E-proceedings of the 36th IAHR World Congress
28 June – 3 July, 2015, The Hague, the Netherlands

DYNAMICS OF FLOOD FLOWS AND BED VARIATIONS IN RIVER SECTIONS REPAIREDTO SHIP-BOTTOM SHAPED CHANNELS FROM COMPOUND CHANNLSTAKUMA SASAKI(1) & SHOJI FUKUOKA(2)(1)

Graduate School of Science and Engineering, Chuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo, 112-8551, Japan,
e-mail t-sasaki@civil.chuo-u.ac.jp(2)

Research and Development Initiative, Chuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo, 112-8551, Japan,
e-mail sfuku@tamacc.chuo-u.ac.jp

ABSTRACT
Compound channels have caused some undesirable problems on flood controls and river environments in recent years.
In order to deal with these problems, one of the authors has proposed that the ship-bottom-shaped channel is a desirable
cross-section of rivers for both flood controls and river environments. The Onga River in Japan challenged to change apart of channel sections to ship-bottom shaped channels from compound channels in the view of river environments.
However, there have been no ideas how much the ship-bottom shaped channels improve the discharge capacity andchannel stability against large floods. A large flood occurred in 2010 after completion of the ship-bottom shaped channel
in the Onga River. In this study, effects of river improvements to ship-bottom shaped channels were investigated andclarified by comparing observed data and flood computation results. Velocity distributions of flood flows are improved andlongitudinal bed variations decreased markedly in the ship-bottom-shaped channels compared to compound channel.
Keywords: ship-bottom shaped channel, compound channel, flood flow, bed variation, cross-sectional shape1.

INTRODUCTION

Channel dredgings and riverbed excavations in compound channels have been conducted for the increase in flood
discharge capacity. However, excessive channel dredgings and excavations have made level of the low-water channel
drop, which have resulted in unstable river bank and excessive growth of vegetation on flood channels. Therefore, it isimportant to do research on stable river channels which are desirable for flood controls and river environments. In theprevious studies, there are a lot of regime equations to describe relations between river width, water depth and discharge
on natural stable alluvial rivers (e.g. Singh (2003)). Ikeda et al. (1986) proposed relationship between river width, water
depth and channel forming discharge in straight channels based on sediment hydraulics consideration. However, theseequations couldn’t explain mechanics of width of stable rivers which have various scales, plain forms and grain sizes(Fukuoka (2010)).
Fukuoka (2010, 2012) proposed dynamic relation equations (Fukuoka’s equations) between dimensionless river width,
dimensionless depth and dimensionless channel forming discharge derived by dimensional analysis. Moreover, Fukuoka
proposed river improvement technique using Fukuoka’s equations and numerical simulations of flood flows and bed： Ship-bottom shaped channel (18.4km～20.2km)： Water level and discharge observation station

Akaike： Water level observation station

6.0km
4.0km

Hikosan River

Hinodehashi Nakasima

N

2.0km
16.0km14.0km18.0km

26.0kmKanrokuhashi

24.0km28.0km

22.0kmKarakuma

Inunaki River

Onga River
30.0km32.0kmKawashima
Figure 1 Objective are and observation stations1

E-proceedings of the 36th IAHR World Congress,
28 June – 3 July, 2015, The Hague, the Netherlands

variations in the Satsunai River where the channel was narrowed and eroded by a series of spur dikes and vegetation
growing. This study proved that cross-sectional shapes of dynamic stable channels determined by his river improvements
method became similar to the ship-bottom shape which can be seen in the natural stable alluvial rivers (Fukuoka (2013)).
These cross-sectional shapes are described as follows. The river bed profiles have continuous and gradual changes of
bed boundary and river widths continuously change in correspondence of water level rising. In this study, we call such across-sectional bed profiles as the ship-bottom shaped channel. Gotoh et al. (2014) presented the method of river bankprotection for preserving tidal flats in ship-bottom shaped channels of the Ota River floodway. However, effects of river
improvements from compound channels to ship-bottom shaped channels have not really been evaluated in actual rivers.
The Onga River was repaired from compound channels to ship-bottom shaped channels for river environments (seeFigure 1, 2 and 3). Figure 2 shows cross-sectional bed profiles of the ship-bottom shaped channels at 19.6km and19.8km. We can see in these figures that cross-sectional bed profiles have continuous and gradual changes in bed
boundary (Higuchi et al. (2007)). After the completion of the ship-bottom shaped channels, we have experienced a large
flood in July 2010.

