E-proceedings of the 38th IAHR World Congress
September 1-6, 2019, Panama City, Panama

doi:10.3850/38WC092019-1618

ANALYSIS OF FLOOD FLOWS IN MOUNTAIN STREAMS WITH BOULDERS ANDROCKS BY USING NON-HYDROSTATIC QUASI THREE-DIMENSIONAL MODEL (Q3DFEBS)
SHOJI FUKUOKA (1), YOSHIHARU TAKEMURA (2) & JYUNKI OHNO (3)
(1,2)

(3)

Research and Development Initiative, Chuo University, Tokyo, Japan,
takemura@tamacc.chuo-u.ac.jp

Graduate School of Science and Engineering, Chuo University, Tokyo, Japan,
a12.fd7b@g.chuo-u.ac.jp

ABSTRACTThe value of the Manning’s roughness coefficients (= n) for mountain streams varies considerably in time andspace due to the complex riverbed geometry including boulders and rocks. It reduces the reliability of the flood
flow analysis in mountain streams. In this paper, the 2017 flood of the Takiyama River is analyzed by using 2D
flow model and Q3D-FEBS based on observed water surface profiles for the reach where the detailed riverbed
geometry including the shapes of boulders and rocks was measured. The results show that the value of n canbe almost invariably determined from the flood flows and channel geometry by taking account of the threedimensional flow velocities and non-hydrostatic pressure distributions around the boulders and rocks by using
Q3D-FEBS. This makes it possible to estimate flood discharge hydrographs and flow structures from the reliableflood flow analysis in mountain streams.
Keywords: Mountain streams, boulders, flood flows, water surface profiles, Q3D-FEBS

1

INTRODUCTION
Mountain streams are characterized by steep bed gradients and complex riverbed geometry including
boulders and rocks. The flood flow analysis in the mountain streams is necessary for many aspects, such asprediction of flood water levels and velocities, estimation of channel discharge capacity and formation of stable
habitats. Flow resistance in the mountain streams have been studied by many researchers (e.g. Bathurst 1985,
Ferguson 2007, David 2010 and Asano 2016). However, the values of the roughness coefficients (e.g. theManning’s roughness coefficient and the Darcy-Weisbach friction factor) calculated in these studies varyconsiderably not only sites, but also with time dependent water depths and discharges of floods. Most of thestudies used discharges as input data. However, it should be taken into account that the discharge
measurement in the mountain streams is less accurate than the water level measurement because of the violent
flood flows (e.g. Bathurst 1990).
Fukuoka proposed a comprehensive method of flood flow analysis using time series data of observed water
surface profiles. The Fukuoka’s method enables reliable flood flow analysis by numerically solving the flow
equations in such a way as to coincide with the observed water surface profiles that can be accurately observed
at multiple points during floods (Fukuoka 2017a and 2017b). In lowland rivers where the riverbed materials arecomposed of sand and gravel, flood flow characteristics such as discharge hydrographs and velocity
distributions can be estimated with reasonable accuracy from two-dimensional flood flow analysis based on the
Fukuoka’s method. Then, the values of the Manning’s roughness coefficient 𝑛 can be well predicted from theriverbed materials and they are almost invariant with time in many cases. On the other hand, in the mountainstreams, the values of 𝑛 calculated by the Fukuoka’s method are considerably large compared to the riverbed
materials and they vary with time (Tsukamoto et al., 2016 and Ohno et al., 2018a). This is probably because
the flow resistance arising from complex riverbed geometry of the mountain streams that cannot be measured
by normal topographic surveys. If these can be considered in flood flow analysis suitable for mountain streams,
we can reasonably determine the values of 𝑛 and reliable flood flow analysis will be possible based on the
Fukuoka’s method in the mountain streams, too.
Recently, we developed a new non-hydrostatic quasi three-dimensional model (Q3D-FEBS) that
introduced the flow equations on boundary surfaces (the free surface and the bottom surface) (Ohno et al.,2018b). Q3D-FEBS can calculate the three-dimensional velocity and pressure distributions due to the complex
riverbed geometry of the mountain streams with high computational efficiency compared to 3D flow model. This
paper presents the applicability of Q3D-FEBS for estimating the flood discharge hydrographs and flow structures2588

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September 1-6, 2019, Panama City, Panama

in the mountain streams based on the Fukuoka’s method in comparison with 2D flow model, while targeting
flood flows of the reach where the detailed riverbed geometry including the shapes of boulders and rocks wasmeasured.

