A simple multi-layer finite volume solver for density-driven shallow water flows

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Available online at www.sciencedirect.com

Mathematics and Computers in Simulation 99 (2014) 170–189

A simple multi-layer finite volume solver for density-drivenshallow water flows

Fayssal Benkhaldoun a, Saida Sari a,∗, Mohammed Seaid b

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

Received 5 November 2011; received in revised form 17 November 2012; accepted 3 April 2013Available online 20 May 2013

Abstract

A simple solver is proposed for the numerical solution of density-driven multi-layer shallow water flows. The governing equationsconsist on coupling the multi-layer shallow water equations for the hydraulic variables with suspended sediment transport equationsfor the concentration variables. The layers can be formed in the shallow water model based on the variation of water densitywhich may depend on the water temperature and salinity. At each time step, the method consists of two stages to update thenumerical solution. In the first stage, the multi-layer shallow water equations are rewritten in a non-conservative form and theintermediate solutions are calculated using the modified method of characteristics. In the second stage, the numerical fluxes arereconstructed from the intermediate solutions in the first stage and used in the conservative form of the multi-layer shallow waterequations. The proposed method avoids Riemann problem solvers and it is suitable for multi-layer shallow water equations on non-flattopography. Several numerical results are presented to illustrate the performance of the proposed finite volume method. The computedresults confirm its capability to solve multi-layer shallow water equations for density-driven flows over flat and non-flat bottomtopography.© 2013 IMACS. Published by Elsevier B.V. All rights reserved.

Keywords: Multi-layer shallow water equations; Density-driven flows; Finite volume method; Modified method of characteristics

1. Introduction

Mathematical modelling of water flows in the hydraulics and oceanic systems is based on the formulationand solution of the appropriate equations of continuity and motion of water. In general, water flows represent a

three-dimensional turbulent Newtonian flow in complicated geometrical domains. The cost of incorporating three-dimensional data in natural water courses is often excessively high. Computational efforts needed to simulatethree-dimensional turbulent flows can also be significant. In view of such considerations, many researchers havetended to use rational approximations in order to develop two-dimensional hydrodynamical models for shallow

∗ Corresponding author. Tel.: +33 149403560.E-mail address: [email protected] (S. Sari).

0378-4754/$36.00 © 2013 IMACS. Published by Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.matcom.2013.04.016

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F. Benkhaldoun et al. / Mathematics and Computers in Simulation 99 (2014) 170–189 171

ater flows. Indeed, under the influence of gravity, many free-surface water flows can be modelled by the shal-ow water equations with the assumption that the vertical scale is much smaller than any typical horizontal scale.hese equations can be derived from the depth-averaged incompressible Navier–Stokes equations using appropriate

ree-surface and boundary conditions along with a hydrostatic pressure assumption. The shallow water equationsn depth-averaged form have been successfully applied to many engineering problems and their application fieldsnclude a wide spectrum of phenomena other than water waves. For instance, the shallow water equations havepplications in environmental and hydraulics engineering such as tidal flows in an estuary or coastal regions, rivers,eservoir and open channel flows. Such practical flow problems are not trivial to simulate since the geometry cane complex and the topography irregular. However, single-layer shallow water equations have the drawback ofissing some physical dynamics in the vertical motion. Therefore, during the last decades, multi-layer shallowater models have been attracted more attention and have became a very useful tools to solve hydrodynamicalows such as rivers, estuaries, bays and other nearshore regions where water flows interact with the bed geom-try and wind shear stresses, see for instance [10,7,2]. The main advantage of these models is the fact that theulti-layer shallow water model avoids the expensive three-dimensional Navier–Stokes equations and obtains strat-

fied horizontal flow velocities as vertical velocities are relatively small and the flow is still within the shallow wateregime.

The multi-layer models studied in [10,7] among others, account only for the vertical variation of the densityetween the water layers. This makes their application very restrictive and cannot be used to model density-drivenows where a horizontal variation in the water density is required for their dynamics. Recently, a single-layer modelas been presented in [11] for shallow water flows with variable horizontal density. It has been shown in [11] thathe governing equations form a hyperbolic system of conservation laws and can be used to model dam-break typeroblems where the water dynamics is controlled by the variation of water densities rather than the variation inater heights. The drawback of this model remains the failure to capture the vertical effects in the water dynam-

cs. This water dynamics is refereed to by density-driven flow and it occurs in many applications such as oceanirculation, incursion of salty water in rivers at the sea–river intersections, and lock-exchange hydraulics for waterodies at different temperature. Our objective in this study is therefore, to develop a multi-layer model for shal-ow water flows with variable horizontal or/and vertical density. It should be stressed that a multi-layer model forhallow water flows with variable density has been recently proposed in [1]. The model uses a single variable toepresent the water height in the system. This fact limits the application of the model in [1] since the formation ofhe layers is artificial and has to exactly follow the bed profile. In our model we use different water density andater height at each layer, and it can be applied to arbitrary number and profile of the layers in the considered water

ystem.Numerical treatment of the multi-layer shallow water equations often presents difficulties due to their nonlinear

orm, presence of the source terms, coupling between the free-surface equation and the equations governing theater flow, compare [7,1] among others. In addition, the difficulty in these models comes from the coupling terms

nvolving some derivatives of the unknown physical variables that make the system non-conservative and possiblyon-hyperbolic. Due to these terms, a numerical scheme originally designed for single-layer shallow water equationsill lead to instabilities when it is applied to each layer separately. In the present work we extend the finite volumeodified method of characteristics developed by the authors in [3] to solve the density-driven multi-layer shallow waterows. The method avoids the solution of Riemann problems and it belongs to the predictor–corrector type methods.he predictor stage uses the method of characteristics to reconstruct the numerical fluxes whereas; the correctortage recovers the conservation equations. The proposed method is simple, conservative, non-oscillatory and suitableor multi-layer shallow water equations for which Riemann problems are difficult to solve. Numerical examples areresented to verify the considered multi-layer shallow water model. We demonstrate the model capability of calculatingateral and vertical distributions of velocities for density-driven multi-layer shallow water flows on flat bottom and over

hump.The organization of the present paper is as follows. In Section 2 we first give a brief description of the model

mployed for the multi-layer shallow water equations in density-driven flows. We then formulate the finite volume

odified method of characteristics for the governing equations in Section 3. This section includes the reconstruction of

he numerical fluxes and the discretization of the source terms. Numerical results are presented in Section 4 for severalest examples in density-driven multi-layer shallow water flows. Section 5 contains concluding remarks and remarksbout future work.

