River Sedimentation – Wieprecht et al. (Eds)© 2017 Taylor & Francis Group, London, ISBN 978-1-138-02945-3

Quasi-three dimensional computations for flows and bed variations incurved channel with gently sloped outer bank
T. Sasaki
Graduate School of Science and Engineering, Chuo University, Tokyo, Japan

S. Fukuoka
Research and Development Initiative, Chuo University, Tokyo, Japan

ABSTRACT: Secondary flows in curved channels cause bed scouring around outer banks. It is supposedthat making slope of outer banks gentler is one of useful measures for mitigating scouring and bank erosion
in curved and meandering channels. Therefore flow structures and bed variations around the gently sloped
outer banks have to be understood to design slope of outer banks in actual rivers. However most of models
proposed in curved channels were applied for the channels having a vertical wall. In this paper, applicability ofthe quasi-three-dimensional calculation model (the bottom velocity computation (BVC) method) is discussed
for the experiments on flows and bed variations in curved channels with gently sloped outer banks (1V:2H
and 1V:3H). From the comparison between calculation and experiment results, it is found that the BVC method
reproduces vertical distributions of horizontal velocities and scouring depth around the gently sloped outer banks
and provides a proper design of curved channels with gently sloped outer banks.

1

INTRODUCTION

Flow structures and riverbed topographies in curved
and meandering reaches have been studied by many
researchers. Secondary flows in curved reaches cause
high velocities and bed scouring around outer banks
and deposition at the inner banks. For preventing bed
scouring and bank erosion around outer banks, various measures such as spur dikes, submerged vanes
and footings have been implemented (e.g. Fukuoka &Watanabe 1993, Roca et al. 2007, Abad et al. 2008).
Making the slope of outer banks gentler is oneof measures for outer bank protection (Fukuoka
et al. 1995). Fukuoka et al. (1983) demonstrated
that erodible-curved channels established stable crosssections with gently sloped outer banks. Hence gently
sloped outer bank is supposed to be a useful means for
mitigating bed scours and bank erosions in curved andmeandering reaches.
Flow structures and bed variations around the gently sloped outer banks have to be understood to design
appropriate slope of outer banks in actual rivers.
For the above reason, a numerical model which canestimates three-dimensional flows and bed variations
around the gently sloped outer banks is required.
However, most of the models proposed in curved or
meandering channels (e.g. Engelund 1974, Nishimoto
et al. 1992, Zeng et al. 2008) were applied for thechannels having a vertical wall.
Otsuki et al. (2001) conducted experiments andnumerical simulations for a curved channel withsloped outer banks in order to estimate forces acting on
the side slope for designing bank revetments. Maynord(1992) carried out experiments to study riprap stability by using a straight-meandering channel with gently
sloped banks (1V:2H and 1V:3H). Finnie et al. (1999)
developed the calculation method based on a depthintegrated model that takes into account development
and decline of secondary flows and applied the model
to Maynord’s (1992) experiments.
3D turbulence models were adopted for flows incurved or meandering flumes with gently sloped banks(Ye & McCorquodale 1998, Wang et al. 2009). 3D
turbulence models are useful to investigate threedimensional flow structures in channels. However,it takes much computational time for applying toactual rivers.
Fukuoka et al. (1995) conducted research on theeffects of slope of outer banks on flow structures andbed variation for a uniformly curved laboratory channel by the quasi-three dimensional model (Fukuoka
et al. 1992). Their model is useful for design of equilibrium cross-sectional form in curved channel, but
not of non-equilibrium changes in flow structures andbed of channel with gently sloped outer banks in actual
rivers.
Fukuoka & Uchida (2011, 2013) have developed
a quasi-three dimensional model which can estimate
the bottom velocities and three-dimensional velocities
due to depth-averaged transport and deformation ofvorticities (the Bottom Velocity Computation (BVC)
method).

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In this paper, applicability of the BVC method is
examined to Fukuoka et al.’s (1995) flows and bed variations experiments in the uniformly curved channel
with gently sloped outer banks.

