TWO DIMENSIONAL ANALYSIS OF SHALLOW WATER FLOW WITHSUBMERSIBLE LARGE ROUGHNESS ELEMENTS
Tatsuhiko UCHIDA, Graduate student, Department of Civil Engineering, Graduate School ofEngineering, Hiroshima University, 1-4-1 Kagamiyama, Higashi-hiroshima 739-8527, Japan, E-mail:utida@hiroshima-u.ac.jp
Shoji FUKUOKA, Professor, Department of Civil Engineering, Graduate School of Engineering,
Hiroshima University, Kagamiyama, Higashi-hiroshima 739-8527, Japan, Telephone
081-824-24-7821, fax number 0824-24-7821, E-mail: sfuku@hiroshima-u.ac.jp
ABSTRUCTIn our previous study, we developed a numerical analysis method of flow with non-submersible
large roughness elements in group. However, the study on the flow with submersible large roughness
elements has not been made enough. The purpose of this paper is to develop a two-dimensional model
for shallow water flow with submersible large roughness elements. For this analysis, it is necessary toestimate drag force of roughness elements. Drag forces acting on the submersible elements aremeasured for various roughness arrangements and water depth. Then, a numerical model is developed
through addition of resistance terms due to roughness elements in the two-dimensional shallow waterequations using the general curvilinear coordinate system. The applicability of this model is verified
by experiments. Additionally, the model is applied to the simulation of flow flooding in an urban
district having submersible and non-submersible houses.
Keywords: Submersible large roughness element, drag force, two-dimensional analysis, urban district
inundation
1. INTRODUCTION
There are many hydraulic problems concerning flows with large roughness, such as foot protection
work, bed protective works locating at downstream of weirs, and fishways, and houses in urban
inundation areas, and mountainous rivers. In order to design these structures and predict these flows, itis necessary to estimate resistance characteristics of large roughness elements.
Researches has been conducted on the flow resistance resulting from large roughness, using
equivalent roughness ks1) 2), Chezy coefficient3), Manning roughness coefficient4) and friction
coefficient f 5). These coefficients are known to vary by arrangements of the roughness elements andflow conditions. In our study6), it was shown that drag force acting on non-submersible largeroughness elements could be estimated with enough accuracy under the assumption of hydrostaticpressure distribution using water depth at front and back faces of the element and that
two-dimensional shallow water analysis could represent flow with various non-submersible elements.
The purpose of this paper is to develop a two-dimensional model for shallow water flow withsubmersible large roughness elements. Drag forces acting on submersible large roughness elements
and their estimating method are investigated for various arrangements of the roughness elements anddifferent water depth. Thereafter, a numerical model is developed through addition of resistance terms
due to submersible roughness elements in the two-dimensional shallow water equations. This model isapplied to simulate inundation of an urban district having submersible and non-submersible houses.
2. DRAG FORCE OF LARGE ROUGHNESS ELEMENTS
Drag forces acting on large roughness elements were measured in a 10m long and 2.5m widechannel, as shown in Fig.1. The channel bed had 1/500 slope. Drag force measurement instrument
was installed inside a pit at 5.7m distance from upstream end of the channel. A 0.2m wide and 0.03m
high roughness element, which is rectangular block, was used in the experiment. Fig.2 shows the0.5m

2.5m

10m
5.7m

3.8m

Pit

Fig1 Layout of the experimental channel
Flow

Submersible Large Roughness Element

Force Measurement

Movable Platform

Fig.2 Layout of the drag force measurement
instrument
Table1 Hydraulic condition of the measurement
Lx/d
Case1
Case2

