EXPERIMENTS AND ANALYSES ON THE DEFORMATION OF THE BED
PROTECTIVE WORKS DOWNSTREAM OF A WEIRTATSUHIKO UCHIDA1 and SHOJI FUKUOKA2
1

Research Associate, Department of Social and Environmental Engineering,
Graduate School of Engineering, Hiroshima University,
1-4-1 Kagamiyama, Higashi-hiroshima, 739-8527, Japan,(Tel: 081-82-424-7847, Fax: 081-82-424-7847, E-mail: utida@hiroshima-u.ac.jp)
2

Professor, Research and Development Initiative, Chuo University,
1-3-1 Kasuga, Bunkyo-ku, Tokyo, 112-8551, Japan,(Tel: 081-3-3817-1625, Fax: 081-3-3817-1625, E-mail: sfuku@tamacc. ac.jp)
Abstract
Bed protective works are installed on the bed downstream of a weir to prevent failureand collapse of the structure due to bed scouring. Because failures of bed protective works
expose the main structure in danger, it is important for river technology and management tomaintain the safety of the bed protective works. So, it is required to establish the method forevaluating the hydrodynamic forces acting on the bed protective blocks and estimating thedeformation range of the structure. In this paper, we measure the hydrodynamic forces acting
on the last row elements in the threshold of block movement and developed the thresholdmodel of block movement. The model of a vertical two-dimensional analysis for the flow withcomplex boundary shape of the blocks is presented, in order to evaluate hydrodynamic forces
acting on the blocks. Then, we developed the numerical model for deformation of the bed
protective works downstream of a weir using the threshold model. It is clarified through thecomparison between measured and computed results that the model can explain theexperimental deformation process.
Keywords: bed protective work, threshold of block movement, hydrodynamic force,
deformation and failure mechanism, last row elements, vertical two-dimensional model
1. INTRODUCTION
Bed protective works are installed on the bed downstream of a weir to prevent failures andcollapses of the structure due to bed scouring. Because failures of bed protective works
endanger the main structure, it is important for designing a weir to maintain a safety of the
bed protective works downstream of the structure.
In general, the bed downstream of bed protective works is eroded. So, many researchers
have been concerned with the local scour downstream of the bed protective works. For
example, Kanda et al. (1993) and Revelli et al. (2001) investigated the scour downstream of abed protective works. Hoffimans and R. Booij (1993) presented the two-dimensional analysis
for the local scour. On the contrary, the bed protective blocks should have enough weightagainst flood flow (Fukuoka et al. (1988)). We indicated the drag force acting on the last row
elements of the bed protective works are greater than that of any other elements (Uchida andFukuoka, (2001)). This indicates that the downstream end of the structure is a weak point due
to not only bed scouring but also large hydrodynamic forces and the completely protection of
the last low block of the bed protective works is difficult. So, it is required to establish themethod for evaluating the stability of the bed protective blocks and estimating thedeformation range of the structure.
In this paper, we present the deformation process of the bed protective blocks and thehydrodynamic forces acting on the last row elements in the threshold of block movement.
And we developed the threshold model of block movement. The model of a vertical
two-dimensional analysis for the flow with bed protective blocks is presented to evaluatehydrodynamic forces acting on the blocks. Then, we developed the model for deformation of
the bed protective works downstream of a weir using the threshold model. It is clarified
through the comparison between measured and computed results that the model can explain
the experimental deformation process.
2. EXPERIMENTS ON THE DEFORMATION OF THE BED PROTECTIVE WORKS
AND THRESHOLD MODEL OF A BLOCK MOVEMENT
The experimental channel is 10m long, 0.45m wide, has a slope of 1/500. The bed
protective blocks are installed downstream of a groundsill of 0.24m height, as shown in Fig.1.
We examined the deformation process of the bed protective blocks, degrading the water level
downstream of the structure hb for the condition of a constant discharge. Table 1 shows theexperimental condition. The blocks deformation breaks out downstream end of the blocks onwhich the large hydrodynamic forces were acted. Then the deformation goes to upstream
direction by the degradation of water level downstream of the structure, as shown in Fig. 9 (a).
To design the bed protective works, whether the blocks survive is important rather than the
motion of the blocks. The threshold model of block movement is discussed below before theevaluation of drag forces and the development of the deformation model of the blocks.
First, we assume that the block got into rotating motion at the outset. To formulate thethreshold of a block movement, we should consider the moment equilibrium of a block
overlapping between the blocks. The threshold of a circular object shown in Fig.1 is written
by Eq. (1).

