Velocity Profile and Flow Resistance Models for DevelopingChute FlowOscar Castro-Orgaz1
Abstract: A developing boundary layer starts at the spillway crest until it reaches the free surface at the so-called inception point, wherethe natural air entrainment is initiated. A detailed reanalysis of the turbulent velocity profiles on steep chutes is made herein, includingmean values for the parameters of the complete turbulent velocity profile in the turbulent rough flow regime, given by the log-wake law.Accounting both for the laws of the wall and the wake, a new rational approach is proposed for a power-law velocity profile within theboundary layer of turbulent rough chute flow. This novel approach directly includes the power-law parameters and does not require for aprofile matching, as is currently required. The results obtained for the turbulent velocity profiles were applied to analytically determine theresistance characteristics for chute flows. The results apply to the developing flow zone upstream of air inception in chute spillways.
DOI: 10.1061/�ASCE�HY.1943-7900.0000190
CE Database subject headings: Air inception; Boundary layers; Flow resistance; Open channel flow; Velocity.
Author keywords: Air inception; Boundary layer; Chute flow; Flow resistance; Open channel flow; Turbulent velocity profiles.
Introduction
Large floods are usually released from a water reservoir up to theenergy dissipator using man-made concrete channels, referred toas chutes. If the chute flow is discharged from a spillway �Fig. 1�,a developing boundary layer is generated at the spillway crest�Chanson 1997�, and grows in thickness until it reaches the freesurface at the so-called inception point, where the natural �freesurface� air entrainment starts �Hager and Blaser 1998�. Thesevelocity profiles are currently assumed to follow the classic loga-rithmic law of the wall �Halbronn �1952a,b,c� 1955; Chanson2004�. However, as experimentally observed by Bauer �1951,1954�, this approximation is inadequate within a developingboundary layer. It appears that matching this type of velocityprofile requires an additional term to be added to the law of thewall, taking into account wake strength. This additional term in-volves the “law of the wake” �Krogstad et al. 1992; Montes1998�. Both Bauer �1951� and Cain and Wood �1981� noted thattheir experimental velocity profiles were better described bypower-law equations, but no definitive results were proposed todetermine the power-law parameters. Bauer �1951� reported amean power-law exponent of 1/4.5, whereas the exponent for theCain and Wood data are 1/6.3. Chen �1991� approximated thepower-law velocity profile assuming that the real velocity profileis well-described by the law of the wall, with no departures due toa wake term. He determined the power-law velocity profile pa-
1Research Engineer, Dept. of Agronomy, Univ. of Cordoba,c/Fernando Colón No. 1, 3 izq., E-14002, Cordoba, Spain. E-mail:[email protected]
Note. This manuscript was submitted on September 10, 2008; ap-proved on December 2, 2009; published online on December 4, 2009.Discussion period open until December 1, 2010; separate discussionsmust be submitted for individual papers. This technical note is part of theJournal of Hydraulic Engineering, Vol. 136, No. 7, July 1, 2010.
rameters by satisfying the law of the wall using a least-squares fit.The test data of both Bauer �1951, 1954� and Cain and Wood�1981� for developing boundary layers were considered correctfrom an engineering standpoint, and current inception point com-putations are performed in terms of the power-law velocity profile�Cain and Wood 1981; Montes 1998; Chanson 2004�.
The objective of this research is to reanalyze the available dataon turbulent velocity profiles in steep chute flow, to determinetheir general law by taking into account both the laws of the walland the wake. Once this velocity profile is defined, an equivalentpower-law velocity approximation is proposed, with generalizedcoefficients determined by a rational approach. Finally, the chuteflow resistance law is analyzed based on the previous results.
Velocity Defect Profiles
The complete time-averaged velocity profile of a turbulent bound-ary layer is composed of a logarithmic term and a wake term�White 1991�, which may be expressed for the turbulent roughregime �Dean 1976; Montes 1998�
where y=distance normal to spillway face measured from the topplane of the roughness elements plus the distance to the virtualorigin of the law of the wall; ks=equivalent roughness height; u=velocity parallel to spillway face at distance y; u�= ��o /��1/2
=shear velocity; �o=boundary shear stress; �=water density; �=von Karman constant; B=constant of integration for the law ofthe wall; �=boundary layer thickness; and �=wake strength pa-rameter. Eq. �1� was originally proposed to describe an equilib-rium boundary layer, with restrictions on how the freestreamvelocity U may vary in the streamwise direction �White 1991�.The first two terms of Eq. �1� are referred to as the law of thewall, whereas the terms involving � correspond to the law of thewake. Eq. �1� accounts for a wake term resulting from a best fit totest data over the entire boundary layer thickness in the turbulentrough regime �Dean 1976; Montes 1998�. The simple polynomialfor the wake effect in Eq. �1� satisfies the boundary conditions forthe velocity profile, including zero slope at the outer edge of theboundary layer, a condition not considered in the classic “cosine”law of the wake by Coles �White 1991�.
