Chapter 3 Model Validation
4.2 Numerical Results
4.2.1 Coexistence of free and forced bars
The coexistence of free and forced bars in the variable-width channels is shown in Figures 4-3(a). In the case of B15A01W06, the height of free bars is much greater than that of forced bars. Numerical simulations revealed that as the train of free bars moved over the forced bars and then migrated downstream, the original characteristics of the forced bars that were left behind would recover.
Figure 4-3 In B15A01W06 (a) Coexistence of free and forced bars; (b) Free bar components
(a)
(b)
The coexistence of free and forced bars gives rise to the nonlinear interactions [Blondeaux and Seminara, 1985; Tubino and Seminara, 1990; Whiting and Dietrich, 1993].
Such interactions may suppress the development of free bars [Kinoshita and Miwa, 1974;
Tubino and Seminara, 1990]. To quantify this forcing effect on free bars, the equilibrium forced bar components were subtracted from the mixed free-forced bar patterns (Figure 4-3 (b)). The resulting free-bar components were compared with those forming in the straight channels. The discussions here include the height, wavelength, and celerity of the free bars. The bar height is defined as the difference between the extreme bed elevations in the left and right halves of a cross section, the profile of bar height is shown in Figure 4-4.
Downstream of the initial disturbance area, peaks of the bar height profile are labeled as Bar01, Bar02, and so on. The first five peaks were used as the target bars in the present study because they emerged at early stages of the numerical simulations and exhibited sufficient temporal and spatial evolutions. The wavelength is defined as the distance between the negative peaks immediately up- and down-stream of a target bar. These definitions offer an advantage of translating the 3D bar configurations into 2D bar characteristics.
Figure 4-4 Definition sketch of bar height profile
4.2.2 Evolution of free bars in channels with variable width
To investigate the forcing effect of width variation on free bars, the feature of the free bars in the straight channel was first captured as a reference state for comparison. Shown in Figure 4-5 are the free bar component in channels with variable-width. In Figure 4-5(a) the distribution of free bars relate to the wave number of channels with variable width.
The peaks of bar height obviously appear in the narrow section of channels and much milder in the wide section. It represents that the variation of channel width have ability to compress and slacken the formation of free bars. Fixed the wave number of channels with variable-width, the deformation of free bars are proportional to the amplitude of channels, as shown in Figure 4-5(b). Observed the free bars patterns at 10m in Figure 4-5(b), the degree of bar height decrease when the amplitude of channels increasing. It implies that amplifying the amplitude of channels result in the suppression of free bars.
The evolution of the characteristic of free bars, ex: the bar height, wave length and celerity, in straight channel (A00W00) reach to the equilibrium stage after 10 hours, as shown in Figure 4-6. The equilibrium bar height, wave length and celerity are 1.18cm, 3.2m and 1.8 cm/min, respectively, which are mostly equal to Bernini’s simulation. The bar height reach to the equilibrium stage first, then wave length and celerity achieve.
Defina (2003) described the analogous phenomena in her numerical simulation.
Evolutions of free bars in the variable-width channels, however, exhibited wavy patterns of bar growth in response to the local variations of channel width. The trends of bar growth in the variable-width channels are similar to those in the straight ones when the forcing effects are small.
To quantify the effect of width variation on free bars, the bar height (BH) and wavelength (BL) of the target bars were normalized by the corresponding values in the straight channels such that the ratios represent the relative effect of width variation.
Moreover, the evolution is expressed using the number of cycles experienced by the target
bars given that the variation of channel width is periodic. These treatments are used in the subsequent analyses. For example, in Figure 4-7 the bar height of target bar in A02W06 was divided by the same target bar in straight channel, as a result the Y-axis in Figure 4-7 represent the ratio of bar height at the same moment. The location of the target bar can be transformed from the corresponding time axis, thus the X-axis in Figure 4-7 was labeled by the number of cycle. The maximum bar height of target bar in each cycle become convergent when the development of target bars reaches equilibrium. It is worthwhile to mention that shifting the target bar by cycle the equilibrium stage of all target bars will overlap in sequence, just like a resonant. It implies that all target bars in channels with variable-width have the same characteristic in the equilibrium stage.
