114664-201112-2

中

Numerical Simulation of Longitudinal and Lateral ChannelDeformations in the Braided Reach of the

Lower Yellow RiverGuangqian Wang1; Junqiang Xia2; and Baosheng Wu3

Abstract: Bank erosion frequently occurs in the Lower Yellow River �LYR�, playing an important role in the evolution of this braidedriver. A two-dimensional �2D� composite model is developed herein that consists of a depth-averaged 2D flow and sediment transportsubmodel and a bank-erosion submodel. The model incorporates a new technique for updating bank geometry during either degradationalor aggradational bed evolution, allowing the two submodels to be closely combined. Using the model, the fluvial processes in the braidedreach of the LYR between Huayuankou and Laitongzhai are simulated, and the calculated results generally agree with the field measure-ments, including the water-surface elevation, variation of water-surface width, and variations of cross-sectional profiles. The calculatedaverage water-surface elevation in the study reach was 0.09 m greater than the observed initial value, and the calculated mean bedelevation for six cross sections was 0.11 m lower than the observed value after 24 days. These errors are attributed to the large variabilityof flow and sediment transport processes. Sensitivity tests of three groups of parameters are conducted, and these groups of parameters arerelated to flow and sediment transport, bank erosion, and model application, respectively. Analysis results of parameter sensitivity testsindicate that bank erodibility coefficient and critical shear stress for bank material are sensitive to the simulated bank erosion process. Thelateral erosion distance at Huayuankou will increase by 19% as the value of bank erodibility coefficient changes from 0.1 to 0.3, and itwill decrease by 57% as the value of critical shear stress for bank increases from 0.6 to 1.2 N /m2. Limited changes of other parametershave relatively small effects on the simulated results for this reach, and the maximum change extent of calculated results is less than 5%.Because the process of sediment transport and bank erosion in the braided reach of the LYR is very complicated, further study is neededto verify the model.

DOI: 10.1061/�ASCE�0733-9429�2008�134:8�1064�

CE Database subject headings: Rivers; China; Bank erosion; Channels; Deformation; Simulation.

http://www.paper.edu.cn国科技论文在线

Introduction

Alluvial rivers adjust themselves in response to the environmentalchanges that may occur naturally or be the result of human ac-tivities such as river training, damming, and diversion �Chang1988; Zhang and Xie 1993�. Channel adjustment can occur in twospatial directions. One is longitudinal adjustment, characterizedby morphological changes such as bed scour or sediment deposi-tion on the bed; and the other is lateral adjustment, distinguishedby river width adjustments, involving bank retreat or advance�Xie 2004�.

1Professor, State Key Laboratory of Hydroscience and Engineering,Tsinghua Univ., Beijing 100084, P.R. China. E-mail: [email protected]

2Associate Professor, State Key Laboratory of Hydroscience andEngineering, Tsinghua Univ., Beijing 100084, P.R. China. E-mail:[email protected]

3Professor, State Key Laboratory of Hydroscience and Engineering,Tsinghua Univ., Beijing 100084, P.R. China. E-mail: [email protected]

Note. Discussion open until January 1, 2009. Separate discussionsmust be submitted for individual papers. To extend the closing date byone month, a written request must be filed with the ASCE ManagingEditor. The manuscript for this paper was submitted for review and pos-sible publication on February 25, 2003; approved on November 5, 2007.This paper is part of the Journal of Hydraulic Engineering, Vol. 134,No. 8, August 1, 2008. ©ASCE, ISSN 0733-9429/2008/8-1064–1078/
$25.00.
1064 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / AUGUST 2008

According to observations and analyses of field data, seriousbank erosion is frequently found along the Lower Yellow River�LYR�, especially in its braided reach shown in Fig. 1. In thebraided reach, scouring usually occurs in the main channel,whereas deposition occurs on the floodplain during the flood sea-son �July–October each year�. This leads to the formation of nar-row and deep cross sections after a flood event. During thenonflood season �November–June�, deposition occurs in the mainchannel and the floodplain banks retreat due to bank erosion,explaining why a narrow and deep cross-sectional profile does notpersist for a long time �Zhao et al. 1998�. Therefore, lateral ero-sion and failure of the floodplain banks play an important role inthe fluvial processes of the LYR.

The Sanmenxia Reservoir, built in 1960, is located at the lowerpart of the Middle Yellow River. During the period of water im-poundment and sediment detention in the reservoir, the river beddownstream was scoured because the flow released from the res-ervoir carried less sediment than its transport capacity. Along withbed degradation, the downstream river channel also graduallywidened by almost 1,000 m four years after the operation of thereservoir �Zhao et al. 1998�. The annual average amount of sedi-ment scoured from the downstream channel was approximately5.58�108 t during the four-year period, of which nearly 35% wasyielded from bank erosion. Analysis of field data in dry seasonsindicates that the daily average rate of bank retreat reached22 m /day, with a maximum value of 270 m /day at Huayuankou
�Wang et al. 2004�.
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Such substantial bank erosion rates in the braided reach of theLYR not only affect the use of riparian resources, but also endan-ger the safety of embankments. Another consequence of bankfailure is that a large amount of bank soil enters the channel,increasing the quantity of sediment deposited in downstreamreaches. It is, therefore, very important to simultaneously simulatethe longitudinal and lateral channel deformations in a braidedriver. However, existing numerical models for alluvial rivers, es-pecially those used in the LYR, seldom simulate both the bed andbank deformation processes. In this study, a two-dimensional�2D� composite model was developed, which can simultaneouslysimulate the longitudinal and lateral channel deformations. Thecomposite model was applied to simulate the braided reach of theLYR.

Development of a 2D Composite Model

The goal of this research is to combine a depth-averaged 2Dsubmodel for the flow and sediment transport process with a sub-model for the process of bank retreat, thereby generating a 2Dcomposite model to simulate changes in river morphology.

Two-Dimensional Flow and Sediment TransportSubmodel

Governing EquationsIn order to fit the irregular geometric configuration of the studyregion, the governing equations of the depth-averaged 2D modelare described in an orthogonal curvilinear coordinate system. Itshould be pointed out that the grid generated in the developedmodel becomes approximately orthogonal by numerically solvinga set of Poisson’s equations. A curvilinear grid that is as nearlyorthogonal as possible is advantageous for the numerical solutionof the governing equations. According to the analysis of observeddata by Long and Zhang �2002�, the average ratio of bed load tototal load of sediment in the LYR is only about 0.5%. Therefore,the bed load is ignored and only the transport of suspended load isconsidered in the model.

