Experimental observations on bar development in cohesionless channels S. Lanzoni M. Tubino Dipartimento di Ingegneria Idraulica, Marittima e Geotecnica, Universit` a di Padova. via Loredan, 20 35131 Padova. fax ++39 049 8275446, e-mail [email protected]Dipartimento di Ingegneria Civile e Ambientale, Universit` a di Trento. via Mesiano 77 38050 Trento. fax ++39 0461 882672, e-mail [email protected]Abstract The formation of migrating bars in straight river reaches has been conclusively explained in terms of an inherent instability of a cohesionless bed subject to a turbulent flow. Theo- retical predictors of the occurrence of bars and of their geometrical characteristics, which have been mainly developed in the eighties, are based on several simplifying assumptions. In particular they refer to infinitely long straight channels and assume the sediment to be uniform and mainly transported as bedload. According to the results of recent theoreti- cal investigations the process may change considerably when the above assumptions are removed and further ingredients are included in the analysis, namely sediment hetero- geneity, transport in suspension and channel non-uniformities, like width and curvature variations. In the paper we review some recent experimental work aimed at investigating the effect of these factors on bar development. It is found that they induce an overall sup- pressing effect on bar instability and inhibit the development of stationary regular trains of alternate bars migrating along the channel. Some as yet unpublished results on bar dynamics in erodible channels which are free to evolve both altimetrically and planimetrically are also presented. 1. Introduction In the last decade the appearance of several refined predictive models of the dynamical response of river systems and the progressive refinement of measuring techniques moti- vated a renewed interest for the experimental investigation of basic processes characteris- ing river dynamics. It is quite well established that river morphology is the result of a highly nonlinear system, whose response mainly depends on the continuous development and interaction of free and forced features. The above process gives rise to a large variety of patterns. To simplify the analysis it is common to distinguish between bed- and channel-forms as the results of erosion-depositional processes characterised by two different timescales. 1
Transcript
Experimental observations on bar developmentin cohesionless channels
S. Lanzoni M. Tubino
Dipartimento di Ingegneria Idraulica, Marittima e Geotecnica,Universita di Padova. via Loredan, 20 35131 Padova.fax ++39 049 8275446, e-mail [email protected] di Ingegneria Civile e Ambientale,
Universita di Trento. via Mesiano 77 38050 Trento.fax ++39 0461 882672, e-mail [email protected]
Abstract
The formation of migrating bars in straight river reaches has been conclusively explainedin terms of an inherent instability of a cohesionless bed subject to a turbulent flow. Theo-retical predictors of the occurrence of bars and of their geometrical characteristics, whichhave been mainly developed in the eighties, are based on several simplifying assumptions.In particular they refer to infinitely long straight channels and assume the sediment to beuniform and mainly transported as bedload. According to the results of recent theoreti-cal investigations the process may change considerably when the above assumptions areremoved and further ingredients are included in the analysis, namely sediment hetero-geneity, transport in suspension and channel non-uniformities, like width and curvaturevariations. In the paper we review some recent experimental work aimed at investigatingthe effect of these factors on bar development. It is found that they induce an overall sup-pressing effect on bar instability and inhibit the development of stationary regular trainsof alternate bars migrating along the channel.
Some as yet unpublished results on bar dynamics in erodible channels which are freeto evolve both altimetrically and planimetrically are also presented.
1. Introduction
In the last decade the appearance of several refined predictive models of the dynamicalresponse of river systems and the progressive refinement of measuring techniques moti-vated a renewed interest for the experimental investigation of basic processes characteris-ing river dynamics.
It is quite well established that river morphology is the result of a highly nonlinearsystem, whose response mainly depends on the continuous development and interactionof free and forced features. The above process gives rise to a large variety of patterns.To simplify the analysis it is common to distinguish between bed- and channel-forms asthe results of erosion-depositional processes characterised by two different timescales.
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According to this approximation bed development is typically investigated for a given(fixed) channel geometry. Furthermore, on a much slower timescale the topographicallydriven flow which forces channel development is generally modelled with reference to theequilibrium bed configuration corresponding to an instantaneous adjustment to channelgeometry.
Though many river systems do not conform to the above approximation, namely thosealluvial channels whose banks may be easily eroded (like single branches of braiding net-works), the above decoupling procedure has been widely adopted to describe success-fully several river processes (see among others the recent contribution of Seminara et al.,2001). A recent attempt to account for the full coupling between bed and bank evolutionin channels with constant width is due to Darby and Thorne (1996).
If we concentrate our attention to large scale bed features, whose planimetric scale isof the order of few channel widths, the above viewpoint leads to a distinction betweenfree bars which can develop spontaneously in straight channels with constant width andbars which are forced by channel nonuniformities, like channel axis sinuosity and widthvariations.
Free bars form both in gravel and sandy-bed streams and consist of repetitive se-quences of depositional diagonal fronts with deep pools at the downstream face and gen-tler riffles along the upstream face. For typical values of the aspect ratio of the chan-nel these topographic sequences display an alternate structure (alternate bars); in widerreaches, however, bed configuration may become more complicate and higher order trans-verse structures, like central or multiple rows bars, are more likely to occur. It is worthnoticing that the vertical scale of bottom changes associated with free bars is of order ofthe flow depth. This implies that the prediction of their occurrence and of their equilib-rium characteristics has a profound impact on several aspects of river engineering suchas the safe conveyance of water, sediment and ice, navigation, bank protection, design offluvial structures, water supply, fishery and environmental protection.
A state of the art review of recent theoretical studies on free bar formation can befound in Tubino et al. (1999) (see also references therein). Bar occurrence has beensuccessfully explained in terms of an inherent instability of an erodible bed subject toa turbulent flow. The process is quite well established when the channel is straight andnot exceedingly wide, the width is constant, the flow is steady, the sediment transportmainly occurs as bed load and the sediment is uniform. Under these conditions varioustheoretical investigations (e.g. Blondeaux and Seminara, 1985; Colombini et al., 1987;Struiksma and Crosato, 1989; Shimizu and Itakura, 1989; Colombini and Tubino, 1991;Schielen et al., 1993), which have been validated through the results of several flumeexperiments (Kinoshita, 1961; Ashida and Shiomi, 1966; Chang et al., 1971; Sukegawa,1971; Muramoto and Fujita, 1978; Ikeda, 1982; Jaeggi, 1984; Fujita and Muramoto,1985; Garcia and Nino, 1993), suggest the following picture:
- in straight indefinitely long channels free bars form above some critical value of the aspect ratio of the channel, depending on both flow and sediment parameters,namely the Shields parameter and the relative bed roughness ; free bars are tempo-rally growing disturbances which migrate downstream at a speed determined mainly bytheir size and by the local sediment flux;
- predicted and observed values of the longitudinal wavelength of bar perturbations fallin the range of 5-12 channel widths: the instability process is not strongly size-selective,different longitudinal modes within the unstable range being characterised by almost sim-
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ilar growth rate;- the transverse mode selected by the instability process depends strongly on the aspect
ratio, as reported in Figure 1 where Parker’s (1990) bedload transport relationship hasbeen employed; as a result in gravel bed rivers free bar instability generally displays analternating structure (mode 1), while central bars (mode 2) or higher order transversemodes are not likely to form spontaneously, in the absence of some forcing mechanism,unless the channel is fairly wide;
- provided the aspect ratio of the channel falls within a convenient neighbourhood ofthe critical value nonlinear interactions lead to periodic bar patterns which display thetypical asymmetries characteristics of river bars, namely diagonal depositional fronts anddeeper pools; both experimental investigations and numerical models suggest that withinthis range nonlinearity is weak in that the growth of higher harmonics is passively drivenby the development of the fundamental alternate-bar mode (see Figures 2 and 3);
- the resulting equilibrium bar amplitude is proportional to the square root of the ex-cess value of the aspect ratio relative to the threshold value (Colombini et al., 1987):comparison of predicted bar height with flume data appears quite satisfactory as reportedin Figure 4;
- according to the results of Schielen et al. (1993) the periodic solutions of Colombiniet al. (1987) may be unstable and lead to quasi periodic solutions, though a straight reachwith a length of few hundred widths is required to appreciate the associated modulationin space and time of bottom configuration;
- as the aspect ratio of the channel increases the nonlinear competition betweendifferent modes becomes stronger and may lead to the occurrence of complex transversestructures (Fujita, 1989; Colombini and Tubino, 1991); in particular, Fujita’ s (1989)experiments suggest that in channels with very large values of aspect ratio multiple rowbars may form; however, the subsequent development of bar pattern, which invariablydisplays a decrease of the order of transverse mode due to the clustering of bedforms, andlocal occurrence of bar emergence generally prevent the establishment of an equilibriumconfiguration.
