Meandering – like River Channel Evolution due to the Formation of Fluvial Bars Kristine D. SANCHEZ Candidate for the Degree of Doctor of Engineering Supervisor: Prof. Norihiro Izumi Division of Field Engineering for the Environment ________________________________________________________________________________________________ Introduction + + = 0 푏푥 푏푦 (4) Meandering has been studied as one of the most where 휕푍 휕푄 휕푄 interesting processes among river evolution phenomena. 휕푡 휕푥 휕푦 Many theoretical studies have been done on the = ( ( + ) ) ( ( + ) 0.7 ) (5a) development of meandering. If a small sinusoidal 2 2 2 2 1⁄2 푄푏 퐾 휃푛 푈 푉 − 휃푐 �휃푛 푈 푉 − 휃푐 disturbance is imposed to the planar shape of river , = ( , ) (5b) channels, the curvature of the river channel causes a �푄푏푥 푄푏푦� 푄푏 표푠휙 푠푖푛휙 deviation of the flow towards the outer bank, and = (5c) accumulation on the inner bank occur, and meandering + 푉 푟 휕푍 makes further development under some conditions. The non-푠푖푛휙dimensional2 bed 2shear− stress1⁄2 components are Ikeda, Parker, Sawai1 provided a first theoretical √푈 푉 훽휃 휕푦 explanation on the mechanism of meandering as a shear = + (6a) instability on both banks in terms of linear stability 2 2 analysis. Afterwards, the influence of interaction and 푏푥 푓 푇 = 퐶 푈 �푈 + 푉 (6b) resonance with fluvial bars has been taken into account 2 2 2 from the condition satisfying the continuity of sediment The boundary푇푏푦 conditions퐶푓 푉 �푈 used푉 are the vanishing and the influence of secondary flow due to meandering flow velocities at the sidewalls, and that the sediment 3 has been included into the linear stability analysis . transport rate evaluated at the junction is equal to the According to these theories, the development of sediment being eroded or deposited at the banks. The meandering and the formation of sand bars interact with process-based bank erosion model states that the time each other. While meandering is caused by the bank variation of bank junction is proportional to the lateral instability due to the curvature of the planar geometry of sediment transport rate at the junction between the bed a river channel, however, the formation of sand bars is and bank regions, which is evaluated by the lateral caused by the bed instability between flow and the sediment transport rate in the bank region and the time riverbed. It follows that they are basically different variation of the bed elevation at the junction5. It is instability. If all meandering is to be explained by the derived to its simple form following the assumptions of bank instability, how should we understand the the study. The channel is assumed to have erodible bed development of meandering originated from the sand and banks, with a constant width being maintained even 4 bars that Kinoshita observed? when the channel is eroding; also, a quasi-steady This study aims to perform linear stability analysis approximation is employed. In addition, an expansion of meandering induced by the formation of sand bars on parameter ϵ is introduced in order to provide a the riverbed with the use of a new bank erosion model. protection that inhibits bank erosion, such as slump Furthermore, the study aims to determine the parameters block armoring or the presence of vegetation. that influence the formation of bars, and to clarify the The process-based bank erosion models are effect of bank erosion into the analysis by determining 1 + 1 (7a) its effect on bar wavelength. = , 1 + 휕푅 퐶푅 푠푅 푗 휕푍 푦=푅 − 1 + 푄 − 1 푅 ⃒ 휕푡 = 퐶푅� 훽푆 휕푡 (7b) Formulation 1 + , 휕퐿 퐶퐿 휕푍 The governing equations,푠퐿 푗 boundary푦= 퐿conditions and The two-dimensional St. Venant shallow water − 푄 − 퐿 ⃒ bank 휕푡erosion models퐶퐿� are expanded훽푆 휕푡 to form a solution of equations, continuity equation, and Exner equation are linearized, homogeneous equations. By performing used as non-dimensional governing equations, as linear stability analysis, the conditions for incipient bar 1 formation, with their corresponding wavelengths and + = + wave speed can be obtained. A disturbance is introduced 푏푥 (1) 휕푈 휕푈 휕퐻 휕푍 푇 to the base state in the streamwise direction. In this 푈 푉 − 12 � � − 훽 휕푥 + 휕푦 = 퐹 휕푥 + 휕푥 퐻 manner, the dependent variables , , , , are (2) 휕푉 휕푉 휕퐻 휕푍 푇푏푦 expanded in the form 푈 푉 − 2 � � − 훽 푈 푉 퐻 푅 퐿 ( , ) = (1, 0) + ( , ) ( ) 휕푥 휕푦 +퐹 휕푦= 0휕푦 퐻 (8a) (3) 푖 푘푥−휔푡 휕푈퐻 휕푉퐻 1 1 푈 푉 퐴 푈 푉 푒 휕푥 휕푦 of slump blocks or the presence of vegetation. As shown ( ) ( ) , = 1, + ( , ) (8b) in Figure 4, the value of is varied to clarify the effect 2 푖 푘푥−휔푡 ( , ) = ( 1푓, 1) + ( , 1 ) 1( ) of bank erosion. It is revealed that the unstable region 퐻 푍 � −훽퐶 퐹 푥� 퐴 퐻 푍 푒 (8c) 휖 푖 푘푥−휔푡 shifts to a range of smaller wavenumbers when From linear푅 퐿 stability− analysis,퐴 