discussion of “finite-depth seepage below flat aprons with equal end cutoffs” by arun k. jain...
TRANSCRIPT
Discussions and Closures
Discussion of “Finite-Depth Seepage belowFlat Aprons with Equal End Cutoffs” byArun K. Jain and Lakshmi N. ReddiDecember 2011, Vol. 137, No. 12, pp. 1659–1667.DOI: 10.1061/(ASCE)HY.1943-7900.0000459
Mahender Choudhary, Ph.D.11Associate Professor, Dept. of Civil Engineering, MNIT, Jaipur, India.
E-mail: [email protected]
The authors have given a solution for seepage below flat apronswith equal end cutoffs under finite-depth conditions, but a more gen-eral solution already exists. Fil’chakov and others (Polubarinova-Kochina 1962) gave solutions for weirs having finite depth, unequalpiles, depressed floor, u=s blanket, and intermediate pile. Further-more, Swamee et al. (1996, 1997) considered unequal piles, multipleintermediate piles, depressed floor, and sloping floor but did not con-sider finite depth. Both these solutions used conformal mappingalong with the Schwarz-Christoffel transformation. Therefore, theproposed solution is a particular case of the existing Fil’chakov’ssolution using a similar technique. The authors have referred toFil’chakov’s work in the Russian language but ignored a readilyavailable solution in English (Polubarinova Kochina 1962) and alsohave not compared the present solution with the particular case ofFil’chakov’s work. Such results are easily reproducible. Thus, thework neither presents an answer to the unsolved seepage problemnor does it involve a new solution technique. Hence, the authorsneed to justify the requirement of another solution to a particularcase of an already existing general solution by using an identicaltechnique.
Solutions obtained using conformal mapping along with theSchwarz-Christoffel transformation are generally implicitly nonlin-ear in terms of transformation parameters. Such a set of implicitnonlinear equations can easily be solved using computer algebrain software (MATLAB 2007; Mathematica 2000) or optimiza-tion techniques (Swamee et al. 1997). For example, Chahar andVadodaria (2008a, b), Choudhary and Chahar (2007), and Swameeet al. (1997) solved nonlinear implicit equations for 4 to 10 trans-formation parameters. The authors claimed that the solution of twononlinear equations was extremely difficult and hence adopted anindirect method of graphs and tables that requires interpolation.Actually, in the present era of computing and the availability ofsoftware involving routines for computer algebra (MATLAB2007; Mathematica 2000), the simultaneous solution of the twononlinear Eqs. (12) and (13) can directly be executed to get αand γ (Chahar 2007, 2009), and then the remaining parameterscan easily be obtained. Therefore, the indirect method presentedin the original paper to get the solution in terms of graphs and tablesis outdated, requires interpolation, and is of limited use; and thus,Figs. 2–4 and Table 1 are redundant.
The authors also developed the simplified Eqs. (30a)–(30c) forPE, PD, and GE, respectively, to avoid the use of design charts(Figs. 5–8) and interpolation in them. In fact, Eqs. (18)–(21)and (25) are straightforward for already computed values of α,β, and γ; therefore, Figs. 5–8, Table 3, and the sections “Develop-ment of Interference Formulae” and “Development of SimplifiedEquations for PE, PD, and GE” are also unnecessary.
References
Chahar, B. R. (2007). “Analysis of seepage from polygon channels.”J. Hydraul. Eng., 133(4), 451–460.
Chahar, B. R. (2009). “Seepage from a special class of a curved channel withdrainage layer at shallow depth.” Water Resour. Res., 45(9), W09423.
Chahar, B. R., and Vadodaria, G. P. (2008a). “Drainage of ponded surfaceby an array of ditches.” J. Irrig. Drain. Eng., 134(6), 815–823.
Chahar, B. R., and Vadodaria, G. P. (2008b). “Steady subsurface drainageof homogeneous soil by ditches.” J. Water Manage., 161(6), 303–311.
Choudhary, M., and Chahar, B. R. (2007). “Recharge/seepage from anarray of rectangular channels.” J. Hydrol., 343(1–2), 71–79.
Mathematica Version 4.1.2.0 [Computer software]. (2000). WolframResearch, Inc., Champaign, IL.
MATLAB Version 7.4 [Computer software]. (2007). The MathWorks, Inc.,Natick, MA.
Polubarinova-Kochina, P. Y. (1962). Theory of groundwater movement.Trans. J. M. R. de Wiest, Princeton Univ., Princeton, NJ, 81–106.
Swamee, P. K., Mishra, G. C., and Salem, A. A. S. (1996). “Optimal designof sloping weir.” J. Irrig. Drain. Eng., 122(4), 248–255.
Swamee, P. K., Mishra, G. C., and Salem, A. A. S. (1997). “Effectivity ofmultiple sheet piles inweir design.” J. Irrig.Drain.Eng., 123(3), 218–221.
Closure to “Finite-Depth Seepage belowFlat Aprons with Equal End Cutoffs” byArun K. Jain and Lakshmi N. ReddiDecember 2011, Vol. 137, No. 12, pp. 1659–1667.DOI: 10.1061/(ASCE)HY.1943-7900.0000459
Arun K. Jain, A.M.ASCE1; and Lakshmi N. Reddi, F.ASCE2
1Research Scholar, Dept. of Civil, Environmental, and ConstructionEngineering, Univ. of Central Florida, Orlando, FL 32816. E-mail:[email protected]
2Dean of Graduate School, Florida International Univ., Miami, FL 33199(corresponding author). E-mail: [email protected]
The authors thank the discusser for the comments. The authorsagree that a more general solution of the case containing an em-bedded dam with unequal end cutoffs and an upstream blankethas been solved by Fil’chakov (1959, 1960). However, for the casein the original paper, no design charts are available in the aforemen-tioned references. Though not explicitly stated, the current paperhas been written to appeal to design engineers/practitioners. Theoriginal paper provides design charts and simplified equations,which serve as handy tools for calculating the seepage character-istics. King (1967) presented a solution for a case of an embeddeddam without any cutoffs, which is a particular case of the solutionprovided by Fil’chakov. The present solution has been satisfactorilycompared with the work done by Khosla (1936) and Malhotra(1936); hence, no other comparisons were needed.
The writers agree with the discusser that MATLAB (2007) andMathematica (2000) are available and can be used for solving non-linear equations. In fact, any method can be used if it gives accurateresults. For the original paper, the authors coded a computerprogram to solve the resulting equations. Because the computerprogram was not provided in the paper, Figs. 2–4 were providedto show the readers the intermediate steps of computations. Also,these figures show the relationship of floor profile ratios with σ andγ, which is of academic interest. The scheme of computation has
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been mentioned in the “Results” section of the original paper.For selected values of B=D and B=c, the variables σ and γ werecomputed with the help of the computer program. No interpolationwas made. Hence, the results are accurate and comparable with thesolution provided by any available software.
The design charts, Figs. 5–8, are useful for the design engineerswho have the values of the floor dimensions—B, D, and c—andwish to avoid the calculation of σ and γ. Similarly, the simplifiedequations provide design engineers easy access to the seepage char-acteristics without losing any significant accuracy.
Pavlovsky (1956) provided a solution for finite-depth seepageunder a flat apron with a downstream end cutoff. The solution isamenable to direct solution. The “Development of InterferenceFormulae” provides Table 3, in which the effect of providing an up-stream pile of the same length is shown. Hence, knowing the valuesof factors from Table 3 and using Pavlovsky’s equation, one can skipsolving the equations. The interference formulae are in excellentagreement with the current analytical solution and hence, are useful.
References
Fil’chakov, P. F. (1959). The theory of filtration beneath hydrotechnicalstructures, Vol. 1, Izd-vo Akademii nauk Ukrainskoi SSR, Kiev,Ukraine.
Fil’chakov, P. F. (1960). The theory of filtration beneath hydrotechnicalstructures, Vol. 2, Izd-vo Akademii naukUkrainskoi SSR, Kiev, Ukraine.
Khosla, A. N., Bose, N. R., and Taylor, E. M. (1936). “Design of weirson permeable foundations.” Publication No. 12, Central Board ofIrrigation, New Delhi, India.
King, G. J. W. (1967). “Seepage under a rectangular dam.” J. Soil Mech.Found. Div, 93(SM2), 45–64.
Malhotra, J. K. (1936). “Appendix to Chapter VII: Mathematical investiga-tions of the subsoil flow under two standard forms of structures.”Publication No. 12, Central Board of Irrigation, New Delhi, India,85–90.
Mathematica Version 4.1.2.0 [Computer software]. (2000). WolframResearch, Inc., Champaign, IL.
MATLAB Version 7.4 [Computer software]. (2007). The MathWorks, Inc.,Natick, MA.
Pavlovsky, N. N. (1956). Collected works, Izd. AN SSSR Moscow–Leningrad, USSR.
Discussion of “Experimental Study ofSubcritical Dividing Flow in anEqual-Width, Four-Branch Junction” byLeonardo S. Nania, Manuel Gómez,José Dolz, Pau Comas, and Juan PomaresOctober 2011, Vol. 137, No. 10, pp. 1298–1305.DOI: 10.1061/(ASCE)HY.1943-7900.0000423
Nicolas Rivière1; Gilbert Travin2; and Richard J. Perkins31Professor, Université de Lyon, Laboratoire de Mécanique des Fluides
et d’Acoustique (LMFA, CNRS UMR5509), INSA de Lyon, Bât.Jacquard, 20 avenue A. Einstein, 69621, Villeurbanne Cedex, France(corresponding author). E-mail: [email protected]
2Assistant Professor, Université de Lyon, Laboratoire de Mécanique desFluides et d’Acoustique (CNRS UMR5509), INSA de Lyon, Bât.Jacquard, 20 avenue Einstein, 69621, Villeurbanne, France. E-mail:[email protected]
3Professor, Université de Lyon, Laboratoire de Mécanique des Fluides etd’Acoustique (LMFA, CNRS UMR5509), Ecole Centrale de Lyon, 36avenue Guy de Collongue, 69134, Ecully Cedex, France. E-mail:[email protected]
The original paper provides a useful contribution to the topicof subcritical open-channel flow in a four-branch intersection.Coincidentally, a second paper on the same topic was publishedin another journal (Rivière et al. 2011) and there are sufficientpoints in common between those two papers to justify a compari-son of the data. In attempting this comparison, the discussers haveencountered several problems with the interpretation of the exper-imental conditions and the data presented by the authors.