The above scenario changes significantly when further ingredients, like sediment het-erogeneity, small scale bedform effects, suspended sediment and non-uniform channelgeometry, are included in the analysis to simulate bar development in actual rivers. Thisis suggested by some recent theoretical contributions (Lanzoni and Tubino, 1999; Tubinoet al., 1999; Repetto and Tubino, 1999; Repetto, 2000). Results of the above analyseswill be briefly reviewed in the following chapters. In the present paper the emphasis isplaced on the results of experimental observations which have been carried out to gain abetter understanding of the above processes and to validate theoretical findings. Results offlume tests performed by Lanzoni (2000a,b) and Repetto et al. (2000) are summarised anddiscussed. Furthermore, some as yet unpublished results concerning a set of experimentson the effect of width variations in channels with erodible banks are also presented.
The paper is organized as follows. In section 2 we describe the experimental facilitiesalong with data acquisition procedures and data processing. Section 3 is devoted to thediscussion on the effect of suspended load and small scale features on free bars. Therole of sediment heterogeneity is treated in section 4. Finally, results of experiments inchannels with variable width are presented in section 5.
The following notations are repeatedly employed in the paper: is the half width of the channel;
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is the average flow depth;
is the mean diameter of sediment; is the water discharge;
is the volumetric solid discharge;
is the average flow velocity;
is the longitudinal coordinate; is the wavenumber of bars;
is the relative bed roughness;
is the initial channel slope;
is the aspect ratio of the channel;
is Shields parameter; is the dimensionless wavenumber of bars.
Furthermore, an asterisk denotes dimensional quantities while the subscript denotesaverage (reference) values of dimensionless parameters. Further notations will be intro-duced in the text.
2. Materials, experimental apparatus and methods
The first and second series of experiments were aimed at investigating the effect of smallscale bedforms and of sediment heterogeneity on free bars (Lanzoni, 2000a,b). The ex-periments were performed at the De Voorst River, Navigation and Structure Division ofDelft Hydraulics in a rectangular flume 55 m long, 1.5 m wide and 1 m deep.
The flume was equipped to recirculate water and sediment separately. At the down-stream end of the flume was placed a V-shaped, 10.45 m long and 3.9 m wide, sedimenttrap that caught virtually 100% of the passing sediment. The material settling on thebottom of the sediment trap was suitably sucked out and conveyed with a small amountof water to a hydrocyclone installed at the upstream end of the flume, where water andsediment were separated. The sediment was collected in a small storage tank below thehydrocyclone and the increase in weight of the tank was continuously recorded by a bal-ance system. When the increase in weight had reached a preset value the storage tankautomatically opened and the sediment was dropped into the flume via a diffuser. Waterslope in the channel was established and/or maintained by adjusting the level of an auto-matically controlled tail gate located at the downstream end of the sediment trap. Waterdischarge was measured continuously through a Rehbock-weir.
In the first series of experiments a nearly uniform sand (MUNI) with mean diameterof 0.19 mm was used. The sediment mixture (FC70) employed in the second series ofexperiments was prepared by suitably mixing two nearly unisize fine (FUNI) and coarse(CUNI) mixtures: in particular, FC70 was composed with 67% of FUNI and 33% ofCUNI. The mixture FC70 has been chosen such that its geometric mean diameter was thesame of that of the nearly uniform sand used in MUNI’s experiments.
The characteristics of the various mixtures used in the experiments are reported inTable 1, where is the grain diameter such that i-th percentage of the material is finerthan ,
and are the geometric mean diameter and geometric standard deviation,respectively. The grain size distributions of the various mixtures are reported in Figure 5.It is worth noticing the poorly-sorted strongly bimodal character of FC70 sediment. Thiskind of mixture, which is common in both gravel rivers and within the transition between
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gravel and sand beds, exhibits a peculiar behaviour in terms of the various fractionaltransport rates (Wilcock and McArdell, 1993). In fact, depending on the ratio betweenthe effective bed shear stress acting on bedload particles (the so called ’skin fiction part’) and the reference (critical) shear stress for incipient motion of the i-th size fractionin the mixture , four different transport regimes can be identified. When fractional transport rates are negligibly small. For fractional transportrates of coarser particles are substantially smaller than those of smaller sizes, and at leastsome of the coarser particles are immobile. For the transport rates ofall fractions tend to be equal. Finally, as the friction velocity
approaches the settling
velocity
of the i-th size fraction a substantial proportion of the transport of this fractionoccurs as suspension.
The ranges of bed slope and water discharge explored in MUNI and FC70 tests werealmost similar. A summary of the experimental conditions is given in Tables 2 and 3,where is water surface slope and
is the average value of the skin friction part of
total bed shear stress
. Following Wilcock (1993)
was estimated using the drag-partition procedure of Einstein (1950) and assuming
and of the CUNI mixture to
be representative of grain roughness of MUNI and FC70 sediments, respectively.In the experiments with heterogeneous sediment the bed material in the flume was
thoroughly mixed by hand prior to the start of the run in such a way that vertical sortingwas prevented as much as possible. The bed was then levelled according to a selectedslope in order to reproduce similar initial conditions for each run. At the beginning ofeach experiments the tail gate located at the end of the sand trap was adjusted to obtainthe required energy slope while the discharge was kept constant. During the experimentsthe bottom profile was measured periodically at every centimeter along three differentlongitudinal sections, using three profile indicators which were installed along the flumeaxis and at 20 cm from each wall. Simultaneously, a water level indicator was used tomeasure the free surface elevation along the flume axis. The instruments were mountedon a carriage, driven by a motor and riding on two rails parallel to the floor of the channel.The measuring reach of the flume ranged between 7.9 and 51.7 m. The experimental runwere stopped when ’quasi’ equilibrium conditions were achieved, i.e. when the bed slopewas almost equal to the water surface slope and the solid discharge and bar features wereobserved not to vary significantly in time.
In some FC70 runs (i.e., P0109, P0609, P0709, P1309) the transported sediment wassampled during the equilibrium phase by collecting into a bucket the sediment flowingover the diffuser. The samples were then saved for size analysis and the sediment was notreturned into the flume, the dropping frequency being high enough to prevent any effectof sampling on sediment supply. The fractional transport rate of the i-th size fractionwas then calculated as , being the proportion of the i-th fraction in the transportand the measured total transport rate per unit width. At the end of run P2009 the bedmaterial in the flume was sampled at several locations using the technique developed byRibberink (1987).
The third series of experiments, performed in the Hydraulic Laboratory of TrentoUniversity, was aimed at investigating the formation of free bars in channel with nonuniform width and fixed banks (Repetto et al., 2000).
