Three-dimensional flow modelling and sediment transport in the River Klarälven

THREE-DIMENSIONAL FLOW MODELLING 821

Copyright © 2004 John Wiley & Sons, Ltd. Earth Surf. Process. Landforms 29, 821–852 (2004)

Earth Surface Processes and Landforms

Earth Surf. Process. Landforms 29, 821–852 (2004)Published online 17 June 2004 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/esp.1071

THREE-DIMENSIONAL FLOW MODELLING AND SEDIMENTTRANSPORT IN THE RIVER KLARÄLVEN

BIJAN DARGAHI

Division of Hydraulic Engineering, The Royal Institute of Technology, Stockholm, Sweden

Received 11 November 2002; Revised 14 July 2003; Accepted 11 November 2003

ABSTRACT

A three-dimensional flow model that uses the RNG k-ε turbulence model and a non-equilibrium wall function was appliedto the River Klarälven in the southwest part of Sweden. The objectives were to study the nature of the flow in the riverbifurcation and to investigate the short-term sediment transport patterns in the river. The effectiveness of three-dimensionalflow models depends upon: (1) how well the river geometry and it surface roughness are modelled; and (2) the choice ofthe closure model. Improvements were obtained by modelling the river in two parts: the entire river reach, and a selectedpart. Composite Manning coefficients were used to account for roughness properties. The method requires a calibrationprocess that ensures the water surface profiles match the field data. The k-ε model under-predicted both the extent of flowseparation zones and the number of secondary flow regions having a spiral motion, in comparison with the RNG k-ε model.The 3-D model could predict with good accuracy both the general and secondary flow fields in the river. The results agreedwell with the 3-D velocity measurements using an acoustic Doppler current profiler. A conceptual model was developed thataccounts for the development of secondary flows in a river bifurcation having two bends. The main flow feature in the rivercross-sections was the existence of multiple counter-rotating spiral motions. The number of spiral motions increased as theriver bends were approached. The river bends also caused vorticity intensification and increased the vertical velocities. Theapplication of the 3-D flow model was extended by solving the sediment continuity equation. The sediment transport patternswere related to the secondary flow fields in the river. The sediment transport patterns at the river bifurcations are charac-terized by the growth of a sandbank. Copyright © 2004 John Wiley & Sons, Ltd.

KEY WORDS: numerical modelling; 3-D models; CFD modelling; river flows; sediment transport; river bifurcation; Klarälven; flooding

INTRODUCTION

This study focuses on the engineering aspects of river morphology applied to the lower reach of the RiverKlarälven. The river is in the southwest part of Sweden and is one of the largest regulated rivers in the country.The present study is concerned mainly with the delta region, where the river bifurcates into a west and an eastchannel at the city of Karlstad. Over a period of 30 years, the sediment transport capacity of the west channelhas gradually diminished. This has caused a growing concern over the risk of flooding in the city. In time, thewest channel may completely silt up if siltation continues. In this paper 3-D numerical flow modelling is usedto investigate the flow features in the river with focus on the river bifurcation. Previous studies can be groupedinto studies of secondary flow features in rivers, and numerical flow and sediment modelling of general flowfeatures in rivers. Regarding the first group, few references can be found in the literature on secondary flowpatterns at channel bifurcations, with the exception of Ashmore (1982), Thompson (1986) and Richardson(1997). They have studied the secondary flow and channel changes in braided rivers that have certain similaritieswith a river bifurcation. However, the main focus of most studies has been on river bends. A full review of theseworks is not possible in view of the limited space available. Richardson (1997) gives an excellent summary ofsignificant contributions to the subject. Here the author wishes to cite some results on the formation and thecharacteristics of the secondary flows in rivers that are relevant to the present study. Secondary flows aredeveloped both in the straight and bend river channels having different mechanisms. In straight channels,secondary flows are stress induced due to the anisotropic distributions of boundary shear stresses (Gessner and

* Correspondence to: B. Dargahi, Division of Hydraulic Engineering, The Royal Institute of Technology, Stockholm, Sweden.E-mail: [email protected]

822 B. DARGAHI


Jones, 1965; Perkins, 1970). Conversely, in river bends strong secondary flows are caused by skewing of cross-stream vorticity, where there is a change in the channel geometry such as a channel bend or a rapid change incross-sectional shape (Squire and Winter, 1951; Perkins, 1970). A common view held until 1975 on secondaryflows in river meanders was the existence of a single helical cell. Hey and Thorne (1975) and Bridge and Jarvis(1982) found a small cell of reversed rotation (an outer bank cell) in addition to the main helical cell. Theyreasoned that the presence of the outer cell depended on two factors: the shape of the outer bank – banksperpendicular to the water surface were more likely to lead to an outer bank cell; and the strength of the flowcomponent towards the outer bank. Further research in the late 1970s and 1980s showed that the skew-inducedsecondary cell does not extend to the inner bank (Dietrich et al., 1979; Thorne et al., 1985; Markham andThorne, 1992). Major theoretical studies on river meanders have been made by Odgaard (1986a, 1986b, 1989a,1989b) and Odgaard and Bergs (1988). They have modelled the flow and the bed topography in meanderchannels by solving the governing equations using an analytical approach. Several equations were suggested forcalculations of flow characteristics in river meanders.

Numerical models have been widely applied to problems in hydraulic engineering. River modelling applica-tions can be grouped into 2-D depth-averaged hydrodynamic models and 3-D models. In comparison with 2-D models, there are fewer application of 3-D models to rivers. Some recent research works are: Weerakoon andTamai (1989); Olsen and Stokseth (1995); Weiming et al. (1997); Sinha et al. (1998); Lane et al. (1999);Bradbrook et al. (2000b); Chau and Jiang (2001). Olsen and Stokseth (1995) carried out a 3-D simulation ofa natural river flow. They modelled an 80 m long reach of the river Sokna in Norway. The model successfullypredicted the flow features and the results were in good agreement with observed data. Sinha et al. (1998)undertook a comprehensive numerical study of the flow through a 4 km reach of the Columbia River. Theysucceeded in modelling both rapidly varying bed topography and the presence of multiple islands. The resultsagreed well with experiments and field measurements. In a recent study, Bradbrook et al. (2000a) modelled theflow in a natural river channel confluence using a Large Eddy Simulation. The model simulated periodic flowfeatures in river confluences and agreed well with experimental data. There are also several recent applicationson use of 3-D sediment transport models, and some examples are: Gessler et al. (1999); Holly and Spasojevic(1999); Fang (2000); Rodi (2000); Weiming et al. (2000); Nicholas (2001). Gessler et al. (1999) applied the USArmy Corps Engineering code CH3D-SED to the Deep Draft Navigation project on the lower Mississippi River.The code predicted the sediment deposition in the river with an accuracy of less 13 per cent in comparison withthe observations. Holly and Spasojevic (1999) applied and verified the CH3D code to study water and sedimentdiversion at the Old River Control complex on the lower Mississippi river.