The governing flow equations are, therefore, written as

�Z+

1 ��UhC�� +

1 ��VhC�� = 0 �1�

Fig. 1. Sketch of

�t C�C� �� C�C� ��

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�U

�t+

U

C�

�U

��+

V

C�

�U

��+

UV

C�C�

�C�

��−

V2

C�C�

�C�

��+

g

C�

�Z

��

+ gn2�U2 + V2

h4/3 U =�t

C�

�A

��−

�t

C�

�B

��2�

�V

�t+

U

C�

�V

��+

V

C�

�V

��+

UV

C�C�

�C�

��−

U2

C�C�

�C�

��+

g

C�

�Z

��

+ gn2�U2 + V2

h4/3 V =�t

C�

�B

��+

�t

C�

�A

��3�

where

A =1

C�C��

��C�U� +

�

��C�V��

B =1

C�C��

��C�V� −

�

��C�U��;

� and ��orthogonal curvilinear coordinates in the streamwiseand transverse directions, respectively; Z�water-surface eleva-tion that is equal to �h+Zb�; h�water depth; Zb�riverbed eleva-tion; U and V�velocity components in the � and � directions,respectively; C� and C��Lami coefficients, which can be ex-pressed as C�=�x�

2+y�2=��x /��2+ ��y /��2 and C�=�x�

2 +y�2

=��x /��2+ ��y /��2, respectively; x and y are Cartesian coor-dinates; �t is turbulent viscosity coefficient calculated by �t

= 1 / 6�u*h, in which ��Von Karman constant; u

*�friction ve-

locity calculated by u*

=nu�g /h1/6, in which u=�U2+V2,g�gravitational acceleration, n�Manning’s roughness coeffi-cient; and t is time.

The nonequilibrium transport equation for each fraction of thenonuniform suspended sediments is then written as

�

�t�hSk� +

1

C�C��

��C�UhSk� +

�

��C�VhSk��

=1

C�C��

��

C�

C�

�

��hSk�� +

�

��

C�

C�

�

��hSk��

+ �*kksksk�S*k − Skfsk� + Sok �4�

where �� and ��turbulent diffusivity coefficients of sediment in

wer Yellow River
the Lo
the � and � directions, respectively; and the relations of ��=�t and

AL OF HYDRAULIC ENGINEERING © ASCE / AUGUST 2008 / 1065


��=�t are used approximately under the conditions of fine-grainsediments and low concentrations �Chien and Wan 1999�; Sok, Sk,S

*k, and sk represent, respectively, lateral input term of sedimenttransport, sediment concentration, sediment transport capacityand effective settling velocity for the kth grain size, respectively;and �

*k, fsk, and ksk�empirical coefficients calibrated using thedata observed in the LYR, their detailed expressions being avail-able in related publications �Zhang et al. 2001; Ni et al. 2004�.

Finally, the longitudinal channel deformation equation due tothe nonequilibrium transport of suspended load is written as

��Zb

�t= �

k=1

N

�*kksksk�fskSk − S

*k� �5�

where ��dry density of the bed material, which is usually equalto 1,400 kg /m3 in the LYR and N�total number of fractions usedto represent the gradation of nonuniform sediments.

Treatment of Relevant ProblemsIn any river with bends, secondary flow has a substantial effect onthe transverse distributions of flow velocity and sediment concen-tration. This effect can be accounted for in the depth-averaged 2Dmodel through adding the corresponding action terms to Eqs.�2�–�4� resulting from the secondary flow �Duan and Julien 2005�.The channel in the study reach generally has a more or lessstraight alignment in a plane view, and its sinuosity coefficient isslightly greater than 1.15. Due to water impoundment and sedi-ment detention of the Sanmenxia Reservoir, large-scale sand barsin the channel were scoured greatly, which lowered the influenceof secondary flow on the calculated results. Therefore, the influ-ence of secondary flow in the braided reach was not as strong asthat in bends of meandering rivers, and this effect was not con-sidered in the developed model.

In the flow and sediment transport submodel, certain key pa-rameters and problems must be carefully treated, such as repre-sentation of initial and boundary conditions, moving boundarytechnique, movable-bed roughness coefficient, parametrization ofsediment transport capacity, and adjustment of the size distribu-tion of bed material during bed deformation. It is important tonote here that treatments of these issues are proposed for thespecial case of the simulation of the LYR. The complex morphol-ogy and channel deformations in the braided reach require manyparameters to be introduced for the developed model. Fortunately,the majority of these parameters, such as the sediment transportcapacity by Zhang and Zhang �1992�, the movable-bed roughnessformula by Zhao and Zhang �1997�, and the empirical coefficientsof �

*k, fsk, and ksk by Zhang �1999�, were calibrated and used inother numerical models for the LYR �Zhang et al. 2001; Ni et al.2004�.1. Initial and boundary conditions. At the upstream boundary,

the inflowing water discharge, the concentration of sus-pended load, and the corresponding grain size distributionare required for each time interval of the simulation. At thedownstream boundary, the water-surface elevation or thestage-discharge rating curve must be provided. For sediment,�Sk /��=0 is assumed. At the channel sidewalls, no-slipboundary conditions are used for the governing flow equa-tions and the zero-sediment flux normal to the wall is speci-fied for the transport equation of suspended load.Furthermore, the initial values of flow and sediment aregiven at each computational node.

2. Treatments of dry nodes. A moving boundary technique is


needed to deal with the changing computational domain.When the water level varies, some nodes can switch fromdry to wet, or vice versa. A drying or wetting process makesthe computational domain complex, especially for velocity.The developed model applies a technique, named “conden-sation” by Chen and Wang �1988�. This approach determinesthe whole computational domain by the highest possiblewater stage. As the riverbed at one node is exposed to thewater surface, the value of friction term at this node is arti-ficially set to be a very large positive number, which makesthe velocity components around this node approximatezero. Then, a false water depth of 0.005 m is assumed forthis dry node in order to avoid a mathematical overflow incomputation.

3. Movable-bed roughness coefficient. In this study, the empiri-cal formula �Zhao and Zhang 1997� is used to estimate themovable-bed roughness coefficient. This relation can be ex-pressed as

n =h1/6

�gcn

�*

h�0.49�

*

h0.77

+3�

81 −

�*

h�sin�

*

h0.2�5�−1

�6�

where cn�vortex coefficient calculated by cn=0.375�;�

*�friction thickness, defined as

�*

= D50 exp�1 + 108.1−13F0.5�1−F3��

and F�Froude number of flow. This empirical formula isvalid for F less than 0.80. The validity of this formula wasverified by Zhao and Zhang �1997�, and it has been usedin other numerical models for the LYR �Zhang et al. 2001; Niet al. 2004�.

4. Sediment transport capacity. The model uses an empiricalformula for sediment transport capacity based on the gravi-tational energy concept proposed by Zhang and Zhang�1992�. This formula has been accepted by researchers and ithas been widely used to compute the sediment transport ca-pacity in the LYR �Hui et al. 2000�, being expressed as

S*

= 2.5� �0.0022 + SV�u3

�ghm� s − m�/ mln h

6D50�0.62

�7�

where s and m denote specific weights of sediment andturbid water, respectively; SV�volumetric sediment con-centration calculated by SV=S /s, in which S �=�k=1

N Sk� isdepth-averaged concentration of suspended sediment at anode, and s is density of sediment; Von Karman coefficient� in muddy water is expressed as a function of SV by�=0.4−1.68�SV�0.365−SV�, and the value of � is close to0.4 as the depth-averaged sediment concentrations are rela-tively low; m�group settling velocity of nonuniform sedi-ments, and is equal to �k=1

N �P*ksk; �P

*k�percentage of thesediment transport capacity for the kth grain size; andD50�median diameter of the bed material. The accuracy ofthis formula has been verified through a series of field andlaboratory data �Shu 1993; Hui et al. 2000�. This formulawas specially proposed for the Yellow River, and can beapplied to both low- and hyper-concentrated sediment-ladenflows, by accounting for the influence of the sediment con-centration on the transport capacity. The reason behind theincrease of the sediment transport capacity in the case ofoncoming high sediment concentrations can be partly attrib-
uted to the changes in the kinematic viscosity, fall velocity,


and Von Karman coefficient. However, the sediment concen-tration in this case study was relatively low, and the influenceof SV on the sediment transport capacity could be ignored.

