Journal of Hydraulic Engineering, Vol. 135, No. 11, November 1, 2009This experimental study focuses the influence of bed material mobility and initial downstream water level on maximum water level and time to peak of dam-break waves. It covers horizontal bed conditions on fixed bed, sand bed, and pumice bed. Results include water surface level time evolution, maxima wave levels and time to peak. The influence of bed material mobility and downstream water level was identified and characterized, stressing the importance of using mathematical models with appropriate sediment transport formulations instead of purely hydrodynamic models to simulate dam-break waves on mobile bed channels

Cardoso, António H.

Ferreira, Rui M. L.

Leal, J. B.

Journal of Hydraulic Engineering

English

Repositório da Universidade Nova de Lisboa

Maximum Level and Time to Peak of Dam-Break Waves onMobile Horizontal BedJoão G. A. B. Leal1; Rui M. L. Ferreira2; and António H. Cardoso3Abstract: This experimental study focuses the influence of bed material mobility and initial downstream water level on maximum waterlevel and time to peak of dam-break waves. It covers horizontal bed conditions on fixed bed, sand bed, and pumice bed. Results includewater surface level time evolution, maxima wave levels and time to peak. The influence of bed material mobility and downstream waterlevel was identified and characterized, stressing the importance of using mathematical models with appropriate sediment transportformulations instead of purely hydrodynamic models to simulate dam-break waves on mobile bed channels.DOI: 10.1061/ASCEHY.1943-7900.0000099CE Database subject headings: Dams; Water waves; Bed materials; Sediment transport.IntroductionThe most important features of dam-break flows DBF are themaximum wave level and time to peak. Lauber and Hager 1998have characterized experimentally these variables for fixed, hori-zontal, initially dry bed downstream the dam, taking into consid-eration the influence of the upstream reservoir length.Recent events and observations point out that DBF can interactstrongly with the mobile bed, causing major impacts in its mor-phology. This was the case, for instance, downstream of Lake Ha!Ha! due to the rupture of a dyke Lapointe et al. 1998. Theoccurrence of intense sediment transport in DBF as well as theinteraction between the flow and the bed can significantly affectthe flow dynamics, usually contributing to increase the water sur-face levels and to decrease the wave-front celerity as comparedwith purely hydrodynamic solutions cf. Leal et al. 2006. Never-theless, such solutions are frequently used due to difficulties inthe implementation and exploitation of models accounting for thebed mobility.The role of bed mobility in DBF was scarcely studied in thepast. For lightweight granular bed material, Capart and Young1998 observed the excavation of a scour hole near the dam crosssection, leading to the onset of an hydraulic jump. Spinewine andZech 2007 concluded that the scour hole and the hydraulic jumpwere significantly attenuated for heavier, less mobile bed. Theeffect of the water level downstream the dam was rarely studied1Assistant Professor, Dept. of Civil Engineering, Faculty of Sciencesand Technology, Universidade Nova de Lisboa, Quinta da Torre, Caparica2829-516, Portugal corresponding author. E-mail: jleal@fct.unl.pt2Assistant Professor, Dept. of Civil Engineering and Architecture,Instituto Superior Técnico, UTL, Av. Rovisco Pais, Lisbon 1049-001,Portugal. E-mail: ruif@civil.ist.utl.pt3Full Professor, Dept. of Civil Engineering and Architecture, InstitutoSuperior Técnico, UTL, Av. Rovisco Pais, Lisbon 1049-001, Portugal.E-mail: ahc@civil.ist.utl.ptNote. This manuscript was submitted on November 16, 2007; ap-proved on May 19, 2009; published online on May 21, 2009. Discussionperiod open until April 1, 2010; separate discussions must be submittedfor individual papers. This technical note is part of the Journal of Hy-draulic Engineering, Vol. 135, No. 11, November 1, 2009. ©ASCE,ISSN 0733-9429/2009/11-995–999/$25.00.JOURNADownloaded 09 Dec 2009 to 193.136.126.36. Redistribution