Discussion of “Modeling the Evolution ofIncised Streams. II: Streambank Erosion”by Eddy J. Langendoen and Andrew SimonJuly 2008, Vol. 134, No. 7, pp. 905–915.DOI: 10.1061/�ASCE�0733-9429�2008�134:7�905�
Patric Rousselot11Research Assoc., Laboratory of Hydraulics, Hydrology and Glaciology
�VAW�, ETH Zurich, Switzerland.
The authors are to be congratulated on having successfully imple-mented a conceptual model for two bank failure mechanisms intotheir numerical model CONservational Channel Evolution andPollutant Transport System �CONCEPTS� and on having vali-dated it against long-term field data of streambank erosion. Acomparison of measured and simulated cross sections at the studysite indicates reasonable behavior of their numerical model ofbank stability for streambank failure.
As noted by the authors, CONCEPTS does not predict theincreased shear stress along an outer riverbank caused by thehelical flow pattern within a bend. Curved channel flow is asso-ciated with a helical motion—often termed “secondary flow”—generating a transverse velocity component in addition to themain streamwise velocity. Yalin and da Silva �2001� mention theprominence of the secondary flow that progressively decreases asthe width-to-depth ratio B /h of a curve increases.
The authors’ treatment of the missing helical motion withintheir field validation is, however, questionable. The increasedshear stresses at the bend apex due to secondary flow are ac-counted for by a reduction of the critical shear stress. However,the range of critical shear stresses used, between 1 Pa up- anddownstream of the bend and 8 Pa at the bend apex, is surprising.This is nearly a one-order-of-magnitude difference, while themagnitude of the secondary flow velocity is typically some 10%of the streamwise velocity magnitude �Blanckaert and de Vriend2004�. The range of the selected critical shear stresses appearstherefore to be extremely large. This is also reflected by the re-sults presented. Cross sections with a small critical shear stress of1 Pa underlie stronger and temporally increased streambank re-treat than those considered using a higher critical shear stress of 4Pa. The discusser therefore questions the sensitivity of the au-thors’ approach relative to the selected model to account for sec-ondary flow. Were simulations also made with a larger or smallerratio of critical shear stresses at cross sections along a typicalbend? Finally, the discusser would like to ask the authors how thecritical shear stresses can be selected in terms of bend hydraulics,topography and geotechnical data?
References
Blanckaert, K., and de Vriend, H. J. �2004�.“Secondary flow in sharpopen-channel bends.” J. Fluid Mech., 498, 353–380.
Yalin, M. S., and Ferreira da Silva, A. M. �2001�. Fluvial processes,
IAHR, Delft, The Netherlands.
JOURNAL
J. Hydraul. Eng. 2009.1
Closure to “Modeling the Evolution ofIncised Streams. II: Streambank Erosion”by Eddy J. Langendoen and Andrew SimonJuly 2008, Vol. 134, No. 7, pp. 905–915.DOI: 10.1061/�ASCE�0733-9429�2008�134:7�905�
Eddy J. Langendoen1 and Andrew Simon2
1Research Hydraulic Engineer, Agricultural Research Service, NationalSedimentation Laboratory, U.S. Dept. of Agriculture, P.O. Box 1157,Oxford, MS 38655. E-mail: [email protected]
2Research Geologist, Agricultural Research Service, National Sedimenta-tion Laboratory, U.S. Dept. of Agriculture, P.O. Box 1157, Oxford,MS 38655. E-mail: [email protected]
The writers appreciate the comments by the discusser. The origi-nal paper showed that streambank erosion can be satisfactorilypredicted by one-dimensional channel evolution computer modelssuch as CONCEPTS if both the resistance to erosion of the basalbank materials and the applied shear stresses by the flow areadequately characterized and calculated, respectively. The appliedbasal shear stresses can be satisfactorily predicted in straight,prismatic channels using simple geometric methods �Khodash-enas and Paquier 1999�. However, as noted by both the writers inthe original paper and by the discusser, a one-dimensional modelcannot predict the increased hydraulic shear stresses on the outerbank in a meander bend such as the Goodwin Creek Bendway�GCB� study site. Consequently, the rate of basal erosion may beunderpredicted.
CONCEPTS calculates basal erosion using an excess shearstress equation �Eq. �24�, Langendoen and Alonso 2008�
E = M��/�c − 1� �1�
where E=erosion rate; M =erosion rate coefficient; � � appliedshear stress; and �c=critical shear stress. Therefore, to determinethe enhanced outer-bank erosion, the model can either estimatethe increased shear stress � or use a reduced critical shear stress �c
such that the shear stress ratio � /�c is correctly reproduced. Thelatter approach was used for the GCB validation study in theoriginal paper.
By validating against observed retreat rates the writers reducedthe median value of the measured critical shear stress of the basalmaterial at the GCB study site, which is approximately 8 Pa, to 1Pa near the bend apex �20-m-long section� and to 4 Pa along20-m-long sections upstream and downstream of the bend apex.The discusser erroneously states that the largest critical shearstresses were used at the bend apex, with reduced critical shearstresses away from the apex. Using the above values of criticalshear stress, the writers obtained excellent agreement betweenpredicted and observed outer-bank retreat rates. The discusser ex-presses surprise at the rather large two- to eightfold reduction incritical shear stress, since the secondary flow is typically about10% of the streamwise velocity magnitude. However, the in-creased shear stresses are not caused by the secondary flow di-rectly, but rather by a shift of the high-velocity core toward the
outer bank near the bend apex due to topographic steering by the
point bar �Nelson and Smith 1989� and advection by the second-ary flow �Blanckaert and de Vriend 2003�.
Fig. 1 shows simulated distributions of streamwise velocityand boundary shear stress at a cross section near the apex and across section 30 m upstream of the apex of the GCB study site atpeak flow �about 70 m3 /s� during the flood event of April 6, 2005�Jia et al. 2008�. The modeling results clearly show the shift of thehigh-velocity core toward the outer bank �Figs. 1�a and c�� result-ing in a four- to eightfold increase in boundary shear stress withinthe basal area of the streambank �Figs. 1�b and d� at a distance of33 m from the left bank�. This agrees very well with the decreasein critical shear stress used in the CONCEPTS simulation.
