Bed Evolution Under Rapidly Varying Flows by a New Method for Wave Speed Estimation

water

Article Bed Evolution under Rapidly Varying Flows by a New Method for Wave Speed Estimation

Khawar Rehman and Yong-Sik Cho *

Department of Civil and Environmental Engineering, Hanyang University, 222 Wangsimni-ro, Seongdong-gu, Seoul 04763, Korea; [email protected] * Correspondence: [email protected]; Tel.: +82-2-2220-0393

Academic Editor: Jeffrey McDonnell Received: 9 April 2016; Accepted: 16 May 2016; Published: 20 May 2016

Abstract: This paper proposes a sediment-transport model based on coupled Saint-Venant and Exner equations. A finite volume method of Godunov type with predictor-corrector steps is used to solve a set of coupled equations. An efficient combination of approximate Riemann solvers is proposed to compute fluxes associated with sediment-laden flow. In addition, a new method is proposed for computing the water depth and velocity values along the shear wave. This method ensures smooth solutions, even for flows with high discontinuities, and on domains with highly distorted grids. The numerical model is tested for channel aggradation on a sloping bottom, dam-break cases at flume-scale and reach-scale with flat bottom configurations and varying downstream water depths. The proposed model is tested for predicting the position of hydraulic jump, wave front propagation, and for predicting magnitude of bed erosion. The comparison between results based on the proposed scheme and analytical, experimental, and published numerical results shows good agreement. Sensitivity analysis shows that the model is computationally efficient and virtually independent of mesh refinement.

Keywords: sediment transport process; numerical modeling; Saint Venant-Exner equations; ﬁnite volume method; dam-break

1. Introduction Numerical techniques for modeling morphodynamics are preferable over field and laboratory observations at large scales, due to reduced computational costs and time [1–4]. Mathematical modeling of morphodynamics can be performed using depth-integrated shallow-water equations (continuity and momentum equations) representing the hydrodynamics, and an Exner equation (continuity of sediment) representing the morphological evolution. These equations can either be modeled using a coupled approach in which the hydrodynamic and morphodynamics flows are solved simultaneously, or using an uncoupled (quasi two-phase) approach, in which flow hydrodynamics are modeled first and then the obtained velocity is used for modeling morphological evolution [5]. According to [6,7], the coupled approach is best suited for analysis of rapidly varying flow conditions, where flow unsteadiness strongly affects sediment evolution; while the uncoupled approach is mostly used for slowly varying flows. The applicability of the uncoupled approach is limited due to the assumption of constant sediment concentration in the lower layer [8]. Coupled shallow-water and Exner equations have been widely used to model sediment transport in rivers, bed evolution in dam-break flows, siltation of reservoirs, and sediment flushing in sewers [4,9–14]. In this study, the authors used a coupled model based on approaches adopted by Goutière, Soares-Frazão et al. (2008) and Murillo and García-Navarro (2010) to simulate a bed load sediment-transport initiated by water flow. Different equations are used in the literature to model solid transport sediment flux. Commonly used sediment-transport formulas include the Grass formula [15],

Water 2016, 8, 212; doi:10.3390/w8050212 www.mdpi.com/journal/water Water 2016, 8, 212 2 of 20

Meyer-Peter and Muller’s (MPM) equation [16], Van Rijn’s equation [17–19], and Nielsen’s formula [20]. The performance of various transport formulas is reported in [21]. For simplicity, the authors have adopted Grass’s model in this study to define a solid transport discharge. This is a simple sediment formula that combines a global case dependent calibration and an easily differentiable power law of velocity [22]. We use the volume method of Godunov-type to solve coupled shallow-water and Exner equations models. Two approaches are usually employed for the computation of interface fluxes in coupled models, the Roe approximate Riemann solver [5,10,22–27], and the Harten-Lax-van Leer-Contact (HLLC) approximate Riemann solver [7,14]. A predictor-corrector step is adopted for evolving the solution to get second-order temporal accuracy. In the predictor step, the solution is updated by half a time step using a non-conservative method; while in the corrector step, the conservation is recovered over a full time step by solving local Riemann problems using the data obtained from the predictor step. The fluxes at the interface are computed using the HLLC approximate Riemann solver for the hydrodynamic part, and by the Harten-Lax-van Leer (HLL) scheme for the morphological changes. Since this study employs the Riemann problem, the general solution for the hydrodynamic part of the solution consists of three waves, a left and right moving wave and a middle wave (a shear wave). In such a problem, the most important feature of flux computation with the HLLC scheme is the choice of relations for approximating the water depth and velocity along the shear wave. The usual practices of computing these variables are either based on the assumption that the two non-linear waves are either shocks or rarefactions, or by using depth positivity condition (readers are referred to [28] for more details). However, in cases of coupled shallow-water and Exner equations, these equations are not applicable due to differences in wave structure. Wave speed estimation in HLLC solver requires the values of Roe-averaged states (water depth, velocity, energy states) [29]. These states are obtained from Roe approximations (see details in [30]). Given that the Roe-averaged states are not uniquely defined [31], different approached are adopted for estimating Roe-averaged states. Most of the numerical models for sediment transport studies [6,13,14,22,23] follow Glaister’s approach [32] for computing Roe averages for a coupled shallow-water Exner equations system. However, Hu et al. [33] argued that artificially introduced intermediate states, like that of Glaister’s, may not be admissible for some cases. Therefore, they proposed an extension of the Roe averages for estimation of wave speeds by avoiding artificial states and relying only on intermediate states for state averages. The HLLC studies [7,14] used the average of left and right Riemann states for computation of variables along the shear wave, but this method makes the solution too dependent on reconstructed values, which can be erroneous for cases in which the grids are skewed or when the problem presents strong discontinuities, and when the reconstruction procedure is not accurate. We propose a new method for approximating the values along the shear wave by Roe averages in which the variable values do not exclusively depend upon reconstructed values. The basic concept is to predict average states directly from the intermediate states by utilizing vertex values at the interface along with reconstructed variables. To make the dependence more even, the variable values at vertex are included in finding the Roe averages. This method avoids potential problems of inaccuracy and instability, which can be caused by meshes with skewed elements, problems with strong discontinuities, and may address the shortcomings of the reconstruction procedure to some extent. The aim of this work is to develop a simple, but robust, coupled shallow-water flow model that can accurately handle the discontinuities and capture shocks associated with dam-break flows on erodible beds. The objective is to develop a model that can help us study the bed evolution under the sudden release of large volume of water and to accurately capture the wave structure associated with such flows. Water 2016, 8, 212 3 of 20

2. Governing Equations The governing equations describe the Saint-Venant system commonly known as shallow-water equations coupled to the Exner equation for sediment-transport and forming a non-linear system of equations.