subject totoo. Stansby et al. 1998 and Leal et al. 2002 verified that thewater surface level behind the wave-front raises as compared withthe pure hydrodynamic solution, creating a jump whose heightincreases with the initial downstream water level. The presentexperimental study intends to contribute to further characterizethe influence of bed mobility and initial water level downstreamthe dam on maximum wave level and time to peak in horizontal,mobile bed channels.ExperimentsThe data used in this study were collected by Leal 2005. Testswere carried out in a 19.2-m long, 0.5-m wide, and 0.7-m highrectangular flume with horizontal bed. For simulating dam-breakevents, a vertical lift-gate, installed in the middle cross section ofthe flume, was opened instantaneously. Three types of bed wereadopted: fixed bed, sand bed, and pumice bed. The sand diameterwas ds=0.8 mm; sand specific gravity was s=2.65. For pumice,these variables were 1.2 mm and 1.40, respectively. The initialwater depth upstream the gate was the same in all tests hu=0.40 m; for mobile bed tests, the initial bed elevation was al-ways kept equal to 0.07 m, both up and downstream the gate; theinitial water depth downstream, hd, ranged from 0.000–0.210 m.The main features of the initial conditions are summarized inTable 1, where =hd /hu is the dimensionless initial downstreamwater depth. Assuming hydrostatic pressure distribution, the timeevolution of water surface levels was measured with seven pres-sure transducers installed downstream the gate, at X=0.5, 2.5, 5.0,7.5, 12.5, 17.5, and 22.5, where X=x /hu and x=longitudinal co-ordinate measured from dam cross section.Time Evolution of Water Surface LevelFig. 1 presents the time evolution of dimensionless water surfacelevels, Zs, of fixed bed tests, =0.00, 0.06, 0.14, and 0.52. In Fig.1, Zs=zs /hu; zs=measured water surface elevation above the ini-tial water level downstream the gate; T= t / hu /g0.5=dimensionless time; t=time; g=gravitational acceleration. Themaxima wave levels are marked with circles; the instants ofL OF HYDRAULIC ENGINEERING © ASCE / NOVEMBER 2009 / 995 ASCE license or copyright; see http://pubs.asce.org/copyrighttime—time limits—above which the limited upstream reservoirlength starts to influence the wave levels at the measuring crosssections are also identified with vertical dashed lines. Time limitswere calculated by assuming that the negative wave-front celeri-ties measured by Leal et al. 2002 also apply downstream. For agiven location, X, the time limit is therefore computed as 2r+X /CNF, where r=Lr /hu=dimensionless reservoir length; Lr=length of the upstream reservoir; CNF=dimensionless negativewave-front celerity.For the lower values of  =0.00, Fig. 1a, and =0.06, Fig.1b, the water surface level rises abruptly, in each section, im-mediately after the arrival of the wave-front; afterwards, it in-creases slowly until reaching the maximum; afterwards, the waterlevel decreases slowly as the upstream reservoir empties. Thisbehavior was also observed by Lauber and Hager 1998 for fixedhorizontal bed. For higher values of  =0.14, Fig. 1c, and=0.52, Fig. 1d, water levels are influenced, after some time,by reflection waves induced by the downstream weir used to guar-antee the water level downstream the gate. Whenever this oc-curred the “corrupted” parts of the records were ignored in theanalysis. As  increases, the maxima wave levels are attainedTable 1. Tests Main FeaturesTest Type of bedhdmT.1 fixed bed 0.000 0.00T.2 0.023 0.06T.3 0.049 0.12T.4 0.056 0.14T.5 0.071 0.18T.6 0.082 0.20T.7 0.094 0.23T.8 0.114 0.28T.9 0.142 0.35T.10 0.212 0.52Fig. 1. Time evolution of dimensionless water level, Zs, at X=0.5,=0.00; b =0.06; c =0.14; and d =0.52996 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / NOVEMBER 200Downloaded 09 Dec 2009 to 193.136.126.36. Redistribution subject tosooner and tend to become independent of the reservoir length,i.e., they occur before the calculated time limit. For the highestvalue of  =0.52, Fig. 1d the maxima wave levels are at-tained just after the arrival of the wave-front due to the fact thatthe