The channel had a length of 15 m and a maximum width of 60 cm. The experimentswere performed using a uniform sediment (WUNI) with a mean diameter of mm. Thehydraulic circuits used to recirculate water and sediment were quite similar to the ones
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used in MUNI and FC70 tests. At the downstream end of the flume water and sedimentwere collected in a 1.3 m long settling basin. The water level inside the basin was con-trolled by adjusting a vertical gate. A first hydrocyclone pump was used to convey bothwater and sediment to the upstream section of the flume where the sediment was sepa-rated from the water through a second hydrocyclone and fed into the flume. The amountof recirculated water was controlled through a valve. In order to avoid localized scourphenomena, a 1 m long reach of the flume bed was covered with gravel just downstreamof the inlet section. Water discharge was measured through an electromagnetic dischargemeter with a relative precision of 2%.
The experiments consisted of two distinct set of runs. A first set of experiments wasperformed in a channel with a constant width of 40 cm, for different values of water dis-charge and slope. The development of free bars in channels with variable width was theninvestigated with identical hydraulic conditions and average channel width. In particu-lar, periodic width variations were constructed inside of the channel by attaching stripsof PVC to wooden profiles, to form vertical banks varying sinusoidally according to therelationship:
(1)
where
is the average half width of the channel, is the wavenumber of width varia-
tions and is a dimensionless coefficient weighting the amplitude of width variations (seeFigure 6). The values of and of the dimensionless wavenumber
are reported
in Table 4 along with a summary of the experimental conditions.In WUNI experiments water surface elevation was monitored through eleven piezome-
ters, with longitudinal spacing of 1.2 m. During each run the shape, location and migrationspeed of bars were periodically surveyed. At intermediate stops and at the end of the runthe bed configuration was measured in detail, in the absence of water flow, using a laserprofiler mounted on a carriage which was automatically driven along the longitudinal andtransverse direction: this allowed to measure 50 points in a cross section, with longitudi-nal spacing of 0.1 m. Bottom elevation data were then analyzed through a FFT procedure.The carriage ran along two rails whose slope could be regulated at a prescribed value forlevelling the bed at the beginning of each run.
In a fourth series of experiments (BUNI), performed in the Hydraulic Laboratory ofTrento University, the altimetric and planimetric evolution of a channel cut into a cohe-sionless flat sloping surface was investigated (Bertoldi et al., 2001). The experimentswere designed to trace the development of laterally unconstrained cohesionless channelsfrom the initial straight configuration until the incipient bifurcation of the flow.
Experiments of this type are the premise for interpreting the formation of braidedrivers (see also Ashmore, 1991, and Seminara et al., 2000): in fact, channel bifurcationcan be viewed as the most important unit process which controls the production of newchannels in braided networks. The division of a stream largely depends on bar dynam-ics. Though different triggering mechanisms have been observed in braided networksreproduced in the laboratory (Ashmore, 1982, 1991), channel bifurcation is typically theconsequence of streamline divergence due to the development of steady central bars. Thisinvolves a transition from previously formed alternate bars (chute cut-off), which occursin a channel which has undergone a relatively fast and almost uniform widening in theinitial stage of the process and whose planform has been modelled by the migration offree bars (Figure 7).
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The experiments were performed in the 60 cm wide, 15 m long flume used in WUNIexperiments, which was filled with well sorted quartz sand with a mean diameter of 0.5mm. A further experimental run was carried out with uniform sand with mean diameterof 1.3 mm. The initial configuration of the bed was plane with a constant slope; a narrowchannel of trapezoidal shape was cut into the surface, with base width equal to 6 cm; theinitial free surface width ranged between 8-12 cm. The channel was supplied with con-stant water discharge, measured through an electromagnetic meter on the delivery pipe.The sediment was supplied at a given constant rate through a volumetric sand feeder anddropped into the channel via a diffuser. At the downstream end of the flume a transversegroundsill was placed into the bed to prevent local scour and minimize upstream influencefrom the outlet condition.
Different hydraulic conditions were tested in the experiments as shown in Table 5:notice that in some cases channel widening led channel banks to reach the fixed walls ofthe flume before the occurrence of the bifurcation. The planimetric development of thechannel was continuously monitored and documented through a digital camera mountedon a carriage that ran along longitudinal rails. The bed topography was surveyed period-ically with the laser profiler used in WUNI experiments, in the absence of the flow. Thefollowing procedure was adopted. At first each experimental run was performed with-out intermediate stops until the final stage and the planimetric evolution was measured.Then each run was repeated, starting from the same initial condition: the experiment wasrestarted after any intermediate stop required for bed survey. The effect of bar dissectioninduced by the withdrawal of the water was always found to be negligible. Additionalmeasurements of surface flow velocity were made at fixed locations along the channelusing a high-speed video camera and light particles as flow tracers.
3. Suspended load and small scale features
According to the results of the linear theory of Tubino et al. (1999) the formation offree bars in sandy streams displays several distinctive features with respect to the case ofgravel beds. At large values of Shields parameter the threshold value tends to vanishas a result of the vanishing role of the stabilizing contribution of gravity. Hence, whensuspended load is dominant the aspect ratio is no longer the control parameter of barformation and bar stability crucially depends on the longitudinal wavelength: shorter barsare damped while longer bars are enhanced. This implies a distortion of marginal curvesfor bar formation as shown in Figure 8.
A simple physical explanation of the latter effect can be given in terms of the longitudi-nal distribution of suspended load on a perturbed bed. Both concentration and suspendedload exhibit a delay with respect to bottom shear stress which may lead the peak of lon-gitudinal transport to exceed bar crests: under this condition sedimentation occurs at barpools and bar topography is damped. The phase lag of suspended load is mainly relatedto the convective term in the advection-diffusion equation for the sediment, whose effectis larger for shorter wavelengths. Hence, as Shields stress increases and suspended loadbecomes dominant, bar perturbations falling within the most unstable range of bedloaddominated gravel rivers, which corresponds to longitudinal wavelengths ranging between6-8 channel widths (i.e. =0.39-0.52), are damped while alternate bars as long as 25-30(i.e. =0.10-0.12) channel widths are expected to form. This is shown in Figure 9 where
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the wavenumber of bar perturbation characterised by the largest value of the growth rateis plotted versus the aspect ratio, for different values of the average Shields parameter
and given values of particle Reynolds number and grain roughness .These results can be interpreted as an indirect suppressing effect on free bars in that
a quite long straight reach is required to allow for their development: for widths of theorder of few hundred meters a river reach should keep straight and uniform over a lengthof several kilometers. Also notice that, while in the case of dominant bed load the alternatebar mode is the fundamental transverse mode in a wide range of values of (see Figures2 and 3), the linear theory of Tubino et al. (1999) suggests that in sandy streams varioustransverse modes are simultaneously unstable and characterised by almost similar growthrates, as shown in Figure 10. This implies that bar dynamics, rather than being driven bythe growth of the fundamental as in the case of Figure 2, is more likely to be controlled bythe nonlinear competition and possible clustering of several unstable transverse modes.
According to the above results the formation of stationary regular trains of fully devel-oped free bars in sandy streams seems rather rare. Also notice that the quasi-uniformityof channel geometry seems an essential requirement for the development of free bars:the results of several theoretical and experimental investigations (Kinoshita and Miwa,1974; Tubino and Seminara, 1990; Whiting and Dietrich, 1993; Bittner, 1994; Repettoand Tubino, 1999; Repetto et al., 2000) suggest that the spatial variability of the curva-ture of channel axis and of channel width strongly inhibit bar formation and their regulardevelopment and migration along the channel.
The above scenario conforms to the results of the laboratory investigations of Lanzoni(2000a) though the experiments were not properly designed to reproduce transport condi-tions dominated by suspended load. Indeed, in all MUNI experiments the sediment wasmainly transported as bedload; however, in those runs characterized by larger dunes a cer-tain amount of particles was put into suspension by the intermittent and intense vorticesarising from the crests of small scale bedforms or just downstream of bar fronts. In suchruns the formation of free bars turned out to be considerably altered and the developmentof regular trains of bedforms was inhibited. MUNI experiments also suggest that in sandystream bar development is likely to be affected by mutual interactions between large scalebedforms and small scale features (ripples and/or dunes).
