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

doi:10.3850/38WC092019-0706

DEVELOPMENT AND APPLICATION OF CALCULATION METHOD FOR AMOUNT OFSUSPENDED SEDIMENT ENTRAINMENT UNDER NON-EQUILIBRIUM CONDITIONSOF FLOWS AND SEDIMENT TRANSPORTS
TAKAHISA GOTOH (1), SHOJI FUKUOKA(2)(1,2)

Research and Development Initiative Chuo University 1, Tokyo, Japan,
e-mail goto510@tamacc.chuo-u.ac.jp

ABSTRACT
Large amounts of suspended sediments are transported by flood flows in sandy rivers. The suspended
sediment concentration is generally calculated by the three-dimensional advection diffusion equation
assuming that the flow and sediment transport near the bed are in equilibrium conditions. In this study, thequasi-3D flood flow and bed variation calculation model is developed by considering non-equilibrium twophase motion of flow and sediment near beds. The developed model is applied to the experiment of local
scouring around a cylinder by Fukuoka et.al. (1997). The calculation model is able to explain the bed scouring
in front and side of the cylinder compared with the conventional model which the flow and sediment transport
is in equilibrium conditions. It is examined that how the thickness of the bed load layer influences theestimation of flow and bed variation.
Keywords: Non-equilibrium flow and sediment movement, Quasi-3D flow analysis, Turbulent intensity, Bed load layer,
Suspended sediment

1

INTRODUCTION
Large amounts of suspended sediments are transported by flood flows in sandy rivers. It is necessary toappropriately calculate suspended sediments entrainment and transportation during a flood. Shugar et.
al.(2010) investigated effects of flow turbulence near beds produced by sand dunes on the sedimententrainments in sandy rivers. McLean (1994) et al. measured in detail the turbulence structures over thedunes in the experimental channels. They showed that vertical components of the turbulence increased on thestoss slope and lee of the sand dunes and it induced sediment entrainments.
In general, the flood flow and sediment transport in sandy rivers has been calculated by the two-dimensional
flood flow and bed variation calculation model. The most of these models assumed the equilibrium bed load
movement. And the suspended load was calculated by the two-dimensional advection diffusion equation. The
vertical distribution of the suspended sediments was assumed to be in an equilibrium condition (Lane-Kalinske,
1941, etc.). The amount of sediment entrainments was estimated by using equilibrium sediment concentrationnear the beds.
Van Rijin (1984) evaluated equilibrium sediment concentration near the bed by the representing the relation
between shear stress, saltation height and sediment concentration by a dimensionless parameter. Itakura et al.
(1984) derived a vertical equation of motion of sand particles on the bed, taking into account the fluid forces
caused by turbulence in the equilibrium conditions. And they evaluated the vertical velocity of the sand particle
at the moment when the particle leaves from beds. The sediment entrainment was evaluated by using thevertical particle velocity. Nakagawa et al. (1986) defined suspended sediments when sand particles havetransitioned from saltation motion to random motion due to turbulence, and calculated probabilistically the rateof the transition. However, those studies assumed that flow and sediment transport near the bed were in
equilibrium conditions. The assumption of the equilibrium condition is not valid during a flood.
Chauchat et al. (2017) proposed the 3D calculation model which treated the flow and sediment transport as a
two-phase motion. This calculation model was able to calculate the bed load and suspended load in a unified
manner without the assumption of the equilibrium conditions near beds. However, it is difficult to apply thiscalculation model in the widespread area of rivers because this model has to perform three-dimensional
calculation in consideration of interfered action between the flows and sediments.
Uchida and Fukuoka (2014) developed the quasi-3D flood flow calculation model which was the depth
integrated model and was capable of calculating three-dimensional flow in a widespread area of rivers.
However, their calculation model could not sufficiently evaluate the suspended sediments due to turbulencedeveloped near the beds, although the suspended sediment concentration is calculated by a threedimensional advection diffusion equation using the three-dimensional flow velocity.
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E-proceedings of the 38th IAHR World Congress
September 1-6, 2019, Panama City, Panama

In this study, a quasi-3D flood flow and bed variation calculation model based on the depth integrated model
was developed to evaluate sediment entrainments and movements by considering non-equilibrium two-phasemotion of flow and sediment near beds.
2

FLOW AND BED VARIATION CALCULATION MODEL CONSIDERING NON-EQUILIBRIUM
TWO-PHASE MOTION OF FLOW AND SEDIMENT IN BED LOAD LAYER

Figure 1. Definition of variables in the calculation model.

