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Journal of Hydraulic Research
ISSN: 0022-1686 (Print) 1814-2079 (Online) Journal homepage: http://www.tandfonline.com/loi/tjhr20
A coupled two-dimensional numerical model forrapidly varying flow, sediment transport and bedmorphology
Xin Liu, Julio Sedano Ángel Infante & Abdolmajid Mohammadian
To cite this article: Xin Liu, Julio Sedano Ángel Infante & Abdolmajid Mohammadian (2015): Acoupled two-dimensional numerical model for rapidly varying flow, sediment transport andbed morphology, Journal of Hydraulic Research, DOI: 10.1080/00221686.2015.1085919
To link to this article: http://dx.doi.org/10.1080/00221686.2015.1085919
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Journal of Hydraulic Research, 2015http://dx.doi.org/10.1080/00221686.2015.1085919© 2015 International Association for Hydro-Environment Engineering and Research
Research paper
A coupled two-dimensional numerical model for rapidly varying flow, sedimenttransport and bed morphologyXIN LIU, PhD Student, Department of Civil Engineering, University of Ottawa, Ottawa, CanadaEmail: [email protected] (author for correspondence)
JULIO SEDANO ÁNGEL INFANTE, Assistant Professor, Department of Civil Engineering, University of Ottawa, Ottawa, CanadaEmail: [email protected]
ABDOLMAJID MOHAMMADIAN (IAHR Member), Associated Professor, Department of Civil Engineering, University ofOttawa, Ottawa, CanadaEmail: [email protected]
ABSTRACTThis paper presents a coupled two-dimensional model that can produce a more stable numerical simulation of rapid bed evolution than the conven-tional decoupled model. To solve the coupled bed-load sediment transport terms using a Godunov-type central-upwind method, a novel scheme toestimate the bed-load fluxes which can produce more accurate results than the previously reported coupled model is proposed using a pair of localwave speeds different from those used for the flow. The two-dimensional shallow water equations are used to solve the flow velocities and waterdepth. The bed level is solved by an Exner-based equation containing bed-load sediment transport as numerical flux terms and sediment entrainmentand deposition as source terms. Analytical formulas to compute the eigenvalues of the Jacobian matrix are developed. The linear reconstruction ofvariables with a multi-dimensional slope limiter, and the second-order Runge-Kutta scheme are employed to achieve higher accuracy in space andtime. For the case of rapid bed-erosion, the accuracy and stability of the proposed coupled model are verified by several numerical tests.
Keywords: Bed-load transport; coupled model; direct numerical simulation methods; finite volume method; shallow flows; two-dimensional models
1 Introduction
Conventional decoupled models for simulating shallow waterequations (SWEs) and bed erosion have normally been devel-oped assuming that the rate of bed morphological evolution is ofa much lower order of magnitude than the flow changes. In suchmodels, the hydrodynamic module is assumed to be identical tothat of a fixed-bed flow, and the bed evolution equation is treatedseparately from the hydrodynamic model. Such approaches areaccurate and stable when dealing with slow water flows witha very slow bed erosion process over a long period of time.However, when considering faster flows with a relatively fastbed evolution process, decoupled approaches might not be suit-able any more, as the interactions between water and sedimentvariables become stronger.
Two fundamentally different directions are considered toaccount for strong interactions between water flow and bed
erosion (Cao, Day, & Egashira, 2002). The first direction isrecognizing that the momentum equation for flow mixtureover an erodible bed differs from that for fixed-bed clearwater flow in the sense that the former contains terms relatedto the density of the mixture and bed mobility. It requiresa different momentum equation to be built for an intensesediment transport layer to account for inertial effects whenlarge amounts of sediments are transported by the flow. Sev-eral morphological models were successfully developed con-sidering this direction. A two-layer model was developedby Fraccarollo and Capart (2002) assuming that the water–sediment mixture is in the thick bed-load layer under clearwater, in which the conservation laws are built for both lay-ers. This model was later improved by considering velocitydecoupling (Capart & Young, 2002), granular phase dilatation(Spinewine & Zech, 2005; Zech, Soares-Frazão, Spinewine,Savary, & Goutiere, 2009), equations coupling (Spinewine,
Received 4 June 2014; accepted 19 August 2015/Currently open for discussion.
ISSN 0022-1686 print/ISSN 1814-2079 onlinehttp://www.tandfonline.com
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Guinot, Soares-Frazão, & Zech, 2011; Swartenbroekx, Soares-Frazão, Spinewine, Guinot, & Zech, 2010), sediment inertiaeffect on flows (Castro-Díaz Fernández-Nieto, González-Vida,& Parés-Madroñal, 2011), large shear stress between two lay-ers (Castro-Díaz et al., 2011), hyperbolicity of the system(Castro-Díaz et al., 2011; Savary & Zech, 2007; Spinewineet al., 2011; Swartenbroekx et al., 2010) and turbulence (Fer-reira, Franca, Leal, & Cardoso, 2009). The two-phase modelshave gained popularity in modelling intense sediment trans-port under sheet-flow conditions, in which the sediment phasecan be written in either Eulerian or Lagrangian form. The“Euler–Lagrange” two-phase models (Drake & Calantoni, 2001;Gotoh & Sakai, 1997; Yeganeh-Bakhtiary, Shabani, Gotoh, &Wang, 2009) can better account for particle–particle interactionsin the hyper-concentrated layer. The “Euler–Euler” two-phasemodels (Bakhtyar, Yeganeh-Bakhtiary, Barry, & Ghaheri, 2009;Greco, Iervolino, Leopardi, & Vacca, 2012; Hsu, Chang, &Hsieh, 2003; H. Liu & Sato, 2006) consider sediment phase asa continuum that follows different constitutive laws from thosefor the clear water, and the momentum transfer between twophases is taken into account. The above models lead to rela-tively expensive computational costs for building the mass andmomentum conservation laws separately for two layers or twophases. To solve the general cases of rapid bed erosion undershallow water flows, the conventional single-layer models werekept having been widely studied. Cao, Pender, Wallis, and Car-ling (2004) modified the classic SWEs by adding componentswhich account for mass and momentum exchange betweenerodible beds and water-mixture flows. The induced entrain-ing sediment was considered as a depth-averaged suspendedload, and a “non-capacity” erosion model was developed intheir study. Yue, Cao, Li, and Che (2008) extended the modeldeveloped by Cao et al. (2004) to two-dimensional (2-D) cases.This modified system was applied by Wu and Wang (2007) andXia, Lin, Falconer, and Wang (2010) with different bed-load andsuspended-load formulas.
