods mass conservative.


Fig. 1. Numerical and analytical solutions for the tidal wave problem at t = 9117.5 s. Water free-surface (left) Water height (middle) and water
velocity (right).
3.2. Discretization of the source terms

The treatment of the source terms in the shallow water equations presents a challenge in many numerical methods,compare for example [4–6]. In our FVC scheme, the discretization of the source terms Qn

i+1/2 in the predictor stage(19) is performed using an upwind method as

hni+1/2 = hn

i+1/2 − t

2xhn

i+1/2(uni+1 − un

i ),

uni+1/2 = un

i+1/2 − t

2xg((hn + B)i+1 − (hn + B)i

)− t

2

Tni

hni

,

cni+1/2 = cn

i+1/2 − t

2xcni+1/2(un

i+1 − uni ) + t

2

Eni

hni

,

(20)

where Eni denotes the discretization of the erosion–deposition term E − D and Tn

i represents the discretization of bedand surface stresses defined as

Tni = n2

b(hn

i

) 43

uni

∣∣uni

∣∣− Cωω |ω| . (21)

The source term approximation Sni in the corrector stage is reconstructed such that the C-property is satisfied, see [2]

for the details. Hence, the corrector stage in the FVC method gives

hn+1i = hn

i − t

x

((hu)ni+1/2 − (hu)ni−1/2

),

(hu)n+1i = (hu)ni − t

x

((hu2 + 1

2gh2)

n

i+1/2− (hu2 + 1

2gh2)

n

i−1/2

)

− t

8xg(hn

i+1 + 2hni + hn

i−1

)(Bi+1 − Bi) − tTn

i ,

(hc)n+1i = (hc)ni − t

x

((huc)ni+1/2 − (huc)ni−1/2

)+ tEn

i .

(22)


Ac

wFc

SS

S

SS

it

4

aimAC

wbd

Fig. 2. Initial conditions for the bedload transport problem. Bedload (left) water height (middle) and water velocity (right).

detailed analysis of convergence and stability has been presented in [2] for nonlinear scalar problems. The bedloadan be recovered using the upwinding techniques as

Bn+1i = Bn

i − t

x(Qn

i+1/2 − Qni−1/2) − t

1 − pEn

i , (23)

here Qni±1/2 = Q(hn

i±1/2, uni±1/2) is calculated using either Eq. (9) or Eq. (10). In summary, the implementation of

VC algorithm to solve the sediment transport Eq. (11) is carried out in the following steps. Given (hni , q

ni , c

ni ), we

ompute (hn+1i , qn+1

i , cn+1i ) via:

tep 1. Compute the departure points Xi+1/2(tn) using the iterations (17).tep 2. Compute the approximations hn

i+1/2 = h(tn, Xi+1/2(tn)), uni+1/2 = u(tn, Xi+1/2(tn)) and cn

i+1/2 =c(tn, Xi+1/2(tn)) employing an interpolation procedure.

tep 3. Evaluate the intermediate states hni+1/2, un

i+1/2 and cni+1/2 from the predictor stage (19).

tep 4. Update the solutions hn+1i , qn+1

i and cn+1i using the corrector stage (12).

tep 5. Update the new bedload Bn+1i using Eq. (23).

Note that other interpolation procedures in Step 2 can also be applied. In our simulations we have used a linearnterpolation since for this type of interpolations the obtained solution remains monotone and the FVC method preserveshe exact C-property at the machine precision, compare [2].

. Numerical results

In this section, the accuracy and performance of the proposed FVC method are tested. Four test examples are used:n accuracy test problem, a tidal flow problem, a bedload transport problem, and simulation of sediment transportn an oscillating lake. The former first tests have analytical solutions that can be used to quantify error in the FVC

ethod while the latter is used to qualify FVC results for more complicated sediment transport and free-surface flows.s with all explicit time stepping methods the theoretical maximum stable time step t is specified according to theourantFriedrichsLewy (CFL) condition

t = Crx

maxk=1,2,3(∣∣λn

k

∣∣) , (24)

here λk is the eigenvalues of the sediment transport system (11) calculated for example in [4,9] and Cr is a constant to
e chosen less than unity. Note that these eigenvalues correspond to the coupled system (11) and (8) with the sedimentischarge (9) which has also been used for the sediment transport flux (10).
In all our simulations, the fixed Courant number Cr = 0.8 is used and the time step is varied according to (24).


Table 1Errors in the water height and water discharge for the accuracy problem at t = 0.1.