α 


α 
1 + x  ⋅ Dx + tan φ 1 − z  ⋅ Dz
cos φ  sin φ γ =(tan φ ⋅ cos θ − sinθ ) ⋅ WW

(1)

In which, (Dx , Dz ) =x,z direction drag forces, (α x ,α z ) = shift coefficients of point of action
by velocity distribution, φ =limit incline for a block stability.
To consider the shift of fulcrum point of rotating motion of a block by protrusion of theblocks, we represented φ by Eq.(2) using a protrusion coefficient C.
cos φ = 1 − C

k0
k

(2)

In which, k0= length of a block, k0= overlapping length between blocks.
We established measurement method of threshold drag forces of a block to discuss theadequacy of the threshold model of block movement. Fig.2 shows the instrument of threshold
drag forces of a bed protective block. In which Fx and Fz are output forces of X and Z
direction, respectively. We put a block for measuring the drag force on blocks connected to
the force measure. The block for measuring the drag force is not fixed with the under blocks
and we call it movable block. Fig.3 shows the change of Fx and Fy by time included the
movement period of a movable block. Fx and Fz include the hydrodynamic force acting on the
movable block when a movable block is on the under fixed blocks. However, if the movableblock leaved the fixed blocks, Fx and Fz will change by the force acting on the movable block∆Fx , ∆Fz . From principle of action and reaction, the force acting on the movable block Dx
and Dz are derived by Eq.(3).
D x = − ∆Fx , Dz = W − ∆Fz(3)
To obtain the adequate force, we checked the movement period of the movable block andextracted the maximum and minimum forces from the output in the period. ∆Fx and ∆Fz
are computed by the difference between the maximum and minimum forces. To check on theaccuracy of this method, we calculate a weight W and fined that the error of this method is
5 %.
In here, we do not discuss Dz but Dx, because Dz is quite a little compared with Dx and W inthis experiment condition. Fig.4 shows the relationship between non-dimensional overlapping
length between blocks k/k0 and the ratio of Dx to W. Dx /W increase with increasing k/k0.
Because, increasing k/k0 rise the fulcrum point of the block rotating motion to the action pointof Dx. A line on Fig.4 is computed Dx/W by Eq.(1) with assumptions of neglecting Dz and α x ,using C=1.6 that was obtained by Eq.(2) using measured limit incline for a block stability φ .
Although some measured Dx/W have varied, the entire measured Dx/W are fit on thecomputation line.
In above discussion, we can determine if blocks can move by the threshold model of Eq.(1)if hydrodynamic forces are given.

3. HYDRODYNAMIC MODEL
A flow with complex boundary shape as bed protective blocks can be computed inprinciple, if all the boundary shapes are taken into the flow model. Hirt (1992) factored the
proportion of fluid part in control volume and its faces into the computation to compute theflow with the complex boundary shape. However, the computation method in which all the
shape and arrange of the blocks are taken into account will be not suitable for practical use,
because the blocks are randomly arranged and the protrusion scales of the blocks are
considerably small compared with water depth of the typical scale of flood flow. In this case,we know that it is better to add the resistance terms to the momentum equations of flow (e.g.
Uchida et al., 2001). However, for using this method, we should clarify how individualhydrodynamic force acting on a bed protective block is taken into the flow model to estimate
the stability of bed protective block.
3.1 GOVERNING EQUATIONS FOR FLOW WITH BED PROTECTIVE WORKS
Fig.6 is a control volume including n blocks. In this paper, ∆y is the channel width in orderto develop the two-dimensional model. A drag force of x direction acting on a block Dx isdivided the components of the form drag of a block Dx’ and the other (volume drag). We
define Dx’ in Eq.(4) using a drag coefficient CD.
1
D x ' = C D ρ u xUk 2
2(4)

In which, ρ = fluid density, u x = averaged x-direction fluid velocity for a control volume,

U =

ui

2

(i = 1,2 ) . The sum of Dx in a control volume is represented by Eq.(5).

nD x = p ⋅∆y∆z (1 − A' x )x + ∆xx− ∆x∆y∆z (1 − V ' )∂p+ nD x '∂x(5)In which, p = averaged pressure intensity for a control volume (D’x is drag force due to thedeviation pressure distribution from p ), A’x, V’= proportion of fluid part in x-section and thecontrol volume. z-direction force is derived in a similar way. Applying a law of conservationof momentum for a control volume including n blocks (Fig. 5) with Eq.(5), we can obtain the
averaged flow equations for a control volume as below.
∂ui ∂A' j ui u jF
1 ∂ p ∂A' j u'i u j '+= − gδ 2 i − i −+∂tV ' ∂x j
ρV ' ρ ∂xi
V ' ∂x j∂A' j u j
∂x j

=0

(6)
(7)