The second direction is coupling the Saint-Venant–Exnerequations and solving them in a synchronous procedure. Thestrategy of coupling the Exner-based equation with the gov-erning hydrodynamic system and approximating the bed-loadfluxes using a Riemann solver has attracted increasing atten-tion from researchers. Cao et al. (2004) coupled the SWEs witha suspended sediment transport equation in their 1-D study.Similar coupled models have been used in Simpson and Castell-tort (2006) and Hu and Cao (2009). Fagherazzi and Sun (2003)reformulated the bed-evolution equation by introducing newconservative variables in order to build a fully coupled sys-tem and correctly capture discontinuous solutions. Li andDuffy (2011) extended this approach to 2-D cases. Soulis (2002)and X. Liu, Landry, and Garcia (2008) solved the coupledsystem of hydrodynamic and morphodynamic equations by afinite volume method, and the bed-load transport was consid-ered in their coupled system. However, the bed-load transportwas not considered in the above-mentioned models, and the
bed-evolution equation is not coupled with the governing sys-tem. Hudson and Sweby (2003) investigated the accuracy anddetermined the validity of both steady and unsteady approachesfor five different types of governing formulations, coupling themorphology continuity equation with hydrodynamic equationson rectangular meshes. They concluded that one of the con-servative reformulated forms, based on the unsteady approach,was the best overall. However, a drawback of this form is itsdependence on z in the Jacobian matrix, which was shown anddiscussed in Murillo and García-Navarro (2010) and Serrano-Pacheco, Murillo, and Garcia-Navarro (2012). Castro-Díaz,Fernández-Nieto, and Ferreiro (2008) studied the numericalapproximation of bed-load sediment transport caused by shal-low layer flows on unstructured meshes with a second-orderMUSCL (monotonic upstream-centred scheme for conservationlaws)-type reconstruction. Murillo and García-Navarro (2010)studied an Exner-based coupled model for 2-D transient flowsover erodible beds on triangular unstructured meshes, and devel-oped a general “Grass formula” format expressing several com-monly used empirical bed-load formulas. However, in the studyby Murillo and García-Navarro (2010), the same eigenvaluesand eigenvectors were used to estimate the fluxes for both thehydrodynamic and morphological models, which may lead toexcessive diffusion for the bed-evolution process. In addition,although considering the bed-load coefficient as a constant in theJacobian matrix can significantly reduce the computational costand difficulties for solving the system, ignoring the variabilityof the bed-load coefficient may lead to inaccurate estimationsof the eigen-structure. X. Liu, Mohammadian, Kurganov, andInfante Sedano (2015) proposed a coupled strategy which usingthe “Lagrange theorem” to estimate the bounds of the eigenval-ues in order to avoid difficulty in calculating those eigenvalues.They focused on very rapid erosion processes, which need ahigher diffusion level. Their approach may need to be adjustedfor real-world applications. Soares-Frazão and Zech (2011) builta coupled system using the HLLC method and chose differentpairs of eigenvalues to estimate the numerical fluxes. However,based on the fact that they only used two specific eigenvaluesduring the simulation process, excessive diffusion might still beintroduced in simulating the bed profile.
For most real-world applications, even for rapid bed-loadtransport cases, the rates of morphological evolution caused bybed-load transport are still considerably smaller than the rate ofthe flow changes. Excessive numerical diffusion may be intro-duced when approximating bed-load transport fluxes using thesame directional local speeds from the hydrodynamic system inthe flux-solver. Accordingly, in Godunov-type flux approxima-tions, it is necessary to consider smaller directional local speedsin the solver in order to accurately approximate the bed-loadsediment flux terms.
In this paper, we focus on the second direction described,and we propose a numerical scheme which uses different pairsof eigenvalues of the Jacobian matrix to estimate the numericalfluxes of flow and bed-load transport. A central-upwind method
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Journal of Hydraulic Research (2015) Coupled model for rapid bed-evolution and flow 3
with novel estimators of local wave-speeds is developed basedon the analytical expressions of the eigenvalues of the Jacobianmatrix. A single-layer model is adopted and it is demonstratedthat with a suitable numerical scheme, the single-layer modelcould lead to an efficient and accurate scheme for a wide rangeof applications. Using the single-layer model, a coupling strat-egy is proposed, and a coupled model for rapid bed erosionis developed. The sediment concentrations are also updated atevery time-step in the system, and sediment entrainment anddeposition are included in the source terms, as they are effec-tive in various practical tests. Multiple test cases are presentedin order to demonstrate the advantages of the proposed schemein terms of numerical performance such as better accuracy andnon-oscillatory behaviour.
