Meandering – Like River Channel Evolution Due to the Formation of Fluvial Bars

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 𝑐𝑐표푠휙 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 2 푈 푉 − � � − 훽 ( 푈 )푉 퐻 푅 퐿 휕푥 휕푦 +퐹 휕푦= 0휕푦 퐻 ( , ) = (1, 0) + ( , ) (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. This indicates that the effect of bank Lanzoni for free bars are used. For the case of both erosion is significant at = 0.06, and as it is erodible banks with no slump blocks and non-erodible maximized, the effect of bank erosion becomes less banks, the values are approximated from the work of 푛 relevant. Hence, the amount휃 of sediment supplied from Carrasco-Milian et al. While the number of runs and the bank to the bed is reduced for as long as is experimental conditions for the experimental studies are increased. not sufficient to provide reliable comparison of the 푛 When = 0, bank erosion vanishes; when the value휃 theoretical results, only the general trend is considered is 1, the banks are erodible such that there is no herein. protection 휖that inhibits bank erosion, either in the form A comparison of the predicted values of bar wavelength to the values of the observed wavelength is examined, as plotted in Figure 9. The theoretical values predicted by the analysis fall within 1.4 m - 1.6 m, and their obtained experimental values range within 2.0 m - 3.4 m6. On the other hand, the theoretical values range from 5.5 m - 6.7 m, while the experimental values fall within 11.6 m – 6 m7. It is revealed that the experimental values are twice as large as the calculated values of bar wavelength. However, the agreement is relatively good for the comparison to the experiment of Lanzoni as long as the experimental values fall within around 6 m – 8 m; at observed values higher than this range, the calculated values still tend to underestimate the observed values of bar wavelength. It is worth noting data points in Figure 9 where the observed wavelength is zero but the theoretical results record otherwise. Such is the case where no bars formed during the experimental runs of Watanabe et al. As referred to their study, no bars formed for runs S-5 to S- Figure 1. Plot for = 0.5, = 0.06. 66. Yet, basing from the theoretical analysis, all of the 푛 six experiments yielded an unstable condition. This 퐹 휃 means that using the flow and sediment parameters of the study, bars theoretically form in all of the experimental set-up, which is not the case for the experiment, as observed that alternate bars formed only when the flow depth is less than 2.5 cm6. On the contrary, there is good agreement between the theoretical results with the experimental results of Lanzoni. In this case, based on the hydraulic conditions used in the experiment, the model predicts that base state is unstable for all the experimental conditions. As for the experiment, all of the 11 runs were able to detect the growth of bars. The predicted meander wavelength is thought to provide a crude estimate of wavelengths observed in flume studies and field conditions1. However, inerodible banks do not allow for meandering to develop; hence, for meandering to progress, banks must erode as well. Herein, the role of bank erosion to the formation of alternate bars is investigated. The results of the experiment of Carrasco-Milian et al. are Figure 2. Plot for = 1.2. used, particularly the values observed at the initial time of the experiment, or when the bars are starting to 퐹 develop. It is to be noted, however, that data are approximated and do not reflect the actual values themselves. As found from their results8, wavelength values for non-erodible banks range from 0.6 m - 3.7 m, while the values of erodible banks range from 0.4 m - 2.55 m. Apparently, the results for the case of non- erodible banks are slightly larger than the case of erodible banks. However, this observation could not provide explanation for how bank erosion affects bar formation. Furthermore, as shown in Figure 10 and Figure 11, the theoretical analysis predicts the base state of the experiment to be stable, for the case of erodible and non-erodible banks. The experimental results lie within the stable region, in which bars do not develop. The unstable region starts to initially develop at a critical aspect ratio = 6.4, whereas the experiment used = 5.7. This elaborates the role of the aspect ratio 훽푐푟 as a governing parameter for bars to initially develop. It is suggested훽 that the aspect ratio of the experiment be Figure 3. Plot for = 0.5, = 0.10. increased to yield comparable results with the analysis. 퐹 휃푛

Figure 7. Plot for = 20, = = 0.50.

훽 푆푅 푆퐿

Figure 4. Instability diagram for variation of . Thin solid contour: pure bar instability, from outside to inside, = 0, 0.3, 0.7, 1. Only neutral curves shown.휖

휖

Figure 8. Plot for = 20, = 0.10.

훽 휃푛 14 12 10 8 Figure 5. Plot for = 20, = 0.005. (m)

푓 6 훽 퐶 λ 4 Watanabe et al. 2 Lanzoni 0 0 2 4 6 8 10 12 14

(m) Figure 9. Plot for predicted wavelength vs. observed wavelength.𝜆𝜆𝑐𝑐

Figure 6. Plot for = 20, = 0.2.