2.1. Hydrodynamic Component Flow evolution can be represented by the shallow-water equations. For an inviscid and incompressible ﬂow, these equations can be expressed as:

Bh Bphuq ` “ 0 Bt Bx Bphuq B 1 (1) ` hu2 ` gh2 “ ghpS ´ τq Bt Bx 2 ˆ ˙ where t is the time, h is the water depth, u is the velocity component, g is the gravitational acceleration, and S and τ are the bed slope and bottom frictional terms, respectively, estimated as:

Bb n2u |u| S “ ´ , τ “ (2) Bx h4{3 where b is bed elevation, and n is Manning’s roughness coefﬁcient.

2.2. Morphodynamic Component The sediment dynamics can be expressed by a simple sediment continuity or Exner equation. Mathematically, this equation is given as [12]:

Bb Bq ` ξ s “ 0 (3) Bt Bx where ξ “ 1{1 ´ p, p is the sediment porosity and qs is sediment-transport discharge. Physically, Equation (3) implies that time variation of a sediment layer in a control volume is due to the net variation of solid transport through the boundaries of the control volume [22]. In this study, the Exner equation is used in its reduced form, as the main focus is bed evolution and ﬂow dynamics. A sediment discharge transport formula is used to close Equation (3). To check the performance of the proposed model, we compute bed load discharge using two approaches, the bed load equation of Grass [15] and the MPM sediment transport formula.

2.2.1. Grass Formula Sediment transport ﬂux in the Grass formula follows a power law, given as:

2 qspGrassq “ Au u (4) ˇ ˇ ˇ ˇ where A is the intensity of interaction between ﬂow andˇ sedimentˇ particles, the value of which ranges between 0 and 1 with increasing interaction strength, and u is the velocity. This equation does not take critical shear stress into account, and it relates the sediment discharge to the ﬂow velocity as in [12]. For the cases modeled in this study, A is either adopted from previous studies, or calculated using Equation (5) for cases with detailed sediment parameters;

0.2 d ´0.6 d 0.005 h ` 0.012D˚ A “ ˆ ˙ (5) ´ ¯ 1{3 ρs gd ρ ´ 1 ´ ´ ¯¯ Water 2016, 8, 212 4 of 20

where d is the local sediment medium diameter; ρs and ρ are the sediment and water density, respectively; and the particle diameter D* is estimated as

1{3 g ρs D˚ “ d ´ 1 (6) υ2 ρ ˆ ˆ ˙˙ where υ is the kinematic viscosity of water.

2.2.2. MPM Formula The MPM formula is given as [16]:

3 3{2 qspMPMq “ 8 g ps ´ 1q d pθ ´ θcq (7) b where s is the speciﬁc gravity of sediment, θ is the Shield’s parameter and θc is threshold Shield’s parameter, set equal to 0.047 in our study, following [14]. Inserting the expression of Shield’s parameter into Equation (7) forms

2 2 3{2 3 n u qspMPMq “ 8 g ps ´ 1q d ´ 0.047 (8) ps ´ 1q dh1{3 b ˆ ˙ 3. Numerical Scheme In conservative form, Equations (1) and (3) can be written as:

BU BF ` “ S (9) Bt Bx where U is the vector of conserved variables, F is a ﬂux vector functions, and S is a vector of source terms. In vector form, the coupled equations can be written as [13]:

h hu 0 2 2 U “ » hu fi , F “ » hu ` 1{2gh fi , S “ » ghpS ´ τq fi (10) b ξqs 0 — ffi — ffi — ffi – fl – fl – fl Integration of Equation (9) over the cell gives

BU dΩ ` ∇ ¨ F “ SdΩ (11) Bt Ωż Ω¿ Ωż where Ω is the volume of a control volume. Equation (11) can be expressed as:

BU dΩ ` FpUq ¨ n dΓ “ pS ` τq dΩ (12) Bt Ωż ¿Γ Ωż where n denotes the unit outward normal to the boundary of the cell, and F(U)¨ n is the ﬂux vector normal to the interface, expressed as:

hu 2 FpUq “ 2 gh (13) » hu ` 2 fi ξq — s ffi – fl Discretization of Equation (12) takes the form: Water 2016, 8, 212 5 of 20

∆t Un`1 “ Un ´ F ´ F ´ S (14) ∆x i`1{2 i´1{2 i ”´ ¯ ı 1 x`1{2 Un “ U px, tnq dx (15) ∆x żx´1{2 where ∆t and ∆x are step sizes in space and time, respectively, and xi´1{2 and xi`1{2 are the boundaries of the cell. The explicit nature of this scheme demands that the time step is limited by the CFL (Courant-Friedrichs-Lewy coefﬁcient) condition.

∆tmax |λ | ν “ k , k “ 1, 2, 3 (16) ∆x where ν is the CFL number, set equal to 0.6 for all cases modeled in this study. Equation (14) is evolved in two steps: the predictor step and the corrector step. In the predictor step, the solution evolves by half a time step employing reconstructed cell center variables. This step is given as:

n` { ∆t U 1 2 “ Un ´ Fn ´ Fn ´ Sn (17) i i 2∆x i`1{2 i´1{2 i ”´ ¯ ı where Fi`1{2 and Fi´1{2 are ﬂuxes at the right and left sides of the cell interface, respectively, n indicates n the initial values of variables, and Ui is the vector of conserved variables at the cell center at initial time n. Fluxes Fi˘1{2 are computed by the HLLC and HLL schemes for the hydrodynamic and morphodynamic parts, respectively. The results from Equation (17) are advanced to a full time step in the second stage by the corrector scheme given as:

∆t n` { n` { n` { Un`1 “ Un ´ F 1 2 ´ F 1 2 ´ S 1 2 (18) i i 2∆x i`1{2 i´1{2 i ”´ ¯ ı 3.1. Flux Computation Flux calculation using coupled shallow-water Exner equations is complicated due to the additional wave, related to sediment information in the wave structure. Approximate Riemann solvers (especially Roe and HLLC schemes) can accurately estimate fluxes for shallow-water flows, as these solvers have high resolution and shock capturing capability. However, for cases with strong interface interactions, like dam-break flows over erodible beds, complications arise due to the additional wave (associated with the sediment layer) issued from an initial discontinuity in the coupled equations. We use HLLC and HLL schemes in this study to compute hydrodynamic and morphological fluxes, respectively. The wave speeds for the coupled system of equations, Equation (7), are estimated from the approximate eigenvalues derived in [13] by analysis of the Jacobian matrix for the coupled system of equations. 2 2 λ1 “ 0.5pu ´ c ´ pu ´ cq ` 4ξdc q (19) b 2 2 λ2 “ 0.5pu ´ c ` pu ´ cq ` 4ξdsc q (20) b λ3 “ u ` c (21) where c “ gh is the wave celerity, and d is a the term involving the normal derivative of sediment transport rate.a Differentiating the sediment discharge, given in Equations (4) and (8), gives us:

BqspGrassq d “ “ 3αu2 (22) s Bu Water 2016, 8, 212 6 of 20

Bq 2 2 1.5 spMPMq B 3 n u ds “ “ 8 g ps ´ 1q d ? ´ 0.047 Bu B u ps ´ 1q d 3 h ˜ ˆ ˙ ¸ (23) 2 a 2 2 1{2 24n 3 n u ds “ u g ps ´ 1q d ´ 0.047 p ´ q 1{3 p ´ q 1{3 s 1 dh ˆ s 1 dh ˙ a The largest eigenvalue λ3 accounts for hydrodynamic effects and λ1,2 give eigenvalues carrying sediment characteristic information. The HLLC scheme used for computing hydrodynamic mass and momentum ﬂuxes is given as [34]:

FL if 0 ď SL hllc $ F˚L “ FL ` SLpU˚L ´ ULq if SL ď 0 ď S˚ Fi`1{2 “ hllc (24) ’ F “ FR ` SRpU˚R ´ URq if S˚ ď 0 ď SR &’ ˚R FR if 0 ě SR ’ ’ The speed of the middle wave% is λ˚, and is approximated as

SLhRpUR ´ SRq ´ SRhLpUL ´ SLq S˚ “ (25) hRpUR ´ SRq ´ hLpUL ´ SLq

1 SK ´ uK U˚K “ hK S˚ (26) S ´ S˚ » fi ˆ K ˙ v — ffi – fl where K = L, R; SL “ min λ1L, λ1R, u˚ ´ gh˚ , SR “ max λ3L, λ3R, u˚ ` gh˚ , h˚ and u˚ are variable values along the shear wave estimated by Equations (29) and (30), respectively. ` a ˘ ` a ˘ The eigenvalue analysis in [13] shows that sediment flow information (λ1,2) is not affected by the fast moving wave (λ3), since the speed with which sediment information is transmitted is very slow compared to the evolution of other hydrodynamic information. We choose the HLL scheme for computing sediment fluxes because of the constant intermediate state between λ1 and λ2, which is the assumption of HLL scheme. Application of the HLL scheme is simple, as intermediate waves between the slowest and fastest moving waves are not present in this approach. The speed of these waves is estimated by eigenvalues of the coupled equation; the speed of the slowest and fastest travelling waves is estimated by Equation (21). The HLL flux is calculated as [34]:

FL if 0 ď SL F “ Fhll “ SR FL´SL FR`SLSRpUR´ULq if S ď 0 ď S (27) i`1{2 $ SR´SL L R &’ FR if 0 ě SR ’ % S “ minpλ , λ , u ´ gh q L 1L 1R ˚ ˚ (28) S “ pλ λ u ` gh q R max 2L, 2R, ˚ a ˚ The important step in arriving at the solution is to approximatea the water depth and velocity across the shear wave. Their constant values, in Equations (29) and (30), are computed by a new method in which the contribution of vertex values is also accounted for

2 hL ` hR ` hv v“1 h˚ “ (29) 4 ř

2 ? hLuL ` hRuR ` hvuv v“1 u˚ “ (30) a a 2ř? hL ` hR ` hv v“1 a a ř Water 2016, 8, 212 7 of 20 where v represents the two vertices of the interface through which the ﬂux is being computed. The vertex values can be computed by the pseudo-Laplacian relation proposed by [35]. This new method for approximation of variable values in the star region also gives weight to the variable values at the vertices. This method is effective for highly skewed control volumes, as well as for structured grids, as the variable approximation does not strongly depend upon the reconstructed values at cell interfaces. In previous studies, the estimation of shear waves is based solely on Roe averages that are obtained by solving locally linearized Riemann problem at each cell interface [12,30,36]:

hL ` hR h˚ “ (31) 2

hLuL ` hRuR u˚ “ (32) a hL ` ahR This method might make the solution susceptiblea toa error for cases in which the mesh is composed of highly distorted grids, which lead to cell-centered variables that are not reconstructed exactly at the mid-point of an interface. We show the performance of the proposed method in Section3 by modeling various test cases on different meshes.

3.2. Source Term Treatment Accuracy of the numerical solution is greatly inﬂuenced by the source term treatment. In the present study, source terms include variable bed slope and shear stress terms. The scheme is well-balanced if its ﬂux gradient is balanced by the slope source terms for a “lake at rest” condition

η “ h ` b “ constant; u “ 0 (33) where η is water surface elevation. We extend and modify the source term treatment, the revised surface gradient method (RSGM), proposed by [37], for shallow water ﬂows on non-erodible beds. We extend and modify RSGM for erodible beds by introducing reconstructed values of water surface elevation to estimate ﬂow depth average and bed level difference.

Bb ηi`1{2 ´ b ` ηi´1{2 ´ bi´1{2 S “ gh dxdy “ g i`1{2 b ´ b ∆y (34) x Bx 2 i`1{2 i´1{2 ∆x∆y ´ ¯ The proposed scheme satisﬁes well-balanced conditions and gives stable and accurate solutions as shown in application of the model. Friction terms were treated explicitly.

3.3. Wetting and Drying Inaccurate treatment of wetting and drying fronts results in signiﬁcant errors in solutions, including negative water depths and unphysical high velocities. A simple treatment is adopted in this study, which involves setting a tolerance depth hmin (0.0001 m in our model), below which cells are considered dry, and consequently the momentum ﬂux across the cell interface becomes zero. This treatment of wet-dry fronts is similar to applying solid boundary conditions on the interface between wet and dry cells.

3.4. Boundary Conditions Wall and open boundary conditions are imposed in the proposed model. Unlike the pure hydrodynamic case, there are three characteristic waves entering and leaving the domain at the inlet and outlet. The additional wave accounts for the presence of sediment. Solid wall conditions are imposed on side walls making the velocity normal to the walls equal to zero. Physical boundary conditions are required at the inlet. For cases in which some physical conditions are not available, Water 2016, 8, 212 8 of 20 Water 2016, 8, 212 8 of 20 conditions are required at the inlet. For cases in which some physical conditions are not available, the theorytheory ofof characteristicscharacteristics isis used.used. The inlet boundary conditions are implemented by the same procedure asas literature literature references, references, including including [23 ,[23,38].38]. A free A free outﬂow outflow condition condition is imposed is imposed at the outletsat the tooutlets enable to wavesenable towaves pass throughto pass thro theugh boundary the boundary without without reﬂection. reflection.