The difficulty then is to determine the reduction factor thatmust be applied to the critical shear stress. The reduction factorwill be a function of bend hydraulics and topography. The writerspropose the following method as a first approximation. First, weassume that the position of the high-velocity core is determinedby the local bed topography in the bend; that is, the highest bedshear stress and flow velocities occur where bed erosion is largest�thalweg and pool locations�. Further, we assume that the shearstress on the basal area of the streambank is proportional to thevelocity gradient normal to the bank
� �U
L�2�
where U is a representative value of the streamwise velocity inthe high-velocity core; and L=distance from the high velocitycore to the streambank. The reduction factor then equals the
0 5 10 15 20 25 30 3578
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ELEVATION,INMETERS
VELOCITY,IN M/S
2.662.301.941.581.210.850.490.13
(a)30 METERS UPSTREAM OF BENDAPEX
Fig. 1. Simulated distributions of streamwise velocity �a and c� andboundary shear stress �b and d� for a cross section 30 m upstream ofthe apex �a and b� and a cross section near the apex �c and d� of theGoodwin Creek Bendway study site �adapted from Jia et al. 2008�
change in normal distance L. The distance L can be obtained from
the surveyed reach geometry. In case of the GCB study site Lreduces approximately from 14.5 m at the bend entrance, to 7.7 m30 m upstream of the bend apex, and to 2.6 m at the bend apex�cf. Fig. 3 in the original manuscript�. As a first approximation theresulting reduction factors at the apex and immediately upstreamof the apex of the bend are 6 and 2, respectively. These reductionfactors agree well with those found, respectively 8 and 2, byvalidating the rate of bank retreat.
References
Blanckaert, K., and de Vriend, H. J. �2003�. “Nonlinear modeling of meanflow redistribution in curved open channels.” Water Resour. Res.,39�12�, 1375.
Jia, Y., Alonso, C. V., Simon, A., Wells, R. R., and Wang, S. S. Y. �2008�.“Modeling flow and vegetation effects in a curved channel.” Proc.,World Environmental and Water Resources Congress 2008 Ahupua’a,R. W. Babcock Jr. and R. Walton, eds., ASCE, Reston, Va., 13.
Khodashenas, S. R., and Paquier, A. �1999�. “A geometrical method forcomputing the distribution of boundary shear stress across irregularstraight open channels.” J. Hydraul. Res., 37�3�, 381–388.
Langendoen, E. J., and Alonso, C. V. �2008�. “Modeling evolution ofincised streams: I. Model formulation and validation of flow and stre-ambed evolution components.” J. Hydraul. Res., 134�6�, 749–762.
Nelson, J. M., and Smith, J. D. �1989�. “Evolution and stability of erod-ible channel beds.” River Meandering, Water Resources Monograph12, S. Ikeda and G. Parker, eds., American Geophysical Union, Wash-ington, D.C., 321–377.
Discussion of “Automatic Calibration of theU.S. EPA SWMM Model for a Large UrbanCatchment” by J. Barco, K. M. Wong, andM. K. StenstromApril 2008, Vol. 134, No. 4, pp. 466–474.DOI: 10.1061/�ASCE�0733-9429�2008�134:4�466�
James E. Ball, M.ASCE1
1School of Civil and Environmental Engineering, Univ. of Technology-Sydney, Sydney, Australia.
The authors have contributed to and illustrated the need for con-tinued research into calibration methodologies for complex catch-ment modeling systems. The search for suitable approaches forparameter evaluation has resulted in the development of manynew techniques and concepts. The authors have continued thetraditional approach of identifying a unique optimal parameter setor near optimal parameter set that is assumed to represent thegeneric catchment characteristics. There are concerns with thisapproach and outlined herein is one of these concerns, namely theidentification of a globally optimal set of parameters that does notrepresent the generic catchment characteristics due to the follow-ing:• Data errors in the input to the model and that used to assess the
performance of the parameter values.• Uniformity in the performance of alternative parameter sets.
Many studies have demonstrated the difficulties, if not the im-possibility, of finding a unique optimal parameter set due to un-certainty of model structure, errors associated with input and
observed data, and interactions between parameters �Kuczera
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1983; Sorooshian et al. 1983; Beven and Binley 1992; Gan andBiftu 1996�. As a result of these sources of error, an optimalparameter set for one set of events may not be optimal for otherevents. Searching for a unique optimal value may lead to theparameter evaluation being based on the best “curve fitting”rather than the best representation of the catchment processes.This was explored by Choi and Ball �2002� who proposed the useof monitoring data to define the point where further parametermodification does not result in additional information being ex-tracted from the available data. The conceptual basis of this ap-proach is based on dividing the available data into calibration,monitoring, and validation data sets and using the monitoring dataat each �or a predefined number of iterations� iteration of theparameter modification as the optimal set is obtained to ensurethat the objective function used to measure the improvement inperformance decreases for events other than those used for thecalibration. As shown in Fig. 1, the process ends when furtheriterations result in a decrease in the performance of the monitor-ing data even though the performance of the calibration data con-tinues to improve. Results from an application of this approachwith SWMM to a catchment in Sydney, NSW �Choi and Ball2002� are shown in Table 1; for this application of the approach81.25% of the calibrations were concluded prior to reaching themaximum number of iterations. Choi and Ball �2002� found thatcontinuing the search beyond the “early stop point” resulted in adecrease in the performance of the parameter values when appliedto different events to those being used for the calibration and,hence, they postulated that the perceived improvement in perfor-mance past the “early stop point” was due to the model perform-ing as a “curve fitting” transformation rather than one where themodel was replicating the major catchment processes.
Fig. 1. Early stopping technique �adapted from Choi and Ball 2002�
Table 1. Occurrence Rate of the Minimum Function Value �Adapted fro
EventSP�%�
Nov. 01, 1994 0
Dec. 22, 1994 0
Jan. 04, 1995 0
Feb. 28, 1995 25
Mean 6.25
Note: SP=start point; ESP=early stop point; and EP=end point.
JOURNAL
J. Hydraul. Eng. 2009.1
The authors have not tested the parameter values developedduring their optimization process with storm events not used aspart of the calibration process. If the authors applied a monitoringapproach to their data set, it would be interesting to see if thesame parameter values were obtained.
