DISCUSSIONS AND CLOSURES

Discussion of “Diagnostic Curve forEstimating Soil Dispersivity andInstantaneously Injected Mass” bySushil K. SinghMay/June 2006, Vol. 132, No. 3, pp. 281–283.

DOI: 10.1061/�ASCE�0733-9437�2006�132:3�281�

Rajesh Srivastava1

1Associate Professor, Dept. of Civil Engineering, Indian Institute of Tech-nology Kanpur, Kanpur 208016, India. E-mail: rajeshs@iitk.ac.in

The author has proposed a method for estimating the dispersivityof soil and the amount of injected mass from column break-through data. The discusser would like to mention the followingpoints regarding this technical note:1. The suitability of the proposed methodology to field applica-

tion is not brought out clearly in the note. For a columnexperiment, typically, the injected mass would be availableand only dispersivity needs to be estimated. For field appli-cation, the assumption of one-dimensional, constant velocityflow may not be valid in most situations.

2. The statement, “The dual coordinates of a selected point onthe matched portion of the graph yield the estimates of dis-persivity and injected mass” creates the false impression thatthe selected point has to be on the diagnostic curve. How-ever, it becomes clear that any point may be selected whenthe author mentions, just before Eq. 9, “If the selected pointis so chosen that �m=1 and f��m�=1.”

3. Although any consistent set of units would work, it is cus-tomary to mention the units, especially if an experimentaldata set is used. It would be useful for the reader if it werementioned that the variables were expressed in centimeter,minute, and milligram. Also, in the Notation section, the au-thor should mention the dimension of Dirac delta as T−1.

4. Although data from Wang et al. �1987� have been used in thisstudy, it has not been mentioned that they have proposedanother graphical method to estimate the dispersivity. Themethodology adopted by them is listed below:

ln�ct3/2� = ln� Mx

2Q���u� −

�x − ut�2

4�ut�1�

which, on differentiation, results in:

d ln�ct3/2�dt

= −u

4�+

x2

4�ut2 �2�

A plot of the slope of the ln�ct3/2� vs t relationship against2

1 / t would, therefore, directly provide the value of the dis-

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persivity, provided u is known. Wang et al. �1987�, however,assumed the porosity �and therefore, u� to be unknown andused the slope �S� and intercept �I� of the d ln�ct3/2� /dt vs1 / t2 curve to compute both u and � as

� =x

4�− SIu = x�−

I

S�3�

Using this technique and computing the derivative ofln�ct3/2� by central difference, Wang et al. �1987� obtained adispersivity value of 0.57 cm and porosity of 0.372 for dataset 1 of the note. The two dispersivity values obtained by theauthor are close to this value, but it is not clear why a poros-ity value of 0.33 was used by the author.

5. Once the porosity and dispersivity are estimated using Eq.�3�, the injected mass could be estimated by plotting Eq. �1�.

However, if porosity is assumed to be known �as in the note�,

Fig. 1. Variation of ln�ct1.5� with X2 for Set 1. � X�0, �: X�0.

Fig. 2. Variation of ct1.5 with �X� for Set 1 for different porosityvalues. Squares: 0.33; circles: 0.334; --------: 0.34.

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one could directly plot ln�ct3/2� vs X2 and obtain � from itsslope and M from the intercept. There is no need to use thediagnostic plot proposed by the author. For set 1, using aporosity of 0.33, the plot is shown in Fig. 1 and the values of� and M were obtained as 0.49 cm and 470 mg, respectively,for X�0; and 0.80 cm and 638 mg, for X�0. �The first andthe last values in Table 1 of Wang et al. �1987� have not beenconsidered, as they deviate from the trend shown by otherpoints.� These values are quite close to those obtained by theauthor and the procedure is much simpler than the use of thediagnostic curve. A note of caution must be added here thatobtaining two significantly different values of the injectedmass clearly indicates that there is a flaw in the methodologybecause it contradicts the experimental conditions �we knowthat the injected mass is the same for a given data set!�.