푅1 the퐿1 solutions푒 of banks that increases from 0 to 1. This indicates that when bank are in-phase are obtained for the case of bar instability erosion is incorporated into the analysis, the wavelength with and without bank erosion. The growth rates of of휖 the bars is longer than when there is no bank erosion. perturbation are solved using ϵ - expansion to consider In linear theory, there exists a maximum growth rate the effect of the slump block armoring or the presence of perturbation [ ] that corresponds to the dominant wavenumber. The dominant wavenumber, of vegetation. The Chebyshev polynomials are also 푚푎푥 employed for numerical solution. If the growth rate is or , which퐼푚 is 휔inversely proportional to the wavelength by 2 , is thought to be the representative positive, the base state is unstable and it is considered 푚푎푥 that bars theoretically form; otherwise, the flat bed is wavelength푘 that serves as an estimate of the wavelength stable and no bars grow. observed in experiments휋 of bar studies. The influence of the variability of on [ ] and the shift of wavenumbers are examined in Figures 5 to 8. It is 푚푎푥 Results generally observed 휖for the퐼푚 aforementioned휔 figures that for = 20, the maximum growth rates of perturbation Analytical Results tend to decrease considerably as increases. In addition, the 훽contours shift to a range of smaller wavenumbers. The analytical model accounts for the variability of the Meanwhile, although the figures휖 are not shown parameters to investigate the effect on both cases of herein, the variation of = 10, 20, 30 and 40 for = 0.08 is also investigated. It is revealed that the pure bar instability and bar instability with bank erosion. 푛 The results obtained from linear stability analysis are maximum growth rates훽 of perturbation are amplified휃 presented herein. In the first subsection, the growth rates with respect to an increase in aspect ratio. This confirms of perturbation [ ] are plotted in the plane. the influence of the aspect ratio as one of the governing The contour for zero growth rate, or the neutral curve, parameters of bar instability; at high aspect ratios, bars refers to the case퐼푚 where휔 the perturbation neither푘 − 훽 grows are more likely to form, and multiple bars may exist. nor decays. It is the curve that delineates the stable However, multiple bars are beyond the scope of this region ( [ ] < 0) and the unstable region ( [ ] > study. It is also worth mentioning that as β approaches 0). In the stable region, the base state that is initiated as 40, the contours for = 0, 0.3, 0.7 and 1.0 overlap. It the flat bed퐼푚 is휔 stable; on the other hand, in the퐼푚 unstable휔 seems that as the aspect ratio increases, the contours region, the pertubation grows, and it can be considered shift to a range of 휖larger wavenumbers, or smaller that in this region, bars initially develop. In the second wavelength values. In this case, at high aspect ratios, the subsection, the values of [ ] are plotted against . effect of is significantly reduced. The flow and sediment parameters used in the 휖 theoretical analysis are as 퐼푚follows:휔 = 7.6, = 0.5, 푘 = Comparison of theoretical results with 0.5, = 0.010, = 0.06, = 0.05, = = 1.0, and experimental results 퐾 푟 퐹 = 1.0.푓 푛 푐 푅 퐿 퐶Figure 1 and휃 Figure 휃2 report 푆the variability푆 of The validation of the theoretical results from linear Froude휖 number. It is observed that as the Froude stability analysis is performed through a comparison of number is increased from 0.5 to 1.2, the unstable region the theoretical bar wavelength values with experimental shifts to a range of smaller wavenumbers. On the other values obtained from studies of alternate bars. In hand, when the Shields number is increased from addition, the computed maximum growth rates are 0.06 to 0.10, as shown in Figure 1 and Figure 3, the obtained from the stability analysis following the flow 푛 critical aspect ratio increases from휃 approximately 6.0 to and sediment characteristics employed in the approximately 9.0. The critical aspect ratio is the experimental set-up of the works of Watanabe et al., minimum value of the range of aspect ratios beyond Lanzoni and Carrasco-Milian and Vionnet. If the 푐푟 which bars initially develop. As such, the 훽threshold maximum growth rate is positive, bars are theoretically value of the aspect ratio increases with respect to an formed; it is then verified into the experimental data if increase in Shields number. Furthermore, as the Shields bars have indeed formed. number increases, the growth rate contours of bar For the case of non-erodible banks, the results of Watanabe et al.6 for steady flow conditions and of instability with bank erosion coincide with that of pure 7 bar instability.