This problem of identifying the point where further modifica-tions to parameter values does not result in the extraction of ad-ditional information from the available data suggests that thereare many alternative combinations of parameter values that resultin similar performance. This has lead to development of tech-niques for estimating the parameter uncertainty for simple catch-ment modeling systems. Simple modeling systems can becategorized as those systems where evaluation of only a few pa-rameters is required for application. Examples of these ap-proaches are:• Bayesian methodology first explored by Kuczera �1983�,
whereby parameter uncertainty is described by the posteriordistribution, which expresses the probability of the parametervalues given the observed data. Marshall et al. �2004�, how-ever, claim that while the Bayesian frameworks are widelyused, the implementation of Bayesian procedures has beenhindered due to difficulties in summarizing and exploring theposterior distribution of parameters for complex catchmentmodeling systems.
• Markov Chain Monte Carlo �MCMC� approaches as presentedby Kuczera and Parent �1998�, Bates and Campbell �2001�,Marshall et al. �2004�, and Gallagher and Doherty �2007�.While these approaches provide computationally feasibleimplementations of Bayesian inference with the aim of gener-ating samples of parameter values from the posterior distribu-tion with reasonable efficiency, a priori knowledge about theproposal distribution of parameters is crucial for effectiveimplementation of a MCMC algorithm.
• The generalized likelihood uncertainty estimation �GLUE�method as presented by Beven and Binley �1992�. Applicationof a GLUE methodology usually involves making a largenumber of Monte Carlo �MC� simulations with different setsof parameter values, generated randomly from uniform distri-butions within the feasible parameter space. While the GLUEmethodology is capable of exploring the whole search space, itis computationally inefficient when very large numbers of ini-tial parameter sets are required �Spear et al. 1994; Bates andCampbell 2001�. To mitigate this problem, a number of studieshave investigated methods for improving the efficiency ofMC-based techniques. An approach commonly adopted �Hel-ton and Davis 2003� has been to use a more efficient samplingalgorithm, such as Latin hypercube sampling. Another alterna-tive was presented by Khu and Werner �2003� who used ahybrid genetic algorithm and artificial neural network, knownas GAANN to improve the efficiency of a GLUE approach.
Building on these studies into the parameter uncertainty, Fangand Ball �2007� used a genetic algorithm �GA� within a GLUEframework to investigate the parameter uncertainty associatedwith the application of SWMM for flow prediction in an urbancatchment. In this case the approach was not limited to a simplecatchment modeling system but rather to a complex catchmentmodeling system with a significant number of spatially variableinterrelated parameters. Defining a behavioral set as being a set ofparameters where the RMSE in discharge was less than 0.1 m3 /s,Fang and Ball �2007� found, after 50 generations with 1,000-parameter sets per generation, approximately 900 alternative setsof parameter values meeting the criterion. Shown in Table 2 arethe mean and standard deviation of the RMSE for these behav-ioral parameter sets. Using these values gives a coefficient ofvariation of approximately 1.5%, which can be interpreted as sug-gesting that there is minimal difference in the performance of anyone of the approx. 900 behavioral parameter sets highlighting thedifficulty of selecting one set of parameter values as the mostdesirable.
Using the concept that there are multiple alternative sets ofparameter values that result in similar performance, it would beinteresting if the authors could provide information about thevariability in the predicted flows of the best-performing sets ofparameter values. Inclusion of the concept of monitoring the cali-bration process in determining the best performing set of param-eter values would be useful also.
References
Bates, B. C., and Campbell, E. P. �2001�. “A Markov chain Monte Carloscheme for parameter estimation and inference in conceptual rainfall-runoff modeling.” Water Resour. Res., 37�4�, 937–947.
Beven, K., and Binley, A. �1992�. “The future of distributed models:Model calibration and uncertainty prediction.” Hydrolog. Process.,6�3�, 279–298.
Choi, K. S., and Ball, J. E. �2002�. “A generic calibration approach:Monitoring the calibration.” Proc., 2002 Hydrology and Water Re-
sources Symp., I. E. Aust.Fang, T., and Ball, J. E. �2007�. “Evaluation of spatially variable param-
eters in a complex system: An application of a genetic algorithm.” J.Hydroinform., 9�3�, 163–173.
Gallagher, M., and Doherty, J. �2007�. “Parameter estimation and uncer-tainty analysis for a watershed model.” Environ. Modell. Software,22�7�, 1000–1020.
Gan, T. Y., and Biftu, G. F. �1996�. “Automatic calibration of conceptualrainfall-runoff models: Optimization algorithms, catchment condi-tions, and model structure.” Water Resour. Res., 32�12�, 3513–3524.
Helton, J. C., and Davis, F. J. �2003�. “Latin hypercube sampling and thepropagation of uncertainty in analyses of complex systems.” Reliab.
Table 2. Performance of Behavioral Sets of Control Parameter Values�Adapted from Fang and Ball 2007�
Khu, S. T., and Werner, M. G. F. �2003�. “Reduction of Monte-Carlosimulation runs for uncertainty estimation in hydrological modeling.”Hydrology Earth Syst. Sci., 7�5�, 680–692.
Kuczera, G. �1983�. “Improved parameter inference in catchment models:1. Evaluating parameter uncertainty.” Water Resour. Res., 19�5�,1151–1162.
Marshall, L., Nott, D., and Sharma, A. �2004�. “A comparative study ofMarkov chain Monte Carlo methods for conceptual rainfall-runoffmodeling.” Water Resour. Res. 40�2�, W02501.
Sorooshian, S., Gupta, V. K., and Fulton, J. L. �1983�. “Evaluation ofmaximum likelihood parameter estimation techniques for conceptualrainfall-runoff models: Influence of calibration data variability andlength on model credibility.” Water Resour. Res., 19�1�, 251–259.
Spear, R. C., Grieb, T. M., and Shang, N. �1994�. “Parameter uncertaintyand interaction in complex environmental models.” Water Resour.Res., 30�11�, 3159–3169.
Closure to “Automatic Calibration of theUS EPA SWMM Model for a Large UrbanCatchment” by Janet Barco, Kenneth M.Wong, and Michael K. StenstromApril 2008, Vol. 134, No. 4, pp. 466–474.