Error in h Error in hu

N L1-error L2-error L∞-error L1-error L2-error L∞-error

50 0.811E−04 0.957E−04 0.222E−03 0.130E−02 0.161E−02 0.292E−02100 0.217E−04 0.276E−04 0.731E−04 0.383E−03 0.531E−03 0.984E−03200 0.553E−05 0.742E−05 0.210E−04 0.102E−03 0.153E−03 0.324E−03400 0.138E−05 0.186E−05 0.564E−05 0.258E−04 0.412E−04 0.932E−04
800 0.337E−06 0.464E−06 0.141E−05 0.646E−05 0.104E−05 0.250E−04
4.1. Accuracy problem

To check the second order accuracy of the FVC scheme we consider the problem of unsteady flow over a sinusoidalbump proposed in [20]. Here we solve the canonical shallow water equations on a fixed bed profile defined by

B(x) = sin2(πx), x ∈ [0, 1],

equipped with the following initial conditions

h(x, 0) = 5 + ecos(2πx), (hu)(x, 0) = sin(cos(2πx)),

and periodic boundary conditions. This problem cannot be solved analytically and therefore a numerical solutioncomputed in a very fine mesh with 25,600 gridpoints, is adopted as the reference solution. This reference solution isused to quantify the results obtained by the FVC method. We compute the L∞-, L1- and L2-error norms as

L∞ = max1≤i≤N

∣∣eni

∣∣ , L1 =N∑

i=1

∣∣eni

∣∣x, L2 =√√√√ N∑

i=1

∣∣eni

∣∣2x, (25)

where eni = wn

i − w(xi, tn) is the error between the numerical solution, wi, and the reference solution, w(xi, tn), attime tn and gridpoint xi. Here, N refers to the total number of gridpoints and the results are displayed at time t = 0.1.

In Table 1 we summarize the error-norms for the water height and the water discharge using different meshes. Itis clear that increasing the number of gridpoinds in the space domain results in a decrease of all error-norms. Similar
results, not reported here, are obtained focusing the attention on the water velocity. As expected the FVC method showsa second-order accuracy for this nonlinear example of shallow water equations.
Fig. 3. Results for the bedload transport problem at t = 238 s using As = 1. Bedload (left) water height (middle) and water velocity (right).


F

4

TL

T

a

tnut

TR

Q

ρ

ρ

ρ

gb

ig. 4. Results for the bedload transport problem at t = 238, 000 s using As = 0.001. Bedload (left) water height (middle) and water velocity (right).

.2. Tidal wave problem

In this example we solve the shallow water equations on fixed bed and without accounting for suspended sediment.his example was used in [6], in which an asymptotic analytical solution was obtained. Here, a channel with length= 14 km is considered with a fixed bed defined by

B(x) = 10 + 40x

L+ 10 sin

(π

(4x

L− 1

2

)).

he bottom friction and wind stresses were neglected in this test. If we take the initial and boundary conditions as

h(x, 0) = 60.5 − B(x), u(x, 0) = 0,

h(0, t) = 64.5 − 4 sin

(π

(4t

86400+ 1

2

)), u(L, t) = 0,

n analytical solution, based on the asymptotic analysis, can be given by [6]

h(x, t) = 64.5 − Z(x) − 4 sin

(π

(4t

86, 400+ 1

2

)),

u(x, t) = (x − L) π

5400h(x, t)cos

(π

(4t

86, 400+ 1

2

)).

Fig. 1 presents the numerical and analytical solutions for the water height and velocity at the simulation time= 9117.5 s using x = 140 m. It is clear the good agreement between the asymptotic analytical solution and the
umerical results obtained by the FVC method using the coarse mesh. The FVC method performs well for thisnsteady shallow water problem and produces accurate solutions without requiring special treatment of the sourceerms or complicated upwind discretization of the gradient fluxes as in [6] among others.
able 2eference parameters used for sediment transport in the oscillating lake.

uantity Reference value Quantity Reference value

w 1000 kg/m3 ν 1.2 × 10−6 m2/s

s 1650 kg/m3 d50 0.23 × 10−3 m

a 1200 kg/m3 nb 0.0157 s/m1/3

9.81 m/s2 p 0.411/32 a 0.000841


Fig. 5. Results for sediment transport in the oscillating lake with erosion–deposition effects at calm situation. Solid lines correspond to initial
conditions.
4.3. Bedload transport problem

We consider the test example of non-erodible bedload transport studied in [9]. The channel is of length 1000 m andthe initial bed is defined as

B(0, x) =⎧⎨⎩ sin2

((x − 300) π

200

), if 300 ≤ x ≤ 500,

0, elsewhere.

Initial water level and initial velocity are given as

h(0, x) = 10 m − B(0, x) and u(0, x) = 10 m2/s

h(0, x).

No suspended sediment is accounted for in this example, the parameter m = 3, the porosity p = 0.4 and two valuesAs = 0.001 and As = 1 are used in the sediment transport flux (9). To obtain realistic initial data for this test problem,we first run the FVC scheme keeping the channel bed fixed until a steady state solution is reached. The computed
solutions h and u, shown in Fig. 2, are then used for moving bed simulations. In all our computations, the space domainis discretized in 100 gridpoints. For comparison reasons, we have included the analytical solutions calculated basedon an asymptotic analysis reported in [9].