This study had three main objectives: (1) to study the flow structure in river bifurcations; (2) to increaseunderstanding of sediment transport in river bifurcations; and (3) to investigate the cause of siltation in riverbranches. The flow structure at a river confluence or a river bifurcation is important in river engineering.However, in contrast to many studies related to river confluences (e.g. Bradbrook et al., 2000a,b), there has beenlimited investigation of river bifurcations (e.g. Ashworth et al., 1992; Richardson, 1997). In line with the firstobjective, the present work is also an attempt to close this gap. The results presented in this study should beuseful in supporting river restoration works and flood alleviation schemes.

METHOD

In modelling river flows several difficult problems must be addressed. These are the choices of the computationalgrid, boundary conditions, turbulence models and wall roughness. Many rivers have a large aspect ratio (width/depth ≈ 5–200) and there is also a significant variation in the flow depth. A common problem is cell skewness,which could cause solution inaccuracy or diversion. The choice of boundary condition and the turbulence modelrequires an understanding of the physics of the flow. To include wall roughness can be a difficult task for riverswith gravel beds or in cases where roughness is heterogeneous. Several commercial computational fluid dynam-ics (CFD) codes are available that can be used to model complex flow problems. In this study a general-purposeCFD code known as Fluent (for details see Fluent, 1995, <http://www.fluent.com>; Kim et al., 1997; Engelmanet al., 2001) is used. Fluent models fluid flow, heat transfer and chemical reactions. It solves the full 3-Dequations in general curvilinear coordinates, for both laminar and turbulent flows. The free surface problems are



modelled using the volume of fraction model (VOF). VOF is based on the method outlined by Hirt and Nichols(1981). The chosen code gives extensive possibilities to control grid generation, boundary conditions and closuremodels.

The numerical models were built and verified using extensive field data. Based on model outputs, observationsare made about the velocity vectors and the secondary flow patterns. The main sediment transport patterns inthe river were also investigated by solving the sediment continuity equation for the total sediment load.

STUDY SITE AND FIELD MEASUREMENTS

The River Klarälven enters Sweden in the north of the county of Värmland. Its course in Värmland is southdown to the river mouth on lake Vänern at the city of Karlstad. Vänern is the largest freshwater lake in Sweden.The course of the river is a sequence of regular meanders that are unique with respect to size and regularity.

A 6 km long reach of the river was modelled (Figure 1) that extends 4·9 km upstream of the bifurcation. Toensure uniform inlet boundary conditions, the upper limit of the reach was placed in a straight river section. Thechoice was motivated by the fact that within a distance of 1 km downstream of the inlet the mean flow depthsand the river width did not change significantly with the distance. The river topography was measured in May

Figure 1. Plan view of the lower reach of the River Klarälven showing the extents of Model A and Model B

824 B. DARGAHI


Figure 2. (a) Plan view of the measured cross-sections in the lower part of the river. (b) Examples of the measured river cross-sections

1977 along 70 cross-sections using an acoustic Doppler current profiler (ADCP). Of these, 15 cross-sectionswere placed in the first 3 km of the river where the topography did not change significantly. The distancebetween the cross-sections depended on the nature of the channel geometry. In the straight parts of the river,the distance varied from 140 m to 250 m. The cross-section spacing ranged from 45 m to 55 m for sectionswithin 110 m upstream of the river bifurcation and the downstream outlet sections. Depending on sectionproperties, the river cross-sections were divided into 15–30 cross-stream segments. Figure 2a shows the locationof measured sections in the lower part of the river. Examples of the river cross-sections are given in Figure 2b.There is a significant variation both in depth and width across the river reach.

The 3-D velocity fields were measured by the same instrument (ADCP) at 11 cross-sections. Seven sectionswere in the main river channel (C20, C19, C18, C16, C9, C2 and C1), two sections in the west-channel (W25and W1) and two sections in the east channel (E1 and E17). The cross-stream spacings (∆z) of the cross-sectionswere the same as used to measure the river topography. The flow depths ( y) at each XZ position (X = streamwisedirection and Z = cross-stream direction) were divided into 18 equally spaced points. At each point 3-D veloci-ties were collected for 1 min and the time mean values were then computed. The closest vertical distance thatthe velocities could be measured was 0·78 m below the water surface and above the river bed. The mean flow



Figure 2. (continued )

discharge (750 m3 s−1) was computed using velocity measurements. During the measurement period, the dis-charge remained constant (steady state) for 48 h. Later, the flow discharge reduced to 285 m3 s−1 over a two-weekperiod. The discharge of 750 m3 s−1 is a medium-high discharge in the river that corresponds to the bank fulldischarge. At each river cross-section, the water surface profiles where also measured. The sediment character-istics were obtained from soil samples.

3-D FLOW SIMULATIONS

Governing equations

The governing equations used in the numerical model are those of the Reynolds Averaged Navier–Stokesequations. Using index notation and assuming incompressible steady flow, the momentum and continuity equa-tions read

826 B. DARGAHI


u

u p uj

i

j i

i

j j j

i jx x x x x

u u∂∂

∂∂

∂∂ ∂

∂∂

′ ′ ( )= − + −1 2

ρν (1)

∂ ′∂u

x

i

j

= 0 (2)

where ui = velocity components, ρ = density, ν = kinematic viscosity and p = pressure. An overbar denotes themean, and a primed quantity is a fluctuating part. The Reynolds stresses are related to mean flow quantities viaone of the three turbulence models, i.e. the k-ε model (Launder and Spalding, 1972), the linear renormalizationgroup (RNG) k-ε model (Yakhot and Orszag, 1986) and the Reynolds stress model (RSM) (Wilcox, 1998). Thek-ε model is a two-dimensional equation model, which means that two additional variables – turbulent kineticenergy (k) and the dissipation rate (ε) – are introduced to model the Reynolds stresses. The RNG model issimilar to the standard k-ε model except for the values of the constants C1ε = 1·42, C2ε = 1·68, Cµ = 0·0845 andα = 1·39, and an additional source term in the ε equation, which is absent from the standard k-ε model. Acommon disadvantage of both k-ε and RNG models is the assumption of isotropic viscosity. The k-ε model isa semi-empirical model proven to give good results in many engineering problems. However, the model doesnot work very well for some complex flows such as separated flows and flows in curved geometries (Bradshawet al., 1981; Versteeg and Malalasekera, 1995; Wilcox, 1998). As opposed to being empirical, the RNG modelis derived from mathematical methods. It gives improved performance for separated and curved flows aswell as flows with moderate swirl (Yakhot and Orszag, 1986; Versteeg and Malalasekera, 1995). In the RNGmodel non-equilibrium effects are treated by including a rate of strain term in the dissipation rate transportequation. In the present study, the k-ε and RNG models were used.