Figure 2. Frame work of the calculation model.
2.1 Framework of the calculation model.
Figure 1 shows the definition of variables in the calculation model and Figure 2 shows the framework of thecalculation model. The calculation region was divided into three layers, bed load layer, transition layer andmain flow layer. In bed load layer which was the lowest layer, the flow and the sediment transport were
calculated by treating as the two-phase motion. The transition layer which was the second lowest region
connected with the main flow and the bed load layer. And, three-dimensional flow in the main flow layer was
calculated by the quasi-3D calculation model (the GBVC method, Uchida & Fukuoka, 2014). The arrows in
Figure 2 indicate the exchange terms of mass and momentum of fluids and the shear stress terms acting oneach layer in the equations.
The three-dimensional flows in the main flow layer and pressure distribution near the beds were calculated
by applying the GBVC method. The surface velocities of the transition layer uti were evaluated by depth
integrated definition of vorticity in Eq.[1], in which were calculated by the depth integrated continuity Eq.[2], thedepth integrated momentum Eq.[3] and the depth integrated vorticity equations.
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September 1-6, 2019, Panama City, Panama

𝜕𝑊ℎ
𝜕𝑧𝑠
𝜕𝑧𝑡
𝑢𝑡𝑖 = 𝑢𝑠𝑖 − ℰ𝑖𝑗3 Ω𝑗ℎ − (
− 𝑤𝑠
+ 𝑤𝑡
)
𝜕𝑥𝑖
𝜕𝑥𝑖
𝜕𝑥𝑖
𝜕ℎ
𝜕𝑡

+

𝜕𝑈𝑗 ℎ
𝜕𝑥𝑗

[2]

− 𝑤𝜎𝑡 =0𝑓
𝜕𝑈𝑖 ℎ 𝜕𝑈𝑖 𝑈𝑗 ℎ𝜕𝑧𝑠 1 𝜕∆𝑝ℎ ∆𝑝𝑡 𝜕𝑧𝑡 𝜏𝑡𝑖 1 𝜕ℎ𝑇𝑖𝑗+− 𝑢𝑡𝑖 𝑤𝜎𝑡 = −𝑔ℎ
−
−
−
+
𝜕𝑡
𝜕𝑥𝑗
𝜕𝑥𝑖 𝜌 𝜕𝑥𝑖𝜌 𝜕𝑥𝑖
𝜌 𝜌 𝜕𝑥𝑗′ 𝑢 ′ − 2𝜌𝜈 𝑆 ,̅̅̅̅̅̅̅
Τ𝑖𝑗 = −𝜌𝑢
𝑖 𝑗
𝑡 𝑖𝑗

[1]

𝜕𝑈𝑗
1 𝜕𝑈𝑖
+
)
2 𝜕𝑥𝑗
𝜕𝑥𝑖

𝑆𝑖𝑗 = (

[3]

[4]

Where i,j=1,2 (x,y direction), Ui is depth averaged flow velocity in i direction, h is water depth, uti is thesurface velocity of the transition layer in i direction, usi is water surface velocity in i direction, W is depth
averaged vertical flow velocity, j is depth averaged vorticity in j direction, wt is vertical flow velocity of thetransition layer, zs is water level, zt is surface elevation of the transition layer, Δpt is the non-hydrostaticpressure component of the pressure at the surface of the transition layer, tt is kinematic eddy viscosity
coefficient on the surface of transition layer, is density, 𝜏𝑡𝑖𝑓 a is shear stresses acting on the bottom of the main
flow. The shear stresses were evaluated in Eq.[5] by the vertical gradients of velocity distributions in Eq.[6].
The velocity distribution in the vertical direction of the transition layer and the bed load layer shown in Eq.
[6] were determined as the third order polynomial by calculating the surface velocities of the transition layer𝑓𝑓uti , the bed load layer velocity 𝑢1𝑖 and the transition layer velocity 𝑢2𝑖 . And the vertical velocity distribution in
the main flow was also assumed the third order polynomial in Eq.[7]. It was determined by using the mean
velocity Ui, the water surface velocity usi, the surface velocities of the transition layer uti and the verticalgradient of velocity at the water surface.
𝑓

