Computation of Flow and Bed Morphology at River Confluences
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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: 2 . Governing equations (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 A h cos 2 u s A hsin 2 S C n n strong instability of this model. Therefore, in the following, cy sn 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/Wy/Wy/Wy/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.