4. Applications Applications

4.1. Dam-Break over a Rigid Bed The proposedproposed schemescheme is is ﬁrst first tested tested for for a benchmarka benchmark hydrodynamic hydrodynamic case case of dam-break of dam-break on a rigidon a non-erodiblerigid non-erodible bed widely bed widely employed employed to verify to model’s verify performancemodel’s performance on dry and on wet dry beds. and Thewet dam-breakbeds. The isdam-break instantaneous is instantaneous on a ﬂat bottom on a flat with bottom negligible with friction.negligible Based friction. on theBased water on levelthe water in the level right in half the ofright the half domain, of the there domain, are twothere cases are two for this cases problem; for this ifproblem; water level if water is over level zero is theover case zero is the said case to beis dam-breaksaid to be dam-break on a wet domain, on a wet and domain, if it is zero and theif it case is zero is dry the bed. case The is dry initial bed. conditions The initial for conditions this problem for constitutethis problem a Riemann constitute problem. a Riemann For problem. initially wet For bed initially the conditions wet bed the are conditions given as are given as

≤≤ 10mifxx1m i f 0 ď x ď xo hphxx()q “= (35)(35)  ≤≤ #0.10.1mifxxLm i f xo oď x ď L

The initial conditions for dry bed areare

≤≤ 10mifxx1m i f 0 ď x ď x o hxp()q “= o (36) h x  ≤≤ (36) #0.00.0 i f xifo xďo x ď xL L = where LL isis the the total total length length of of the the channel channel taken taken equal equal to 200to 200 m, andm, andxo “ x100o m100mis the is position the position of dam. of Thedam. results The results are compared are compared with analytical with analytical solution solution [39] at t [39]= 1, at 12, t 21= 1, and 12, 2721s and for flow27 s for over flow wet over and bedwet inand Figure bed in1. Figure The numerical 1. The numerical results show results excellent show agreementexcellent agreement with the analytical with the solution;analytical the solution; transition the oftransition wet/dry of front wet/dry is accurately front is captured accurately showing captur thated theshowing scheme that is well-balanced. the scheme Proposedis well-balanced. solution showsProposed excellent solution agreement shows excellent with analytical agreement solution with analytical where the solution results arewhere almost the results indistinguishable are almost overindistinguishable both wet and over dry bedsboth inwet Figure and1 a,b,dry whereasbeds in theFigure scheme 1a,b, without whereas modification the scheme also without gives verymodification satisfactory also predictions gives very with satisfactory slight difference predictions from with the analyticalslight difference solution from when the flow analytical changes fromsolution subcritical when toflow supercritical; changes from the difference, subcritical however, to supercritical; is indistinguishable the difference, on the plottedhowever, scale. is Weindistinguishable observe that the on water the plotted level estimated scale. We by observ proposede that scheme the water gives slightlylevel estimated better results by proposed than the scheme withoutgives slightly modification. better results than the scheme without modification.

(a)

Figure 1. Cont.

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(b)

Figure 1. Comparison of water levels for dam-break overover rigidrigid bedbed for:for: (a)) wetwet bed;bed; andand ((bb)) drydry bed.bed.

4.2. Bed-Slope Aggradation This testtest casecase modelsmodels aa sediment-ladensediment-laden flowflow over a slopingsloping bedbed thatthat emulatesemulates flowsflows inin mountainous terrainsterrains triggered triggered by torrentialby torrential rains. rains. A uniform A uniform supercritical supercritical flow with flow constant with sedimentconstant supplysediment enters supply the domainenters the from domain the left from boundary. the left Bed boundary. evolution isBed observed evolution until is the observed equilibrium until state the equilibrium state is reached by gradual aggradation of the bed. Boundary conditions at x = 0 and L is reached by gradual aggradation of the bed. Boundary conditions at x = 0 and L are set equal to qs q andare set 0, respectively. equal to s In and order 0, respectively. to compare our In resultsorder to with compare [13]—based our results on similar with governing[13]—based equations on similar to thisgoverning study—we equations used the to this MPM study—we formula to used calculate the MP sedimentM formula discharge to calculate with a sediment mean sediment discharge diameter with ofa mean 1.65 mm sediment and porosity diameter (p) of 0.42.1.65 mm The analyticaland porosity solution (p) of of 0.42. this problemThe analytical was presented solution byof [this40], givenproblem as: was presented by [40], given as: 8 ∞ π pL ´ xq  t{T0 π ()Lx−  b px, tq() “ =+Ane tT0sin ()p2n ` 1q (37) bxt,sin21 Aen 2L n  (37) n“0 ÿ n„=0  ˆ2L  ˙ where 8 denotes the equilibrium state; An is the Fourier Series coefficient, estimated by Equation (39); where ∞ denotes the equilibrium state; An is the Fourier Series coefficient, estimated by Equation (39); and T0 is the response time for the problem given as: and T0 is the response time for the problem given as: 8 2 1 Bqs ∞ 2L2 T0 “ 12∂q L (38) =1 ´ p BS s π T0« 0 q“cstff (38) 1−∂pˆ S ˙ ˆπ ˙ 0 qcst= n p´1q 8 8 A “ n L pS ´ S q n ()−1 2 2 0 0 (39) =−p2n ` 1q π8 ∞ An LS()00 S (39) ()21n + 22π The channel characteristics are: length (L) = 6.90 m; width = 0.5 m; bed roughness = 0.0165 s¨ m´1/3; 3 and slopeThe channelS0 = 2.4%. characteristics The flow parameters are: length are (L fixed) = 6.90 as: waterm; width discharge = 0.5 qm;= 7.98bed mroughness/s and sediment = 0.0165 3 discharges·m−1/3; and upstream slope S0q s= =2.4%. 0.147 The m /s. flow These parameters parameters aregive fixed a responseas: water time discharge of T0 = q 1670 = 7.98 s and m3/s a finaland equilibriumsediment discharge slope, S upstream, equal to q 3.03%.s = 0.147 The m3 results/s. These are parameters compared give with a the response analytical time and of T numerical0 = 1670 s solutionand a final in Figureequilibrium2. After slope, start S, of equal the flowto 3.03%. at t =The 0, bedresults evolution are compared goes through with the a transientanalytical state and (observednumerical at solutiont = 900 andin Figure 1800 s) 2. until After the start final of equilibrium the flow at statet = 0, isbed reached evolution at t = goes 3600 through s. Good a agreement transient isstate observed (observed between at t = the900 computedand 1800 s) and until analytical the final solutions equilibrium at all state time is instants.reached at Our t = scheme3600 s. Good gives slightlyagreement better is observed approximations between for the aggradation computed than and theanalytical results ofsolutions [13]. This at applicationall time instants. shows Our the applicabilityscheme gives of slightly our scheme better forapproximations modeling sloping for aggradation rivers. This than also the demonstrates results of [13]. the capabilityThis application of the modelshows tothe compute applicability the equilibrium of our scheme slope basedfor modeling on aggradation sloping and rivers. degradation This also of channels,demonstrates which the is usefulcapability for limitingof the model sedimentation to compute of waterthe equilibrium bodies. slope based on aggradation and degradation of channels, which is useful for limiting sedimentation of water bodies.

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Figure 2. Bed aggradation from sediment overloading. Figure 2. Bed aggradation from sediment overloading.