DOI: 10.1061/�ASCE�0733-9429�2008�134:4�464�
Janet Barco1; Kenneth M. Wong2; andMichael K. Stenstrom, F.ASCE3
1Postdoctoral fellow, Civil and Environmental Engineering Dept., 5732Boelter Hall, Univ. of California-Los Angeles, Los Angeles, CA90095. E-mail: [email protected]
2Civil Engineer, City of Los Angeles Los Angeles World Airports, LosAngeles, CA 90045.
3Professor, Civil and Environmental Engineering Dept., 5714 BoelterHall, Univ. of California-Los Angeles, Los Angeles, CA 90095.E-mail: [email protected]
The writers are thankful to the discusser for suggesting alternativeways of identifying optimal or near-optimal parameter sets. Wenote that we divided the ten storm data sets into calibration andvalidation data sets by ranking the data sets with respect to rain-fall and selecting every other data set for calibration. The valuesare reported as “Pred/Valid” in Table 3 of the original paper. Themean value of parameters from these calibration data sets werethen used to simulate the other five storms. The results from thesefive simulations can be compared to the results when the param-eters for each storm are identified one by one �“Pred/Storm bystorm”�.
If we understand the discusser’s proposal, we would need tomodify our procedure by running two copies of SWMM, oneidentifying the parameters using a calibration data set, and thesecond using the identified parameters in a validation data set, atevery iteration. The procedure ends when the parameters from thecalibration data set no longer improve the error for the validationdata set. This procedure would be somewhat more tedious andrequire greater modifications �but feasible in our opinion� to theSWMM source code. Applying this technique to other models forsimilar purposes, such as HSPF �Ackerman et al. 2005�, would besignificantly more difficult. The benefits of this alternative proce-dure would have to be compared to the benefits of similar effort in
other aspects of the optimization or data acquisition.
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The results shown in Table 3 of the original paper can be usedto estimate the alternative calibration method proposed by thediscusser by comparing the optimal results between “Pred/Valid”and “Pred/Storm by storm.” Table 4 of the original paper showsthe relative differences between calibration method and data, butnot between calibration methods. The mean error differences be-tween storm-by-storm calibration for the five validation stormsand using the average parameters from the five calibration stormsis �7.8% for total volume and �11% for peak flow. Storms 5, 7,and 8 had 5% or less difference in total flow error. Storm 4 hadthe greatest difference with 28% difference, while storm 2 had10% difference in flow. The optimal parameter sets had a similartrend in error. These results suggest that there may be little dif-ference between the method suggested by the proposer andstraightforward optimization, at least for this work.
We acknowledge that there can be data sets and models whereadditional iterations in calibration data sets may produce moreerror in validation data sets. We believe the problem may be moresignificant with “synthetic datasets” where a model solution isperturbed with white noise and the parameters are identified as ademonstration of an optimization technique. Such data sets mayhave error structures quite different than actual data. An alternatemethod for identifying a global optimal set of parameters for theten storm data sets might be accomplished by sequential solutionof SWMM at every iteration for every data set. This would befeasible with the existing complex method and the existing modi-fied SWMM code. Weighting functions would need to be appliedto normalize the error for the different storms to compensate fortheir magnitudes. Early in this project we did this for several ofthe data sets shown in Table 3 of the original paper, but concludedthe additional benefits did not justify the effort.
As computing speeds increase, time constraints of slower con-verging methods are becoming less costly. This may allow sim-pler codes that are often easier to reliably implement, or may bemore robust with difficult objective functions, to be used by prac-titioners for a wide range of real-world problems. We view thecomplex method as one candidate. Our group has implemented itfor a wide range of problems �Stenstrom et al. 1981; Hwang andStenstrom 1985; Yuan et al. 1993; Tzeng et al. 2003�, and it couldbe used with the discusser’s technique.
References
Ackerman, D., Schiff, K. C., and Weisberg, S. B. �2005�. “EvaluatingHSPF in an arid, urban watershed.” J. Am. Water Resour. Assoc.,41�2�, 477–486.
Hwang, H. J., and Stenstrom, M. K. �1985�. “Evaluation of fine-bubblealpha factors in near-full scale equipment.” J. Water Pollut. ControlFed., 57�12�, 1142–1151.
Stenstrom, M. K., Brown, L. C., and Hwang, H. J. �1981�. “Oxygentransfer parameter estimation.” J. Envir. Engrg. Div., 107�2�, 379–397.
Tzeng, C. J., Iranpour, R., and Stenstrom, M. K. �2003�. “Modeling andcontrol of oxygen transfer in the HPO activated sludge process.” J.Environ. Eng., 129�5�, 402–411.
Yuan, W. B. W., Okrent, D., and Stenstrom, M. K. �1993�. “Model cali-bration for the high-purity oxygen activated sludge process—Algorithm development and evaluation.” Water Sci. Technol. 28�11–
12�, 163–171.
JOURNAL
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Discussion of “Numerical and ExperimentalStudy of Dividing Open-Channel Flows”by A. S. Ramamurthy, Junying Qu, andDiep VoOctober 2007, Vol. 133, No. 10, pp. 1135–1144.DOI: 10.1061/�ASCE�0733-9429�2007�133:10�1135�
Kamal El Kadi Abderrezzak1 and André Paquier2
1Research Engineer, Cemagref, Hydrology-Hydraulics Research Unit, 3bis quai Chauveau CP220, 69336 Lyon, Cedex 09, France; presently,Electricité de France �EDF�-R&D-LNHE, Quai Waiter, F-78400Chatou, France. E-mail: [email protected]
2Research Engineer, Hydrology-Hydraulics Research Unit, 3 bis quaiChauveau CP220, 69336 Lyon, Cedex 09, France. E-mail:[email protected]
The authors contributed to the study of dividing flows in openchannels both by experimental and numerical means. Detailedlaboratory measurements of 3D mean velocity components andwater levels were undertaken for subcritical dividing flows in a90° sharp-edged, rectangular junction formed by horizontal open-channels of equal width. The authors proposed a free-surface 3Dtwo-equation turbulence model and tested it against laboratoryexperiments, with reasonable agreement being obtained betweenthe numerical predictions and experimental data. The discusserswould like to comment on the numerical modeling aspects. Thenotations used in this discussion are the same as those defined bythe authors. Hereafter, Exp.1 and Exp.2 refer to the experimentwith Qr=0.409 and 0.838, respectively, of the original paper.