Literature Review
In the interval between the initial submission of this article and itspublication, several other papers on the flow in four-branch inter-sections have been published, and it might have been useful to citethese, if only for completeness. Supercritical flows have been in-vestigated using experimental (Mignot et al. 2008b, 2009), theoreti-cal (Mignot et al. 2011), and numerical approaches (Mignot et al.2008a). For the subcritical regime, Rivière et al. (2006) showed alinear relationship between the ratios of inlet and outlet discharges,and the data from that study were compared with three-dimensional(3D) numerical simulations by Li and Zeng (2010). Rivière et al.(2011) provided data from a range of conditions, and used thatdata to derive a semiempirical model to compute the outlet flowdistribution.
Experimental Facility
Although the installation is of a reasonable size [the channelsare five times wider than those used by Rivière et al. (2011),for example], the width-to-length ratio is very small (Table 1),so the recirculation zones downstream of the intersection influencethe flow over the weirs, as already indicated by the authors. It istherefore unlikely that the flow in the downstream channels couldbe considered as one-dimensional (1D), and, in particular, the depthmeasurements just downstream of the intersection are probably notrepresentative of the flow conditions because they were made in aregion where the flow is strongly influenced by these recirculations,with a lateral component and strong local accelerations. Finally,and this is perhaps the most important point, the installationappears to be strongly asymmetrical, although there are no geomet-rical measurements to confirm this. There are eight data points forwhich the outlet conditions are identical, and the inlet conditionsare close to identical (to within 2.7% for the greatest difference); inall these cases the flow Qoy leaving the branch y is systematicallyless than the flow rate Qox in the branch x, where the flow ratesshould be identical, and the variation from symmetry ranges from−10% to −31% (see Fig. 2 of the original paper). This indicatesclearly some departure from symmetry in the experimental configu-ration, but although the discussers have tested various hypotheses(e.g., difference in channel slopes and error in setting the height ofthe outlet weirs), an explanation consistent with the entire data sethas not been found.
Choice of the Dimensionless Parameters Influencingthe Flow
In the model developed by Rivière et al. (2011), the x-direction issystematically defined as parallel to the channel with the highestinlet discharge (the main flow Qim) and the y-direction is by def-inition orthogonal to the x-direction, transporting the tributary flowQit. In the original article, the x-direction can be aligned either withthe main or the tributary flow, depending on the flow configuration
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being studied; bearing this in mind, the range of flow parameters iscompared in Table 1. The correlations proposed in the two papersrequire the same number of dimensionless parameters, but theseparameters are based on different variables. The authors assumethat the upstream depths yix and yiy depend on the downstreamdepths yox and yoy. They nevertheless retain yix to define the x-inletdischarge through the Froude number Fix ¼ Qix=ðyix3=2g1=2bÞ.Rivière et al. (2011) adopt the widely used assumptionyix ¼ yiy, but the link between yix and yox is derived explicitly us-ing a momentum balance adapted from Ramamurthy et al. (1990).To avoid using dependent variables in the analysis (which risksintroducing spurious correlations between dimensionless parame-ters), the dimensionless form of the inlet discharge is defined asRg ¼ Qim=ðb5=2g1=2Þ. The main difference between the two papersconcerns the downstream conditions. The authors use the dimen-sionless water depths b=yox and b=yoy, whereas Rivière et al.(2011) prefer the dimensionless heights Cm=b and Ct=b of thecontrol weirs at the ends of the outlet channels. There are two ad-vantages to this approach: First, each couple (Qo, C) corresponds toa 1D water depth at the end of the channel, through the stage dis-charge relationship for the weir, thereby avoiding uncertainty aboutthe representativity of point-depth measurements. Second, Cm=band Ct=b can be varied independently of each other and of the othervariables; this is not possible with the dependent variables b=yoxand b=yoy. Rivière et al. (2011) show how this downstream con-dition can be applied to practical situations in which the outlet flowis not controlled by a weir.
Momentum Balance
Ramamurthy et al. (1990) provided a momentum balance equa-tion for the three-branch intersection, which can be adapted tothe four-branch case by taking account of the momentum in thetributary inflow (Rivière et al. 2011). This equation provides theratio of the upstream and downstream depths in the y-channels,Ry ¼ yix=yox, in the form of an implicit function Ry ¼fðRg;Rq;Rqi; yxo=bÞ. The theoretical values of Ry are plotted inFig. 1 against the experimental values for the complete data set ob-tained by the authors. This shows that the measured values do notsatisfy the 1D conservation of momentum because the departurefrom the theoretical values varies between −17% and þ27%. Thisis probably because the flow was not 1D where the depths weremeasured, as already indicated by the authors. What is more dis-concerting is that the results separate into two clear groups—thedepth ratio is consistently less than the theoretical value whenthe main flow is in the x-channel and consistently greater whenthe main flow is in the y-channel. The discussers believe that thisindicates a fundamental asymmetry in the installation, which ex-plains why symmetrical inlet and outlet conditions do not generatea symmetrical flow distribution as they ought to. The discussershave tested various hypotheses that might explain this dissymmetryusing the original data reported by Comas-Pelegrí (2005), but they
have not been able to find any consistent and coherent explanationfor this. This effect has therefore not been able to be correctedfor, and this has limited the ability to compare the data withthe empirical correlation derived by Rivière et al. (2011), whichis based on 1D flow variables. Nevertheless, a simple comparisonbetween the two is provided in this discussion in an attempt toassess the possible importance of scale effects for the derivedcorrelation.
Discharge Distribution
As shown in Table 1, the range of experimental conditions studiedby the authors is different from that used by Rivière et al. (2011) toderive their correlation. In particular, the term in the correlationinvolving Rb becomes negative when Rb < 0.03, but the data ofthe original article includes values of Rb between 0.01 and 0.03,so it has been necessary to modify the initial correlation to accountfor this; the revised correlation then becomes
Rq ¼ ð1þ 0.0936 − 0.4624 − 0.0936RcÞRqi þ 0.4624
− 0.1031ðRc − 1ÞðRb − 0.01Þð1=Rg þ 10.07Þ ð1Þ
with a correlation coefficient R2 ¼ 0.9956 for the original data ofRivière et al. (2011). In Eq. (1), the discharge distribution Rq isdefined as Rq ¼ ðQot=QimÞ; the authors use an equivalent param-eter RqT ¼ Qot=ðQimþQitÞ ¼ Rq=ð1þ RqiÞ. This expression forRq has been applied to the data using the method described inRivière et al. (2011): starting with an arbitrary initial value of
Table 1. Range of Parameters the Original Paper and Rivière et al. (2011)
ReferenceQim(L=s)
Qit(L=s)
b(cm)
Cx(cm)
Cy(cm)
Rqi ¼QiM=Qit
Rb ¼Cx=b
Rc ¼Cy=Cx
Rg ¼Qim=ðb5gÞ1=2
Rq ¼Qot=Qim
RqT ¼Qot=ðQim þQitÞ Li=b Lo=b
Originalpaper
12.3–75 5.57–74.5 150 2–8 2–8 0.11–0.995 0.013–0.053 0.25–4 0.000068–0.0087 0.1685–1.5195 0.108–0.83 1.33 3.33
Rivièreet al. (2011)
3.5–10.05 0–9 30 3–11 2.6–10.5 0–1 0.1–0.37 0.4–2.5 0.0226–0.065 0.05–1.645 0.04–0.91 6.67 8.7
Fig. 1. Theoretical and experimental upstream-to-downstream ratios ofwater depth
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Rq, values of Cx and Cy are computed, corresponding to theexperimental values of yox and yoy. A new value of Rq is then de-rived from Cx, Cy and the other dimensionless parameters charac-terizing the flow. This procedure is repeated until the value of Rq isstable. The results of applying this correlation to the data of theauthors are shown in Fig. 2 in terms of RqT for consistency withthe original article.
The agreement is poor, with the difference between the theoreti-cal and the experimental values exceeding −40% and þ60% in ex-treme cases. For the flow configurations with identical weirs(Cx ¼ Cy; closed symbols in Fig. 2), the discharge in the y-branchis always underestimated irrespective of whether it is the tributaryor the main channel. The opposite effect is observed whenCx ≠ Cy. This is consistent with the previous observations thatthe experimental setup is asymmetrical and the local water depthmeasurements cannot account for this asymmetry of downstreamconditions. The comparison does not improve if the measuredwater depths are replaced by stage discharge relationships appro-priate for the outlet weirs. Similarly, the agreement is not improvedby using the water depths at the ends of the channels [obtainedfrom Comas-Pelegrí (2005)]—either directly or in the form ofstage-discharge relationships.
In a final attempt to account for the difference in flow rates,the discussers have increased the weir height at the end of they-channel by a constant amount, determined so as to match themeasured flow distributions as closely as possible for the eightconfigurations for which Rqi ¼ 1 and Cx ¼ Cy (Fig. 2 in theoriginal article). A least-squares fitting procedure yields an esti-mate of δ ¼ 1.1 cm for this additional height; these results areshown as the open symbols in Fig. 3. The other data points, cor-rected using the same offset for the weir height, are shown asfilled symbols in Fig. 3. This modification yields much betteragreement between all the measured flow distribution ratiosand the correlation proposed by Rivière et al. (2011), suggestingthat the correlation can be applied at scales larger than those usedin its derivation.