wave propagates downstream as a solitary wave or as a groupof waves cf. Leal 2005.For sand bed, the time evolution of the water surface elevationis practically as described above. Still, the maxima wave levels atX=0.5 are attained slightly sooner cf. Leal 2005.Fig. 2 presents the time evolution of the dimensionless watersurface levels of pumice bed tests, =0.03, 0.11, 0.20 and 0.51.Again, the water surface behavior presents qualitative similaritieswith the one described for fixed bed. Important discrepancies areobserved, however, at the cross section the nearest to the gate forthe lower  values Figs. 2a and b, where the water surfacelevel does not rise slowly until reaching its maximum. Instead, itexhibits an oscillatory behavior; the maximum level is attainedmuch sooner than for the other sections and is independent of thereservoir length. It can be concluded that, for locations near thegate, the increase of bed mobility induces the increase of thewater surface levels. Although this can be attributed to the addedType of bedhdmSand ds=0.8 mm s=2.65 0.003 0.010.012 0.030.042 0.100.073 0.180.106 0.260.204 0.51Pumice ds=1.2 mm s=1.40 0.013 0.030.014 0.040.044 0.110.078 0.200.107 0.270.207 0.51.0, 7.5, 12.5, 17.5 and 22.5 right to left for fixed bed tests: a TestTs.1Ts.2Ts.3Ts.4Ts.5Ts.6Tp.1Tp.2Tp.3Tp.4Tp.5Tp.62.5, 59 ASCE license or copyright; see http://pubs.asce.org/copyrightsediment inertia and to the increase of flow resistance, it shouldbe mostly dictated by morphological changes—scour hole—observed in the bed profile cf. Leal 2005.Maxima Wave LevelsIt was found that maxima wave levels, Zsmax, can be fitted bythe following relation1/Zsmax = a1 + a3X + a2 + a4 1where a1, a2, a3 and a4=regressions coefficients which depend onthe bed type, according to Table 2. Fig. 3 plots Eq. 1 against theexperimental data. The equation proposed by Lauber and Hager1998 for fixed, horizontal dry bedZsmax =491 + 1X−5/4 2where X=rX−2/3, is also included in Fig. 3a. For fixed andsand bed tests,  0.2, as well as for pumice bed tests, 0.3, 1 / Zsmax increases linearly with X, i.e., the maximumwater surface elevation decreases with X. This can be attributed tothe finite length of the reservoir and is in accordance with re-ported real cases. However, for higher values of , waves behaveas solitary waves or groups of waves, the kinetic energy convertsinto potential energy through the increase of the water surfaceelevation and the influence of the upstream reservoir length ulti-mately ceases. The influence of the initial downstream water leveli.e.,  decays with X, since, for high X values, maxima waterFig. 2. Time evolution of dimensionless water level, Zs, at X=0.5, 2=0.03; b =0.11; c =0.20; and d =0.51Table 2. Regression Coefficients of Eq. 1Type of beda1a2a3a4Fixed 0.2233 4.3681 0.0431 2.2354Sand 0.2598 4.4543 0.0630 2.0880Pumice 0.1894 5.1618 0.0744 1.7340.5, 5.0, 7.5, 12.5, 17.5 and 22.5 right to left for pumice bed tests: a JOURNADownloaded 09 Dec 2009 to 193.136.126.36. Redistribution subject toFig. 3. Variation of dimensionless maximum wave level, 1 / Zsmax,with X for: a fixed bed; b sand bed; and c pumice bedL OF HYDRAULIC ENGINEERING © ASCE / NOVEMBER 2009 / 997 ASCE license or copyright; see http://pubs.asce.org/copyrightsurface levels tend to a constant value irrespective of .Although the described behavior applies to the three bed types,pumice bed results are somewhat different from fixed bed andsand bed. For small values of , the water surface level is slightlyhigher, which is in accordance to what was mentioned before, i.e.,the bed morphological changes and the flow-bed interaction in-crease the maxima wave levels, especially near the gate crosssection.Time to PeakFig. 4 presents the variation of the dimensionless time to peak,T=TmaxX−2/3, with X. Tmax= tmaxg /hu1/2; tmax=time to peak, as-sociated to the maximum water level. The equation proposed byLauber and Hager 1998 for fixed, horizontal dry bed is alsoplottedT = 1.71 + X 3From Fig. 4, it can be concluded that the equation of Lauber andFig. 