In which,

Fi =

nDi '
C ρu U  θ = D i  b  ,∆x∆y∆z2k  θ bo 

θ b = 1 − V ' , θ bo = Vbo / k 3 , x1 , x 2 = x , z , u i = averaged xi-direction fluid velocity distribution
for a control volume, δ ij = Kronecker delta， u' i = the deviation of xi-direction fluid velocity
from u i .
In this study, fluid mixing terms in Eq.(6) are represented by Eq.(8), using averaged strain
rate of fluid and kinematic eddy viscosityν t . And, ν t is represented based on Smagorinsky
model to take into the account the flow mixing by the blocks resistance.

u' i u' j = ν t

2
∂u i
uF
2
= 2ν t S ij , ν t = (∆ ⋅ Cs ) 2 Sij + i i∂x jρV ' ν t(8)

In which, ∆ = ∆x∆z , Cs = 0.3 .
Eq.(6) and Eq.(7) are solved in σ coordinate system (Uchida et al., 2004) to take into
account water surface profiles downstream of a weir.

3.2 EVALUATION OF DRAG FORCE AND DEFORMATION MODEL
In this section, we discuss the evaluation method of drag force acting on a block. In Eq.(5),the drag force is represented only the form drag component of second term in Eq.(5), if we
introduce an assumption of a uniform volume density. This means that, in the model, the dragforce acting on a block is not computed directly because the model regard drag forces due toaveraged pressure gradient as the internal stress of the blocks. So, in order to evaluate dragforce acting on a block, we should separate the target block from the other blocks. In otherwords, we should evaluate the drag force by Eq.(5) without the first term of right side. In thiscase, we can obtain Di/W that is the ratio of drag force acting on a block Di to its weight inwater Ww by Eq. (5) as below.
 θ bo ∂p C D ui Di1−= ∫ ∫ ∫ f*i dxdydz =+U dxdydz∫∫∫
Ww k k ksgθ bo k k k  ρ ∂xi
2k


(13)

We can determine the threshold of block movement by the computation, using Eq.(1) andEq.(13). The deformation model is the simple model, in which only the threshold model ofblock movement is used. In the model, as shown in Fig. 7, criterion of start and stop are

γ 1 and γ 2 . And a criterion of finishing washout is γ 3 . Because γ 1 means the threshold of block
movement, γ 1 = 1.0 . However, γ 2 and γ 3 should be smaller than one if we use Eq.(1), because
γ 2 and γ 3 have a strong relationship to speed reduction and acceleration respectively. So, in thisstudy, γ 2 = γ 3 = 1.0 is taken in the deformation analysis.
4. COMPARISONS BETWEEN MEASURED AND COMPUTED RESULTS
Fig.8 shows comparisons of (a) water surface profiles and (b) velocity distributions
between measured and computed results. In the computation, we use CD=7.4 that is obtainedby the experiment. There is scarcely a flow between the blocks except in the thin upper layer,
in which the momentum of the flow is exchanged between the flow between the blocks andthe flow over the blocks. We can see that the flow over the blocks is supper critical in Fig.8(b).
The kinetic energy of the supper critical flow is attenuated downstream of the blocks,
producing the wave jump flow. Although the water surface of two-dimensional analysis has apronounced tendency of the wave jump than that of measurement, we also examined the wave
jump occurred by the experiment. As a general rule, the flow will change to submerged jet
flow from wave jump flow by the degradation of water level downstream of a weir. But, we
can not look see the submerged jet flow downstream of the blocks in both of the experimentand the computation. Because, the more the water level downstream of the blocks degrades,
the more water flows into the void of blocks, in which the flow energy is burned. The
computed results cannot fit in the measured results just downstream of the weir. The model
formulated by simplifies the shape and the resistance of blocks is not fit to express thecomplex flow field as the flow impinging on the blocks. On the other hand, the model can
explain the characteristics of the flow field downstream part of the blocks which controls thepattern of the blocks deformation.
Fig.9 shows comparisons of the blocks deformation processes between measured andcomputed results. The blocks profile by computed results deforms from the downstream-end,
and then the blocks deformation goes upstream by the degradation of water level downstreamof the blocks, as in the similar way of the experiment．Because the motions of the blocks are
not computed in the model, the blocks profiles of moved blocks (downstream part of theblocks) are different between measured and computed results. However, the computed
profiles of the blocks by surviving blocks (upstream part of the blocks) are similar to that of
the experiment.

5. CONCLUSIONS
1)

We established measurement method of threshold drag forces of a block and developed
the threshold model of block movement.
2) The model of a vertical two-dimensional analysis for the flow with bed protective blocks
is presented to evaluate hydrodynamic forces acting on the blocks. And we developed themodel for deformation of the bed protective works downstream of a weir using thethreshold model.
3) The model can explain the measured results of the flow with bed protective works.
4) The computed profiles of the blocks by surviving blocks (upstream part of the blocks) aresimilar to that of the experiment.