In conclusion, then, it appears that there are two problems withthe data provided by the original article; the depths have been mea-sured at positions where the flow is strongly three-dimensional, sothat the depths are not representative of average conditions in thechannel, and the installation contains an unidentified asymmetry,
which means that symmetrical inlet and outlet conditions donot result in a symmetrical flow distribution. It might be pos-sible to correct for these effects by an ad hoc modification ofthe weir height in the y-channel, in which case the data thenappear consistent with those measured by Rivière et al. (2011),but there is no physical justification for this modification. If thevalidity of this modification is accepted, then the modified dataof the original article show that the correlation proposed by Rivièreet al. (2011) is valid for conditions outside those used in itsderivation.
References
Comas-Pelegrí, P. (2005). “Estudi sobre la distribució de cabals en unencreuament de carrers durant sucesos d’inundació, en condicions derègim subcrític.” M.Sc. thesis, Universitat Politècnica de Catalunya,Barcelona, Spain (in Spanish).
Li, C. W., and Zeng, C. (2010). “Flow division at a channel crossing withsubcritical or supercritical flow.” Int. J. Numer. Methods Fluids, 62(1),56–73.
Mignot, E., Paquier, A., and Rivière, N. (2008a). “2D numerical study of asymmetrical 4-branches supercritical cross junction.” J. Hydraul. Res.,46(6), 723–738.
Mignot, E., Rivière, N., Paquier, A., and Perkins, R. J. (2011). “Hydraulicmodels of the flow distribution in a four open channels junction withsupercritical flow.” J. Hydraul. Eng., 137(3), 289–299.
Mignot, E., Rivière, N., Perkins, R. J., and Paquier, A. (2008b). “Flowpatterns in a four branches junction with supercritical flow.” J. Hydraul.Eng., 134(6), 701–713.
Mignot, E., Rivière, N., Perkins, R. J., Paquier, A., and Travin, G. (2009).“Closure to ‘Flow patterns in a four-branch junction with supercriticalflow’ by Emmanuel Mignot, Nicolas Rivère, Richard Perkins, andAndré Paquier.” J. Hydraul. Eng., 135(11), 1023–1024.
Ramamurthy, A. S., Tran, D. M., and Carballada, L. B. (1990).“Dividing flow in open channels.” J. Hydraul. Eng., 116(3),449–455.
Rivière, N., Perkins, R. J., Chocat, B., and Lecus, A. (2006). “Floodingflows in city crossroads: 1D modelling and prediction.” Water Sci.Technol., 54(6–7), 75–82.
Rivière, N., Travin, G., and Perkins, R. J. (2011). “Subcritical open channelflows in four branch intersections.” Water Resour. Res., 47(10),W10517.
Fig. 2. Discharge distribution RqT given by Eq. (1) as a function ofthe experimental values
Fig. 3. Discharge distribution RqT given by Eq. (1) by consider-ing a y-weir shifted by δ ¼ 1.1 cm as a function of the experimentalvalues
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Closure to “Experimental Study ofSubcritical Dividing Flow in anEqual-Width, Four-Branch Junction” byLeonardo S. Nania, Manuel Gómez,José Dolz, Pau Comas, and Juan PomaresOctober 2011, Vol. 137, No. 10, pp. 1298–1305.DOI: 10.1061/(ASCE)HY.1943-7900.0000423
L. Nania1; M. Gómez2; J. Dolz3; P. Comas4; andJ. Pomares51Associate Professor, Universidad de Granada, E.T.S. de Ingeniería de
Caminos, Canales y Puertos, Campus de Fuentenueva, 18071, Granada,Spain (corresponding author). E-mail: [email protected]
2Professor, Universitat Politècnica de Catalunya, E.T.S. de Ingenieros deCaminos, Canales y Puertos, Edifici D1, 08034, Barcelona, Spain.E-mail: [email protected]
3Professor, Universitat Politècnica de Catalunya, E.T.S. de Ingenieros deCaminos, Canales y Puertos, Edifici D1, 08034, Barcelona, Spain.E-mail: [email protected]
4Civil Engineer, AGBAR-Aquaplan, Barcelona, Spain. E-mail: [email protected]
5Technical Engineer, Universitat Politècnica de Catalunya, E.T.S. de Inge-nieros de Caminos, Canales y Puertos, Edifici D1, 08034, Barcelona,Spain. E-mail: [email protected]
The writers thank the discussers for their interest in the paper.In this closure, an attempt to clarify some issues indicated by thediscussers will be made. Due to the lack of more detailed measuresof parameters of flow (e.g., depths and velocities) in the experimen-tal campaign, they have been replaced by results of numericalsimulations made by Nania et al. (unpublished data, 2012) withthe two-dimensional (2D) hydrodynamic software (IBER v. 1.6),which has been validated with experimental data (Bladé 2005).
Experimental Facility
Relating to the experimental facility, although the width-to-lengthratio of the output channels is relatively small and the recircula-tion zones could interact with the weirs, its influence on the flowover the weirs can be considered negligible because of the smallflow velocity and the relative horizontality of the water surfaceeverywhere. In this sense, Nania et al. (unpublished data, 2012)show that for a case with the strongest unbalance of inflows and astrong unbalance in downstream boundary conditions (case: Qix ¼75.05 L=s, Qiy ¼ 8.26 L=s, Qox ¼ 32.71 L=s, Qoy ¼ 50.60 L=s,hwx ¼ 8 cm, and hwy ¼ 4 cm), maximum velocities in outputchannels reach only 0.3 m=s in the x-direction and 0.7 m=s in they-direction. Moreover, water surface in the output channels haselevation differences of only 3.3 mm in the x-direction and near1.6 cm in the y-direction.
On the other hand, an unintended slight lack of horizontalityin the bed of the experimental facility produced by a groundsettlement was detected and measured (see Table 1) after the ex-perimental campaign with subcritical flow finished, which intro-duced a mild asymmetry. The mentioned asymmetry changes theway of incorporating the boundary conditions, giving rise to analternative relationship and being the data of the original articlestill fully usable knowing the actual topography of the experimen-tal facility.
The more meaningful issue relating to the nonhorizontality ofthe experimental facility is that the bed elevation of the endof the y-direction channel is an average of 1.03 cm higher than that
of the x-direction channel. Similarly, taking again as reference thebed at the end of the x-direction channel, the bed immediately afterthe junction is an average of 0.44 cm higher in the x-direction and1.22 cm higher in the y-direction. Given that the flow is subcriticaleverywhere and that the control is downstream, it seems reasonableto use as boundary condition the water elevation instead of thedepth as in the original paper.
Boundary Conditions
In order to describe the boundary conditions, Nania et al.(unpublished data, 2012) conclude that it is more suitable to usethe cross section just upstream of the weirs instead of that locatedimmediately after the junction as used by the original article and assuggested by the discussers. However, the variable to considershould be one related to flow such as flow depth or water surfaceelevation instead of crest height of the weirs, as suggested byRivière et al. (2011) for the following reasons: (1) weir heightis a geometrical variable, not a flow variable; and (2) weirs willnever be present in a real-life street network but only a water eleva-tion as a downstream boundary condition. In this case, weirs areused in the experimental facility with the unique goal of gettinga variety of different water levels as downstream boundary condi-tions. That implies a modification in the dimensionless variableaccounting for the boundary conditions changing the depthimmediately after the junction by the water surface elevation justupstream of the weirs, and hence a derivation of a new relationshipas will be developed subsequently.
Variability of the Depth along the Cross Section
The more questionable issue of considering the flow depth justdownstream of the junction as a boundary condition is its repre-sentativeness as a boundary condition because of the lack of uni-formity of depth and velocity along the cross section. In order tocharacterize the depth variability, a case with the strongest unbal-ance of input discharges and a strong unbalance in the boundaryconditions is taken, so the variability in the depths can be con-sidered the maximum ones. The case is Qix ¼ 75.05 L=s,Qiy ¼ 8.26 L=s,Qox ¼ 32.71 L=s,Qoy ¼ 50.60 L=s, hwx ¼ 8 cm,and hwy ¼ 4 cm. Table 2 summarizes the minimum and maximumwater surface elevations in three cross sections, namely, justupstream of the junction (wix, wiy), immediately after the junction(wox, woy), and just upstream of the weir (wwx, wwx) in the x- andy-directions.
Table 1. Bed Elevations in Experimental Setup
x y z
0 2 0.2010 3.5 0.22 3.5 0.20142 2 0.19622 8.5 0.1953.5 8.5 0.19353.5 3.5 0.1913.5 2 0.18578.5 3.5 0.1878.5 2 0.1813.5 0 0.2012 0 0.201
Note: Values given in meters.
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As indicated in Table 2, the largest differences between thelowest and highest water elevation in every cross section coincidewith the x-direction upstream of the junction (1.3 cm) and they-direction immediately after the junction (1.6 cm), which is under-standable, observing that the larger input discharge comes from thex-direction and the larger output discharge goes in the y-direction,so almost half of the discharge has to turn 90°. Being that theflow depths are of the order of 10–13 cm, this implies that themaximum variations are 10% above and below the average depth.However, the differences near the weirs are only 0.22 cm in thex-direction and 0.66 cm in the y-direction, indicating that thesecross sections would be more suitable to be considered as a boun-dary condition.