4. Variation of dimensionless time to peak, T, with X: a fixedbed; b sand bed; and c pumice bedHager 1998 is a rather good predictor as soon as  0.14 and998 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / NOVEMBER 200Downloaded 09 Dec 2009 to 193.136.126.36. Redistribution subject toX15. Since higher values of X correspond to cross sectionslocated near the gate, it can also be concluded that the morpho-logical changes occurring in the mobile bed influence mostly thetime to peak at those sections. For high  values, the wavemaxima levels are attained shortly after the wave-front arrival.Nevertheless, the influence of the water depth downstream i.e., on time to peak is smaller as X decreases.ConclusionsFrom the previous sections and within the range of covered ex-perimental conditions, the following conclusions can be drawn:1 maxima wave levels near the gate are higher and attainedsooner for highly movable bed material pumice as comparedwith fixed bed, and become independent of the upstream reservoirlength; 2 for small initial water depth downstream the gate, themaxima wave levels decrease in the downstream direction, whilethe opposite trend is observed for high initial water depth; 3 theinfluence of the water depth downstream—expressed by —onthe amplitude of water level variations and on time to peak isattenuated in the downstream direction.These results stress the importance of using mathematicalmodels incorporating appropriate sediment transport formulationsinstead of purely hydrodynamic models to propagate dam-breakwaves on mobile bed channels.AcknowledgmentsThe writers wish to acknowledge the financial support of the Por-tuguese Foundation for Science and Technology through theGrant Nos. POCTI/36069/ECM/99 and PTDC/ECM/70652/2006.NotationThe following symbols are used in this technical note:a1, a2, a3, a4  regressions coefficients of Eq. 1;CNF  dimensionless celerity of the negativewave-front;ds  sediment mean diameter;g  gravitational acceleration;h  flow depth;hd  initial water depth downstream;hu  initial water depth upstream;Lr  upstream reservoir length;s  sediment specific gravity;T  dimensionless time after the dam-break;T  dimensionless time to peak defined asTmaxX−2/3;Tmax  dimensionless time to peak;t  time after the dam-break;tmax  time to peak;X  dimensionless location from the dam;X  dimensionless length parameter defined asrX−2/3;x  location from the dam section;Zs  dimensionless water surface elevationmeasured from the initial water leveldownstream of the dam;zs  water surface elevation measured from theinitial water level downstream of the dam;9 ASCE license or copyright; see http://pubs.asce.org/copyright  dimensionless initial downstream water depth;andr  dimensionless upstream reservoir length.ReferencesCapart, H., and Young, D. L. 1998. “Formation of a jump by the dam-break wave over a granular bed.” J. Fluid Mech., 372, 165–187.Lapointe, M. F., Secretan, Y., Driscoll, S., Bergeron, N., and Leclerc, M.1998. “Response of the Ha! Ha! River to the flood of July 1996 inthe Saguenay Region of Quebec: Large-scale avulsion in a glaciatedvalley.” Water Resour. Res., 34, 2383–2392.JOURNADownloaded 09 Dec 2009 to 193.136.126.36. Redistribution subject toLauber, G., and Hager, W. H. 1998. “Experiments to dambreak wave:Horizontal channel.” J. Hydraul. Res., 363, 291–307.Leal, J. G. A. B. 2005. “Experimental and mathematical modeling ofdam-break waves over mobile bed open channels.” Ph.D. thesis, Uni-versidade da Beira Interior, Covilhã, Portugal in Portuguese.Leal, J. G. A. B., Ferreira, R. M. L., and Cardoso, A. H. 2002. “Dam-break waves on movable bed.” Proc., of the Int. Conf. on FluvialHydraulics, Balkema, Rotterdam, The Netherlands, 981–990.Leal, J. G. A. B., Ferreira, R. M. L., and Cardoso, A. H. 2006. “Dam-break wave-front celerity.” J. Hydraul. Eng., 1321, 69–76.Spinewine, B., and Zech, Y. 2007. “Small-scale laboratory dam-breakwaves on movable beds.” J. Hydraul. Res., 45 Extra Issue, 73–86.Stansby, P. K., Chegini, A., and Barnes, T. C. D. 1998. “The initialstages of dam-break flow.” J. Fluid Mech., 374, 407–424.L OF HYDRAULIC ENGINEERING © ASCE / NOVEMBER 2009 / 999 ASCE license or copyright; see http://pubs.asce.org/copyright