Numerical procedure

Fluent uses a control volume-based method (Launder and Spalding, 1973; Patankar, 1980) to convert thedifferential conservation equations to algebraic equations that can be solved numerically. The method consistsof integrating the differential equations about each control volume, yielding a finite difference equation thatconserves each quantity on a control volume basis (Jameson et al., 1981). Non-staggered grid storage is usedto define the discrete control volumes. To calculate the derivatives of the flow variables three interpolationschemes are available, i.e. power-law, second-order upwind/central difference (Patankar, 1980; Versteeg andMalalasekera, 1995), and Quadratic (QUICK; Leonard, 1979). In the present study, the second-order scheme wasused. The choice is motivated by the greater accuracy of the scheme in comparison with central differencing orfirst-order upwind schemes. The set of simultaneous algebraic equations is solved by a semi-implicit iterativescheme which starts from an arbitrary initial solution and converges to the correct solution doing severaliterations. The iterative calculation was monitored and stopped when the maximum residual value had fallen to1E-03 and the overall mass imbalance was 1 per cent. (In Fluent residual is defined as the error associated withthe discretized equation, summed over all computation cells and normalized by the sum of the centre coefficientsof the equation. The term maximum refers to the residual for the transport equation showing the largest residualvalue.) For pressure velocity coupling three solution procedures are available in Fluent, i.e. SIMPLE, SIMPLECand PISO. A comprehensive review of these algorithms is given by Versteeg and Malalasekera (1995). In thepresent study the PISO algorithm was used. This method is particularly useful for calculation on meshes witha high degree of distortion as in the present river model.

Geometry and grid

The geometry was created from a set of separate volumes in three steps. Each measured cross-section wasdefined by three 3-D spline curves: two at the sides defining the river banks and one river bed in between. Thecurves were connected using six 3-D skin surfaces. Each volume was then created by stitching the six surfaces.The complete river reach consisted of 25 volumes. To construct the grids both structured and unstructured typeswere tested. Using a structured grid was not possible. The main reason was the significant variation in the rivertopography, which caused highly distorted and skewed mesh elements. Instead an unstructured grid consistingof Tet/hybrid elements was used. A Tet/hybrid mesh consists primarily of tetrahedral elements but may include



hexahedral, pyramidal and wedge elements where possible. The initial mesh numbers used to model the rivernumbered 105. The mesh numbers were successively increased to 106 in four steps that nearly reached thecapacity of the computer used. The results presented here are based on the fine mesh simulations (106). Thesmallest volume mesh that could be created was about 0·67 m3 (∆x = 2 m, ∆y = 0·084 m, ∆z = 4 m). To increasethe accuracy of the simulations, two models were used (see Figures 1 and 2a): (1) a coarse Model A thatcorresponded to the entire 6 km reach of the river; and (2) a separate finer mesh Model B to simulate thedownstream 2 km of the reach. Figure 2a shows the extent of Model B, with the inlet section marked with athick line. It should be emphasized that the only purpose of Model A was to find the inlet boundary conditionsfor Model B.

Boundary conditions

Free surface. Two approaches were tested to define the free-surface boundary conditions. In the first ap-proach an attempt was made to use the free-surface model VOF. However, using the model was not possibleowing to the limited computer capacity. In the second approach the model was titled by the slopes computedfrom the measured water surface evaluations along the river reach. The water surface was defined as a planeof symmetry. Symmetry implies that the normal velocity and the normal gradients of all variables are zero atthe plane of symmetry.

Inlets. The definition of an inlet requires the values of the velocity vectors and turbulence properties. The inletconditions were defined differently for Models A and B. In the former case, the inlet was described by themeasured flow velocities at section C20 that corresponded to a discharge of 750 m3 s−1. The inlet turbulenceproperties for Model A were defined in terms of the turbulent kinetic energy and the dissipation rate. Theturbulent kinetic energy was calculated by assuming a linear variation from a near wall value of

k

u

Cw *= ⋅

2

0 5µ

(3)

to a free-stream value

kns = TuU2f (4)

The dissipation rate was calculated from Equation 5

ε *=

u

l

3

(5)

In which Tu = turbulence intensity, u* = shear velocity, l = mixing length, Cµ = 0·0845, and Uf = free-streamvelocity. The shear velocities were computed by fitting the universal log-law velocity profile to ADCP data. Thefree-stream velocities were also obtained from ADCP data. The value of l was taken as 0·085 times the hydraulicradius (hr) at section C20. In Equation 4, a value of 2 per cent for free-stream turbulence intensity (Tu) was used.This value is consistence with the free-stream values reported in the literature for turbulent boundary layersalong flat walls (Hinze, 1975). For a flat wall Tu varies from about 15 per cent at the wall to 2 per cent nearthe surface.

The inlet conditions, i.e. the velocity profiles and the turbulence properties for Model B, were defined usingthe results of Model A.

Outlets. Two outlets of pressure type were defined at sections W1 and E17 (Figure 2a). The reason for thechoice of pressure type instead of the velocity type was that the latter is not a well-posed boundary conditionfor a model with one inlet and two outlets. The estimated values of the relative static pressures were used atthe outlets.

Wall regions. To estimate the effect of the wall on the turbulent flow empirical wall functions known asstandard equilibrium and non-equilibrium wall functions were used. The standard wall function employs Laun-der and Spalding’s (1973) log-law of the wall to compute the wall shear stress:

828 B. DARGAHI


uC k

uE

C k y

v

µ µ

κ

1 4 1 2 1 4 1 21/ /

*

/ /

ln=

(6)

in which u = the fluid velocity at a point in the log-law layer, E = a constant equal to 9·8 for hydraulically smoothwalls and y = distance from the wall. The non-equilibrium function (Kim and Choudhury, 1995) uses the samelogarithmic law for the velocity as the standard wall function, but it relaxes the local equilibrium assumption(production = dissipation) which is adopted by the standard wall function in computing the budget of turbulentkinetic energy. This makes the function more suitable for complex flows than the standard function. The non-equilibrium function employs the two-layer concept in computing the budget of turbulent kinetic energy at thewall-adjacent cells. The wall neighbouring cells are assumed to consist of a viscous sublayer and a fullyturbulent layer. The wall roughness effects are accounted for by replacing E with a modified function E′:

E

E

KK

K

hs

ss

r

h′

, Re=+ ⋅

=++

1 0 3 8

λ(7)

In which Ks = roughness height, hr = hydraulic radius, Reh = Reynolds number based on hr and λ = frictionfactor. The friction factors were found from the Moody diagram using roughness heights. The river bed and theside slopes were divided into separate wall regions having different values of sediment size. The roughnessheights were found from the Stricker equation and the Manning’s coefficients estimated from Equation 4 (Chow,1959):

n = (n0 + n1 + n2 + n3 + n4)n5 (8)

where n0 = basic n value for a straight uniform smooth channel, n1 = a value for the effect of surface irregular-ities, n2 = a value for channel-cross section shape and size, n3 = a value for obstructions, n4 = a value for vegeta-tion and flow conditions, and n5 = a correction factor for meandering of the channel.

Specification of roughness values. To specify the various n values given in Equation 8, the following pro-cedure was used:

(1) An initial set of n values were selected from published data (e.g. Chow, 1959, p. 109, Table 5.5; andMannings n Reference Online, published by USGS) and data obtained through comparison with similarrivers. As a further guideline the basic n values were also checked using a roughness height equal to 3D90.In sediment transport literature 3D90 is the recommended choice to account for natural random arrangementsof sand particles (Rijn, 1990).