훽 퐹 selected aspect ratio in the unstable region where bars grow. This is helpful by selecting a set of flow and sediment parameters to be employed for an experimental set-up. Furthermore, a plot of the contour of growth rate of perturbation vs. wavenumber can provide the dominant wavenumber corresponding to the maximum growth rate for a given set of flow and sediment parameters. The dominant wavenumber provides the wavelength that serves as a crude estimate of the wavelengths observed in experiment.

Notation

. , : coordinates in the streamwise and lateral directions, respectively . 푥: time푦 . , : flow velocities in the and directions . 푡 , : flow depth and bed elevation 푈 푉 푥 푦 Figure 10. Instability diagram for erodible banks. . , : right bank and left bank junctions . 퐻: aspect푍 ratio . 푅 퐿: critical aspect ratio . 훽: Froude number 푐푟 . 훽: channel planform curvature . 퐹: and channel slope . 퐶 , , : right bank and left bank slopes . 푆 , + + 푅 :퐿 /2; /2 . 푆 , 푆 : width of the right bank and left bank 푅 퐿 . 푅� , 퐿� 푅, 퐵: non-퐿dimensional퐵 bed shear stress, 퐵푅 퐵퐿 and푏 its푏푥 components푏푦 in the and directions . 푇 푇, 푇 : sediment transport rates in the 푥 푦 central푏푥 bed푏푦 region in the and directions . 푄 , 푄 : sediment transport rates in the right bank and left bank regions푥 푦with equation 푠푅푗 푠퐿푗 푄similar푄 to , but evaluated at the right bank and left bank junctions, respectively 푏푥 푏푦 . : parameter푄 used푄 to describe the effect of protection against bank erosion, e.g. slump blocksϵ or presence of vegetation . , : empirical constants Figure 10. Instability diagram for non-erodible banks. . : bed friction coefficient . 퐾 :푟 ______푓 Shields number . 퐶 : non-dimensional critical bed shear stress 푛 . 휃: 푐 non-dimensional wavenumber Conclusion . 휃 : dominant wavenumber . 푘 : angular frequency 푚푎푥 The study concludes that the aspect ratio, Froude . 푘 [ ]: imaginary part of angular frequency or number, Shields number, bank slopes and bed friction 휔the growth rate of perturbation coefficient are the governing parameters that influence . 퐼푚[휔 ] : maximum growth rate of the formation of bars. In addition, when bank erosion is perturbation 푚푎푥 incorporated into the analysis, the wavelength of the . 퐼푚: observed휔 wavelength in meters bars become longer and the maximum growth rates are . : predicted wavelength in meters decreased. 𝜆𝜆 푐 A comparison of the theoretical and experimental 𝜆𝜆 results is performed. It is revealed that the observed References wavelength is twice as large as the calculated wavelength values. Hence, the predicted results from the 1. Ikeda, S., G. Parker, and K. Sawai: Bend analysis generally underestimate the results of the theory of river meanders, part 1, Linear experiment. The relevance of the results is evaluated development, J. Fluid Mech., 112, 363-377, herein. Instability diagrams in the plane can serve 1981. as basis for providing a range of wavenumbers for a 푘 − 훽 2. Tubino, M., and G. Seminara: Free-forced interactions in developing meanders and suppression of free bars, J. Fluid Mech., 214, 131-159, 1990. 3. Johanneson, H. and G. Parker: Linear theory of river meanders, In: River Meandering, (Eds. S. Ikeda and G. Parker), pp. 181--213, AGU, 1989. 4. Kinoshita, R., On the formation of sand bars in river channels, Observation of meandering channels, Proceedings of JSCE, 42, 1--21, 1957. 5. Parker, G., Y., et al.: A new framework for modeling the migration of meandering rivers, Earth Surf. Process. Landforms, 36, 70-86, 2011. 6. Watanabe, Y., M. Tubino, G. Zolezzi and K. Hoshi: Behavior of alternate bars under unsteady flow conditions, Available online: http://river.ceri.go.jp/contents/archive/docs/h13 -303-312.pdf. 7. Lanzoni, S.: Experiments on bar formation in a straight flume 1. Uniform sediment, Water Resources Res., 36(11), 3337-3349, 2000. 8. Carrasco-Milian, A., C. A. Vionnet, and E. M. Valentine: Experimental study on the formation of alternate bars with fixed and loose banks, RCEM 2009 Proceedings, Argentina, 773-778, 2010.