6. The biggest concern the discusser has about this note �and aprevious note, see Srivastava 2006� is the hysteresis in thedispersion coefficient. One should not presume hysteresis inthe dispersion coefficient just because the data for X�0 andX�0 result in different values of D. Fig. 2 shows the diag-nostic plot of the data for three different values of porosity.For the value assumed by the author �0.33� the rising andfalling limbs of the breakthrough curve are apparently differ-ent leading to the hypothesis of hysteresis. However, a smallchange in the porosity �0.334� causes the data to lie on asingle curve. The behavior for a porosity of 0.34 indicatesthe extreme sensitivity of the diagnostic curve to the value ofporosity. The author may wish to address this issue beforethe diagnostic curve can be used in field applications. A plotof Eq. �1�, shown in Fig. 3, depicts a similar sensitivity to thevalue of porosity and, therefore, seepage velocity, u. Themethod employed by Wang et al. �1987�, using Eq. �3�,avoids this sensitivity and appears to be robust as it estimatesthe porosity also.

In conclusion, the discusser believes that the method proposedby Wang et al. �1987� for estimation of dispersivity using Eq. �3�is likely to be better than the diagnostic curve proposed by theauthor, as it estimates both the dispersivity and the seepage ve-locity �if the injected mass is also to be estimated, a separate

Fig. 3. Variation of ln�ct1.5� with X2 for Set 1 for different porosityvalues: circles: 0.334; -----: 0.34

graph may be plotted using Eq. �1��. Also, the discusser believes

112 / JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING © ASCE

that the hysteresis in dispersion coefficient, as envisaged in thenote, should be critically examined.

References

Srivastava, R. �2006�. “Discussion of ‘Master diagnostic curve for dis-persion coefficient of soils.’” J. Environ. Eng., 132�9�, 1082–1083.

Wang, H. Q., Crampon, N., Huberson, S., and Garnier, J. M. �1987�. “Alinear graphical method for determining hydrodispersive characteris-tics in tracer experiments with instantaneous injection.” J. Hydrol.,95�1/2�, 143–154.

Closure to “Diagnostic Curve forEstimating Soil Dispersivity andInstantaneously Injected Mass” bySushil K. SinghMay/June 2006, Vol. 132, No. 3, pp. 281–283.DOI: 10.1061/�ASCE�0733-9437�2006�132:3�281�

Sushil K. Singh1

1Scientist, National Institute of Hydrology, Roorkee-247 667, Uttarak-hand, India. E-mail: sukusi1@yahoo.com

The writer thanks the discusser for his interest in the technicalnote and raising certain points. The following are the clarifica-tions.1. The testing of the writer’s diagnostic curve method using the

laboratory data was carried out with the objective of demon-strating its possible suitability to field applications in assess-ing the pollutant load from point sources. It is true that theassumptions considered in devising the method may not ap-plicable in the case of certain field situations. However, theproposed method can be applicable where the field situationsacceptably fulfill the assumptions.

2. The discusser seemed to follow what the writer wanted toconvey; there was no apparent misconstruction. However, toavoid any confusion, the “matched portion” could read“overlapped portion.”

3. The missing units in the Notation section have been added asErrata at the end of this closure.

4. The reference of Wang et al. �1987� was mentioned in theoriginal technical note. The Péclet number being large, theseepage velocity can be easily estimated by dividing thesample length by time to the peak of the breakthrough curve�Singh 2002b�. Thus the value of the porosity obtained is0.33.

