CHARACTERISTICS OF WATER SURFACE PROFILES OF
OPEN CHANNEL FLOW WITH LEVEE OVERTOPPINGSHINSUKE NAMOTO 1, TADAHARU ISHIKAWA 2 & TAKASHI KOJIMA 1
1
TOKEN C.E.E. Consultants Co., Ltd., Tokyo, Japan
2Tokyo Institute of Technology, Tokyo, Japan

ABSTRACTThe risk of levee overtopping has increased because of the frequency of severe rain storm due toclimate change. Flow depth over levees is a design parameter that must be considered, but standards
have not been agreed on. In this study, the mechanisms of levee overtopping are investigated byanalyzing the water surface profile in an open channel with side-overflow using a 2D shallow watermodel on an unstructured triangular mesh system. Scenarios in which the channel capacity decreases
downstream were considered; the first when the channel slope decreases and the second a “Y-shape”
confluence of two river channels. The computation for the slope changing channel exhibited thefollowing results: (1) levee overtopping occurred not only in the downstream channel but also in theupstream channel due to the backwater effect, (2) the overflow depth was much smaller than the valuepredicted for normal flow without side-overflow, and (3) the water stage at the levee crown was lower
than the transversely averaged one around the slope change section due to the 2-D flow effect. The
computational results for the confluent channel had the tendencies common to those for the slopechanging channel, but further showed: (4) the overflow depth anomaly appeared at the inner corner ofthe confluence with an increase in the confluent angle, which suggested that the delta area between
the confluent channels could be used as a detention basin by armoring the short reach at the corner.
Keywords: levee overflow depth, numerical experiment, 2‐D flow simulation, channel slope change,confluence, levee reinforcement

1 INTRODUCTION
Global climate change has increased the frequency of rain runoff exceeding the capacity of
conventional flood control measures, and several recent flood disasters have been caused bylevee overtopping. Increasing river channel capacity is often difficult because of intensive
land use, so reinforcement of earth levees by surface armoring is considered for mitigation(JSCE report, 2008). However, standards of overflow depth for levee armoring design havenot been defined.
Suga et al. showed that a simple pavement on the levee crown would prolong the lifeof earth levees significantly when the overflow depth is less than 20 cm based on the resultsof hydraulic model tests [1]. Ishikawa presented a 1-D perturbation analysis of the water
surface profile for prismatic channel flow with side-overflow and suggested that the levee
overflow depth is significantly lower than the value predicted for normal flow without sideoverflow [2]. According to a recent flood report [3], a newly constructed earth levee with auniformly paved levee crown endured against failure for a long duration during overtopping
because the overflow was evenly distributed over a long distance. This suggests that a
relatively light levee crown pavement can mitigate against overtopping if the overflow
depth is not large.
In this study, the basic characteristics of levee overtopping were numericallyinvestigated on open channel flow with side-overflow using a 2D shallow water model.
Two conditions in which the channel capacity decreases downstream due to an abrupt
decrease in channel slope (A-series) and a “Y-shape” confluence of two river channels (Bseries) were considered.

2 METHODOLOGY
2-1 Numerical model
A shallow water flow model developed for tsunami run-up simulation [4] was used. In themodel formulation, the hyperbolic differential equations were converted to finite difference
equations by the finite volume method on an unstructured triangular mesh system [5], andthe Eulerian method was adopted for calculating the flood front to avoid so-called “Cproperty collapse” [6].
The flow rate over the levees was calculated using the following formula proposedby [7]:

 0.35 h1 2 gh1q
 0.91 h2 2 g ( h1  h2 )if h2 / h1  2 / 3otherwise(1)