𝑓

𝑓

𝜏𝑡𝑖
19𝑢𝑡𝑖 + 8𝑢2𝑖 − 24𝑢1𝑖
= 𝜈𝑡𝑡
𝜌
6𝛿𝑏
𝑢𝑖 (𝜂𝑏 ) =

1
1
𝑓
𝑓
𝑓
𝑓
(16𝑢𝑡𝑖 + 32𝑢2𝑖 − 32𝑢1𝑖 )𝜂𝑏 3 − (16𝑢𝑡𝑖 + 56𝑢2𝑖 − 40𝑢1𝑖 )𝜂𝑏 2
3
3
1
𝑓
𝑓
+ (3𝑢𝑡𝑖 + 24𝑢2𝑖 − 8𝑢1𝑖 )𝜂𝑏3

𝑢𝑖 (𝜂) = 𝑈𝑖 + ∆𝑢𝑖 (12𝜂 3 − 12𝜂 2 + 1) − 𝛿𝑢𝑖 (4𝜂 3 − 3𝜂 2 )

[5]

[6]

[7]

Whereui is i direction flow velocity of any height, Δ𝑢𝑖 = 𝑢𝑠𝑖 − 𝑈𝑖 , δ𝑢𝑖 = 𝑢𝑠𝑖 − 𝑢𝑡𝑖 , zs-z)/h, bz/b, b isthickness of the bed load layerThe thickness of the bed load layer b is set to times the diameter of the bed
materials. Since the thickness of the bed load layer is originally determined by the interaction between the flow
and the sediment movements near the beds, it is inappropriate to make the thickness of the bed load layer
constant. Therefore, the effects of the differences in the thickness of the bed load layer on the calculation
results of flows and bed variations were examined in section 4.
The flow and sediment transport in the bed load layer were calculated as two phase motion. Thosevelocities in the bed load layer were estimated by the Reynolds-averaged continuity equations and horizontal
momentum equations taking into account the interaction between flows and sediment movements. And,turbulence intensities in the horizontal and vertical directions were computed by their transport equations. The
amount of sediment entrainments was evaluated by the vertical momentum equation of the sediment particles
considering fluid forces caused by the Reynolds-averaged velocity and the turbulence components in thevertical direction.
The sediment transports in the transition layer and main flow layer were calculated as suspended sediments
by a three-dimensional advection diffusion equation. The amount of sediment entrainments from the bed load
layer was given as the bottom boundary condition of the equation. And, bed variations were calculated by thecontinuity equation of bed load and suspended load.
2.2 Non-equilibrium two-phase motion of flow and sediment calculation model near the bed
The equations of motion and the continuity equation of flow in the transition layer are expressed by Eq. [8]and Eq. [9], respectively. And, the equation of motion and the continuity equation of flow in the bed load layer
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September 1-6, 2019, Panama City, Panama

are also expressed by Eq. [10] and Eq. [11], respectively. Here, the horizontal shear stress term in Eq. [8] andEq. [10] are ignored on the assumption that those layer thicknesses are thin. In addition, the non-hydrostaticpressure component of pressure pt is assumed to be uniformly in the vertical direction in the transition layer
and the bed load layer.
𝑓

𝑓

𝑓

𝑓

𝑓

𝜕𝑢2𝑖 𝛿𝑏 𝜕𝑢2𝑖 𝑢2𝑗 𝛿𝑏𝜕𝑧𝑠 1 𝜕∆𝑝𝑡 𝛿𝑏 𝜏𝑡𝑖 𝜏1𝑖
𝑓
𝑓𝑓++ 𝑢2𝑖 𝑤𝜎𝑡 − 𝑢2𝑖 𝑤𝜎1 = −𝑔𝛿𝑏
−
+
−
𝜕𝑡
𝜕𝑥𝑗
𝜕𝑥𝑖 𝜌 𝜕𝑥𝑖
𝜌
𝜌