Sediment transport capacity by size fraction, �P*k, also

needs to be determined. In this study, the approach by Li�1987� is applied to calculate �P

*k. This formula directlyuses flow parameters and gradation of bed material to calcu-late the fractional sediment transport capacity, and its de-tailed description can be found in Zhang and Xie �1993�.

5. Gradation adjustment of bed material during bed deforma-tion. In order to simulate the phenomenon of armoring orsorting of bed material during degradation or aggradation,the bed material at each computational node is divided intotwo layers in the vertical direction: the upper one is calledthe mixing or active layer and the lower one is called thememory layer. The thickness of the mixing layer is denotedby Hb, and its gradation is represented by �Pbk. The memorylayer is further divided into m smaller sublayers, the thick-ness and gradation of each sublayer being represented by�Hm and �Pmk, respectively. �Hsk is assumed to be thethickness of scour or deposition for the kth grain size at anode, and the total thickness of bed deformation at this node�Hs is equal to �k=1

N �Hsk. The adjustment procedure of thesize distribution of surface bed material can be classified intothe following two cases.

If �Hs exceeds zero, bed deposition will occur. In this case,the formula for the gradation adjustment of the mixing layer canbe written as

�Pbkt+�t =

�Hsk + �Pbkt �Hb

t − �Hs�Hb

t+�t �8a�

where �Pbkt and �Pbk

t+�t�gradations of the mixing layer atthe time levels t and t+�t, respectively, and Hb

t andHb

t+�t�thicknesses of the mixing layer with respect to these twotime levels, respectively.

If �Hs is less than zero, bed scour will occur. In this case, theformula for the gradation adjustment of the mixed layer can bewritten as

�Pbkt+�t =

�Hsk + �Pbkt Hb

t + �Hs �Premk

Hbt+�t �8b�

where �Premk�average gradation of the top several memory su-blayers, whose total thickness is �Hs . In this case, the totalnumber of memory sublayers decreases, and their gradations areadjusted correspondingly.

The thickness of mixing layer depends not only on the inflow-ing discharge condition and the size distribution of bed material,but also on the magnitude of channel deformation. Karim andHolly �1986� and Borah et al. �1982� showed that it is difficult tosimulate the variation of mixing layer thickness. Observed dataon the size distribution of bed material are usually insufficient inthe vertical direction, so that an excessively complicated treat-ment of the mixing layer does not necessarily improve the com-putational accuracy �Xie 2004�. Therefore a constant mixing layerthickness of 2.0 m is used for the sand-bed braided river in thismodel, following the experience of previous researchers �Han1979; Wang 1994�.

Numerical Solution ProcedureThe procedure for numerically solving the governing flow and
sediment equations includes the following three steps:
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First, the governing flow Eqs. �1�–�3� are split into two sets ofequations in the � and � directions using the method of fractionalsteps in space �Yanenko 1971�. Then the “time marching”alternating-direction-implicit scheme is applied to solve the twosets of discretized equations in a staggered grid, with the detaileddiscretization method provided in Leendertse �1970�. Second, themethod of fractional steps in space is also applied to split the 2Dsuspended load transport equation into two one-dimensional �1D�advection-diffusion equations in the � and � directions. Then atime increment is divided into two half-steps. In the first half-timestep, with the known value of Sl at the time level l, the splitequation in the � direction is solved first and the intermediatevalue Sl+1/2 at the time level l+1 /2 can be obtained using anexponential explicit finite-difference scheme. In the second half-time step, the split equation in the � direction is solved, and thevalue of Sl+1 at the time level l+1 can be obtained using a Crank–Nicholson implicit finite-difference scheme. Results of previousnumerical experiments showed that when the advection process isweak in transport, it is proper to use an implicit difference schemeboth in the � and � directions. In the sediment transport in theLYR, the advection process is very important, and, thus, it is morereasonable to use an exponential explicit difference scheme in the� direction and the Crank–Nicholson implicit scheme in the �direction, respectively. Therefore, different schemes in differentdirections are used in the solution of sediment transport equationin order to avoid producing a nonphysical oscillation phenom-enon in the calculation resulting from the strong advection pro-cess. A detailed description of this hybrid scheme can be found inXia and Wang �2001�. Third, the bed elevation at each node at theend of time level l is obtained by solving Eq. �5� with the explicitscheme.

Bank-Erosion Submodel

Bank advance occurs as a result of deposition in the near-bankzone where the local sediment supply exceeds its transport capac-ity. For the braided reach in the LYR, bank retreat occurs prima-rily due to lateral erosion and failure of banks �Chien and Zhou1965; Chien et al. 1989�. This paper only considers the bankretreat process by lateral erosion and bank failure. Thus, the bank-erosion model proposed by Osman and Thorne �1988� can beimproved herein.

In Osman and Thorne �1988�, the eroded and failed bank ma-terial was calculated by assuming planar slip failure. In reality,some of the eroded bank material is transported within the chan-nel as either suspended load or bed load, whereas a portion of thecoarser material remains deposited in the bank-toe zone. Theformer is taken as the lateral input term in Eq. �4�, and the latteris used to determine the deposition thickness in the near-bankzone. The bank-erosion model by Osman and Thorne �1988� was,therefore, modified as follows. First, the influence of the near-bank bed deformation on bank stability in the current time inter-val is not taken into consideration. Rather, this effect is accountedfor in the next time interval by modifying the bed elevation nearthe bank at the end of the time step. The advantage of this treat-ment is that the lateral input term in Eq. �4� can be obtainedbefore the equation is solved. Furthermore, this effect can be ac-counted for indirectly in the next time step. Secondly, in theOsman and Thorne �1988� model, only bank stability induced bybed scour is analyzed. But in reality, bank failure may occurduring either bed scour or sediment deposition on the bed. There-fore, the bank stability during both bed scour and sediment depo-
sition is accounted for in the enhanced submodel.


Computation of Lateral Erosion DistanceA formula similar to the one proposed by Osman and Thorne�1988� is applied to compute the lateral erosion distance �B �m�in a time increment �t �s�. This can be expressed as

�B = Cl�t�� f − �c�

bke−1.3�c �9�

where bk�specific weight of bank soil �kN /m3�; � f�flow shearstress acting on the banks in the near-bank zone �N /m2�;Cl�erodibility coefficient related to bank soil properties, which isa user-specified parameter determined by calibration; and�c�critical shear stress for the bank material �N /m2�. Tang�1963� proposed a simple formula to calculate the value of �c,which can be expressed as

�c = 4.13 � 10−2� s − �d50 +cf

d50�10�

in which cf�coefficient equal to 2.842�10−4 N /m; �specificweight of the flowing fluid �N /m3�, and d50�median diameter ofbank soil �m�. This formula is valid when the median diameterranges from 0.001 to 200 mm. Because the cohesive and fine-grained bank material is usually eroded by the entrainment ofaggregates or crumbs of soil rather than individual particles�Thorne et al. 1997�, a relationship between mean critical shearstress and median diameter of bank soil is used to calculate thelateral erosion distance. In the calculation, the slope angle of in-fluence on the critical shear stress can be neglected because themagnitude of boundary shear stress is one order greater than thevalue of critical shear stress of bank soil �Wang et al. 2004�.