2

GORVERNING EQUATIONS OF Q3D-FEBS
The definition sketch of Q3D-FEBS is shown in Figure 1. We define the bottom surface 𝑧𝑏 slightly above
the bed surface 𝑧0 . A vertical coordinate is given as 𝜂 = (𝑧𝑠 − 𝑧)/ℎ and approximates vertical distributions of thehorizontal velocities by the third-order polynomial.
𝑢𝑖 = ∆𝑢𝑖 (12𝜂3 − 12𝜂2 + 1) + 𝛿𝑢𝑖 (−4𝜂3 + 3𝜂2 ) + 𝑈𝑖

[1]

Where, 𝑖 = 1, 2 (𝑥1 = 𝑥, 𝑥2 = 𝑦), 𝑢𝑖 : velocity in 𝑖 direction, 𝑈𝑖 : depth-average velocity in 𝑖 direction, 𝑢𝑠𝑖 : water
surface velocity in 𝑖 direction, 𝑢𝑏𝑖 : bottom surface velocity in 𝑖 direction, ∆𝑢𝑖 = 𝑢𝑠𝑖 − 𝑈𝑖 , 𝛿𝑢𝑖 = 𝑢𝑠𝑖 − 𝑢𝑏𝑖 , 𝑧𝑠 : waterlevel, ℎ: water depth. The boundary conditions used in the derivation of Eq. [1] are as follows.
∂𝑢𝑖 ⁄𝜕𝑧 = 0 𝑎𝑡

𝑢𝑖 = 𝑢𝑠𝑖 ,𝑢𝑖 = 𝑢𝑏𝑖1𝑎𝑡𝜂 = 0 (𝑧 = 𝑧𝑠 )

𝜂 = 1 (𝑧 = 𝑧𝑏 )𝑧𝑠

1
∫ 𝑢𝑖 𝑑𝜂 (= ∫ 𝑢𝑖 𝑑𝑧) = 𝑈𝑖
ℎ 𝑧𝑏
0

[2]
[3]
[4]

Figure 1. Definition sketch of Q3D-FEBS.
Q3D-FEBS solves the equations of motion on a free surface (𝜂 = 0) and a bottom surface (𝜂 = 1) in addition
to the depth-integral flow equations shown by Eq. [5] and Eq. [6] to calculate the unknown quantities of Eq. [1].
𝜕ℎ 𝜕𝑈𝑖 ℎ
+
=0
𝜕𝑡
𝜕𝑥𝑖

[5]

𝜕𝑈𝑖 ℎ 𝜕𝑈𝑗 𝑈𝑖 ℎ𝜕𝑧𝑠 1 𝜕ℎ𝑝̅′ 𝑝𝑏′ 𝜕𝑧𝑏 𝜕ℎ𝑢𝑖′ 𝑢𝑗′𝜕𝜕𝑈𝑖 𝜕𝑈𝑗𝜏̂𝑏𝑖) − 𝑆̂
+
= −𝑔ℎ
−
−
−
+
𝜈 ℎ(
+
𝜕𝑡
𝜕𝑥𝑗
𝜕𝑥𝑖 𝜌 𝜕𝑥𝑖𝜌 𝜕𝑥𝑖
𝜕𝑥𝑗
𝜕𝑥𝑗 𝑡 𝜕𝑥𝑗 𝜕𝑥𝑖𝜌[6]Where, 𝜌: density of water, 𝑢𝑖′ = 𝑢𝑖 - 𝑈𝑖 , 𝑔: acceleration of gravity, 𝑝′: non-hydrostatic pressure, 𝜈𝑡 : eddy viscosity,𝜏̂𝑏𝑖 : bottom shear stress in 𝑖 direction, 𝑆̂ = √1 + (𝜕𝑧𝑏 ⁄𝜕𝑥𝑖 )2. Overbar “–“ represents a depth-averaged value.
The form resistance of boulders and rocks can be calculated from the third term of Eq. [6] that represents thenon-hydrostatic pressure received from riverbeds. The non-hydrostatic pressure in Eq. [6] are calculated by Eq.
[7] derived from the depth-integral equation of motion in the vertical direction.
𝑝𝑏′
𝜕𝑊𝜏̂𝑏𝑧 𝑝′ 1 𝑝𝑏′ 𝑈𝑖 ℎ 𝜕(𝑤𝑠 − 𝑤𝑏 )= 𝑈𝑖 ℎ
+ 𝑆̂
, =
+
𝜌
𝜕𝑥𝑖
𝜌 𝜌 2𝜌
12
𝜕𝑥𝑖[7]