Eq. �1� includes the “free” parameters �, B, and � which aredetermined based on test data. The data of Bauer �1951� for flowover rough chutes of angles �=20°, 40°, and 60°, and those ofBormann �1968� for �=33.5° and several roughness patterns willbe used. Both Bauer �1951� and Bormann �1968� determined theskin friction coefficient Cf with the integral streamwise momen-tum equation for the turbulent boundary layer, given by von Kar-man as �Bauer 1954; White 1991�
1
2Cf =
d�m
dx+ �1 +
1
2S� �m
U2
dU2
dx�2�
with Cf =�o / ��U2 /2�=skin friction coefficient; U=free stream ve-locity; �m=boundary layer momentum thickness; x=streamwisedistance; and S=shape factor of boundary layer. All these termson the right-hand side of Eq. �2� were determined by Bauer�1951� and Bormann �1968� using test data, thereby avoiding theprofile matching method as proposed by Krogstad et al. �1992� todetermine Cf. The velocity defect law is obtained from Eq. �1� as
U − u
u�= −
1
�ln� y
�� +
1
��2� − �1 + 6��� y
��2
+ �1 + 4��� y
��3�
�3�
allowing a determination of � and � by comparison with theexperimental data. For simplicity, the standard value of �=0.41was assumed, in agreement with Tominaga and Nezu �1992�. Thetest data of Bormann �1968� for the velocity defect profiles areplotted in Fig. 2, resulting in a mean value of �=0.2. A feature ofchute spillway flow is a small pressure gradient dh /dx→0 �Mon-tes 1998�, related to the acceleration dU /dx. Thus, the pressuredistribution normal to the spillway face is assumed to follow thehydrostatic law. If this is accepted and the log-wake law is appli-cable, then it would be expected ��0.5–0.6, as in a classiczero-pressure gradient boundary layer flow. However, in develop-ing chute flow the term dU /dx is small, but it cannot be neglectedneither in the momentum balance nor in the energy equation,indicating that gravity flow in a chute spillway is different fromthe zero-pressure gradient boundary layer flow. If Fig. 2 is exam-ined substantial scatter is found in the outer region. It appears toindicate that different � values should be used to characterize the
velocity profile at different stations or for different discharges.
Thus, as expected in an equilibrium boundary layer �=��dU /dx�. However, to the lowest order, the log-wake law witha constant � value is reasonable approximation for practical pur-poses. The fit also corroborated that the mean value �=0.41adopted herein reasonably agrees with the trend of the test data.
The test data of Bauer �1951� are considered in Fig. 3. Thevelocity defect profile is plotted for �=0.41, but �=0.4 wasfound to improve the data trend. Based on Fig. 3, the chute slopedoes not significantly affect �. However, substantial scatter isfound in the outer region too. The velocity defect data of Bauer�1951� and Bormann �1968� �Figs. 2 and 3� involve differences in�, however. The complete mechanism by which � is influencedby the wall conditions is not yet fully understood. Recent researchindicates that the wake characteristics are affected by free streamturbulence, the roughness pattern �Krogstad et al. 1992; Bal-achandar et al. 2002� and the chute surface geometry �Krogstadand Antonia 1999; Tachie et al. 2000�. Based on test data,Krogstad and Antonia �1999� concluded that the surface rough-ness affects the turbulence characteristics significantly, even ifroughness patterns produced the same effect on the velocity pro-file. Bauer �1951� used wire mesh screen fixed to the chute bot-tom, whereas Bormann �1968� simulated roughness with sandgrains. The main difference between these two studies is in theroughness characteristics. The wake effect appears to be larger forroughness simulated with wire mesh than with sand, suggestingcaution if up-scaling chute model data to prototype scale. Tachieet al. �2000� reported a similar trend between wire mesh and sand
Fig. 2. Velocity defect profiles from �—� Eq. �3� with �=0.2,Bormann �1968� test data for different discharges �� � � � � ���=33.5°
Fig. 3. Velocity defect profiles from �—� Eq. �3� with �=0.4, Bauer�1951� test data ��� �=20°, ��� �=40° and ��� �=60°
.136:447-452.