Elevation (T.P.m)

The objective of our study is to clarify effects of the river improvements from compound channels to ship-bottom shaped
channels. Therefore, we conducted the quasi-three-dimensional flow analysis and bed variation analysis in order tounderstand dynamics of flood flows and bed variations in ship-bottom shaped channel. Flood flows and bed variations in
the compound channel before the river improvement works are compared with those in ship bottom shaped channels byusing the calculation results and Fukuoka’s equations.

12

Ship-bottom shaped channel

Flood marks (2010’s flood)

Compound channel

Design high water level

Design high water level

10
8
6
4
2
0

50

100

150

200

150

200

Elevation (T.P.m)

Lateral distance (m)(a) 19.6 km
12Design high water level

10
▽

▽

8
6
4

2
0

50

100

Lateral distance (m)(b) 19.8 km
Figure 2 Cross-sectional profiles of ship-bottom shaped channel at 19.6km and 19.8km

Flow
Figure 3 Ship-bottom shaped channel in the Onga River

2.

Objective area and flood

2.1 Objective area and characteristics of ship-bottom shaped channel
Figure 1 shows a plan form and observation stations in the objective study area. River improvements to the ship-bottom
shaped channels were conducted near the confluence with the Hikosan River. Revetments of the main channel were2

E-proceedings of the 36th IAHR World Congress
28 June – 3 July, 2015, The Hague, the Netherlands

removed for the purpose of river environments and river-landscape (see Figure 3). The cross-sectional shapes of thecompound channel were repaired so that the flood channels have mild slopes longitudinally and laterally (see Figure 3).
Figure 4 shows contour-lines of the flood channel in the ship-bottom shaped channel. The flood channel of the left bank
was designed to have longitudinal and lateral undulations for considering visitor’s behavior and river-landscape (Higuchi
et al. (2007)).

The Hikosan RiverThe Onga River

Flood channel
19.8km

19.6km
19.4km

19.2kmFigure 4 Undulations on flood channel of the Onga River

Figure 5 shows comparisons between cross-sectional bed profiles in stable straight channels described by Eq. [1] and theship-bottom shaped channels at 19.6km and 19.8km. This equation was derived on the basis of experiments of a straightchannel with bank erosions (Fukuoka and Yamasaka (1984))  b yh b y 2b 
 1  exp   exp   exp  
H
D 
D 
 D  [1]Where h: water depth, b: half of river width, y: distance from the center of channel, H and D: coefficients which are
determined cross-sectional area and lateral slope of riverbed. From these figures, cross-sectional bed profiles of the shipbottom shaped channels in the Onga River seem to be described by Eq. [1].

Ship-bottom shaped channel (after river improvements)
Compound channel (before river improvements)

Height (m)Eq. [1] (Fukuoka and Yamasaka’s equation)

0
-120

-100

-80

-60

-40

-20

0
-5
-10

Height (m)Lateral distance (m)(a) 19.6 km (left bank)

5
0
-120

-80

-40

0

40

80

-5
Lateral distance (m)(b) 19.8 km
Figure 5 Comparison between cross-sectional shape of stable straight channel and ship-bottom shaped channel

2.2 Objective flood
After the completion of improvement works, large-scale flood occurred in July 2010. For evaluating effects of changes toship-bottom shaped channel, we used the time series data of the water levels measured at five observation stations anddrainage pump stations (see Figure 1). Flood marks were also obtained and almost comparable with design high waterlevel (see Figure 2). Discharge hydrographs were observed at five observation stations. Figure 6 shows the flood3

E-proceedings of the 36th IAHR World Congress,
28 June – 3 July, 2015, The Hague, the Netherlands

discharge hydrographs observed by floats and H-Q curves at Hinodebashi (18.7km) observation station. The flood
discharge hydrographs had three peaks. The flood duration time was about three days. The peak discharge recorded at3
about 3000 m /s.
Discharge observed by H-Q curves

Discharge observed by floats

3500

Discharge (m3/s)

3000
2500

2000
1500
1000
500
0
2010/7/12 0:00

2010/7/13 0:00

2010/7/14 0:00

2010/7/15 0:00

Time (hour)
Figure 6 Discharge hydrographs at Hinodebashi observation station3.