The authors stated that, at the end of each computational itera-tion, the mean velocities at the two outlets are adjusted to ensurethe measured flow rates Qd in the main channel and Qb in thebranch channel. This means that the proposed 3D model requiresa priori knowledge of the discharge ratio �Qr=Qb /Qu�, which isnot often available in various practical cases, including river bi-furcations and street intersections in urban areas. Since the flowdischarge distribution has importance in terms of the velocity anddepth of flow in each branch of the junction �Zanichelli et al.2004�, the 3D model proposed by the authors has then a seriouslimitation for practical situations. As an alternative, the discussersperformed a numerical study using a free-surface 2D depth-averaged model, in which the outlet flows Qd and Qb were notprescribed. Rather, they were considered as unknown parametersthat should be predicted by the 2D model accurately. The 2Dmodel relies on the depth averaged Saint-Venant equations, whichare solved using an explicit second-order scheme �Mignot et al.2006�. The effect of 3D turbulence is implicitly taken into con-sideration and the eddy viscosity �T characterizing its effect isgiven by Eq. �1�, in which u*�bottom friction velocity beingrelated to the Manning roughness nr �Eq. �2��, h�flow depth, and��dimensionless coefficient. The dimensionless coefficient��flow and geometry-dependent parameter �Rodi 1993�. Shionoand Knight �1991� proposed a value of 0.07 for regular channels
To avoid uncertainty in the choice of � value, Exp. 1 was used toadjust its value. The calibrated model was then used to simulateExp. 2. The Manning roughness nr was set at 0.01, a commonlyvalue used for Plexiglas plates �Rivière et al. 2007�. In the nu-merical computations, a Cartesian mesh of 1947 cells was usedwith a uniform space step of 0.06 m. For Exp. 1, � was adjustedso that the computed Qr matched the measured value of 0.409.Results were found to be not very sensitive to the � value �Table1�. Of particular interest, in the calculation without the eddy vis-cosity effect �i.e., ��0� the 2D model predicts Qr with a relativeerror of 2.4%. Among all the trials ��0.07 seemed optimal andwas chosen, as it provides a relative error less than 1%. Thecorresponding water surface profiles are shown in Fig. 1�a�, to be
Table 1. Calibration of the Model Using Exp. 1
� value Measured Qr Computed Qr Relative error
0 0.409 0.4188 2.40%
0.07 0.4122 0.78%
0.10 0.4168 1.91%
Fig. 1. Contours of water surface profiles: �a� Qr=0.409; �b� Qr
compared with Fig. 9�a� of the original paper, and the overallagreement between the numerical predictions and measurementsis reasonable. The calibrated model was then applied to simulateExp. 2. The measured flow discharge ratio of 0.838 was predictedwith a relative error of 1.8%, which is a modest and acceptableerror for practical purposes. The predicted water surface profilesare plotted in Fig. 1�b�, to be compared with Fig. 9�b� and Fig. 20of the original paper. Again, fairly good agreement between the2D numerical predictions and laboratory measurements is ob-tained.
The experimental measurements of the authors point out thatdividing flows is three-dimensional in their detailed characteris-tics, but a 2D model can be used for predicting well at least theflow discharge distribution and water surface profiles. The effectof 3D turbulence can be related to the bottom friction throughEqs. �1� and �2�, in which � has to be calibrated. For the authors’experiments, ��0.07 was found. Finally, the discussers are inter-ested to know the performance of the 3D model in terms of flowdischarge distribution and water levels if the numerical results ateach time iteration are not adjusted to ensure the measured outletflows Qd and Qb.
References
Mignot, E., Paquier, A., and Haider, S. �2006�. “Modeling floods in adense urban area using 2D shallow water equations.” J. Hydrol.,327�1–2�, 186–199.
Rivière, N., Travin, G., and Perkins, R. J. �2007�. “Transcritical flow inopen channel intersections.” Proc., 32nd IAHR Congress �CD-ROM�,IAHR, Venice.
Rodi, W. �1993�. Turbulence models and their application inhydraulics—A state of the art review, IAHR Monograph, 3rd Ed.,Balkema, Rotterdam, The Netherlands.
Shiono, K. and Knight, D. W. �1991�. “Turbulent open-channel flowswith variable depth across the channel.” J. Fluid Mech., 222, 617–646.
Zanichelli, G., Caroni, E., and Fiorotto, V. �2004�. “River bifurcationanalysis by physical and numerical modeling.” J. Hydraul. Eng.,130�3�, 237–242.
Closure to “Numerical and ExperimentalStudy of Dividing Open-Channel Flows” byA. S. Ramamurthy, Junying Qu, andDiep VoOctober 2007, Vol. 133, No. 10, pp. 1135–1144.DOI: 10.1061/�ASCE�0733-9429�2007�133:10�1135�
A. S. Ramamurthy1; Junying Qu2; and Diep Vo3
1Prof., Dept. of Civil Engineering, Concordia Univ., 1455 de Maison-neuve W., Montreal, Quebec, Canada H3G 1M8. E-mail: [email protected]
The writers thank the discussers for their interest in our paper.The writers are pleased to know that the discussers have devel-oped a two-dimensional numerical model to predict the flow dis-charge and the water surface profiles. Actually one can also use
Eq. �1� in the paper to estimate the discharge and the correspond-
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ing water depths for dividing open channel flows.The purpose of the original paper was to investigate the de-
tailed three-dimensional characteristics of dividing open-channelflow, such as secondary flows and flow recirculation using experi-mental and numerical methods. It was not the main emphasis inthe paper to find the discharge distribution.
Barring the extremely restrictive case where the branch flow isrelatively very low, dividing open-channel flows are inherently3D in character since the flow separates both in the branch and inthe main. The latter is due to the deceleration experienced in themain section downstream of the branch. As such, only very ap-proximate solutions can be obtained for the characteristics of di-viding open-channel flow problems, if the model is not threedimensional.
