Computation of Flow and Bed Morphology at River Confluences
Thanh Mung DINH Candidate for the Degree of Doctor of Philosophy Supervisor: Prof. Yasuyuki Shimizu Division of Field Engineering for the Environment
Introduction M muM m vM (h zb ) bx gh ghsin 1 (2) t x y x River channel confluences are commonly encountered in u'2 h u'v'h u u h h Scx nature as well as in engineering. These are critical x y x x y y interfaces where intense changes in physical processes within drainage or fluvial networks occur. These changes N uN vN (h z ) by v'u'h m m gh b (3) influence both the local and downstream characteristics of t x y y x flow dynamics and the bed morphology. Flow features in v'2 h v v h h S these regions are often known to be complex and highly cy y x x y y three-dimensional. Although there are a number of studies on flow at these regions conducted in the past, there are still where (x, y): spatial coordinate, (u, v): depth-averaged gaps to be studied for this complex flow. Furthermore, velocity components in (x, y) directions, t: time, h: water behavior of bed morphology at river confluences has not depth, (M, N): discharge fluxes in (x, y) directions defined still been clearly understood. Besides, although local as (hu, hv) respectively, g: gravity acceleration, (u’, v’): configuration of the confluent river mouths may influence turbulence velocities in (x, y) directions, z : bed level, (τ , flow and bed morphology, this aspect has not been paid b bx τ ): bottom shear stress vectors, : dynamic viscosity attention at all. In addition, effect of bank vegetation and by bar vegetation on bed morphology within these regions has coefficient, sin1: bed slope, f: friction coefficient (function not been obviously understood. of Reynolds number), ρ: water density, m: momentum Besides the above problems, numerical modeling of coefficient, u2 , uv, v2 : depth-averaged Reynolds flow dynamics and bed morphology at river confluences stress tensors, and Scx , Scy: additional terms caused by still requires much more attempts. The 3D nonlinear k - secondary currents and defined later. model are proved to be successful in simulating complex The depth-averaged Reynolds stress tensors are flow. However, this model has not been developed for river evaluated based on the zero-equation turbulence model. For confluence problems. Although a 3D model is broadly linear model, this term is evaluated as adopted to simulate flow at a small part of river course, it is 2 u u (D )S k (4) difficult to predict flows and sediment transports during i j h ij 3 ij floods from the upper region of a river to the river mouth. but for nonlinear one, a non-linear term is added to the In addition, at river confluences, adding a sediment- Reynolds stress tensor as transport module to a 3D flow model remains a challenge 2 due to difficulties in designing a suitable numerical mesh. u u (D )S k i j h ij 3 ij Furthermore, 3D modeling is still a time-consumed and (5) costly work. Therefore, development of depth-averaged 2D h 3 1 2 p Dh c Sij Sij Cij u* models, which are corrected for effect of secondary current, u* 1 3 are still a reasonable compromise for the problems of flow i, j 1,2 dynamics and bed morphology up to now. D is eddy viscosity and is evaluated based on an 0- The purpose of this PhD thesis is to address the above h equation turbulence model with considering reduction of problems through computations of flow, bed morphology at eddy viscosity near the wall. In Eq. (4), D is evaluated these regions. h This paper summarizes the main contents of this thesis. Dh fDhu* (6) while Dh in Eq.(5) considers contribution of strain and spin and is evaluated as
Computation of flow at an open-channel Dh fDcDhu* (7) confluence with depth-averaged 2d models k is depth-averaged turbulent kinetic energy evaluated by the empirical formula as follows: . Governing equations 2 (8) k 2.07u*
Here, u is local friction velocity; is calibrated constant ( The governing equations used in this study are depth- * = 0.80 is used in this study). averaged 2D shallow water flow equations described in the Cartesian coordinate as follows: cD is the coefficient of eddy viscosity and is a function of Continuity equation: strain and rotation parameters as follows 2 2 h M N (1) 1 c S c 0 c ns n t x y D 2 2 4 4 2 2 (9) 1 cdsS cd cdsS cds1S cd1 cd1S Momentum equations: Here S and are strain and rotation parameters. c is the coefficient of the non-linear quadratic term and including conservativeness of physical quantities and evaluated as computational stability. The QUICK scheme with second 1 (10) order accuracy in space is employed for convective inertia c c 0 1 m S 2 m 2 terms. The Adams Bashforth method with second order ds d accuracy in time is used for time integration.