Where,𝑊: depth-average velocity in the vertical direction, 𝑤𝑠 : water surface velocity in the vertical direction, 𝑤𝑏 :bottom surface velocity in the vertical direction. 𝑊 is obtained from Eq. [1] and the equation of continuity.
𝑊=

1 𝜕(𝑧𝑠 + 𝑧𝑏 ) 1 𝜕(𝑧𝑠 + 𝑧𝑏 ) 1 𝜕 2 𝛿𝑢𝑖 ∆𝑢𝑖
)
+ 𝑈𝑖
+
ℎ (
+
2
𝜕𝑡
2
𝜕𝑥𝑖
ℎ 𝜕𝑥𝑖
20
10

[8]

𝑤𝑠 and 𝑤𝑏 are given from the kinematic boundary conditions at the water surface and the bottom surface.
𝑤𝑠 =

𝜕𝑧𝑠
𝜕𝑧𝑠
𝜕𝑧𝑏
𝜕𝑧𝑏
+ 𝑢𝑠𝑖
, 𝑤𝑏 =
+ 𝑢𝑏𝑖
𝜕𝑡
𝜕𝑥𝑖
𝜕𝑡
𝜕𝑥𝑖

[9]
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Eq. [10] and Eq. [11] are the equations of motion on the water surface (𝜂 = 0) and the bottom surface (𝜂 = 1).
𝜕𝑢𝑠𝑖
𝜕𝑢𝑠𝑖
𝜕𝑧𝑠 1 𝜕𝑝′ 𝜕𝑧𝑠 𝜈𝑡𝑠 𝜕 2 𝑢𝑖
|
|
+ 𝑢𝑠𝑗
= −𝑔
+
+
𝜕𝑡
𝜕𝑥𝑗
𝜕𝑥𝑖 𝜌 𝜕𝑧 𝑠 𝜕𝑥𝑖𝜌 𝜕𝑧 2 𝑠𝜕𝑢𝑏𝑖
𝜕𝑢𝑏𝑖𝜕𝑧𝑠 1 𝜕𝑝𝑏′ 1 𝜕𝑝′ 𝜕𝑧𝑏𝜕𝜕𝑢𝑏𝑖 𝜕𝑢𝑏𝑗𝑆̂ 𝜏̂𝑏𝑖 − 𝜏̂0𝑖
|
)+
+ 𝑢𝑏𝑗
= −𝑔
−
+
+
𝜈𝑡𝑏 (
+
𝜕𝑡
𝜕𝑥𝑗
𝜕𝑥𝑖 𝜌 𝜕𝑥𝑖 𝜌 𝜕𝑧 𝑏 𝜕𝑥𝑖 𝜕𝑥𝑗
𝜕𝑥𝑗
𝜕𝑥𝑖
𝜌 𝛿𝑧𝑏

[10]
[11]

Where, 𝜏̂0𝑖 : bed shear stress in 𝑖 direction, 𝛿𝑧𝑏 = 𝑐𝑧𝑏 ℎ, 𝑐𝑧𝑏 = 0.03. The shear stress and pressure at the water
surface are assumed 0. The gradient of the non-hydrostatic pressure in the vertical direction at the water surface
and the bottom surface are calculated from Eq. [12] derived from the equations of motion in the vertical direction
on the water surface and the bottom surface.
1 𝜕𝑝′
𝜕𝑤𝑠
| = −𝑢𝑠𝑖,𝜌 𝜕𝑧 𝑠
𝜕𝑥𝑖

1 𝜕𝑝′
𝜕𝑤𝑏
| = −𝑢𝑏𝑖𝜌 𝜕𝑧 𝑏
𝜕𝑥𝑖[12]

The bottom shear stress and the bed shear stress are evaluated by Eq. [13] and Eq. [14], respectively.
𝜏̂0𝑖 = 𝜌𝑐𝑏2 𝑢𝑏𝑖 |𝒖𝒃 |，𝜏̂ 0𝑧 = 𝜌𝑐𝑏2 𝑤𝑏 |𝒖𝒃 |𝜏̂𝑏𝑖 = 𝜈𝑡𝑏

𝑐𝑏 =

[13]

𝜕𝑢𝑖
𝜏̂ 0𝑧
| ，𝜏̂ 𝑏𝑧 =𝜕𝑧 𝑏
𝑐𝑧𝑏 + 1𝐶0
√1 + 𝑐𝑧𝑏 ,1 − 2𝐶0⁄𝜅

[14]