The train of free bars migrates downstream meanwhile induces new bars further downstream. The induced free bars have characteristics similar to upstream ones and transmit those characteristics to downstream. Shown in Figures 4-8 are the evolutions of bar characteristic ratio as a function of the number of width variation cycles, where the number of cycles experienced by each target bar is shifted such that the resonant stages are all overlapped. In Figure 4-8 (a), for example, all bars in B15A01W06 reached the final resonant stages. The resonant state of Bar01 is the longest, indicating that Bar01 reached the resonant stage first, followed by Bar02, and so on. Variations of the bar height (BH) and wavelength (BL) of the free bars are highly related to the wave length of channels with variable-width. The sensitivities of the wavelength and celerity to the forcing effect are, however, different. In Figures 4-8, for example, the bar height ratios range between 0.85~1.15 in A02W02 while in A02W08 range between 0.9~1.0. The variation of bar height ratio is inversely proportional to the wave number of channels with variable-width.
There is minimum variation of bar wavelength ratio when the wave number of channels is 0.6 (W06 series). It may be due to that the wave length of free bars in straight channel which is equal to 3.2m, which is twice of the wave length of W06 series (1.6m). The
variation of channel width and wave length of free bars in W06 series are in phase, as a result bar wave length ratio in W06 are nearly equal to unity.
Figure 4-5 Free bar component in (a) A04 which the amplitude of channels is fixed and the wave number is altered and (b) W04 series at 8 hour. W00 and A00 represent the same straight channel run.
(a)
(b)
Figure 4-6 Bar height (BH), celerity (BC) and wavelength (BL) evolves with time in B15 series.
Figure 4-6 (continued)
Figure 4-6 (continued)
Figure 4-6 (continued)
Figure 4-7 Acquire the equilibrium stage of target bars.
Figure 4-8 Bar height (BH), wavelength (BL) and celerity (BC) reach to the equilibrium state.
Figure 4-8 (continued)
Figure 4-8 (continued)
Figure 4-8 (continued)
4.2.3 Quantitative forcing effect on equilibrium stage
The forcing effect of width variation can be quantified using the amplitude and wavelength of the wall sinuosity. The forcing effect of the variable-width channel can be represented by two factors. The first is the wavy factor (WN), defined as the dimensionless wave number of the variable-width channel, representing the waviness frequency of the channel wall. The second is the amplitude factor (Amp), defined as the ratio of amplitude to channel width, representing the degree of width perturbation. Shown in Figure 4-9 are the ratios of equilibrium bar characteristics varying as a function of wavy and amplitude factor, which display a suppression trend. The combination of the wavy factor and amplitude factor defines the forcing factor. The expression of forcing factor FF is taken to be :
FF Amp e WN (4-1)
The equilibrium bar heights in B15 series are used as the outcomes corresponding to the forcing effect of width variation. The regression relation between the ratios of equilibrium bar height (BH), wavelength (BL) and celerity (BC) and the forcing factor are
2.16
Figure 4-10 shows a satisfactory coefficient of determination R2=0.95, 0.93, and 0.87, respectively. The power of the forcing factor is 2.16 in all regression which implies the defined forcing factor is consistent with bar evolution. Figure 4-10 also demonstrates the suppression of free bars by the forcing effect as the equilibrium bar characteristics declines with the increasing forcing factor.
Figure 4-9 Ratio of equilibrium (a) bar height, (c) wavelength and (e) celerity vs. Amp.
factor, (b) bar height, (d) wavelength and (f) celerity vs. WN factor.
(a) (b)
(c)
(e) (f) (d)
Figure 4-10 Ratio of equilibrium (a) bar height, (b) wavelength and (c) celerity vs.
forcing factor
Forcing Factor Ratio of BH
RBH = -0.38FF2.16 + 1 R2 = 0.95
Forcing Factor Ratio of BL
RBL = -0.13FF2.16 + 1 R2 = 0.93
Forcing Factor Ratio of BC
RBC = -0.16FF2.16 + 1.041 R2 = 0.87
(a)
(b)
(c)
Chapter 5 Conclusions
5.1 Conclusions
5.1.1 2D morphodynamic model
1. A general 2D morphodynamic model was developed here. The most important improvement made in this study is applying the Streamline Upwind Petrov Galerkin (SUPG) scheme to the sediment continuity equation. The resulting bed evolution model has the ability to simulate the translation dominated bedform, such as the free migrating alternate bars.