The lateral erosion distance �B is equal to the horizontal dis-tance between points A and B in Fig. 2�a� or between points B and

Fig. 2. Sketch map of bank failures: �a� Initial bank failure; �b�subsequent bank failure

C in Fig. 2�b�. �c in Eq. �10� is the critical shear stress for the


bank material at the subsection between nodes �i , j� and �i , j+1�,and d50 in Eq. �10� can be represented by the mean value of themedian diameters at these two nodes. In the case of Fig. 2�a�, � f in

Eq. �9� can be calculated as � f =u*2 , in which is density of

turbid water; and u*

is friction velocity at a near-bank node �i , j�.In the case of Fig. 2�b�, the shear stress at location B is assumedto be equal to its value at location A.

Analysis of Bank StabilityIn this study, stability relations developed in soil mechanics areapplied directly to analyze riverbank stability, as well as to predictother relevant parameters. The analysis of bank stability is di-vided into the following two steps �Osman and Thorne 1988�:1. Initial bank failure. Fig. 2�a� shows the geometry of a steep

riverbank after lateral erosion. With the known values of theinitial riverbank height, H1, the initial riverbank slope, i0, andthe near-bank shear stress, � f, the lateral erosion distance,�B, is obtained from Eqs. �9� and �10�. As a result, the bankheight of the uneroded part, H2, is also obtained, and theratio �H1 /H2�m can be determined. The angle, �, between thefailure surface and the horizontal plane is given by �Osmanand Thorne 1988�

� = 0.5�tan−1��H1/H2�m�1.0 − K2�tan�i0�� + �� 11�

where K�ratio of the depth of the tension crack, Ht, to theheight of the riverbank, H1 and ��internal friction angle ofbank material. As Ht is a function of the geotechnical prop-erties of bank soil, K varies with soil properties. A constantvalue of K=0.5 is used as an approximation as proposed byOsman and Thorne �1988�, which maybe artificially destabi-lizes the simulated river banks with respect to mass failuredue to the use of the maximum value of K=0.5. Due to thelack of detailed data on bank soil properties, the geotechnicalparameters at all computational nodes are assumed to beidentical. The angle � is obtained with the known values of�H1 /H2�m, i0, K, and �. Then according to the stability analy-sis derived from soil mechanics, the analytical value of thebank height ratio �H1 /H2�c is expressed as �Osman andThorne 1988�

�H1/H2�c = 0.5��2/�1 + ��2/�1�2 − 4��3/�1�� 12�

where �1= �1−K2��0.5 sin 2�−cos2 � tan ��: �2=2�1−K�C /� bkH2�; �3= �sin � cos � tan �−sin2 �� / tan�i0�; and C�co-hesion of soil.

The following approach indicates whether or not the ri-verbank will fail. If �H1 /H2�m after lateral erosion is less than�H1 /H2�c computed from Eq. �12�, then the bank slope isstable and the bank height has not reached the critical heightfor failure. If �H1 /H2�m��H1 /H2�c, then the riverbank willfail and H1 is critical bank height. In the case of bank failure,the failed block width BW and the failed volume per unitlength VB for any subsection are computed using the corre-sponding geometrical relation. If �H1 /H2�m� �H1 /H2�c, thenthe bank is unstable and bank failure must have already takenplace. In this case, the computed lateral erosion distance �Bis unrealistically large, and the time increment in the modelshould be reduced.

2. Subsequent bank failure. After initial bank failure, it is usu-ally postulated that the subsequent bank failure is a parallelbank retreat, which implies the bank-slope angle � remainsunchanged in subsequent failures, as shown in Fig. 2�b�.
Using the above-presented approach, the ratio �H1 /H2�m is


obtained. But, the formula for the analytical bank height ratio�H1 /H2�c is modified as �Osman and Thorne 1988�

�H1/H2�c = 0.5�2/1 + ��2/1�2 + 4� �13�

where 1=sin � cos �−cos2 � tan � and 2=2�1−K�C /� bkH2�. For subsequent bank failures, the bank stability isanalyzed using the same procedure as the one previouslyproposed for the initial bank failure.

Lateral Input Term in Eq. (4)Applying the computational procedure of Osman and Thorne�1988�, the volume of lateral erosion and bank failure per unitchannel length for the jth subsection of the ith section Vbk�i , j�can be calculated using the procedure of bank stability analysis.Then the volume of bank erosion per unit length for the ith sec-tion Vbk�i� can be calculated using Vbk�i�=� j=1

jmax−1Vbk�i , j�, wherejmax−1�total number of subsections for the ith section. With theknown values of bk �density of bank soil�, e �water content ofsoil�, and Vbk�i�, the lateral input term in Eq. �4� can be deter-mined. In addition, the longitudinal distance and the averagewater-surface width between sections i−1 and i are �x�i� andB�i�, respectively; and the probability of bank erosion, P2, has tobe introduced because the length of the short reach, �x�i�, islarger than the length where the process of bank erosion happens.Therefore, the total volume of bank erosion in the short reachcan be expressed as Vi−1,i=0.5�Vbk�i−1�+Vbk�i��x�i�P2, and thecorresponding sediment mass from the eroded and failed banksoil is Mi−1,i=Vi−1,ibk / �1+e�. The ratio of the suspended loadresulting from the eroded bank soil to the total eroded bank soil isassumed to be P1. Thus, the lateral input term for the kth grain

Fig. 3. Sketch of the bank geometry modification procedure: �a� initafter failure; and �d� bank geometry after lateral and longitudinal bed

size in Eq. �4� is obtained as

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Sok =1

�t

Mi−1,i · P1

�x�i� · B�i��Pbk �14�

where Sok�sediment input in unit area and time for the kth frac-tion �kg /m2 s� and �Pbk�percentage of the kth fraction of thebank material. The remaining portion of eroded and failed banksoil is transformed into bed material and is deposited in the near-bank zone.

According to the analysis of observed data in the LYR, a largeportion of the material from bank erosion is deposited in the mainchannel by the lateral transport of suspended load �Zhao et al.1998�. Therefore, the value of P1 is usually very low, and theresulting value of Sok is lower than the other terms in Eq. �4�. Theparameter P2 is an indication of the possibility that riverbanksmay suffer from bank erosion, the value of which is usually esti-mated from the observed data. Sensitivity analysis of these pa-rameters is conducted in a later section.

