Journal of Applied Fluid Mechanics, Vol. 13, No. 5, pp. 1611-1622, 2020. Available online at www.jafmonline.net, ISSN 1735-3572, EISSN 1735-3645. DOI: 10.36884/jafm.13.05.30842

A Semi-Coupled Model for Morphological Flow Simulation in Bend

K. Bora and H. M. Kalita†

Department. of Civil Engineering., National Institite of Technology, Meghalaya, Shillong,793003, India

†Corresponding Author Email: [email protected]

(Received August 9, 2019; accepted February 8, 2020)

ABSTRACT

This paper presents a new efficient and robust numerical model for morphological flow simulation in river bends. The hydrodynamic model is developed by solving the two dimensional (2D) shallow water equations using a total variation diminishing (TVD) MacCormack predictor corrector scheme. The present TVD method is very simple and provides accurate results free from numerical oscillations near sharp gradient. The effective stresses are modeled by using a constant eddy viscosity model. The transport model solves the using a simple forward time and central space (FTCS) finite difference algorithm. The model incorporates the helical flow and the transverse bed slope effects on the sediment direction computation. These models are coupled using the semicoupled approach. The present semicoupled model is used to replicate two popular experimental test cases of both tight and loose bends. The obtained results in terms of bed level variation reveal that the model can accurately simulate several features of the bed changes such as oscillations of the transverse bed profile with the formation of point bars and pools along the banks. The values obtained for three widely used statistical parameters show the applicability of the present model for this type of complex scenarios.

Keywords: Shallow water equations; Exner equation; TVD MacCormack scheme; FTCS scheme; Semi- coupled model; Sediment transport model.

NOMENCLATURE

C chezy coefficient rc radius of curvature of streamlines Cr local courant number S specific gravity of sediment D50 median grain size u velocity components in x direction fs sediment shape factor v velocity components in y direction g gravitational acceleration Z bed elevation h depth of flow zexp experimental non dimensional relative bed J jacobian determinant variation znum experimental non K von karman constant dimensional relative bed variation 푘 퐿푖,푗 spatial and temporal grid level lξ finite difference operators in ξ direction α angle of sediment transport direction lη finite difference operators in η direction αt empirical coefficient n mannings roughness parameter υt eddy viscosity nn number of grids λ porosity of the bed material qs total per unit width ξ computational domain in x direction qsx bed load discharge/unit width in x η computational domain in y direction direction θ shields parameter qsy bed load discharge/unit width in y δ direction of bed shear stress direction φ cylindrical coordinates of bed shear stress r cylindrical coordinates

1. INTRODUCTION (or helical flow), turbulence, river bed form change, river plan form change, etc. In a river bend, scour Flow in a river bend is a complex phenomenon as it generally occurs along the outer while the involves several processes such as, occurs along the inner one, due to the