4.3. Dam-Break in an Erodible Channel 4.3. Dam-Break in an Erodible Channel We simulate a dam-break case on a mobile bed to assess the performance of proposed scheme We simulate a dam-break case on a mobile bed to assess the performance of proposed scheme for high speed flows associated with rapid bed deformation and high speed wave propagation. This for high speed flows associated with rapid bed deformation and high speed wave propagation. case was previously also tested by [41,42] to verify their models which were also based on coupled This case was previously also tested by [41,42] to verify their models which were also based on coupled shallow-water and Exner equations. shallow-water and Exner equations. The computational domain consists of a frictionless erodible channel bed of 50 m length with a The computational domain consists of a frictionless erodible channel bed of 50 m length with dam at the middle of the channel. Following [41,42] the initial water depth of 1 m is fixed upstream a dam at the middle of the channel. Following [41,42] the initial water depth of 1 m is fixed upstream of the dam and downstream is considered dry. Computations of sediment discharge by Grass of the dam and downstream is considered dry. Computations of sediment discharge by Grass formula formula and MPM formula have no significant effects on the numerical results for bed evolution, and MPM formula have no significant effects on the numerical results for bed evolution, water depths, water depths, and velocity [41,42]. We used Grass formula to compute sediment discharge with and velocity [41,42]. We used Grass formula to compute sediment discharge with sediment porosity (p) sediment porosity (p) of 0.4 and A is taken equal to 0.004. At time t = 0, the dam is instantaneously of 0.4 and A is taken equal to 0.004. At time t = 0, the dam is instantaneously removed; the consequent removed; the consequent water level, bed deformation, and velocity is compared with exact water level, bed deformation, and velocity is compared with exact Riemann solution of the problem in Riemann solution of the problem in Figure 3 (readers are referred to [42] for details on exact Figure3 (readers are referred to [42] for details on exact Riemann solution). Riemann solution). For comparison with the exact solution, the results for water level and bed deformation For comparison with the exact solution, the results for water level and bed deformation are are non-dimensionalized with respect to initial water depth in the channel, and velocity is non-dimensionalized with respect to initial water depth in the channel, and velocity is non-dimensionalised by wave celerity. The results are plotted at 2 s interval up to 5 s. The numerical non-dimensionalised by wave celerity. The results are plotted at 2 s interval up to 5 s. The numerical results show the correct propagation of wave-front, bed evolution, and velocity. Shocks introduced results show the correct propagation of wave-front, bed evolution, and velocity. Shocks introduced by the bed discontinuity are well captured at all instants and the numerical solution is free of by the bed discontinuity are well captured at all instants and the numerical solution is free of oscillations. Numerical solution (proposed and without modification) show lag to exact solution oscillations. Numerical solution (proposed and without modification) show lag to exact solution at at all time instants, which was also observed by [41]. In Figure3b, we observe that numerical solution all time instants, which was also observed by [41]. In Figure 3b, we observe that numerical solution slightly overestimates bed deformation with time advancement, which is a characteristic feature of slightly overestimates bed deformation with time advancement, which is a characteristic feature of computing sediment discharge by Grass formula due to neglecting water depth in flux estimation. computing sediment discharge by Grass formula due to neglecting water depth in flux estimation. Overall, flow variables show very good agreement with exact solution with the proposed scheme Overall, flow variables show very good agreement with exact solution with the proposed scheme giving better results in all cases. giving better results in all cases.

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(a)

(b)

(c)

Figure 3. Comparison of (a) water level; (b) bed deformation;deformation; and (c) velocity with exact solution for dam-break over an erodibleerodible bed.bed.

4.4. Dam-Break over Sand Bed Experimental studies for sand bed evolution under dam-break waves were conducted at the Experimental studies for sand bed evolution under dam-break waves were conducted at the Civil Civil Engineering Department of Universite Catholique de Louvain (UCL), Belgium [43]. The Engineering Department of Universite Catholique de Louvain (UCL), Belgium [43]. The experimental experimental results were later validated by numerical simulations in references [8,22,44,45] as well results were later validated by numerical simulations in references [8,22,44,45] as well as by others. as by others. The tests were conducted for idealized dam-break cases over an erodible bed. The The tests were conducted for idealized dam-break cases over an erodible bed. The experimental set-up experimental set-up included a flume 6 m long, 0.25 m wide and 0.70 m deep. A thin steel gate, fixed in the center of flume, was used as a dam in the experiments. In the original experiments [43], the erosion was observed for two bed materials, uniform coarse sand and Polyvinyl Chloride (PVC)

Water 2016, 8, 212 12 of 20 included a flume 6 m long, 0.25 m wide and 0.70 m deep. A thin steel gate, fixed in the center of flume, was used as a dam in the experiments. In the original experiments [43], the erosion was observed for Water 2016, 8, 212 12 of 20 two bed materials, uniform coarse sand and Polyvinyl Chloride (PVC) pellets. For brevity purposes, pellets.this study For simulates brevity thepurposes, experiment this withstudy a sandsimulates bed, extendingthe experiment on both with sides a ofsand the dam.bed, extending The thickness on bothof the sides sand of bed the varied dam. forThe different thickness configurations of the sand bed to accountvaried for for different bed discontinuity configurations at dam to location,account forand bed these discontinuity configuration at dam were location, modeled and in the these study. configuration were modeled in the study. The following [43] [43] four different cases are simulated in this study, which primarily differ in upstream bed layer thickness and initial water depth. Following previous studies [22], [22], the values of the dimensional constant ( A),), sediment porosity porosity (p), (p), and and Manning’s coefficient coefficient ( (n)) are set equal to 0.01, 0.4, and 0.0165, respectively, for all configurations.configurations. Figure 44 shows shows thethe initialinitial configurationsconfigurations forfor which the dam-break flowflow is simulated using the proposedproposed scheme. The domain was discretized into 400 cells.cells. TheThe modelmodel waswas run run using using a refineda refined mesh, mesh, but but no no significant significant improvement improvement in the in resultsthe results was wasobserved observed (discussed (discussed further further in Section in Section 4.5). A4.5). CFL A numberCFL number of 0.6 wasof 0.6 used was due used to due the explicitto the explicit nature natureof the scheme.of the scheme. The dam The was dam removed was removed instantaneously instantaneously at t = 0, at and t = the0, and results the results were obtained were obtained for the forfour the water four surface water and surface bed profiles.and bed Model profiles. results Mode werel results compared were to thecompared experimental to the measurements experimental measurementsof Spinewine and of Spinewine Zech (2007) and and Zech to the (2007) numerical and to schemethe numerical without scheme the modification without the introduced modification in introducedEquations (31) in Equations and (32). (31) and (32).

(a) (b)

Figure 4. InitialInitial Configurations Configurations for for dam-break dam-break over over sand sand bed: bed: ( (aa)) Configuration Configuration 1; 1; ( (bb)) Configuration Configuration 2; 2; (c) ConfigurationConfiguration 3;3; andand ((d)) ConfigurationConfiguration 4.4.