[8]

𝑓

𝜕𝑢2𝑗 𝛿𝑏𝜕𝑥𝑗

𝑓− 𝑤𝜎𝑡 + 𝑤𝜎1 = 0
𝑓

[9]

𝑓

𝑓
𝜕(1 − 𝐶1 )𝑢1𝑖 𝛿𝑏 𝜕(1 − 𝐶1 )𝑢1𝑖 𝑢1𝑗 𝛿𝑏𝑓++ (1 − 𝐶1 )𝑢1𝑖 𝑤𝜎1
𝜕𝑡
𝜕𝑥𝑗
𝑓
𝑓
𝜕𝑧𝑠 1 𝜕∆𝑃𝑡 𝛿𝑏 𝜏1𝑖 𝜏0𝑖 𝑓𝑖 𝛿𝑏= −𝑔𝛿𝑏 (1 − 𝐶1 )−+−−𝜕𝑥𝑖 𝜌 𝜕𝑥𝑖
𝜌
𝜌
𝜌

[10]

𝑓

𝜕𝑢1𝑗 (1 − 𝐶1 )𝛿𝑏𝑓− 𝑤𝜎1 (1 − 𝐶1 ) = 0
[11]
𝜕𝑥𝑗
𝑓
𝑓
Where 𝑢1𝑖 and 𝑢2𝑖 denotes velocities in the bed load layer and the transition layer in i direction, respectively.
𝑓And, 𝑤𝜎1 denotes the flow velocity in the direction perpendicular to the bed load layer. The shear stress acting𝑓𝑓on the surface of the bed load layer 𝜏1𝑖 and the shear stress acting on the bottom of the bed load layer 𝜏0𝑖 areexpressed by Eq. [12] and Eq. [13] using the vertical distribution of the flow velocities in the transition layer
and the bed load layer expressed by Eq. [7]. t1, t0 are kinematic eddy viscosity coefficient on the surface of
bed load layer and near the bed, respectively. The sediment concentration C1 in the bed load layer is set to a
constant value of 0.6 which was around maximum value of the bed load (Van Rijin, 1984).
𝑓

𝑓

𝑓

5𝑢𝑡𝑗 − 8𝑢2𝑖 + 8𝑢1𝑖
𝜏1𝑖
= 𝜈𝑡1
𝜌
6𝛿𝑏
𝑓

[12]

𝑓

𝑓
3𝑢𝑡𝑗 + 24𝑢2𝑖 − 8𝑢1𝑖
𝜏0𝑖
= 𝜈𝑡0
[13]
𝜌
6𝛿𝑏
The horizontal particle velocity of the bed load is determined by the equation of motion shown in Eq. [14].
The hydrodynamic forces acting on the sand particles 𝑓𝑖 are evaluated as drag forces shown in Eq. [15] and𝑠
the drag coefficient is evaluated by Eq. [16] (Clift et.al, 1978). The shear stress 𝜏𝑐𝑖
to which the bed load is
subjected from the bottom of the bed load layer is evaluated by Eq. [17] using the dynamic friction coefficient𝑠
𝜇𝑘 . Here, 𝑢1𝑖
represents the particle velocity in i direction of the bed load layer. 𝑅𝑒 is Reynolds number, d isdiameter of sediment particle, υ is kinematic viscosity coefficient. And, 𝐴2 , 𝐴3 are a two-dimensional and threedimensional shape factor, respectively.
𝑠 𝑠𝑠𝑢1𝑗 𝛿𝑏
𝜕𝐶1 𝑢1𝑖
𝛿𝑏 𝜕𝐶1 𝑢1𝑖𝑠++ 𝐶1 𝑢1𝑖
𝑤𝜎1
𝜕𝑡
𝜕𝑥𝑗
𝑠
𝜌𝑔𝛿𝑏 𝜕𝑧𝑠 1 𝜕∆𝑃𝑡 𝛿𝑏 (𝜌𝑠 − 𝜌)𝑔𝛿𝑏 𝜕𝑧𝑏 𝜏𝑐𝑖
𝑓𝑖 𝛿𝑏
=−
−
−
−
+
𝜌𝑠 𝜕𝑥𝑖 𝜌𝑠 𝜕𝑥𝑖𝜌𝑠𝜕𝑥𝑖 𝜌𝑠𝜌𝑠𝑓𝑖 =𝐶𝐷 (𝑅𝑒 ) =