K. Bora and H. M. Kalita / JAFM, Vol. 13, No. 5, pp. 1611-1622, 2020. difference of centrifugal forces between the upper channel bend is truly a three dimensional (3D) and the lower layer of flow (Kassem and Chaudhry process in nature and therefore it is advantageous to 2002). can undermine or destroy the river use 3D models in this regard. This has leads to training structures, while deposition may obstruct development of many 3D models for this purpose. water intakes or the navigations (Vasquez Wang and Adeff (1986) developed a 3D model for 2005). water flow and sediment transport using finite element method. Demuren (1991) proposed a 3D Modelling can provide some prior idea regarding numerical model for morphological simulation in scouring, deposition, etc., which is of significant channel bend employing k-ε turbulence model. Wu practical concern. Models may be classified into et al. (2000) presented a 3D numerical model for analytical (or theoretical), experimental or hydrodynamic and morphological flow simulation of numerical, based on the approach used. El-Khudairy a channel bend. They solved the Reynolds-averaged (1970) developed one of the pioneer analytical Navier-Stokes equations using finite volume method models for determining the steady state transverse for this purpose. Olesen (1985) developed 3D bed profile in a channel bend. After that several numerical model for computing the analytical models (Engelund 1974; Kikkawa, Ikeda, planform, meander wave-length and downstream and Kitagawa 1976; Zimmerman and Kennedy 1978; meander migration in a meandering channel. Odgaard 1981; Parker, Sawai, and Ikeda 1982; Khosronejad et al. (2007) developed a 3D model for Seminara, Zolezzi, Tubino, and Zardi 2001) are numerical simulation of flow and sediment transport reported in the literature for bend flow simulation. in channel bend and observed that k-ω model These models however, are applicable only for performs better than k-ε model. Ramamurthy et al. simple bends due to several assumptions associated (2012) proposed a 3D model for morphological flow with it. Moreover, they are also restricted to steady simulation in sharp channel bend and observed that analysis due to the same reason. Experimental Volume of Fluid approach is excellent in this regard. analysis can overcome these issues to a great extent. Khosronejad et al. (2015) developed a 3D numerical Hooke (1974) did one of the earliest experimental model in curvilinear coordinates for studying the bed studies in the University of Uppsala, Sweden. He elevation changes in a channel bend equipped with observed that the region of maximum bed shear in- rock structures. Khosronejad et al. (2018) stress and maximum sediment discharge is on the extended the same model to determine the optimal point in the upstream of the bend. It crosses the location of rock-vane in channel bends. channel centerline in the middle part of the bend and follows the concave bank to the next point bar Despite the fact that 3D models are superior for downstream. Sutmuller and Glerum (1980) morphological flow simulation in channel bends, the conducted experiments in 180◦ channel bend and computational cost of 3D models is very high and found out the point bar height and pool depth. often prohibitive for engineering analysis and design Struiksma (1983) did similar experiments for 140◦ studies (Begnudelli, Valiani, and Sanders 2010). channel bend considering three different discharge Under certain considerations and limitations, it is values. For each discharge value, he found out the feasible to use 2D depth averaged models to conquer velocity and bed profiles at different cross sections this problem, specifically for certain engineering throughout the channel length. Garcia and Ni ñ o applications (Lane and Ferguson 2005). This has (1993) tried to determine the height, wavelength, and motivated many researchers to develop 2D celerity of alternate bars in an experimental channel numerical models for flow and sediment transport in bend. Rahman et al. (1996) performed experimental channel bends. Flokstra and Koch (1981) proposed studies to analyze the morphological changes of one of the pioneer 2D numerical models for flow and meandering channel with special attention to bank sediment transport in channel bends. They applied erosion. Roca et al. (2007) developed physical their model to simulate the bed elevation changes in models for simulating the bend scour and its a 180◦ channel bend and found that their model could prevention by outer bank footing. Abad and Garcia reproduce the point bar near the the inner bank and (2009) conducted experimental studies to examine also the pool near the outer bank. However, their the effect of bend orientation on river bed model could not replicate the asymmetry of the bed morphodynamics. Uddin and Rahman (2012) topography. In addition, their model is applicable conducted an experimental study for observing the only for steady flow. Struiksma et al. (1985) flow and scour pattern in a bend of River Yamuna, proposed a 2D steady numerical model for Bangladesh. In a more recent study, Lee and Dang morpholgical simulation in channel bends. However, (2018) did experimental studies to determine the their model can only be applied for channel geometry depth in a 90◦ channel bend. which can be represented as a series of circular bends of constant width. Shimizu and Itakura (1989) Even though experimental model can easily be used developed another steady 2D morphological model to analyze the scouring, deposition and all related in cylindrical coordinate sys-tem. However, it is phenomenon in a channel bend, it has some inherent valid only for definite geometry. In order to limitations. Some of such are high investment and overcome this steady flow and specific geometry maintenance cost, scale issues, repetitive approach, issues, Kassem and Chaudhry (2002), for the first etc. With the advent of computer powers in last few time presented a 2D boundary fitted numerical model decades, researchers have started giving effort for for simulation of bed deformation in channel bends. numerical simulation of morphological processes in The problem with their model is that it is based on a channel bend. The morphological process in a numerical model, which is subjected to artificial