To solve the flow equations, discharge and flow . Secondary current model velocity at the upstream end of the main channel and the branch channel, and water level downstream are specified The additional terms expressing effects of secondary as boundary conditions. currents, Scx and Scy, in Eq. (2) and Eq. (3) are defined as Computational results are compared to the u s A hsin 2 u s A hcos2 S C n n experimental results of Weber et al. (2001). It should note cx sn (11a) x y that attempts to apply Model 2 are done during the process 2 2 2 of implementation of this study, but are not successful. The An hsin An hcos sin Cn2 reason for this may be attributed to very sharp streamline x y curvature of flow downstream of the junction causing u s An h cos 2 u s An hsin 2 strong instability of this model. Therefore, in the following, Scy Csn x y (11b) only the results obtained from the models 1, 3 and 4 are 2 2 2 reported. An hsin cos An h cos Cn2 x y . Computational results and discussions Here is angle between a streamline and x-axis. Csn and Cn2 are model coefficients defined by Eq. (12) using the Comparison of performance of Model 1 with the linear and similarity functions of velocity profile in the stream-wise nonlinear turbulence models carried out shows that the and transverse directions and , respectively. f s f n nonlinear is suitable for further simulations in this study. The followings are some of the main results obtained. u is depth-average velocity in the stream-wise direction s Figs. 1 and 2 show the results predicted by the models and is defined by Eq. (13). in comparison with the experimental one for the case of low 1 1 , 2 (12) discharge ratio (q*). As seen from Figs. 1 and 2, all three Csn fs ( ) fn ( )d Cn2 fn ( ) d 0 0 models reproduce important flow patterns at the vicinity of z (13) the junction, that is, a separation zone immediately us ( ) u s f s ( ) , h downstream of the junction and a contracted flow with higher velocity. However, it can be seen that there is a Here, u is the stream-wise velocity profile in the s difference in separation generated by the models for q* = vertical direction. 0.25 in which secondary current is strong (Fig. 1). Model 1 The coefficient An in Eq. (11a) and Eq. (11b) means the under-predicts the length of this region, while both Model 3 magnitude of the secondary current, which is induced by and Model 4 fairly well reproduce it. streamline curvature, and is defined as In contrast, for a high q*, that is, weak secondary (14) un ( ) An fn ( ) current, all the models generate the results (not be shown here) agreeing well with the measured one and there is no Here un ( ) is the transverse velocity profile in the vertical significant difference in the results by the models. direction and z is the direction perpendicular to the bottom bed. In this chapter as well as throughout this thesis, four types of depth-averaged 2D models are proposed to simulate the flow at an open-channel confluence. Model 1 a) Exp. is a conventional 2D model using a non-linear 0-equation turbulence model without effects of secondary currents; 1 Model 2 is also a 2D model with effects of secondary 0.5 0
current without consideration of lag between the streamline Series2 y/W -0.5 curvature and development of secondary currents; Model 3 b) Model 1 is a 2D model with effects of secondary currents and lag -1 -4 -3 -2 -1 0 1 between a streamline curvature and development of x/W secondary current; and Model 4 is a 2D model that consider effects of secondary currents, lag between a streamline curvature and development of secondary currents as well as change of mainstream velocity profile influenced by c) Model 3 secondary currents. When Model 2 and Model 3 are used, An is evaluated by Engelund (1974) and Hosoda et al. (2001), respectively, while the similarity functions fs and fn are those proposed by
Engelund (1974). However, when Model 4 is employed, the d) Model 4 similarity functions are those derived by Onda et al. (2006) and the strength of secondary current is evaluated using Hosoda et al. (2001). Fig. 1 Comparison of dimensionless velocity vector field The fundamental equations are solved numerically for q* = 0.25 (q* defined as the ratio of the upstream main using the finite volume method with a full staggered grid channel flow (Qm) to the total flow (Qt)) i a) x/W = -1.33 d) x/W = -2.33 V 1.8 i j j i j j i 1.6 1.8 V (V W )V W V W 1.4 1.6 j j j 1.2 1.4 (16) 1 1.2 exp. Exp. t 0.8 1
* 0.8 * u 0.6 0.6 Model 1 Model 1 0.4 u 0.4 0.2 0.2 i 1 ij i j ij 0 0 Model 3 Model 3 -0.2 -0.2 F g j p j v v 2 j S -0.4 -0.4 -0.6 -0.6 Model 4 Model 4 -0.8 -0.8 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 y/W y/W k and equations:
b) x/W = -1.67 e) x/W = -2.67 k j j j l j i 1.8 1.8 k(V W ) k W g v v V - 1.6 1.6 1.4 1.4 j j il j 1.2 1.2 t (17) 1 Exp.1 Exp.