(2) A public domain program developed by the US Army Corps of Engineers known as River Analysis Systemor HEC-RAS was used to calculate the water surface profiles. The water surface profiles were calculatedusing the initial roughness values (n) and the results were compared with the field measurements. Theoptimal values found were: n0 = 0·025, n1 = 0 and 0·005, n2 = 0·005, n3 = 0·01, n4 = 0·01 to 0·025 and n5 = 1·15.These values correctly specified the known water surface locations.

The above procedure ensured that the various roughness effects were accounted by the wall function throughEquation 6.

Solution procedure. A usual procedure to judge solution convergence is to use a residual definition. Thedefinition implies that convergence is reached when the normalized changes in variables between successiveiterations are equal to or less than a certain limit. In the present study the limit was taken as 10−3. To obtaina converged solution for each case, six weeks of computations were necessary on a powerful double-processorPC. The models were first run as laminar flow with reduced under-relaxation factors with a first-order approxi-mation (momentum equation). Once a solution was obtained, the turbulence models were started and second-order approximations were used. Model sensitivity to the choice of turbulence models and wall functions andgrid (mesh) size was also investigated.



2-D SEDIMENT TRANSPORT MODEL

Governing equations

The two-dimensional sediment continuity equation was used to find the general sediment transport patternsin the river reach. The continuity equation for sediment transport written in curvilinear coordinate is

∂∂

∂∂

∂∂

∂∂

Y

t

Y C

t

q

s

q

n

b m tr tn ( )

+ +−

+

=1

10

χ(9)

where Yb = the bed level variation, t = time, Ym = mean flow depth, C = depth averaged (normalized) sedimentconcentration, χ = porosity, and qts and qtn = s and n components of the total transport load per unit width. Thesecomponents are induced by the flow velocities along (s) and perpendicular (n) to the streamline. The rates ofaverage total transport loads were calculated using the Ackers-White (1973) method given by Equations 10–16:

q K D

u

Y Y

Yt m

n

cr

cr

m

=

−

*

uu

(10)

Y

u

s gD Y D

n

m m m

n

(( ) )

log( / )

*=−

⋅

⋅

−

1 566 100 5

1u

(11)

Y

Dcr

*

=⋅

+ ⋅⋅

0 23014

0 5for 1 < D* < 60 and Ycr = 0·17 for D* ≥ 60 (12)

D D

s gm*

/

( )

=−

12

1 3

ν(13)

m

D

*

=⋅

+ ⋅6 83

167 for 1 < D* < 60 and m = 1·78 for D* ≥ 60 (14)

n = 1 − 0·5log(D*) for 1 < D* < 60 and n = 0 for D* ≥ 60 (15)

K = 10−3·46+2·79log(D*)−0·98(logD*) for 1 < D* < 60 and K = 0·025 for D* ≥ 60 (16)

where qt = total load transport, u = depth-averaged velocity, Dm = characteristic sediment diameter(= D35), ands = sediment specific density. Ackers-White equations were based on analysis of 925 sets of flume and field data.The K and m coefficients are the revised forms given by H. T. Wallingford. The method is known to give goodresults for flume data and large rivers. One disadvantage of the method is the relatively strong influence of bedroughness on the transport rate (Rijn, 1990). The second term in Equation 9 was ignored, since according to Rodi(2000) it can be neglected for quasi-steady conditions.

Solution procedure

The partial differential Equation 9 was solved numerically using an explicit first-order finite difference schemealong s and n grid lines given by Equation 17:

Y Y

t

q q

s

q q

n

b

j

b

j

ts

j

ts

j

tn

j

tn

j

i i i i k k

+ −+

−−

+−

=+ +

11

101 1

( )

∆ ∆ ∆λ(17)

830 B. DARGAHI


in which Yb = bed level variation, i and k = grid lines in s and n directions, respectively, ∆t = time step, ∆s = gridspacing in s direction, ∆n = grid spacing in n direction and j = time index. To create the computational grid analgebraic grid generator scheme was used (Knupp and Steinberg, 1994). Two advantages of the method are rapidcomputation and control over grid point locations. The river was modelled with 2000 rectangular cells in ahorizontal plane.

The computational parameters were T = 48 h (total time), ∆t = 60 s. A 48 h period was chosen because theflow discharge remained constant during this period. At each time step the mean flow depth (Ym) at each locationwas calculated from Equation 18:

Ymj+1 = Ym

j + (Ybi

j+1 − Yjbi) (18)

To allow for the dynamic nature of the sediment transport the flow velocities were corrected at each time step.The mean flow velocities were reduced at each time step by the ratio of the mean flow depth at time j to thedepth at time j + 1. It was assumed that the shapes of the velocity profiles remained unchanged during theerosion process. This method was considered adequate for the sediment modelling. The small variations in thebed elevations (4–20 mm) did not justify the use of more refined methods.

To compute the sediment loads the values of the mean, shear velocities and sediment size were needed. Themean velocities were calculated by depth integrating the computed 3-D velocities. The velocities were firsttransformed into the curvilinear coordinate and mean values were calculated. The initial shear velocities wereestimated using the computed wall shear stresses. The sediment sizes were obtained from the soil samples.

To define the inlet sediment load, the depth-averaged velocities were first computed. These velocities werethen used to compute the sediment load from Equations 10–16.

Model validation

The flow field. The validation of the numerical results involved both comparisons with the measured flowvelocities and the flow structures. For this purpose the 3-D velocity measurements at 11 cross-sections wereused. The numerical velocity data were extracted from the computation results at the same locations as the fieldmeasurements. No comparison was possible at 0·78 m below the water surface and 0·78 above the bed sincethese depths were outside the ADCP device range. To quantify the comparison two statistical parameters werecalculated, i.e. relative errors and correlation coefficients. The relative errors were obtained by taking thedifference between numerical and measured values and then dividing the results by the measured values. Thecorrelation coefficients (R2) were calculated by comparing the computed velocity profiles with the measured datapoints. The measured velocity profiles were predicted with a relative accuracy within the range 1·5–10 per cent.The correlation coefficients varied in the range 0·7–0·93. The lower values corresponded to the vertical veloci-ties. Examples are shown in Figures 3a and 3b. Figure 3a shows the velocity components in the vertical directionat the deepest points of sections C2, E1 and W25. The relative errors are also plotted in the figure. Figure 3bshows the three velocity components at eight cross-sections at 1·03 surface metres below the water surface. Inthis figure the solid lines are the computed results and the velocity components are marked by letters u, v andw. The numerical results for the velocity components u and w agreed better with the field data than the verticalvelocity component v. This could be due to insufficient mesh resolution of the numerical model in the verticaldirection and the response of the ADCP device to low flow velocities. Regarding the flow structures comparisoncould only be made for the cross-sections where the velocities were measured. The flow structures at these cross-sections were deduced by studying the velocity vectors. A reasonable agreement was found between the flowstructures predicted by the model and those deduced from field data. The agreement was defined in terms of thenumbers, locations, strengths, the rotation signs of the secondary circulation cells and the length of surface flowseparation zones. The numbers of cells predicted by the numerical model was higher than those obtained fromthe field data. The numerical model predicted additional cells near the river banks. There are two possiblereasons for the disagreement. One reason is in flow regions close to the river banks, and the flow depths andvelocities were low and therefore more difficult to measure by the instrument (ADCP). The second reason is thatin comparison with the field measurements, the numerical model had a higher mesh resolution near solidboundaries and the inflection points that marked the slope change between the river bed and the banks. However,