Modification of the Bank GeometryIn the composite model, the bank geometry after erosion is modi-fied according to the calculated results of the longitudinal andlateral channel deformations. The computational domain is repre-sented by Imax points in the longitudinal direction, and by Jmax

points in the lateral direction. In the lateral direction, the Jmax

points result in Jmax−1 subsections for a given cross section.According to the hydraulic conditions calculated by the flow

submodel, the line connecting nodes �i , j� and �i , j+1� comprisesthe side slope of a floodplain bank as the jth subsection of the ithsection. The topographic profile represented by the thick dashedline A–F in Fig. 3�a�, denoted as the sketched bank slope, is usedin the calculation of longitudinal riverbed deformation. The origi-

k geometry; �b� lateral erosion and bank failure; �c� bank geometrymations

ial bandefor

nal bank is the line A–D–F because banks with cohesive soil



usually are steep. The broken line in Fig. 3�b�, is assumed to bethe bank surface after lateral erosion because of the hydraulicforce. If the mean degree of bank slope increases due to lateralerosion, it consequently becomes less stable. As a result, bankfailure occurs. Thus, the bank surface after failure is the brokenline A–B–H–E–F in Fig. 3�c�. In Fig. 3�c� the node �i , j� issubmerged, and its scour or deposition thickness is determineddirectly by Eq. �5�.

The computational grid remains unchanged in calculation,which ensures that the horizontal positions at all computationalnodes remain unchanged, but allows the bed elevation at eachnode to be changed. In this aspect, the developed model differsfrom other models �Nagata et al. 2000; Duan and Wang 2001;Darby et al. 2002�. Two sets of 1D arrays are provided to recordthe bank geometry during bank erosion for the jth subsection ofthe ith section, such as the broken lines A–B–C–D–F andA–B–H–E–F. One 1D array is used to record the horizontaldistance, and its original location is the dry node �i , j+1�. Another1D array is used to record the bed elevation of the bank geometry,and its elevation system is the same as the observed one. As such,2�jmax−1� sets of arrays are necessary to record all the subsec-tions for a given section. As a result, the bed elevation of eachsubsection becomes variable, by which the process of bank ero-sion and the corresponding bank line shifting can be indirectlyrepresented. It is assumed that the deformation experienced at thebank toe B is identical to that at the submerged node A in Fig. 3.Thus, the bank surface at the end of a time interval can be deter-mined according to the deformation result at node A. If depositionoccurs at node A at the end of a given time interval, the bankgeometry will be changed into the broken line A1–B1–H–E–F;otherwise, it will be the broken line A2–B2–B–H–E–F, asshown in Fig. 3�d�. During the calculation of bank deformation, ifthe tension crack E–H develops toward node F and exceeds thisnode, the bed elevation at node �i , j+1� should be modified im-mediately and this node will be submerged at the next time inter-val. Therefore, in the next time interval, the deformationmagnitude at this node should be determined by Eq. �5�. In thisway, the process of bank retreat or advance during bed scour andsediment deposition on the bed can be simulated easily with thecomputational grid remaining unchanged.

In subsequent time intervals, if the water-surface elevation

Fig. 4. Computational grid a

falls in the channel, and the nodes �i , j� and �i , j+1� are trans-


ferred to those above the water surface, the calculation of bankdeformation at this subsection becomes unnecessary, but the ge-ometry of this bank slope must still be recorded. When the stagerises and the nodes �i , j� and �i , j+1� are below the water surface,the calculation of bank deformation again becomes unnecessary,and the deformation results at these nodes can be estimated byEq. �5�.

Solution Procedure for the 2D Composite Model

The following steps are taken when the 2D composite model isapplied to simulate the longitudinal and lateral channel deforma-tions in the LYR �Xia 2002�:1. Input initial and boundary conditions, including the initial

bed topography, initial flow and sediment conditions, inflow-ing water and sediment hydrographs at the upstream bound-ary, and water-surface elevation hydrograph at thedownstream boundary;

2. Compute the two-dimensional hydraulic variables using the2D flow submodel;

3. Simulate the process of lateral erosion and bank failure usingthe bank-erosion submodel, and calculate the lateral inputterm in Eq. �4�;

4. Compute the concentrations of suspended load for k grainsizes using the 2D sediment transport submodel;

5. Calculate the longitudinal channel deformation, and deter-mine the bed elevation at the end of the time interval usingEq. �5�;

6. Modify the bank geometry according to the results of bedand bank deformations; and

7. Update the gradation of bed material after bed scour or sedi-ment deposition.

By repeating steps �2�–�7�, the changes of bed morphologyboth in the longitudinal and lateral directions can be obtained.

Simulation of the Braided Reach in the LYR

A general description of the braided reach in the LYR is given inthis section, and an approximately orthogonal curvilinear grid isgenerated for the study region. With known initial and boundaryconditions, the channel deformation process is then simulated

tour of initial bed elevation
nd con
using the developed 2D composite model.


General Description of the Study Region

The braided reach is wide and shallow with an average channelwidth of 4–8 km, a relatively steep longitudinal channel slope of1.5 m /104 m, and numerous sand bars and complex branches ex-posed during low flow seasons. Frequent channel shifting mainlyresults from the nonequilibrium transport of suspended load andserious bank erosion �Zhang and Xie 1993�.

Measurements indicate that bed material in the main channel isfine sand, and the material on the floodplain is composed of finesand and silt. The median diameter of bed material in the mainchannel ranges from 0.092 to 0.077 mm along the study reach,and correspondingly, the variance of the particle size distribution��g� varies from 1.84 to 2.76. According to the classification stan-dard, bank soil on the floodplain is cohesive soil �Chen et al.1994�.

Grid Generation

The braided reach between Huayuankou and Laitongzhai was se-lected for this study. The reach had a length of 15.7 km with 19measured sections. The simulation period was from June 1 toJune 24 in 1961, which was a dry season. During this period, thewater-surface elevation was relatively low and the channel widthin the short reach ranged from 1.3 to 3.0 km. This computationaldomain was divided into 80�20 cells with a maximum grid spac-ing of 355 m and a minimum one of 50 m, as shown in Fig. 4.Fig. 4, the locations of some observed sections in the domainwere also indicated.

Computational Conditions

It is preferable to approximate the real unsteady inflowing waterand sediment hydrographs by a series of stepped hydrographswith constant flow and sediment properties for the purpose ofsaving computer time. Fig. 5�a� shows the hydrographs of thedaily average inflowing water discharge and sediment concentra-tion at the upstream boundary. A daily average water-surface el-evation hydrograph was used at the downstream boundary, asshown in Fig. 5�b�. The uncoupled and steady solution methodwas used to simplify the computation, and this simplified methodwas assumed to be valid and feasible because sediment movementand changes in river morphology are relatively slow in compari-son to the changes in hydraulic variables. The flow submodelneeds to be run for a sufficiently long time to achieve a steadystate, as does the sediment submodel. In the calculation, a timestep of one day was used for each time interval. All unmeasuredcross-sectional topography in the domain was obtained by linearinterpolation based on the observed data, and it can also be seenfrom Fig. 4 that the topography is indeed complex in this braidedreach. In the model, the nonuniform sediments of bed materialwere divided into six fractions, and they were 0.002, 0.007, 0.016,0.035, 0.071, and 0.158 mm, respectively. Using this linear inter-polation, the size distribution of bed material at each node forwhich no measurements were available was obtained.