Maximum level and time to peak of dam-break waves on mobile horizontal bed

New University of Lisbon's Repository

Maximum Level and Time to Peak of Dam-Break Waves on
Mobile Horizontal Bed
João G. A. B. Leal1; Rui M. L. Ferreira2; and António H. Cardoso3
Abstract: This experimental study focuses the influence of bed material mobility and initial downstream water level on maximum water
level and time to peak of dam-break waves. It covers horizontal bed conditions on fixed bed, sand bed, and pumice bed. Results include
water surface level time evolution, maxima wave levels and time to peak. The influence of bed material mobility and downstream water
level was identified and characterized, stressing the importance of using mathematical models with appropriate sediment transport
formulations instead of purely hydrodynamic models to simulate dam-break waves on mobile bed channels.
DOI: 10.1061/ASCEHY.1943-7900.0000099
CE Database subject headings: Dams; Water waves; Bed materials; Sediment transport.Introduction
The most important features of dam-break flows DBF are the
maximum wave level and time to peak. Lauber and Hager 1998
have characterized experimentally these variables for fixed, hori-
zontal, initially dry bed downstream the dam, taking into consid-
eration the influence of the upstream reservoir length.
Recent events and observations point out that DBF can interact
strongly with the mobile bed, causing major impacts in its mor-
phology. This was the case, for instance, downstream of Lake Ha!
Ha! due to the rupture of a dyke Lapointe et al. 1998. The
occurrence of intense sediment transport in DBF as well as the
interaction between the flow and the bed can significantly affect
the flow dynamics, usually contributing to increase the water sur-
face levels and to decrease the wave-front celerity as compared
with purely hydrodynamic solutions cf. Leal et al. 2006. Never-
theless, such solutions are frequently used due to difficulties in
the implementation and exploitation of models accounting for the
bed mobility.
The role of bed mobility in DBF was scarcely studied in the
past. For lightweight granular bed material, Capart and Young
1998 observed the excavation of a scour hole near the dam cross
section, leading to the onset of an hydraulic jump. Spinewine and
Zech 2007 concluded that the scour hole and the hydraulic jump
were significantly attenuated for heavier, less mobile bed. The
effect of the water level downstream the dam was rarely studied
1Assistant Professor, Dept. of Civil Engineering, Faculty of Sciences
and Technology, Universidade Nova de Lisboa, Quinta da Torre, Caparica
2829-516, Portugal corresponding author. E-mail: jleal@fct.unl.pt
2Assistant Professor, Dept. of Civil Engineering and Architecture,
Instituto Superior Técnico, UTL, Av. Rovisco Pais, Lisbon 1049-001,
Portugal. E-mail: ruif@civil.ist.utl.pt
3Full Professor, Dept. of Civil Engineering and Architecture, Instituto
Superior Técnico, UTL, Av. Rovisco Pais, Lisbon 1049-001, Portugal.
E-mail: ahc@civil.ist.utl.pt
Note. This manuscript was submitted on November 16, 2007; ap-
proved on May 19, 2009; published online on May 21, 2009. Discussion
period open until April 1, 2010; separate discussions must be submitted
for individual papers. This technical note is part of the Journal of Hy-
draulic Engineering, Vol. 135, No. 11, November 1, 2009. ©ASCE,
ISSN 0733-9429/2009/11-995–999/$25.00.
JOURNA
Downloaded 09 Dec 2009 to 193.136.126.36. Redistribution subject totoo. Stansby et al. 1998 and Leal et al. 2002 verified that the
water surface level behind the wave-front raises as compared with
the pure hydrodynamic solution, creating a jump whose height
increases with the initial downstream water level. The present
experimental study intends to contribute to further characterize
the influence of bed mobility and initial water level downstream
the dam on maximum wave level and time to peak in horizontal,
mobile bed channels.
Experiments
The data used in this study were collected by Leal 2005. Tests
were carried out in a 19.2-m long, 0.5-m wide, and 0.7-m high
rectangular flume with horizontal bed. For simulating dam-break
events, a vertical lift-gate, installed in the middle cross section of
the flume, was opened instantaneously. Three types of bed were
adopted: fixed bed, sand bed, and pumice bed. The sand diameter
was ds=0.8 mm; sand specific gravity was s=2.65. For pumice,
these variables were 1.2 mm and 1.40, respectively. The initial
water depth upstream the gate was the same in all tests hu
=0.40 m; for mobile bed tests, the initial bed elevation was al-
ways kept equal to 0.07 m, both up and downstream the gate; the
initial water depth downstream, hd, ranged from 0.000–0.210 m.
The main features of the initial conditions are summarized in
Table 1, where =hd /hu is the dimensionless initial downstream
water depth. Assuming hydrostatic pressure distribution, the time
evolution of water surface levels was measured with seven pres-
sure transducers installed downstream the gate, at X=0.5, 2.5, 5.0,
7.5, 12.5, 17.5, and 22.5, where X=x /hu and x=longitudinal co-
ordinate measured from dam cross section.
Time Evolution of Water Surface Level
Fig. 1 presents the time evolution of dimensionless water surface
levels, Zs