2

NUMERICAL ANAYSIS METHOD

2.1 The bottom velocity computation (BVC) method
The BVC method is a practical numerical model
which can calculate bottom velocities and vertical
distributions of horizontal velocities in the framework of the two-dimensional approach. Since pressure
distributions in uniformly curved channels are hydrostatic except near the wall, hydrostatic distributionsof the BVC method is applied in the present analysis although the BVC methods are used for bothflows with hydrostatic and non-hydrostatic pressure
distributions (Uchida & Fukuoka 2011).
The water depth and depth-averaged horizontal
velocities are calculated by depth-integrated continuity equation (Equation 1) and depth-integrated horizontal momentum equations considering momentum
transport due to vertical distributions of horizontal
velocities (Equation 2).

are obtained by the Equation 5, which is derived bydepth-integrated horizontal vorticity.

where usi = horizontal velocities at water surface;
εij = Levi-Civita symbol; and i = depth average ofhorizontal vorticity.
The BVC method is composed of depth integrated
horizontal vorticity equation (Equation 6), water surface velocity equation (Equation 7) in addition todepth-integrated continuity equations (Equation 1)
and depth-integrated horizontal momentum equations(Equation 2).

where ER = rotational term of vertical distributionof vorticity; Pω = production term of vorticity frombottom thin vortex layer; and Dωij = horizontal vorticity flux due to convection, rotation, dispersion andturbulence diffusion.

where Psi = production term of surface velocities(shear stress under the water surface layer).
The vertical distributions of horizontal velocities
are described by the cubic functional form (Equation 8) using depth-averaged horizontal velocities Ui ,horizontal velocities at water surface usi , and bottom
velocities ubi with the assumption that vertical gradientof velocity at water surface is zero.
where h = water depth; Ui = depth-averaged horizontal velocities; g = gravitational acceleration;
zs = water level; τbi = bed shear stress; τij = horizontal
shear stress due to turbulence; and ui , uj = deviation
velocity components from depth-averaged velocity.
τbi and τij are described as follows, respectively.

where, ubi = bottom velocities; δzb = (e3 − 1)/h; δz0 :the origin height of log law; ks = equivalent roughness height; and νt = eddy viscosity coefficient. νt is
calculated from a local equilibrium model that takes
into account the vertical distributions of horizontal
velocities.
One of the characteristics of the BVC method
is to calculate bottom velocities which are important for bed variation analysis. The bottom velocities

where ui = usi − Ui ; δ = usi − ubi ; and η = (zs − z)/h.
The momentum transport due to the vertical distribution of horizontal velocities in Equation 2 is calculated
by integrated Equation 8 respect to depth. The vertical
distributions of vorticity are defined as the differential
form of Equation 8.

Figure 1. Plan form of the flume.

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2.2

Bed variation computation method

Experiments (Fukuoka et al. 1995) were conductedunder the condition of no suspended sediments. The
bed variation computation took only bed load into
account. The temporal variation in bed elevations areevaluated by the two-dimensional continuity equation
for sediment. The bed load transports rate is calculatedby the Ashida & Michiue bed load formula (Ashida &Michiue 1972). The tractive force is evaluated bybottom velocities solved from Equation (5). The critical tractive force is calculated by Iwagaki formula(τ∗c0 = non-dimensional critical tractive force is 0.034
for d = 0.8 mm of the experiments). Effect of bed slopeon the tractive forces and critical tractive forces of sediment particle are considered by Fukuoka &Yamasaka
equations (Fukuoka & Yamasaka 1983).
3 ANALYSIS TO FLOWS AND BED
VARIATIONS IN A CURVED CHANNELWITH GENTLY SLOPED OUTER BANKS
3.1 Experimental and computational conditions

Figure 2. Initial cross-sectional form of flume.
Table 1.

Experimental conditions (Fukuoka et al. 1995).

Channel length
Channel width
Water discharge
Sediment discharge
Sediment diameter
Bed slopeCentral radius of curveture

24 m
1m
1.8 × 10−2 m3 /s
1.7 × 10−6 m3 /s
0.8 × 10−3 m
1/500
4.5 m

Fukuoka et al. (1995) investigated flow structuresand bed variations for a uniformly curved channel
with gently sloped outer banks as shown in Figure 1 and 2. The experimental conditions are summarized in Table 1. The sand was filled and leveled asshown in Figure 2. The steady discharge and sedimentsupply were given at the upstream end of the channel. After the equilibrium condition of the flow wasestablished, longitudinal water level, bed topographiesand vertical distributions of streamwise velocities andsecondary flow velocities were measured. The experiments were conducted in four cases where the outer

Figure 3. Comparisons of longitudinal water surface profiles and deepest level of channels bed between the experiments(Fukuoka et al. 1995) and the calculation results.

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Figure 4. Comparisons of the depth-averaged velocity distributions along the cross-section between the experiments andcalculations.

Figure 5. Comparisons of vertical distributions of the streamwise velocity (m/s) between the experiments and calculations.
(η = 0 : Water surface, η = 1 : Bed surface).