This paper is organized as follows: In Section 2, the gov-erning equations are explained. The bed-load transport formulaand the empirical formulas for sediment entrainment and depo-sition are also described in Section 2. In Section 3, the governingequations are discretized using the finite volume method (FVM),and the interface fluxes of each cell are calculated using thecentral-upwind scheme with novel local wave estimators. More-over, linear reconstructions of variables, treatment of sourceterms, the time integration scheme, and boundary conditionsare also introduced in Section 3. Several practical tests arethen presented in Section 4, where the proposed model is veri-fied by comparing the simulated results with experimental data,conventional decoupled models and previously reported cou-pled model. Some concluding remarks complete the study inSection 5.
2 Governing equations
2.1 Description of hydraulic and sediment transport system
In the current study, the governing equations, consisting ofthe shallow water system, sediment transport, and Exner-basedequation, are given by:
∂h∂t
+ ∂
∂x(uh) + ∂
∂x(vh) = 0 (1)
∂
∂t(uh) + ∂
∂x
(u2h + 1
2gh2
)+ ∂
∂y(uvh) = −gh
∂z∂x
− ghSfx
(2)
∂
∂t(vh) + ∂
∂x(uvh) + ∂
∂y
(v2h + 1
2gh2
)= −gh
∂z∂y
− ghSfy
(3)
∂
∂t(hc) + ∂
∂x(uhc) + ∂
∂y(vhc) = E − D (4)
∂z∂t
+ ξ∂qbx
∂x+ ξ
∂qby
∂y= D − E
1 − p(5)
in which t is time, h is the water depth, x and y are hori-zontal coordinates, u and v are the depth-averaged velocity-components in the x and y directions, respectively, z is the bed
level, g is the gravitational acceleration, p is the bed porosity,ξ is a coefficient equal to (1 − p)−1, and c is the volumetricsediment concentration. In addition, Sfx and Sfy are the frictionslope terms in x and y directions, respectively, which can bedetermined by conventional formulas involving the Manningroughness coefficient nb, i.e. Sfx = n2
bh−4/3u√
u2 + v2 and Sfy =n2
bh−4/3v√
u2 + v2, qbx and qby are components of bed-loaddischarge qb in x and y directions, respectively.
2.2 Bed-load formulas
In the current study, the bed-load sediment transport rate isestimated by the Meyer-Peter and Müller (MPM) formula(Meyer-Peter & Müller, 1948). The formula is as follows:
qb = 8√
(s − 1)gd3(θ − θc)32 (6)
in which s is the specific gravity of sediment given by s =ρs/ρf , where ρs is the density of sediment particles, ρf is thedensity of water, d is the mean particle diameter, θc is thecritical bed shear stress and θc = 0.047 (Smart, 1984), and θ
is the dimensionless shear stress which can be calculated asθ = hS′/[(s − 1)d]. Using S′ = Sf =
√S2
fx + S2fy , we arrive at:
θ = nb2
(s − 1)dh1/3|u|2 (7)
in which |u| = √u2 + v2.
2.3 Sediment deposition and entrainment fluxes
Sediment entrainment E, due to turbulence, and sediment depo-sition D, due to gravitational action, are two distinct mecha-nisms involved in sediment exchange between the fluid and theerodible bed. The relationships in Cao et al. (2004) are used inthis study:
E =⎧⎨⎩ϕ(θ − θc)|u|d−0.2h−1 if θ ≥ θc
0 otherwise
D = ωo(1 − Ca)mCa
(8)
where ωo is the settling velocity of a single parti-cle in tranquil water, which can be calculated as ωo =√
(13.95ν/d)2 + 1.09sgd − 13.95ν/d (Zhang & Xie, 1993),where ν is the kinematic viscosity of water, m is an expo-nent representing the effect of hindered settling due to sedimentconcentration, and Ca is the local near-bed sediment volu-metric concentration, which can be determined by Ca = αc,in which the coefficient α is usually larger than unity and cmust not exceed (1 − p). Cao et al. (2004) suggested usingα = min[2, (1 − p)/c], and ϕ as a coefficient to control the ero-sion force, which may be obtained using empirical formulas
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for intense and strong turbulent bursting entrainment processes.Parameter ϕ is calibrated for a relatively weak sediment entrain-ment process in this paper, which is required to extend thisformula for situations of bed erosion with a less turbulentbursting effect.