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172 F. Benkhaldoun et al. / Mathematics and Computers in Simulation 99 (2014) 170–189

Fig. 1. Schematic of a multi-layer shallow water equations.

2. Equations for density-driven multi-layer shallow water flows

In the current study we are interested on density-driven flows occurring on the water free-surface where assumptionsof shallow water flows applied. We consider the one-dimensional multi-layer shallow water equations written in aconservative form as

∂t(ρjhj) + ∂x(ρjhjuj) = 0,

∂t(ρjhjuj) + ∂x

(ρjhju

2j + 1

2gρjh

2j

)= −gρjhj∂xZ − gρjhj

j−1∑k=1

∂xhk − ghj

M∑k=j+1

∂x(ρkhk),(1)

where j = 1, . . ., M, with M is the total number of layers, ρj is the water density of the jth layer, hj(t, x) is the water heightof the jth layer, uj(t, x) is the local water velocity for the jth layer, Z(x) is the bottom topography and g the gravitationalacceleration, see Fig. 1 for a simplified representation. It is evident that for two layers with constant density ρ1 and ρ2(ρ2 < ρ1), Eq. (1) reduce to the standard two-layer shallow water equations studied for example in [7]

∂th1 + ∂x(h1u1) = 0,

∂t(h1u1) + ∂x

(h1u

21 + 1

2gh2

1

)= −gh1∂xZ − gh1

ρ2

ρ1∂xh2,

∂th2 + ∂x(h2u2) = 0,

∂t(h2u2) + ∂x

(h2u

22 + 1

2gh2

2

)= −gh2∂xZ − gh2∂xh1.

(2)

In the current work, we assume that a sediment transport takes place such that the density depends on space and timevariables, i.e., ρj = ρj(t, x). This requires additional equations for its evolution. Here, the equations used to close thesystem are given by

ρj = ρw + (ρsj − ρw)cj, j = 1, . . . , M, (3)

where ρsj is the sediment density with ρsj > ρw, and cj is the depth-averaged concentration of the suspended sedimentfor the jth layer. The equation for mass conservation of species is modeled by

∂t(ρsjhjcj) + ∂x(ρsjhjujcj) = 0, j = 1, . . . , M. (4)

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F. Benkhaldoun et al. / Mathematics and Computers in Simulation 99 (2014) 170–189 173

ote that for the single-layer problem (i.e. M = 1), the governing equations (1) and (4) reduce to the canonical model

∂t (ρh) + ∂x (ρhu) = 0,

∂t (ρhu) + ∂x

(ρhu2 + 1

2gρh2

)= −gρh∂xZ,

∂t(ρshc) + ∂x(ρshuc) = 0,

(5)

here ρ = ρw + (ρs − ρw)c. The single-layer model (5) has been investigated in [11] among others. The authorspplied a Roe-type finite volume method to approximate numerical solutions to the system (5). In the current study,e also apply the proposed finite volume method to solve this class of problems.For simplicity in presentation we rewrite Eqs. (1) and (4) in a compact conservative form as

∂tW + ∂xF(W) = Q(W), (6)

here W is the vector of conserved variables, F the vector of flux functions and Q is the vector of source terms.nother multi-layer shallow water system with variable density has also been presented in [1]. This model does not

ccount for conservation of species (4) and uses instead a Boussinesq approximation for dependence of the density onhe temperature.

An equivalent system to the water flow equations (1) and the suspended sediment equations (4) can be obtained bysing the physical variables as

D(j)t (ρjhj) + ρjhj∂xuj = 0, D

(j)t uj + g∂x

⎛⎝Z + 1

2hj +

j−1∑k=1

hk

⎞⎠ = − g

ρj

∂x

⎛⎝1

2ρjhj +

M∑k=j+1

ρkhk

⎞⎠ ,

D(j)t ρj = 0, j = 1, . . . , M, (7)

here D(j)t denotes the total derivative defined as

D(j)t ω = ∂tω + uj∂xω, j = 1, . . . , M. (8)

ote that D(j)t ω measures the rate of change of the function ω following the trajectories of the flow particles in the

th layer. We should also emphasize that it is not easy to confirm the hyperbolicity of the system (6). In the case of itsingle-layer counterpart, the authors in [11] have calculated the three eigenvalues of the system. The two-layer systemith constant density is only conditionally hyperbolic, see for example [10,7] whereas, the multi-layer system with

onstant density in [1] is proven to be hyperbolic only for the two-layer case. It is worth remarking that the finite volumeodified method of characteristics proposed in this paper does not require the explicit calculation of the eigenvalues

f (6) and can be applied for arbitrary number M of the layers. In what follows we describe the different steps of theroposed finite volume modified method of characteristics.