Calculation method

3.1 Procedure of calculation
In recent years, Fukuoka (2011) proposed general method calculating flood flows and bed variations using time series
data of water surface profiles during a flood. The calculation method is based on the idea that influences of crosssectional forms, plane forms and bed variations during floods appear clearly in the observed temporal changes in thewater surface profiles. Moreover, Uchida and Fukuoka (2011) developed the quasi-three-dimensional flow analysis
method (the Bottom Velocity Computation (BVC) method) which can estimate vertical velocity distributions and bed
surface velocities. And, Fukuoka (2011) has clarified characteristics of flood flows and bed variations in various rivers by
the calculation method using both observed temporal changes in water surface profiles and the BVC method.
It is necessary to clarify three-dimensional structures of flood flows and stability of river beds in the ship-bottom shaped
channel in order to evaluate effects of river improvements. First, the BVC method is applied to clarify flood flows and bed
variations in the ship-bottom shaped channel of the Onga River in July 2010’s flood. Second, effects of the ship-bottom
shaped channels are investigated by comparing velocity distributions and bed variations in the ship-bottom shaped
channels with those of compound channels, which was channel before the river improvement works.
3.2 Governing equations
The BVC method (Uchida and Fukuoka (2011)) enable to estimate vertical velocity distributions and bed surface
velocities by calculating Eq. [2]~[6]. The bed surface velocity is obtained by the Eq. [2], which is derived by depth-integral
horizontal vorticity.
ubi  usi   j h[2]

Where ubi: bed surface velocity, usi: water surface velocity, Ω: depth-averaged vorticity and h: water depth.
To calculate Eq. [2], it is required to solve the depth-integral continuity equations (Eq. [3]), depth-integral horizontal
momentum equations (Eq. [4]), the depth-integral horizontal vorticity equations (Eq. [5]) and water surface velocity
equations (Eq. [6]).

h U j h

0
t
x j

h ijU i h U iU j hz  gh s  bi 
t
x j
xi

x j

[3]
[4]

Where, Ui : depth-averaged horizontal velocity, g: gravitational acceleration, zs: water level, τbi: bed shear stress, τij:
horizontal shear stress due to turbulence and vertical velocity distribution.

hDij i h ERi  P i 
t
x j

[5]

Where ER: rotational term of vertical velocity, Pw: production term of vorticity from bottom thin vortex layer and Dwij:
horizontal vorticity flux due to convection, rotation, dispersion and turbulence diffusion.

4E-proceedings of the 36th IAHR World Congress
28 June – 3 July, 2015, The Hague, the Netherlands

u si
u
z
 u sj si   g s  Psi
t
x j
xi

[6]