3 Numerical scheme
3.1 Coupled governing system
Equations (1)–(5) constitute the governing system which can bepresented in the following vector form:
∂U∂t
+ ∂F∂x
+ ∂G∂y
− Sb = S (9)
where U is the vector of conserved variables, F and G are theconvective fluxes in the x and y directions respectively, Sb isthe bed-slope term and S is the source term. These vectors aredefined as:
U =
⎛⎜⎜⎜⎜⎜⎝h
uhvhchz
⎞⎟⎟⎟⎟⎟⎠ , F =
⎛⎜⎜⎜⎜⎜⎝uh
u2h + 0.5gh2
vuhuchξqbx
⎞⎟⎟⎟⎟⎟⎠ , G =
⎛⎜⎜⎜⎜⎜⎝vh
uvhv2h + 0.5gh2
vchξqby
⎞⎟⎟⎟⎟⎟⎠(10)
Sb =
⎛⎜⎜⎜⎜⎜⎝0
−gh ∂z∂x
−gh ∂z∂y
00
⎞⎟⎟⎟⎟⎟⎠ , S =
⎛⎜⎜⎜⎜⎜⎝0
−ghSfx
−ghSfy
E − DD−E1−p
⎞⎟⎟⎟⎟⎟⎠ (11)
In this study, following the coupled approach developed inMurillo and García-Navarro (2010), a new flux term associ-ated to the bed slope Sb, normal to a direction given by the unitvector n, is defined by:
T · n =(
0, −ghδzδx
nx, −ghδzδx
ny , 0, 0)T
(12)
Accordingly, one can get:
J nδU = δFn(U) − Sb = δ(Fnx + Gny) · n − T · n (13)
where n is the outward unit vector normal to the cell interfaces,nx and ny are components of unit normal vector n in the x andy directions, respectively. The analytical form of the Jacobianmatrix J n is written as:
Jn =
⎛⎜⎜⎜⎜⎜⎜⎜⎝
0 nx ny 0 0
−(uh)q⊥/(h2) + ghnx q⊥/h + uhh nx (uh)ny/h 0 ghnx
−(vh)q⊥/(h2) + ghny (vh)nx/h q⊥/h + (vh)ny/h 0 ghny
−(ch)q⊥/(h2) (ch)nx/h (ch)ny/h q⊥/h 0
A B C 0 0
⎞⎟⎟⎟⎟⎟⎟⎟⎠(14)
in which q⊥ = (uh)nx + (vh)ny and A, B and C are:
A = −28ξnb2√gd√
(s − 1)
q⊥
|q|
[nb
2
(s − 1) d|q|2h7/3 − θc
]1/2 |q|2h10/3
B = 24ξnb2√gd
h7/3√
(s − 1)(uh)
[nb
2
(s − 1) d|q|2h7/3 − θc
]1/2
×q⊥
|q| − 8ξ√
g (s − 1) d3
×[
nb2
(s − 1) d|q|2h7/3 − θc
]3/2
[nx
|q| − (uh)q⊥
|q|3]
C = 24ξnb2√gd
h7/3√
(s − 1)(vh)
[nb
2
(s − 1) d|q|2h7/3 − θc
]1/2
×q⊥
|q| − 8ξ√
g (s − 1) d3
×[
nb2
(s − 1) d|q|2h7/3 − θc
]3/2 [nx
|q| − (vh)q⊥
|q|3]
where |q| =√
(uh)2 + (vh)2.The eigenvalues of J n are the roots of the polynomial
f (l, U) of the determinant∣∣∣ ∂Fn(U)
∂U − lI∣∣∣, and the f (l, U) is
calculated as:
f (l, U) = −(
l − q⊥
h
)2{
l3 − 2q⊥
hl2
+[(
q⊥
h
)2
− g(h + hD)
]l + gh
(q⊥
hD − E
)}(15)
where D = Bnx + Cny and E = [(uh)B + (vh)C
]/(6h). Two of
the eigenvalues of J n are l1 = l2 = q⊥/h from Eq. (15), theother three eigenvalues can be obtained by solving the followingcubic:
P (l) = l3 + a1l2 + a2l + a3 (16)
in which a1 = −2q⊥/h, a2 = [(q⊥/h
)2 − g(h + hD)]
and a3 =gh
(q⊥D/h − E
). We define:
Q = 19
(3a2 − a1
2) , R = 154
(9a1a2 − 27a3 − 2a1
3) (17)
The governing system is strictly hyperbolic when condition|q⊥/h| < 6
√gh is held (Cordier, Le, & De Luna, 2011), all
three roots are real and unequal, and can be determined using
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the following definitions (Spiegel, 1999):
l3 = 2√
−Q cos(γ
3
)− a1
3,
l4 = 2√
−Q cos(
γ + 2π
3
)− a1
3,
l5 = 2√
−Q cos(
γ + 4π
3
)− a1
3
(18)
in which cos γ = R/√
−Q3.
3.2 Discretization of governing equations
In the present study, the coupled system comprising five gov-erning equations is discretized on unstructured triangular gridsusing the finite volume method. Integrating the system over thecontrol volume, one obtains:∫
Ai
∂U∂t
dA +∮
[Fn(U) − Tn(U)
]d =
∫Ai
S(U)dA (19)
in which denotes the boundary of control volume Ai,Fn(U) = Fnx + Gny and Tn(U) = T · n. The line-integral termsin Eq. (20) can be discretized using the divergence theorem:
∂Ui
∂t= − 1
Ai
3∑k=1
Fiklik + Si (20)
in which lik is the length of the kth edge of Ai, and Fik isnumerical flux across lik:
Fik = FA[(UL)ik, (UR)ik] (21)
in which FA is a hyperbolic solver estimating the numericalfluxes through the cell interfaces, and (UL)ik and (UR)ik arethe reconstructed values of U on the left and right sides of likrespectively.
3.3 Central-upwind method
Central-upwind schemes on general triangular grids for solvingtwo-dimensional systems of conservation laws were developedin Kurganov and Petrova (2005); they offer the main advantagesof the Godunov-type central schemes while keeping simplic-ity, universality, and robustness, and can be applied to prob-lems with complicated geometries. Applying the central-upwindmethod, Eq. (21) can be written by:
Fik =[Sr �F(UL)ik + Sl �F(UR)ik
] · n − SrSl[(UR)ik − (UL)ik](Sr + Sl)
(22)
in which �F(UL)ik and �F(UR)ik are normal fluxes on the left andright sides of lik, respectively, and Sr and Sl are the one-sidedlocal speeds of propagation on the right and left sides of likrespectively.
Wave speed estimator for the hydrodynamic model
For the hydrodynamic model, the conventional central-upwindmethod is used. The one-sided local speeds Sr and Sl aredetermined by seeking the absolute values of both the maxi-mum non-negative eigenvalue and the maximum non-positiveeigenvalue:
Sr = max{lN [J n(UL)]ik , lN [J n(UR)]ik , 0},Sl = − min{l1 [J n(UL)]ik , l1 [J n(UR)]ik , 0}
(23)
where l1 [J n(U)]ik ≤ · · · ≤ lN [J n(U)]ik being the N eigen-values of the Jacobian matrix J n(U) using the reconstructedvariables (UL)ik or (UR)ik.