𝑔𝑛2
𝐶0 = √ 1⁄3ℎ[15]

Where, 𝜕 2 𝑢𝑖 ⁄𝜕𝑧 2 |𝑠 = (−24∆𝑢𝑖 + 6𝛿𝑢𝑖 )/ℎ2 , 𝜕𝑢𝑖 ⁄𝜕𝑥|𝑠 = (−12∆𝑢𝑖 + 6𝛿𝑢𝑖 )/ℎ , 𝑛 : the Manning’s roughness
coefficient, 𝜅 = 0.41. The eddy viscosities are assumed constant along a vertical direction (𝜈𝑡 =𝜈𝑡𝑠 = 𝜈𝑡𝑏 ) andgiven as follows.
11
𝜈𝑡 = 𝜅𝑢∗ ℎ[16]
6√1 + 𝑐𝑧𝑏
3

USEFULNESS OF Q3D-FEBS FOR FLOOD FLOW ANALYSIS IN MOUNTAIN STREAMS WITH
BOULDERS AND ROCKS
3.1 Studied reach and flood
The Takiyama River is located in Hiroshima Prefecture, Japan. The studied reach is 4.58 km to 6.3 km (theKurotaki observation station) upstream from the Nukui Dam (catchment area of 253km2) in the Takiyama River(see Figure 2). The average riverbed gradient is about 0.02. The detailed riverbed geometry including theshapes of boulders and rocks was measured by aerial photography survey using a drone. In addition, pressure
type water level gauges have been installed at the point of ◯ in Figure 2. They were fastened to very large
rocks as shown in Figure 3.
Figure 4 shows aerial photographs of Area 1, Area 2 and Area 3 shown in Figure 2. Boulders and rocks of
1 to 3 m in height are distributed on the riverbed. Figure 5 shows the inflow and outflow discharge hydrographsof the Nukui Dam and the water level hydrograph in the reservoir observed during the 2017 flood in the Takiyama
River. The 2017 flood is a record-breaking flood after the dam construction. The water level gauges installed at
4.9 km and 5.8 km were unfortunately washed away during the flood, but we could obtain water level data atother points. We performed flood flow analysis with target to Jul. 2:00 to Jul. 17:00 in Figure 5.
Elevation (T.P.m)
4.58k

Area 34.9k (missing)

5.8k (missing)

Area 2
Area 1

5.46k

Kurotaki（6.3k）

direction of flow

Figure 2. Location and observation system of the studied reach.
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Water level gauge
Water level gauge(a) 4.9 km(b) 5.46 km

Figure 3. Water level gauges at 4.9 km and 5.46 km points.

6.0km

direction of flow

6.2km

About 3m in height and about
5m in longitudinal length(a) Area 1

5.0km
5.2km

5.4km

direction of flow(b) Area 2(c) Area 3

Figure 4. Aerial photographs of Area 1, Area 2 and Area 3 shown in Figure 2.

(T.P.：Tokyo Peil)

Figure 5. Inflow and outflow discharge hydrographs of the Nukui Dam and water level hydrograph in thereservoir observed during the 2017 flood of the Takiyama River.
3.2 Computational gridRiverbed geometry of the studied reach was measured at 30 cm intervals by the aerial photographic survey.
Figure 6 shows the cross-sectional shapes at 5.4 km point based on the normal topographic survey at 200 m
intervals and the aerial photography survey. The normal water level and the maximum water level for the 2017
flood at 5.46 km are shown in Figure 6. The riverbed elevations of the aerial photographic survey tend to behigher than that of the normal topographic survey at a location lower than the normal water level. On the other
hand, riverbed elevations of these surveys show good agreements at a location above the normal water level.
This indicates that the aerial photographic survey may overestimate the riverbed elevation under the water
surface. However, the shapes of boulders and rocks projecting from the water surface can be correctly
measured by the aerial photographic survey.

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Figure 6. Cross-sectional shapes by normal topographic survey and aerial photographic survey at 5.4 km
point.
Q3D-FEBS can consider both skin-friction and form drag of boulders and rocks by incorporating theirshapes into the computational grid. Therefore, computational grid size is an important factor to estimate the flow
resistance. We used computational grid size of 1 m with reference to our previous study (Ohno et al., 2018b).
The riverbed elevations of the computational grid were given by averaging the aerial photographic survey data
within each grid cell. Figure 7 shows comparison of the 3D riverbed elevation contours by the aerial photographic
survey and the computational grid. The computational grid can almost reproduce the shapes of boulders androcks of several meters in size, which are considered to be the main source of flow resistance in the studied
reach. The flow resistance due to the small roughness which cannot represent the shape by the computational
grid is evaluated by the Manning’s roughness coefficient 𝑛.