2. The Vertical Average model (VA) was used in the morphodynamic model and compared with the linear theory of regular perturbation method [Wu and Yeh, 2005]. A comparison between numerical model and linear theory reveals that both of them have similar results, but the numerical model obtained more accurate results in the simulation.
3. We are the first to successfully simulate the evolution of free bars using the FE scheme without degrading the revolution of sediment transport in a unit discrete element, attributable to the application of SUPG scheme in solving the sediment continuity equation. The translation and dispersion of the bedform are captured and validated with the cases of forced bars (side bar and central bar), free bars in straight channel, coexistence of free and forced bar (low and high average aspect ratio) which involve the general cases of bed deformation.
5.1.2 Influences of forcing effect on free bars
1. Numerical simulations revealed that as the train of free bars moved over the forced bars and then migrated downstream, the original characteristics of the forced bars that were left behind would recover. Evolution of free bars will reach an equilibrium state, which are the same as those developing in a straight.
2. The time evolutions of bar characteristic (bar height, wavelength, and celerity) reach a equilibrium state during a sufficient period. Shifting the target bar by cycle the equilibrium stage of all target bars will overlap in sequence, just like a resonant. It is implies that all target bars in channels with variable-width have the same characteristic in equilibrium stage.
3. Evolutions of free bars in the variable-width channels, however, exhibited wavy patterns of bar height, wavelength, and celerity in response to the local variations of channel width. The trends of bar characteristic (height, wavelength, and celerity) in the variable-width channels are similar to those in the straight ones when the forcing effects are small.
4. The variation of bar height ratio is inverse proportion to the wave number of channels with variable-width. There is minimum variation of bar wavelength ratio when the wave number of channels is 0.6 (W06 series). It may be due to that the wave length of free bars in straight channel which is equal to 3.2m is twice of the wave length of W06 series (1.6m). The variation of channel width and wave length of free bars in W06 series are in phase, as the result bar wave length ratio in W06 are nearly equal to unity.
5. The forcing effect of the variable-width channel can be represented by two factors, which are the wavy factor (WN) and the amplitude factor (Amp). the ratio of equilibrium bar characteristics varying as a function of wavy and amplitude factor and display a suppression trend.
6. The expression of forcing factor is defined as: FF = Amp x eWN. The regression relation between the ratio of equilibrium bar height (BH), wavelength (BL) and celerity (BC) and the forcing factor have a satisfactory coefficient of determination R2=0.95, 0.93, and 0.87, respectively. The free bars suppressed by the forcing effect with the power of 2.16 demonstrate the equilibrium bar characteristics declines with the increasing forcing factor.
5.2 Suggestions
1. The ranges of flow conditions (e.g., average aspect ratio) may be expanded to obtain more general and broadly validated conclusions regarding the influences of forcing effect on free bars.
2. To explain the suppression of free bars in channels with periodic width variations, theoretical models that employ the perturbation [Murdock, 1999] theory may be developed in the future.
3. The channel forcing effects include the width variation and channel curvature.
Only the effect of width variation is investigated in this study. The forcing effect of channel curvature on free bars, such as the effect of point bars in river bends on alternate bars, may be included in future work.
References
Bernini A., Caleffi V. and Valiani A., 2006, Numerical modeling of alternate bars in shallow channels, International Association of Sedimentologists in Braided rivers: process, deposits, ecology and management, edited by Gregory H.
Sambrook Smith et al., Blackwell publishing, Malden
Bittner L.D., 1994, River Bed Response to Channel width Variation, Master Thesis, University of Illinois.
Blanckaert K. and de Vriend H.J., 2003, Nonlinear modeling of mean flow redistribution in curved open channels, Water Resources Research, 39(12):
6.1-6.14.
Blanckaert K. and Graf W.H., 2004, Momentum transport in sharp open channel bends, Journal of Hydraulic Engineering, 130(3): 186-198.
Blondeaux P. and Seminara G., 1985, Aunified bar-bend theory of river meanders, Journal of Fluid Mechanics, 157:449-470
Brooks A.N., and Hughes T.J.R., 1982, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, 32, 199-259.