Discharge Distribution
The equation presented by Rivière et al. (2011) for predicting thedischarge distribution in a four-branch junction with subcriticalflow and invoked by the discussers has another important weakpoint related to the number of dimensionless parameters represent-ing the phenomenon. Such an equation [Eq. (1) in the discussion]can be rewritten as
Rq ¼ 0.452þ 0.6312Rqi − 0.0936RcRqi − 0.1031RcRb
Rg
þ 0.001031Rc
Rgþ 0.1031
Rb
Rg− 0.001031
1
Rg− 1.0382RcRb
þ 0.010382Rc þ 1.0382Rb ð1Þ
where
Rq ¼Qoy
Qix; Rqi ¼
Qiy
Qix; Rc ¼
cycx
; Rb ¼cxb;
1
Rg¼ b5=2g1=2
Qix; RcRqi ¼
cycx
Qiy
Qix;
RcRb
Rg¼ cyb3=2g1=2
Qix;
Rc
Rg¼ cy
cx
b5=2g1=2
Qix;
Rb
Rg¼ cxb3=2g1=2
Qix; RcRb ¼
cyb
where Qix, Qiy = the input discharges in the x- and y-directions,respectively; Qoy = the output discharge in the y-direction; cx,cy = weir heights in the x- and y-directions, respectively; andb = the channel width. Eq. (1) is a linear relationship between10 dimensionless parameters of which it is apparent that Rq,Rqi, Rc, Rb, and 1=Rg could be the basic ones, while the remainingfive are combinations of the former ones. The original article, in thedimensional analysis of the problem of dividing flow in a four-branch junction with two input flows and two output flows, statesthat flow distribution can be represented by
Qox
QT¼ f1
�Qix
QT;byox
;yoxyoy
;Fix
�ð2Þ
where Qix=QT = inflow ratio; Qox=QT = outflow ratio; b=yox =aspect ratio of the outcoming flow in the x-direction; yox=yoy =ratio between outflow depths just downstream of the junction;and Fix ¼ Qix=ðby1.5ix
ffiffiffig
p Þ. In turn, Rivière et al. (2011) state forthe same objective the following function:
Qoy
Qix¼ f2
�Qiy
Qix;cxb;cycx
;Qix
b2ffiffiffiffiffigb
p�
or Rq ¼ f2ðRqi;Rb;Rc;RgÞ
ð3Þ
Both coincide in that the phenomenon should be represented bya relationship between five dimensionless parameters, so Eq. (1)has five redundant dimensionless parameters that introduce a spu-rious correlation and is in disagreement with the Π Theorem ofdimensional analysis. It is known that the more parameters usedto adjust a function to a set of data, the larger correlation coeffi-cient can be obtained. Adapting Eq. (1) to have five dimensionlessparameters and changing Rc ¼ cy=cx with Ry ¼ ywy=ywx, usingthe experimental data by the authors results in the followingequation:
Rq ¼ 5.221þ 0.7427Rqi − 0.00046581
Rg− 4.5884Ry
− 3.8751Rb ð4Þ
Fig. 1 shows the predictions with Eq. (4) in terms of (1) Rq and(2) Qox=QT . Although a high correlation coefficient r2 of 0.92is obtained in terms of Rq, it decreases to 0.27 when presentedin terms of Qox=QT and some cases take values greater than 1,implying that Qox>QT , which has no physical meaning.
Improved Relation to Estimate Flow Distribution in aFour-Branch Junction with Subcritical Flow
In order to be able to use the data taken with the nonhorizontalexperimental facility, given that the control of flow distributionis downstream and that the more important issue is the differencebetween water surface elevations in every output channel thatconstitute the boundary conditions, as concluded by Nania et al.(unpublished data, 2012), it is convenient to change the dimension-less parameter yox=yoy by one including the previously mentioneddifference and also changing the cross section used as reference,Δw=yox, with Δw ¼ wwx − wwy, where wwi is the water elevationjust upstream of the weirs in the i-direction, which in turn is ob-tained by zwiþywi, bed elevation plus depth in the i-direction, suchthat the relation proposed in the original paper becomes
Qox
QT¼ Aþ B �Δw
ywxþ C �Qix
QTþD � b
ywxþ E � Fix ð5Þ
A new analysis of data indicated again the convenience of takingnine groups of data according to the limits of Table 3 that results inthe coefficients of Table 4 for Eq. (5).
Table 2. Maximum and Minimum Water Elevation in a Cross Section
Feature wix wiy wox woy wwx wwy
Maximum 0.3131 0.3140 0.3165 0.3124 0.3154 0.3034Minimum 0.3002 0.3135 0.3132 0.2966 0.3132 0.2968Maximum minus minimum 0.0129 0.0005 0.0033 0.0158 0.0022 0.0066
Note: Case: Qix ¼ 75.05 L=s, Qiy ¼ 8.26 L=s, Qox ¼ 32.71 L=s, Qoy ¼ 50.60 L=s, hwx ¼ 8 cm, and hwy ¼ 4 cm. Datum = 0.181 m below the lowest pointof experimental facility.
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The global correlation coefficient r2 is 0.90 (see Fig. 2), but it isa remarkable fact that when the same data are presented in terms ofRq ¼ Qoy=Qix as is done by Rivière et al. (2011) instead ofQox=QT , r2 increases up to 0.98. If the analysis is performed byreplacing the parameterΔw=ywx withΔo=yox, whereΔo is the pre-viously mentioned difference immediately after the junction andyox is the depth measured in the same location, the correlation co-efficient decreases to 0.76 if represented in terms of Qox=QT and0.98 if it is done based on Rq ¼ Qoy=Qix. That demonstrates theconvenience of taking the cross section just upstream of the weir asthe boundary condition.
Using a unique set of coefficients for Eq. (5) for the whole rangeof the dimensionless parameters, a better fit is obtained withA = 0.28082, B = −1.7231, C = 0.3126, D = 0.00068, andE = 0.32579, with the correlation coefficient being 0.59 in termsof Qox=QT and 0.92 in terms of Rq ¼ Qoy=Qix.
In the application of Eq. (5), water elevations used as boundaryconditions ought to be taken at a distance of 10=3 channel widthsdownstream of the junction and the values of the dimensionlessparameters should fall inside the range of the data used for thecoefficient determination.
Conclusions
Depths used in the analysis by the authors were measured in re-gions where the flow has mainly a 2D behavior and a deeper analy-sis made by Nania et al. (unpublished data, 2012) concluded thatthe cross section just upstream of the weir would be a more suitablelocation considering the boundary conditions.
A slight nonhorizontality was detected in the bed of theexperimental facility, but because flow is subcritical everywhere,velocities are very low, and the flow control is at the downstreamend, it seems reasonable to take water elevation as the boundary
y = 0.9297x + 0.2341
R2 = 0.925
0
2
4
6
8
10
0 2 4 6 8 10Observed Rq
Pre
dict
ed R
q
y = 0.5325x + 0.3207
R2 = 0.2716
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
Observed Qox/QT
Pre
dict
ed Q
ox/Q
T
(a) (b)
Fig. 1. (a) Observed and predicted Rq ¼ Qiy=Qix; (b) observed and predicted Qox=QT using Eq. (4) and experimental data from the original article
Table 3. Dimensionless Parameters Used in the Regression Analysis andIntervals Assumed
Dimensionlessparameter Full range Subinterval 1 Subinterval 2 Subinterval 3
a3 ¼ b=ywx 9.3–28.1 9.3–13 13–17 17–28.1a4 ¼ Δw=ywx −0.27–0.24 −0.11–0.0 0.0–0.05 0.05–0.24
Table 4. Coefficients Used in Eq. (5) for Cases Defined for a Given Combination of Ranges of b=ywx and Δw=ywx
Interval Number A B C D E r2
a3.1–a4.1 17 0.47081 −1.62205 0.31860 −0.00377 −0.20965 0.38a3.1–a4.2 17 0.78785 −1.62910 0.22884 −0.03806 0.10716 0.68a3.1–a4.3 16 1.16398 0.15566 0.71411 −0.07571 −1.27362 0.99a3.2–a4.1 14 0.22539 −0.50448 1.01628 0.01274 −1.17807 0.61a3.2–a4.2 17 −0.02159 −4.04528 0.59503 0.01493 0.31782 0.90a3.2–a4.3 23 0.41803 −2.76015 −0.36832 −0.00636 1.66882 0.95a3.3–a4.1 19 0.24224 0.03328 0.75319 0.01063 −0.72029 0.46a3.3–a4.2 20 0.16041 −3.70514 0.59618 0.00933 −0.06677 0.77a3.3–a4.3 16 0.04958 −1.80785 0.02041 0.01046 0.88309 0.90Total 159 Global r2 0.90
y = 0.8995x + 0.048
R2 = 0.8995
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
Observed Qox/QT
Pre
dict
ed Q
ox/Q
T
Fig. 2. Observed versus predicted Qox=QT with Eq. (5) and coeffi-cients of Table 4, using experimental data from the original article
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condition instead of depth, avoiding the inaccuracies related tothe nonhorizontal bed. This is done easily, knowing the actualtopography. A revised relation to describe flow distribution in afour-branch junction with subcritical flow is presented, taking thedifference of water elevation just upstream of the weirs in eachdirection as the boundary conditions.
The relation proposed by Rivière et al. (2011) introduces theparameter Rq ¼ Qoy=Qix, which gives better correlation coeffi-cients than using Qox=QT for the same data; however, becauseRq ranges theoretically from 0 to infinity, a high correlation coef-ficient does not mean necessarily that physically inconsistentresults cannot appear (i.e., Qox>QT). An attempt was made adapt-ing the relation by Rivière et al. (2011) using five dimensionlessparameters and the data from the original article. The result showsa lower correlation coefficient than using Eq. (5) even if a single setof coefficients is taken.