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K. Bora and H. M. Kalita / JAFM, Vol. 13, No. 5, pp. 1611-1622, 2020. viscosity for stability. The model is very much where S is elevation of water surface, h is depth of sensitive to the artificial viscosity coefficient. flow, u, v are velocity components in x and y Vasquez et al. (2008) developed a 2D depth averaged directions, respectively, g is gravitational model for simulating the scour and the depsotion in aceleration, and n is Mannings roughness parameter. a meandering channel. For this purpose, they solved In the momentum equations (Eqs. 2 and 3), υt is the the 2D shallow water equations using unstructured eddy viscocity due to turbulence and is calculated triangular elements. Abad et al. (2008) developed a here using the depth averaged parabolic eddy numerical model for morphological flow simulation viscosity model (Wu 2007) given as in channel bends using curvilinear coordinates and promising results are obtained. However, it is again gn 2 22 (4) restricted to steady flow and could not locate the tth1 () u v point bar downstream of the curve when applied in a h 3 channel bend. Begnudelli et al. (2010) proposed another 2D model in the same regard by using where αt is an empirical coefficient between 0.3 and Godunov-type finite volume method. Their model 1.0 (Elder 1959). This value is kept as 0.7 throughout also could not replicate the point bar and the pool the study. downstream of the curve, when applied to Sutmuller The planform of natural is mostly irregualr in and Glerum (1980) test channel. shape and application of the above governing The present work proposes a simple but effective equations in that case becomes very difficult. In numerical model for 2D simulation of hydrodynamic order to overcome this issue, the equations are and morphological flow simulation in river bend. For transformed from the physical domain (x, y) to a the hydrodynamic portion, it solves the unsteady computational domain (ξ, η) and given as (Anderson shallow water equations in a boundary fitted et al. 1984) coordinate system using a TVD based MacCormack ()JS   predictor corrector scheme. Therefore, it may be (huy  hvx  )  ( hvx   huy  )  0 applied to complex river planform as well as for t   unsteady cases. The advantage of present TVD (5) version is that it does not require any calculation of    eigen values and eigen vectors. For the 22 ()()()Jhu hu y  huvx   huvx   hu y  morphological modelling, the Exner sediment t   continuity equation is solved using a simple finite S  S gn2 u u 2  v 2 difference scheme. Finally, both the models are  ghy  ghy  J  D12  D  1 coupled using the simple semicoupled manner. h 3 (6) 2. NUMERICAL MODEL    ()()()Jhv huvy  hv22 x  hv x  huvy t      2.1 Hydrodynamic Equations and S  S gn2 u u 2  v 2 Numerical Solution ghx  ghx  J  D34  D  1 Assuming a hydrostatic pressure distribution, the 2D h 3 shallow water equations can be derived by vertically (7) integrating the Reynolds-Averaged Navier Stokes equations (Chaudhry 2007). Neglecting wind shear, where the diffusive terms D1, D2, D3 and D4 are ignoring Coriolis acceleration and for small bottom given as slope, these hydrodynamic governing equations can 22 be written as (Alauddin and Tsujimoto 2012)  1122 D1 vt  y22 ( hu ) x ( hu ) JJ S ()() hu  hv (8)    0 (1) 22 t  x  y 11 y y ()() hu x  x  hu  JJ        ()()()hu  hu2  huv    t  x  y 22  1122 D2 vt  y ( hu ) x ( hu ) S gn2 u u 2  v 2  ()() hu   hu JJ22 hg  [][]   x1  xtt  x  y  y (9) h 3 1122 (2) y y ()() hu x  x  hu  JJ        ()()()hv  huv  hv 2    22 t  x  y  1122 D3 vt  y22 ( hv ) x ( hv ) S gn2 v u 2  v 2  ()() hv   hv JJ hg  [][]   y1  xtt  x  y  y h 3 1122 y y ()() hv x  x  hv  (3) JJ        (10)

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22Falconer 2007).  1122 D4 vt  y22 ( hv ) x ( hv ) JJ The 2D problem is splitted into a sequence of two one dimensional (1D) problems and is solved at each 1122 time step. Using the operator splitting given by y y()() hv x x hv      Strang (1968), Eq. 13 maybe splitted to JJ        (11) LM P (15) In Eqs. 5-11, xξ,yξ,xη,yη are some coefficients t  obtained due to grid transformation and are determined here using central difference LN Q (16) approximations. J is the Jacobian determinant of grid t  transformation and given as where, J x y x y (12)       In matrix form, Eqs. 5-7 can be written as 0 2 2 2 LMN   S gn u u v    R (13) P  ghy  J  D1 t    1 h 3  where, S ghx  D 3  JS huy hvx