0.8 0.8 * * 0.6 0.6
u Model 1 u 0.4 0.4Model 1 D 0.2 0.2 0 Model0 3 Model 3 t ij -0.2 -0.2 g k -0.4 -0.4 -0.6 -0.6Model 4 Model 4 j i -0.8 -0.8 k 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 y/W y/W j j j j (V W ) W C g vlv V i c) x/W = -2.00 f) x/W = -3.00 t j j 1 k il j (18) 1.8 1.8 1.6 1.6 1.4 1.4 1.2 1.2 1 1Exp. Exp. 2 D 0.8 0.8 * t ij * 0.6 0.6 u Model 1 Model 1 u 0.4 0.4 C g 0.2 0.2 0 0Model 3 Model 3 2 k j i -0.2 -0.2 -0.4 -0.4Model 4 -0.6 -0.6 Model 4 -0.8 -0.8 In the linear turbulence model, Reynolds stress is 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 y/W y/W evaluated as Eq. (19), while it is calculated using Eq. (20) Fig. 2 Comparison of dimensionless longitidunal velocity in the nonlinear one. 2 vi v j 2D S ij k i g sj (19) t 3 s i j ij 2 i sj k (20) v v 2Dt S k s g Dt c* 1Q1 2Q2 3Q3 3 2 k (21) Dt C Eddy viscosity coefficient is evaluated by Eq. (21). In
the linear turbulence model, C=0.09. However, in the nonlinear model, the coefficients c* and C are not constant, but the functions of strain parameter, S and rotation one, as follows. 2 2 1 cns S cn (22) C C 0 2 2 4 4 1 cds S cd cds S cds1S cd1 1 c (23) * 1 m S 2 m 2 ds d Fig. 3 Cross-section view of vector field for q* = 0.25 . Kinematic boundary condition Using the models with effect of secondary current, this flow pattern is able to be visualized as shown in Fig. 3. This For the junction flow in this study, the free-surface also explains why the results with the models with elevation changes apparently, increasing before and secondary current effect (Model 3 and Model 4) are decreasing after the junction. Therefore, a treatment of the improved in comparison with that by Model 1. free-surface which uses the kinematic boundary condition is employed. H H H (24) V1 V 2 w Computation of flow at an open-channel t 1 2 confluence with 3D models where H is the height of the free surface; t is time; V1 and V2 are contra-variant components of flow velocity in 1 and . Governing equations 2 directions, respectively; and w is the Cartesian component of velocity in the z-direction. The superscripts 1 In regions of open-channel flow where its solution domain and 2 indicate the directions and , respectively. changes in time due to the movement of boundaries, for example, in free-surface flows, this movement must be . Dynamic boundary condition calculated as part of the solution. In these cases the grid has to move with the boundary. In this study, the unsteady In the previous studies, the hydrostatic pressure is often Reynolds-Averaged Navier-Stokes equations are used and assumed as considering dynamic boundary condition for the basic equations are written in a moving boundary-fitted cases with insignificant deformation of free surface. coordinate system using the covariant derivative instead of However, for the cases where the curvature of free surface the partial differential as follows: is large, surface tension may affect pressure gradient in the Continuity equation: interfacial region. Therefore, non-hydrostatic pressure is 1 V G 0 (15) used in the present work with effect of the surface tension G considered. The approach employed calculates the Momentum equation: dimensional dynamic boundary condition as Eq. (25). 3 DP 23V 2M C (25) where DP is dynamic pressure; Mc is the mean curvature of the free surface and is defined for the 3 surface; is the
3 y/W Highest turbulence region 3V surface tension; and is covariant derivative of the a) Exp. contra-variant velocity component, V3 with respect to the 3 direction and the superscript 3 indicates the vertical coordinate.