References
Bladé, E. (2005). “Modelación del flujo en lámina libre sobre caucesnaturales. Análisis integrado con esquemas en volúmenes finitosen una y dos dimensiones.” Ph.D. thesis, Instituto Flumen, UniversitatPolitècnica de Catalunya, Barcelona, Spain (in Spanish).
IBER v. 1.6 [Computer Software]. Grupo de Ingeniería del Agua y delMedio Ambiente (GEAMA), Universidad de A Coruña, A Coruña,Spain; Instituto FLUMEN, Universitat Politècnica de Catalunya,Barcelona, Spain and Centro Internacional de Métodos Numéricosen Ingeniería (CIMNE), Barcelona, Spain, ⟨http://www.iberaula.es/⟩.
Rivière, N., Travin, G., and Perkins, R. J. (2011). “Subcritical open channelflows in four branch intersections.” Water Resour. Res., 47(10),W10517.
Discussion of “Uncertainty Model forIn Situ Quality Control of Stationary ADCPOpen-Channel Discharge Measurement”by Hening HuangJanuary 2012, Vol. 138, No. 1, pp. 4–12.DOI: 10.1061/(ASCE)HY.1943-7900.0000492
M. Muste, M.ASCE1; and K. Lee21Research Engineer and Adjunct Professor, IIHR—Hydroscience &
Engineering, Univ. of Iowa, Iowa City, IA 52242. E-mail: [email protected]
2Graduate Student, IIHR—Hydroscience & Engineering, Univ. of Iowa,Iowa City, IA 52242 (corresponding author). E-mail: [email protected]
Hydroscience is dealing with processes that are variable in timeand space, making the measurement process difficult, prone to un-certainties, and, at times, even impossible to be captured withcommon instrumentation. For a long time, the hydroscience com-munity has been screening and evaluating uncertainty analysis(UA) methodologies and frameworks that can be uniformly appliedacross the myriad of instruments and measurement situationsencountered in routine and custom applications. The search forrelevant UA methods is challenging because there is a multitudeof UA approaches developed by various communities that arenot fully harmonized in term of concepts, terminology, and proto-cols (Muste et al. 2012). Furthermore, the ISO TC 113 technicalcommittee on hydrometry (the closest to the hydroscience area)
continues to produce new standards related to uncertainty andspecific instruments at a pace that is difficult to be evaluated,assimilated, and implemented (29 hydrometry-related standards in1973–1983). Given the situation common users willing to applyUA are facing a daunting task (Thomas 2002).
With these considerations in mind, the publication of the presentcontribution is of interest for many readers as uncertainty is increas-ingly evoked in the hydrometry area and the search for robustand credible analyses is in high demand. The author should becommended for bringing the most recent concepts in UA, topicsquite scarcely covered in technical journals. The UA concepts usedin the presented analysis are those of the ISO 5168 (ISO 2005) andISO 748 (ISO 2007). The two standards have been revised toharmonize with the Guide to the Expression of Uncertainty inMeasurement (GUM) [Joint Committee for Guides in Metrology(JCGM) 2008], which is considered “the most authoritative docu-ment in all aspects of terminology and evaluation of uncertainty”(ISO 2005, 5168). While the author’s intention is laudable, thereare several limitations of the paper, mostly stemming from alack of rigor in using the terminology of the new standards, sub-jective interpretations, and mixture of old and new UA conceptsthat might confuse rather than provide users with a good recipefor UA implementation. The present discussers consider that it isappropriate to bring into discussion some concerns about the val-idity of some of the interpretations and the significance of the prac-tical UA demonstration presented in the paper under discussionwith the realization that there is a need for clarity and rigor to con-vince the community to adopt UA methodology. The discussion isbased on the GUM (JCGM 2008) guidelines, which is the basis forthe ISO standards used in this paper.
Terminology Issues
The hallmark distinction of GUM terminology compared withprevious UA standards is the classification of uncertainties inType A and Type B based on their method of evaluation.Acknowledging the importance of these terms in general andfor the context of the present discussion especially, the discussersreplicate herein their definition as provided in GUM (JCGM2008, p. 7):
Type A standard uncertainty is obtained from a probabilitydensity function derived from an observed frequency distribu-tion, while a Type B standard uncertainty is obtained froman assumed probability density function based on the degreeof belief that an event will occur. Both approaches employrecognized interpretations of probability.
And further on in GUM (JCGM 2008, p. 11):
For an estimate of an input quantity that has not been obtainedfrom repeated observations, the associated estimated standarduncertainty is evaluated by scientific judgment based on all ofthe available information on the possible variability of inputquantity
such as (1) previous measurement data; (2) experience with orgeneral knowledge; (3) manufacturer’s specifications; (4) calibra-tion data; and (5) handbooks.
In the opinion of the present discussers, the analysis presentedin the paper does not contain Type A uncertainty (as defined by theGUM-compliant standards) because there are no repeated mea-surements to create a probability density function for any of thevariables used in the calculation of the subsection discharges or
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for the total discharge. The author considers the repeated stationarymeasurements collected with acoustic Doppler current profiler(ADCP) to obtain the mean velocities as a sample statistics forthe estimation of Type A uncertainties. This is not the case asthe first part of Eq. (1) for determining discharges in subsectionscontains the mean velocity measured along a vertical profile andnot the instantaneous velocity. The concept of “instantaneous” dis-charge implicitly assumed by the author for the second form ofEq. (1) and in Eq. (10) is not consonant with hydraulic terminol-ogy as discharge is a mean flow quantity defined by the meanvelocity. The latter is obtained by averaging measurements ac-quired under the same conditions to create a sample large enoughto capture the scales of the turbulence (Nezu and Nakagawa 1993).In the context of UA, the standard deviation of the mean velocityassumes the existence of a sample statistics of repeated meanvelocities collected at the same location, with the same instrument,and over enough measurement duration (exposure time) (Musteet al. 2004). Even if all these requirements are fulfilled, uncertaintywill still affect the final results. It is important to mention that UAis applied after all the measurement biases have been corrected(JCGM 2008, Section 3.2.4) and the measurement protocolsimplemented with rigor (JCGM 2008, Section 3.4.8). From thisperspective, is intriguing that the author presents in this paper aset of repeated total discharge measurements (pp. 8–10), but theyare not used for evaluating Type A uncertainty, rather to “validate”the proposed model.
Another departure from the language of these new standardsis the use of the word combination “estimation of randomuncertainty.” The expression “random uncertainty” is avoided byGUM-based ISO references. For example, Annex I of ISO 5168(2005) referenced in the paper dedicates ample space to make surethat the proper distinction is made when using the previouslymentioned term:
this International Standard contains significant changes inthat the concepts and terminology of random and systematiccomponents of uncertainty are no longer the preferred catego-ries. There are two main reasons for this: a) in conformancewith GUM, components of uncertainty arising from randomand systematic causes, after they have been evaluated, aretreated identically; b) the terms can be used in ways thatare ambiguous and confusing.
The present discussers are concerned that the imprecise use of thestandard language might raise questions in the minds of the users,especially those who are new to the UA area.
Procedural Issues
The overall goals of the paper are developing an UA “model” and“validating” it. Model validation is defined as (p. 8): “multiple mea-surements at a site (30 or more) under a steady flow condition sothat the approximate true uncertainty may be obtained and used tovalidate an uncertainty model.” There are two observations aboutthe targeted goals. The first observation is that the author’s vocabu-lary is quite rare in the UA literature as validation assumes that thetrue values of the measurements are available. It is the true valuesthat are not known in UA in the first place. The UA methodologiesattempt to estimate the departure from a hypothetical truth usingthe best statistical and mathematical tools available. The secondobservation is related to the author’s motivation for introducingthe model: “ : : : discharge measurement using the stationary ADCPmethod is in practice often conducted only once at a site. One maynot have confidence in the quality of such a single measurement
unless the uncertainty associated with the measurement can beobtained and evaluated. Therefore, an immediate estimate of theuncertainty when a discharge measurement at a site is completedis desired for measurement quality control and assurance.” It is notclear for the present discussers why we need an additional modelfor UA as all GUM-based standards contain procedures to deal withsingle measurement situations (e.g., ISO 5168 Section D.6 andAnnex C). What is here referred to as “a model” is simply the in-formation that goes along with that single measurement analysis, soan “immediate” and rigorous analysis can be conducted with justone measurement (obviously the reported uncertainty will be largerthan situations where repeated measurements are available). As willbe explained below, the author’s proposed model is generic, exclud-ing important factors involved in a new in situ measurement situa-tion (flow characteristics, measurement environment, operationalconsiderations); hence the described analysis has no capacity togive confidence or to be rigorous.
The author chooses for the presented analysis the relative stan-dard uncertainty (RSU) as a model for the functional relationship ofthe measurement process. It is not explained why this selection waspreferred. Most of the examples provided in the GUM-relatedliterature use the dimensional standard uncertainties in the func-tional relationships as there are uncertainties that are not estimatedas percentages of the measured quantities (e.g., the resolution ofan instrument), and this approach better exposes the sensitivitycoefficients that affect each uncertainty component. The presentedanalysis follows closely the example G.4 from ISO 5168, p. 56. Inthis example it is clearly specified (p. 59): “The entire uncertaintycalculation then becomes a Type B evaluation of uncertainty sincethe components of the uncertainties quoted in the analysis are basedon previous measurements and calibration data.” In contrast,the author states that the paper’s example includes both Type Aand Type B components. The uncertainty qualifiers used in thepaper are not rigorously documented and, as explained earlier,nonconform with GUM (JCGM 2008) definitions. Specifically, onp. 5 of the original paper the author states: “The Type A uncertaintycomponent accounts for the site-specific measurement conditionssuch as ambient turbulence; ADCP pitch, roll, and headingvariations/errors; and the system noise of the ADCP used at the site.”Similarly, on p. 7 is stated, “The Type B uncertainty componentsin the model consider error sources such as width-measurementerrors, calibration errors, and the limited number of verticals.” Aclarification of the basis of these associations using the conceptualdefinitions provided by the standards is needed to illuminate thereader on how to implement the UA model in future applications.This loose use of the terminology might be confusing for the readerswho are new to the area and clearly cannot contribute to concertedprogress to promote uniform and robust uncertainty analyses.