5. The writer agrees to the discusser’s view that the injectedmass can be estimated using Eq. 3 of the original technicalnote, provided the porosity and dispersivity are known. ForWang et al.’s �1987� estimates of the dispersivity and poros-ity �0.57 cm and 0.372, respectively�, the estimates of theinjected mass for different values of time using the writer’sEq. 3 vary from 17943 g to 148 g, with an average of3583 g. This huge variation shows that Wang et al.’s esti-mates of the dispersivity and injected mass are not correct.The simulated concentrations using Eq. 3 of the originaltechnical note �for dispersivity�0.57, porosity�0.372, and

injected mass�3583 g� are compared to the observed con-

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centrations in Table 1 of this closure. The average estimatesof the dispersivity, porosity, and injected mass obtained are0.651 cm, 0.33, and 560 g respectively, using the writer’smethod. It is observed from Table 1, that the parameters es-timated using the writer’s diagnostic curve, are reliable. Ifthe parameters estimated considering the hysteresis are usedseparately for X�0 and X�0, a more accurate reproductionof the observed concentrations is obtained.

The utility of the writer’s diagnostic curve method hasbeen clarified above. It has all the advantages of the diagnos-tic curve proposed by Singh �2005� and is applicable for anyvalue of the Péclet number. The additional advantages of thewriter’s diagnostic curve method are: �a� It can identify andquantify the hysteresis in the dispersivity, because the diag-nostic curve is fully utilized for X�0 and for X�0. Thisjustifies the separate analysis of the concentration data forX�0 and X�0, thus making the method suitable for ad-vanced study on the subject. �b� The diagnostic curve doesnot change with the different data sets to be analyzed; in theother method, a new straight line needs to be drawn witheach new data set. �c� It does not require smoothing of thebreakthrough curve and computation of numerical derivative.Only two values from the matched �overlapped� graphs areneeded for simultaneously estimating the dispersivity and in-jected mass, and the estimation equations are simple. �d� Thediagnostic curve serves as a guide, which, when comparedwith the diagnostically plotted points, gives a simultaneousvisualization of errors, enabling the data with large errors tobe identified easily by the considerable departure of the di-agnostically plotted points from the diagnostic curve.

Further analysis of data by the discusser is based on theuse of Eqs. 3 and 5 of the original technical note. The dis-cusser has used the concept of hysteresis identification pro-