Bank soil properties affect the channel deformation processes,and the key effects can usually be expressed by parameters suchas �c, C, �, bk, and Cl. Changes in any of these parameters mayaffect the simulated results. Among them, �c mainly affects thelateral distance of bank erosion and the thickness of bed scour orsediment deposition. In the braided reach of the LYR, the magni-tude of the shear stress of flow, � f, is one order higher than the
value of �c according to research by Wang et al. �2004�. As � f is
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much higher than �c, the influence of �c on channel deformationmay be ignored. Comparisons of computed results using variousvalues of C, �, and bk indicated that these three parameters havelittle influence on the simulated results, as they vary in limitedranges �Wang and Xia 2001�. According to the available observeddata, a specific weight, bk, of 20 kN /m3, an internal frictionangle, �, of 18°, a cohesion, C, of 12 kN /m2, and a water con-tent, e, of 0.15 were used in the model �Chien and Zhou 1965; Yeet al. 1990�. In addition, P1=0.25, P2=0.50, and Cl=0.18 weredetermined by calibration using the observed data. Consideringthe high erodibility of the bank soil in the braided reach, a rela-tively higher value of the erodibility coefficient, Cl, is believed tobe reasonable. The complete computational process took aboutone hour using a personal computer with the AMD CPU proces-sor of 2.01 GHz.

Analyses of Simulated Results

Water-Surface Elevation along the ReachA comparison between calculated and observed water-surface el-evations on June 1, 1961 along the study reach is presented inorder to verify the applicability of the flow submodel, as shown inFig. 6. It indicates that the proposed flow submodel can accu-rately simulate the observed water-surface profile. On the aver-

Fig. 5. �a� Daily average discharge and sediment concentration at theinlet; �b� daily average water-surface elevation at the outlet

age, the calculated water-surface elevation in the study reach was



0.09 m greater than the observed value. The maximum error be-tween the computed and measured values is about 0.3 m nearDongliubao. This error may be attributed to the fact that therewere several center bars above the surface in this reach. In a crosssection, the observed water-surface elevation was obtained by cal-culating the average value of water-surface elevations at only afew observed points, whereas the simulated water-surface eleva-tion is equal to the average value of water-surface elevations at allsubmerged nodes of the cross section.

Lateral Distributions of Velocity and SedimentConcentrationFig. 7 shows the lateral distributions of depth-averaged velocityand Fig. 8 shows the concentration of suspended sediment at Hua-yuankou for different times. At the first time, the calculated ve-locity and sediment concentration agreed closely with theobserved data. The velocity at the central part of this section wasrelatively low due to the shallow water depth; and the velocitiesnear the banks were relatively high, which corresponded with therelatively high water depth �Fig. 7�a��. The concentration distri-bution was similar to that of the velocity at the first time �Fig.8�a��. However, there was a larger error between the calculatedresults and the observed data at the last time �Figs. 7�b� and 8�b��.At this time, the calculated cross-section-averaged velocity wasabout 25% lower than the observed value, which could be a resultof the relatively high calculated water level, which was 0.10 mhigher than the observed value.

Fig. 6. Water-surface elevation along the study reach on June 1, 1961

Fig. 7. Comparison between calculated and observed depth-averagedvelocities at Huayuankou: �a� June 1, 1961; �b� June 24, 1961


Variations of Cross SectionsFig. 9 shows the variations of six cross-sectional profiles alongthe reach, with a comparison of calculated and observed profilesat the end of the calculation. Fig. 9�a� shows the variation of thecross section at Huayuankou. The sediment deposition on the ri-verbed at the center continued from June 1 to 24 in 1961, and thewater-surface width gradually increased with time. Due to lateralerosion by the near-bank flow, the left and right banks of thefloodplain continuously retreated during this period. The rightbank retreated about 170 m after 24 days, whereas the left bankretreated about 200 m. The corresponding retreat rates of the rightand left banks were 7.4 and 8.8 m /day, respectively, which gen-erated a total eroded and failed soil volume per unit length of390 m3, much greater than the longitudinal deposition volume perunit length of 230 m3. Therefore, the magnitude of channel widthadjustment was significantly greater than that of bed scouring orsediment deposition, indicating that lateral erosion and failure ofthe floodplain banks play an important role in the evolution of theLYR during dry seasons. The calculated cross-sectional geometrywas very similar to the observed cross-section geometry after 24days. As for the longitudinal bed deposition in this section, thecalculated values were greater than the observed ones at the cen-ter zone of the section, but the reverse was true in the near-bankzone.

At Dongdaba, sediment deposition occurred in the near-right-bank zone, and the right and left banks of the center bar failed andretreated. The right pool shifted about 210 m toward the leftbank; and the left pool shifted about 100 m toward the right bank.Predictions at this section were in good agreement with the ob-served data �Fig. 9�b��. At Wuchang, the right pool shifted about270 m toward the center as deposition occurred in the near-right-bank zone �Fig. 9�c��. The result indicates that it is necessary tosimultaneously account for bank erosion processes of both exter-nal boundaries of a computational domain and internal boundariesof center bars or islands when simulating the channel deformationin the LYR. At Shiqiaoxi �Fig. 9�d��, the cross sections changedvery little with time, and there were discrepancies between thecalculated results and the observed data. At Liuyao �Fig. 9�e��, thedeep channel shifted about 150 m toward the right bank, whereasthe sediment deposited on the bed. The right bank of the flood-plain at Laitongzhai retreated backward 130 m along with thelongitudinal deposition thickness of about 1.0 m in the mainchannel �Fig. 9�f��. It seems that the shift direction of the pooldepends on the scour-resistant property of bank soil and the localsediment transport capacity.

In summary, Fig. 9 shows that the deformation processes of

Fig. 8. Comparison between calculated and observed depth-averagedsediment concentrations at Huayuankou: �a� June 1, 1961; �b� June24, 1961

bank retreat and main channel scour or sediment deposition in the


Huayuankou-Laitongzhai braided reach during the dry seasonwere simulated with the proposed model. The calculated meanbed elevation for the six cross sections was 0.11 m lower than theobserved value at the end of the calculation. This error is accept-able considering the large changes in the river morphology of thestudy reach.

Variation of Water-Surface Width at HuayuankouEven if the water-surface elevation remains unchanged in thechannel, the water-surface width varies due to lateral erosion andfailure of banks. The mean increase rate of water-surface widthreached 21.4 m /day in the first period up to and including June14, much greater than the mean rate of 1.0 m /day in the secondperiod after June 14. The wetted cross-sectional area of 832 m2

on June 1, a small value corresponding to a narrow river width,increased to 1,270 m2 on June 14. The lateral erosion resultingfrom the near-bank flow was very intense and the river widthincreased very quickly from June 1 to 14. As the width of thewater surface increased to 1,090 m, its mean increase rate de-creased to 1.0 m /day due to the increase of the wetted cross-sectional area and the decrease of velocity in the channel after

Fig. 9. Calculated and observed cross-sectional profiles at different sCS-14 Shiqiaoxi; �e� CS-16 Liuyao; and �f� CS-19 Laitongzhai

June 14.