Ft

ttcpofNa

4

d

ig. 6. Results for sediment transport in the oscillating lake with erosion–deposition effects for blowing wind from the east. Solid lines correspondo initial conditions.

In Fig. 3 we present the numerical results for the test case using As = 1 at time t = 238 s. The results obtained for theest case using As = 0.001 at time t = 238, 000 s are illustrated in Fig. 4. As can be seen, for large values of As, fasterransport has been detected in the considered example. In both cases, a reasonable agreement is observed between theomputed results and the analytical solutions. The proposed FVC scheme performs well for this sediment transportroblem since it does not diffuse the moving bed and no spurious oscillations have been observed when the water flowsver the bed. Furthermore, the obtained results using the FVC scheme compare favorably with those reported in [9,10]or the same sediment transport problem using numerical methods based on solving the Riemann problem solvers.ote that the performance of the proposed FVC method is very attractive since the computed solutions remain stable

nd oscillation-free even for coarse grids without solving nonlinear systems or Riemann problems.

.4. Sediment transport in an oscillating lake

Our final test example consists on solving sediment transport in an oscillating lake accounting for erosion andeposition effects. The lake is of length 1000 m and the initial conditions are given by

B(x, 0) = 3

2

(1 − cos

(π

x − 500

500

)), u(x, 0) = 0,

h(x, 0) = max

[0, 4 − B(x, 0) + 0.3

(sin

(500 − x

250

)+ 0.3 max (0, −5 + B(x, 0))

)],


Bed

B a

nd fr

ee−

surf

ace

B+

h

Fig. 7. Results for sediment transport in the oscillating lake with erosion–deposition effects for blowing wind from the west. Solid lines correspondto initial conditions.

c(0, x) ={

0.6, if x ≤ 300,

0, elsewhere.

The bedload evolves according to Eq. (10) and the selected values for the evaluation of the present method aresummarized in Table 2. Depending on the wind conditions, three situations are simulated namely:

(i) Calm situation corresponding to (ω = 0 m/s).(ii) Wind blowing from the east corresponding to (ω = 3.3 m/s).

(iii) Wind blowing from the west corresponding to (ω = 3.3 m/s).

The computational domain is discretized in 200 gridpoints and the computed water free-surface, bedload andsediment concentration are illustrated at three different instants t = 75 s, t = 150 s and t = 300 s.

In Fig. 5 we present numerical results obtained using calm wind conditions. Those obtained for the wind blowingfrom the east and the wind blowing from the west are displayed in Figs. 6 and 7, respectively. In these figures, wealso show the initial conditions depicted with solid lines. It is clear that using the conditions for the sediment transportand the considered wind situations, the flow exhibits a hydraulic jump with different order of magnitudes near thedownstream of the lake. At the beginning of simulation time, the water flow enters the lake from the eastern boundary
and flows towards the eastern exit of the lake. At later time, due to tidal waves, the water flow changes the directionpointing towards the eastern coast of the lake. A deposition of the suspended sediment on the bottom bed is alsodetected on the eastern exit of the lake for all the test case with the wind blowing from the east.

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The effects of wind conditions are obviously observed in the distributions of the flow field and sediment concentrationresented in Figs. 6 and 7. A faster decay rate is clearly seen for suspended sediment subject to the wind blowing fromhe west. For the considered flow and wind conditions, the deposition force has been seen to play a very weak roleor driven the free-surface water in the lake which results in a thinner layer of sediment accumulated on the lake bed.he proposed FVC method performs very satisfactorily for this nonlinear coupled problem since it does not diffuse

he moving fronts and no spurious oscillations have been detected near steep gradients of the flow field and sedimentoncentration in the computational domain.

. Conclusions

In this paper, we have proposed a new finite volume method for solving the coupled system of sediment transporty free-surface water flows. The proposed finite volume method consists of two stages which can be viewed as aredictor–corrector procedure. In the first stage, the scheme reconstructs the numerical fluxes using the method ofharacteristics. This stage results in an upwind discretization of the characteristic variables and avoids the Riemannroblem solvers. In the second stage, the solution is updated using a special treatment of the bed bottom in order to obtainwell-balanced discretization of the flux gradient and the source term. The proposed method does not require eitheronlinear solution or special front tracking techniques. Furthermore, it has strong applicability to various conservativeumerical schemes as shown in the numerical results.

The methods have been numerically examined for several test examples in morphodynamic problems without andith suspended sediment. We have also compared the numerical results to analytical solutions. In the considered testroblems, the proposed method has exhibited accurate prediction of both, the free-surface and the bedload with correctonservation property, and stable representation of free surface response to the movable bed and suspended sediment.he results make it promising to be applicable also to real situations where, beyond the many sources of complexity,

here is a more severe demand for accuracy in predicting the morphological evolution, which must be performed forong simulation times.

cknowledgments

This work was partly performed while the second author was a visiting professor at LAGA, Université Paris 13.inancial support provided by LAGA, Université Paris 13 is gratefully acknowledged.

eferences

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