 2  L Jhu ,, M   hu y  huvx   Jhv   huvy hv2 x    0  hvx huy S 2 Q ghy D 2 N huvx hu y and  2 hv x huvy 2 2 2  S gn u u v ghx  J1  D 4 2 2 2  S  S gn u u  v h 3 ghy  ghy  J 1   The finite difference sequence followed for the 1D  h 3    problems in Eqs. 15 -16 are given as  DD12  R    Lkk1  l l l l L (17)  S  S gn2 v u 2  v 2  i,, j    i j  ghx ghx J 1   where lξ and lη are the finite difference operators  h 3    along ξ and η directions, respectively. The subscript  DD34  and superscript of L represents the spatial and (14) temporal grid levels, respectively. It can be easily seen from Eq. 17 that values of the primitive flow The above hydrodynamic equations representing the variables are computed at next step by employing 2D unsteady open channel flow are a set of mixed four consecutive sweeps. Equation 15 is solved first hyperbolic partial differential equations. Therefore, in ξ direction using the initial values of the flow it is very difficult to solve them analytically and parameters (sweep 1). After that Eq. 16 is solved in mostly they are targeted with numerical approaches. η direction using the latest updated values (sweep 2). MacCormack predictor corrector scheme After that Eq. 16 is again solved in η direction using (MacCormack 2003) is one of the excellent and the latest updated values (sweep 3). Finally, Eq. 15 widely used numerical techniques in this regard. is solved in ξ direction using the latest updated Many researchers (Liang, Ö zgen, Hinkelmann, Xiao, values, to get the final values at next time step (sweep and Chen 2015; Bellos and Tsakiris 2015; Bellos, 4). It may be mentioned here that sweep 1 and sweep Kourtis, Moreno-Rodenas, and Tsihrintzis 2017) 4 are exactly same except that the initial values are have applied this technique for the shallow water different. Similarly, sweep 2 and sweep 3 are also equations in diverse situations. This second order similar to each other except the initial values. Only accurate method is composed of two steps namely sweep 1 is discussed here in detail to save the space. predictor and corrector steps. Even though the Sweep 2 can easily be implementated by considering method is very simple and efficient, it faces the finite difference approximation in η direction (using dispersion error (Anderson, Tannehill, and Pletcher index j). The first sweep of Eq. 17 in ξ direction can 2016) when applied to hyperbolic equations. A TVD be written as approach is used here to remove the dispersion error. The TVD method proposed by Davis (1984) is p kt k k k L L () M  M   tP (18) utilized here due to its independency on i i i i1 i characteristic transformation (Liang, Lin, and

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Where Z is the bed elevation, qsx and qsy are c kt p p p Li L i () M i1  M i   tP i (19) components of bed load discharges per unit width in  x and y directions, respectively and λ is the porosity of the bed material (a default value of 0.4 is k1 1 p c k considered here). If α is the angle of sediment LLLLTVDi, j() i  i i  i (20) 2 transport direction, the components of bed load transport can be calculated from Where,

k qsx q scos( ), q sy q s sin( ) (30) TVDi G()() r i  G r i 11  L   i  2 (21) in which qs is the total bed load discharge per unit k G()() rii11  G r  L width and is calculated here using the formula given   i  2 by Engelund (1974) as

Equation 21 represents the TVD step added in ξ 2 5 3 C 2 direction. ∆ξ and ∆t are the grid spacings in ξ and qs 0.05 (( S 1) gD50 )  (31) time directions, respectively. Superscripts p and c g stands for predicted and corrected values, respectively. Subscript i is the index in ξ direction. where S is specific gravity of sediment, D50 is median grain size, C is Chezy coefficient and θ is In Eq. 21, Shields parameter. The expression of Shields

k k k parameter is given as LLL11 ii  i  22 2 (22) uv k k k  = 2 (32) LLL11 ii   CSD( 1) i  50 2