Other debatable interpretations are related to the attribution ofsome uncertainty and of the correlated/uncorrelated uncertainties inuncertainty propagation Eq. (2). The author states on p. 5 that thesubsection discharges obtained with the midsection method areuncorrelated, while the mean-section method is correlated. Theseinterpretations are correct when applied to the estimation of sub-section discharges, qi. The present discussers, however, argue thatas the total discharge subjected to UA in this paper is a sum ofsubsection discharges as defined by Eq. (1), the correlation betweenvariables occurs in this summation rather than in the individualsubsection discharges. Consequently, the subsection dischargesare correlated irrespective of the discharge estimation algorithm.Accordingly, Eq. (2) has to account for these correlations throughappropriate terms that currently are missing. For more details theusers are directed to Section F.1.2 Correlations in GUM (JCGM2008, pp. 62–64) where the reasoning for the appearance of these
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correlations is explained. Another questionable formulation inEq. (2) is related to the uncertainties components associated withthe width measurements where they are double-counted, namely, inthe third and fourth terms of the equation.
The paper introduces new types of uncertainty classification andan abundance of information that makes the paper needlessly com-plex instead of providing a clear and clean demonstration of UAimplementation. First, the author splits the total uncertainty in TypeA and B uncertainty components in Eqs. (4)–(6). Second, in thevalidation section, the author distinguishes the ub 0—random uncer-tainty component of the Type B and the uQ 0—random uncertaintycomponent of Type B of the overall relative combined uncertainty.A justification for these uncertainty groupings should be in orderas they are not used in the GUM-based standard practices. Last,extended space is used for details about various sources of errorsaffecting the mean velocity (flow angle, speed of sound, pitch-rollheading, velocity distribution) that are not used in the analysis asthese error sources are subsequently lumped into one Type A un-certainty that affects the discharge (rather than the velocity). Whatis the added value of presenting information that is not used in thepaper, especially when the paper’s subject is complex enough andmost probably new to the readers? Instead, more details shouldhave been provided on the relationships between velocity in bins,verticals, and discharges in subsections. For these important rela-tionships the reader is directed to manufacturer’s manuals, whichoften do not disclose the relevant information under the banner ofthe constraints of the proprietary algorithms.
In summary, the UA conducted in this paper is based only onEqs. (3), (9), (12), and (13) and two estimates provided in Table 1.The estimation of ucal [in Eq. (8) and Table 1] is based on frugal andquestionable assumptions (e.g., “ : : : ucal is assumed to be 1% forthe current-meter : : : . This value may also be assumed for station-ary ADCP for convenience.”). Furthermore the data set assembledby the author for validation purposes presents multiple problems asbenchmark data for UA. First, it contains a variety of flow condi-tions (open- and ice-covered channels) that are characterized bydistinctly different velocity profiles. Second, the data are acquiredwith three instruments that differ through configuration, operationmodes, and bin sizes. The flow regime and the duration of themeasurements in verticals in the five data sets are not specified.This amalgam of data is not recommended to be used for UAinferences, and even less for “validation” purposes. Actually theinterpretation of the validation results is uncertain; i.e., the authorconcludes that “further investigation is desired when more andlarger data sets are available.”
Despite the criticism expressed above, these discussers welcomethe paper, as there are very few contributions and researchersinvolved in this important aspect of hydroscience practice. Thepublished paper and the thoughts expressed herein are part of aprocess that was also experienced in the adoption of UA by othercommunities where a long time was spent on independent develop-ments, discussion of various terminology, and agreements onprotocols to conduct UA in practice. The author’s effort is impres-sive and includes subjects that invite more evaluation and dialogue.UA is in its infancy in this area, and researchers need an open andtransparent dialogue on the frameworks, terminology, and proto-cols for conducting UA. This effort is considerable and requiresinvolvement of instrument manufacturers, operators, and riverscientists in order to properly address all sources of uncertaintiesinvolved in the measurement process. Expert knowledge, practicalconsiderations, engineering judgment, and familiarity with UA ex-pertise need to guide the dialogue. Reaching a common perspectiveon UA will allow researchers to compare results, use complemen-tary data obtained by others, and continue to improve all aspects of
measurements. Such an effort is currently undergoing under aWorld Meteorological Organization project where UA frameworkshave been analyzed and examples of UA implementation have beenprovided (Pilon et al. 2007).
References
International Organization for Standardization (ISO). (2005). “Measure-ment of fluid flow—Procedures for the evaluation of uncertainties.”ISO 5168, Geneva.
International Organization for Standardization (ISO). (2007). “Hydrometry—Measurement of liquid flow in open channels using current meters orfloats.” ISO 748, Geneva.
Joint Committee for Guides in Metrology (JCGM). (2008). “Evaluation ofmeasurement data - Guide to the expression of uncertainty in measure-ment (GUM 1995 with minor corrections).” Geneva, Switzerland.
Muste, M., Lee, K., and Bertrand-Krajewski, J.-L. (2012). “Standardizeduncertainty analysis frameworks for hydrometry: Review of relevantapproaches and implementation examples.” Hydrolog. Sci. J., 57(4),643–667.
Muste, M., Yu, K., Pratt, T., and Abraham, D. (2004). “Practical aspects ofADCP data use for quantification of mean river flow characteristics.Part II: Fixed-vessel measurements.” Flow Meas. Instrum., 15(1),17–28.
Nezu, I., and Nakagawa, H. (1993). Turbulence in open-channel flows,Balkema, Totterdam, The Netherlands.
Pilon, P. J., Fulford, J. M., Kopaliani, Z., McCurry, P. J., Ozbey, N., andCaponi, C. (2007). “Proposal for the assessment of flow measurementinstruments and techniques.” Proc., XXXII IAHR Congress, Corila,Venice, Italy.
Thomas, F. (2002). “Open channel flow measurement using internationalstandards: Introducing a standards programme and selecting a stan-dard.” Flow Meas. Instrum., 13, 303–307.
Closure to “Uncertainty Model for In SituQuality Control of Stationary ADCPOpen-Channel Discharge Measurement”by Hening HuangJanuary 2012, Vol. 138, No. 1, pp. 4–12.DOI: 10.1061/(ASCE)HY.1943-7900.0000492
Hening Huang, P.E.11Principal Hydraulic Engineer, Teledyne RD Instruments, 14020 Stowe Dr.,
Poway, CA 92064. E-mail: [email protected]
The writer would like to thank the discussers for their interest in thepaper. The following addresses the discussers’ comments.
The definitions of the Type A and Type B evaluation of uncer-tainty are clearly written in the Guide to the Expression of Uncer-tainty in Measurement (GUM) [Joint Committee for Guides inMetrology (JCGM) 2008]. In practice, whether an uncertaintycomponent is considered to be Type A or Type B dependson how its standard uncertainty is estimated. If its standard uncer-tainty is estimated by the statistical analysis of a series ofobservations, the uncertainty component is Type A. Otherwise, itis Type B. In the proposed model, the standard uncertainty of thesubsection discharge is estimated by the statistical analysis ofsubsection discharge data derived from the acoustic Doppler currentprofiler (ADCP) measurements. Therefore, it is a Type A uncertaintycomponent. Note that the original form of Eq. (1) in the originalpaper is the summation of subsection discharges. Thus, a direct
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analysis of the subsection discharge is valid. As a matter of fact, usingthe subsection discharge as a variable in the uncertainty analysis is animportant feature of the proposed model, which makes the uncer-tainty analysis much easier and cleaner as compared to using thesubsection velocity, width, and depth as variables.
The concept of instantaneous discharge may originate from themoving-boat ADCP discharge-measurement method that wasfirst introduced in the early 1980s. A general equation for riverdischarge Q through an arbitrary surface, s, is defined as
Q ¼Zs
→ u •→ n ds ð1Þ
where → u = water velocity vector; ds = differential area;and → n = unit vector normal to ds.
In the moving-boat ADCP method, an ADCP collects ensemble(single ping or multiple ping) data for water velocities at depthcells (bins), water depth, and boat velocity along the boat’s track.Accordingly, Eq. (1) can be rewritten in the following form(Christensen and Herrick 1982):
Q ¼ZT
0
Zd
0
ð→ u ×→ VbÞ •→ k · dz dt ð2Þ
where T = total transect time; d = water depth; → Vb = boat velo-city vector;→ k = unit vector in the vertical direction; dz = verticaldifferential depth; and dt = differential time.
Eq. (2) can be rewritten as
Q ¼ZT
0
qðtÞ dt ð3Þ
where
qðtÞ ¼Zd
0
ð→ u ×→ VbÞ •→ k · dz ð4Þ
is the instantaneous subsection discharge at time t, which is calcu-lated by an ADCP software (for example, WinRiver II for TeledyneRD Instruments ADCPs) using the ADCP ensemble (single ping ormultiple ping) data for water velocity profile, water depth, and boatvelocity. The integral is replaced by a summation in the calculationby the ADCP software.
Similarly, in the stationary ADCP method, the subsectiondischarge q̄ can be written as
q̄ ¼ZTd
0
qðtÞ dt ð5Þ
where Td = measurement duration at a vertical and
qðtÞ ¼ wZd
0
→ u •→ n · dz ð6Þ
is the instantaneous subsection discharge at time t, which is calcu-lated by an ADCP software (for example, SxS Pro for Teledyne RDInstruments ADCPs) using the ADCP ensemble (single ping ormultiple ping) data for water velocity profile and water depth.Again, the integral is replaced by a summation in the calculationby the ADCP software. Note that the subsection width w isassumed to be a constant, which is to be entered by an operatorinto the ADCP software during the field measurement.