. Modified finite volume method of characteristics

Let us discretize the spatial domain into control volumes [xi−1/2, xi+1/2] with uniform size �x = xi+1/2 − xi−1/2 andivide the temporal domain into subintervals [tn, tn+1] with stepsize �t. Here, tn = n�t, xi−1/2 = i�x and xi = (i + 1/2)�xs the center of the control volume. Integrating the system (6) with respect to space and time over the time-space contrololume [tn, tn+1] × [xi−1/2, xi+1/2] we obtain the following discrete equations

Wn+1i = Wn

i − �tFn

i+1/2 − Fni−1/2

�x+ �tQn

i , (9)

here Wni is the space average of the solution W in the control volume [xi−1/2, xi+1/2] and at time tn, i.e.,

Wni = 1

�x

∫ xi+1/2

xi−1/2

W(tn, x) dx,

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174 F. Benkhaldoun et al. / Mathematics and Computers in Simulation 99 (2014) 170–189

and Fni±1/2 = F(Wn

i±1/2) are the numerical fluxes at cell interfaces x = xi±1/2 and time tn. In (9), Qni is a notation for

the discretized source terms Q(Wi) in (6). It should be pointed out that as with all explicit time stepping methods thetheoretical maximum stable time step �t is specified according to the Courant–Friedrichs–Lewy (CFL) condition

�t = Cr�x

maxj=1,...,M(∣∣∣λn

j

∣∣∣ , ∣∣∣μnj

∣∣∣ , ∣∣∣νnj

∣∣∣ ) , (10)

where Cr is a constant to be chosen less than unity and λj, μj and νj are the eigenvalues associated with each layerseparately defined as

λj = uj −√

ghj, μj = uj, νj = uj +√

ghj, j = 1, 2, . . . , M.

The spatial discretization of Eq. (9) is complete when a numerical construction of the numerical fluxes Fni±1/2 and source

terms Qni is chosen. In general, the construction of the 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 restrict the applicationof the method for which Riemann solutions are not available. Our objective in the present work is to extend a FiniteVolume modified method of Characteristics (FVC) proposed by the authors to solve canonical single-layer shallowwater equations in [3] to the density-driven flow system (6). The FVC method is simple, easy to implement, andaccurately solves the conservation equations without relying on Riemann problem solvers. The central idea of theFVC method consists of reconstructing the numerical fluxes Fn

i±1/2 by integrating the advective system (7) along thecharacteristics defined by the water velocity at each layer. In this section we give a brief description of the FVC methodfor solving the system (6) and the reader is urged to see our previous work in [3] for detailed formulation and analysisof the FVC method.

3.1. Discretization of the flux gradients

To reconstruct the numerical fluxes Fni±1/2 in (9), we consider the method of characteristics applied to the advective

version of the system (7). The main idea behind the method of characteristics is to impose a regular grid at the newtime level, and to backtrack the flow trajectories to the previous time level. At the old time level, the quantities that areneeded are evaluated by interpolation from their known values on a regular grid, see for example [14,13]. Thus, thecharacteristic curves associated with Eq. (7) are solutions of the initial-value problem

dXj,i+1/2(τ)

dτ= uj,i+1/2(τ, Xj,i+1/2(τ)), τ ∈

[tn, tn + �t

2

],

Xj,i+1/2

(tn + �t

2

)= xi+1/2, j = 1, 2, . . . , M. (11)

Note that Xj,i+1/2(τ) is the departure point at time τ of a particle that will arrive at point xi+1/2 in time tn + �t/2. Themethod of characteristics does not follow the flow particles forward in time, as the Lagrangian schemes do, instead ittraces backward the position at time tn of particles that will reach the points of a fixed mesh at time tn + �t/2. By doingso, the method avoids the grid distortion difficulties that the conventional Lagrangian schemes have, see for instance[14,13]. The solutions of (11) can be expressed as

Xj,i+1/2(tn) = xi+1/2 −∫ tn+�t/2

tn

uj,i+1/2(Xj,i+1/2(τ))dτ = xi+1/2 − δj,i+1/2, j = 1, 2, . . . , M. (12)

It is worth remarking that the departure points in (12) are calculated in the interval [tn, tn + �t/2] instead of [tn, tn+1].This is motivated by the idea of reconstructing a predictor–corrector scheme where the predictor stage is computed atthe fractional time tn + �t/2 completed by a corrector stage computed at the end time tn+1. This fractional time steppingis also supported by the analysis reported in [3].

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F. Benkhaldoun et al. / Mathematics and Computers in Simulation 99 (2014) 170–189 175

Fi

T

i

H

No

wc

wf

w

ig. 2. A schematic diagram showing the control volumes and the main quantities used in the calculation of the departure points. The exact trajectorys represented by a solid line and the approximate trajectory with a dashed line.

To compute the displacement δj,i+1/2 in (12) we consider the following iteration

�(0)j,i+1/2 = �t

2uj,i+1/2(tn, xi+1/2),

�(m)j,i+1/2 = �t

2uj,i+1/2

(tn, xi+1/2 − �(m−1)

j,i+1/2

), m = 1, 2, . . . .

(13)

he iterations (13) are terminated when the following criteria∥∥∥δ(m)j − δ

(m−1)j

∥∥∥∥∥∥δ(m−1)j

∥∥∥ < ε, (14)

s fulfilled for the L∞-norm ‖· ‖ and a given tolerance ε. It is also known [12] that∥∥∥�j − �(m)j

∥∥∥ ≤ �t

8

∥∥∥�j − �(m−1)j

∥∥∥ maxj=1,...,M

(∣∣∂xuj

∣∣) , m = 1, 2, . . . . (15)

ence, a necessary condition for the convergence of iterations (13) is that the velocity gradient satisfies

maxj=1,...,M

(∣∣∂xuj

∣∣)�t ≤ 8. (16)

ote that the condition (16) is sufficient to guarantee that the characteristics curves do not intersect during a time stepf size �t/2. A schematic representation of the quantities involved in computing the departure points is shown in Fig. 2.