Figure 3. Comparison between computed (lines) and measured velocity profiles in (a) three vertical sections and the corresponding plotsof relative errors, and (b) eight cross-sections at 1·03 m below the surface. •, field data u component; o, field data w component; +, field

data v component

832 B. DARGAHI





Figure 4. Velocity contours in cross-section W25. (a) 3-D model; (b) field data. The numbers on contour lines give velocities in m s−1

good agreement was obtained regarding the locations and the signs of the cells at all 11 cross-sections.Figure 4 shows the velocity contours at section W25 found from the numerical model and field data. Thelocations and the signs of three secondary circulation cells are correctly predicted by the 3-D model. To comparethe strengths of the secondary circulation cells, line integrals of the tangential components of the velocitiesaround each cell were computed. The relative errors in values of circulations between the model and the fieldvalues were in the range 5–15 per cent. The disagreement is due to the relative accuracy range (1·5–10 per cent)of the flow velocities. There were also some uncertainties in defining closed curved regions for calculations ofcell circulations. Table I compares the length of separation zones obtained from the 3-D model and fieldobservations at the surface. The relative error between the RNG fine mesh model and field data are also givenin the table. The error range is 7–12 per cent. However, the real error can be higher due to the difficulties inmeasuring the field values.

To summarize the validation of the numerical model, the possible reasons for the discrepancies are:

(1) inaccuracies in location of measuring points;(2) point velocity measurement errors;(3) errors in modelling the flow;(4) errors in modelling the geometry.

The first category is related to the field velocity measurements taken from a boat. Considering the fact thata boat cannot maintain an absolute fixed position due to the flow velocity and wind, errors are introduced invelocity measurements. A deviation of ±20 cm from the fixed position can be assumed. This deviation couldcause large errors if there is a steep velocity change in the plane of measurements. The magnitude of this errorcould not be estimated accurately. A rough estimate of the nearby velocities within a distance of ±20 cm at themeasuring point gave an error in the range 3–5 per cent.

The second category consists of errors related to the instrument, its volume resolution, the range of operationand the sampling time. The ADCP device could measure instantaneous three-dimensional velocity vectors with5 per cent accuracy. The vertical resolution of the instrument was 0·05 m, which is less than the vertical meshspacing (∆z) used in the numerical model, i.e. 0·084 m. Thus, the instrument resolution error can be ignored.

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The other error source was the velocity range. The lowest measurable velocity was 0·01 m2 s−1. No data existedon the actual value of this error. The vertical velocities (v) in many parts of the model were close to the0·01 m2 s−1 limit. One can conclude that the apparent good agreement between the measured and computed v-velocity component may be erroneous. The randomness of the field data may be due to insufficient sampling time.

The third category is related to the numerical methods (discretization and iteration errors), boundary condi-tions and the closure models. For a carefully modelled problem that has well-posed boundary conditions, theseerrors are relatively low in comparison with other errors.

The fourth category is how the model geometry was built. The modelled geometry was an approximation ofthe river topography as it was based on measurements of discrete cross-sections. The regions between the cross-sections were interpolated by 3-D spline surfaces and may not represent the true topography of the river. A rapidvariation in the topography significantly affects the flow velocity distributions. The spacing used in the presentstudy were selected with special attention to the section properties of the river. However, they may not havecaptured important changes of the river bed.

Based on the above discussion a total error of about 10 per cent (Figure 3a) is a reasonable assumption.However, the real error can be higher if error values could be allocated to the third and fourth categories. Thefact that there is a good agreement with the field data suggests that the modelling efforts have been successful.Regarding the relative importance of the errors, the author believes that misinterpolation of river bed topographyand its roughness properties are the main error sources in the present study.

Sediment transport. To validate the sediment transport model, comparison was made between the com-puted and measured river cross-sections. For this purpose, the river cross-sections were accurately measured atsections C2, W25 and E1 at a discharge of 285 m3 s−1 immediately after the passage of the high flow discharge(750 m3 s−1). Figure 5 shows the results, where the solid curves are field measurements and the points are valuespredicted by the model. The model appears to give a good prediction of the bed level variations. However, theevidence is not conclusive since the bed variations were small and therefore difficult to measure. An importantresult is the fact that the model predicted continuous sedimentation upstream of the river bifurcation and in thewest river channel. The prediction was made for the situation in the year 1997. Six years later, in March 2003,this trend is now confirmed by the appearance of a large sandbank upstream of the river bifurcation. Thesandbank can be observed in Figure 6 (figure orientation could be not changed) in which the various regionsare marked with text. It is about 200 m long and 100 m wide, with a maximum height 0·6 m above the riverbed level recorded in 1977. Further discussion regarding the sandbank is presented in the Results section. Themodel also predicted the classical erosion and deposition patterns observed in river bends. This is illustrated inFigures 11b and 11c.

Considering the simplifications made to model the sediment transport in the river, a reasonable agreementexists with the limited field data. The model has also been successful in predicting the response of the river reachto a high flow discharge. The main error sources in the model are the use of the semi-empirical transportequations, the use of mean sediment sizes which were based on limited soil samples, and the assumption of adepth average two-dimensional sediment transport. To study how the choice of sediment transport equationsaffected the results, additional calculations were made using Rijn’s equations (see Rijn, 1990). A generalincrease of ±15 per cent in the bed elevation was observed. However, due to the semi-empirical nature oftransport equations and the lack of extensive data on sediment transport, stating which equations were moresuitable is not possible. The author believes that the kind of problem reported here will remain as long astheoretically based transport equations are not available.

Grid issues

Three-dimensional grid generation for a complex geometry is not a straightforward task. Many researchershave shown that the grid type has a significant influence upon the results (e.g. Bates et al., 1998; Lane et al.,1999). The main issues are the type of grid, i.e. structured and unstructured grids, grid spacing and gridskewness. False diffusion can occur if the grid lines are not aligned with the flow direction (Patankar, 1980) asin unstructured grids. The amount of false diffusion was reduced by using as many hexahedral elements aspossible and using small mesh elements. The region upstream of the river bifurcation was particularly difficultto mesh due to the flow split and highly irregular river topography. An excessive grid skewness (maximum value



Figure 5. Comparison between the computed and measured cross-section W25. —, field data; • sediment model

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Figure 6. Photograph showing a newly formed large sandbank upstream of the river bifurcation (March 2003)

for 3-D problems is 0·8) can give either unrealistic results or solution divergence. These problems appear inregions where the geometry has sharp angles and the flow depth is shallow. An example is the region markedby the intersection of the river banks with the river bed.