, of fixed bed tests, =0.00, 0.06, 0.14, and 0.52. In Fig.
1, Zs

=zs
 /hu; zs=measured water surface elevation above the ini-
tial water level downstream the gate; T= t / hu /g0.5
=dimensionless time; t=time; g=gravitational acceleration. The
maxima wave levels are marked with circles; the instants of
L OF HYDRAULIC ENGINEERING © ASCE / NOVEMBER 2009 / 995
 ASCE license or copyright; see http://pubs.asce.org/copyright
time—time limits—above which the limited upstream reservoir
length starts to influence the wave levels at the measuring cross
sections are also identified with vertical dashed lines. Time limits
were calculated by assuming that the negative wave-front celeri-
ties measured by Leal et al. 2002 also apply downstream. For a
given location, X, the time limit is therefore computed as 2r
+X /CNF, where r=Lr /hu=dimensionless reservoir length; Lr
=length of the upstream reservoir; CNF=dimensionless negative
wave-front celerity.
For the lower values of  =0.00, Fig. 1a, and =0.06, Fig.
1b, the water surface level rises abruptly, in each section, im-
mediately after the arrival of the wave-front; afterwards, it in-
creases slowly until reaching the maximum; afterwards, the water
level decreases slowly as the upstream reservoir empties. This
behavior was also observed by Lauber and Hager 1998 for fixed
horizontal bed. For higher values of  =0.14, Fig. 1c, and
=0.52, Fig. 1d, water levels are influenced, after some time,
by reflection waves induced by the downstream weir used to guar-
antee the water level downstream the gate. Whenever this oc-
curred the “corrupted” parts of the records were ignored in the
analysis. As  increases, the maxima wave levels are attained
Table 1. Tests Main Features
Test Type of bed
hd
m


T.1 fixed bed 0.000 0.00
T.2 0.023 0.06
T.3 0.049 0.12
T.4 0.056 0.14
T.5 0.071 0.18
T.6 0.082 0.20
T.7 0.094 0.23
T.8 0.114 0.28
T.9 0.142 0.35
T.10 0.212 0.52
Fig. 1. Time evolution of dimensionless water level, Zs, at X=0.5,
=0.00; b =0.06; c =0.14; and d =0.52996 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / NOVEMBER 200
Downloaded 09 Dec 2009 to 193.136.126.36. Redistribution subject tosooner and tend to become independent of the reservoir length,
i.e., they occur before the calculated time limit. For the highest
value of  =0.52, Fig. 1d the maxima wave levels are at-
tained just after the arrival of the wave-front due to the fact that
the wave propagates downstream as a solitary wave or as a group
of waves cf. Leal 2005.
For sand bed, the time evolution of the water surface elevation
is practically as described above. Still, the maxima wave levels at
X=0.5 are attained slightly sooner cf. Leal 2005.
Fig. 2 presents the time evolution of the dimensionless water
surface levels of pumice bed tests, =0.03, 0.11, 0.20 and 0.51.
Again, the water surface behavior presents qualitative similarities
with the one described for fixed bed. Important discrepancies are
observed, however, at the cross section the nearest to the gate for
the lower  values Figs. 2a and b, where the water surface
level does not rise slowly until reaching its maximum. Instead, it
exhibits an oscillatory behavior; the maximum level is attained
much sooner than for the other sections and is independent of the
reservoir length. It can be concluded that, for locations near the
gate, the increase of bed mobility induces the increase of the
water surface levels. Although this can be attributed to the added
Type of bed
hd
m