Once the characteristics curves Xj,i+1/2(tn) are known, a solution at the cell interface xi+1/2 is reconstructed as

�nj,i+1/2 = �j

(tn + �t

2, xi+1/2

)= �j(tn, Xj,i+1/2(tn)) := �n

j,i+1/2, (17)

here �nj,i+1/2 is the solution at the characteristic foot Xj,i+1/2(tn) computed by interpolation from the gridpoints of the

ontrol volume where the departure point resides i.e.

�nj,i+1/2 = P(�j(tn, Xj,i+1/2(tn))), (18)

here P represents the interpolating polynomial. For instance, a Lagrange-based interpolation polynomials can beormulated as

P(�j(tn, Xi+1/2(tn))) =∑

k

Lk(Xj,i+1/2(tn))�nj,k, (19)

ith Lk are the Lagrange basis polynomials given by

Lk(x) =∏q=0q /= k

x − xq

xk − xq

.

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176 F. Benkhaldoun et al. / Mathematics and Computers in Simulation 99 (2014) 170–189

Fig. 3. Numerical results for density dam-break problem with a single initial discontinuity in the single-layer model.

Note that other interpolation procedures in (18) can also be applied. It should also be stressed that in general, themethod of characteristics fails to conserve mass, compare [13] and further references are therein. However, in ourFVC method the mass lost in the predictor step for calculating the intermediate stages Wn

i±1/2 will be recovered in the

corrector step (9). It is evident from the formulation (9) that the FVC method is mass conservative.

Fig. 4. Numerical results for density dam-break problem with two initial discontinuities in the single-layer model.

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F. Benkhaldoun et al. / Mathematics and Computers in Simulation 99 (2014) 170–189 177

F

3

csmth

F

ig. 5. The water heights h1 and h2 at time t = 5 (left plot) and at time t = 10 (right plot) for the internal dam-break problem in the two-layer model.

.2. Discretization of the source terms

The treatment of source terms in the shallow water equations presents a challenge in many numerical methods,ompare [4] and further references are therein. In our FVC scheme, the source term approximation Qn

i in the correctortage is reconstructed such that the still-water equilibrium (C-property) is satisfied. Here, at the equilibrium the sedimentedium is assumed to be saturated. Furthermore, from the state equation (3), a constant concentration cj is equivalent

o a constant density ρj for each layer j = 1, 2, . . ., M. Borrowing ideas from the analysis reported in [5], the systemas the steady states at rest

uj = 0, ∂x

⎛⎝Z +

j∑hk +

M∑ ρkhk

⎞⎠ = 0, j = 1, 2, . . . , M. (20)

k=1 k=j+1ρj

ig. 6. Results for the interface propagation problem using ρ1/ρ2 = 0.7 at time t = 1 (left plot) and using ρ1/ρ2 = 0.98 at time t = 5 (right plot).

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178 F. Benkhaldoun et al. / Mathematics and Computers in Simulation 99 (2014) 170–189

Fig. 7. Initial conditions (left plot) and results at steady-state time (right plot) for the lock exchange problem on non-flat bottom using (26) andSection 4.2.

Hence, a numerical scheme is said to satisfy the C-property for the multi-layer model (6) if the conditions

uj = 0, Z +j∑

k=1

hk +M∑

k=j+1

ρk

ρj

hk = H, ρj = Cj, (21)

hold for stationary flows at rest. In (21), H and Cj are arbitrary nonnegative constants. Remark that, if ρ1 < ρ2 < · · · < ρM,the above condition yields

uj = 0, Z + hM = HM, hj = Hj, ρj = Cj, j = 1, 2, . . . , M − 1, (22)

with Hj (j = 1, 2, . . ., M − 1) are nonnegative constants, whereas if ρ1 = ρ2 = · · · = ρM, the condition (21) reduces to

uj = 0, Z + h1 + h2 + · · · + hM = H, j = 1, 2, . . . , M, (23)

Therefore, the treatment of source terms in (9) is reconstructed such that the condition (23) is preserved at the discretizedlevel.

Applied to Eq. (7), the characteristic solutions are given by

rnj,i+1/2 = rn

j,i+1/2 − ν

2rnj,i+1/2(un

j,i+1 − unj,i),

unj,i+1/2 = un

j,i+1/2 − ν

2g

⎛⎝⎡⎣Z + 1

2hj +

j−1∑k=1

hk

⎤⎦

i+1

−⎡⎣Z + 1

2hj +

j−1∑k=1

hk

⎤⎦

i

⎞⎠

ν g

⎛⎝⎡⎣1 M∑ ⎤

⎦⎡⎣1 M∑ ⎤

⎦⎞⎠

−

2 ρnj,i+1/2 2

ρjhj +k=j+1

(ρkhk)

i+1

−2ρjhj +

k=j+1

(ρkhk)

i

,

ρnj,i+1/2 = ρn

j,i+1/2, j = 1, . . . , M, (24)

where rj = ρjhj, ν = �t/�x, rnj,i+1/2, un

j,i+1/2 and ρnj,i+1/2 are the solutions at the characteristic foot computed by

interpolation from the gridpoints of the control volume where the departure points Xj,i+1/2(tn) are located. The numericalfluxes Fi±1/2 in (9) are calculated using the intermediate states Wn

i±1/2 recovered accordingly from the characteristic

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F. Benkhaldoun et al. / Mathematics and Computers in Simulation 99 (2014) 170–189 179

s

Ffl

Fig. 8. Configuration for the flow system used in the simulations.

olutions in (24). Hence, the FVC method (9) reduces to

rn+1j,i = rn

j,i − ν((rjuj)ni+1/2 − (rjuj)ni−1/2),

qn+1j,i = qn

j,i − ν

([ρjhju

2j + 1

2gρjh

2j

]n

i+1/2−[ρjhju

2j + 1

2gρjh

2j

]n

i−1/2

)

− 1

2νgrn

j,i

([Z +

j−1∑k=1

hk

]i+1

−[

Z +j−1∑k=1

hk

]i−1

)

− 1

2νghn

j,i

⎛⎝[ M∑

k=j+1

(ρkhk)

]i+1

−[

M∑k=j+1

(ρkhk

)]i−1

⎞⎠ ,

σn+1j,i = σn

j,i − ν((σjuj)ni+1/2 − (σjuj)ni−1/2), (25)

ig. 9. Initial water densities (top) and water heights (bottom) for the 5-layer model (left) 10-layer model (middle) and 20-layer model (right) on aat bottom.