Where Psi: the production term of surface velocity (shear stress under the water surface layer).
The variations in river bed elevations are evaluated from two-dimensional continuity equation for sediment transport. The
bed load transport rate of each grain size is calculated by Ashida and Michiue formula (Ashida and Michiue, 1972). The
critical tractive forces for each particle size and mean particle size are calculated by modified Egiazaroff formula (Ashida
and Michiue, 1972) and Iwagaki formula (Iwagaki,1956), respectively. Effects of bed slope on tractive forces and criticaltractive forces of sediments particles are evaluated by Fukuoka and Yamasaka equations (Fukuoka and Yamasaka(1986)). The transports of suspended sediments are computed by two-dimensional advection and diffusion equations.
The vertical distributions of suspended load, pickup rate of suspended load from river bed and sedimentation velocity arecalculated by using Lane-Kalinske formula (Lane and Kalinske, (1941)), Kishi and Itakura formula (Itakura and Kishi(1980)), Rubey formula (Rubey,1933), respectively.
3.3 Calculation conditions
3.3.1 After river improvements (the ship-bottom shaped channel)
Boundary conditions at the upstream and downstream ends of the Onga River are given by observed water-level
hydrographs at Kawashima (30.5km) and Karakuma (13.7km) observation stations shown in Figure1. The upstream
boundary condition of the Hikosan River is given by observed water level hydrograph at Akaike (7.5km) observation
station. Initial conditions of bed forms are the cross-sectional bed forms measured at intervals of 200 meters in 2009. The
undulations contour lines of flood channels shown in Figure4 are drawn by using cross-sectional bed profiles measured atintervals of 50 meters. The Manning’s roughness coefficients and vegetation permeability coefficients are determined so
as to agree with observed temporal changes in water surface profiles during 2010’s flood. These coefficients are shown in
Table1 (a). Figure7 shows grain size distributions of observed data and calculation.
Table 1 The Manning’s roughness coefficient and vegetation permeability coefficient of objective rivers(a) After river improvements
Manning roughness coefficient (m-1/3・ s)

main channel

flood channel

Vegetation permeability
coefficient (m/s)

Onga River (12.6km~18.8km,19.4km~31.2km)

0.029

0.04～0.043

30～70

Hikosan River (0.0km~7.8km)

0.029

0.040

30～70

Near the confuluence

0.033

0.040

25～70

Section(b) Before river improvements
Manning roughness coefficient (m-1/3・ s)

main channel

flood channel

Vegetation permeability
coefficient (m/s)

Onga River (12.6km~18.8km,19.4km~31.2km)

0.029

0.04～0.043

30～70

Hikosan River (0.0km~7.8km)

0.029

0.040

40～70

Near the confuluence

0.030

0.040

30～70

Volume percentage finer than (%)

SectionCalculation

16.0km25.2km

13.0km

20 .0km

29.4km

1

10

100
90
80
70
60

50
40
30
20
10
0

0.01

0.1

100

Grain size (mm)Figure 7 Particle grain size distributions of observed data and used for the calculation

3.3.2 Before river improvements (the compound channel)
Boundary conditions at the upstream end of the Onga River and Hikosan River are given by discharge hydrographs
determined above the calculation at Kawashima (30.5km) and Akaike (7.5km) observation stations. Boundary condition at
downstream end is given by observed water level hydrographs at Karakuma (13.7km) in 2010 flood. Initial bed profiles5E-proceedings of the 36th IAHR World Congress,
28 June – 3 July, 2015, The Hague, the Netherlands

are made by cross-sectional profiles measured in 2002. Table 1 (b) shows the Manning’s roughness coefficients andvegetation permeability coefficients. These coefficients are determined so as to reproduce time series of observed water
surface profiles during 2003’s flood.
4.

Calculation results

4.1 Calculation results of the ship-bottom shaped channel
Figure 8 shows comparisons between observed and calculated temporal changes in water surface profiles and mean bed
elevation in the Onga River. The calculated water surface profiles and mean bed elevations almost agree with observed
ones. However, the observed water levels at Hinodebashi (18.7km) observation station are higher than the calculated
ones due to water level increased by bridge pier. From observed data and calculated results shown in Figure 8, changes
in mean bed variations in the ship-bottom shaped channel are understood to be small.
Flood marks(right bank)Observed water level

Flood marks(left bank)Calculated water surface profiles

Elevation (T.P.m)

Initial average bed elevations

17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0

Average bed elevations after flood

Average bed elevations after flood

Kawashima 7/14 10:00
7/14 07:00

Hikosan River
confluence

7/14 04:00
7/14 01:00

Kanrokubashi
Hinodebashi
KarakumaShip-bottom shaped channel
13.5

15.5

17.5

19.5

21.5

23.5

25.5

27.5

29.5

Longitudinal distance (km)Figure 8 Comparisons between observed and calculated water surface profiles and mean bed elevation