24
𝑅𝑒

1 𝐴2
2
2
2
𝑓
𝑓
𝑓
𝑓
𝑠 √
𝜌𝐶1 𝐶𝐷 (𝑅𝑒 )(𝑢1𝑖 − 𝑢1𝑖) (𝑢1 − 𝑢1𝑠 ) + (𝑣1 − 𝑣1𝑠 ) + (𝑤1 − 𝑤1𝑠 )2 𝐴3 𝑑1
1
𝐴2 = 𝜋, 𝐴3 = 𝜋
46𝑓(1 + 0.15𝑅𝑒 0.687 ), 𝑅𝑒 =
𝑠
𝜏𝑐𝑖
=

1801

[14]

|𝑈1 |𝑑
𝜐

𝑓

𝑓

2

𝑓

2

𝑓

[15]

2

, |𝑈1 | = √(𝑢1 − 𝑢1𝑠 ) + (𝑣1 − 𝑣1𝑠 ) + (𝑤1 − 𝑤1𝑠 )𝜇𝑘 (𝜌𝑠 − 𝜌)𝑔𝛿𝑏 𝑢1𝑖,|𝑈1 |𝜇𝑘 = 0.4

[16]

[17]

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

The bed elevation 𝑧𝑏 can be expressed by Eq. [18] from the relationship between difference in inflow andoutflow of the bed load 𝑞𝑏𝑗 and the amount of the sediment entrainment 𝑞𝑠𝑢 and the sedimentation by
assuming the thickness and sediment concentration in the bed load layer to be constant. Here, 𝑞𝑏𝑗 is bed loadrate in the j direction. The sedimentation amount is expressed by using the falling velocity 𝑤𝑠 of the sedimentparticles and the sediment concentration 𝑐𝑠2 of the suspended load in the transition layer.
𝜕𝑧𝑏 𝜕𝑞𝑏𝑗𝑠+= 𝑤𝑠 𝑐𝑠2 − 𝑞𝑠𝑢 , 𝑞𝑏𝑗 = 𝐶1 𝑢1𝑗
𝛿𝑏
[18]
𝜕𝑡
𝜕𝑥𝑗
The amount of the sediment entrainment from bed per unit area and unit time is calculated by the Eq. [19]using the sediment concentration in the bed load layer and the particle velocity in the vertical direction. The
vertical particle velocity is calculated from the vertical equation of motion (Eq. [20]) for the particles taking into𝑓account turbulence in the bed load layer. The flow velocity 𝑢1𝑖 in Eq. [20] is divided into the two, the Reynolds
𝑓𝑓averaged value 𝑢̅1𝑖 and the turbulence components 𝑢′1𝑖 . It is assumed that the distribution of the components
𝑓of the turbulence 𝑢′ 𝑖 follows a normal distribution, and that a flow velocity value above 25% is important forthe sediment entrainment from the beds. The Eq. [20] is simplified to Eq. [21] by the assumption.
(1 − 𝜆)𝑞𝑠𝑢 = 0.25𝐶1 𝑤1𝑠 , 𝑤1𝑠 > 0
𝐴3 (𝜌𝑠 + 𝐶𝑀 )𝜌𝑠 d3

𝑑𝑤1𝑠

[19]

1

𝑓

𝑓

= 𝐹𝑧 + 𝐹′𝑧 − 𝐴3 (𝜌𝑠 − 𝜌)𝑔d3 = 𝐴2 𝐶𝐷 𝜌d2 (𝑤̅1 + 𝑤 ′1 − 𝑤1𝑠 )・

𝑑𝑡

2

[20]