Callander R.A., 1969, Instability and river channels. Journal of Fluid Mechanics, 36:165-480
Colombini M., Seminara G. and Tubino M., 1987, Finite-amplitude alternate bars, Journal of Fluid Mechanics, 181:213-232
Colombini M. and Tubino M, 1991, Finite-amplitude free bars : A fully nonlinear spectral solution, in Sand Transport in Rivers, Estuaries and the sea, 163-169, edited by Soulsby R. and Bettes R., published by Balkema A.A. and Brookfield Vt.
Cui Y., Parker G., and Paola C., 1996, Numerical simulation of aggradation and downstream fining, Journal of Hydraulic Research, 34(2): 185-204.
Cui Y., and Parker G., 2003a, Sediment pulses in mountain rivers: 1. Experiments, Water Resources Research, 39(9), 1239.
Cui Y., and Parker G., 2003b, Sediment pulses in mountain rivers: 2. Comparison between experiments and numerical predictions, Water Resources Research, 39(9), 1240.
Defina A., 2003, Numerical experiments on bar growth, Water Resources Research, 39(4) : ESG 2
de Vriend H.J., 1976, A mathematical model of steady flow in curved shallow channels, Journal of Hydraulic Research, 18(4): 327-341.
Engelund F., 1974, Flow and bed topography in channel bends, Journal of Hydraulic Division, 100: 1631-1648.
Engelund F. and Skovgaard O., 1973, On the origin of meandering and braiding in alluvial streams. Journal of Fluid Mechanics, 57:289-302
Fredsoe J.,1978, Meandering and braiding of rivers. Journal of Fluid Mechanics, 84:609-624
Fujita T. and Muramoto T., 1985, Studies on the process of development of alternate bars, Bull. Disaster Prev. Res. Inst. Kyoto Univ., 35 : 55-86
Fukuoka S., 1989, Finite amplitude development of alternate bars. in River Meandering, Water Resources Monograph, 12:237-266, edited by Ikeda S. and Parker G., AGU, Washington D.C.
Garcia M. and Nino Y., 1993, Dynamics of sediment bars in straight and meandering channels : Experiments on the Resonance Phenomenon, Journal of Hydraulic Researches, 31(6) : 739-761
Ghamry H.K., 1999, Two dimensional vertically averaged and moment equations for shallow free surface flows, PhD thesis, University of Alberta, Canada.
Ghamry H.K., and Steffler P.M., 2002, Two dimensional vertically averaged and moment equation for rapidly varied flows, Journal of Hydraulic Research, 40(5):
579-587.
Ghamry H.K., and Steffler P.M., 2005, Two dimensional depth averaged modeling of flow in curved open channel, Journal of Hydraulic Research, 43(1): 44-55.
Giraldo F.X., 1995, A space marching adaptive remeshing technique applied to the 3D Euler equations for supersonic flow, PhD thesis, University of Virginia.
Hicks F.E., and Steffler P.M., 1992, Characteristic dissipative Galerkin scheme for open-channel flow, Journal of Hydraulic Engineering, 118(2): 337-352.
Hoger A., and Carlson D.E., 1984, Determination of the stretch and rotation in the polar decomposition of the deformation gradient, Quarterly of Applied Mathematics, 42(1): 113-117.
Hughes T.J.R. and Mallet M., 1986, A new finite element formation for computational fluid dynamics: III. The generalized streamline operator for multidimensional advective-diffusive systems. Computational Methods in Applied Mechanics and Engineering, 58(3): 305-328.
Ikeda S., Parker G., and Sawai K., 1981, Bend theory of river meanders. Part 1.
Linear development, Journal of Fluid Mechanisms, 112: 363-377.
Kassem A.A. and Chaudhry M.H., 2002, Numerical modeling of bed evolution in channel bends. Journal of Hydraulic Engineering, 128(5): 507-514.
Kinoshita R. and Miwa H., 1974, River channel formation which prevents downstream translation of transverse bars, Shinsabo, 94 : 12-17 (in Japanese) Koch F.G., and Flokstra C., 1981, Bed level computations for curved alluvial channels.
Proceedings of the XIX Congress of the IARH, New Delhi, India, 2: 357-388.