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180 F. Benkhaldoun et al. / Mathematics and Computers in Simulation 99 (2014) 170–189

Fig. 10. Water densities (top), water heights (middle) and velocity fields (bottom) for the 5-layer model on a flat bottom. From left to right t = 30 s,120 s, and 240 s.

where rj = ρjhj, qj = rjuj and σj = ρsjhjcj , with j = 1, . . ., M. In our FVC method, the reconstruction of the terms hnj,i

and rnj,i in (25) are carried out such that the discretization of the source terms is well balanced with the discretization

of flux gradients using the same concept as in [3] by

hnj,i = 1

4(hn

j,i+1 + 2hnj,i + hn

j,i−1), rnj,i = 1

4(rn

j,i+1 + 2rnj,i + rn

j,i−1).

In summary, the implementation of FVC algorithm to solve the density-driven multi-layer shallow water equations (6)is carried out in the following steps. Given (hn

j,i, qnj,i, σn

j,i), we compute (hn+1j,i , qn+1

j,i , σn+1j,i ) via:

Step 1. Calculate the departure points Xj,i+1/2(tn), with j = 1, . . ., M using the iterative procedure (12) and (13).Step 2. Compute the approximations

hnj,i+1/2 = hj(tn, Xj,i+1/2(tn)), un

j,i+1/2 = uj(tn, Xj,i+1/2(tn)), ρnj,i+1/2 = ρj(tn, Xj,i+1/2(tn)),

employing an interpolation procedure.Step 3. Evaluate the intermediate states rn

j,i+1/2, unj,i+1/2 and ρn

j,i+1/2 from the predictor stage (24).

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F. Benkhaldoun et al. / Mathematics and Computers in Simulation 99 (2014) 170–189 181

S

S

Nipi

4

otd

Fig. 11. The same as Fig. 10 but for the 10-layer model on a flat bottom.

tep 4. Update the species concentration cnj,i+1/2 using Eq. (3) as

cnj,i+1/2 = ρn

j,i+1/2 − ρw

ρsj − ρw

.

tep 5. Compute the conservation solutions rn+1j,i , qn+1

j,i and σn+1j,i using the corrector stage (25).

ote 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 methodreserves the exact water equilibrium at the machine precision, compare [3]. For the treatment of wetting/drying areasn the implementation of our FVC method, we have used the same techniques proposed in [6].

. Numerical results and applications

In this section we present numerical results obtained for several test examples for density-driven shallow water flows

ver both a flat and a non-flat bottom. The main goals of this section are to illustrate the numerical performance ofhe FVC method described above and to verify numerically its capabilities to solve shallow water flows with variableensity on a non-flat bottom. In all the computations reported herein, the Courant number Cr is set to 0.5 and the time

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Fig. 12. The same as Fig. 10 but for the 20-layer model on a flat bottom.

−0.4 −0.2 0 0.2 0.4 0.6 0.80

1

2

3

4

5

6

7

8

9

10

Velocity

Elevation

5−layer7−layer10−layer15−layer20−layer25−layer50−layer

0.2 0.3 0.4 0.5 0.6 0.70

1

2

3

4

5

6

7

8

9

10

Concentration

Elevation

5−layer7−layer10−layer15−layer20−layer25−layer50−layer

Fig. 13. Water velocity (left) and species concentration (right) at the mid channel x = 50 m for the multi-layer model on a flat bottom.

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F. Benkhaldoun et al. / Mathematics and Computers in Simulation 99 (2014) 170–189 183

Fa

sa

4

tfdwchraadis

dtρ

clhdwtt

ig. 14. Initial water densities (top) and water heights (bottom) for the 5-layer model (left) 10-layer model (middle) and 20-layer model (right) over hump.

tepsize �t is adjusted at each step according to the CFL condition (10). Hereafter, unless otherwise stated, the gravitycceleration g = 9.81 m/s2,

.1. Results for single-layer models

We first demonstrate the performance of the proposed FVC method for solving the single-layer system (5) by solvinghe test examples studied in [11]. Hence, we consider a density dam-break problem with a single initial discontinuityrom [11]. The problem consists of solving Eq. (6) in a flat channel of length 500 m filled with two liquids withensity ρ = 10 kg/m3 in the left section and ρ = 1 kg/m3 in the right section. Initially, the system is at rest with constantater height h = 1 m and g = 1 m/s2. In Fig. 3 we display the time evolution of the density, water height, velocity and

oncentration variables using a mesh with 250 gridpoints. It is clear from these results that at the initial time, theydrostatic pressure difference at the interface of the two liquids drives a flow of higher density liquid towards theight, pushing the lower density liquid ahead. To conserve mass, the free surface of the lower density liquid rises and

rightward propagating shock-like bore forms. This flow features have been accurately captured by our FVC schemend they are in good agreement with the results reported in [11]. It should be stressed that the mechanisms of theensity dam-break problems are similar to that of the standard dam-break induced by change in free-surface depth,n that a leftward rarefaction, a rightward shock and a contact wave are formed. Similar wave structures also occur inhock tube gas dynamics.