Table 1. Comparison of Observed and Simulated Concentrations

TimeObserved

conc

Wang et al. �1987�method

Diagnostic curvemethod

Simulatedconc

Percenterror

Simulatedconc

Percenterror

18.00 1.85E-01 3.69E-02 80.0 2.64E-01 42.4

18.33 2.56E-01 7.53E-02 −70.6 3.49E-01 36.4

18.67 3.45E-01 1.46E-01 −57.7 4.41E-01 27.7

19.00 4.39E-01 2.59E-01 −41.0 5.24E-01 19.4

19.33 5.32E-01 4.31E-01 −19.0 5.93E-01 11.5

19.67 6.20E-01 6.85E-01 10.5 6.42E-01 3.6

20.00 6.67E-01 1.013 51.9 6.63E-01 −0.5

20.33 6.68E-01 1.420 112.5 6.57E-01 −1.7

20.67 6.43E-01 1.904 112.5 6.24E-01 −3.0

21.00 6.05E-01 2.408 298.0 5.71E-01 −5.7

21.33 5.53E-01 2.907 425.6 5.03E-01 −9.0

21.67 4.80E-01 3.368 601.6 4.27E-01 −11.1

22.00 3.98E-01 3.720 834.8 3.51E-01 −11.7

22.33 3.15E-01 3.946 1152.8 2.81E-01 −10.9

22.67 2.52E-01 4.026 1497.6 2.15E-01 −14.5

23.00 2.17E-01 3.953 1721.6 1.62E-01 −25.3

24.00 1.25E-01 3.027 2321.6 5.79E-02 −53.7

posed in the original technical note for separately analyzing

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the data for X�0 and X�0 with the writer’s estimate of theporosity. The writer’s method is simpler and more justifiedthan the straight-line procedure employed by the discusser. Inaddition to its advantages over a straight-line procedure dis-cussed above, �a� It requires values of ct1.5 and X to be plot-ted, while logarithmic values of ct1.5 and X2 are needed in thestraight-line procedure. A few data points can yield reliableestimates of the parameters, while larger number of points isrequired for drawing a reliable straight line. �b� The outliersin the data can easily and justifiably be identified; and theseoutliers are automatically not considered during matching. Inthe straight-line procedure, identifying and deleting an out-lier may be unjustified in many cases, however, it is better toremove the outliers successively �Singh 2002a�, rather thansimultaneously and this requires repetitive drawing of astraight line for the same data set. However, the straight-lineprocedure can be used to obtain the approximate values ofthe parameters.

There is no flaw in the methodology because it is men-tioned in Eqs. 11 and 12 of the original technical note thatthe injected masses considered for X�0 and X�0 are dif-ferent, which explains the hysteresis in the dispersivity. Theinjected mass considered for X�0 is less than that for X�0. The actual injected mass is given by the sum of thesetwo masses �Eq. 13 of the original technical note.�

6. The discusser’s concern about the hysteresis has been ad-dressed above. The hysteresis in dispersivity has been ob-served and discussed for the several data sets in the originaltechnical note and Singh �2005, 2006a�. The reason to avoidpresumption of hysteresis is not clear. The concept of hyster-esis has been dealt with in more detail by Singh �2005�. Theutility of the diagnostic curve is not affected by the discuss-er’s exercise with different values of the porosity. The con-cept of the hysteresis is justified in the case of dispersivitybecause the dispersivity cannot be physically measured, onlyestimated. Since porosity can be physically measured, it doesnot seem logical to take a different value of the porosity or ahysteresis in the porosity. The writer’s Eq. 3 is applicableonly for the correct value of the porosity �=0.33 for the ex-ample discussed�. However, it seems that the discusser did

1.5

Fig. 1. Diagnostically plotted points �assumed porosity�0.334�

not change the values of ct , but changed X for different

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values of the porosity, which show a nonideal condition. Itseems that the porosity value of 0.334 has been adopted bythe discusser from Singh �2006b�. This value of the porositywas estimated considering no hysteresis in the dispersivity.Singh �2006b� method can be used for a normal study, whilethe writer’s diagnostic curve method is also suited to an ad-vanced study on the subject. The diagnostic points are plot-ted in Fig. 1 for this value of porosity �0.334�, which showsthat the diagnostic curve cannot be matched to all data withthe same accuracy as done in the original technical note. Iftwo separate matching of the plotted points for X�0 and X�0 are considered, a larger scatter than that observed in theoriginal technical note is noticed.

Diagnostic curves have also been proposed in other fields �seeSingh 2007a; 2007b; 2007c�. The proposed diagnostic curvemethod outperforms the Wang et al. �1987� method. The hyster-esis in the dispersivity as substantiated in the original technicalnote is valid and may form a basis for an advanced study on thesubject.

Errata

The following are the errata for the original technical note.

1. The following units have been used in the original technical note andthis closure.

�M��milligram;

�L��centimeter; and

�T��minute.

2. Page 282:

The titles of Fig. 2 and 3 should be interchanged.

3. Page 283, Table 1, Set 2, second column:

The “Positive” and “Negative” should be interchanged.

References

Singh, S. K. �2002a�. “Well loss estimation: Variable pumping replacingstep drawdown test.” J. Hydraul. Eng., 128�3�, 343–348.

Singh, S. K. �2002b�. “Estimating dispersion coefficient and porosityfrom soil-column tests.” J. Environ. Eng., 128�11�, 1095–1099.

Singh, S. K. �2005�. “Master diagnostic curve for dispersion coefficientof soils.” J. Environ. Eng., 131�6�, 988–993.

Singh, S. K. �2006a�. “Closure to ‘Master diagnostic curve for dispersioncoefficient of soils’ by Sushil K. Singh.” J. Environ. Eng., 132�9�,1083–1084.