y/W Highest turbulence region
. Numerical procedure b) Calculated with the linear
The momentum equations and transport equations of k and are solved with the conservative finite-volume method based on a fully-staggered grid system. A TVD-MUSCL y/W Highest turbulence region scheme is applied to the convective terms in the momentum c) Calculated with the nonlinear equations and the transport equations of k and . The Adams-Bashforth scheme with second-order accuracy in time is used for time integration in each equation. Calculation of pressure field is implemented using a Fig. 4 Comparison of dimensionless turbulence energy (k*) fractional step method coupling with the Highly Simplified by the models with the experimental one at z/W = 0.278. Mark and Cell (HSMAC) method. 1 0.5 . Computational results and discussions 0.309 0 Series2
y/W One case of the experiment of Weber et al. (2001), q* = - -0.5 a) Measured -1 -4 -3 -2 -1 0 1 0.25, is used for validation of the linear and nonlinear x/W models proposed above. Kinetic turbulence energy: In the literature, there has been no validation of b) Calculated with turbulence kinetic energy at an open-channel confluence the linear model found so far. In this study, this quantity is first validated with the present 3D model using the linear and nonlinear turbulence models. The result is illustrated in Fig. 4. It can be seen that the nonlinear model reproduces turbulence c) Calculated with the distribution better than the linear does in comparison with nonlinear model the experiment.
Water surface elevation: Fig. 5 Comparison of water surface contour Fig. 5 displays water surface contours measured and a) Exp. predicted by the linear and nonlinear turbulence models, a) Exp. respectively. It can be observed that the predicted results agree well with the measured ones. Secondary current: Secondary current predicted by the linear and nonlinear 3D models and the measured one at the section of x/W = - b) Calculated with the linear 2.00 are shown in Fig. 6. The important points which can be observed from Fig. 6 are as follows: i) both the linear and nonlinear models can reproduce the large vortex near the outer bank well; and ii) the nonlinear predicts correctly the c) Calculated with the nonlinear small vortex in terms of its position and rotation direction, thus the distortion of the separation zone boundary, while the linear incorrectly does. This great difference is attributed to consideration of anisotropy of Reynolds stresses in the nonlinear model. Border of the separation zone (u=0) Fig. 6 Comparison of v*-w* velocity field at x/W = -2.00 . Effect of discharge ratio on secondary currents, bed shear stress, and flow constriction decreases. There is a change in the direction of the transverse bed shear stress component, thus potential The nonlinear 3D model is used for this study purpose. The tendency of sediment transport as discharge ratio exceeds total discharge is kept constant as 0.17m3/s, while only 0.50. discharge ratio changes. The brief results realized as In the current study, a more accurate approach based follows (figures are not shown here). on an effective area is presented to estimate coefficient of Strength of secondary currents decreases as discharge flow contraction at an open-channel confluence. This ratio increases. There is a change in the rotation direction of approach generates the results agreeing well with the the large vortex as discharge ratio is larger than 0.50. experimental ones, while the conventional approach which As discharge ratio increases, bed shear stresses uses the concept of the effective width over-predicts this quantity. As discharge ratio increases, flow constriction In the above relations, is angle between a streamline substantially decreases. and x-axis; q x and q y denote Cartesian components of the b b s n bed-load vector in the x- and y directions; and qb and qb Computation of bed morphology at a channel are components of the bed-load vector in the stream-wise confluence with depth-averaged 2D models and transverse directions. is calculated by Yamaguchi