The distinctive features of free bars in MUNI experiments are summarized in Table 6,where
denotes bar wavelength,
is bar height calculated as the difference betweenthe maximum and the minimum bed elevation within a bar unit, and is bar celerity,estimated by comparing the plots of the longitudinal bed profiles measured at differenttimes during the experiments. The values of bar height and wavelength measured inMUNI experiments are plotted in dimensionless form in Figures 11a,b, where the resultsof WUNI experiments in the constant width channel are also included (see Table 8).
In MUNI experiments the presence of small scale bedforms made it difficult to cor-rectly estimate bar features. In fact, the longitudinal bottom profiles were significantlydistorted by high frequency components. In particular, the interaction between bars andsmall scale bedforms was extremely strong in runs P2403, P0404, P1204 and P2804. Aclear evidence of such interaction is exhibited by the power spectra of left side and rightside bed profiles (taken 0.2 m from the flume walls) which were characterised by thepresence of two main peaks, with comparable intensities and wavelengths differing by afactor of about two (see Figure 12). A third minor peak was also detected displaying awavelength similar to that associated with the dominant peak of the longitudinal axis pro-
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file. The latter peak was related to the presence of small scale bed forms with wavelengthand height ranging between 40-60 cm and 2-4 cm, respectively.
Also notice that in runs P2304, P0404, P1204, and P2804 the bar pattern displayed adistinct modulation (oscillation) in time even after a quasi-steady condition was achieved.Periods in which a rather regular train of bars was present in almost the entire flume werefollowed by intervals characterized by a very irregular bar topography. The existence of aspatial sequence of wave groups was less evident, owing to the relatively low number ofbar wavelengths which can be contained within the measuring reach of the flume. Some-times the detected fluctuations of bottom topography were caused by poorly developedbar fronts which formed in the upstream part of the flume. These fronts migrated down-stream with larger wavespeed of previously formed bars until they were absorbed by thefully developed train of bars in the downstream part of the flume. Merging of these frontscaused periodic fluctuations of bar structure. This behaviour seems to conform to the the-oretical results of Schielen et al. (1993) which suggest that, for a given value of the rateof change of bedload function with respect to Shields parameter, a critical value of thefriction coefficient exists below which a modulation in time and in space of the bottomconfiguration may develop. However, in Lanzoni’s (2000a) experiments measured valuesof the above parameters fall outside of the range theoretically predicted for the onset ofmodulation.
4. The effect of sediment heterogeneity on free bars
In a recent contribution (Lanzoni and Tubino, 1999) we have incorporated the effect ofsediment heterogeneity within the framework of a linear bar-stability analysis. The novelfeature in this case is that bed deformation and the spatial variation of size distributionaffect each other.
When the bed is not plane, bottom deformation induces spatial variations of sedimenttransport which are associated to the spatial distribution of bed shear stress and to gravita-tional effects. Since different grain sizes exhibit different responses to variations of shearstress and gravity this implies a sorting pattern. Gravitational effects tend to pull coarserparticles downward, while selective transport of different grain sizes fractions tends todisplace coarser particles where the peak of bottom shear stress occurs.
On the other hand, a spatial distribution of grain size modifies bottom developmentsince it affects sediment transport. The effect is twofold. There is a direct effect onsediment transport related to different mobility of grain sizes within the mixture: thelocal response of sediment transport to shear stress and gravity changes from place toplace when bed composition is not uniform. Furthermore, a sorting pattern is felt by thefluid as a perturbation of local roughness: this effect induces a contribution to the shearstress which cumulates with that of bed topography. The latter effect is negligible in caseof bar stability since flow variability is mainly a consequence of bottom development.However, the effect may be relevant to explain the formation of small scale features, likebedload sheets, which are inherently associated with the heterogeneous character of thesediment (Seminara et al., 1996).
Theoretical results of Lanzoni and Tubino (1999) suggest that the selective transportof different size fractions within the mixture affects significantly the balance between sta-bilizing and destabilizing contributions which governs bar stability. More specifically,
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within a fairly wide range of values of the dimensionless control parameters the variouscontributions related to sediment heterogeneity act simultaneously to produce an over-all stabilising effect on bottom development, which results in a reduction of bar growthrate, shortening of bar wavelength and damping of migration speed. These effects areenhanced at small values of the Shields parameter and for increasing values of thestandard deviation of the mixture (see Figure 13). Furthermore, the resulting spatialdisplacement of particles appears to be mainly associated to selective transport, whichimplies a progressive coarsening along the upstream face of bars.
The overall damping effect of sorting on free bars can be mainly explained in termsof a reduction of the destabilising role of longitudinal transport, due to coarsening alongbar profile, coupled with an increase of the stabilising contribution of gravity whose roleis reinforced by sediment sorting. Theoretical findings agree with the results of the ex-perimental observations (Lanzoni and Tubino, 1999; Lanzoni, 2000b). Also notice thatthe suppressing effect on bar wavespeed associated to coarsening of bar front was alsodocumented by Lisle at al. (1991).
FC70 experiments were properly designed to investigate the effect on free bar mor-phology induced by longitudinal and vertical sorting. A summary of measured character-istics of free bars is given in Table 7. In these experiments bar development was stronglyaffected by the behaviour of various fractional transport rates. In several runs (i.e., P0806,P2006, P2906, P0307 P0507, P0807 and P1207) the ratio
was falling in range (1
- 2.1) corresponding to the region of partial transport. In general the bed remained nearlyplane and very long and quasi-steady bars formed; migrating bars with small height wereobserved to grow episodically in the downstream reach of the flume. Moreover, pavedsuperficial layers (5-20 cm wide) typically formed near the side walls of the flume a fewhours after the beginning of the run.
Further increasing the bed shear stress, the condition of fully mobilized transport wasapproached (P0109, P0509, P0609, P0709, P0809, P1309, P2009). A regular train ofwell formed alternate bars migrating along the full length of the channel was observedonly in early stages of the experiments. These initially formed bars gradually migratedout of the flume, and later on only irregularly shaped alternate bars were observed to growrather sporadically in the final reach of the flume (Figure 14). The height of these barswas found to increase as they migrated downstream, though it was always lower than thecharacteristic height of the initially formed regular bars.
This suppressing mechanism on free bars seems mainly related to the process of scourand fill associated with the migration of the initial train of bars, which caused intensevertical and longitudinal patterns of sorting. The pool at the downstream face of a bar frontacted as a trap for the coarse fraction and a thick layer of coarse sediment formed abovethe average level of bar troughs. Moreover, the selective transport of different grain sizesalong the gentler riffles upstream of each bar front led to the accretion of coarser particlestowards bar crests. As a bar front migrated downstream the coarse matrix at the bar troughwas progressively covered by finer sand and a winnowing process initiated through whichthe coarse matrix was slowly filled with finer particles. Therefore, as a given run wenton, the bed was substantially reworked by bar migration and the composition of the activelayer was likely to differ significantly from the initial composition.
The above picture is confirmed by the results of sieving analysis of samples of bedand transported sediments collected during the experiments. The longitudinal and verti-cal patterns of sorting induced in run P2009 by the initial migration of bars are shown
10
in Figure 15, where the vertical distributions of sediment grain size at different locationsalong a bar unit are reported. Computed average values of grain size distributions suggestthat the overall bed composition did not change appreciably during the initial stage of therun, the portion of bed analyzed resulting only slightly coarser than the bulk mixture. Onthe contrary, the samples of the transported sediment collected in various runs (P0109,P0609, P0709 and P1309) during the initial stage characterised by the migration of regu-lar train of bars indicate a lack of both finer and coarser fractions, which was quite largefor the finer fraction. This implied a significant coarsening of the transported sedimentwith respect to the bulk mixture. This is shown in Figure 16 where the mean cumulativegrain size distributions of the transported and bulk mixture are reported. The above resultssuggest that in a recirculating feeding system both the dynamical armour and the rework-ing of the bed due to bar migration may affect significantly the composition of transportedsediment, leading to an irregular surface texture and vertical grain size distribution whichcauses the bars to grow episodically only in the final reach of the flume.