2
2
2
𝑓
𝑓
√(𝑢̅1𝑓 + 𝑢′1𝑓 − 𝑢1𝑠 ) + (𝑣̅1𝑓 + 𝑣 ′ 1𝑓 − 𝑣1𝑠 ) + (𝑤
̅1 + 𝑤 ′1 − 𝑤1𝑠 ) − 𝐴3 (𝜌𝑠 − 𝜌)𝑔d3

1

𝑓

𝑓

≈ 𝐴2 𝐶𝐷 𝜌d2 (𝑤̅1 − 𝑤1𝑠 + 0.67√̅̅̅̅̅̅̅
𝑤 ′ 𝑤 ′1 )・
2

2

2

2

𝑓
𝑓
𝑓
′ 𝑢 ′ 𝑓 + ̅̅̅̅̅̅
√(𝑢̅1𝑓 − 𝑢1𝑠 ) + (𝑣̅1𝑓 − 𝑣1𝑠 ) + (𝑤̅̅̅̅̅̅
̅1 − 𝑤1𝑠 ) + 0.45(𝑢𝑣 ′ 𝑣 ′1 + ̅̅̅̅̅̅̅
𝑤 ′ 𝑤 ′1 ) − 𝐴3 (𝜌𝑠 − 𝜌)𝑔d3
1[21]

The suspended sediment transport is calculated by the advection diffusion equation of the sediment in themain flow layer and the transition layer in Eq. [22] and in Eq. [23].
𝜕𝐶𝑠 ∆𝑧 𝜕𝐶𝑠 𝑢∆𝑧 𝜕𝐶𝑠 𝑣∆𝑧 𝜕𝐶𝑠 𝑤∆𝑧
𝜕
𝜕𝐶𝑠
𝜕
𝜕𝐶𝑠
𝜕
𝜕𝐶𝑠
+
+
+
=
(𝜈𝑡 ∆𝑧)+(𝜈𝑡 ∆𝑧) + (𝜈𝑡 ∆𝑧
)
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑥
𝜕𝑥
𝜕𝑦
𝜕𝑦
𝜕𝑧
𝜕𝑧
𝑓

[22]

𝑓

𝜕𝐶2 𝛿𝑏 𝜕𝐶2 𝑢2𝑖 𝛿𝑏 𝜕𝐶2 𝑣2𝑖 𝛿𝑏𝑓+++ 𝐶2 𝑤1𝑖
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕
𝜕𝐶2
𝜕
𝜕𝐶2
𝜕
𝜕𝐶2
[23]
=
(𝜈 𝛿
)+
(𝜈 𝛿
) + (𝜈𝑡𝑡
) + 𝑞𝑠𝑢 − 𝐶2 𝑤𝑠
𝜕𝑥 𝑡2 𝑏 𝜕𝑥
𝜕𝑦 𝑡2 𝑏 𝜕𝑦𝜕𝑧𝜕𝑧
The horizontal and vertical components of the turbulence intensity in the bed load layer and the transition
layer are calculated by the transport equations of the turbulence intensity shown in Eq. [24]. Here, the kinetic
eddy viscosity coefficients used for the calculation of the shear stresses in Eq. [12] and Eq. [13] are evaluatedby the Eq. [26] using the turbulence intensity. The dissipation rate of the turbulence is evaluated by Eq. [26],and yp is the dimension of length.
𝑓′ 𝑢′ 𝑓 𝛿′ 𝑢′ 𝑓 𝛿̅̅̅̅̅̅̅
̅̅̅̅̅̅̅𝜕𝑢𝜕𝑢
𝜕 2 ̅̅̅̅̅̅̅
𝑢′ 𝑖 𝑢′ 𝑖𝑘𝜕𝑢̅𝑖
𝜀 ′ ′𝑓 2𝑖 𝑖𝑘 𝑏𝑖 𝑖𝑘 𝑏′ 𝑤′ 𝑓̅̅̅̅̅̅̅
̅̅̅̅̅̅̅
+ 𝑢̅𝑗 𝑘𝑓+ 2𝑢=
𝜈() − 𝐶1 (𝑢𝑖𝑖 𝑢 𝑖 𝑘 − 𝜅) − 𝜀
𝑘
2
𝜕𝑡
𝜕𝑥𝑗
𝜕𝑧
𝜕𝑥𝑗
𝜅
3
𝑓
̅̅̅̅̅̅̅
̅ 𝑓,
𝑢′𝑖 𝑤′𝑘 = 2𝜈𝑡𝑘 𝑆𝑖𝑧
𝑘