Figure 9 shows observed and calculated discharge hydrographs at Hinodebashi (18.7km), Kanrokubashi (19.8km) andNakashima (1.2km) observation stations. In this figure, our numerical method can explain observed dischargehydrographs at three observation stations, although calculated peak discharge at Hinodebashi (18.7km) observation
station is lower than observed data.
Hinodebashi
Kanroubashi
Nakashima

Solid line：Calculated discharge
Plot：Observed discharge

3500

Discharge (m3/s)

3000

2500
2000
1500
1000
500
0
2010/7/12 0:00

2010/7/13 0:00

2010/7/14 0:00

2010/7/15 0:00

Time (hour)
Figure 9 Discharge hydrographs at Hinodehashi observation station
6

E-proceedings of the 36th IAHR World Congress
28 June – 3 July, 2015, The Hague, the Netherlands

Figure 10 shows the contour map of water depth and the temporal changes in shore lines of the ship-bottom shaped
channel in the Onga River and compound channel in the Hikosan River. The water surface width of compound channels
in the Hikosan River quickly spreads to the levee from 06:00 to 08:00, but that of ship-bottom shaped channels in theOnga River gradually spreads with water level increasing. Lateral and longitudinal undulations on the flood channel in the
ship-bottom shaped channel create divers changes in the shore lines and present beautiful landscape during the flood.
Water depth (m)7/13/4:00

CompoundchannelCompoundchannelCompoundchannelShip-bottom
shaped channel

Ship-bottom
shaped channel

7/13/6:00

Ship-bottom
shaped channel

7/13/8:00

Figure 10 Temporal changes in shore lines

4.2 Comparison hydraulic characteristics between the ship-bottom shaped channel and the compound channel
Fukuoka (2010, 2012) has indicated that the relations between dimensionless cross-sectional profiles and dimensionlessdischarge in stable alluvial rivers are descried by Eq. [7], [8] (Fukuoka’s equation).


 Q 
2.80
5
 gId 
r 


0.40



 Q 

 6.33
5
 gId 
dr
r 


0.40

B

 Q 
h

 0.14
 gId 5 
dr
r 


[7]

0.38

[8]

Where Q = channel-forming discharge, B = river width, h = water depth, g = gravitational acceleration, I = energy gradientand dr = representative grain diameter. We compare relationships between dimensionless river width, dimensionlessdepth and dimensionless channel forming discharge in the ship-bottom shaped channel with those in the compoundchannels before the river improvements. Figure 11 shows temporal changes in the relationships between dimensionlesswidth, dimensionless depth and dimensionless channel-forming discharge at 19.6km and 19.8km in the flood rising period.
Upper limit
Average line

Ship-bottom shaped channel (after river improvements)
Compound channel (before river improvements)

Lower limit

106

106

105
B
dr

105
B
dr

104

104

103
h
dr

103
h
dr

102
1010

1011

1012

Q

102
1010

1011

1012

Q
gId

5
r

(a) 19.6km

gId

5
r

(b) 19.8km

Figure 11 Relationships between dimensionless width, dimensionless depth, and dimensionless channel-forming discharge

7E-proceedings of the 36th IAHR World Congress,
28 June – 3 July, 2015, The Hague, the Netherlands

From these figures, the dimensionless width of the compound channel (yellow solid line) becomes smaller than the lower
limit line (red solid line) around the bank-full condition of the main channel and rapidly changes with increasing discharge.
On the other hand, the dimensionless width of the ship-bottom shaped channel (green solid line) increases in almost
parallel to average line (black solid line) with increasing dimensionless channel-forming discharge. It is expected fromabove results that cross-sectional profiles of ship-bottom shaped channels make velocity distribution within a crosssection improve and a stable channel against flood flows.
Figure 12 shows longitudinal distributions of cross-sectional area at the peak discharge in the Onga River. At the river
sections from 19.6km to 20.2km, the cross-sectional area of the ship-bottom shaped channels are larger than those of
the compound channels. Therefore, discharge capacity of the improved river channels increases from that of the previous
channels.
Ship-bottom shaped channel (after river improvements)
Compound channel (before river improvements)