The comparison between uniform (MUNI) and graded (FC70) sediment tests is notstraightforward. In fact, the presence of small-scale bed forms appeared to be a distinc-tive feature of all MUNI experiments. On the contrary, sediment heterogeneity stronglyinhibited the formation of ripples and dunes in FC70 experiments except for those carriedout with smaller slopes (runs P0206 and P0606) where a condition of partial transportwas achieved. Therefore, while in MUNI experiments the friction coefficient was mainlycontrolled by drag resistance, in FC70 runs the flow resistance was essentially dominatedby bed friction. Hence, the values of the friction coefficient measured in MUNI exper-iments were invariably lower than those characteristics of FC70 runs, as shown in Figure17. As a consequence, even though two given experiments were performed under similarhydraulic conditions, the friction coefficients were rather different.
Hence, the comparison can only be pursued for the experiments characterised by simi-lar values of the average Shields parameter , aspect ratio and relative grain roughness. Two sets of experiments can then singled out. The first set includes the uniform sed-iment tests P1801, P0404 and P2403 and the bimodal sediment runs P0606, P2006 andP2906. The second set consists of runs P1605, P1505, P2709, P2809 and P2909 (MUNI)and P0807, P0109, P1309 and P2009 (FC70).
Observed values of bar height and wavelength are reported in Figures 18a,b, whereexperimental data of Lanzoni et al. (1994) are also included: they were obtained in aflume 18 m long, 0.36 m wide, using a uniform sediment made up of glass spheres, withdiameter 1.5 mm, and a weakly bimodal mixtures of glass spheres with roughly the samemean geometric grain diameter. Figure 18a suggests that an overall damping effect on barheight is induced by sediment heterogeneity: notice that the experimental points fallingon the x-axis refer to experiments with non uniform sediments in which migrating barsdid not form. It also appears that sorting effects become weaker as the Shields stressincreases and the condition of equal mobility is approached.
The effect of sorting on bar wavelength is less clear. In Lanzoni et al.’s (1994) experi-ments sediment heterogeneity was always found to induce a reduction of bar wavelength.On the contrary, observed values of bar wavelength in FC70 experiments, when comparedwith the results of MUNI experiments, do not exhibit significant changes.
Theoretical findings for values of relevant dimensionless parameters corresponding toexperimental runs are reported in Figure 19. For the experiments with heterogeneous sed-iment the values of the maximum growth rate of bars and of their wavelength have been
11
calculated through the dispersion relationship obtained by Lanzoni and Tubino (1999),with a hiding exponent of 0.2. In case of uniform sediment the dispersion relationshipgiven in Lanzoni (2000b) has been used. In both cases Parker’s (1990) surface based for-mula has been employed to evaluate bedload transport; moreover, the friction coefficientin MUNI tests has been calculated through the roughness predictor proposed by Richard-son and Simons (1967) to account for the effect of small scale bed features. The observeddamping effect of sorting on free bars is reproduced by the theory through a reductionof the maximum growth rate of bar perturbations (see Figure 19a). Note that theoreticalresults would predict the growth of alternate bars for some bimodal runs (P0606, P2006and P2906) in which the bed actually remained plane. However, the experimental value of was very close to the threshold value below which bars are not expected to develop.Theoretical predictions of bar wavelength reported in Figure 19b agree with experimentalfindings: sediment heterogeneity induces a reduction of bar length in case of Lanzoni etal.’s (1994) experiments, while predicted values of do not change significantly whenconsidering either MUNI or FC70 sediments.
5. Free bars in channels with variable width
In natural rivers spatial non uniformities of the planimetric configuration, like width varia-tions and curvature distribution, give rise to forcing effects which lead to quasi-steady beddisturbances. The transverse structure of these forced responses essentially depends onthe characteristics of the forcing mechanism: in meandering channels an alternate (pointbar) configuration arises as a consequence of the antisymmetric forcing associated to theperiodic change of sign of channel curvature; on the contrary, a central bar configurationcan be the result of a symmetrical forcing like that induced by streamline convergence-divergence associated with repetitive width variations.
Coexistence of free and forced bedforms is often observed when the forcing mecha-nisms are relatively weak, as in self formed channels in their initial stage of development(e.g., Fujita and Muramoto, 1985). In many cases it has been observed that when theforcing effect is large enough freely migrating forms are suppressed and a steady bot-tom pattern may develop, which in turn affects the subsequent planimetric evolution ofthe channel. In case of meandering channels a threshold value of channel curvature ex-ists, depending on flow and sediment characteristics, above which free migrating bars aresuppressed. This was first observed experimentally by Kinoshita and Miwa (1974) andlater interpreted theoretically by Tubino and Seminara (1990). Further experimental ev-idence of the suppression of migrating bars in large amplitude meanders was also givenby Whiting and Dietrich (1993). An analogous suppressing effect on free bars seemsto be associated with channel width variations: recent theoretical results suggest that inchannels with sinusoidally varying width a threshold value of the amplitude of width vari-ations can be determined above which migrating alternate bars are suppressed in favourof a steady central-bar configuration (Repetto and Tubino, 1999). The threshold value isfound to decrease for increasing values of Shields stress and to depend on the wavelengthof width variations.
The two processes display strong similarities. However, in case of meandering chan-nels the forced steady topography displays an alternate structure similar to that of free barswhich would spontaneously develop in the channel in the absence of the forcing mech-
12
anism. On the contrary, width variations force the development of central bar patternswhich would not form spontaneously unless the channel be fairly wide (see Figure 1).Hence, width variations seem to provide a suitable mechanism to explain the stabilisationof alternate bars and their transition to central bar-scour couplets whose development isan essential feature of braiding rivers. Also notice that, as pointed out in chapter 2, widthvariations are typically generated in single branches of braided networks by the initialmigration of free alternate bars.
The suppressing effect of width variations on free bars predicted by the nonlineartheory of Repetto and Tubino (1999) has been confirmed by the results of WUNI experi-ments. In the first set of experiments, performed in channel with constant width of m,the aspect ratio was always large enough to allow for the development of regular trainsof free migrating alternate bars, with dimensionless wavenumber falling within the range . In the experiments performed in channels with sinusoidally varying width the al-timetrical response was significantly different from that found in constant width channel:two different regimes were observed depending on the value attained by the amplitude ofwidth variations defined in equation (1).
For relatively low values of free migrating bars formed in the channel. However,their development was irregular and highly unsteady, like in the case of the heterogeneoussediment discussed in previous section. More specifically, central bars were observed toform at the wider sections during the initial stage of each run. These bars slowly migrateddownstream and were stopped before reaching the narrower sections. Hence, a steady beddeformation with the same length of the imposed width variations was established, withsediment transport mainly occurring at the channel sides within the narrowest sectionsand at the centerline within the widest sections. This configuration, however, turned outto be unstable and finally alternate bars developed in the channel superposed on the steadyconfiguration. However, regular trains of bars were not able to migrate along the fulllength of the channel. Rather, it was observed a recursive process of growth and decayof bars characterised by variable values of length, height and migration speed. Due tothe interaction with the steady configuration the values of both bar height and wavelengthwere invariably smaller than those measured in the constant width case: this is shown inFigures 20a,b where measured values of bar characteristics in the first (constant width)and second set (variable width) of WUNI experiments are reported. Experimental resultsare summarised in Table 8, where
is the amplitude and is the wavelength of the
leading component of alternate bar type of the Fourier representation of bed topography.Experimental data for the variable width channel are averaged values which are based onmeasurements taken on different sequences of fairly developed migrating bars.