In the paper, it is stated that at the end of iteration, the meanvelocities at the two outlets were adjusted to match the previouslymeasured flow rates recorded in earlier tests. The strategy was toimprove the simulation efficiency and also to keep the flows con-servative in the computational domain. When the numerical studywas conducted, all experimental study had been completed. The“in-house” 3D numerical model was developed by the secondwriter. Therefore, the flow velocities could be adjusted at eachoutlet to match the known measured flow rates and save time forsimulation which yielded the general flow characteristics.
Discussion of “Transcritical Flow due toChannel Contraction” by O. Castro-Orgaz,J. V. Giraldez, and J. L. AyusoApril 2008, Vol. 134, No. 4, pp. 492–496.DOI: 10.1061/�ASCE�0733-9429�2008�134:4�492�
Hubert Chanson1
1Prof., School of Civil Engineering, The Univ. of Queensland, BrisbaneQLD 4072, Australia. E-mail: [email protected]
The authors presented a useful contribution to the topic of tran-scritical flow in open channels. The design of channel contractionis not obvious when transcritical or near-critical flows take place.The concept of critical flow conditions was first developed byBélanger �1828� and later expanded by Bakhmeteff �1912, 1932�.Bélanger and Bakhmeteff both defined the concept of critical flowin relation to the singularity of the backwater equation; that is, asthe singularity of Eq. �1� presented by the authors �Montes 1998�.Herein further applications of transcritical flow in channel con-traction are discussed and a solution of the critical flow singular-ity is presented.
A classical example of transcritical flow in channel contractionis the minimum energy loss �MEL� waterway design. The conceptof �MEL� structure was developed by late Professor GordonMcKay �1971, 1978�. Both MEL culverts and weirs were builtand operated successfully for over 30 years �Chanson 2004�. TheMEL waterways are designed with the concept of minimum headloss, transcritical flow along the entire waterways, and nearly-constant total head along the waterway at design flow conditions�Apelt 1983; Chanson 2004�. Fig. 1 presents a MEL spillwaydesigned to operate at transcritical flow conditions for a designdischarge of 850 m3 /s. Fig. 2 shows some experimental measure-ments in a MEL culvert model in operation for the design flowrate �McDonald 1996�. The data include the free-surface elevation
Y and total head H, while the bed elevation zo, critical depth dc,
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J. Hydraul. Eng. 2009.1
and channel width B are also shown. Note the free-surface undu-lations in the culvert barrel and the relatively smooth transitionfrom transcritical flow to subcritical flow at the downstream endof the culvert outlet. The presence of free-surface undulations iscommonly observed in man-made channels as well as naturalwaterways operating at transcritical flow �Apelt 1983�. This flowsituation cannot be solved by the backwater equation and theexample illustrates a limitation of the authors’ method. Undularflow calculations require a more advanced modeling approach�Montes and Chanson 1998�.
The singularity of critical flow conditions can be resolved interms of the concept of minimum specific energy, including forflow situations with nonhydrostatic pressure distributions �Chan-son 2006�. For a rectangular channel, the continuity and Bernoulliequations give two equations in terms of the critical flow depthand depth-averaged velocity. When the minimum specific energyEmin and flow rate per unit width q are known, the critical flowdepth dc�solutions of a third-order polynomial equation
� dc
Emin�3
−1
�� dc
Emin�2
+1
2�2
3�3�CD
2
�= 0 �1�
where the pressure coefficient � accounts for the nonhydrostaticpressure distribution; ��Boussinesq coefficient; and CD
=dimensionless discharge coefficient, defined respectively as
� =1
2+
1
dc�
0
dc P�y�gd
dy �2�
� =1
V2dc�
0
dc
v�y�2dy �3�
CD =q
�g�2
3Emin�3
�4�
where P�pressure at a distance y from the bed; �fluid density;v=local velocity; and V=depth-averaged velocity. For a givendischarge per unit width and minimum specific energy, the num-ber of real solutions to the critical flow condition equation �Eq.�1�� depends upon the sign of the dimensionless discriminant ∆
=4
729
�CD2 �2
�6 ��CD2 �2 − 1� �5�
There is one real solution for ∆�0, two real solutions for ∆�0, and three real solutions for ∆�0. Above a broad-crestedweir, the discriminant ∆ is zero and the analytical solution yieldsthe classical result: dc=�3 q2 /g.
Experimental measurements of pressure and velocity distribu-tions above circular- and rounded-crested weirs suggested that ∆� 0 for that geometry and that the only stable solution for thecritical flow conditions satisfies
dc
Emin=
1
3��1 − cos
3+�3�1 − �cos
3�2�� Solution S3
�6�
where
cos = 1 − 2�CD2 �2 �7�
and dc�critical flow depth for a nonhydrostatic pressure-distribution flow situation. Detailed experimental measurements
were reanalyzed and are shown in Fig. 3, where the analytical
solution of the dimensionless critical flow depth �dc /Emin is com-pared with experimental data. The analytical results include thethree real solutions for ∆�0, and the unique real solution for ∆�0. The solution for ∆�0 is �dc /Emin=2 /3.
The solution S1 occurs for 2 /3��dc /Emin�1 �Fig. 3� since
dc
Emin=
2
3��1
3+ cos
3� Solution S1 �8�
It implies the existence of convex streamlines as observed in un-dular flow situations �Montes and Chanson 1998; Chanson 2005�.Fig. 3 includes undular flow data based upon detailed pressureand velocity measurements at both wave crests and troughs. Theexperimental results demonstrate the existence of the solution S1for ∆�0.
For ∆�0, the analytical solution of the critical flow depth is
dc
Emin=
1
��1
3+�3 2
27�2 − �CD
2 �2� + ��6
+�3 2
27�2 − �CD
2 �2� − ��6� �9�
Note the point of inflexion of the solution ∆ � 0 for �CD2 �2
=4 /3 and �dc /Emin=0.78 �Fig. 3�. Altogether Eqs. �6�, �8�, and�9� show a range of analytical solutions for the critical flow depth.
References
Apelt, C. J. �1983�. “Hydraulics of minimum energy culverts and bridgewaterways.” Australian Civil Engrg Trans., 25�2�, 89–95.
Bakhmeteff, B. A. �1912�. “O neravnomernom wwijenii jidkosti v otkry-tom rusle �Varied flow in an open channel�.” St Petersburg, Russia �inRussian�.