REFERENCES
Fukuoka, S., Fujita, K. and Morita, K., 1988. Hydraulic roles of revetment, Civil Engineering
Journal vol.30, pp115-120 (in Japanese).
Kanda, K., Muramoto, Y. and Fujita, Y., 1996. Local scour and its reduction method in
downstream of bed protection work, Journal of hydraulic, coastal and environmental
engineering, No.551/II-37, pp.21-36 (in Japanese).
Hoffmans, G. J. C. M. and Booij, R., 1993．Two-Dimensional Mathematical modelling ofLocal-Scour Holes, Journal of Hydraulic Research, IAHR, Vol.31, No.5, pp.615-634.
Revelli, R., Ridolfi, L. and Sordo, S., 2001. On the effects of roughness changes on local
scour in rivers, XXIX IAHR Congress Proceedings, Theme D Vol.1, pp.572-577.
Hirt, C. W., 1992. Volume-fraction techniques: powerful tools for wind engineering, Journalof Wind engineering, No.52, pp.333-344.
Uchida, T. and Fukuoka, S. , 2001. Two dimensional analysis of shallow water flow withsubmersible large roughness elements, XXIX IAHR Congress Proceedings, Theme D Vol.1,
pp.364-370.
Uchida, T. , Fukuoka, S. and Watanabe, A., 2004. Vertical two-dimensional analysis for local
scour just downstream from a groundsill, Journal of hydraulic, coastal and environmental
engineering, No.768/II-68, pp.45-54 (in Japanese).
Table 1 Experimental conditions
Flow

Discharge

7.5×10-2 (m2/s)

A ground sill

Channel wide

0.45 (m)Channel length

10.0 (m)Characteristics
length of the blockk

9.1×10-2 (m)Volume of theblock Vbo

2.4×10-3 (m3)Specific gravity
of a block

2.3

Bed protective blocks

1.0m
0.45m

Fig.1 Bed protective blocks downstreamof a groundsill

Flow

Dz

A movable block

Dz

k

Zb
φ0
W

Dx

Fixed blocks

Dx

φb
k0

Fulcrum point
Fig.2 Moment equilibrium of a circular object

ForcemeasureFz

W
Fx

Fig. 3 An instrument of threshold drag force
acting on a bed protective blockHydrodynamic force（Ｎ）

14

Fx

12

Fz

∆Fz

10
8
6

∆Fx

4

A movement
movable block

2
0
7

8

9

periodofa10Time (s)Fig. 4 An example of hydrodynamic changes by time included a movement period of a
movable block
2.0

Dx/W

1.5

1.0

0.5

4 layers
2 layers
C=1.6

0.0
0.1

0.2

0.3

0.4

0.5

k0/k
Fig. 5 Relationships between non-dimensional overlapping length k0/k and x-direction
drag forces Dx/W

∆x

Flow

Washout point
Start: γ > γ 1
Finish: γ < γ 3

z

Dx

∆z

Running passγ > γ2

Deposit point

y

∆y

x

x+Δx

x

Fig.6 A control volume including n bed
protective blocks

Bed protective works

γ < γ2

Fig.7 Simple computation method for the
bed protective works deformation

0.4
Experiment

Computation

Z(m)

0.3
0.2
0.1
0

0

1

X(m)

(a) Water surface
1

1

ο
ο

ο
ο Surface of bed0ο
ο protective works

0

-1

-1

-2

-2

-3
-0.5 0

u

ο
ο
ο

ο

ο

3

ο
ο
ο

2

ο
ο

ο
ο
ο

ο
1

X=0.1m

X=0.6m

0.5

0.5

1

4

ο
ο

-3
1.5 -0.5 0

1

ο
ο

1.5

uο ghcghc

X=1.2m0-0.5 0

0.5

u

1

Bed

1.5

ghc

(b) Vertical velocity distributions (hc: Critical depth (0.083m))
Fig.8 Comparisons of flow fields between measured and computed results

Z(m)

0.3
0.93<h0/hb
0.76<h0/hb<0.93
0.59<h0/hb<0.76
h0/hb<0.59

0.2

0.1

0

0

0.5

1

(a) maeasured

1.5 X( m)0.3
h0/h b>1.00
h0/h b=0.91
h0/h b=0.71
h0/h b=0.50

Z(m)

0.2

0.1

0

0

0.5

1

(b) computed

1.5 X( m)Fig. 9 Comparisons of blocks deformation process between measured and computed results