where q is the unit width flow rate over the levee, and h1 and h2 are the water surface
heights on the upstream and downstream sides of overflow from the levee crown,respectively.
2-2 Experimental condition
Fig. 1 shows the experimental conditions for the A-series experiment. The channel is astraight rectangular sectional channel having a constant width of 100 m and a constantlevee height of 3 m, and the channel slope changes from θ1 to θ2 (the values of which arelisted in Table 1). Manning’s roughness was assumed as 0.03. The upstream discharge wasassumed as bankful discharge, and the downstream water surface level was assumed to beequal to the levee crown level. Levee overtopping was expected to occur around the slopechanging section, as illustrated in the figure.
Fig. 2 shows the experimental conditions for the B-series experiment. The channelslope θ3 and the levee height H0 were assumed to be constant throughout the entire channel
reach. Three conditions were considered by changing the confluent angle θ4, as listed inTable 2. The upstream discharge and the downstream water surface level were the same as
those for the A-series experiment. Levee overtopping was expected to occur around theconfluence section.
Table 1: A-series calculation conditions
Upstream
Downstream
Case #
slope θ1slope θ2
A-1
1/200
1/300
A-2
Do.
1/400
A-3
Do.
1/500
Table 2: B-series calculation conditions
ChannelConfluent
Case #
slope θ3
angle θ4B-1
1/300
30 °
B-2
Do60 °
B-3
Do90 °

Fig. 1: ExperimentalEconditions (AA-series: slope change)

Fig. 2:: Experimentaal condition (BB-series: conflluence)

3 RESULTTS AND DISCCUSSIONS
3-1 A-seriess experiments
Fig. 3 shows the water su
urface profiless at the channeel center line H(x),Hand Fig.. 4 shows thelongitudinall variation of volume flux Q
Q(x). H(x) annd Q(x) decreaased downstreeam from thesection in whichw
slope deecreases becaause of side-ov
verflow and approachedatoo the bankfulflow conditiion. In the up
pstream reachh, side-overflo
ow occurred due
d to the bacckwater fromthe section of channel slope
schange although thee discharge wasw less than the bankfuldischarge.
Fig. 5 shows the noormal flow deepths correspo
onding to the locallflow ratee Q(x) whichwere calculaated by the Maanning’s form
mula without consideringcthee side-overfloow effect:1



H N ( x)  (n Q( x)) /( B  2 2 )35(2)where HN (xx) is the normaal flow depth,, n is Manning
g’s roughness, Q(x) is locall flow rate, B
is the channnel width, andd θ2 is the channnel slope. Beecause Eq. (2) gave valuess much larger
than those forf the flow withw side-overfflow, it can be concluded thatt a simple pprediction of
overtoppingg depth basedd on the norm
mal flow deptth would resu
ult in an exceessive armor
design.

Fig. 3: Water surfacee profile alongg channel center line.

g. 4: Longituddinal variationn of volume flux.
FigF 5: Water surface profille estimated frrom normal flow conditionss.
Fig.

Fig. 6(a)6 shows the dimensionlesss overflow deepth normalized with the chhannel depth,h(x)=H(x)/HH0, near the seection of channnel slope chaange, while Fig.
F 6(b) show
ws the excess
discharge normalized
nwiith bankful ddischarge, q(xx)={Q(x) - Q0}/Q0. The hh(x) and q(x)decreased withw θ2, but affter scaling thhem by 1/θ2 and (1/θ2)1/2, respectively, each can be
expressed byy single curvees as shown inn Fig. 7(a) andd Fig. 7(b).
Fig. 8 illustrates the relationshipp between q(x)) and h(x), in which the arrrows indicate
the directionn from the upsstream to the ddownstream. Note
N that the bankfulbdischa
harge changes
across the seection of channnel slope chaange. The bannkful flow con
ndition at the uupstream end
section is pllaced at the orrigin of the diiagram. The fllow condition moves upperr left because
the dimensionless depth h increases ddue to the bacckwater fromm the slope-chhange section
while q decreases due to side-overflow
w. Because of the abrupt change
cof Q0 at the slopechange sectiion, the dimennsionless exc ess dischargee, q, jumps to the positive sside with thesame h. Aftter then the flow condition approaches thet bankful coondition at thee origin. The
black brokenn line in the figure
fiindicatees the relationsship between q and h obtainned from Eq.
(2) which iss located higher than the sim
mulation resu
ults (colored liines). This facct means thatthe actual ovverflow depth
h is much smaaller than the value predictted from a rellationship fornormal flow
w without side--overflow.