Bakhmeteff, B. A. �1932�. Hydraulics of open channels, 1st Ed.,McGraw-Hill, New York.
Bélanger, J. B. �1828�. “Essai sur la solution numérique de quelquesproblèmes relatifs au mouvement permanent des eaux courantes�Essay on the numerical solution of some problems relative to steadyflow of water�.” Carilian-Goeury, Paris �in French�.
Chanson, H. �2004�. The hydraulics of open channel flows: An introduc-tion, 2nd Ed., Butterworth-Heinemann, Oxford.
Chanson, H. �2005�. “Physical modeling of the flow field in an undulartidal bore.” J. Hydraul. Res., 43�3�, 234–244.
Chanson, H. �2006�. “Minimum specific energy and critical flow condi-tions in open channels.” J. Irrig. Drain. Eng., 132�5�, 498–502.
Fawer, C. �1937�. “Etude de quelques ecoulements permanents à filetscourbes �Study of some steady flows with curved streamlines�.” The-sis, Imprimerie La Concorde, Lausanne, Switzerland �in French�.
McDonald, I. J. A. �1996�. “Analysis of a model minimum energy lossculvert.” Undergraduate thesis, Dept. of Civil Engineering, The Uni-versity of Queensland, Brisbane, Australia.
McKay, G. R. �1971�. “Design of minimum energy culverts.” ResearchReport, Dept. of Civil Eng., Univ. of Queensland, Brisbane, Australia.
McKay, G. R. �1978�. “Design principles of minimum energy water-ways.” Proc., Workshop on Minimum Energy Design of Culvert andBridge Waterways, Australian Road Research Board, Melbourne, Aus-tralia, Session 1, 1–39.
Montes, J. S. �1998�. Hydraulics of open channel flow, ASCE, Reston,Va.
Montes, J. S., and Chanson, H. �1998�. “Characteristics of undular hy-draulic jumps: Results and calculations.” J. Hydraul. Eng., 124�2�,192–205.
Vo, N. D. �1992�. “Characteristics of curvilinear flow past circular-crested
Fig. 1. View from downstream of the MEL weir spillway, LakeKurwongbah, Sideling Creek Dam, Petrie QLD 1958-69 �dam height:25 m; design flow: 850 m3 /s; inlet width: 107 m; throat width: 30.5m�
Fig. 2. Experimental observations of a MEL culvert model operationat design flow �after McDonald 1996; throat width: Bmin=0.1 m; flowrate: 0.010 m3 /s�
Fig. 3. Dimensionless critical flow depth �dc /Emin as a function of�CD
2 �2 at critical flow conditions for ∆ � 0 and for ∆ � 0; compari-son between analytical solution and experimental observations�Fawer 1937; Vo 1992; Chanson 2005�
weirs.” Ph.D. thesis, Concordia Univ., Canada.
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Closure to “Transcritical Flow due toChannel Contraction” by O. Castro-Orgaz,J. V. Giráldez, and J. L. AyusoApril 2008, Vol. 134, No. 4, pp. 492–496.DOI: 10.1061/�ASCE�0733-9429�2008�134:4�492�
O. Castro-Orgaz1; J. V. Giráldez2; and J. L. Ayuso3
2Dept. of Agronomy, Univ. of Cordoba, Edif. da Vinci, Cra Madrid km396, 14071 Cordoba, Spain. E-mail: [email protected]
3Dept. of Agricultural Engineering, Univ. of Cordoba, Edif. da Vinci, CraMadrid km 396, 14071 Cordoba, Spain. E-mail: [email protected]
The writers thank Professor Chanson for his valuable contributionas well as for his interest in our approach. The concept of criticalflow was first established by Bélanger �1828� as a singularity inthe backwater equation, but, as quoted by Montes �1998�, in agradually weakly curved bed, the pressure distribution remainshydrostatic and the vertical flow profile of Bélanger’s approachunder critical flow conditions does not appear in observations.Massé �1938� removed the contradiction in Bélanger’s approachby defining critical flow as a singular point of the backwaterequation. This is the main argument adopted by the authors intheir approach. It was recently proved �Castro-Orgaz et al. 2008a�that the singular point concept is a generalized definition of criti-cal flow conditions, which remains valid for any velocity andpressure distribution across the flow depth, and is not limited tothe gradually varied flow approach. The singular point of thebackwater equation for gradually varied flows
dy
dx=
sen � − Sf +Q2
gA3
�A
�x
cos � −Q2
gA3
�A
�y
�1�
where y=flow depth normal to channel bed; Sf =friction slopeA=flow section; Q=discharge, and ��channel bed angle with thehorizontal, is consistent with the minimum specific energy�E /�y=0
E = y cos � +Q2
2gA2 �2�
where E=specific energy head; thus, with Bakhmeteff’s �1932�concept of critical flow. Eq. �1� was first developed by Bélanger�1828�, but the first application of the concept of singular point tothis equation is originally due to Iwasa �1958�. Further develop-ments of the theory are provided by Castro-Orgaz �2008a�. Eq. �2�is the lowest-order approach for the specific energy, as it is basedon the hydrostatic pressure and uniform velocity distributions. Eq.�1� can be used in the development of a design approach of theMEL structures, quoted by Chanson in his discussion. Let us sup-pose an MEL culvert of a rectangular cross section, in which thewidth B is measured along the normals �Chanson 2004�. Along anormal, the velocity is assumed to be uniform and the pressuredistribution follows the hydrostatic approach �Apelt 1983�, con-ditions under which Eq. �1� can be applied. Thus, Eq. �1� remainsapproximately valid in a MEL structure provided that the flowsection A=By is measured along the equipotential curve or nor-mal. If the flow is assumed to be frictionless, the singular point of
Eq. �1� yields
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J. Hydraul. Eng. 2009.1
sen � −1
B
�A
�x= 0 �3�
A particular solution of Eq. �3� is sen� � 0 and �A /�x=0 �i.e.,the top of a weir with a minimum width�. For a rectangular crosssection, if the bed slope So�x� is a design parameter �Chanson2004�, the general integration of Eq. �3� yields the law of varia-tion of the length of the normal B�x� in the streamwise directionas
B�x� = �Bmax−2/3 +
3
2�Q2
g�−1/3
z�x��−3/2
�4�
where Bmax=maximum width at the beginning of the inlet fan andz=difference in elevations between the beginning of the inletand the normal at distance x along the main channel axis. Eq. �4�is in agreement with the developments for the classical design ofminimum energy loss culverts �Apelt 1983; Chanson 2004�. Thetraditional design of MEL culverts is based on critical flow allover the culvert. Obviously, this may produce unstable flow atdesign conditions, typical of near critical flows �Chanson 2004�.To overcome this unsatisfactory design, the MEL structure is usu-ally designed for subcritical flow conditions �Chanson 2004� �i.e.,F is typically at around 0.7 as a mean, with F=Q / �gA3 /B�1/2 asthe Froude number�. Eq. �1� can be rewritten as a function of F toyield
dy
dx=
sen � −f
8F2 +
F2
B
�A
�x
cos � − F2 �5�
where f�friction factor �Chanson 2004�. A new design concept isassumed, F=constant�1, to avoid free-surface undulations atnear-critical flow conditions, and quasi-normal flow conditionsfrom Eq. �5� yields
sen � −f
8F2 +
F2
B
�A
�x= 0 �6�
Integration of �6�, adopting a mean friction factor f , results in
B�x� = Bmax−2/3 + �3
2�Q2
g�−1/3�z�x�F−2 −
f
8x��−3/2
�7�
which is the new design equation. For critical flow conditions�F�1� and frictionless fluid �f =0� Eq. �7� degenerates into Eq.�4�, which is associated with critical flow design all over thewaterway. Eq. �7� avoids the free surface undulations under de-sign conditions and also includes, directly, the frictional losses inthe fan. This permits a direct design of the structure in only onestep, as no additional readjustments for frictional losses are re-quired when using Eq. �7�.