In Figures 21a,b a comparison is presented in terms of the Fourier spectrum of bedtopography measured in the constant and variable width channel, with identical hydraulicconditions and average width. The amplitudes of different transverse modes are reportedseparately: mode 0 and 1 denote purely longitudinal bed deformations and alternate bars,respectively. In the constant width case (Figure 21a) free alternate bars with dimension-less wavenumber formed in the flume; furthermore, second order modes weredriven through nonlinear interactions by the development of the fundamental. In the vari-able width channel (Figure 21b) alternate bars with smaller amplitude and shorter wave-length ( ) were observed to migrate over the steady topography forced by theimposed width variation with .
13
For larger values of free bars were suppressed and a steady bottom configurationdeveloped in the channel (Figure 22). According to the theoretical results of Repetto andTubino (2000) the equilibrium bed configuration was in this case characterised by twoleading components: a purely longitudinal component, almost in phase with bank profile,leading to deposition at the widest sections, and a central bar component, whose relativeposition with respect to bank profile was found to depend on the wavelength of widthvariations (Figure 23).
A similar suppressing mechanism on migrating bars was detected in BUNI experi-ments and was typically observed to precede flow bifurcation. In these experiments thebed configuration was found to change continuously due to channel widening and plani-metric development. During the initial stage of the experiments bank erosion induced analmost uniform widening of the straight channel cut into the flat sediment surface. Hence,the aspect ratio of the channel increased and the critical condition for bar formationwas rapidly approached such that alternate bars formed and migrated downstream. Mea-sured characteristics of these bars are given in Figures 24a,b in terms of the dimensionlessvalue of bar wavenumber , scaled with the instantaneous average width of the channel.Notice that the initial values of reported in Figure 24a fall within the most unstablerange typical of experiments with fixed walls. However, as channel widen-ing proceeded the length of bars kept almost fixed: hence, the values of dimensionlesswavenumber increased during the experiments as documented in Figure 24b, shiftingtowards a less unstable range. Also notice that larger values of were attained in theexperiments characterised by higher flow discharges, which correspond to larger valuesof the initial Shields stress .
The development of bar pools in the initial stage of experiments provided preferen-tial sites for bank erosion, leading to a sequence of bumps along both banks as shownin Figure 7. The resulting periodic variation of channel width and the incipient channelcurvature then led to the suppression of migrating alternate bars in favour of a steady cen-tral bar pattern. The above transition process is documented by Figures 25a,b where thetime development of the amplitude of alternate () and central bars () measuredin the whole set of BUNI experiments, as obtained from the Fourier analysis of bottomtopography, is reported: experimental results are given in terms of the aspect ratio of thechannel which was invariably found to increase during the experiments. The figures showthat the amplitude of alternate bars invariably decreased during the experiments while thecentral bar pattern was developing.
Notice that according to the theoretical and experimental results for fixed wall chan-nels (see references in section 1) the amplitude of alternate bars should increase with ,until the aspect ratio is not large enough to lead to the spontaneous development of centralbars. In erodible channels which are free to develop planimetrically the damping effecton the amplitude of alternate bars and their transition to central bars seem to be enhancedby planimetric forcing. This is shown in Figures 26a,b where some examples of the timedevelopment of geometrical characteristics of alternate bars and of bank profile observedin BUNI experiments are given. It appears that the decay process of the amplitude ofalternate bars is associated with a simultaneous increase of the amplitude of bank oscil-lations, as obtained from the Fourier analysis of bank profiles. Measured values of thewavenumber of both alternate bars and bank oscillations suggest the close relationshipbetween the alternate perturbation of bank stress induced by free bars and the bumpedconfiguration observed in the early stage of development of the braiding pattern. It is
14
worth noticing that the inception of flow bifurcation in all the experiments was marked bya sudden decrease of the amplitude of bank oscillations: this invariably occurred, when acentral bar configuration started to develop, as a result of the tendency of the flow patternto concentrate on both sides of the channel.
15
This work has been developed within the framework of the National Project Cofin’97 ”Fluvial and Coastal Morphodynamics” jointly supported by the Italian Ministry ofUniversity and Scientific Research and by the Universities of Trento and Padova. Wethank Walter Bertoldi, Rodolfo Repetto, Gianluca Vignoli and Guido Zolezzi for theirkind assistance in the preparation of several tables and figures.
16
References
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Ashmore, P. E., 1982. Laboratory modelling of gravel bed stream morphology. Earth Surf. Proc.and Landforms, 7, 201-225, .
Ashmore, P. E., 1991. How do gravel-bed rivers braid. Can. J. Sci., 28, 326-341.
Bertoldi, W., Tubino, M., & Zolezzi, G., 2001. 2nd IAHR Symposium on River, Coastal andEstuarine Morphodynamics, Obihiro, Japan, September.
Bittner, L., 1994. River bed response to channel width variations. M.S. Thesis, Univ. of Illinois.
Blondeaux, P., & Seminara, G., 1985. A unified bar-bend theory of river meanders. J. FluidMech., 157, 449-470.
Chang, H., Simons, D. B., & Woolhiser, D. A., 1971. Flume experiments on alternate barsformation. J. Waterways, Harbours, Coastal Eng. Div. Am. Soc. Civ. Eng., 97, 155-165.
Colombini, M., Seminara, G., & Tubino, M., 1987. Finite-amplitude alternate bars. J. FluidMech., 181, 213-232.
Colombini, M., & Tubino, M., 1991. Finite-amplitude free bars: a fully nonlinear spectral solu-tion. In Sand Transport in Rivers, Estuaries and the Sea, edited by R. Soulsby and R. Bettes,Proc. Euromech. 262 Colloquium, Wallingford, UK, 26-29 June, 163-169, Balkema.
Darby, S. E., & Thorne, C. R., 1996. Numerical simulation of widening and bed deformation ofstraight sand-bed rivers. I: Model development. J. Hydraul. Eng. ASCE, 122(4), 184-193.
Einstein, H. A., 1950. The bedload function for sediment transport in open channel flows. Tech.Bull., 1026, Soil Conserv. Serv., U.S. Dep. of Agric., Washington, D.C.
Fujita, Y., 1989. Bar and channel formation in braided streams. In River Meandering, edited byS. Ikeda and G. Parker, AGU Water Resources Monograph 12, 417-462.
Fujita, Y., & Muramoto, Y., 1985. Studies on the process of development of alternate bars.Disaster Prevention Research Institute Bull., Kyoto Univ., 35, Part 3, No. 314, 55-86.
Garcia, M. H., & Nino, Y., 1993. Dynamics of sediment bars in straight and meandering chan-nels: experiments on the resonance phenomenon. J. Hydraul. Res., 31(6), 739-761.
Ikeda, S., 1982. Prediction of alternate bar wavelength and height. Rep. Dept. Found. Eng. andConst. Eng., Saitama Univ., 12, 23-45.
Jaeggi, M., 1984. Formation and effects of alternate bars. J. Hydraul. Div. Am. Soc. Civ. Eng.,110, 1103-1122.
Kinoshita, R., 1961. Investigation of channel deformation in Ishikari River. Rep. Bureau ofResources, Dept. Science and Technology, Japan, 1-174.
Kinoshita, R., & Miwa, H., 1974. River channel formation which prevents downstream transla-tion of transverse bars. Shinshabo, 94, 12-17 (in japanese).
17
Lanzoni, S., Tubino, M., & Bruno, S. 1994. Formazione di barre alternate in alvei incoerenti agranulometria non uniforme. XXIII Convegno di Idraulica e Costruzioni Idrauliche, Napoli,20-25 settembre (in italian).
Lanzoni, S., & Tubino, M., 1999. Grain sorting and bar instability. J. Fluid Mech., 393, 149-174.
Lanzoni, S., 2000a. Experiments on bar formation in a straight flume 1. Uniform sediment.Water Resour. Res., 36(11), 3337-3349.