Figures 5–8 show the comparisons between the simulated results and experimental Figures5–8 show the comparisons between the simulated results and experimental measurements. measurements. Results for configuration 1 are plotted in Figure 5a–c at 0.25 s, 0.75 s, and 1.25 s, Results for conﬁguration 1 are plotted in Figure5a–c at 0.25 s, 0.75 s, and 1.25 s, respectively, after the respectively, after the dam-break. At all time intervals, the results using the proposed scheme are in dam-break. At all time intervals, the results using the proposed scheme are in very good agreement very good agreement with the experimental results. We observe that the model without with the experimental results. We observe that the model without modiﬁcation overestimates bed modification overestimates bed erosion particularly immediately after the dam-break, and is not erosion particularly immediately after the dam-break, and is not able to follow the experimental able to follow the experimental wave-front, whereas the proposed scheme gives very good wave-front, whereas the proposed scheme gives very good agreement with the erosion and the agreement with the erosion and the wave-fronts observed in the experiment. The position of wave-fronts observed in the experiment. The position of wave-fronts given by the proposed scheme wave-fronts given by the proposed scheme at t = 0.25 s and 0.75 s are in good agreement with at t = 0.25 s and 0.75 s are in good agreement with previous results, while at t = 1.25 s it is slightly previous results, while at t = 1.25 s it is slightly above the measured, showing slow movement of the above the measured, showing slow movement of the wave, due to bed friction. As water propagates wave, due to bed friction. As water propagates downstream, more particles come in contact with the downstream, more particles come in contact with the bed surface compared with the initial time bed surface compared with the initial time intervals resulting in slower propagation of the intervals resulting in slower propagation of the wave-front. However, considering human error in wave-front. However, considering human error in experimental measurements, and considering the experimental measurements, and considering the limitations of numerical modeling, we can conclude limitations of numerical modeling, we can conclude that, overall, the agreement between numerical that, overall, the agreement between numerical results and experimental results using the proposed results and experimental results using the proposed scheme is very satisfactory. scheme is very satisfactory.

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(a)

(b)

(c)

Figure 5.5. NumericalNumerical and and experimental experimental results results of of dam-break for configuration conﬁguration 1: ( (aa)) tt = 0.25 0.25 s; (b) tt = 0.75 s; and ( c) t = 1.25 s.

Figure 6 shows the comparison of water surface and bed profiles for configuration 2 at t = 0.5 s. Figure6 shows the comparison of water surface and bed profiles for configuration 2 at t = 0.5 s. Although the bed profile is in fair agreement with experimental results, the wave-front in the Although the bed profile is in fair agreement with experimental results, the wave-front in the present present study slightly leads the measured. The leading wave-front may be due to the excessive study slightly leads the measured. The leading wave-front may be due to the excessive erosion of erosion of the bed step, which, according to Evangelista et al. (2013), occurs due to a greater angle of the bed step, which, according to Evangelista et al. (2013), occurs due to a greater angle of internal internal friction than a vertical slope of the bed step. The angle of internal friction is not considered friction than a vertical slope of the bed step. The angle of internal friction is not considered in the in the Grass formula, which may be the reason for the slightly lower bed scour estimates and the leading Grasswave fronts. formula, which may be the reason for the slightly lower bed scour estimates and the leading wave fronts.

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Figure 6. Numerical and experimental results of dam-break for configuration 2 at t = 0.5 s. FigureFigure 6.6. Numerical andand experimentalexperimental resultsresults ofof dam-breakdam-break forfor configurationconfiguration 22 atatt t= = 0.50.5 s.s. Results for configuration 3 at t = 0.25 s, 0.75 s and 1.25 s after dam-break are presented in Figure 7. Results for configuration 3 at t = 0.25 s, 0.75 s and 1.25 s after dam-break are presented in Figure7. InResults Figure 7a,for theconfiguration numerical studies 3 at t = under-predict 0.25 s, 0.75 s andthe bed1.25 scour; s after this dam-break is due to theare suddenpresented removal in Figure of 7. In Figure7a, the numerical studies under-predict the bed scour; this is due to the sudden removal of In Figurethe gate 7a, in theexperiments, numerical which studies causes under-predict excessive soil th ecollapse bed scour; at the this start is dueof dam-break. to the sudden However, removal at of the gate in experiments, which causes excessive soil collapse at the start of dam-break. However, at thelater gate instants,in experiments, when the which bed causeslayer smoothens, excessive soilthe collapseresults of at the the proposed start of dam-break. scheme are However,in good at laterlateragreement instants,instants, whenwith when the the experimentalthe bed bed layer layer smoothens, results smoothens, as compared the results the toresults of the the scheme proposedof the without proposed scheme modification, arescheme in good asare agreementshown in good withagreementin the Figure experimental with 7b,c. the The experimental differences results as compared in results position as to of compared the wave scheme fronts to without likelythe scheme occur modification, forwithout the same modification, as shown reason indiscussed Figure as shown7 b,c. Thein Figurefor differences configuration 7b,c. The in position differences1. of wave in position fronts likely of wave occur fronts for the likely same occur reason for discussed the same for reason configuration discussed 1. for configuration 1.

(a)

(b)

Figure 7. Cont. Figure 7. Cont.

(b)

Figure 7. Cont.

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Water 2016, 8, 212 15 of 20

(c)

Figure 7. Numerical and experimental results of dam-break for Configuration 3: (a) t = 0.25 s; Figure 7. Numerical and experimental results of dam-break for Configuration 3: (a) t = 0.25 s; (b) t = 0.75 s; and (c) t = 1.25 s. (c) (b) t = 0.75 s; and (c) t = 1.25 s. Figure 7. Numerical and experimental results of dam-break for Configuration 3: (a) t = 0.25 s; Figure(b) t = 80.75 shows s; and the (c) tresults = 1.25 s. for configuration 4 after t = 0.25 s, 0.50 s and 1.0 s after dam-break. TheFigure formation8 shows of the the hydraulic results for jump configuration is observed at 4 afterx = 3.2t m= and 0.25 x s, = 0.503.6 m s at and t = 1.00.25s s afterand t dam-break.= 0.50 s, Therespectively, formationFigureof and8 shows the occurs hydraulic the whenresultsjump the for flow configuration is observed is in transition at4 afterx = from 3.2t = 0.25 m super and s, 0.50 criticalx = s 3.6 and to m 1.0sub at st critical.after= 0.25 dam-break. sHowever, and t = 0.50 s, respectively,theThe jump formation vanishes and of occurs the at hydraulict = when 1.0 s and thejump flowthe is observedwave is in transitionfront at xbecomes = 3.2 from m andflatter super x = due3.6 critical m to at the t to= 0.25inertia sub scritical. and of tthe = 0.50 However,water s, thedownstream jumprespectively, vanishes of and the at occursdam.t = 1.0Following when s and the thetheflow previous wave is in transition front trend, becomes thefrom proposed super flatter critical scheme due to to givessub the critical. inertiabetter predictionsHowever, of the water downstreamforthe bed jump evolution of vanishes the dam. and at wave-front Followingt = 1.0 s and propagation the the previous wave frontfor trend, this becomes case the also. proposed flatter Overall due scheme tocomparisons the givesinertia better inof Figuresthe predictions water 5–8 for bedshowdownstream evolution reasonably of and the good wave-frontdam. agreement Following propagation with the previousexperiment for trend, thisal measurements. casethe proposed also. Overall scheme These comparisons givesresults better demonstrate predictions in Figures the 5 –8 for bed evolution and wave-front propagation for this case also. Overall comparisons in Figures 5–8 showproposed reasonably scheme’s good ability agreement to accurately with experimental predict bed evolution measurements. and wave-front These results transition demonstrate under dam-breakshow reasonably flows, with good effi agreementcient handling with experiment of shocks. al measurements. These results demonstrate the the proposed scheme’s ability to accurately predict bed evolution and wave-front transition under proposed scheme’s ability to accurately predict bed evolution and wave-front transition under dam-breakdam-break flows, flows, with with efficient efficient handling handling of of shocks. shocks.