Kurotaki (6.3 km)

Kurotaki (6.3 km)

direction of flow(a) Computational grid.

direction of flow(b) Aerial photographic survey.

Figure 7. 3D riverbed elevation contours by aerial photographic survey and computational grid.
3.3 Usefulness of Q3D-FEBS for flood flow analysis in mountain streams with boulders and rocks
At first, we analyzed the 2017 flood in the Takiyama River based on the time series data of the observed
water surface profiles in the studied reach by using following two calculation methods.
(1) Conventional 2D method: a computational grid is created from the normal topographic survey at 200 m
intervals and 2D flow model is used for flood flow analysis.
(2) Present method: a computational grid is created in the manner described in 3.2 and Q3D-FEBS is used forflood flow analysis.
The boundary conditions at upstream and downstream ends are given by the observed water level
hydrographs at the Kurotaki observation station (6.3 km) and at 4.58 km point, respectively. Figure 8 shows thecomparison of the observed and calculated water surface profiles in the rising period of the 2017 flood. Figure
9 shows comparison of calculated discharge hydrographs at the Kurotaki observation station (6.3km) and the
inflow discharge hydrograph of the Nukui Dam. The calculations results can almost explain the observation
results. In Figure 8 (b), the water levels calculated by the present method tends to be slightly higher than theobserved data at 5.46 km point. It is probably because that the riverbed elevations of the computational gridaround 5.46 km point may contain errors as pointed out in 3.2 since there are few boulders and rocks projectingfrom the water surface (see Figure 4 (b)).

25
92E-proceedings of the 38th IAHR World Congress
September 1-6, 2019, Panama City, Panama(a) Conventional 2D method.

(b) Present method.

Figure 8. Observed and calculated water surface profiles in rising period of the 2017 flood.

Figure 9. Calculated discharge hydrographs at the Kurotaki observation station (6.3 km) and inflow dischargehydrograph of the Nukui Dam for the 2017 flood.
The values of the Manning’s roughness coefficient 𝑛 calculated by the conventional 2D method and thepresent method are compared in Table 1. They are determined in a way as to explain the temporal changes inobserved water surface profiles and the inflow discharge hydrograph of the Nukui Dam as mentioned above. In
the conventional method, the values of 𝑛 are in the range of 0.04 to 0.072, and it is necessary to change thevalues over time in order to coincide with observed water surface profiles. On the other hand, in the present
method, 𝑛 has a constant value of 0.032, which is a reasonable value from the engineering sense about
mountain streams. It can be noticed that the water surface profiles calculated by the present method shown in
Figure 8 (b) are clearly affected by the longitudinal mean riverbed profiles in comparison with the conventional
2D method (Figure 8 (a)). This indicates that observation errors of the mean riverbed elevations have influenceson the calculated 𝑛 values. In addition to this, as discussed on the following, the flow structures in the mountainstreams are three-dimensional due to the influence of complex riverbed geometry including boulders and rocks.
These are the reasons why the 𝑛 value of the conventional method varies in time and space. Although such aproblem remains, the most important fact is that the flood discharge hydrographs by the conventional 2D method
as well as the present method can be calculated in a sense that the flood flow analysis based on the time series
data of the observed water surface profiles as shown in Figure 8 and Figure 9 is performed.
Table 1. The Manning’s roughness coefficient 𝑛 calculated by the conventional 2D and present methods.
2:00 – 6:00 7/5
0.06

6:00 – 9:00 7/5

9:00 – 12:00 7/5

6.4 – 5.6 km

0.06 → 0.072

0.072 → 0.06

12:00 – 15:00 7/5
0.06

Conventional
method

5.6 – 5.25 km

0.04

0.04 → 0.048

0.048 → 0.04

0.04

5.25 – 4.5 km

0.06

0.06 → 0.072

0.072 → 0.06

0.06

Present
method

6.4 – 4.5 km

0.032

In the following, the applicability of 2D flow model and Q3D-FEBS to the estimation of flow structures inmountain streams is discussed. For this purpose, we analyze the 2017 flood by the following method.
(3) 2D method: a computational grid is created in the manner described in 3.2 and 2D flow model is used forflood flow analysis.
Figure 10 show the comparison of calculation results by the present method and the 2D method in the flood
peak for the reach of 6.0 km to 6.2km where the boulders and rocks in several meters are densely distributed.
Figure 11 and Figure 12 are calculated velocity distributions by the present method and the 2D method in A-A’
and B-B’ cross-sections shown in Figure 10 (a). In the surrounding of the large boulders and rocks seen in2593