Fig. 6(a): Dimensionlesss overflow deepth

Fig.. 6(b): Dimenssionless excesss discharge

Fig. 7(a): Overflow
O
deptth scaled by 1 /θ2

Fig. 7((b): Excess discharge scaledd by (1/θ2)1/2

Fig. 8: Corrrelation betw
ween q and hFig. 9 shows the velocityvvectoors on a colo
or counter maap of water ssurface levelmeasured frrom the leveee crown. Notee that the callculation resu
ult is shown ffor a half ofchannel fromm the center liine (y > 0) beccause of the flow
fsymmetryy. There is a w
water surface
rise at the chhannel center just below thhe slope-changge section (x > 0), which mmeans that thewater stage at the levee crrown is lowerr than the transsversely averaaged one.

Fiig. 9: Flow fieeld near slope--change sectioon.

3-2 B-seriess experiments
Fig. 10 show
ws the longitu
udinal water suurface profilees at the channnel center linees. The water
surface risess sharply arou
und the conflueent section ov
ver the levee crown (3 m), aand decreases
gradually too the bankful condition,c
as w
was observed in the A-seriees experimentss (see Fig. 3).
Fig. 11
1 shows the velocity vecttors on a color counter map
m of water ssurface levelmeasured frrom the levee crown. Theree was a signifficant rise in waterw
surface at the insidecorner just upstream
ufromm the conflueent section, whichw
caused overflow
oto thhe delta area
between thee two confluen
nt channels.
Fig. 12
1 summarizees the resultss of the threee calculationss (B-1, B-2 aand B-3) for
maximum overflow
odepthh at the inner corner of the confluence. TheT water surfface anomaly
over the levvee crown inccreases with ssin(θ4/2). The value at θ4=0 (red dot) w
was estimatedfrom the ressult of the A-series experim
ments for the saame channel capacity
creducction.

1 Water surfface profiles aat the channel center line of confluent chaannels.
Fig. 10:Fig
g. 11: Flow fieeld near the coonfluence secttion.

Fig. 12: Overrflow depth att inner corner..

Fig. 13 shows an aerialaphotograaph of floodin
ng that occurred in the wateershed of theYoshida Rivver in 2015 whhen the leveess were not brooken except att the upstream
m reach of theZen River. TheT majority of the floodinng was caused
d by levee oveerflow at the iinner cornersof the river confluences. The delta areea between Taakebayashi annd Yoshida R
Rivers is used
for paddy fiields because it is prone too flooding, whhile the delta area between the Yoshida
and Zen Rivvers was recen
ntly developeed for industriial use after an
a expresswayy interchange
was located there.
Consiidering the floow characterisstics obtainedd from the num
merical analyssis, the leveenear the innner corner of thet latter conffluence should be higher thhan the outsidde bank. The
delta area of the former, on the other hhand, can be used as a detention basin tto reduce thedischarge doownstream byy reinforcing tthe levees agaainst overtopp
ping. The leveee height anddesign againnst overtoppiing for both cases will neeed to be dettermined usinng numericalsimulation ofo the flow witth side-overfloow for the acttual river topo
ography.

Fig. 13: Aerial photographh of flooding ini Yoshida Rivver watershedd

CONCLUSION
A series of 2-D numerical experiments on open channel flow with side-overflow were
performed for conditions in which the channel capacity was decreased downstream due toan abrupt channel slope decrease and a “Y-shape” confluence. The conclusions are as
follows:(1) Levee overtopping occurs not only in the downstream reach from the section with thechannel capacity reduction but also in the upstream reach where the discharge is less thanthe bankful discharge due to the backwater effect.
(2) The levee overflow depth is much smaller than the value predicted by the relationshipfor normal flow without side-over flow in most of the river reach except for a short reacharound the section incorporating the channel capacity reduction caused by the riverconfluence.
(3) A significant rise in water surface occurs at the inner corner just upstream from theconfluent section, which induces a large overflow to the delta area between the confluent
channels. The water surface anomaly over the levee crown increases with sin( θ 4/2)
where θ4 is the angle of confluence.
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[3] Ministry of Land, Infrastructure and Transport, 2017,
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