(a)

(b) (b) Figure 8. Cont. Figure 8. Cont. Figure 8. Cont.

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((cc))

FigureFigure 8. 8.Numerical Numerical and and experimental experimental resultsresults ofof dam-breakdam-break for Configuration 4:4: (a(a) ) t t == 0.250.25 s; s; Figure 8. Numerical and experimental results of dam-break for Configuration 4: (a) t = 0.25 s; (b()b t) = t 0.5= 0.5 s; s;and and (c ()c t) =t =1.0 1.0 s. s. (b) t = 0.5 s; and (c) t = 1.0 s. 4.5.4.5. Sensitivity Sensitivity Analysis Analysis 4.5. Sensitivity Analysis A Asensitivity sensitivity analysis analysis is is performed performed toto studystudy thethe effects of mesh resolutionresolution onon computationalcomputational time,Atime, sensitivity flow flow variables variables analysis (discharge), (discharge), is performed and and bed tobed study elevatioelevatio then.n. effects TheThe model of mesh is simulated resolution forfor on fivefive computational consecutively consecutively time, flowrefined variablesrefined mesh mesh (discharge), systems; systems; the the and number number bed elevation.of of elementselements The fromfrom model thethe coarsest is simulated toto finestfinest for meshmesh five consecutively isis 400,400, 1000, 1000, 2000, 2000, refined ® mesh3500,3500, systems; and and 5000. 5000. the The number The analysis analysis of elements was was performed performed from the onon coarsest anan IntelIntel® to Core finest TMTM mesh i7-3770i7-3770 is 400, (8(8 MM 1000, Cache,Cache, 2000, 3.4 3.4 GHz) 3500,GHz) and 5000.computercomputer The analysis with with 8 was8GB GB performedDDR3 DDR3 RAM.RAM. on FigureFigure an Intel 99 ®showsshowsCore thethe TM increase i7-3770 inin (8 simulationsimulation M Cache, time 3.4time GHz) withwith computermeshmesh withrefinementrefinement 8 GB DDR3 for for configurations RAM. configurations Figure 91 1 showsand and 4. 4. It theIt cancan increase bebe observedobserved in simulation in Figure time99 thatthat with therethere mesh areare nono refinement significant significant for changes in water depth and velocity with mesh refinement. This shows that the proposed scheme configurationschanges in water 1 and depth 4. It canand bevelocity observed with inmesh Figure refinement.9 that there This are shows no significantthat the proposed changes scheme in water satisfies the primary requirement of robust solvers to be computationally efficient and independent satisfies the primary requirement of robust solvers to be computationally efficient and independent depthof and the velocitymesh resolution. with mesh refinement. This shows that the proposed scheme satisfies the primary requirementof the mesh of resolution. robust solvers to be computationally efficient and independent of the mesh resolution.

CPU time Config 1 CPU time Config 4 CPUMax time Erosion Config Config1 1 CPUMax timeErosion Config Config4 4 MaxMax Erosion Water dischargeConfig1 Config1 Max WaterErosion discharge Config4 Config4 1.00 Max Water discharge Config1 Max Water discharge Config4 0.25 1.00 0.25 0.20 0.20 0.15 0.15 0.10 0.20 0.10 0.20 0.05 0.05 CPU Time (seconds)CPU Time 0.00

CPU Time (seconds)CPU Time 0.00 -0.05 -0.05 0.04 -0.10 0.04 300 3000 -0.10 300Number of cells 3000 Number of cells Figure 9. CPU-time performance and Sensitivity of hydraulic and morphologic parameters to mesh refinement. FigureFigure 9. CPU-time 9. CPU-time performance performance and Sensitivity and Sensitivity of hydrauli ofc and hydraulic morphologic and morphologicparameters to mesh parameters refinement. to meshSensitivity refinement. of the sediment parameters, including Manning’s roughness (n) and porosity (p) are alsoSensitivity analyzed forof the the sedimentlaboratory paramete scale testrs, discusse includingd in Manning’sSection 4.4. roughnessWe consider (n )configurations and porosity 1(p) and are also4 for analyzed comparison for the purposes. laboratory Figure scale 10 test shows discusse the varid ination Section in results4.4. We as consider each parameter configurations varies with1 and Sensitivity of the sediment parameters, including Manning’s roughness (n) and porosity (p) are 4 forthe comparisonrest kept constant. purposes. In FigureFigure 10a,b,10 shows no signthe ificantvariation changes in results are observedas each parameter in bed erosion varies with with also analyzed for the laboratory scale test discussed in Section 4.4. We consider configurations 1 and 4 thechanges rest kept in sedimentconstant. parameters,In Figure 10a,b, which no again sign ificantcan be changesattributed are to observedthe presence in bed of downstreamerosion with forchanges comparisonwater. Figurein sediment purposes. 10b shows parameters, Figure that with 10 whichchangesshows again thein be variation dcan roughness, be attributed in results the hydraulic to as the each jumppresence parameter extends of downstream variesbeyond withthe the restwater. keptmeasured constant.Figure value 10b In shows(shown Figure that in 10 Figure a,b,with no changes 8b), significant which in be is ddue changes roughness, to the are lagging the observed hydraulic wave infront bedjump causing erosion extends a withdelay beyond changesin the the in sedimentmeasuredformation parameters, value of the (shown jump. which in againFigure can8b), bewhich attributed is due to the lagging presence wave of downstream front causing water. a delay Figure in the 10b showsformation that with of the changes jump. in bed roughness, the hydraulic jump extends beyond the measured value (shown in Figure8b), which is due to the lagging wave front causing a delay in the formation of the jump. Water 2016, 8, 212 17 of 20 Water 2016, 8, 212 17 of 20 Water 2016, 8, 212 17 of 20

(a)

(b)