Singh, S. K. �2006b�. “Estimating dispersivity and injected mass frombreakthrough curve due to instantaneous source.” J. Hydrol., 329�3–4�, 685–691.

Singh, S. K. �2007a�. “Diagnostic curve method for consolidation coeffi-cient.” Int. J. Geomech., 7�1�, 75–79.

Singh, S. K. �2007b�. “Diagnostic curve for confined aquifer parametersfrom early drawdowns.” J. Irrig. Drain. Eng., in press.

Singh, S. K. �2007c�. “New methods for aquifer parameters from slug testdata.” J. Irrig. Drain. Eng., 133�3�, 272–275.

Wang, H. Q., Crampon, N., Huberson, S., and Garnier, J. M. �1987�. “Alinear graphical method for determining hydrodispersive characteris-tics in tracer experiments with instantaneous injection.” J. Hydrol.,

95�1–2�, 143–154.

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Discussion of “Flow Patterns and EnergyDissipation over Various Stepped Chutes”by Chaiyuth Chinnarasri andSomchai WongwisesJunuary/February 2006, Vol. 132, No. 1, pp.70–76.DOI: 10.1061/�ASCE�0733-9437�2006�132:1�70�

Masayuki Takahashi, A.M.ASCE1;Youichi Yasuda, M.ASCE2; and Iwao Ohtsu, M.ASCE3

1Assistant Professor, Dept. of Civil Engineering, Nihon Univ., College ofScience and Technology, Kanda Surugadai 1-8-14, Chiyoda-ku,Tokyo, 101-8308, Japan. E-mail: masayuki@civil.cst.nihon-u.ac.jp

2Professor, Dept. of Civil Engineering, Nihon Univ., College of Scienceand Technology, Kanda Surugadai 1-8-14, Chiyoda-ku, Tokyo,101-8308, Japan.

3Professor, Dept. of Civil Engineering, Nihon Univ., College of Scienceand Technology, Kanda Surugadai 1-8-14, Chiyoda-ku, Tokyo,101-8308, Japan.

The discussers have systematically investigated the flow cha-racteristics of stepped channels with horizontal steps. Also, ahydraulic design chart for stepped channels has been presented�Ohtsu et al. 2004�.

In this discussion, the flow conditions of stepped channelswith end sills are shown for �=30°. The relative energy loss ofstepped channels with horizontal steps is discussed in skimming,transition, and nappe flow regimes. Also, the relative energy lossof stepped channels with end sills is compared to the relative lossfor horizontal steps.

Flow Conditions of Stepped Channelswith End Sills

Generally, the flow conditions have been classified as: �1� skim-ming flow with corner eddies formed at each step; �2� nappe flowwith an air pocket at each step; and �3� transition flow, in which,a nappe with an air pocket is not always formed and corner eddiesare partly formed at some steps. The existence of the transitionflow was first reported by Ohtsu and Yasuda �1997�.

For stepped channels with end sills shown in Fig. 1, flowconditions of the nappe flow are divided into three types as shownin Fig. 2:1. Type 1 nappe flow:

For large discharge �for small relative step height h /dc�, thenappe flow without surface roller is formed �Fig. 2�a��.

2. Type 3 nappe flow:For small discharge �for large h /dc�, the nappe flow withsurface roller is formed �Fig. 2�c��.

3. Type 2 nappe flow:Between Type 1 nappe and Type 3 nappe flows, there existsa transition flow, in which the nappe flow with the surfaceroller is formed at some steps; whereas in the other, thenappe flow without surface roller is formed.

In Type 2 nappe flow, there exist both steady and unsteadyflows. For unsteady flows, the nappe flows with and without sur-face roller appear on a step alternating with time �Fig. 2�b-1� and2�b-2��. It was found that, in the upper step pool, the deflection ofthe nappe changes periodically. This unsteady flow condition maycorrespond to the wave phenomenon reported by Redeker �2000�

and Chanson �2001�.