Cross-sectional area (m2)2000

Hikosan River confluence

1800
1600
1400
1200
1000

800
18.4

18.6

18.8

19

19.2

19.4

19.6

19.8

20

20.2

Longitudinal distance (km)Figure 12 Longitudinal changes in cross-sectional area

Next, we investigate three-dimensional structures of flood flows and bed variations in the ship-bottom shaped channel
and the compound channels. Figure 13 shows contour lines of velocity distributions in cross-sections at 19.6km and19.8km at the peak discharge. From these figures, it is found that maximum velocity of the ship-bottom shaped channels
is smaller than that of the compound channels. And, lateral velocity gradient between main channel and flood channel ofthe ship-bottom shaped channels is milder than that of the compound channels. In compound channels, momentum
exchanges between high-velocity flow in the main channel and low-velocity flows in the flood channels occur and increase
hydraulic resistances. Therefore, hydraulic resistances of ship-bottom shaped channels are found to be relatively smaller
than those of the compound channels. Moreover, velocities near embankment of the ship-bottom shaped channel remain
low velocities in the peak discharge.

Ship-bottom shaped channel(after river improvements)

Main flow velocity (m/s)

Compound channel(before river improvements)(a) 19.6 km

Ship-bottom shaped channel(after river improvements)

Compound channel(before river improvements)(b) 19.8 km
Figure 13 Main flow velocity contours in cross-sections8

E-proceedings of the 36th IAHR World Congress
28 June – 3 July, 2015, The Hague, the Netherlands

Figures 14(a) and (b) show contour lines of longitudinal bed variations near the confluence of the Onga River and the
Hikosan River after the flood. Bed variations in the ship-bottom shaped channel shown in Figure 14(b) become less thanthose in compound channel shown in Figure 14(a).

Hikosan River

Bed variation(m)18.8kmDeposition

19.0km

19.2km
19.4km20.2km
19.6kmOnga River

19.8km20.0km(a) Compound channel (before river improvements)

Scouring

Hikosan River

18.8km19.0km

19.2km
19.4km20.2km

19.6kmOnga River

19.8km20.0km(b) Ship-bottom shaped channel (after river improvements)
Figure 14 Bed variations contour lines in the main channels after the flood

Figure 15 shows longitudinal distributions of bed load. Longitudinal changes in bed load at the peak discharge and rising
period in the ship-bottom shaped channels (shown by solid lines) become smaller and more uniform than those ofcompound channels. Therefore, it is concluded that river bed of the ship-bottom shaped channel improved in the Onga
River are relatively stable against flood flows.
Peak discharge
Rising period

Solid line : Ship-bottom shaped channel (after river improvements)
Broken line : Compound channel (before river improvements)

Falling period

Amount of bed load (m3/s)

0.04

0.03

0.02

0.01

0
19

19.2

19.4
19.6
19.8
Longitudinal distance (m)

20

20.2

Figure 15 Longitudinal distributions of bed load

9

E-proceedings of the 36th IAHR World Congress,
28 June – 3 July, 2015, The Hague, the Netherlands

5.

CONCLUSIONS

Effects of river improvements to the ship-bottom shaped channels from the compound channels are investigated byapplying the BVC method using observed water surface profiles in the Onga River. Moreover, Calculation results of shipbottom shaped channels are compared with those of the compound channels. The following conclusions were derived inthis study.
(1)

From the relationships between dimensionless width, dimensionless depth and dimensionless channel-forming
discharge in the Onga River, the ship-bottom shaped channels have desirable cross-sectional bed profiles satisfying
Fukuoka’s equations which accord with cross-sectional profiles in stable natural rivers.

(2)

We clarified three-dimensional structures of flood flows in the ship-bottom shaped channels and the compoundchannels by developed numerical model. From these results, it was found that velocity distributions of flood flows in
the ship-bottom shaped channels become more uniform. Therefore, hydraulic resistances of the ship-bottom shaped
channels are less than those of the compound channels.

(3)

Longitudinal changes in bed load transports in the ship-bottom shaped channels become smaller than those incompound channels. Ship-bottom shaped channels improve markedly hydraulic characteristics compared to thoseof compound channels.

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