A moving-boat ADCP or a stationary ADCP typically outputsensemble data at a rate of 1 to 2 Hz. The ADCP ensemble velocitydata at such a high output rate is considered to be instantaneousvelocity as compared to the mean velocity measured by a conven-tional current meter during a duration of 40 s or more. That theconventional equation for a total discharge is defined by a meanvelocity is consistent with the mean velocity measured by a conven-tional current meter. However, as can be seen in Eq. (3) or (5), atotal discharge can be formulated using instantaneous subsectiondischarges that are consistent with either the moving-boat ADCPmethod or the stationary ADCP method.
The proposed model is for estimating the uncertainty associatedwith a single discharge measurement using the stationary ADCPmethod. It is similar to the model presented in ISO 5168 (ISO2005) or ISO 748 (ISO 2007) for estimating the uncertainty asso-ciated with a single discharge measurement using the conventionalcurrent-meter method. If repeated discharge measurements are con-ducted at a site under a steady flow condition, using either theADCP method or the current-meter method, a Type A evaluationof uncertainty can be directly applied to the measured discharges,and there will be no need for an uncertainty model. This is the exactsituation that occurs when using the moving-boat ADCP methodthat usually requires four discharge measurements (transects)according to a measurement quality control policy (e.g., Oberget al. 2005). The available data sets from the repeated stationaryADCP discharge measurements presented in the original paperwere used for the Type A evaluation. The results from the TypeA evaluation were the field relative standard uncertainties (RSUs),which were then used to validate the proposed uncertainty model.
Although ISO 5168 started to use the Type A and Type B ter-minology, it still observed the traditional classification of randomand systematic uncertainty sources and described the relationshipbetween the two terminologies (see Annex I in ISO 5168). In thewriter’s opinion, the random and systematic classification is usefuland will exist along with the Type A and Type B classification inpractice and in the literature. Each classification has its own advan-tages in terms of describing the physics and mathematics involvedin the uncertainty analysis. The use of both terminologies may helpthe readers understand the nature or characteristics of error sourcesfrom different points of view. It is interesting to note that the seconddiscusser used the terms “bias limits” and “precision limits,” in-stead of following the Type A and Type B terminology, in theirpaper published in 2007 (González and Muste 2007).
Again, the proposed model is for estimating the uncertainty as-sociated with a single discharge measurement using the stationaryADCP method. The discussers stated that: “It is not clear for thepresent discussers why we need an additional model for UA as allGUM-based standards contain procedures to deal with single meas-urement situations (e.g., ISO 5168 Section D.6 and Annex C).”However, referring to ISO 5168 Section D.6 and Annex C, neitherof them can be used to estimate the uncertainty associated with astationary ADCP discharge measurement. ISO 5168 Section D.6deals with estimating an uncertainty value for a single measurementby using an external standard deviation derived from past data. Inour case of discharge measurement using an ADCP at a site, theremay be no past data that can be used for the uncertainty analysis.Even if the past data at a site are available, the data may not reflectthe current measurement conditions at the site. The ISO 5168 An-nex C, on the other hand, discusses the coverage factors associatedwith the Student’s t distribution. It has nothing to do with the topicof single measurement uncertainty. The readers would be benefitedif the discussers had presented a numerical example to demonstratehow a GUM-based procedure could be applied to the uncertaintyanalysis of a stationary ADCP discharge measurement.
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The RSU was chosen in the proposed model. This was to beconsistent with the uncertainty model for the current-meter methodpresented in ISO 5168 or ISO 748.
The original paper clearly defined and described the Type A andType B uncertainty components in the proposed model; therefore,they are not described again here. The definitions of these uncer-tainty components conform to GUM (JCGM 2008) definitions. Theproposed model requires field data to estimate the Type A uncer-tainty components. This is a fundamental difference from the un-certainty analysis example for the current-meter method presentedin ISO 5168 or ISO 748, which is a pure Type B evaluation becausenone of the uncertainty components are estimated using field data.
As for how to implement the proposed model in future appli-cations, one is encouraged to conduct a field ADCP measurementusing the ADCP software SxS Pro. The proposed model has beenbuilt into the SxS Pro software, which outputs uncertainty values assoon as the measurements at all verticals are completed.
Like conventional current-meter measurements, the ADCPvelocity profile measurements at verticals are assumed to beindependent. Thus, the subsection discharges calculated by themidsection method (that is used for the sake of discussion here)are also independent. The summation of the independent subsec-tion discharges used to estimate the total discharge cannot make theindependent subsection discharges become correlated. In thisregard, the discussers’ statements, “ : : : the correlation betweenvariables occurs in this summation rather than in the individualsubsection discharges. Consequently, the subsection dischargesare correlated irrespective of the discharge estimation algorithm.”are confusing, and their citation of Section F.1.2 Correlations inGUM (JCGM 2008) does not support the statements.
The writer confirms that the uncertainty component associatedwith the width measurement is not double counted in the proposedmodel. The width of a subsection is a constant (user input) incalculating instantaneous subsection discharges by the ADCPsoftware. Therefore, the standard uncertainty of the subsectiondischarge does not contain the contribution from the widthmeasurement error. The contribution is considered by the third termon the right side of Eq. (2) or (3) in the original paper.
The proposed uncertainty model for the stationary ADCPmethod is the counterpart of the uncertainty model for thecurrent-meter method presented in ISO 5168 or ISO 748. Theproposed model is, in fact, less complex than its counterpartbecause it directly uses the subsection discharge as a variable inthe uncertainty analysis.
Regarding the use of estimated random uncertainty compo-nents in the model validation, the original paper has already statedthat: “The bias in discharge cannot be determined from the field databecause the true discharge is unknown.” Therefore, the field RSUwas assumed to account for random errors only, and the estimatedoverall random uncertainty u 0
Q was compared to the field RSU.The detailed descriptions of various error sources were re-
quested by the reviewers during the review process of the originalpaper. The magnitudes of some of the errors or their contribution tothe discharge uncertainty were quantitatively estimated. As a result,some of the errors were found insignificant and negligible; there-fore, they were not considered in the proposed model. Although thedescriptions took an extended space, the readers received completeinformation related to the uncertainty analysis.
The proposed model avoided using the relationship betweenvelocities in depth cells (bins). The correlation between velocitiesin depth cells cannot be satisfactorily estimated or formulated. Thewriter analyzed 15 sets of ADCP velocity profile data. The resultsindicated that the coefficient of correlation between velocities inadjacent depth cells ranged from 6% to 75%, depending on cell
size, site (turbulence), and ADCP model. In addition, the coeffi-cient of correlation between velocities in nonadjacent depth cellswas not found to be zero.
The field data sets used in the validation of the proposed modelcovered a variety of flow and site conditions (open- and ice-coveredchannels) and three ADCP models. This was done in order toexamine the model’s capability in handling different flow and siteconditions and different ADCP models. One of the advantages ofthe proposed model, as compared to a pure Type B model, is that itaccounts for site specific conditions, including ambient turbulence,and the type of ADCP model used at the site. As expected, theuncertainty analysis results were not the same for different sitesand different ADCP models.
The validation of the proposed model involved two aspects. Onewas to examine the robustness of the proposed model. That is, theestimated uncertainties should be consistent among the repeateddischarge measurements at a site under a steady flow condi-tion and with the same ADCP. The model was demonstrated to berobust because the relative difference between the estimateduncertainty of any discharge measurement and the mean of theestimated uncertainties of all discharge measurements at a sitewas found to range from 0 to 1.06% (see Tables 3 to 7 in the origi-nal paper). The other aspect of the model validation was to examinethe agreement between the estimated uncertainty and the Type Aevaluation of uncertainty for the repeated discharge measurements(i.e., the field RSU). The model was found to agree reasonably wellwith the field RSU. However, because (1) only five data sets wereavailable and used in the validation, (2) the sizes of the data setswere small, and (3) the estimated uncertainties seemed conservativewith yet identified reason, the writer suggested that further inves-tigation be desired when more and larger data sets are available.
In summary, the proposed model provides ADCP users with asimple, practical tool for estimating the uncertainty associated witha single discharge measurement using the stationary ADCPmethod. The proposed model conforms to GUM (JCGM 2008),ISO 5168, and ISO 748. In the writer’s opinion, GUM (JCGM2008) provides a general guidance for uncertainty analysis. Thefundamental theory, the law of propagation of uncertainties, isthe basis of GUM (JCGM 2008) and other uncertainty analysisdocuments. However, the application of the theory to a specificproblem must be customized. The model presented in ISO 5168or ISO 748 is a customized model for the current-meter method.The proposed model is a customized model for the stationaryADCP discharge-measurement method.
References
Christensen, J. L., and Herrick, L. E. (1982). “Mississippi River test:Volume 1: Final report.” Rep. DCP4400/300, prepared for the U.S.Geological Survey, AMETEK/Straza Division, El Cajon, California.
González-Castro, J. A., and Muste, M. (2007). “Framework for estimatinguncertainty of ADCP measurements from a moving boat by standard-ized uncertainty analysis.” J. Hydraul. Eng., 133(12), 1390–1410.
ISO. (2005). “Measurement of fluid flow—Procedures for the evaluation ofuncertainties.” ISO 5168, Geneva.
ISO. (2007). “Hydrometry—Measurement of liquid flow in open channelsusing current meters or floats.” ISO 748, Geneva.
Joint Committee for Guides in Metrology (JCGM). (2008). “Evaluation ofmeasurement data—Guide to the expression of uncertainty in measure-ment (GUM 1995 with minor corrections).” Geneva, Switzerland.