Lanzoni, S., 2000b. Experiments on bar formation in a straight flume 2. Graded sediment. WaterResour. Res., 36(11), 3351-3363.
Lisle, T. E., Ikeda, H., & Iseya, F., 1991. Formation of stationary alternate bars in a steep channelwith mixed size sediment: a flume experiment. Earth Surf. Proc. Landforms., 16, 463-469.
Muramoto, Y., & Fujita, Y., 1978. The classification of meso-scale river bed configuration andthe criterion of its formation. Proc. 22nd Japanese Conf. on Hydraulics, ISCE, 275-282.
Parker, G. 1990. Surface-based bedload transport relation for gravel rivers. J. Hydraul. Res, 28,417-436.
Repetto, R., 2000. Unit processes in braided rivers. PHD Thesis, University of Genova.
Repetto, R., & Tubino, M., 1999. Transition from migrating alternate bars to steady centralbars in channels with variable width. IAHR Symposium on River, Coastal and EstuarineMorphodynamics, Genova, Italy, September 1999, 605-614.
Repetto, R., & Tubino, M., 2000. Topographyc expressions of bars in channel with variablewidth. Journal of Physics and Chemistry of the Earth, part B, 26(1), 71-76.
Repetto, R., Tubino, M., & Volcan, C., 2000. La risposta altimetrica di correnti a fondo mobilein canali a larghezza variabile: osservazioni sperimentali. XXVII Convegno di Idraulica eCostruzioni Idrauliche, Genova, Italy, September 2000, 333-342 (in Italian).
Ribberink, J. S., 1987. Mathematical modelling of one-dimensional morphological changes inrivers with non-uniform sediment. Com. on Hydraul. and Geotec. Eng., TU DELFT, 200pp..
Richardson, E. V., & Simons, D. B., 1967. Resistance to flow in sand channels. 12th IAHRCongress., Fort Collins, Colorado.
Schielen, R., Doelman, A., & Swart H.E. de, 1993. On the nonlinear dynamics of free bars instraight channels. J. Fluid Mech., 252, 325-356.
Seminara, G., Colombini, M., & Parker, G., 1996. Nearly pure sorting waves and formation ofbedload sheets. J. Fluid Mech., 312, 253-278.
Seminara, G., Tubino, M., & Paola, C., 2000. The morphodynamics of braiding rivers: exper-imental and theoretical results on unit processes. Gravel-bed Rivers 2000, Christchurch,New Zealand, August 2000.
Seminara, G., Zolezzi, G., Tubino, M., & Zardi, D., 2001. Downstream and upstream influencein river meandering. Part two: planimetric development. J. Fluid Mech., in press.
18
Shimizu, Y., & Itakura, T., 1989. Calculation of bed variation in alluvial channel, J. Hydraul.Eng., 115(3), 367-384.
Struiksma, N., & Crosato, A., 1989. Analysis of a 2-D bed topography model for rivers. InRiver Meandering, edited by S. Ikeda and G. Parker, AGU Water Resources Monograph12, 153-180.
Sukegawa, N., 1971. Study on meandering of streams in straight channels. Rep. Bureau ofResources, Dept. Science and Technology, Japan, 335-363.
Tubino, M., Repetto, R., & Zolezzi, G., 1999. Free bars in rivers. J. Hydraul. Res., 37(6),759-775.
Tubino, M., & Seminara, G., 1990. Free-forced interactions in developing meanders and sup-pression of free bars. J. Fluid Mech., 214, 131-159.
Vignoli, G., & Tubino, M., 2000. La risposta altimetrica di correnti a fondo mobile in presenzadi dominante trasporto in sospensione: teoria non lineare. XXVII Convegno di Idraulica eCostruzioni Idrauliche, Genova, Italy, September 2000, 377-386 (in Italian).
Whiting, P.J., & Dietrich, W.E., 1993. Experimental studies of bed topography and flow patternsin large-amplitude meanders. 1. Observations. Water Resour. Res., 29(11), 3605-3622.
Wilcock, P. R., 1993. Critical shear stress of natural sediments. J. Hydraul. Eng., 119, 491-505.
Wilcock, P. R., & McArdell, B. W., 1993. Surface-based fractional transport rates: Mobilizationthresholds and partial transport of a sand-gravel sediment. Water Resour. Res., 29, 1297-1312.
19
0 0.05 0.1 0.15 0.20
50
100
Θ
β
no bars
alternate bars: transverse mode 1
central bars: transverse mode 2
multiple row bars: transverse mode 3
Figure 1: Regions of occurrence of different free transverse modes in the plane (, ): = 0.01.
0 5000 10000 15000 200000.0
0.1
0.2
0.4
0.5 A 11 A 22 A 02
Am
plitu
de
Time
0.3
Figure 2: Time development of the amplitude of Fourier components of bottom topogra-phy in case of dominant bedload as predicted by the numerical model of Vignoli & Tubino(2000): = 0.1; = 0.1, = 18. is the fundamental alternate bar component,
and are second order harmonics.
20
1
2
3
4
5
6
7
8
S1
S2
S3
S4
S5
S6
S7
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
x modes
y modes
Figure 3: The Fourier spectrum of the equilibrium bar topography in case of dominantbedload as predicted by the numerical model of Vignoli & Tubino (2000): = 0.1, =0.1, = 18. and denote the longitudinal and transverse direction, respectively.
0
1
2
3
4
5
0 1 2 3 4 5
BM
(H
)
th
BM(H ) exp
123456789
Figure 4: The maximum bar height as predicted by Colombini et al. (1987)is compared with experimental data of various authors. Data falling betweensolid lines are such that : 1 Jaeggi (1984) PVC;2 Jaeggi (1984) sand; 3 Sukegawa (1971); 4 Kinoshita (1961); 5 Muramoto & Fuijta(1978); 6 Ashida & Shiomi (1966); 7 Ikeda (1982); 8 Fujita & Muramoto (1985); 9Garcia & Nino (1993).
21
10−2
10−1
100
101
0
50
100
d*i (mm)
% f
iner FUNI
MUNI CUNI
FC70
Figure 5: Particle size distributions of the sediment used in MUNI and FC70 experimentsof Lanzoni (2000a,b): the mixture FC70 was obtained by mixing 67% of FUNI and 33%of CUNI.
Figure 6: WUNI experiments: sketch of channel geometry and notations.
22
Figure 7: Planimetric development of a channel cut into a cohesionless flat sloping surface(run B12 in Table 5).
0
5
10
15
20
25
30
0 0.2 0.4 0.6 0.8 1 1.2λ
β
bedload and weak suspended loadbedload only
bars
plane bed
bars
Figure 8: The effect of suspended load on the marginal curve for bar formation on theplane (, ).
23
5
10
15
20
25
30
0 0.1 0.2 0.3 0.4λ | Ω max
β
θ0=.75θ0=1.0θ0=1.5θ0=2.0
Figure 9: The wavenumber of the fastest growing bar is plotted versus the width ratio fordifferent values of the Shields parameter (dune-covered bed, , ).
0
1
2
3
4
5
10 20 30 40 50 60β
mode 1mode 2mode 3
Ωmax
10 x -4
0
Figure 10: The growth rate of the fastest growing bar perturbation is plottedversus the aspect ratio for different transverse modes (dune-covered bed, , ).
24
0 0.1 0.2 0.3 0.40
1
2
3
4
Θg
Hb
(a)
0 0.1 0.2 0.3 0.40
10
20
30
40
Θg
L b
(b)
Figure 11: Measured values a) of bar height (scaled with the average depth
) and b) of
bar wavelength (scaled with the half width
) are plotted versus Shields stress based
on the geometric grain size for experiments in straight flumes (MUNI and WUNI runs).The triangles refer to MUNI runs P1801,P0404, P2403; the circles refer to MUNI runsP1605, P1505, P2709, P2809, P2909; the filled circles refer to WUNI runs.