In the second example for the single-layer system (5) we solve a density dam-break problem with two initialiscontinuities from [11]. Here, a flat channel of length 100 m is filled at the left-hand side and right-hand side ofhe channel with a liquid with density ρ= 1 kg/m3. At the centre of the channel there is a liquid column of density= 10 kg/m3 and width of 1 m. Initially, the system is at rest with constant water height h = 1 m and g = 1 m/s2. Theomputed results are illustrated in Fig. 4 for the t − x phase space. As can be seen, the sudden collapse of the denseriquid in the central column causes primary shock waves to be created and propagate as bores in the direction fromigh to low density. Two outward propagating bores are generated, traveling in opposite directions. Each primary boreecreases in strength with time, which can be seen from the curved shock path. On the other hand, a pair of rarefaction

aves travels inward from the interfaces. The rarefaction waves are almost immediately reflected at the center, and

hen move outward, weakening rapidly. The accuracy of the proposed finite volume is highly achieved in reproducinghese physical features for shallow water flows with variable horizontal density.

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184 F. Benkhaldoun et al. / Mathematics and Computers in Simulation 99 (2014) 170–189

4.2. Results for two-layer models

Next we examine the performance of the FVC method for solving the two-layer shallow water flows with variablehorizontal density. All the test example considered in this section have been studied in the literature using the two-layershallow water equations with constant density (2). For all the test example considered, a good agreement has been foundbetween the numerical results obtained by solving our two-layer model with variable density (1) and those numericalresults obtained using the two-layer model with constant density (2).

In the first example, we consider an internal dam-break problem modelled by the two-layer problem (1) with M = 2,on flat bottom and subject to the following initial conditions

(h1(x, 0), h2(x, 0))T ={

(0.5, 0.5)T , if x ≤ 50,

(0.45, 0.55)T , elsewhere,u1(x, 0) = u2(x, 0) = 2.5.

Notice that this test example has been solved using the so-called Q-scheme in [8]. Here we use the same density ratioρ2/ρ1 = 0.98 and a mesh of 100 gridpoints as in [8]. To preserve this density ratio in our two-layer model, the initialdensities ρ1(0, x) = 1163.3 and ρ2(0, x) = 1140 are used in the simulations. Fig. 5 presents the water heights h1 andh2 at times t = 5 and t = 10. As expected the water free-surface remains flat during the simulation times. The FVCmethod captures the correct dynamics for this test example and exactly preserves the water equilibrium at the machineprecision. As can be seen, there is no instability like in the results reported in [8] which are related to the fact thatthe entropy inequality is only semi-discrete. The proposed FVC method accurately resolves this internal dam-breakproblem without exhibiting nonphysical oscillations.

In this second example we solve the interface propagation problem studied in [5,9]. Here we solve the two-layershallow water equations with variable density (1) in the domain [0, 10] with flat bottom. Initially,

(h1(x, 0), h2(x, 0))T ={

(0.2, 1.8)T , if x ≤ 5,

(1.8, 0.2)T , elsewhere,u1(x, 0) = u2(x, 0) = 0.

Note that the above initial conditions correspond to a dam-break problem where initially a jump at the interface,while still having the total water height constant. In Fig. 6 we illustrate the water free-surface and the interface fortwo situations namely, for ρ2/ρ1 = 0.7 at time t = 1 (corresponding to the initial densities ρ1(0, x) = 1480 and ρ2(0,x) = 1036) and for ρ2/ρ1 = 0.98 at t = 5 (corresponding to the initial densities ρ1(0, x) = 1480 and ρ2(0, x) = 1450.4).For all considered schemes the computational domain [0, 10] is discretized into 300 control volumes. As expectedthe numerical solution of this problem consists of three constant states connected by two rarefaction waves and twoshocks. Observe that increasing the density ratio results in the elimination of shocks in the water free-surface. As canbe seen from the numerical results presented in Fig. 6, no oscillations or smearing of shocks have been detected inthe computed results. The results depicted in Fig. 6 also show good agreement with those reported in [5,9] for thetwo-layer model with constant density (2). As can be seen, good behavior is recovered by the FVC method for theconsidered flow conditions in the two-layer system (1) without any significant loss of accuracy.

As third test example for two-layer shallow water problems, we consider the lock exchange problem proposed in[10]. The bottom topography is considered to be a Gaussian-shape function defined as

Z (x) = e−x2 − 2. (26)

The two layers are initially separated and the lighter water is on the left while the heavier one is on the right i.e.,

(h1(x, 0), h2(x, 0))T ={

(0, −Z(x))T , if x ≤ 0,

(−Z(x), 0)T , elsewhere,u1(x, 0) = u2(x, 0) = 0.

The bed and initial water heights are depicted in the left plot of Fig. 7. As in [10], the density ratio ρ1/ρ2 = 0.98 ispreserved by setting the initial densities ρ1(0, x) = 1341.2 and ρ2(0, x) = 1140. The computational domain is [− 3, 3]

and the boundary conditions are imposed on the water discharges q1 = h1u1 and q2 = h2u2 as at each end of the interval.In this initial-boundary value problem the heavier water propagates to the left, while the lighter one moves to theright. The solution is expected to converge to a smooth steady state. We compute a numerical steady-state solutionon a uniform grid with �x = 0.02. The obtained results, shown in the right plot of Fig. 7, are very similar to the ones

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F1

ofdd

4

ss

Fi

ig. 15. Water densities (top), water heights (middle) and velocity fields (bottom) for the 5-layer model over a hump. From left to right t = 30 s,20 s, and 240 s.

btained in [10]. We note that no interface instabilities have been observed in this example, even though initially h2 = 0or x > 0 and h1 = 0 for x < 0 and at small times, either h1 or h2 is (almost) zero in a significant part of the computationalomain. One of the key stability factors here is the ability of our FVC method to preserve the positivity of each layerepth.