2

*

F = Fx/F0

layout of the instrument and placement oflarge an element. The instrument had a
movable platform inside the pit. Theplatform is used to adjust height of theroughness element for accurate force
measurement. Drag forces were measured
for an arrangement of large roughness
elements in rows and columns in the channel
to clarify a relationship between drag forces
and spacing of the large roughness elements.
Hydraulic conditions of the experiment are
shown in Table 1. Here, Lx and Ly arespacing between roughness elements in thelongitudinal and transverse direction
respectively. B is width and d is height of theroughness elements. Relative depth h* isdefined by a ratio of water depth h at 1.0m
from upstream end of the channel to theelement height d and is an importantparameter for the flow with large roughness
elements.
Fig.3 shows longitudinal distributions ofdrag forces for groups of submersible large
roughness elements. Here, relative distance
X* is a ratio of distance of an element fromupstream end of a group elements to total
longitudinal length of the group elements.
On the other hand, relative force F* isdefined by F*=Fx/F0. Where, Fx is a force onan element for its group arrangement and F0
is a force on the element only for single
block in the channel, both having same
hydraulic condition. Fig.3 shows eventhough, Ly is different in Case1 and Case2 as
seen in Table1, drag forces for the inside of
the group are almost similar for both thecases. This result indicates that Lx rather than
Ly affects drag force. Drag forces are
extremely smaller for the condition of
Lx/d=3.3 compared with other two cases of
Lx/d. Because, the pressure acting on theupstream face of an element is reduced for
Lx/d=3.3 due to smaller length of Lx
compared to length of vortex shedding froma just upstream element. As the spacing Lx
increases for Lx/d>3.3, velocity approaching
to the element recovers toward normal flow
condition. Therefore, drag forces for thesecases increased with increasing Lx as shownin Fig.3. The drag force acting on the6.7
20.0

Discharge
Q(m3/s)0.0089～0.007850.0089～0.00587

RelativeDepth h*
1.2～3.1
1.0～2.4

Case1
Lx/d=3.3
Lx/d=6.7
Lx/d=13.3
Case2
Lx/d=3.3
Lx/d=6.7
Lx/d=13.3

10

3.3～
20.0
3.3～
13.3

Ly/d0

0.2

0.4

0.6

0.8

1X*
Fig.3 Longitudinal drag force distribution forgroups of submersible large roughness elements

roughness elements locating at extreme downstream of a group (X*=1.0) is greater than that of any
other roughness elements. This is because of large difference in water depth between front and back of
the last roughness elements, which occurs due to difference in resistance inside and outside of thegroup elements. Therefore, for Case1, in which transverse spacing length Ly is 3 times shortercompared to Case2, the drag force of the last element is greater than that of Case2. The similar
phenomenon occurred in case of non-submersible roughness elements 6).
3. ESTIMATING METHOD OF DRAG FORCE ACTING ON SUBMERSIBLE LARGEROUGHNESS ELEMENTDrag force acting on non-submersible roughness element can be evaluated by equation (1) ofhydrostatic distribution 6). With the similar assumption of hydrostatic distribution, drag force acting on
submersible large roughness element is given by equation (2).
122
ρgB h 1 − h 22F x = ρgBd(h 1 − h 2 )Fx =() (1)60( 2)Drag coefficient CD

Measured drag force (gf)50
X=Y
Lx/d=13.3Where, Fx=drag force, ρ =water density,
g=gravity acceleration, h1= water depth at
40
Lx/d=6.7
front face of a roughness element, h2=water30
depth at back face of the roughness element.
Fig.4 shows relationships between
Lx/d=3.3
20
computed drag forces using actual water
depth from equation (1) or (2) and measured
10drag forces acting on large roughness
elements. Triangular symbols indicate
0
non-submergence 10)
submergence(Case2)
non-submersible roughness elements andblack round symbols indicate submersible-10-1001020
30405060roughness elements of Case2. The black
Computed drag force (gf)round symbol has different size depending on
Fig.4 Relationships between computed with an assumption
discharge, and hence relative depth h*. A sizeof hydrostatic and measured drag force acting on theof the black round symbol increases with thesubmersible large roughness elements
increasing of relative depth h*, and the black
2.5
round symbols having same Lx are connected
Case1
Case2
by a solid line. For the case of theLx/d=3.3
Lx/d=3.3
2
Lx/d=6.7
Lx/d=6.7
non-submersible
roughnesselements,
Lx/d=13.3
Lx/d=13.3
calculated results and measured results are
1.5
almost similar. For the submersible elements,*however, as relative depth h increases,1
difference between computation andmeasurement becomes greater. Here, below is
0.5
an explanation of the above result. As water
depth h increases above the roughness
0element height, a flow over the elements
become established. Under this flow condition,
downward component of the flow become
-0.5
0.5
11.5
2
2.5
3
3.5
large at the downstream area of theRelative depth h*submersible roughness element. That means,pressure distribution acting on the back faceFig.5 Relationship between drag coefficient and relativeof the element is smaller than hydrostatic
depth h*distribution. Therefore, computed result isgreater than that of the measurement for submersible roughness element. Above mensioned result