Figure 6. Comparisons of vertical distributions of the secondary flow/cross sectional average velocity between theexperiments and calculations. (η = 0 : Water surface, η = 1 : Bed surface).

banks have the vertical wall and slope of 1V:2H, 1V:3H
and 1V:4H. BVC method is applied to the case of
1V:2H and 1V:3H experiments.
Steady discharge (1.8 × 10−2 m3 /s) and sediment
discharge (1.7 × 10−6 m3 /s) are supplied at the
upstream end of the channel in the experiments andcalculations. The bed material is a uniform sand of
d = 0.8 mm. The calculation grid size is 0.035 m in thelongitudinal direction and 0.020 m in the lateral direction. The equivalent roughness height ks is given by 3dso as to agree with longitudinal water surface profilesof the experiments.

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Figure 7. Comparisons of cross-sectional form between measurements and calculation results.

3.2

Calculation results

The measurement (Fukuoka et al. 1995) and calculation results of longitudinal water level and deepest
bed level are compared in Figure 3. The calculation
results of the 1V:2H and 1V:3H experiments explain
well longitudinal water level profiles and deepest bed
levels in the reach from 11 m to 13 m where velocity
measurements were conducted. On the other hand, calculated bed levels in the upstream reach are deeper than
measured ones and calculated water levels are slightly
lower than the measured data. This reason would beconsidered as follows: A partition is installed at the
upstream end of the channel to prevent the sand to
be washed away by the discharge inflowing from the
upstream pond. However, it made flows and sediments
transports near the upstream end disturb although thechannel cross-section is uniform. The partition was notconsidered in the present analysis.
Figure 4 shows measured and calculated distributions of the depth-averaged streamwise velocities in
the cross-sections. They are average values of velocities measured at 11.0 m, 12.0 m, 12.5 m and 13.0 m
cross-sections. Vertical distributions of streamwise
velocities and secondary flows are also averaged in asimilar manner. In the 1V:2H experiment (Figure 4(a)),
a high-velocity zone is seen near the outer bank. On
the other hand, the high-velocity zone in the 1V:3H
experiment (Figure 4(b)) shifted toward the channel
center compared with the 1V:2H experiments. Moreover, maximum velocity in the 1V:3H experimentbecome smaller than that of the 1V:2H experiment. It
shows that gentler slope of outer banks makes velocity distributions uniform and decreases velocities near
the outer bank. The calculation results explain well thecharacteristics of velocity distributions in the 1V:2H
and 1V:3H experiments.
Figure 5 compares the results of experimental andcalculated vertical distributions of streamwise velocities (0.1 m, 0.2 m, 0.3 m, 0.4 m and 0.5 m from theouter bank).Vertical distributions of streamwise velocities become uniform near outer banks because large
momentum at the water surface is transported to thebottom by the secondary flows. Calculation results
almost agree with experimental results.
Figure 6 shows comparisons of secondary flow
velocities between the experiments and calculations.
The secondary flow velocities are shown in dimensionless form by cross-sectional averaged-velocities.

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Calculation results reproduce measured data aroundthe outer bank. On the other hand, calculation results in
the inner bank are in rather poor agreement with measured data compared near outer bank. It is caused byformation of sand bars around the inner banks whichwere not seen in experiments (to be discussed later),because secondary flow velocities calculated in the
fixed bed condition made by measured bed topographies almost agreed with measured secondary currents
in the inner bank (Sasaki et al. 2015).
Figure 7 shows measured and calculated equilibrium cross-sections at 9.0 m, 10.0 m, 11.0 m, 12.0 m,
13.0 m and 14.0 m. Brown lines in Figure 7 are initial
cross-sections at each section. As shown by black broken lines of Figure 7 (a) and (b), scouring positions in
the 1V:3H experiment moved towards the center of thechannel and scouring depth decreased compared to the
1V:2H experiment. Calculation results shown by blacksolid lines almost reproduced scouring positions anddepth around the gently sloped-outer banks. However,sand bars which had not be seen in the experiment wereformed around the inner bank in the both 1V:2H and1V:3H calculations. It can be presumed that the sandbars were generated by large sediment transports dueto the excessive scouring around the upstream reach(see Figure 3 (a) and (b)) in the calculation.
These results show that the BVC method can calculate flows and bed variations in curved channels withgently sloped-outer banks.
4

CONCLUSIONSThe bottom velocity computation method (BVC
method) was applied to Fukuoka et al.’s (1995) experiments. The main conclusions derived from this study
are shown below.
1. The BVC method reproduces vertical distributionsof streamwise velocities and secondary flow velocities in the uniformly curved channels with gently
sloped-outer banks (1V:2H and 1V:3H).
2. The BVC method can explain scouring positions
and depth around the gently sloped-outer banks.
3. Above results suggest that the BVC method might
be applied for designing actual rivers having gently
sloped-outer banks.
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