k k k k k k Whenever flow takes place in channel bend, a S1.().()().()  S 1   hu 1  hu 1   hv 1  hv 1 i i  i  i  i  i  secondary (or helical) flow is generated due to  2 2 2 2 2 2 ri  k k k k k k centrifugal acceleration. This leads to movement of S1.().()().()  S 1   hu 1  hu 1   hv 1  hv 1 i i  i  i  i  i  bed sediment from the outer bank to the inner bank. 2 2 2 2 2 2 Therefore, the bed changes in a channel bend is (23) directly related to the helical intensity. This helical k k k k k k flow causes the direction of bed shear stress (δ) to S1.().()().()  S 1   hu 1  hu 1   hv 1  hv 1 i i  i  i  i  i  deviate from the mean flow velocity. Employing a  2 2 2 2 2 2 ri  k k k k k k secondary flow correction (Struiksma, Olesen, S1.().()().()  S 1   hu 1  hu 1   hv 1  hv 1 i i  i  i  i  i  Flokstra, and De Vriend 1985), δ can be found out 2 2 2 2 2 2 (24) from The expression for G () in Eq. 21 is u Ah  arctan( ) arctan( ) (33) vr G( f ) 0.5  c  [1  ( f )] (25) c

where rc is the radius of curvature of streamlines and where the flux limiter function φ( f ) is given by is defined as

(ff ) max{0,min(2 ,1)} (26) 3 ()uv22 2 r  (34) and the value of the variable c is given as c u  u  v  v u  22  u  u  Cr(1  Cr ), if Cr  0.5 x  y  x  y c   (27)  0.25,if Cr>0.5 Parameter A in Eq. 33 is considered here to vary with cases and is given as Where Cr is the local courant number given as 2 g ()u gh t A (1 ) (35) Cr  (28) K 2 KC x in which K is Von Karman constant (=0.4). 2.2 Sediment Transport Equation and Numerical Solution Whenever there is some lateral slope in the channel bed, the sediment transport direction deviates from Neglecting grain sorting and considering only bed the bed shear stress (δ) due to gravity force on the load transport, the 2D sediment continuity equation sediment particle. The sediment transport direction can be written as (Struiksma et al. 1985) (α) is then calculated from the below equation (Flokstra and Koch 1981; Struiksma, Olesen, Z 1 q q (sx sy )  0 (29) Flokstra, and De Vriend 1985): t1   x  y