Oberg, K. A., Morlock, S. E., and Caldwell, W. S. (2005). “Quality-assurance plan for discharge measurements using acoustic Dopplercurrent profilers.” Scientific Investigations Rep. 2005-5183, U.S.Geological Survey, Reston, VA.
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Discussion of “Influence of Entrapped AirPockets on Hydraulic Transients in WaterPipelines” by Ling Zhou, Deyou Liu,Bryan Karney, and Qinfen ZhangDecember 2011, Vol. 137, No. 12, pp. 1686–1692.DOI: 10.1061/(ASCE)HY.1943-7900.0000460
Samba Bousso1 and Musandji Fuamba, Ph.D.21Ph.D. Student, Dept. of Civil, Geological and Mining Engineering,
Polytechnique Montreal, C.P. 6079, succursale Centre-ville Montréal,QC H3C 3A7, Canada (corresponding author). E-mail: [email protected]
2Professor, Dept. of Civil, Geological and Mining Engineering,Polytechnique Montreal, C.P. 6079, succursale Centre-ville Montréal,QC H3C 3A7, Canada. E-mail: [email protected]
The authors propose a simplified model simulating the pressurevariations associated with a filling undulating pipeline containingan entrapped air pocket. Interested in the impact of the initial voidfraction on the observed maximum pressure, the authors comparetheir numerical results to experimental results. They simplify re-sults on the liquid inertia and energy loss of a short water columnnear the air-water interface. Results indicate that their model is ableto accurately predict peak pressures.
The discussers wish to share comments regarding the following:1. Assumption of the inertial force;2. Assumption of the inertial energy loss and impact;3. Instability due to changes on slope;4. Validity of the assumption of the vertical interface; and5. Attenuation of oscillations by air and energy dissipation.
Assumption of the Inertial Force
The authors assume that the inertia is negligible. The validity of thishypothesis is questionable in the presence of a vertical pipe becausethe inertial force becomes dominant over frictional forces. The dis-cussers suggest this hypothesis would be limited to a certain rangeof slope threshold and pressure data.
Assumption of the Inertial Energy Loss and Impact
To take into account the effect of slope change, the authorsconsidered the term Zc as continuity to determine the functionof the head of the interface. This function occurs only in the cal-culation of the air-water interface. Regarding local energy losses,the model neglects them in the tiny water column. However, theauthors fail to demonstrate how the energy losses produced by highslope changes are taken into account. The presence of bends withangles of up to 90° generates relatively high energy losses. Howdo the authors justify the noninclusion of these energy losses?The discussers suggest including them and considering the slopechanges as a boundary condition.
Instability due to Changes in Slope
In the presence of low head, there may be discontinuity in thezone of slope changes, particularly at bends of 90°. Eq. (6) fromthe original paper can be used to take slope change into accountwhen calculating flow conditions at the interface zone, but notthe pressurized column. In the case of low pressure head, numerical
instability may appear if a grid point coincides with the point ofchange of direction. The discussers wonder why the authors de-cided to omit the slope term in the momentum equation.
Validity of the Assumption of the Vertical Interface
The validity of the air-water interface in this type of model is ques-tionable when the reservoir pressure is low. This hypothesis shoulddepend on the pressure in the reservoir, the air pressure, and pos-sibly the pipe diameter and the opening time of the valve. Testingthe rapid filling of low piezometric head, Guizani et al. (2006)showed that the water-air interface tends to bow when the pressuredecreases. The discussers think the hypothesis of water-air interfaceperpendicular to the centerline of the pipe might not be valid at thepoint at which the pipe changes direction in vertical pipes or insteep pipe slopes with lower head. Especially during slow filling,the lower region of pipe will first fill up and trap air pockets in theupper parts.
Attenuation of Oscillations by Air and EnergyDissipation
In different configurations pointed out by the authors, there aresignificant changes in slope with deviations of up to 90°. Theseslope variations affect the energy dissipation by attenuating thepressure oscillations. However, it seems unclear whether the modeltakes into account the attenuation effect of the pressure. It would beinteresting to distinguish the effect of energy dissipation by the dif-ferent slope variations and the attenuation effect due to the initialfraction of air.
References
Guizani, M., Vasconcelos, J. G., Wright, S. J., and Maalel, K. (2006).“Investigation of rapid filling of empty pipes.” CHI Intelligent Modelingof Urban Water Systems, Monograph 14, W. James, K. N. Irvine,E. A. McBean, and R. E. Pitt, eds., Guelph, Ontario, Canada.
Closure to “Influence of Entrapped AirPockets on Hydraulic Transients in WaterPipelines” by Ling Zhou, Deyou Liu,Bryan Karney, and Qinfen ZhangDecember 2011, Vol. 137, No. 12, pp. 1686–1692.DOI: 10.1061/(ASCE)HY.1943-7900.0000460
Ling Zhou, Ph.D.1; Deyou Liu2; Bryan Karney, M.ASCE3;and Qinfen Zhang, Ph.D.41Lecturer, College of Water Conservancy and Hydropower Engineering,
Hohai Univ., 1 Xikang Rd., Nanjing, China 210098 (correspondingauthor). E-mail: [email protected]
2Professor, College of Water Conservancy and Hydropower Engineering,Hohai Univ., 1 Xikang Rd., Nanjing, China 210098. E-mail:[email protected]
3Professor, Dept. of Civil Engineering, Univ. of Toronto, 35 St. George St.,Toronto, ON, Canada M5S 1A4. E-mail: [email protected]
4R&D Associate, Oak Ridge National Laboratory, 1 Bethel Valley Rd.,P.O. Box 2008, Oak Ridge, TN 37831-6038. E-mail: [email protected]
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The writers would like to thank the discussers for their interestin the discussed paper. The discussion provides valuable sugges-tions and recommendations for further research on this topic.The aim of this closure is to clarify some comments raised in thediscussion.
Assumption of the Inertial Force
In the discussed paper, the authors assume that the liquid inertia andenergy loss of a tiny water column ΔLwi adjacent to the air-waterinterface can be ignored, and 0.5Δx ≤ ΔLwi < 1.5Δx (Δx isthe method of characteristics grid length). Namely, only the watercolumn with a length ΔLwi neglects inertia, and all other waterregions consider inertia. A very short Δx (Δx ¼ 0.001–0.01 m) isused in the proposed model for numerical stability. Thus, the watercolumn neglecting the inertia (0.0005 m ≤ ΔLwi < 0.015 m)accounts for quite a small portion of the whole water column(the experimental pipeline is 10.4 m in length). It can be foundin Fig. 6 of the discussed paper that the proposed model with0.5Δx ≤ ΔLwi < 1.5Δx can get the consistent air pressure oscil-lation pattern with Lee’s elastic model considering all the inertia(Lee 2005). Consequently, the proposed model has the samenumerical accuracy as the complete elastic water model, and theassumption regarding the inertial force of ΔLwi is not limited toa certain range of slope threshold and pressure data.
Assumption of the Inertial Energy Loss and Impact
As discussed previously, because the water columnΔLwi is such asmall portion of the water column, the effect of inertia and localenergy loss of ΔLwi on the pressure oscillation can be negligible.The proposed model only ignores the energy loss of ΔLwi, andconsiders all energy loses of the other water portions. Moreover,the energy losses produced by high slope changes have beenconsidered in the pipe friction f.
Instability due to Changes in Slope
The slope term is considered in the momentum equation in the dis-cussed paper. Actually, the air-water interface falls into disorderwhen the water column reaches the bend of pipeline. In order torealize the simulation, the one-dimensional (1D) model is devel-oped with the assumption of the vertical air-water interface. Based
on the assumption, the proposed model can result in numerical sta-bility, even at the pipe bend.
Validity of the Assumption of the Vertical Interface
The authors agree with the discussers’ opinion on this problem. Theauthors (Liu et al. 2011; Zhou et al. 2011) pointed out that thisassumption could cause calculation errors, especially for the secondand following peak pressures and cycles. The volume of fluid(VOF) model was introduced by Zhou et al. (2011) for the simu-lation of the water filling a pipeline with an entrapped air pocket.Results indicate the VOF model can accurately simulate the move-ment of an air pocket and pressure surge. Although the 1D modelfails to track the real air-water interface, it can effectively predictthe maximum pressure (relevant for most practical applications).In addition, the 1D model has the obvious advantages of lesscalculation, high velocity, and high efficiency. Therefore, the devel-opment of the 1D model is still beneficial for the practical pipelinesystems.
Attenuation of Oscillations by Air and EnergyDissipation
In the proposed model, the air pocket is treated as an adiabaticprocess. The polytropic coefficient has no effect on pressure attenu-ation because the compression/expansion at a fixed polytropiccoefficient is reversible and does not dissipate energy. Pipe frictionand turbulence production at the air-water interface are the mostlikely explanations for pressure attenuation. Due to the assumptionon the vertical interface, the proposed model just includes pipefriction and local energy loss, which results in slight energy dissi-pation (as shown in Fig. 7 of the discussed paper).
References
Lee, N. H. (2005). “Effect of pressurization and expulsion of entrapped airin pipelines.” Doctoral dissertation, Georgia Institute of Technology,Atlanta.
Liu, D. Y., Zhou, L., Karney, B., Zhang, Q. F., and Ou, C. Q. (2011).“Rigid-plug elastic-water model for transient pipe flow with entrappedair pocket.” J. Hydraul. Res., 49(6), 799–803.
Zhou, L., Liu, D. Y., and Ou, C. Q. (2011). “Simulation of flow transients ina water filling pipe containing entrapped air pocket with VOF model.”Eng. Appl. Comput. Fluid Mech., 5(1), 127–140.
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