2Fx
CD =
ρAU 2(3)Water level (cm)

15indicates that drag foces acting on thesubmersible roughness elements can not
be estimated by equation (2).
Therefore, this study investigates dragcoefficient CD through the equation (3).

10Measured
Calculated
CD=0.0CD=1.250

Velocity (cm/s)

02468
10Where, A= a projected area (=dB), U= aLongitudinal distance from upstream end of the channel (m)velocity at 1.0m from upstream end of theFig.6 Longitudinal distribution of water depthchannelFig.5 shows relationships between
Measured
4.5m from upstream end of the channel
relative depth h* and drag coefficient
Calculated
60
CD=0.0computed at the center of the submersible
CD=1.2roughness groups. Drag coefficients CDvaries with the elements arrangement and40
relative depth h* as shown in Fig.5. Dragcoefficient CD is large when relative depth
h* is small (h*<2.0). The greater relative
20
depth h* becomes, however, the smaller
drag coefficients CD become, and then,
Backward area of elements
for h*>2.0, drag coefficient CD does not
000.5
11.5
22.5
change with h*. It is considered that, for h*> 2.0, the shape and size of vortex motion
Transverse distance from right bank of the channel (m)occurring at a back of roughness elements
Fig.7 Traverse distribution of streamwise velocity
becomes stable without direct influence
by water depth. And variation of drag coefficient CD with elements arrangement is relatively small forh*>2.0. So, drag force acting on submersible roughness element can be calculated by equation (3),using a proper drag coefficient CD in each roughness arrangement.

4. TWO-DIMENSIONAL NUMERICAL ANALYSIS FOR FLOW WITH SUBMERSIBLE
LARGE ROUGHNESS ELEMENTS
A two-dimensional shallow water model is developed by employing general curvilinear coordinatesystem 6). The momentum equations include resistance terms for submersible large roughness
elements. The resistance terms, which mean Fx in equation (3), are given at immediate upstream
meshes of the submersible elements. Bed level of computation meshes at roughness element is raisedby the element height (and hence d). For the computation, experimental discharge at upstream end andcritical depth at downstream end were given as boundary condition. The slip-condition is applied forboth the sidewall of the channel. The calculation continues routinely by using forward difference oftime until discharge becomes stable, and gets the final output of the computation.
The following are comparisons between computed and experimental result for Lx/d=13.3 of Case1.
Fig.6 shows a longitudinal distribution of water depth. The water level computed without resistanceterms is lower than measured water level due to underestimation of resistance of the elements. The
water level computed with resistance terms, however, is almost similar to that of the experiment. Fig.7
shows comparison of transverse distribution of streamwise velocity between computed and measured
result. Computed velocity without resistance terms is not reduced for backward area of the elements
compared with that of the experiment. With resistance terms, however, computed velocity is almost
similar to that of the experiment. The above results indicate that the model, which involves theresistance terms with a proper drag coefficient CD, is appropriate for a numerical analysis of flow withsubmersible large roughness elements.