[24]

[25]

𝐶1 = 1.8
3

𝑓

𝜈𝑡𝑘 =

𝐶𝜇 𝑘𝑘
𝜀

1 ′ ′ 𝑓𝑓
, 𝑘𝑘 = ̅̅̅̅̅̅̅
𝑢 𝑖𝑢 𝑖𝑘
2

𝜀=

1
𝜅𝑦𝑝

𝑓
𝑘𝑘 2

( )
3.3

[26]

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3

EXPERIMENTAL AND CALCULATION CONDITION OF LOCAL SCOURING AND DEPOSITIONAROUND A CYLINDER

Figure 3. Longitudinal view of the experimental channel (Fukuoka et.al., 1997).
3.1 Experimental conditions

Table 1. Experimental condition.

Channel width
Channel length
Cylinder diameter
Discharge
Initial bed slope

1.5 m Mean water depth
27.5 m Mean velocity
0.2 m Sediment diameter
90 ℓ /s Specific gravity
1/600 Sediment supply

0.137 m0.442 m/s
0.8 mm
2.59
0.84 ℓ /min

The calculation model was applied to the experiment on the bed variation around the cylinder in order toclarify the validity of the model. Figure 3 shows a longitudinal view of the movable bed experiment channel(Fukuoka et.al. ,1997). In this experiment, the horizontal and vertical distribution of the flow velocity around thecylinder, the bed variation and the fluid forces acting on the cylinder were measured. Table 1 shows the
specifications of the experimental channel and the hydraulic quantity of the experiment. The channel lengthwas 27.5m, the channel width was 1.5m, and the initial bed slope was 1/600. The cylinder was 20cm indiameter and was located at a position of 10.5m from the downstream end of the channel. The particle size ofthe bed material was 0.8 mm. The discharge from the upstream end was 90 ℓ/s. And, the sediment supply
was carried out from the upstream end of the channel.
3.2 Calculation condition
The size of the calculation mesh was 2.5cm square. The initial bed gradient was 1/600 as same as theexperimental conditions. After the experiment, the channel beds were lowered by around 2 to 3cm. The initial
bed elevations in 5 meshes from the downstream end were lowered by 3cm in order to taking into account of
the bed degradations in the experimental results. The uniform bed material of 0.8mm was set. The discharge
and sediment volume were given as shown in Table1. The dimension length yp of the dissipation rate shownin Eq. [26] was set to 1.5b so that the mean water depth and mean flow velocity of the calculation result
almost agreed with the experimental results shown in Table 1.
4
CALCULATION RESULTS
Figure 4 shows the experimental and calculation results of the flow velocity distribution at a height of 0.07m
below the water surface. The calculation results almost explained experimental results of the flow velocity
upstream of the cylinder and the range of separation area downstream of the cylinder. Figure 5 compares thevelocity at near the bed of the experiment results with the calculated velocity on surface of the transition layer.
Although the calculation results could explain the range of the reverse flow at near the beds in front of thecylinder, the calculated range of the separation in the lee of the cylinder was wider than the measured one
Figure 6(a) indicates the bed variation contour of the experimental results. And Figure 6(b) and Figure 6(c)show the calculation results of bed variation and the velocity vector in the bed load layer. The bed load layer
thicknesses of the calculations were set in 2 and 5 times the particle size, respectively. Figure 6(d) indicatesthe calculation result of the bed surface velocity vector and bed variation by the conventional sedimenttransport model in Japan. The conventional model was conducted by using the equilibrium sediment
discharge formula (Ashida, et.al, 1972) and sediment entrainment formula (Itakura, et.al, 1984). Those1803

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calculation results were compared with the experiment, after the bed scouring in front of the cylinder becamealmost stable which was 20 minutes after start of the calculation. The developed model explained that thechannel bed was scoured from the front of the cylinder toward the rear, although the scour depth in the front of
the cylinder was larger than the experimental results (see Figure 6(a), Figure 6(b) and Figure 6(c)). The
conventional model shown in Figure 6(d) explained the scouring in front of the cylinder, but could not
reproduce the scouring of the rear of the cylinder.
On the other hand, the developed calculation model could not explain range of the sediment deposition in thedownstream of the cylinder. This was because the separation area behind the cylinder and its velocity
distribution were not to be properly calculated.