Wave speed estimator for bed-load sediment transport
The conventional decoupled model using the straightforwardcentral-difference method or the simple upwind method toapproximate the bed-load fluxes, will introduce numerical oscil-lations and instabilities. Thus, it requires a coupling strategy anda modern upwind scheme to stably approximate the bed-loadfluxes, instead of the conventional decoupled method. How-ever, in the real-world cases of fast-propagating flows, the wavespeeds of the bed-load sediment transport are much smallerthan the wave speeds of the hydrodynamic flows. Therefore,a lower diffusion-level than hydrodynamic solver is essentiallyrequired to apply the modern upwind method to the Exner-basedequation. Using the same one-sided local speeds Sr of the hydro-dynamic model, which are calculated in Eq. (23), will causeadditional bed deformation deviating from the measured values,which are typically referred to as numerical diffusion.
A typical wave structure is illustrated in Fig. 1. To esti-mate the local wave-speeds of the bed-load sediment transport,in this paper we propose a novel wave-estimator using a dif-ferent characteristic wave with a smaller shock wave speed toapproximate the numerical fluxes of bed-load transport termsusing the central-upwind method. The corresponding one-sidedlocal speeds for bed-load sediment transport S∗
r and S∗l are
determined by:
S∗r = max{lK [J n(UL)]ik , lK [J n(UR)]ik , 0},
S∗l = −min{l1 [J n(UL)]ik , l1 [J n(UR)]ik , 0}
(24)
with l1 [J n(U)]ik ≤ · · · ≤ 0 ≤ lk [J n(U)]ik ≤ · · · ≤ lN [J n(U)]ik
being the N eigenvalues of the Jacobian matrix J n(U) using thereconstructed variables (UL)ik or (UR)ik. Applying the central-upwind method with this novel local wave-estimator, the fluxesof bed-load sediment transport in the Exner-based equation (5)can be estimated by:
Fik =[S∗
r�F(UL)ik + S∗
l�F(UR)ik
] · n − S∗r S∗
l [(UR)ik − (UL)ik](S∗
r + S∗l )
(25)
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l1l2 l3
l4
0 x
S*l
S*r
Sr
t
Figure 1 Typical characteristic waves of the system
In order to handle the situation where both S∗r and S∗
l are zero orvery close to zero, in this paper the flux computed in Eq. (25) isreplaced by Fik = 0.5
[ �F(UL)ik + �F(UR)ik], if S∗
l + S∗r < 10−6.
3.4 Linear reconstruction and multi-dimensional slopelimiter
In the current study, the following reconstruction of unknownsis employed to acquire second-order spatial accuracy:
U(x, y) = Ui + (Ux)i(x − xi) + (Uy)i(y − yi), (x, y) ∈ Ai
(26)
where U(x, y) is the reconstructed value of variables at point(x, y) inside of cell i, xi and yi are the coordinates of the centre ofcell i, and (Ux)i and (Uy)i are component-wise approximationsof numerical gradient ∇Ui which are computed via the multidi-mensional limiter (Jawahar & Kamath, 2000), which is used tominimize the oscillations of the reconstructions:
∇Ui = �a∇Ua + �b∇Ub + �c∇Uc (27)
where ∇Ua, ∇Ub and ∇Uc are the three unlimited gradients cho-sen following Pan and Cheng (1993) (the calculation of theseunlimited gradients is given in Section 3.5); the weights �a, �b
and �c are calculated by:
�a(ga, gb, gc) = (gbgc + ε2)
(ga2 + gb
2 + gc2 + 3ε2)
�b(ga, gb, gc) = (gagc + ε2)
(ga2 + gb
2 + gc2 + 3ε2)
�c(ga, gb, gc) = (gagb + ε2)
(ga2 + gb
2 + gc2 + 3ε2)
(28)
where ga, gb and gc are given by ga = ‖∇Ua‖22, gb = ‖∇Ub‖2
2,and gc = ‖∇Uc‖2
2. The parameter ε is a small number which isintroduced to prevent indeterminacy caused by the vanishing ofthe three gradients, in regions of uniform flow.
3.5 Calculation of derivatives
In order to calculate the derivative in a cell i, e.g. the derivativesof Ui, a triangular zone formed by the centres of three adjacent
x
Ui1
Ui2
Ui3
Ui
i
y
Figure 2 A sketch of the derivatives calculation
cells is considered (the shaded zone in Fig. 2 with surface areaA′
i). Using the divergence theorem:
∂Ui
∂x= 1
A′i
∫A′
∂Ui
∂xdAi
i
≈ (Ui3 + Ui2) �y1 + (Ui1 + Ui3) �y2 + (Ui2 + Ui1)�y3
2A′i
(29)
∂Ui
∂y= 1
A′i
∫A′
∂Ui
∂ydAi
i
≈ − (Ui3 + Ui2)�x1 + (Ui1 + Ui3)�x2 + (Ui2 + Ui1)�x3
2A′i
(30)
where:
�x1 = (xi3 − xi2) , �x2 = (xi1 − xi3) , �x3 = (xi2 − xi1)
�y1 = (yi3 − yi2) , �y2 = (yi1 − yi3) , �y3 = (yi2 − yi1)
This approach is used to calculate the unlimited gradients ofvariables.