0 5 10 15
0 5 10 15
0 5 10 15
20
15
10
5
0
20
15
10
5
0
20
15
10
5
0wavenumber
pow
er d
ensi
typo
wer
den
sity
pow
er d
ensi
ty
(a)
(b)
(c)
wavenumber
wavenumber
Figure 12: Example of the power spectrum of a) right, b) centre and c) left bed profilesmeasured during the equilibrium phase of run P2403 (MUNI experiment).
25
0.5 1 1.50
0.5
1
1.5
α
Ωb/Ω
u
σ0 = 0
0.51.0
1.5
2.0
0.1
Figure 13: The ratio between the growth rate of alternate bars for the bimodalcase and for the uniform case, as predicted by Lanzoni & Tubino (1999), is plotted ver-sus bar wavenumber for different values of the standard deviation of the mixture andgiven values of the dimensionless geometric mean diameter ( ) and of Shieldsparameter based on ( ): .
(a)
(b)
Figure 14: MUNI experiments: examples of longitudinal bed profiles and difference be-tween right side and left side bed elevation ( ). a) Initial phase of P1309; b) Equilib-rium phase of P1309.
26
Figure 15: (a) Longitudinal bed profiles measured during the initial phase of P2009(MUNI experiments). (b) The vertical distribution of the quantity
is plot-ted for each of the 10 different locations sampled along a bar unit, 20 cm from the leftwall of the flume, during the initial phase of P2009. denotes the local geometricalmean grain size of the sampled sediment while is the geometrical mean grain size ofthe initially mixed sediment.
10−2
10−1
100
101
0
50
100
d*i (mm)
perc
enta
ge f
iner
Figure 16: MUNI experiments: cumulative grain size distributions of transported sedi-ment are plotted as a function of the grain size . The symbols are as follows: Æ, P0109;, P0609; , P0709; , P1309. The continuous line refers to the cumulative grain sizedistribution of the initially mixed bed.
27
0 0.2 0.40
0.005
0.01
0.015
0.02
Θg
C0
Figure 17: Comparison between the values of friction coefficient observed in FC70’s() and in MUNI’s tests (Æ).
0 0.1 0.2 0.3 0.40
1
2
3
Θg
Hb
(a)
0 0.1 0.2 0.3 0.40
10
20
30
40
Θg
L b
(b)
Figure 18: Measured values a) of bar height (scaled with the average depth
) and b) of
bar wavelength (scaled with the half channel width
) observed in MUNI (open circles
and triangles) and FC70 (filled circles and triangles) experimental runs are plotted againstthe Shields parameter based on the geometric grain diameter. The triangles refer to runsP1801,P0404, P2403 (MUNI) and to runs P0606, P2006, P2906 (FC70); the circles referto runs P1605, P1505, P2709, P2809, P2909 (MUNI) and to runs P0807, P0109, P1309,P2009 (FC70). Experimental data from Lanzoni et al. (1994) are also included. Openboxes refer to uniform sediment made up of glass spheres with diameter 1.5 mm; filledboxes refer to a weakly bimodal mixture composed by glass spheres with diameters 1 mmand 2 mm in proportion 1:1 (from Lanzoni, 2000b).
28
0 0.1 0.2 0.3 0.410
−5
100
Θg
Gro
wth
rat
e
(a)
0 0.1 0.2 0.3 0.40
10
20
30
40
Θg
L b
(b)
Figure 19: Theoretical values a) of the maximum growth rate and b)of the wavelength ofalternate bars (scaled with the half channel width
), corresponding to the experimental
data reported in Figure 18, as predicted by Lanzoni & Tubino (1999). Open and filledsymbols refer to uniform and bimodal sediments, respectively (from Lanzoni, 2000b).
0 2 4 6 8 10 120
5
10
constant width
varia
ble
wid
th
A1
(a)
0 0.2 0.4 0.6 0.80
0.2
0.4
0.6
0.8
constant width
varia
ble
wid
th
λ
(b)
Figure 20: WUNI experiments: comparison between measured values (a) of the ampli-tude (in mm) and (b) dimensionless wavenumber of the leading Fourier componentof bed topography associated with alternate bars in constant width and variable widthexperiments (from Repetto et al., 2000).
29
0
2
4
6
8
10
0 0.35 0.7 1.05
ampl
itude
longitudinal wavenumber
transverse mode 0
0
2
4
6
8
10
0 0.5 1 1.5longitudinal wavenumber
ampl
itude
transverse mode 0
longitudinal wavenumber
ampl
itude
transverse mode 1
0
2
4
6
8
10
0 0.6 1.2 1.8
(a) (b)
0
2
4
6
8
10
0 0.35 0.7 1.05longitudinal wavenumber
ampl
itude
transverse mode 1
Figure 21: Fourier spectra of the leading components of bottom elevation in run a11 ofWUNI experiments: (a) constant width channel; (b) variable width channel.
30
0
(a)
b
b
b
b
π/λ
2π/λ
3π/λ
4π/λ
x-1
0
1
y
-2
0
2
ζ0
π/λ
2π/λ
3π/λ
4π/λ
x
-2
0
2
0
(b)
b
b
b
b
π/λ
2π/λ
3π/λ
4π/λ
x-1
0
1
y
-202
ζ0
π/λ
2π/λ
3π/λ
4π/λ
x
-202
0
(c)
b
b
b
b
π/λ
2π/λ
3π/λ
4π/λ
x-1
0
1
y
-2
0
2
ζ0
π/λ
2π/λ
3π/λ
4π/λ
x
-2
0
2
Figure 22: Equilibrium bed configuration for different values of the wavenumber of widthvariations: a) , b) , c) ; , , (fromRepetto & Tubino, 2000).
31
Figure 23: Steady bottom configuration in channels with variable width (from Repettoand Tubino, 2000).
0 0.05 0.1 0.15 0.20
0.2
0.4
0.6
0.8
Θ0
λ
(a)
0 0.05 0.1 0.15 0.20
0.2
0.4
0.6
0.8
Θ0
λ
(b)
Figure 24: The initial (a) and final (b) values of the dimensionless wavenumber ofalternate bars observed in BUNI experiments are plotted versus the initial value of Shieldsstress .
32
0 20 40 600.2
0.3
0.4
0.5
β
HB
A(a)
0 20 40 600
0.05
0.1
0.15
0.2
β
HC
A
(b)
Figure 25: BUNI experiments: measured values of the amplitude of (a) alternate bars and (b) central bars (both scaled with the initial flow depth
) are plotted
versus the aspect ratio of the channel .
33
Table 1: Properties of the various sediment mixtures. Measures in mm.
Figure 26: BUNI experiments: time development of alternate bars and channel charac-teristics. : dimensionless bar height, scaled with the initial flow depth
(filled
circles); : dimensionless amplitude of bank oscillations, scaled with the average halfwidth
(open circles); ): dimensionless bar wavenumber, scaled with the average half
width
(filled circles); : dimensionless wavenumber of bank oscillations, scaled with
the average half width
(open circles); ): aspect ratio of the channel. Plots a), b), c)
refer to run B9; plots d), e), f) refer to run B12.
34
Table 2: Summary of hydraulic experimental conditions in MUNI experiments.
Table 7: Alternate bar characteristics observed during the initial stage and the final equi-librium phase (denoted by a $ apex) of FC70 experimental runs. Notes are as follows. 1:Nearly plane bed with very long bars; paved sinuous streaks (about 20 cm wide) formedin the upstream 20-30 m of the flume; in the downstream reach very irregular migratingbars superposed episodically on the forced bars. 2: Bars were observed to form and growirregularly only in the downstream reach on the flume. 3: A very regular train of barsformed in the initial phase of the run; the final, equilibrium phase, on the contrary, wascharacterized by irregular bars growing only in the downstream reach of the flume. 4:Only the final equilibrium phase was monitored.
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