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Fig. 10 shows the variation of water-surface width atHuayuankou. At the first time �on June 1�, the calculated width of750 m was smaller than the observed value of 800 m due toerrors caused by topographic representation in the model. OnJune 14, the calculated water-surface width reached a large value

: �a� CS-1 Huayuankou; �b� CS-5 Dongdaba; �c� CS-8 Wuchang; �d�

Fig. 10. Progression of changing water-surface width at Huayuankou

ections



of 1,045 m, about 45 m less than the observed value. Hereafter,the calculated water-surface width remained nearly unchanged,which was similar to the observed variation.

Because the water-surface elevation at this section only variedover the range of 0.25 m during this period, it can be concludedthat the widening of the water surface was not induced by stagerising, and instead it mainly resulted from lateral erosion andfailure of the banks. In fact, the increased value of water-surfacewidth was identical to the bank retreat distance, which reached avalue of 370 m after 24 days. The final the water-surface width onJune 24 increased by 37% compared with the value on June 1.

Parameter Sensitivity Tests

Because of the complexity of sediment transport and bank erosionin a braided reach, it is difficult to eliminate the empirical param-eters in the proposed 2D model. Among the empirical relationsused in the model, most of them such as sediment transport ca-pacity, movable-bed roughness, were verified by earlier research-ers �Zhao and Zhang, 1997; Zhang and Zhang 1992; Hui et al.2000�. Therefore, only these parameters, such as Cl in Eq. �9� andP1 and P2 in Eq. �14� first introduced for the proposed model areconsidered in sensitivity tests in the following. In addition, theinfluence of different grid dimensions on the calculated results isalso analyzed, and effects of different parameter values on thelateral erosion distance at Huayuankou are also investigated. The

Fig. 11. Water-surface widths at Huayuankou resulting for differentCl values

Fig. 12. Cross-sectional profiles at Huayuankou resulting fromdifferent P1 values


parameter sensitivity tests are done through numerical experi-ments for the case study previously presented. In each set ofnumerical experiments, only one parameter is adjusted, whereasother parameters are fixed at constant values.

Test of Parameter Cl

The parameter Cl in Eq. �9� depends largely on the physical andchemical makeup of the bank soil, and the types and amounts of

Fig. 13. Mean sediment concentrations in section from bank erosionat Huayuankou resulting from different P2 values

Fig. 14. �a� Lateral distributions of depth-averaged velocity atHuayuankou resulting from different grid dimensions; �b� finalcross-sectional profiles at Huayuankou resulting from different griddimensions


salts in the pores and eroding fluids. This parameter can be deter-mined by experiments according to Osman and Thorne �1988�.Three different numerical experimental runs are conducted forCl=0.10, 0.18, and 0.30, respectively, with other parametersbeing fixed. Fig. 11 shows the changes of water-surface widthwith time at Huayuankou resulting from different Cl values. It canbe seen that an increase in the Cl value resulted in an increase inthe rate of bank erosion. The calculated water-surface widthchanges for Cl=0.18 agreed relatively well with the observedchanges. The results of Fig. 11 indicate that the parameter Cl

should be calibrated in model applications.

Test of Parameter P1

The parameter P1 in Eq. �14� indicates the ratio of the erodedbank soil becoming suspended load to the total eroded bank soil.To find the influence of parameter P1 on the calculated results,three numerical experimental runs are conducted for P1=0.25,0.35, and 0.45, respectively, with other parameters being fixed.Fig. 12 shows the cross-sectional profiles at Huayuankou after 24days. It can be seen that similar cross-sectional profiles were ob-tained for different values of P1. This result indicates that thebank erosion process is not sensitive to parameter P1. After 24days, the mean bed elevations at the section were 91.502, 91.493,and 91.489 m for the three runs, respectively, and, therefore, dif-ferent P1 values slightly affect the process of bed scour and sedi-ment deposition. The parameter P1 is usually estimated based onthe bank soil properties, and then calibrated with observed data.

Test of Parameter P2

Another parameter P2 in Eq. �14� represents the probability ofbank erosion along a reach, namely the longitudinal extent of

Table 1. Lateral Erosion Distances at Huayuankou Resulting from Diffe

Para

Number ParametersValues usedin the model

1 Para. in �t 1/6

2 �s in Eq. �4� �s=�t

3 Para. in Eq. �7� 2.5

4 Para. in Eq. �7� 0.62

5 Para. in Eq. �7� 0.0022

6 Hb in Eq. �8� 2.0 m

7 Cl in Eq. �9� 0.18

8 Para. in Eq. �10� 4.13�10−2 4

9 Cf in Eq. �10� 2.842�10−4 2

10 �c in Eq. �10� Eq. �10�

11 K in Eq. �11� 0.5

12 � in Eq. �11� 18°

13 bk in Eq. �11� 20 kN /m3 1

14 C in Eq. �12� 12 kN /m2 1

15 e in Eq. �14� 0.15

16 P1 in Eq. �14� 0.25

17 P2 in Eq. �14� 0.50

18 Time steps 1.0 day

19 Grid dimensions 80�20

Note: Based on the values of parameters used in the model, the calculaslightly greater that the observed value of 310 m.

mass failure within a modeled reach �Darby and Thorne 1996�.

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P2=0 means that the sediment from bank erosion is equal to zero,and P2=1 means that the sediment from bank erosion can reachits maximum. To determine the influence of P2 on the calculatedresults, three different numerical experimental runs are conductedfor P2=0.30, 0.50, and 0.70, respectively, with other parametersbeing fixed. Calculated results indicate that different P2 valueshave little influence on the lateral and longitudinal channel defor-mations, because the value of the source term in Eq. �4� is verysmall. Fig. 13 shows the mean sediment concentrations in sectionfrom bank erosion at Huayuankou. It can be seen that the sedi-ment concentration from bank erosion increased with the increaseof P2 value, but the changes in sediment concentration were verysmall compared with the sediment concentration coming fromupstream. Therefore, parameter P2 is initially estimated based onthe bank soil properties, and then calibrated with observed data.

Test of Grid Dimensions

In the 1960s, only 19 cross-sectional profiles were observed in the15.7 km long study reach at a time interval of ten or more days.Because the mean distance between two adjacent sections canreach 0.8 km, the precision of the observed data is insufficient for2D simulation. In order to simulate the channel deformationsusing a 2D model, the initial bed elevation at each computationalnode needs to be determined by linear interpolation based on theobserved data for 19 sections.