/ JANUARY/FEBRUARY 2008

Fig. 3 shows the hydraulic condition required for the forma-tion of each flow type of nappe flow. For design purposes, Fig. 3can be used to avoid the formation of unsteady flow.

Energy Loss of Stepped Channel Flows

Generally, the following functional relationship has been estab-lished regarding the relative energy loss of stepped channels withhorizontal steps �Yasuda et al. 2001; Boes and Hager 2003; Ohtsuet al. 2004; Takahashi et al. 2004�

EL

E0= F�Ht

dc,

h

dc,�� �1�

where Ht /dc= the relative total drop height of stepped channel,h /dc= the relative step height.

Fig. 1. Definition sketch

Fig. 2. Flow conditions on stepped channel with end sills for �=30Type 3 nappe flow

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The discussers have confirmed that relative energy loss, EL /E0

is independent of Reynolds number R �R=q /�; q= discharge perunit width, �= kinematic viscosity of water� for R�1.6104.

To understand the effect of relative step height h /dc on relativeenergy loss E /E0 from Eq. �1�, it is necessary to analyze the dataunder given relative drop height Ht /dc and angel of stepped chan-nel �. Figs. 7 and 8 in the authors’ paper are not able to show theeffect of dc /h on EL /E0 under given values of Ht /dc and �.

According to the discussers’ study, the values of EL /E0 areindependent of h /dc for h /dc�0.5 �Ohtsu et al. 2004; Takahashiet al. 2004�. Therefore, Eq. �1� can be expressed as follows�Ohtsu et al. 2004; Takahashi et al. 2004�:

EL

E0= F�Ht

dc,�� for h/dc � 0.5 �2�

Fig. 4 is obtained by arranging the experimental data in accor-dance with Eq. �2�. Fig. 4 shows the relative energy loss of skim-ming, transition, and nappe flows in stepped channels with hori-zontal steps. Herein, the residual energy head at the end of thestepped channel was obtained using a non-intrusive method�Ohtsu et al. 2000; Ohtsu et al. 2004�. In Fig. 4, He is the dropheight required to form quasi-uniform flow. In a quasi-uniformflow region, flow conditions remain constant as are the residualenergy and the flow velocity. As shown in Fig. 4, the discussers’data coincide with other researchers’ data �Essery and Horner1978; Pinherio and Fael 2000; Chamani and Rajaratnam 1999�except for the authors’ data ��=45° � and data of Chinnarasri andWongwises �2004� ��=60° �. For h /dc�0.5, the values of EL /E0

under given � and Ht /dc are almost equal to those for skimming,transition, and nappe flows.

Ohtsu et al. �2004� proposed empirical equations for the rela-tive energy loss of stepped channels for a wide range of �, h /dc,and Ht /dc. EL /E0 of skimming flow for h /dc�0.5 is obtained byusing the empirical equations, and EL /E0 of transition and nappeflows can be predicted for both nonuniform and quasi-uniform

ype 1 nappe flow; �b� unsteady flow in Type 2 nappe flow; and �c�

°: �a� T

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116 / JOURNAL OF IRRIGATION AND DRAINAGE ENGINEERING © ASCE

flows. These prediction curves shown in Fig. 4 coincide withexperimental data.

Fig. 5 shows the comparison of EL /E0 between horizontalsteps and steps with end sills for �=30°. As shown in Fig. 5, therelative energy loss of stepped channels with end sills is largerthan that with horizontal steps.

References

Boes, R. M., and Hager, W. H. �2003�. “Hydraulic design of steppedspillways.” J. Hydraul. Eng., 129�9�, 671–679.

Chamani, M. R., and Rajaratnam, N. �1999�. “Characteristics ofskimming flow over stepped spillways.” J. Hydraul. Eng., 125�4�,361–368.