Although in the paper it was justified that Eq. �1� would re-
sures, if Eq. �2� is explicitly used, this application is no longervalid. Therefore, a higher order approach is required in flows withnonhydrostatic pressure distributions. Such is the case of flowsover round-crested weirs and Venturi channels �Fig. 1� with tran-scritical flow. These are short transitions with converging stream-lines, where the irrotational flow theory applies �Rouse 1932�.Castro-Orgaz �2008a� developed a higher-order approach for theenergy equation, which is valid for flows with nonhydrostaticpressure and nonuniform velocity distributions such as
H = z + h +Q2
2gA2 exp� 2hh�
m + 2−
h�2
2m̄ + 1+ hz� − 2
h�z�
m̄ + 1− z�2�
= E + z �8�
where H=total head; z=elevation of the channel bottom from anarbitrary but fixed datum; h=flow depth defined as the verticalprojection of a normal; h�=dh /dx; h�=d2h /dx2; z�=dz /dx; z�=d2z /dx2; and m and m̄ are parameters describing the flow net. Inthe curvilinear flow over weirs and flumes the streamlines arecurved and sloped, and the critical depth is defined as the verticalprojection of the normal passing across the extreme in the channelgeometry �i.e., a bed top elevation in a weir and a minimum widthin a flume, see Fig. 1�, where the extended specific energy givenby Eq. �8� reaches a minimum value. Eq. �8� was successfullyapplied to round-crested weirs �Castro-Orgaz et al. 2008b� andVenturi channels �Castro-Orgaz 2008a,b�, thereby accounting fornonhydrostatic effects in the discharge characteristics. This is thesame case on which the discusser focused his analysis for thecritical depth in curvilinear flow. Eq. �8� can also be applied to theundular flow in the MEL barrel.
Notation
The following symbols are used in this closure:A � cross-sectional area �m2�;B � length of a normal in plant view �m�;E � specific energy head �m�;F � Froude number;f � friction factor;g � acceleration of gravity �m /s2�;H � total energy head �m�;h � flow depth defined as vertical projection of a
m � rate of change of streamline curvature along anormal;
m̄ � rate of change of streamline inclination alonga normal;
Q � flow discharge rate �m3 /s�;Sf � friction slope;So � channel bottom slope;x � streamwise coordinate �m�;y � flow depth defined normal to channel bed
�m�;z � elevation of channel bottom �m�;
z� � dz /dx;z� � d2z /dx2�m−1�; and� � angle of channel bottom with horizontal.
References
Apelt, C. J. �1983�. “Hydraulics of minimum energy culverts and bridgewaterways.” Aust. Civ. Eng. Trans., 25�2�, 89–95.
Bakhmeteff, B. A. �1932�. Hydraulics of open channels, McGraw-Hill,New York.
Bélanger, J. B. �1828�. Essai sur la solution numérique de quelquesproblèmes relatifs au mouvement permanent des eaux courantes[Analysis of the numerical solution of some problems related to steadywater flows], Carilian-Goeury, Paris �in French�.
Castro-Orgaz, O. �2008a�. Critical flow in hydraulic structures, Ph.D.diss., Dept. of Agronomy, University of Cordoba, Spain.
Castro-Orgaz, O. �2008b�. “Hydraulic design of Khafagi flumes.” J. Hy-draul. Res., 46�5�, 691–698.
Castro-Orgaz, O., Giraldez, J. V., and Ayuso, J. L. �2008a�. “Energy andmomentum under critical flow conditions.” J. Hydraul. Res., 46�6�,844–848. .
Castro-Orgaz, O., Giraldez, J. V., and Ayuso, J. L. �2008b�. “Critical flowover spillway profiles.” Water Management, Proc., ICE, 161�2�, 89–95.
Chanson, H. �2004�. The hydraulics of open channel flows: An introduc-tion, 2nd Ed., Elsevier Butterworth-Heinemann, Oxford.
Iwasa, Y. �1958�. “Hydraulic significance of transitional behaviours offlows in channel transitions and controls.” Mem. Fac. Eng., KyotoUniv., 20�4�, 237–276.
Massé, P. �1938�. “Ressaut et ligne deau dans les cours à pente variable�Hydraulic jump and backwater effects in variable slope channels�.”Revue Générale de l�Hydraulique, 4�19�, 7–11; �20�, 61–64.
Montes, J. S. �1998�. Hydraulics of open channel flow, ASCE, Reston,Va.
Rouse, H. �1932�. “The distribution of hydraulic energy in weir flow in
relation to spillway design.” MS thesis, MIT, Boston.