The use of a coarser grid (<105) caused rapid solution divergence. The divergence was caused by growingvalues of the normalized residuals of the pressure term. Converged solutions were obtained for the grid size inthe range 105 to 106. However, the velocity magnitudes were unrealistically high in regions near the bed and theriver banks. Detailed examination of the grid showed high values of skewness (i.e. 1·2) in a few regions. Theproblem was resolved by smoothing the grid. The smoothing process involved a successive increase of the meshnumbers by subdividing the skewed volume meshes into smaller volumes. The upper limit for skewness was setto 0·8, which implies the smoothing operation continued until this limit was reached. The process increased themesh numbers from 105 to 7 × 105.

A grid dependency study was carried out with meshes of 105, 5 × 105, 7 × 105, 106 for each turbulence model.This was done by local mesh refinement using a gradient adaption procedure. The procedure involved successiverefinement of the mesh size in areas of high velocity gradients. The procedure added 105 boundary meshes tothe model. The flow structures, the flow separation lengths and the velocity profiles were affected. Regardingthe flow structure, the coarse mesh (105) did not predict any secondary flow cells in the river cross-sections. Byincreasing the mesh numbers to 5 × 105 some secondary cells became apparent. However, increasing the meshnumbers to106 had no further influence on the numbers and locations of the secondary flow cells. Figure 7illustrates the point in discussion. The velocity contours at section W25 are presented for different mesh numbersand turbulence model. The length of flow separation region at the surface was also affected by mesh refinement.The results presented in Table I indicate an improved agreement with the field observation when the meshnumbers were increased. Mesh refinement also affected the wall shear stress and the velocity profile. The wallshear stresses were changed by ±5% and the shape of the velocity profiles were modified. Near the wall, thevelocity profiles became flatter. Without near wall data, one cannot evaluate the implications of changes in wallshear stresses and velocity profiles. The main results of grid dependence tests are as follows:



Figure 7. Influence of mesh size and turbulence model on the flow structure at section W25. (a) Mesh no. 105, k-ε model; (b) meshno. 5 × 105, k-ε model; (c) mesh no. 5 × 105, RNG model; (d) mesh no. 106, RNG model. The numbers on contour lines are velocities

in m s−1

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Table I. Length of surface flow separation zones. Comparison between k-ε, RNG models and field observations

Flow separation Numerical model Field Relative errorzone Length of the flow separation (m) (1) observtions (m) (2) (%)

Figs 8a, 8b, 8c k-ε model k-ε model RNG model 100*(1–2)/2Mesh = 5 × 105 Mesh = 106 Mesh = 106

S1 92 100 116 125 7·2S2 62 70 79 90 12S3 161 170 186 200 7S4 74 82 93 100 7S5 59 65 80 92 11·3S6 31 33 39 45 13S7 91 102 116 130 10S8 63 65 74 82 9·7S9 140 163 186 200 7S10 187 203 220 245 10

(1) The choice of turbulence model is more important than mesh refinement, provided a reasonably fine meshexists.

(2) Based on the foregoing results and discussion it is a reasonable assumption that the results presented in thisstudy can be considered to satisfy grid independence requirements.

(3) There is also a limit to grid independence tests as the numerical model does not reproduce the exact rivertopography. According to Lane et al. (1999), as the grid size becomes finer, the flow field becomes succes-sively dependent on sampling used to define the topography.

RESULTS

The results obtained for Model A served as input to the Model B, and the present results and discussions arebased on Model B using the RNG model and fine mesh (106).

Flow field

The model successfully reproduced the secondary flows at the river bends besides the recirculation regions.To illustrate some of these features Figures 8a, 8b, 8c, 9a, 9b and 9c are included. The flow field in the riveris characterized by recirculation regions in horizontal plans and secondary flows in cross-sections. Recirculationregions were at the sides of the river and they were the result of turbulent flow separation. Examples are shownin Figures 8a, 8b and 8c, where the contours of the velocity magnitudes are shown in a horizontal plane at thesurface. Ten large flow separation regions marked by letters S1 to S10 were recognized. With the exception ofthe two river bends, the velocities were low in these regions, whereas in the west channel bend the velocitieswere as high as 1·4 m s−1. The higher velocities in the west channel bend were due to higher flow curvature andnarrower cross-sections. To characterize the recirculation regions their streamwise lengths were measured (seeTable I). These lengths were longer in the two river bends than the rest of the river (see Table I). The large flowseparation at the entrance to the west channel, i.e. S3, reduced the effective entrance cross-section by 10 per cent.The decrease in area reduced the portion of flow discharge into the west channel (total discharge constant). Theexamination of the velocity vectors and velocity contours in different horizontal planes showed that the flowseparation regions had a complex three-dimensional form that extended from the water surface to the river bed.The dimensions of the regions decreased as the river bed was approached.

The secondary flows in river cross-sections consisted of multiple counter-rotating spiral motions. Figures 9a,9b, and 9c show the flow structures in some selective cross-sections, in the main channel, west channel and eastchannel, respectively. In these figures the results appear in left and right columns. The left columns are the 3-D model results whereas the right columns are interpolated flow structures. The spiral motions are shown by



Figure 8. Computed velocity contours (3-D model) in (a) main channel and bifurcation, (b) west channel bend, and (c) east channel bend.Numbers on contour lines are velocities in metres per second (S1–S10: flow separation regions)

closed circles with rotation indicated by arrows. The interpolations were based on the velocity vectors and thevorticity components. There are two main findings:

(1) The number of spiral motions is increased in the river bends (Figure 9a: C7b and C7a; Figure 9b: W13; andFigure 9c: E11).

(2) Upstream of the bifurcation, the spiral motions are intensified and their numbers increase (Figure 9a: C4 andC2).

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Upstream of the bifurcation secondary flow regions are induced by the upstream river bend. This secondary flow

diverts the velocity vectors from the right river bank to the left river bank (seen along the main flow direction)

and thereby increases the quantity of discharge that flows into the east channel. The general form of these

secondary flows agrees with those reported in the literature. However, the numbers of spiral motions found in

the present study are higher than those reported in the literature. The existence of two counter-rotating helical

cells in river bends was reported by Ashmore (1982) and Thompson (1986). The present results agree better with

the findings of Richardson (1997). Upstream of a braid bar, he found a single core of primary flow and a number

of weak cells of stress-induced secondary circulation.

The nature of the flow split can be further studied by investigating the distribution of the velocity components

along the thalwegs line (a line that connects the deepest points in the channel). Figures 10a to 10c show the

variation of normalized velocity components at the two planes 0·1 m below the water surface and 0·1 m above

the bed. The mean flow velocity at the inlet (discharge/area) was used to normalize the data. The general features

can be summarized thus:

(1) The magnitudes of the streamwise (u) and cross-stream (w) velocities are comparable.

(2) The vertical velocities (v) increase sharply in both channels downstream of the bifurcation.

(3) In comparison with the west channel, the streamwise velocities are higher in the east channel whereas the

cross-stream velocities are lower.

(4) In both river bends, the vertical velocities sharply increase near the river bed.

(5) A sharp increase in vertical velocities occurs upstream of the river bifurcation.

The higher flow velocities near the river bed is due to the local flow acceleration caused by flow contractions

and curvatures effects.