.3. Results for multi-layer models

We consider a test example of multi-layer density-driven flow problem in a rectangular channel of length 100 m,ee Fig. 8 for a sketch. The total water height in the channel is set to H = 10 m and discretized into M superposed layersuch that the first 4 layers are selected to be within a height b = 3 m close to the bottom i.e.

hj(t, x) =

⎧⎪⎪⎨⎪⎪⎩

b − Z(x)

4, if j ≤ 4,

H − b

M − 4, otherwise.

(27)

or the water density distribution we assume horizontal and vertical variations are taken place in the channel. As shown

n Fig. 8, the discontinuity in the water densities is located at a parabolic interface y(x) defined by

y(x) = 0.03x2 + 0.01x − 8.

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186 F. Benkhaldoun et al. / Mathematics and Computers in Simulation 99 (2014) 170–189

Fig. 16. The same as Fig. 15 but for the 10-layer model over a hump.

Thus, given the left upper density ρUL = 900 kg/m3, the right upper ρU

R = 1010 kg/m3, the left lower densityρL

L = 990 kg/m3 and the right lower density ρLR = 1100 kg/m3, the initial water density ρj at each layer j is given

by

ρj(0, x) ={

ρLL − (j − 1)�ρL, if x < y(x),

ρLR − (j − 1)�ρR, if x > y(x),

(28)

where the density increments �ρL and �ρR are defined as

�ρL = ρLL − ρU

L

M − 1, �ρR = ρL

R − ρUR

M − 1.

This variation in the water density can be interpreted by the variation of water temperature or water salinity with thewater depth in the channel and as a consequence the main flow is governed by the density variation. In our simulations,the computational domain is discretized into 100 control volumes and wall boundary conditions are implemented.The system is assumed to be at rest and at t = 0 the interface collapses and the flow problem consists of a shockwave traveling downstream and a rarefaction wave traveling upstream. These flow features are well-established for thecanonical dam-break flow problems.

4.3.1. Density-driven flow on a flat bottomFirst we present numerical results obtained on flat bottom i.e. Z(x) = 0. In Fig. 9 we display the initial conditions for

water densities and water heights associated with 5-layer, 10-layer and 20-layer shallow water models. We have used

different color scales to differentiate between the layers with a darker color refers to the layers close to the channel bed.We apply the multi-layer shallow equations (6) to these initial conditions and numerical results are displayed at threedifferent instants namely, t = 30 s, 120 s, and 240 s. Fig. 10 presents the results obtained for the 5-layer model. Thoseresults obtained for the 10-layer and 20-layer models are illustrated in Figs. 11 and 12, respectively. In these figures

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F. Benkhaldoun et al. / Mathematics and Computers in Simulation 99 (2014) 170–189 187

wta

slatpsttcpd

wflor

Fig. 17. The same as Fig. 15 but for the 20-layer model over a hump.

e show the snapshots of the water densities, water heights and water velocity fields. It should be stressed that forhe presented velocity fields, the vertical velocity is calculated using the divergence-free condition in the flow systemlong with the method described in [2].

For all considered layers, we observe that the variation in the water density results in moving fronts with differentpeeds and different amplitudes traveling in the channel. Obviously, the free-surface trends in the considered multi-ayer models look similar however, the velocity fields exhibit different flow features. Observe the recirculation zonesppeared in the velocity fields obtained using the 10-layer and the 20-layer models. The 5-layer model fails to capturehe vertical effects for the water dynamics in the considered flow system. We have also noted that the 5-layer modelroduces diffusive water density and free-surface profiles. This diffusion has been reduced in the water density and free-urface results obtained using the 10-layer and 20-layer models. It is clear that the more layers used in the simulationhe more accurate description of the vertical flow effects become. For example, using the 20-layer shallow water modelhe response of the upper free-surface layer to the vertical dynamics is more pronounced than using the 10-layer model,ompare the velocity fields in Fig. 12. The FVC method performs well for this unsteady multi-layer shallow waterroblem and produces accurate solutions without requiring special treatment of the source terms or complicated upwindiscretization of the gradient fluxes.

For visualizing the comparisons, we display in Fig. 13 the water velocity and the species concentration associatedith each layer at the mid of the channel (x = 50 m) for different multi-layer models at time t = 120 s. Under the actual

ow conditions, it is clear that the cross section plots exhibited different behaviors in the channel center and the resultsbtained for the 50-layer model are the most accurate. Similar features have been observed for a comparison, noteported here, of cross sections for water heights and water densities. As expected for low number of layers in the flow

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188 F. Benkhaldoun et al. / Mathematics and Computers in Simulation 99 (2014) 170–189

system, the vertical effects may not be clearly captured. Note that these flow features are impossible to recover using thesingle-layer model studied in [11]. The computed results also verify the stability and the well-balanced properties of theconsidered finite volume modified method of characteristics. The proposed FVC method performs very satisfactorilyfor this multi-layer shallow water flow problem since it does not diffuse the moving fronts and no spurious oscillationshave been detected near steep gradients of the flow field and water density in the computational domain.

Fig. 15

4.3.2. Density-driven flow over a humpNow we turn our attention to the test example of density-driven flow over a hump defined by

Z(x) = e− (x−50)2

200 .

The initial conditions for water densities and water heights are depicted in Fig. 14 for the considered 5-layer, 10-layerand 20-layer shallow water models. In Fig. 15 we present the time evolution of the water densities, water heights andwater velocity fields at t = 30 s, 120 s, and 240 s for the 5-layer model. Those results obtained for the 10-layer and 20-layer models are illustrated in Figs. 16 and 17, respectively. It is clear that using the conditions for the density-drivenflow problem and the considered bottom topography, the flow exhibits a recirculation zone with different order ofmagnitudes over the hump. At the beginning of simulation time, the water flows over the hump and moves towards thechannel walls. At later time, due to the boundary conditions imposed on the walls, the water flow changes the directionand a water soliton is formed and it propagates over the hump. As can be seen, the response of the water free-surfaceto the bottom bed is more pronounced for the 5-layer model that the other 10-layer and 20-layer models.