Lanzoni S., 2000, Experiment s on bar formation in a straight flume: 1. Uniform sediment, Water Resource Research, 36:3337-3349
Lanzoni S. and Tubino M., 2001, Experimental observations on bar development in cohesionless channels, Excerpta, G.N.I., 14:119-152, CUEN-Napoli, Napoli, Italy
Lisle T.E., Pizzuto J.E., Ikeda H., Iseya F., and Kodama Y., 1997, Evolution of a sediment wave in an experimental channel, Water Resources Research, 33(8):
1971-1981.
Lisle T.E., Cui Y., Parker G., Pizzuto J.E., and Dodd A.M., 2001, The dominance of dispersion in the evolution of bed material waves in gravel-bed rivers, Earth Surface Processes and Landforms, 26, 1409-1420.
Molls T., and Chaudhry H.M., 1995, Depth-averaged open-channel flow model, Journal of Hydraulic Engineering, 121(6): 453-465.
Murdock J.A., 1999, Perturbations : Theory and Methods, Wiley, NewYork.
Nelson J.M. and Smith J.D., 1989, Flow in meandering channels with natural topography, in River Meandering, Water Resources Monograph, 12:69-102, edited by Ikeda S. and Parker G., AGU, Washington D.C.
Parker G., 1976, On the cause and characteristic scales of meandering and braiding in rivers, Journal of Fluid Mechanics, 76:457-480
Parker G. and Johannesson H., 1989, Observations of several recent theories of resonance and overdeepening in meandering channels. In River Meandering, Water Resources Monography, 12:379-415, edited by Ikeda S. and Perker G., AGU, Washington D.C.
Repetto R., Tubino M. and Paola C., 2002, Planimetric Instability of Channels with Variable Width , Journal of Fluid Mechanics, 457: 79-109
Schielen R., Doelman A. and de Swart H.E., 1993, On the nonlinear dynamics of free bars in straight channels, Journal of Fluid Mechanics, 252:325-356
Seminara G. and Tubino M., 1989, Alternate bars and meandering : Free, forced and mixed interactions, in River Meandering, Water Resources Monography, 12:267:320, edited by Ikeda S. and Parker G., AGU, Washington D.C.
Seminara G., 2006, Meanders, Journal of Fluid Mechanisms, 554: 217-297
Shimizu Y. and Itakura T., 1989, Calculation of bed variation in alluvial channels, Journal of Hydraulic Engineering, 115(3): 367-384.
Shimizu Y. Tamaguchi H. and Itakura T., 1990, Three-dimensional computation of flow and bed deformation. Journal of Hydraulic Engineering, 116(9): 1090-1108 Struiksma N., 1985, Prediction of 2-D bed topography model for rivers. River
Meandering, Water Resources Monograph, edited by Ikeda S. and Parker G. et al., 8: 151-180, AGU, Washington D.C.
Struiksma N., Olesen K.W., Flokstra C. and De Veriend H.J., 1985, Bed deformation in curved alluvial channels, Journal of Hydraulic Research, 23: 57-79.
Tubino M. and Seminara G., 1990, Free-forced interactions in developing meanders and suppression of free bars, Journal of Fluid Mechanics, 214 : 131-159
Tubino M., Repetto R. and Zolezzi G., 2000, Free bars in Rivers, Journal of hydraulic Research, 37(6) : 759 – 775
Vasquez J.A., 2005, Two-dimensional finite element river morphology model, PhD thesis, University of British Columbia, Canada.
Vasquez J.A., Millar R.G., and Steffler P.M., 2007, Two-dimensional finite element river morphology model, Canada Journal of Civil Engineering, 34: 752-760.
Whiting P.J. and Dietrich W.E., 1993, Experimental Constraints on Bar Migration Through Bends: Implications for Meander Wavelength Selection, Water Resources Research, 29(4) : 1091-1102
Wu F.C., and Yeh T.H., 2005, Forced bars induced by variations of channel width:
Implications for incipient bifurcation, Journal of Geophysical Research, 110, F02009
Zech Y., Soares-Frazao S., Spinewine B., Bellal M., and Savary C., 2005, The morphodynamics of super- and transcritical flow, in River, Coastal and Estuarine Morphodynamics, edited by Parker G. and Garcia M., pp.239-251, Taylor & Francis Group, London.