Sediment transport

The sediment model was used to investigate the general erosion–deposition patterns for a set of given

boundary conditions. The computed sediment transport patterns changed with increasing time steps. During the

initial period, localized sediment transport took place only in the river bends and regions of high bed shear



Figure 9. Left column: computed velocity contours (3-D model) in (a) main channel, (b) west channel, and (c) east channel. Rightcolumn (a–c) shows interpolated flow structures (conceptual model)

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Figure 10. Normalized velocity ratios along the thalwegs line at 0·1 m below the surface and 0·1 m above the bed, 3-D model. (a) u/Um;(b) w/Um; (c) v/Um



stresses. With increasing time steps the rate of transport increased sharply and larger areas were affected. The

result implied that the rate of transport was not uniform during the simulation time (48 h). This is due to the

non-uniform distribution of bed shear stresses and the state of transport. The state of transport depends on

the numerical value of (Y − Ycr) given in Equation 10. Sediment transport will continue until an ‘equilibrium state’

or zero sediment transport is reached, i.e. Y − Ycr = 0 (Equation 10). This implies a dynamic balance between the

rate of sediment transport in the river reach and the supply of sediment at the inlet. It has not been the intention

of the present study to simulate the ‘equilibrium’ state of the transport. The key issue has been the response of

the river to a constant discharge that lasted up to 48 h. The results presented here correspond to the accumulated

sediment transport patterns after 48 h. Figures 11a, 11b, and 11c show the computed sediment transport patterns.

Figure 11. Erosion–deposition patterns in (a) the river bifurcation, (b) the west river bend, and (c) the east river bend. t = 48 h, 2-D sedimentmodel. Units are millimetres, erosion is indicated by a negative sign, and deposition by a positive sign

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In these figures the numbers are in millimetres and erosion and sedimentation are indicated by negative and

positive signs, respectively. The general features are:

(1) Erosion takes place at the entrance to the east channel. The magnitudes are in the range −5 mm to −15 mm.

(2) The sediment transport patterns upstream of the bifurcation are divided into two regions of sedimentation

and erosion (Figure 11a). These regions correspond to west and east channels, respectively. The east channel

in its upper part is more subject to erosion than the west channel.

(3) Upstream of the west channel the outline of a new sand bank is shown (Figure 11a).

(4) The rate of erosion is higher in the west channel bend than the east channel bend.

(5) The first 500 m of the west channel is subject to sedimentation in contrast to the east-channel.

(6) Localized erosion takes place close to the bifurcation section.

The sediment model simulated part of the classical erosion patterns observed in river bends (see Figures 11b

and 11c). The maximum erosion depth was −24 mm, which occurred in the west channel near the concave curve.

A similar trend was found in the east channel bend although the erosion depths were about 30 per cent less

the west channel bend. The considerable increase in erosion activity in the river bends is related to high

bed shear stresses in these regions. Erosion in the region close to the bifurcation section is induced by the sharp

nose of the sandbank that forms the bifurcation. There is an analogy with local erosion upstream of the bridge

piers.

The calculated sediment transport patterns were closely related to the velocity vectors and the distributions

of bed shear stresses. Areas with flow circulations corresponded to sediment depositions. The normalized bed

shear stresses give a general indication of the sediment transport patterns in the river. The computed wall shear

stresses were normalized by use of Shields critical shear stress for the bed materials (see Figure 12). In all

deposition areas the normalized shear stresses were less than unity. In the west channel river, the shear stress

ratios varied in the range 1·7 to 7. Higher values were found at the river bed. The corresponding range in the

east channel was 0·86 to 1·5.

The numerical model suggested the formation of a new large sandbank upstream of the river bifurcation. It

is interesting to compare these results with a recent field observation. The dimensions of this sandbank do not




Figure 12. Contours of normalized wall shear stresses (bed shear stress/critical Shield shear stress) in the lower part of the river reach,3-D model

Figure 13. Comparison between extrapolated 2-D sediment model river cross-section at C2 and the field data in 1997 and 2003

agree with the numerical model results (Figures 6 and 11a). The disagreement was anticipated as the numerical

model is valid for 48 h of simulation. One issue was whether the small sandbank predicted by the model would

continue to grow with time or diminish. To investigate this issue the model was run for additional time steps.

The calculations were stopped when the bed variations were less than ±0·1 mm. This criterion was reached after

288 additional time steps (at 1 h interval). Figure 13 compares the results of calculations at section C2 with field

data in 1977 and 2003. There is a reasonable agreement with the recent field data (2003) that confirms a

continuous growth of the sandbank. However, the agreement is not conclusive since when there is a significant

variation in river bed levels the velocity correction method based on Equation 18 would no longer be adequate

to calculate the rate of sediment transport.

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Figure 14. Variation of normalized vorticity with distance along the thalwegs line, 3-D model

DISCUSSION

Choice of closure model

Using the RNG model here increased the extent of the flow separation regions by 15 per cent (see Table I).

The numbers of secondary vortices were also increased from a single vortex predicted by the k-ε model to

multiply spiral cells (Figure 7). These results agree better with the field observations of the large flow separation

regions, as well as the physics of the secondary flows. The RNG model also improved agreement with the field

velocity data. However, one cannot rule out the possibility of inaccuracies in predicting the flow features in

separated flow regions and regions with strong local flow curvatures. They occur in places where there is a sharp

change in the river topography and the bank slopes. In these flow regions the anisotropy of turbulence affects

the mean flow and higher-order closure models should be implied.

Wall function

The type of wall function did not have a significant influence on the general flow patterns, whereas the

velocities and wall shear stresses were affected. The flow velocities obtained from the non-equilibrium wall

function gave a better agreement with the field data than the standard wall function. The wall shear stresses were

also reduced by 10 per cent when the non-equilibrium wall function was used. This result is not conclusive since

no wall shear stress measurements existed for the river. However, it is reasonable to assume that the non-

equilibrium wall function worked better in the present study.

Secondary flows

The numerical results were used to develop a conceptual model for the development of the secondary flows

near the river bifurcation. The main feature is the existence of multiple counter-rotating spiral motions upstream

of the bifurcation and in the two channels. The number of spiral cells (helical cells) increase as the river bends

are approached. Upstream of the bifurcation, secondary motions are induced by the anisotropic distribution of

wall shear stresses and the unequal approach flow depths. A similar mechanism prevails in the straight parts of

the river. Typical river bend helical cells are observed in the two river bends. The outlines of proposed concep-

tual model for the development of the secondary flows in the river cross-sections are as follows. The schematic

of the model was presented in the right columns of Figures 9a, 9b and 9c. The section numbers refer to these

columns.

• Main channel: Figure 9a

(1) The secondary flow entering Bend 1 (C8) consists of two anti-rotating spiral cells.



(2) The number of spiral cells is increased as the flow turns around the river bend (C7b and C7a). The

additional cells are created in the vicinity of the river slopes.

(3) The influence of river Bend 1 prevails for a about 700 m downstream to section C5. At this section a

flow recovery takes place where the flow resembles section C8.

(4) The number of spiral cells is again increased at section C4 (250 m upstream of the bifurcation) as the

flow enters East Bend and West Bend 1.