Again, the proposed FVC scheme performs well for this density-driven flow problem since it does not diffusethe moving fronts and no spurious oscillations have been observed when the water flows over the hump. Note thatthe performance of the proposed FVC method is very attractive since the computed solutions remain stable andoscillation-free even for coarse grids without solving nonlinear systems or Riemann problems.

5. Conclusions

We have proposed a simple and accurate finite volume modified method of characteristics to solve multi-layershallow water equations for density-driven flows. The proposed finite volume method consists of two stages which canbe viewed as a predictor–corrector procedure. In the first stage, the scheme reconstructs the numerical fluxes usingthe method of characteristics. This stage results in an upwind discretization of the characteristic variables and avoidsthe Riemann problem solvers. In the second stage, the solution is updated using the conservation system. The methodcombines the attractive attributes of the finite volume discretization and the method of characteristics to yield a simplealgorithm for multi-layer density-driven shallow water flows. The method does not require either nonlinear solutionor special treatment of the bed bottom. The proposed method has been numerically examined for the test example ofdensity-driven flow problem on both a flat and a non-flat topography. The obtained results have exhibited accurateprediction of both, the water free-surface and the water velocity field with correct dynamics, and stable representationof free-surface response to the variation in the water density. The results make it promising to be applicable also to realsituations where, beyond the many sources of complexity, there is a more severe demand for accuracy in predictingdensity-driven shallow water flows, which must be performed for long time.

We conclude with some comments on the current development of this finite volume method, in terms of bothphysical and numerical features that will be implemented. In this paper, we have only considered source terms dueto the bottom topography. However, in many hydraulic scenarios, mass exchanges, friction losses and viscous terms,which interact with the hydraulics through the introduction of stress terms in the momentum equations, can be thedominant force in the multi-layer shallow water equations for density-driven flows. Therefore, future work will involveinclusion of viscous coupling a wave model component into the modelling system to include the effects of bottomfriction, wind stresses, eddy viscosity, and mass exchange in the multi-layer density-driven flows. Numerically, the

present scheme is a suite of finite volume modified method of characteristics that are currently being developed. Othermethod components will include application to tidal flows and lock exchange flows. In many situations, these modelswill be solved on large domains and over irregular bathymetries such as coastal scenarios. The proposed finite volumemodified method of characteristics is particularly advantageous for this class of applications.

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A

m

R

[

[

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F. Benkhaldoun et al. / Mathematics and Computers in Simulation 99 (2014) 170–189 189

cknowledgment

The authors would like to thank Prof. E. Audusse for valuable discussions about the modelling aspect of theulti-layer shallow water flows.

eferences

[1] E. Audusse, M.-O. Bristeau, M. Pelanti, J. Sainte-Marie, Approximation of the hydrostatic Navier–Stokes system for density stratified flowsby a multilayer model: kinetic interpretation and numerical solution, Journal of Computational Physics 230 (2011) 3453–3478.

[2] E. Audusse, M.-O. Bristeau, M. Pelanti, J. Sainte-Marie, A multilayer Saint–Venant system with mass exchanges for shallow water flows.Derivation and Numerical Validation, M2AN Mathematical Modelling and Numerical Analysis 45 (2011) 169–200.

[3] F. Benkhaldoun, M. Seaïd, A simple finite volume method for the shallow water equations, Journal of Computational and Applied Mathematics234 (2010) 58–72.

[4] F. Benkhaldoun, I. Elmahi, M. Seaïd, A new finite volume method for flux-gradient and source-term balancing in shallow water equations,Computer Methods in Applied Mechanics and Engineering 199 (2010) 49–52.

[5] F. Bouchut, T. Morales, An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment, M2AN MathematicalModelling and Numerical Analysis 42 (2008) 683–698.

[6] P. Brufau, M.E. Vázquez-Cendón, P. García-Navarro, A numerical model for the flooding and drying of irregular domains, International Journalof Numerical Methods in Fluids 39 (2002) 247–275.

[7] M.J. Castro, J.A. García-Rodríguez, J.M. González-Vida, J. Macías, C. Parés, M.E. Vázquez-Cendón, Numerical simulation of two-layershallow water flows through channels with irregular geometry, Journal of Computational Physics 195 (2004) 202–235.

[8] M.J. Castro, J. Macías, C. Parés, A Q-scheme for a class of systems of coupled conservation laws with source term, application to a two-layer1d shallow water system, M2AN Mathematical Modelling and Numerical Analysis 35 (2001) 107–127.

[9] M. Dudzinski, M.L. Medvidova, Well-balanced path-consistent finite volume eg schemes for the two-layer shallow water equations, Compu-tational Science and High Performance Computing IV (2009) 121–136.

10] D. Farmer, L. Armi, Maximal two-layer exchange over a sill and through a combination of a sill and contraction with barotropic flow, Journalof Fluid Mechanics 164 (1986) 53–76.

11] F.Z. Leighton, A.G.L. Borthwick, P.H. Taylor, 1-D numerical modelling of shallow flows with variable horizontal density, International Journalfor Numerical Methods in Fluids 62 (2010) 1209–1231.

12] J. Pudykiewicz, A. Staniforth, Some properties and comparative performance of the semi-Lagrangian method of Robert in the solution ofadvection–diffusion equation, Atmosphere-Ocean 22 (1984) 283–308.

13] M. Seaïd, On the quasi-monotone modified method of characteristics for transport-diffusion problems with reactive sources, Computer Methodsin Applied Mathematics 2 (2002) 186–210.

14] C. Temperton, A. Staniforth, An efficient two-time-level semi-Lagrangian semi-implicit integration scheme, Quarterly Journal of RoyalMeteorological Society 113 (1987) 1025–1039.