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1 Z corresponding uniform flow depth for that sin  discharge is used at the down-stream boundary. fy  tan  s (36) The solid boundaries are simulated as free slip 1 Z cos  boundary (Anderson, Tannehill, and Pletcher 2016), i.e. normal velocity component is set to zero fxs  and the transverse velocity component is 휕푍 휕푍 extrapolated from interior domain. Regarding the where fs is the sediment shape factor, and are 휕푥 휕푦 morphological model, the sediment transport the bed slopes in x and y directions, respectively. component to solid boundary is set to zero and the In an alluvial channel bend starting from a flat bed, tangential component is extrapolated from the sediment is first transported from the outer bank to interior domain. The bed elevations on the banks the inner bank by the secondary flow. This leads to are also extrapolated from the interior domain. scouring on the outer bank and deposition on the 2.4. Semicoupled Approach inner bank. It produces a transverse slope in the direction of flow. As the transverse slope grows, The hydrodynamic and morphological models sediment starts to move towards the outer bank due presented in the earlier section may be linked with to gravity. Finally, an equilibrium condition is two methodologies such as fully coupled and semi- reached between these two effects; when the coupled approaches. As semicoupled approach has maximum scouring and deposition is observed in the many advantages such as, computationally efficient, bend. ease in incorporating different sediment transport formulas, etc. (Kassem and Chaudhry 2002), the In order to make the present model compatible with present model uses semicoupled approach in this natural rivers, the sediment continuity equation is regard. also transformed from the physical domain (x, y) to a computational domain (ξ, η) and is given as Figure 1 shows the flow chart for the model. The (Anderson, Tannehill, and Pletcher 2016) model starts with setting the initial values for all the flow variables and sediment properties. The 1 ()()JZ q y q x initial bed elevation is also specified. With these t 1   sx sy data, the hydrodynamic model runs and computes (37) the flow variables h, u and v. These flow variables 1  are then used to compute qsx and qsy. The (qsy x  q sx y )  0 1 morphological model is then run and subsequently the new bed elevation (Z) is computed. The model Equation 37 is solved here using a simple finite then checks one convergence criteria difference scheme where the time derivative is (Convergence-A in the flow chart). According to approximated by forward difference and the space this convergence, two conditions are to be fulfilled derivatives are approximated by central difference such as (Soulis 2002), (a) the axial velocity operators (FTCS scheme). The difference equation component averaged over the flow field between after finite difference approximations is given as two successive iterations drops below 0.0005 and 1 (b) the axial sediment transport component ZZZZZZk1  k (1  )( k  k  k  k ) ij, ij ,4 ij , 1 ij ,  1 ijij  1,  1, averaged over the flow field between two t successive iterations drops below 0.0005. If this [{(q y )k  ( q x ) k }  {( q y ) k  ( q x ) k }] 2 (1 )J k sx i1, j sy  i  1, j sx  i  1, j sy  i  1, j convergence is not satisfied, the preceding ij, iterations continue. Else, the model updates the bed t [{(q x )k  ( q y ) k }  {( q x ) k  ( q y )k }] elevation and accordingly new value of h is 2 (1 )J k sx i, j 1 sy  i , j  1 sx  i , j  1 sy  ij,1 ij, computed. These steps will run till Convergence-B (38) is satisfied, if the upstream discharge is constant. The convergence-B is said to be fulfilled if the bed where ω is a diffusion factor and is added here to elevation change averaged over the flow field remove the oscillations during the numerical between two successive iterations drops below simulation (Tsakiris and Bellos 2014). A fixed value 0.00005. In case of unsteady case, the model runs of 0.99 is used throughout the analysis. till the last time step is reached. If this condition of 2.3 Initial and Boundary Conditions Convergence-B/last time step is reached, the model stops. Else, it again starts from In order to numerically solve the above governing hydrodynamic model. hydrodynamic and sediment continuity equations, proper initial and boundary conditions are 3. APPLICATION OF THE required. Constant discharge values are used as initial condition in the hydrodynamic model. Initial NUMERICAL MODEL bed elevation is used as initial condition in the morphological model. Initial value of sediment In order to assess the applicability of the present transport is calculated using the flow and sediment model, it is applied to two experimental test cases of properties. Regarding boundary condition, two known bed deformation. Sutmuller and Glerum types of boundary conditions exist here namely, (1980) experimentally found out the bed deformation open and closed boundaries. A constant discharge in a 180◦ channel bend at Laboratory of Fluid value is used as upstream boundary and the Mechanics (LFM) of the Delft University of

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Technology. The experiment T6 of that study is 3.1 LFM Test Case considered here as the first test case. The second case considered here is the experiment (T2) done by Sutmuller and Glerum (1980) used uniform sediment Struiksma (1983) at the Delft Hydraulics Laboratory in their experiment with standard deviation of σg = (DHL) in a 140◦ channel bend. Figure 2 shows the 1.15. The specific gravity of the sediment particles is schematic diagram for both the channel bends. The 2.65. In numerical simulation, two straight reaches details of experimental conditions for both the test of length 15 m each are considered on the upstream cases are given in Table 1. and downstream of the bend, to nullify the boundary effect on the computed result. The whole domain is divided into 1179 finite difference grids. In the straight portions the grid spacings are ∆x = 0.3335 m and ∆y = 0.2125 m. The grid spacings in the curved portion are ∆r = 0.2125 m and ∆φ = 4.50, where r and φ are cylindrical coordinates. The value of Mannings coefficient is calculated as 0.028 from the given Chezys coefficient. The value of Courant number is 0.91. The sediment shape factor is considered as 2. The upstream flow is considered to be 0.17 cumec and the downstream water depth is considered as 0.2 m. The bed elevation is kept constant at both the upstream and downstream boundaries. The simulation is started from the bed slope as given in Table table1 and run upto steady state. For this purpose, the time is considered as the iteration number.