. Flow model and Izumi (2006) after linearization, in which local slope in the s-direction is incorporated, while is evaluated using For the purpose of development of a model that is able to Eq. by Hasegawa (1984). apply for various geometries as well as for real cases, depth-averaged equations are expressed in a general . Computational results and discussions curvilinear coordinate as follows: Continuity equation: Computational conditions are same as those in Shigeeda et 1 h Mˆ Nˆ (26) al. (2009). Numerical procedures are same as used in 0 J t J J Chapter 2 above. First, validation of the model is carried Momentum equations: out. Then, effect of discharge ratio and confluence angle on
Mˆ Mˆ Mˆ M N (27) bed morphology is investigated using Model 4. x x y y U V U V U V t J J J J J To verify the models, bed morphology is assessed 2 2 z z 2 2 through deviation of bed elevation after an approximate gh x y s x x y y s b x u2h x x u2h y v2h J J J J J J equilibrium reached in comparison with the initial bed (flat 2 y y 2 x y y x x y bed). Fig. 7 shows bed deformation observed by Shigeeda v h uvh uvh x S1 S 2 S 3 S 4 J J J et al. (2009) and predicted by the models. As seen from Fig. S S S S y 1 2 3 4 7, only Model 4 can capture the important features of bed Nˆ Nˆ Nˆ M N (28) morphology at the present confluence including avalanche x x y y U V U V U V t J J J J J faces, scour hole, and scouring range downstream the 2 2 z z 2 2 gh x y s x x y y s b x u2h x x u2h y v2h junction, while Model 1 and Model 3 do not. The reason for J J J J J J this difference is because Model 4 reproduced distinct flow
y y 2 x y y x 2 x y v h uvh uvh x S1 S 2 S 3 S 4 patterns including secondary currents. These secondary J J J currents play an important role in preventing from sediment y S1 S 2 S 3 S 4 movement into the scour, hence maintaining the scour. where (x, y): spatial coordinate, (u, v): velocity in (x, y) direction, (ξ, η): generalized curvilinear coordinate, t: time, . Effect of discharge ratio and confluence angle on h: water depth, (ξx, ξy, ηx, ηy): matrix, J: Jacobian of coordinate transformation, (M, N): discharge flux vectors, bed morphology
Mˆ , Nˆ ( ): contra-variant discharge flux vectors for unit For these purposes, Model 4 is employed. The main width, (U, V): contra-variant components of velocity findings are as follows. As discharge ratio increases, bed ξ η vectors, g: gravity acceleration, zs: water level, (τb , τb ): erosion within and downstream the confluence substantially contra-variant components of bottom shear stress vectors, ρ: decreases, while the penetration of the avalanche face into 2 2 the confluence significantly increases. water density, and u , uv, v :Reynolds stress tensor. The turbulence model and secondary current model are a) Measured by Shigeeda et al.2009 same as those used in Chapter 2 above. Depth-averaged 2D Avalanche face models are same as those in Chapter 2. Y(m)
Scour hole Scouring range 1 . Sediment transport model 1 b) Simulated by Model 1
Evolution of bed morphology is calculated using the Y(m) 0 0 3 4 equation of sediment continuity, the Exner equation. In a 0 11 22 3 4 1 generalized curvilinear coordinate, this equation has the 1 c) Simulated by Model 2
following form: Y(m) z 1 q q 0 b b b (29) 0 3 4 0 0 1 2 3 4 t J 1 J J 1 d) Simulated by Model 3 1 Avalanche face Scouring range
where zb is the time-dependent bed elevation; and are Scour hole Y(m) generalized curvilinear coordinates; is the porosity of the 0 0 1 2 3 4 bed sediment; J is Jacobian of coordinate transformation; 0 1 X(m)2 3 4 -0.03 -0.015 0.0 and qb and qb , which are evaluated by Eqs. (30 – 31), Bed deformation (m) are components of the bed-load vector of sediment flux in Fig. 7 Comparison of bed deformation measured by the - and directions. Shigeeda et al. (2009) and predicted by the models
y x x y x x y y Effects of flow stages and vegetation on bed q Jq Jq ; q Jq Jq (30) b b b b b b morphology at a real river confluence x s n y s n qb qb cos qb sin ; qb qb sin qb cos (31)
. Flow model a) T = 0 h (Initial bed measured in 2001)
The flow model used in this study is similar to that A B C D E presented in Chapter 4 above. However, in order to A B C D E consider effect of vegetation, a new term which includes b) T = 41 h (low flow) this effect is introduced into the right-hand side of Bed elevation momentum equations as follows: (m)