)mc
3=d(degrembuS)
mc
6=d(degrembuS)
mc
9=d(degrembuSnoN)
5m.
(25. APPLICATION OF THE MODEL FOR AN URBAN INNUNDATION WITH
NON-SUBMERSIBLE AND SUBMERSIBLE HOUSESIn our previous study, horizontal two-dimensional model accurately simulated the inundation of anurban district with non-submersible houses. However, houses near probable failure locations of a leveeand low ground level area are in risk of flooding. The urban houses may have both submersed andnon-submersed.. Therefore, a numerical model is required to predict an inundation not only for
non-submersible houses but also for submersible houses. In this chapter, an inundation in an urban
district is assumed, for a location where comparatively large number of submersible houses exist. The
model, which was discussed in the proceeding chapter, is applied for a randomly arrangement ofsubmersible and no-submersible large
roughness elements. Fig.8 shows
computational mesh with arrangement of
the elements. A main street is assumed in
the center of the urban district as shownin Fig.8. All flow velocities entering into
the mesh over the non-submersible
element are given zero as boundary
condition. And the resistance terms for
Fig.8 Computation mesh with the elements arrangement
the non-submersible element arecalculated by using equation (1). For
15submersible element, bed level of
MeasuredComputedmeshes under the element is raised by the10element height, and the resistance terms
are calculated by equation (4). For the5flow with the submersible andBednon-submersible elements, it is not
00246810
appropriate to use velocity at upstream ofLongitudinal distance from upstream end of the channel (m)the group elements for equation (4) asFig.9 Longitudinal water level profile by computation anddiscussed in the proceeding chapter.
measurement
Instead of that, the velocity at a point,
MeasuredComputed
where the resistance terms are given, is50
used in equation (4).
4.5m from upstream end of the channel
40
1
22 db' sin θ30
Fx = − C D u x u x + u y
2A'20(4)1
22 db ' sin θ
10
Fy = − C D u y u x + u y
2A'0Where, Fx and Fy =drag force of x and y
-10
00.5
1
1.5
22.5
direction, ux and uy =velocity of x and yTransverse distanse from right bank of the channel (m)direction, b’= element width in a grid,50
6.0m from upstream end of the channel
A’= area of a grid, θ = angle between U
40and front face of the element ( 0 ≤ θ ≤ π )30
Fig.9 shows longitudinal water level20
profilesforcomputation
and10measurement. The computed result
0differs a little from the measured result-1000.5
1
1.5
22.5
near 7.5m from upstream end of theTransverse distanse from right bank of the channel (m)channel. Distortion of the mesh and themisplacement of the element due to Fig.10 Comparison between computed and measured results for
sidewall of the channel are considered as transverse distribution of streamwise velocity
the reason for that difference. But as a25.
115.
00

Water level (cm)Main velocity(cm/s)
Main velocity(cm/s)

)
m(
98

7

65432

s/m
c
0
55.
225.
1
15.
008

9

8

9

7

6

5

4

3

2

Measured velocity vector
5.
225.
1
15.
00
7

6

5

4

3

2

whole, computed water depth is almost
similar to measured water depth. Since anactual urban district does not have theboundary channel sidewall, these errors cannot occur. It can be considered, therefore,that the numerical model can evaluate
resistance of the elements with highaccuracy. Fig.10 and Fig.11 show thecomparison between computed andmeasured results of transverse distributionof velocity and velocity vector respectively.
Concentrating high flow occurs in the mainstreet. The measured results are almost
reproduced by the computation.
The above indicates that the numericalmodel is applicable to simulate theinundation in urban district with randomlylocated submersible and non-submersible
houses.

Computed velocity vector
Fig.11 Comparison between measured and computed velocity
vector

6. CONCLUSION
1. Drag force acting on a submersible large roughness element inside of a group of the element iseffected by longitudinal spacing Lx rather than transverse spacing Ly. The drag force of theelement locating at extreme downstream of the group elements is greater than that of any otherelements.
2. Drag force acting on submersible large roughness element cannot be estimated by an assumptionof hydrostatic pressure distribution due to establishment of flow over the element. Instead of that,the drag force can be estimated by using a proper drag coefficient CD in each roughness
arrangement.
3. The two-dimensional model describes the flow accurately with the submersible elements becauseof proper estimation of resistance of the elements. It also shows that the model is applicable tosimulate the inundation in an urban district with randomly located submersible andnon-submersible houses.
REFERENCE
1.
2.
3.
4.
5.
6.

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Sayre,J.B. and Albertson,M.L.: Roughness spacing in rigid open channels, Proc.of ASCE, Vol.87, HY3, pp.121-150,
1961
Herbich, J.B. and Shulits, S.: Large-scale roughness in open-channel flow, Proc.of ASCE, Vol.90, HY6, pp.203-230,
1964
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