Figure 3 shows the relative bed (∆푍/퐻0)variations with respect to initial water depth computed by the present model along with the experimental result. The same figure also shows the results of previous numerical models of Abad et al. (2008) and Begnudelli et al. (2010). The positive val ues indicate deposition and the negative values indicate scouring of the channel bed. The transverse bed slope generally oscillates along the bend and leads to a large point bar on the inner bank and a deep pool on the outer bank, downstream of the entrance (Vasquez 2005). Around the centre of the bend these phenomenon reduce and again increase around the exit of the bend. This leads to formation of a point bar on the inner bank and a pool on the outer bank, around the exit of the bend. All these deformations are accurately reproduced in the results obtained Fig. 1. Flow chart for the present semicoupled from the present model (Fig. 3b). In the results model. reported by Abad et al. (2008)(Fig. 3c), size of the pools on the outer bank is observed to be smaller in Table 1Flow and sediment parameters compared to experimental one. Further, the height of the point bar on the inner bank is uniform throughout Parameter Simbol LFM DHL the curve. No point bar is visible around the downstream of the curve. The topographical plot of Discharge Q(m3/s) 0.17 0.061 Begnudelli et al. (2010) (Fig. 3d) does not show any width B(m) 1.7 1.5 pool around the downstream of the curve. Water depth h(m) 0.20 0.10 Eventhough their model could correctly reproduce Flow velocity u(m/s) 0.50 0.41 the size of the bigger pool on the outer bank downstream of the entrance, it fails to mimic the Water slope S0 (%) 0.18 0.203 bigger point bar at the same location on the inner Chezy’s 1 26.4 28.8 bank. co-efficient Q(푚2/s) Median grain Apart from the contour plot of the domain, numerical D50(mm) 0.78 0.45 diameter bed topographies along six cross sections are also compared with the experimental result. Begnudelli et Shields θ 0.28 0.27 al. (2010) did similar comparisons and reported that parameter their predictions are better than predictions of Abad Bend radius Rc(m) 4.25 12 et al. (2008). The present work therefore compares the result of the model with the results of Bend length Lc(m) 13.35 29.32 Begnudelli et al. (2010).

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Fig. 2. Schematic view of the test channels: a) LFM flow channel; b) DHL flow channel.

Fig. 3. Relative bed variations with respect to initial water depth (∆Z/H0) for LFM test case: a) Experiment by Sutmuller and Glerum (1980); b) Prediction by the present model; c) Prediction by Abad et al. (2008); d) Prediction by Begnudelli et al. (2010).

Figure 4 shows the comparison of transverse bed Agreement (IA)(Willmott 1981) and Nash Indicator profiles along the cross sections with experimental (NSE) (Nash and Sutcliffe 1970) are calculated here results and the results obtained by Begnudelli et for all the sections. The expressions for these factors al.(2010). These figures clearly show that the present are as given below model can excellently predict the outer bank nn scouring, inner bank deposition and also the 2 centreline bed height, for all the sections. The results ()zzexp  num i 1 are also found to be better than the results of IA 1 nn (39) Begnudelli et al. (2010). For additional quantitative 2 ()znum  zexp  z exp  z exp analysis, two parameters namely, Index of i 1

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Fig. 4. Comparison by transverse bed profiles for LFM test case: measured (x), predictions by present model (solid line) and predictions by Begnudelli et al. (2010) (+).

nn and ∆y = 0.1875 m. While in the bend portion these 2 ()zznum  exp values are ∆r = 0.1875 m and ∆φ = 2.50. This has NSE 1 i 1 (40) leads to 1035 grids for the whole flow domain. The nn sediment shape factor is considered as 2. The ()zz 2  exp exp upstream discharge is 0.061 cumec and the i 1 downstream water depth is 0.1 m. The bed elevation Where nn is the number of grids in the considered is kept as constant at both the upstream and cross section, znum is the computed non dimensional downstream boundaries. Starting from the initial bed relative bed variation, zexp is the experimental non the model is run upto steady state by considering dimensional relative bed variation and zexp is the time as the iteration parameter. averaged experimental non dimensional relative bed variation. The numerical values obtained for these two parameters are shown in Table 2. The values clearly shows that the present model is appropriate for these type of tight curvature.