Figure 10. Sensitivity analysis for Configuration( 4b )to model parameters at t = 0.75 s: (a) porosity p; Figure 10. Sensitivity analysis for Configuration 4 to model parameters at t = 0.75 s: (a) porosity p; and and (b) Manning’s roughness n. (b)Figure Manning’s 10. Sensitivity roughness analysisn. for Configuration 4 to model parameters at t = 0.75 s: (a) porosity p; and (b) Manning’s roughness n. For configuration 1, the variation in porosity (0.35–0.38) and Manning’s roughness (0.03–0.04) For configuration 1, the variation in porosity (0.35–0.38) and Manning’s roughness (0.03–0.04) showFor significant configuration increases 1, the in bedvariation erosion, in asporosity seen in (0.35–0.38)Figure 11a,b, and respectively. Manning’s Theroughness increase (0.03–0.04) in values showofshow both significant significant parameters increases increases result from in in bed bed increased erosion, erosion, interactioas asseen seen inn Figure of in sediment Figure 11a,b, 11 respectively.anda,b, water respectively. related The increase to Thedirect increasein contact values in valuesof waterboth of bothparameters particles parameters with result sediment, resultfrom increased from due increasedto absence interactio interactionofn tail-water. of sediment of This sediment and scenario water and relatedalso water results to relateddirect in a contacthigher to direct contactresistanceof water of waterparticles offered particles withto the sediment, withflow sediment,in comparisondue to absence due to to configuration absenceof tail-water. of tail-water. This4. We scenario conclude This also scenario that results the also inabsence a resultshigher of in a highertail-waterresistance resistance makesoffered offered theto theerodible toflow the in flowbed comparison more in comparison sensitive to configuration toto configurationchanges 4. in We sediment 4.conclude We conclude parameters, that the that absence theand absence can of ofsignificantlytail-water tail-water makes makes affect thethe the bederodible erodible evolution bed bed moreunder more sensitivewave-f sensitiveront toto propagation,changes changes in in sedimentas sediment observed parameters, parameters,in Figure 11.and and can can significantlysignificantly affect affect the the bed bed evolution evolution under under wave-frontwave-front propagation, as as observed observed in in Figure Figure 11. 11 .

(a)

Figure( 11.a) Cont. Figure 11. Cont. Figure 11. Cont.

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(b)

Figure 11. Sensitivity analysis for Configuration 1 to the model parameters at t = 0.75 s: (a) porosity p; Figure 11. Sensitivity analysis for Configuration 1 to the model parameters at t = 0.75 s: (a) porosity p; and (b) Manning’s n. and (b) Manning’s n. 5. Conclusions 5. Conclusions This paper introduces a simple and robust model for bed evolution and its effects on water surfaceThis paperprofile introduces under rapidly a simple varying and flows robust using model a formulation for bed evolution comprised and itsof effectscoupled on Saint-Venant water surface profileand underExner equations. rapidly varying A finite flows volume using method a formulation of Godunov comprised type is adopted of coupled to obtain Saint-Venant solutions and for Exnerthe equations.coupled system. A finite The volume proposed method method of Godunov for the calculat type ision adopted of variables to obtain in the solutions star region for has the proven coupled system.to give The better proposed results than method the previous for the calculation numerical st ofudies, variables and is in able the to star reproduce region hasthe provenexperimental to give betterresults results very than satisfactorily. the previous The numerical sensitivity studies, analysis and is ableshows to reproducethat the theproposed experimental scheme results is verycomputationally satisfactorily. Thevery sensitivity efficient, which analysis is important shows that for the schemes proposed modeling scheme sediment is computationally flow since the very efficient,wave structure which is involved important is forusually schemes complicated. modeling It sedimentwas observed flow for since all cases the wave that the structure values involvedof bed is usuallyfriction, complicated.porosity, and Itdimensional was observed constant for all ( casesA), which that therepresents values ofthe bed intensity friction, of porosity,interaction and dimensionalbetween water constant and (Asediment), which particles, represents significantly the intensity influence of interaction the stability between and water accuracy and sedimentof the particles,model, and significantly the position influence of the hydraulic the stability jump. and Anal accuracyytical solutions, of the original model, experimental and the position data, ofand the hydraulicnumerical jump. results Analytical are adopted solutions, for comparisons. original experimental The accuracy data, of the and model numerical is tested results for a number are adopted of forcases, comparisons. highlighting The its accuracy efficiency of theand modeladaptability is tested under for various a number conditions. of cases, highlighting its efficiency andAcknowledgments: adaptability under This various research conditions. was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea Government (MSIP) (No. 2015R1A2A1A15054097). Acknowledgments: This research was supported by the National Research Foundation of Korea (NRF) grant fundedAuthor by theContributions: Korea Government Khawar (MSIP)Rehman (No. designed 2015R1A2A1A15054097). and performed the simulations, analyzed the data, and wrote the paper; Yong-sik Cho contributed providing materials/analysis tools and ideas in developing the Authormanuscript. Contributions: Khawar Rehman designed and performed the simulations, analyzed the data, and wrote the paper; Yong-sik Cho contributed providing materials/analysis tools and ideas in developing the manuscript. Conflicts of Interest: The authors declare no conflict of interest. Conflicts of Interest: The authors declare no conflict of interest. References References 1. Carrivick, J.L.; Manville, V.; Graettinger, A.; Cronin, S.J. Coupled fluid dynamics-sediment transport 1. Carrivick, J.L.; Manville, V.; Graettinger, A.; Cronin, S.J. Coupled fluid dynamics-sediment transport modelling modelling of a Crater Lake break-out lahar: Mt. Ruapehu, New Zealand. J. Hydrol. 2010, 388, 399–413. J. Hydrol. 2010 388 2. of aKim, Crater J.; Ivanov, Lake break-out V.Y.; Katopodes, lahar: Mt. N.D. Ruapehu, Modeling New erosion Zealand. and sedimentation, coupled, 399–413. with [CrossRef hydrological] 2. Kim,and J.; overland Ivanov, V.Y.;flow Katopodes,processes at N.D.the watershed Modeling scale. erosion Water and Resour. sedimentation Res. 2013, coupled49, 5134–5154. with hydrological and 3. overlandGuan, M.; flow Wright, processes N.G.; at Sleigh, the watershed P.A.; Carrivick, scale. Water L.J. Assessment Resour. Res. of 2013hydro-morphodynamic, 49, 5134–5154. [CrossRef modelling] and 3. Guan,geomorphological M.; Wright, N.G.; impacts Sleigh, of a P.A.; sediment-charged Carrivick, L.J. jökulhlaup, Assessment at of Sólheimajökull, hydro-morphodynamic Iceland. J. modellingHydrol. 2015 and, geomorphological530, 336–349. impacts of a sediment-charged jökulhlaup, at Sólheimajökull, Iceland. J. Hydrol. 2015, 530, 4. 336–349.Huang, [CrossRef W.; Cao, ]Z.; Carling, P.; Pender, G. Coupled 2D hydrodynamic and sediment transport modeling 4. Huang,of megaflood W.; Cao, due Z.; Carling,to glacier P.;dam-break Pender, G.in CoupledAltai Mountains, 2D hydrodynamic Southern Siberia. and sediment J. Mt. Sci.transport 2014, 11, 1442–1453. modeling of 5. megafloodBenkhaldoun, due toF.; glacier Sahmim, dam-break S.; Seaïd, in M. Altai A Mountains,two-dimensional Southern finite Siberia.volumeJ. morphodynamic Mt. Sci. 2014, 11 model, 1442–1453. on [CrossRefunstructured] triangular grids. Int. J. Numer. Methods Fluids 2010, 63, 1296–1327. 5. 6. 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