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Figure 10 (a), non-hydrostatic pressure head at the bottom is more than 1 m in absolute value and the velocity
distributions become three-dimensionally complicated (see Figure 10 (b) and Figure 12). These threedimensional effects cannot be considered in the 2D method. As a result, the calculated water levels andvelocities vary extremely around the large boulders and rocks as shown in Figure 10 (c), (e) and Figure 11. On
the other hand, the present method shows the plausible results as shown in Figure 10 (d), (f) and Figure 12.
Especially, Figure 12 indicates the tendency that velocities near the riverbeds are reduced under the influenceof the non-uniformly distributed boulders and rocks and the flow velocities near the water surface increase. Thisis consistent with the results reported in the previous study (Bathurst 1990). These results indicate that the flow
structures in the mountain streams are three-dimensional due to the presence of large boulders and rocks.
Therefore, the three-dimensional flow structures such as velocity distributions, local water levels and nonhydrostatic pressure distributions cannot be represented by 2D flow model even if the detailed geometry is taken
into account. The three-dimensional flow structures in the mountain streams can be estimated by performing
Q3D-FEBS based on the observed water surface profiles and the detailed riverbed geometry including theshapes of boulders and rocks. It will also extend the discussion on the sediment transport, riverbed variations,and formations of stable habitats in the mountain streams.
6.0k

6.0k

Elevation (T.P.m)

Elevation (T.P.m)※lines show elevation at 0.2 m intervals

B

A
6.2k

6.2k

B’

A’

Non-hydrostatic pressure head (m)direction of flow

direction of flow(b) Non-hydrostatic pressure head at bottom(present method).

(a) Riverbed geometry.
6.0k

6.0k

Elevation (T.P.m)

Elevation (T.P.m)

6.2k

6.2k

Water level (T.P.m)

Water level (T.P.m)
direction of flow

direction of flow(d) Water level (present method).

(c) Water level (2D method).
6.0k

6.0k

Elevation (T.P.m)

Elevation (T.P.m)
5m/s

5m/s

6.2k

6.2k

Depth average velocity (m/s)

Depth average velocity (m/s)
direction of flow(e) Depth average velocity (2D method).

direction of flow(f) Depth average velocity (present method).

Figure 10. Calculation results of 2D method and present method in the 2017 flood peak for the reach of 6.0
km to 6.2km.

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Elevation (T.P.m)

Elevation (T.P.m)

Velocity (m/s)

Velocity (m/s)

Transverse distance (m)Transverse distance (m)(a) A-A’ cross-section (2D method)(b) B-B’ cross-section (2D method)

Figure 11. Velocity distributions by 2D method in A-A’ and B-B’ cross-sections shown in Figure 10 (a).
Elevation (T.P.m)

Elevation (T.P.m)

Velocity (m/s)

Velocity (m/s)

Transverse distance (m)(a) A-A’ cross-section (present method)

Transverse distance (m)(b) B-B’ cross-section (present method)

Figure 12. Velocity distributions by present method in A-A’ and B-B’ cross-sections shown in Figure 10 (a).

4

CONCLUSIONSIn this paper, the usefulness of Q3D-FEBS for the flood flow analysis based on the time series data of theobserved water surface profiles for the mountain streams with boulders and rocks was investigated from thecomparison with 2D flow model. Main conclusions are shown below.
1.

Flood discharge hydrographs in the mountain streams can be approximately estimated by the conventional
2D method using the normal topographic survey at 200 m intervals if the flood flow analysis is performed
based on the times series data of the observed water surface profiles. In this case, the Manning’s
roughness coefficient has considerably large values compared to the riverbed materials and the values
vary with time.

2.

Flow structures such as velocity distributions and local water levels in the mountain streams are threedimensionally complicated due to the influence of the complex riverbed geometry including boulders androcks and that cannot be represented by 2D flow model even if the detailed riverbed geometry is taken into
account.

3.

The flood discharge hydrographs and three-dimensional flow structures in the mountain streams canbe estimated from the reliable flood flow analysis by using Q3D-FEBS, detailed riverbed geometry andthe observed water surface profiles. Then the roughness coefficients can be determined reasonably
from the flood flows and channel geometry.

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