Table 2 Performance indicators for the LFM test case Position IA NSE Section 8 0.995 0.980 Section 10 0.954 0.796 Section 12 0.980 0.916 Section 14 0.951 0.798 Section 16 0.963 0.845 Section 18 0.961 0.853

3.2 DHL Test Case Fig. 5. Computed relative bed variations with respect to initial water depth (∆Z/H0) for DHL Struiksma (1983) also considered uniform test case. with standard deviation of σg = 1.19 and specific gravity of 2.65 in his experiment. Two straight Figure 5 shows the relative bed change on channel reaches of length 15 m each are also imensionalized with initial water depth (∆Z/H0) . considered in the numerical simulation. The This figure clearly shows the bigger pool on the outer Mannings coefficient is calculated as 0.0237. The bank and the point bar on the inner bank, down- Courant number value is 0.94. The grid spacings stream of the bend entrance. The model can also considered in the straight portions are ∆x = 0.5236 m replicate the smaller pool and the point bar around

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Fig. 6. Comparison of bed level variations along two longitudinal sections for DHL test case.

the exit of the bend. Moreover, the alternate bars on numerical model for morphological flow simulation the downstream of the bend as observed in the in river bend. For this purpose, a hydrodynamic experiment are exactly reproduced by the present model and a sediment transport model are coupled model. For further analysis, numerical bed level with the efficient semicoupled approach. The variations along two longitudinal sections are also hydrodynamic model solves the shallow water compared with the experimental results and are equations in a boundary fitted coordinate system to shown in Fig. 6. Figure 6 clearly shows that the make the model compatible with river bends. A present model compares satisfactorily with the simple but efficient TVD MacCormack scheme is experimental result. The point bar and the pool near used here for solution of the governing the entrance of the bend are clearly reproduced. hydrodynamic equations. The sediment transport However, there is some discrepancy between the model is developed by solving the Exner equation in experimental and the numerical results for the pool a boundary fitted coordinate system. The simple around the exit of the bend. This discrepancy was FTCS scheme is used for discretization of the also observed by Kassem and Chaudhry (2002). governing sediment transport equation. The Kassem and Chaudhry (2002) claimed that it may be sediment transport model incorporates the secondary due to the uncertainties in the sediment discharge flow effects by a pseudo 3D model. The effects of calculation and evaluation of sediment movement transverse bed slope on the sediment transport direction. The statistical parameter Brier Skill Score direction are included by a simple bed slope (BSS) (Davis 1984; Guan, Wright, Sleigh, Ahilan, correction formula. and Lamb 2016) is used here for quantitative analysis of the numerical results. The BSS parameter is given To verify the applicability of present model it is by applied to replicate the bed level variations in two experimental channel bends. These experimental test 푛푛 2 cases include both tight and loose bends to affirm the ∑푖=1(푧푛푢푚(푡)−푧푒푥푝(푡)) 퐵푆푆 = 1 − 2 potential of the model in diverse situations. The ∑푛푛 푖=1(푧푛푢푚(푡)−푧̅̅푒푥푝̅̅̅̅(0)) results show that the model can excellently simulate (41) the point bars near the inner bank and the pools near the outer bank in the bend. The alternate bar The BSS values are found to be 0.93 and 0.83 for the formations downstream of the bend are also inner bank and the outer bank, respectively and are replicated accurately. Three popular statistical well within the excellent range (Guan, Wright, parameters are calculated for quantitative analysis of Sleigh, Ahilan, and Lamb 2016). In order to check the results and the values obtained establish the the effect of grid size on the computed results, potential of the present model for this type of another grid is also considered. For this case the total complex river phenomenon. number of grids is 1287. The numerical results for this grid size is also shown in Fig. 6. The results The present model assumes uniform sediment in the clearly show that both the grids are showing almost river and therefore cannot include the grain sorting similar results. Therefore, it may be concluded that as well as armoring effects of bed sediment. This with further decrease in grid size will not provide any may be a future work of the present model. The effect remarkable change in the results. of suspended sediment on the bed level variation can also be included in the model and is considered as the future scope of the present work. Another scope of 4. CONCLUSION future work may be use of some advanced and complex turbulence models. This study presents a 2D depth averaged unsteady

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