3.6 Time integration
In order to obtain a second-order accuracy in time, the two-stageexplicit total variation diminishing (TVD) Runge-Kutta methodadopted in Song, Zhou, Guo, Zou, and Liu (2011) is appliedin the current study. Since the numerical model is explicit,its stability can be controlled by aCourant–Friedrichs–Lewy(CFL)-type condition. For a triangular grid system, a CFL-typecondition is applied to estimate an appropriate time step. It canbe expressed as:
�t = Cr min(
Ri
Sr
)ik
i = 1, 2, . . . , N (31)
where Cr is the Courant number specified in the range 0 < Cr ≤1, N is the total number of cells, Ri is the minimum distancefrom the centre to the edges of the ith triangular cell, and Sr is
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the right-sided local speed of propagation for the kth edge of celli, which is estimated in Eq. (23).
4 Numerical tests
In this section, several test cases are performed, to evaluatethe performance of the proposed coupled model. Test cases 1and 4 compare the proposed method with decoupled modelsin which the bed-load fluxes in the Exner-based equation aresolved separately by a simple upwind method, i.e.:
Fik ={ �F(UL)ik if p⊥ ≥ 0
�F(UR)ik if p⊥ < 0(32)
4.1 Dam failure due to over-topping flow
This test is chosen to verify the accuracy of the proposed modelin predicting the erosion caused by a rapid flow on an initiallyirregular bed. The corresponding laboratory experiment wascarried out and reported in Tingsanchali and Chinnarasri (2001).The tests were performed in a rectangular flume with a 35 mlength, 1.0 m depth, and 1.0 m width. The height and crest-widthof the dam were fixed at 0.80 m and 0.30 m, respectively. Theupstream slope was fixed at 1V:3H, while the downstream slope(DIS) was varied. Sand, described as Sand I in Tingsanchali andChinnarasri (2001), is used as the bed sediment in the currentstudy. The mean grain size of Sand I is d = 1.13 mm. In thistest, DIS = 1V:2.5H is chosen for the simulation. The reservoirinflow discharge is set at 1.23 l s−1.
Only the bed-load transport is considered in this test. Thecomputational domain is discretized by a grid of uniform cellsize of 0.05 m. Figure 3 shows the simulated bed profiles fromthe proposed coupled model and a conventional decoupledmodel at 30 s and 60 s, respectively. These are also comparedwith the experimental measurements. It can be seen that theproposed model is able to more accurately predict the bed pro-file under a rapid varying flow than the conventional decoupledmodel which does not yield sufficient erosion under the fastover-topping flow.
4.2 Dam failure due to over-topping flow
In this test, an experiment with a dam-break flow over anerodible bed, as reported by Fraccarollo and Capart (2002),is used. The experimental set concerns small-scale laboratorydam-break waves of initial water depth h0 = 10 cm releasedover an erodible channel bed. The sediment consists of cylindri-cal PVC pellets with an average spherical diameter of 3.5 mm,a density of 1540 kg m−3 and a fall velocity ω0 ≈ 18 cm s−1.The testing reach had a length of 2.5 m with an initial flat bed.The Manning roughness coefficient is set to 0.025 s m−1/3. Atthe beginning, a 10 cm initial water depth is set upstream. Thedam-break wave is then released by rapidly lifting the sluice
x (m)
z (m
)
t = 30 s
x (m)
z (m
)
t = 60 s
151
1.8
1
1.8
17 19 21
Initial bed profileProposed modelDecoupled modelMeasured
Initial bed profileProposed modelDecoupled modelMeasured
15 17 19 21
Figure 3 Measured and simulated bed profiles at t = 30 s and 60 s
wat
er le
vel a
nd b
ed le
vel (
m)
x (m)0 0.5–0.5
Bed lelvels
Measured dataThe proposed modelFraccarollo et al. (2003)
Water surfaces
–1
0
0.04
0.08
1
Figure 4 Simulated and measured water levels and bed profiles att = 10t0
gate. In the numerical simulation, the computational domain isdiscretized by grids of size 0.005 m. The total simulated timeis 10t0, in which the hydrodynamic time scale t0 = √
h0/g ≈0.101 s.
Figure 4 shows good agreement of the water surface and bedprofile between the simulated and measured results. Comparedwith the numerical results from Fraccarollo, Capart, and Zech(2003), the current model shows a better agreement with thedata measured upstream of the dam; the current model can pre-cisely simulate the scouring hole at the position of the collapseddam, with a better agreement for the front waves. The currentmodel predicts excessive erosion in the downstream area, how-ever, whereas the results from Fraccarollo et al. (2003) showeda better agreement for bed level than the current model.
4.3 2-D dam-break flows over mobile bed with suddenenlargement
This test on a two-dimensional dam-break flow over an erodi-ble bed with sudden enlargement followed the experiment
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0
0.10
0.14
wat
er le
vel (
m)
0.18
0.22
4
Simulated
P1 Station
Measured
8 12time (s)
0
0.10
0.14
wat
er le
vel (
m)
0.18
0.22
4
Simulated
P2 Station
Measured
8 12time (s)
0
0.10
0.14
wat
er le
vel (
m)
0.18
0.22
4
Simulated
P3 Station
Measured
8 12time (s)
0
0.10
0.14
wat
er le
vel (
m)
0.18
0.22
4
Simulated
P4 Station
Measured
8 12time (s)
0
0.10
0.14
wat
er le
vel (
m)
0.18
0.22
4
SimulatedP5 Station
Measured
8 12time (s)
0
0.10
0.14
wat
er le
vel (
m)
0.18
0.22
4
Simulated
P6 Station
Measured
8 12time (s)
Figure 5 Comparisons between measured and simulated water-level evolutions at different stations
carried out at the Hydraulic Laboratory of the Universitécatholique de Louvain (Belgium) that was reported by Palumbo,Soares-Frazão, Goutière, Pianese and Zech (2008) and Goutiere,Soares-Frazão and Zech (2011). The experiment was performedin a 6 m-long flume presenting a non-symmetrical suddenwidening from 0.25 m to 0.5 m width, located 1 m downstreamof the gate. The channel width is 0.25 m before x = 4 m and0.5 m after x = 4 m.