In order to study the influence of different grid dimensions onthe calculated results, three different numerical experimental runsare done for 80�20, 120�30, and 160�40 grid cells, respec-tively, with all parameters kept unchanged. Figs. 14�a and b�show lateral distributions of depth-averaged velocity on June 1and final cross-sectional profiles at Huayuankou resulting for dif-

rameters

alues

Values for testsLateral erosion distancesresulting from tests �m�

0.7 338.8 338.2

t �s=3�t 338.9 337.8

2.2 314.5 327.7

0.70 342.6 345.0

0 0.0025 337.5 339.9

3.0 m 337.5 336.9

0.30 286.8 342.1

0−2 5.00�10−2 337.7 337.4

0−4 3.50�10−4 339.5 338.9

m2 1.2 N /m2 290.7 124.9

0.4 337.5 338.0

16° 340.1 339.5

m3 22 kN /m3 338.0 339.9

m2 16 kN /m2 338.0 337.5

0.20 339.8 340.2

0.45 339.7 340.7

0.70 339.6 338.0

y 0.25 day 341.4 342.7

30 160�40 337.5 334.9

ral erosion distance at Huayuankou is 339.2 m after 24 days, which is

rent Pa

meter v

0.5

�s=2�

2.0

0.66

0.002

2.5 m

0.10

.50�1

.00�1

0.6 N /0.3

12°

8 kN /4 kN /

0.18

0.35

0.30

0.5 da

120�

ted late

ferent grid dimensions, respectively. It can be seen that similar



results were obtained for different grid dimensions, and an in-crease in the density of the computational grid can slightly im-prove the calculated results.

Effects of Other Different Parameter Values on theLateral Erosion Distance

In addition to the above-presented tests of parameter sensitivity,effects of different parameter values on the lateral erosion dis-tance at Huayuankou are conducted in order to analyze the sen-sitivity of these parameters to the process of bank erosion, and thecalculated results are given in Table 1.

There are three groups of parameters introduced for the pro-posed model in Table 1. The first group of parameters �parametersfrom 1 to 6� is related to the process of flow and sediment trans-port; the second group of parameters �parameters from 7 to 17� isrelated to the process of bank erosion; and the third group ofparameters �parameters 18 and 19� is related to the model appli-cation. Among the first group of parameters, different values forparameters in Eq. �7� have larger effects on the lateral erosiondistance at Huayuankou than other parameters. Among the secondgroup of parameters, the effects of different values for Cl and �c

on the calculated results are very predominant, compared withother parameters. The lateral erosion distance at Huayuankou willincrease by 19% as the value of bank erodibility coefficientchanges from 0.1 to 0.3, and it will decrease by 57% as the valueof critical shear stress for bank increases from 0.6 to 1.2 N /m2.For the third group of parameters, a decrease in the time stepresults in a small increase in the lateral erosion distance, and anincrease in the grid dimensions results in a decrease in the lateralerosion distance at Huayuankou. Therefore, it is necessary tocarefully select some key parameters such as parameters in Eq.�7�, Cl, and �c, which should be calibrated before the applicationof the proposed model.

Conclusions

Bank erosion is a very active fluvial process in the LYR, espe-cially in the braided reach. A new 2D composite model for simu-lating bed and bank deformations was therefore developed. Thiscomposite model consists of two submodels, which are a depth-averaged 2D flow and sediment transport submodel in a nearlyorthogonal curvilinear coordinate system, and a submodel tosimulate the process of bank retreat and advance. A new tech-nique for updating the bank geometry during the bed degrada-tional and aggradational evolution was presented. These measuresallow the developed composite model to simulate the longitudinaland lateral channel deformations in braided rivers with complexbed topography under the condition that the horizontal positionsof grid points remain unchanged in the calculation. The fluvialprocesses in a braided reach of the LYR between Huayuankouand Laitongzhai were simulated in detail, and the calculated re-sults were generally in agreement with the prototype-measureddata.

Parameter sensitivity tests indicated that different values forthe majority of parameters have relatively small effects on thesimulated results for this case study with the exception of param-eters in the formula for erodibility coefficient for bank, and criti-cal shear stress for bank soil. As the current case is nearlyone-dimensional due to the relatively simple bathymetry, this
study does not verify the numerical model for a generally com-

plex three-dimensional geometry. This would be a topic for fur-ther studies.

Acknowledgments

This project was supported by the Natural Science Foundation ofChina �Grant No. 50409002�, by the Program of Strategic Scien-tific Alliances between China and The Netherlands �Grant No.2004CB720402�, and by the Science Fund for Creative ResearchGroups of the Natural Science Foundation of China �Grant No.50221903�.

Notation

The following symbols are used in this paper:B�i� � average water-surface width between sections

i−1 and i;BW � width of the failed bank block;

C � cohesion of the bank soil;Cl � erodibility coefficient related to soil property;

C�, C� � Lami coefficients in the � and � directions,respectively;

D50 � median diameter of the bed material;d50 � median diameter of the bank soil;

e � water content of the bank soil;F � Froude number of flow;

fsk � coefficient in Eq. �4�;Hb � thickness of the mixed layer of bed material;Ht � depth of tension crack;H1 � initial riverbank height;H2 � riverbank height above the turning point;

�H1 /H2�c � calculated bank-height ratio;�H1 /H2�m � measured bank-height ratio;

h � water depth;i0 � initial angle of the bank slope;K � ratio of crack depth to bank height;

ksk � coefficient in Eq. �4�;Mi−1,i � eroded and failed sediment mass in a reach

between i−1th and i th sections;N � number of fractions for nonuniform sediments;n � Manning’s roughness coefficient;

P1 � ratio of the eroded bank soil becomingsuspended load to the total eroded bank soil;

P2 � indication of the probability that riverbanksmay suffer from bank erosion;

Sk � concentration of suspended load for the kthgrain size;

SV � sediment concentration by volume;S

*, S

*k � sediment concentration and sediment transportcapacity for the kth grain size, respectively;

t � time;U � velocity component in the � direction;u � depth-averaged velocity;

u* � friction velocity;V � velocity component in the � direction;

Vbk�i� � total volume of bank erosion per unit length forthe ith section;

Vbk�i , j� � failed volume per unit length for jth subsection
of the ith section �=VB�;


Vi−1,i � total volume of bank erosion in a reachbetween the i−1th and ith sections;

Z � water-surface elevation;Zb � riverbed elevation;

�*k � coefficient in Eq. �4�;� � angle of bank slope after failure; � specific weight of water;

bk � specific weight of the bank soil; s, m � specific weights of sediment and turbid water,

respectively;�B � lateral erosion distance;

�Hm � thickness of each memory sublayer of bedmaterial;

�Hs � total thickness of bed deformation;�Hsk � thickness of scour or deposition for the kth

grain size;�Pbk � gradation of bed material or bank soil;�Pmk � gradation of the mth memory sublayer of bed

material;�Premk � average gradation in the specified memory

sublayers;�P

*k � gradation of sediment transport capacity;�t � time increment;

�x�i� � longitudinal distance between sections i−1 andi;

�g � mean geometric variance of the particle sizedistribution;

��, �� turbulent diffusivity coefficients of sediment in� and � directions, respectively;

� � curvilinear coordinate in the transversedirection;

� � von Karman coefficient;�1, �2, �3 � factors in Eq. �12�;

�t � turbulent viscosity coefficient;� � curvilinear coordinate in the streamwise

direction; � density of turbid water;

� � dry density of the bed material;bk � density of the bank soil;s � density of the sediment;� � flow shear stress acting on the near-bank zone;

�c � critical shear stress for bank material;� � internal friction angle of the bank soil;

m � group settling velocity of nonuniform sediments;sk � effective settling velocity for the kth grain size;

and1, 2 � factors in Eq. �13�.

Subscripts

i � node number in the � direction;j � node number in the � direction;k � size group number of nonuniform sediments;l � time level; and

m � number of the memory sublayers.

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