Sand ds=0.8 mm s=2.65 0.003 0.01
0.012 0.03
0.042 0.10
0.073 0.18
0.106 0.26
0.204 0.51
Pumice ds=1.2 mm s=1.40 0.013 0.03
0.014 0.04
0.044 0.11
0.078 0.20
0.107 0.27
0.207 0.51
.0, 7.5, 12.5, 17.5 and 22.5 right to left for fixed bed tests: a Test
Ts.1
Ts.2
Ts.3
Ts.4
Ts.5
Ts.6
Tp.1
Tp.2
Tp.3
Tp.4
Tp.5
Tp.62.5, 59
 ASCE license or copyright; see http://pubs.asce.org/copyright
sediment inertia and to the increase of flow resistance, it should
be mostly dictated by morphological changes—scour hole—
observed in the bed profile cf. Leal 2005.
Maxima Wave Levels
It was found that maxima wave levels, Zsmax, can be fitted by
the following relation
1/Zs
max = a1 + a3X + a2 + a4 1
where a1, a2, a3 and a4=regressions coefficients which depend on
the bed type, according to Table 2. Fig. 3 plots Eq. 1 against the
experimental data. The equation proposed by Lauber and Hager
1998 for fixed, horizontal dry bed
Zs
max =
4
91 + 1X
−5/4
2
where X=rX−2/3, is also included in Fig. 3a. For fixed and
sand bed tests, 0.2, as well as for pumice bed tests, 
0.3, 1 / Zs
max increases linearly with X, i.e., the maximum
water surface elevation decreases with X. This can be attributed to
the finite length of the reservoir and is in accordance with re-
ported real cases. However, for higher values of , waves behave
as solitary waves or groups of waves, the kinetic energy converts
into potential energy through the increase of the water surface
elevation and the influence of the upstream reservoir length ulti-
mately ceases. The influence of the initial downstream water level
i.e.,  decays with X, since, for high X values, maxima water
Fig. 2. Time evolution of dimensionless water level, Zs, at X=0.5, 2
=0.03; b =0.11; c =0.20; and d =0.51
Table 2. Regression Coefficients of Eq. 1
Type of bed
a1