The data at six locations, which are P1 (3.75 m, 0.125 m), P2(4.2 m, 0.125 m), P3 (4.45 m, 0.125 m), P4 (4.95 m, 0.125 m),P5 (4.2 m, 0.375 m), and P6 (4.95 m, 0.375 m), are chosen inthis test to be compared with the simulated results. In this test,measured cross-section profiles at three locations, which are atx = 4.2 m, x = 4.3 m and x = 4.5 m, are chosen to verify thesimulation.
An initial clear-water depth of 0.25 m is set upstream of thegate at x = 3 m. The initial conditions consist of a 0.1 m-thickhorizontal layer of fully saturated sand along the entire lengthof the flume. The sediment used is uniform coarse sand with amedian diameter of d50 = 1.72 mm and relative density of 2.63,deposited with a porosity of 0.3922.
The domain is discretized into 9168 triangular cells withan average area of 2.32 × 10−4 m2, which are refined at thesection of enlargement. A standard closed boundary condition isapplied to all side-walls, and a free-outflow boundary conditionis applied at the downstream outlet. The Manning coefficientis set to be 0.018 s m−1/3 in our test. In order to compare theproposed method with the method used by Soares-Frazão andZech (2011), a first-order spatial reconstruction is implemented,the suspended sediment is ignored, and only bed-load transportis considered.
Figure 5 shows the comparison between predicted water lev-els and experimentally measured values at the aforementionedmonitoring stations over the test period. The proposed modelshows very good agreement with the measured water surfaceat monitoring stations, from which it can be concluded that theproposed model shows a desirable accuracy and is able to pre-dict the water surface profile of a rapidly varying flow affectedby bed evolution.
In Fig. 6, the results of simulated bed profiles from theproposed model with novel wave speed estimator and a cou-pled model using hydrodynamic wave estimator, in which the
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0
0.08
0.12
0.5x (m)
x = 4.2 mExperimental dataProposed wave estimatorSoares-Frazao and Zech (2011)Simulated using hydrodynamicwave-estimator
z (m
)
0
0.08
0.12
0.5x (m)
x = 4.3 mExperimental dataProposed wave estimatorSoares-Frazao and Zech (2011)Simulated using hydrodynamicwave-estimator
z (m
)
0
0.09
0.13
0.5x (m)
x = 4.5 mExperimental dataProposed wave estimatorSoares-Frazao and Zech (2011)Simulated using hydrodynamicwave-estimator
z (m
)
Figure 6 Simulated and measured bed profiles at different cross–sections (x = 4.1, 4.3 and 4.4 m) compared with the results fromSoares-Frazão and Zech (2011)
maximum shock celerity Sr is used for bed-load sediment fluxes,are shown at measured cross-sections at x = 4.2, 4.3 and 4.5 m,respectively.
It can be seen that the coupled model using Sr (computedfrom Eq. (23) for hydrodynamic system) to compute bed-loadfluxes introduces excessive diffusion and results in a flattenedbed profile. The results fromSoares-Frazão and Zech (2011)are also compared with our proposed model and the measureddata. It can be seen that, comparing with the simulated resultsby Soares-Frazão and Zech (2011), our proposed method can
yield more accurate prediction of bed erosion; the positionsof trenches and dunes predicted by the proposed method aremore accurate. Moreover, it can be observed that the simu-lated bed-profile from the proposed model is accurate and sharp,while the results from Soares-Frazão and Zech (2011) seemmore diffusive. The reason may be explained as follows. In theproposed sediment wave speed estimator, the smallest positivecharacteristic wave from the five characteristic waves is chosen.Therefore, when bed-evolution in the x-direction is dominant,minimum diffusion is introduced in the y-direction. However, inthe wave-speed estimator from Soares-Frazão and Zech (2011),the width of the Riemann fan is larger than is needed, and thusit introduces excessive diffusion in practical cases.
4.4 2-D dam-break flow over an erodible bed
To further examine the numerical performance of the presentscheme, a benchmark test of a 2-D dam-break problem witha rapidly varying unsteady flow is chosen in this paper. Thistest case was first introduced by Fennema and Chaudhry (1990)in a numerical method study, which has been widely usedby many researchers (Anastasiou & Chan, 1997; Jawahar &Kamath, 2000; Wang & Liu, 2001). The original case wasmostly applied on an initially wet and fixed bed. It is modified inthe present study with the aim of testing the ability of the currentmodel to simulate front wave propagation over an initially drybed with wet–dry interface tracking, with particular attentionto the 2-D aspects of the flow motion; and the model’s perfor-mance in predicting rapid erosion under a highly energetic flowon a large scale. The modelling domain area is a 200 m × 200 mbasin with a flat, dry bed. A 10 m-thick dam splits the basininto two equally-sized regions. The clear water depths are 7 mand 0.5 m on the left and right sides of the dam, respectively.At t = 0 s, a 75 m-wide breach centred at y = 125 m is assumedto form instantaneously. The duration of the simulation is 10 s.The initial velocity of the whole modelling area is 0 m s−1 andthe outlet boundary at x = 200 m is specified with a free out-flow boundary condition; meanwhile, all other boundaries areset to a standard wall condition. The computational domain isdiscretized in to 5134 triangles with an average area of 7.53 m2.