(5) A further increase in the number of spiral cells takes place as the flow approaches the river bifurcation

(C2). The cells are reduced in diameter and are pressed more towards the river bed. The flow structure

is due to the combined effect of the bends (i.e., East Bend and West Bend 1) and the rapid change of

in river topography.

• West-channel: Figure 9b

(1) The secondary flow entering the channel at section W27 consists of several anti-rotating spiral

cells.

(2) The number of spiral cells is reduced between sections W25 and W20.

(3) Downstream of section W20, the flow is again influenced by the river bend (West Bend 2).

(4) At the apex of the river bend (West Bend 2) section W13, the number of spiral cells increases

significantly.

(5) Flow recovery takes place at section W7, where there are only two secondary cells (not shown in the

figure).

• East channel: Figure 9c

(1) The secondary flow entering the east channel consists of only two anti-rotating spiral cells (E1). This flow

structure is maintained until section E6.

(2) The number of spiral cells increases as the east bend is approached (E6 and E11).

(3) Flow recovery takes place downstream of section E12 and the number of spiral cells decreases again to

two at section E14.

In comparison with previous studies (e.g. Richardson, 1997; Blanchaert and Graf, 2001), the number of spiral

cells reported in the present study is higher. The conceptual model is also consistent with the ADCP data as

discussed in the ‘Model validation’ section (see also Figure 4).

The vortices were intensified as they were stretched in the streamwise direction in the river bends. The

increase in vorticity is illustrated in Figure 13, where the magnitudes of normalized mean vorticities (in the YZ

plane) are plotted as a function of distance measured along the river boundary. The normalized values were

found by division with the mean vorticity at the river inlet. The vorticity increases as the bifurcation region is

approached. In the west channel bend, the vorticity increases sharply to a maximum value at section W15, and

there after it rapidly decays to the mean value. During the decay period, the loss by viscous action exceeds the

gain by vorticity intensification. In the east channel a similar variation of vorticity is observed, although the

growth and decays of vorticity are slower than in the west channel.

Sediment transport

The secondary flows play an important role in forming the channel perimeter. The influence of classical

spiral motions on the distribution of scour and deposition in river bends is acknowledged in literature. A more

specific feature of the present study is the nature of sediment transport patterns upstream of the river bifurcation.

The flow in this region is dominated by anti-rotating spiral cells. The influence of these cells is evidence from

Figure 9a, sections C4 and C2. The two spiral cells accumulate the sediment into the middle of the channel by

scouring the channel bed at its sides. The channel perimeters are analogous to two back-to-back meanders having

two deep thalwegs and a deposition zone in between.

The sediment transport model simulated only the short-term behaviour of the river in response to the increased

flow discharge during a spring period. The model provided useful results despite the various simplifications.

The main deficiencies of the model, apart from being two-dimensional, are the estimated rates of transport by

use of semi-empirical equations and the inlet loads to the model. During a 48-hour period relatively large

deposition takes place at the entrance to the west channel. This deposition outlines the contours of a new

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sandbank. The sandbank is due to the unfavourable flow field induced by the upstream secondary flows and large

flow separation regions. Upstream of the bifurcation, the spiral motions divert the flow more into the east

channel. The extensive flow separation at the right bank of the west channel contributes further to the deposition.

However, despite the deposition at the entrance to the west channel, more erosion takes place in the lower part

of this channel than in the east channel. By decreasing the portion of flow discharge into the west channel, the

extent of the siltation problem is more likely to increase than decrease. The reduced flow velocities followed

by a sharp drop in discharge will deposit more material into the west channel.

The significant feature of sediment transport upstream of the bifurcation is the growth of a sandbank. The

sandbank is induced by the combined effects of secondary flows close to the bifurcation and the flow field

exerted by the upstream river bend.

Applicability of 3-D models

The study suggests that there are two main limitations to the predictive ability of 3-D models to rivers. One

is a result of the difficulties of specifying the complex variation of the river bed topography and its roughness

characteristics. In practice, building a numerical model that exactly matches the real world is not possible.

Therefore, the numerical model would reflect the deficiencies in the field measurements. Correct estimates of

the surface roughness are needed to model the wall boundaries properly. Without extensive field data, one has

to rely on using various roughness heights in combination with Manning coefficients. For this, good engineering

judgements are necessary to allocate reasonable roughness values to the river. An alternative approach used in

the present study is to first compute the water surface profiles by use of the widely available hydraulic models.

The second step involves a calibration procedure by which the roughness values are adjusted to match

the measured water surface profile. The other limitation is due to the turbulence modelling. The choice of the

turbulence model has a significant influence on the flow details. There are also other limitations imposed by

the numerical methods and procedures. The author believes that the modelling results are more sensitive to the

physical parameters than the numerical parameters. The fact that the model was validated by field data suggests

a good correspondence between the river and the model. The statement is further supported by grid independ-

ence tests and the converged solution. One further problem that limits the application of 3-D flow models to river

flows is the long CPU time needed.

CONCLUSIONS

A 3-D numerical model was successfully applied to the River Klarälven in Sweden. The main flow characteristic

was correctly modelled and the results agreed well with the 3-D field velocities measurements. The use of a

sediment transport model showed that the west channel of the river is subject to sedimentation and the problem

will most likely worsen with time if no engineering measures are taken. The following general conclusions can

be drawn from the present study that should be applicable to similar rivers:

(1) Numerical models are valuable tools to deal with river engineering problems. However, the effectiveness of

these tools depends upon how well the river geometry and its surface roughness are modelled and the choice

of closure model. To improve the modelling of a complex river, the river can be modelled in two parts: one

a coarse grid model of the entire reach and the other a fine grid model of the region of interest. Special care

should be taken to match the correct boundary conditions for the fine model. The use of wall functions to

model the roughness effects would require detailed field data on roughness properties of the river that is

often not available. A compromise is to use composite Manning coefficients based on roughness heights that

account for the various river roughness properties. The optimal roughness heights can then be found by a

calibration process that ensures the water surface profiles match the field data. The choice of closure model

can affect the flow field. The k-ε model under-predicts both the extent of flow separation zones and the

number of vortices in comparison with the RNG k-ε model.

(2) The main flow feature is the existence of multiple counter-rotating spiral motions. The number of spirals

increases as the river bends are approached. The vertical velocities sharply increase near the river bends. The

river bends also cause vorticity intensification.



(3) The nature of a river bifurcation in a delta region and sediment transport patterns are related to secondary

flow fields. Of special importance are the upstream secondary flows that work as input boundary conditions

for the bifurcation. This can cause sedimentation problems at a river bifurcation.

(4) The short-term sediment transport response of a river to a high flow can be studied by solving a 2-D

sediment transport continuity equation in combination with a flow model.

(5) The sediment transport patterns at the river bifurcations are characterized by the growth of a sandbank.

ACKNOWLEDGEMENTS

This research work was supported by the City of Karlstad, represented by the board of planning, Jan-Olof

Seveborg. The writer gratefully acknowledges the financial support of the City of Karlstad and Klas Cederwall

for initiating the research. The author is also thankful to the reviewer for the valuable advice and suggestions

that have improved the manuscript.

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Three-dimensional flow modelling and sediment transport in the River Klarälven