Contour line: bed elevation (m)(a) Experimentresult(b) Calculation b (2d, 1.6mm)

Figure 4. Horizontal velocity distribution at a height of 0.07m below the water surface.
Contour line: bed elevation (m)(a) Experiment(b) Calculation b (2d, 1.6mm)
Figure 5. The experimental velocity distribution near the beds and the calculated surface velocity of thetransition layer.
Figure 7 indicates the contour diagrams of the vertical component of turbulence intensities in the bed load
layer. The bed load thicknesses were 2 and 5 times the sediment particle size, respectively. Figure 8 shows
the contour diagrams of the sediment entrainment amounts and flow velocity in the bed load layer. The
calculation results indicated that the turbulence intensities increased and sediment entrainments occurred in
front and side of the cylinder. And the sediments movements occurred in the rear of the cylinder shown in
Figure 9. Figure 9 shows calculation results of particle velocities of the bed load layer and the contour lines ofthe bed elevations of the in the case that the bed load layer thickness was twice the particle size. The
developed model was able to explain the bed scouring around the cylinder by calculating those nonequilibrium flows and sediment movements near the beds.

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Contour line: bed elevation (m)(a) Experiment(b) Calculation (b =2d, 1.6mm)

Contour line: bed elevation (m)(c) Calculation (b= 5d, 4.0mm)(d) Calculation by conventional
model
Figure 6. The contour of the bed variations in the experiment and calculation by the developed model ( b =2d,b =5d) and the conventional model.
(m/s)

2

(m/s)2(a) b (2d, 1.6mm)(b) b (5d, 4.0mm)

Figure 7. The turbulence intensity in the bed load layer.
The differences of bed variations by the thickness of bed load layer introduced are shown in Figure 6(b) and
Figure 6(c). The bed scouring in the thick bed load layer became deeper than that of the thin bed load layer.
And, the turbulence intensity in the bed load layer was also larger than that of the thin bed load layer (seeFigure 7). Figure 10 shows a longitudinal sectional view of the calculated velocity distributions. The bed load
thicknesses were 2 and 5 times the sediment particle size, respectively. And the flow velocity distributions
were shown along the center line of the channel. The range and intensity of reverse flow in front of thecylinder having the thick bed load layer was larger than those having thin bed load layer. Those results
suggest that the calculation model should consider the changes in the thickness of the bed load layer
associated with non-equilibrium flows and sediment transports near the beds.

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Contour line: bed elevation (m)Figure 8. The amount of the suspended sediment entrainment. (b =2d, 1.6mm)

Contour line: bed elevation (m)Figure 9. The velocity vector of sediment particle in the bed load layer. (b =2d, 1.6mm)(a) b (2d, 1.6mm)(b) b (5d, 4.0mm)

Figure 10. The longitudinal and vertical velocity distribution of flow in front of the cylinder.
5

CONCLUSIONSThe quasi-3D flood flow and bed variation calculation model was developed by considering non-equilibrium
two-phase motion of flows and sediments near beds. The calculation model was able to explain the bed
scouring in front and side of the cylinder.
However, the calculation model was not able to elucidate the range and scale of the sediment depositions
downstream of the cylinder because the separating flow from the cylinder was inaccurately calculated. In
additions, the differences in thickness of the bed load layer introduced in the model affected degrees of flow
and bed variation around the cylinder. It is required to be improved so that the thickness of bed load layer
should vary with flows and sediment transports in the bed load layer.
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REFERENCES
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Uchida, T. and Fukuoka, S. (2014). Numerical Calculation for Bed Variation in Compound-Meandering
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