Continuity equation: c) T = 83 h (Starting flood hour) 1 h Mˆ Nˆ (32) 0 J t J J
Momentum equations: d) T = 102 h (Flood peak) (33) Mˆ Mˆ Mˆ M N x x y y U V U V U V t J J J J J 2 2 2 2 x y z xx y y z y gh s s b v x u2h x x u2h v2h e) T = 161 h (Ending flood hour) J J J J J J J 0 1000 m
y y 2 x y yx 2x y v h uvh uvh x S1 S 2 S 3 S 4 J J J
y S1 S 2 S 3 S 4 f) Measured (2006) (34) Fig. 8 Process of bed morphology evolution predicted by Nˆ Nˆ Nˆ M N x x y y U V U V U V Model 4 t J J J J J 2 2 z z 2 2 gh x y s x x y y s b v x u2h x x u2h y v2h morphology here and discharge ratio seems less significant J J J J J J J in this cases. y y 2 x y yx 2x y v h uvh uvhx S1 S 2 S 3 S 4 J J J
y S1 S 2 S 3 S 4 . Effect of local configuration of the confluent where , are contra-variant components of bottom river mouths and vegetation on bed morphology v v shear stress vectors induced by vegetation and are The local configuration of the confluent river mouths is a calculated by Eq. 35) below. Other symbols are the same as major control on bed morphology within the confluence. the ones in Chapter 4. This aspect is first identified through numerical simulation Sediment transport model is same as used in Chapter 4. here. Through assumed cases with vegetation growing over . Vegetation model the confluence bar, the computational results demonstrate 2 2 2 2 (35) v CDvU min(h,Hv ) u v ; v CDvV min(h,Hv ) u v that bar vegetation greatly affect bed morphology within
Here CD is a bulk drag coefficient of vegetation (=1.0); v is and downstream the confluence. the projected plant area per unit volume; Hv is the height of trees. Vegetation is assumed to be stiff, cylindrical, and regularly distributed for simplicity in this study. Min- References function is evaluated as Engelund, F. (1974). “Flow and bed topography in channel h if h H (emergent vegetation) v (36) bends.” Journal of Hydraulic Division, ASCE, Vol.100, HY11, min(h, H v ) 1631–1648. H v if h H v (submerged vegetatim) Hasegawa, K. 1984. “Hydraulic research on planimetric forms, bed topographies and flow in alluvial rivers”. Thesis presented to . Bank erosion model Hokkaido University, at Sapporo, Japan, in partial fulfillment of the requirements for the degree of Doctor of Engineering (in To calculate bank erosion, the simple approach Japanese). proposed by Hasegawa (1984) is used in the present study. Hosoda, T., Nagata, N., Kimura, I., Michibata, K., and Iwata, M. The model proposed above is applied to the Tokachi- (2001). “A depth averaged model of open channel flows with lag Toshibetsu River confluence locating in Eastern Hokkaido between main flows and secondary currents in a generalized in Japan for investigation of response of bed morphology at curvilinear coordinate systems.” Advances in Fluid Modeling & this confluence to different flow stages and effect of Turbulence Measurements, World Scientific, 6370. vegetation and local configuration of the confluent river Onda, S., Hosoda, T., and Kimura, I. (2006). “Refinement of a depth averaged model in curved channel in generalized curvilinear mouths on bed morphology. coordinate system and its verification.” Ann. J. Hydraul. Engrg, JSCE, 50, 769774. . Computational results and discussions Shigeeda, M., Akiyama, J., and Moriyama, T. (2009). “Numerical simulation of flow and bed topography in a vicinity of a river Fig. 8 shows the result of temporal evolution process of bed confluence by two-dimensional model.” Annual Journal of morphology at the study confluence predicted by Model 4. Hydraulic Engineering, JSCE, Vol.53, 793–798 (in Japanese). Weber, L. J., Eric, D. S., and Nicola, M. (2001). “Experiments on Analysis of this process in details indicates that response of o bed morphology at the present confluence has a periodic flow at a 90 open-channel junction.” J. Hydraul. Eng., ASCE, tendency to different flow rate. In suck a confluence, flow 127(5), 340350. Yamaguchi S., Izumi, N. (2006). “Weakly nonlinear analysis of rate substantially influence flow dynamics and bed dunes by the use of a sediment transport formula incorporating the pressure gradient.” Proceedings of the 4th IAHR Symposium on River, Coastal and Estuarine Morphodynamics, RCEM 2005, Urbana, Illinois, USA, pp. 813–820.