Chanson, H. �2001�. The hydraulics of stepped chutes and spillways,

Balkema, Lisse, The Netherlands.Chinnarasri, C., and Wongwises, S. �2004�. “Flow regimes and energy

loss on chutes with upward inclined steps.” Can. J. Civ. Eng., 31�5�,870–879.

Essery, I. T. S., and Horner, M. W. �1978� “The hydraulic design ofstepped spillways.” CIRIA Report No. 33, 2nd Ed., Construction In-dustry Research and Information Association, London.

Ohtsu, I., and Yasuda, Y. �1997�. “Characteristics of flow conditions

on stepped channels.” Proc., 27th IAHR Congress, IAHR, Madrid,

583–588.Ohtsu, I., Yasuda, Y., and Takahashi, M. �2000�. “Discussion of ‘Hydrau-

lics of skimming flow on modeled stepped spillways.” J. Hydraul.Eng., 126�12�, 950–951.

Ohtsu, I., Yasuda, Y., and Takahashi, M. �2004�. “Flow characteristicsof skimming flows in stepped channels.” J. Hydraul. Eng., 130�9�,860–869.

Pinherio, A. N., and Fael, C. S. �2000� “Nappe flow in steppedchannels—Occurrence and energy dissipation.” Proc., Int. Workshop

on Hydraulics of Stepped Spillways, H. E. Minor and W. H. Hager,eds., Balkema Rotterdam, The Netherlands, 119–126.

Redeker, M. �2000�. “Wave phenomena in the cascade type spillway ofthe Sorpe dam.” Interim Report, Ruhrverband, Dept. of Dams andWater Courses, Essen.

Takahashi, M., Yasuda, Y., and Ohtsu, I. �2004� “Energy loss of super-critical flows in stepped channel flows.” Annual J. Hydraul. Eng.,Japan Society of Civil Engineers, 48, 871–876 �in Japanese�.

Yasuda, Y., Takahashi, M., and Ohtsu, I. �2001�. “Energy dissipation ofskimming flows on stepped-channel chutes.” Proc., 29th IAHR

Fig. 5. Comparison of EL /E0 between horizontal steps and steps withend sills

Fig. 3. Classification of flow conditions in stepped channel withend sills, �Sk� skimming flow; �Tr� transition flow; ��� boundarybetween Type 1 and Type 2; ��� boundary between Type 2 steadyand unsteady flows; ��� boundary between Type 2 unsteady andType 3 flows; ��� boundary between Type 2 steady and Type 3flows; �—� Eq. �3� in the authors’ paper; and �----� Eq. �4� in theauthors’ paper

Fig. 4. Relative energy loss in stepped channel with horizontal steps:�a� 19° �23°, �b� 30° �60°, and �c� �=45°

Congress, IAHR, Madrid, 531–536.

/ JANUARY/FEBRUARY 2008

Closure to “Flow Patterns and EnergyDissipation over Various Stepped Chutes”by Chaiyuth Chinnarasri andSomchai WongwisesFebruary 2006, Vol. 132, No. 1, pp. 70–76.DOI: 10.1061/�ASCE�0733-9437�2006�132:1�70�

Chaiyuth Chinnarasri1 and Somchai Wongwises2

1Associate Professor, Water Resources Engineering Research Lab.�WAREE�, Dept. of Civil Engineering, King Mongkut’s University ofTechnology Thonburi, Bangmod, Bangkok 10140, Thailand. E-mail:chaiyuth.chi@kmutt.ac.th

2Professor, Fluid Mechanics, Thermal Engineering and Multiphase FlowResearch Lab. �FUTURE�, Dept. of Mechanical Engineering, KingMongkut’s University of Technology Thonburi, Bangmod, Bangkok10140, Thailand.

We are indebted to the valuable discussion. The enrichment of thestudy with the additional references is very useful. It also demon-strates the importance of the research on stepped channels. All ofthe discussers’ comments have been divided into two main groupsand addressed.