a2

a3

a4

Fixed 0.2233 4.3681 0.0431 2.2354
Sand 0.2598 4.4543 0.0630 2.0880
Pumice 0.1894 5.1618 0.0744 1.7340.5, 5.0, 7.5, 12.5, 17.5 and 22.5 right to left for pumice bed tests: a JOURNA
Downloaded 09 Dec 2009 to 193.136.126.36. Redistribution subject toFig. 3. Variation of dimensionless maximum wave level, 1 / Zsmax,
with X for: a fixed bed; b sand bed; and c pumice bedL OF HYDRAULIC ENGINEERING © ASCE / NOVEMBER 2009 / 997
 ASCE license or copyright; see http://pubs.asce.org/copyright
surface levels tend to a constant value irrespective of .
Although the described behavior applies to the three bed types,
pumice bed results are somewhat different from fixed bed and
sand bed. For small values of , the water surface level is slightly
higher, which is in accordance to what was mentioned before, i.e.,
the bed morphological changes and the flow-bed interaction in-
crease the maxima wave levels, especially near the gate cross
section.
Time to Peak
Fig. 4 presents the variation of the dimensionless time to peak,
T=TmaxX−2/3, with X. Tmax= tmaxg /hu1/2; tmax=time to peak, as-
sociated to the maximum water level. The equation proposed by
Lauber and Hager 1998 for fixed, horizontal dry bed is also
plotted
T = 1.71 + X 3
From Fig. 4, it can be concluded that the equation of Lauber and
Fig. 4. Variation of dimensionless time to peak, T, with X: a fixed
bed; b sand bed; and c pumice bedHager 1998 is a rather good predictor as soon as 0.14 and
998 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / NOVEMBER 200
Downloaded 09 Dec 2009 to 193.136.126.36. Redistribution subject toX15. Since higher values of X correspond to cross sections
located near the gate, it can also be concluded that the morpho-
logical changes occurring in the mobile bed influence mostly the
time to peak at those sections. For high  values, the wave
maxima levels are attained shortly after the wave-front arrival.
Nevertheless, the influence of the water depth downstream i.e.,
 on time to peak is smaller as X decreases.
Conclusions
From the previous sections and within the range of covered ex-
perimental conditions, the following conclusions can be drawn:
1 maxima wave levels near the gate are higher and attained
sooner for highly movable bed material pumice as compared
with fixed bed, and become independent of the upstream reservoir
length; 2 for small initial water depth downstream the gate, the
maxima wave levels decrease in the downstream direction, while
the opposite trend is observed for high initial water depth; 3 the
influence of the water depth downstream—expressed by —on
the amplitude of water level variations and on time to peak is
attenuated in the downstream direction.
These results stress the importance of using mathematical
models incorporating appropriate sediment transport formulations
instead of purely hydrodynamic models to propagate dam-break
waves on mobile bed channels.
Acknowledgments
The writers wish to acknowledge the financial support of the Por-
tuguese Foundation for Science and Technology through the
Grant Nos. POCTI/36069/ECM/99 and PTDC/ECM/70652/2006.
Notation
The following symbols are used in this technical note:
a1, a2, a3, a4  regressions coefficients of Eq. 1;
CNF  dimensionless celerity of the negative
wave-front;
ds  sediment mean diameter;
g  gravitational acceleration;
h  flow depth;
hd  initial water depth downstream;
hu  initial water depth upstream;
Lr  upstream reservoir length;
s  sediment specific gravity;
T  dimensionless time after the dam-break;
T  dimensionless time to peak defined as
TmaxX−2/3;
Tmax  dimensionless time to peak;
t  time after the dam-break;
tmax  time to peak;
X  dimensionless location from the dam;
X  dimensionless length parameter defined as
rX−2/3;
x  location from the dam section;
Zs
  dimensionless water surface elevation
measured from the initial water level
downstream of the dam;
zs
  water surface elevation measured from theinitial water level downstream of the dam;
9
 ASCE license or copyright; see http://pubs.asce.org/copyright
  dimensionless initial downstream water depth;
and
r  dimensionless upstream reservoir length.
References
Capart, H., and Young, D. L. 1998. “Formation of a jump by the dam-
break wave over a granular bed.” J. Fluid Mech., 372, 165–187.
Lapointe, M. F., Secretan, Y., Driscoll, S., Bergeron, N., and Leclerc, M.
1998. “Response of the Ha! Ha! River to the flood of July 1996 in
the Saguenay Region of Quebec: Large-scale avulsion in a glaciated
valley.” Water Resour. Res., 34, 2383–2392.
JOURNA
Downloaded 09 Dec 2009 to 193.136.126.36. Redistribution subject toLauber, G., and Hager, W. H. 1998. “Experiments to dambreak wave:
Horizontal channel.” J. Hydraul. Res., 363, 291–307.
Leal, J. G. A. B. 2005. “Experimental and mathematical modeling of
dam-break waves over mobile bed open channels.” Ph.D. thesis, Uni-
versidade da Beira Interior, Covilhã, Portugal in Portuguese.
Leal, J. G. A. B., Ferreira, R. M. L., and Cardoso, A. H. 2002. “Dam-
break waves on movable bed.” Proc., of the Int. Conf. on Fluvial
Hydraulics, Balkema, Rotterdam, The Netherlands, 981–990.
Leal, J. G. A. B., Ferreira, R. M. L., and Cardoso, A. H. 2006. “Dam-
break wave-front celerity.” J. Hydraul. Eng., 1321, 69–76.
Spinewine, B., and Zech, Y. 2007. “Small-scale laboratory dam-break
waves on movable beds.” J. Hydraul. Res., 45 Extra Issue, 73–86.
Stansby, P. K., Chegini, A., and Barnes, T. C. D. 1998. “The initial
stages of dam-break flow.” J. Fluid Mech., 374, 407–424.
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Maximum level and time to peak of dam-break waves on mobile horizontal bed

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