Figure 7 shows the meshes generated for this test. Figure 8shows the 3-D views of the simulated water surface on a fixedbed at t = 10 s. In Fig. 8, it can be seen that a shock wave formsand propagates downstream and a depression wave spreadsupstream during the simulation process. The predicted waterprofile with a fixed bed from the proposed model closely agreeswith other researchers’ results (Anastasiou & Chan, 1997; Fen-nema and Chaudhry, 1990; Jawahar & Kamath, 2000; Wang &Liu, 2001).
Figure 9 shows the simulated bed profile at t = 10 s for boththe proposed model and a conventional decoupled model. Itcan be observed that the erosion depths of both models basi-cally agree with each other, since the sediment entrainment Eand deposition D terms in Eq. (5) dominate the erosion depth.
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10 X. Liu et al. Journal of Hydraulic Research (2015)
00
200
200x (m)
y (m
)
Figure 7 Discretized meshes
200
x (m) y (m)
2005.2
7.4
9.6
11.8
waterlevel(m)
0
0
wat
er le
vel (
m)
6
12
Figure 8 Simulated water level on a fixed bed at t = 10 s using theproposed model
200
x (m)
x (m)
y (m)
y (m)
z (m
)
200
200
0
2
50
z (m
)
2
50
0200(a)
(b)
Figure 9 Simulated bed profiles z at t = 10 s from (a) the proposedmodel and (b) the decoupled model
However, since bed-load transport exists as a numerical flux,it actually control the oscillation of the simulated bed-profile.The bed erosion from the proposed model in Fig. 9a shows verygood non-oscillatory form during the simulation process whilethe conventional decoupled model shows clear oscillations atthe corners of the broken dam (Fig. 9b), which is incorrect.
5 Conclusion
In this paper, an accurate and stable two-dimensional couplednumerical model involving the bed-erosion process is proposedto simultaneously predict rapidly varying flow, sediment trans-port and rapid bed evolution using the finite volume methodbased on unstructured triangular grids. To account for the rel-atively strong interaction effects between the water and thebed-evolution under a rapidly varying flow, a coupled strat-egy is adopted, and the bed-load sediment transport terms aresolved as fluxes by the central-upwind method. In the numericaltests, the proposed model shows enhanced performance com-pared with the conventional decoupled model and previouslyreported coupled models.
A novel wave estimator in conjunction with the central-upwind method is proposed and successfully applied to thecoupled water-sediment system involving a rapid bed-erosionprocess. It was demonstrated that, in comparison with the decou-pled model, applying the central-upwind method to approximatethe bed-load fluxes can successfully avoid the numerical oscil-lations caused by simple and less stable schemes, e.g. simpleupwind methods (see section 4.4). This illustrates the advan-tage of a coupled approach for bed-load transport under rapidflows. However, it was also found that excessive bed deforma-tion (shown in Fig. 6) could be introduced by using the sameone-sided local wave speed Sr as in the hydrodynamic modelin the solver of bed-load fluxes (see section 4.3). This impliesthat the largest non-negative eigenvalue of the coupled systemwill cause excessive numerical diffusion to bed-evolution andcan only be used for a hydrodynamic model in the central-upwind method. Based on the fact that the bed-deformationspeed is much smaller than the speed of the fluid, even forfast-propagating flows, we proposed a novel wave speed esti-mator seeking the smallest non-negative eigenvalue of coupledsystem as characteristic wave speed (S∗
r ) in the central-upwindmethod to compute the bed-load fluxes. Based on the results ofthe numerical tests, the proposed scheme can predict accurateresults for both the hydrodynamic system and the morphody-namic system, with better agreement with the experimental datathan the decoupled and coupled models using Sr. Consequently,the proposed scheme has advantages in terms of accuracy andstability which are shown in the numerical tests. In addition,we have provided analytical expressions for calculating theeigenvalues of the coupled shallow-water–Exner system, whichgreatly enhances the efficiency of the proposed method.
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Journal of Hydraulic Research (2015) Coupled model for rapid bed-evolution and flow 11
The current study shows that the proposed scheme pro-vides better accuracy and stability than conventional decoupledmethods and can avoid excessive numerical diffusions andoscillations.
Funding
This publication was made possible by the National Priori-ties Research Programme (NPRP) [Grant number 4-935-2-354]from the Qatar National Research Fund (a member of theQatar Foundation). The statements made herein are solely theresponsibility of the authors.
Notation
Ai = control volume (m2)c = volumetric sediment concentrationd = mean particle diameter (m)D = sediment deposition discharge (m s−1)E = sediment entrainment discharge (m s−1)F, G = vector of convective fluxes in the x and y directionsFik = numerical convective fluxg = gravitational acceleration (m s−2)h = water depth (m)J n = Jacobian matrix of the governing systemlik = length of the kth edge of control volume Ai (m)n = unit normal vectornx, ny = components of n in x and y directionsnb = Manning roughness coefficient (s m−1/3)p = bed porosityqb = bed-load transport discharge (m2 s−1)qbx, qby = components of qb in the x and y directions (m2 s−1)S = vector of source termsSr, Sl = one-sided local speeds of hydrodynamic model
(m s−1)S∗
r , S∗l = one-sided local speeds of morphological model
(m s−1)t = time (s)U = vector of conserved variablesu, v = depth-averaged velocity in the x and y directions
(m s−1)x, y = spatial coordinates (m)z = bed-level (m)θ = dimensionless shear stressρs, ρf = density of solid and fluid (kg m−3)ω0 = settling velocity of a single particle (m s−1)ν = kinematic viscosity of water (m2 s−1)ϕ = a coefficient to control the erosion force (m1.2)
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