Flow Conditions of Stepped Channels with End Sills

Based on the authors’ experimental data, the flow conditions onvarious stepped chutes can be classified as: �1� skimming flowwith corner eddies formed at each step; �2� nappe flow with an airpocket at each step; and �3� transition flow, in which a nappe withan air pocket is not always formed and corner eddies are partlyformed at some steps.

In the nappe flow on the stepped channels with end sills, therecirculation flow could be observed either at the corner of theend sills, or in the middle portion of the steps. It depends on thedirection of the successive jet from the previous steps and thechute slopes. At the brink of the step, a hydraulic jump could beobserved when the height of the end sill is small. However, awater surface roller is formed instead, when the height of end sillsis large. The experimental results of the discussers on the hydrau-lic condition required for the formation of each flow type ofnappe flow strongly supported the equation proposed by the au-thors for maximum critical flow depth for nappe flow regime onvarious stepped chutes.

Energy Loss of Stepped Channel Flows

The relative energy loss of flow over various stepped chutes�EL /Eo� can be expressed herein again as,

EL

Eo= f�dc

h,h

l,m

h,N�

where dc /h is the relative critical flow depth, h / l is the chute

slope �tan �=h / l�, m /h is the relative step height, and N is the

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number of steps in the chute. For a horizontal step, the term m /hbecomes zero and can be neglected. The number of steps can beincluded in the term for relative critical flow depth and becomesNh /dc or Ht /dc, where Ht is the total drop height. The trend of therelationship between the energy loss ratio �EL /Ht� and the relativedrop height �Ht /dc� was first demonstrated by Stephenson �1991�for a small scale stepped channel.

The discussers have tried to compare the relative energy lossof flows through horizontal stepped channels based on a differententrance to the stepped channels and the measurements of theresidual flow energy obtained from the authors, discussers, andChamani and Rajaratnam �1999�. In the discussers’ experiments,the entrance to the stepped channel was the WES-type crest andthe residual energy was computed from the measurement of waterdepths in non-aerated flow regions. The maximum channel slopewas 55°. In the experiments of Chamani and Rajaratnam �1999�,the entrance to the stepped channel was ogee crest type and theresidual energy was computed from the mean velocity of the air-water mixture at the section near the base of the stepped channel.The channel slopes were 51.3° and 59°, and the relative energyloss was in the range of 48-63%. In the authors’ experiments, thestep sizes were uniform throughout the channel and the residualenergy was computed from the measurement of water depths atabout three to four times the step length away from the loweststep face. The maximum channel slope was 60°.

As for the effect of channel slope on the relative energy loss,for a specified relative drop height �Ht /dc�, the experimental re-sults of Ohtsu et al. �2004� showed that the relative energy lossdecreased with increasing channel slope. Therefore, to comparethe relative energy loss among others, the channel slope must bethe same. Fig. 4�b� of the discussers focuses on the case of thesteep stepped channels. However, it is found that the relative en-ergy losses estimated by authors were not much different from theresults obtained by Chamani and Rajaratnam �1999�.

It must be emphasized that the possible reasons for these dif-ferences include the type of entrance to the stepped channels, themeasurements of the residual flow energy, the slope of steppedchannels, the effects of viscosity, surface tension, and air entrain-ment ratio.

Based on the discusser’s experimental data on the effect of endsills on the relative energy loss, it is found that their results con-firm the authors’ results that the steps with end sills form the mostefficient configuration in terms of relative energy dissipation,compared to horizontal steps and inclined stepped channels.

References

Chamani, M. R., and Rajaratnam, N. �1999�. “Characteristics of skim-ming flow over stepped spillways.” J. Hydraul. Eng., 125�4�, 361–368.

Ohtsu, I., Yasuda, Y., and Takahashi, M. �2004�. “Flow characteristics ofskimming flows in stepped channels.” J. Hydraul. Eng., 130�9�, 860–869.

Stephenson, D. �1991�. “Energy dissipation down stepped spillways.” Int.Water Power Dam Constr..

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