Dow
nloa
ded
from
asc
elib
rary
.org
by
V U
L P
ER
IOD
ICA
LS
on 0
5/01
/13.
Cop
yrig
ht A
SCE
. For
per
sona
l use
onl
y; a
ll ri
ghts
res
erve
d.
grain-roughened surfaces. For practical computations, the value�=0.2 appears to be accurate enough, as a sand grain-roughenedsurface describes a concrete chute bottom better than wire mesh.
Power-Law Approximation for Chute Flow
According to the tests of Bauer �1951, 1954�, his velocity profilesare fitted better by a power-law model than the classic logarithmiclaw of the wall for fully developed turbulent pipe and channelflows, as used by Halbronn �1952a,b,c, 1955�. On a log-log plot,Bauer found a straight line within the boundary layer. However,Bauer �1951, 1954� made no detailed analysis of the turbulentboundary layer profiles. George �2007� argued that no universallog-law exists �given by universal constants of � and B� appli-cable to the entire velocity profile of channel, pipe and boundarylayer flows, supporting the existence of different wake terms inthe turbulent velocity profile for each flow type. George �2007�further noted that developing boundary layers are theoreticallybetter described by power-law than by log-law velocity profiles,such as used by Halbronn �1952a,b,c�. Balachandar et al. �2002�indicated the usefulness of the power-law equations for shallowrough channel flows. It therefore appears reasonable to simulatevelocity profiles of steep turbulent developing boundary layerswith power-law functions.
The generalized equation for a power-law velocity profile is�Chen 1991�
u
u�= �� y
ks�1/n
�4�
with �=coefficient and n=exponent. A rational approach is thusrequired to estimate the parameters n and �. In contrast to Chen�1991�, who made a statistical analysis to fit a power-law functionto the law of the wall, a new rational approach to determine n and� is adopted below, including both the laws of the wall and of thewake.
Assume that the mean turbulent velocity profile is accuratelydescribed by Eq. �1�, involving the developing shear layer of skinfriction Cf and displacement thickness ��, obtained from Eq. �1�using boundary layer theory, as explained below. A different ve-locity profile would produce different values of Cf and ��, result-ing in a different shear layer. Therefore, an equivalent shear layerto that of Eq. �1� is considered by using Eq. �4�, in terms of bothCf and ��. The displacement thickness �� according to Eq. �1� is
�� =�0
� �1 −u
U�dy =
�u�
�U�11
12+ �� �5�
The displacement thickness of Eq. �4� is �Chanson 2004; Montes1998�
�� =�
1 + n�6�
Imposing this ��, either using Eqs. �5� and �6�, results in
n = �U
u��11
12+ ��−1
− 1 = �� 2
Cf�1/2�11
12+ ��−1
− 1 �7�
and from Eq. �4�
JO
J. Hydraul. Eng. 2010
� = � 2
Cf�1/2� �
ks�−1/n
�8�
Using Eq. �1� at the outer edge of the boundary layer where u�y=��=U yields
U
u�=
1
�ln� �
ks� + B +
2�
� � 2
Cf�1/2
�9�
which, when inserted into Eqs. �7� and �8�, allows the evaluationof n and �. The standard value of B=8.5 is adopted for turbulentrough channel flow �Kironoto and Graf 1994; White 1991�. Notethat Eq. �1� was carefully constructed to satisfy a zero-slope con-dition, but the power-law model of course cannot satisfy this con-dition. However, this particular feature is not important forinception point computations based on power-law profiles �Cainand Wood 1981�.
The interest of converting Eq. �1� into a power law relies onimproving the actual approach for chute flow. Note that the com-putation of the inception point based on a log-wake law results inmuch more complex computations, and no explicit and simpleresults may be obtained. Only numerical solutions appear to bepossible �White 1991�. However, the power-law profile yieldssimple and explicit equations for inception point computations�Chanson 1997�. The new power-law approach, used in thesecomputations, permits to take into account the real behavior of theturbulent boundary layer in a simple way. Data of velocity pro-files in steep chutes from different sources resulted in differentvalues of n, i.e., for the data of Bauer �1951� n=4.5 as a meanvalue, whereas Cain and Wood �1981� obtained n=6.3. Thesedifferences cannot be explained, or predicted, at the present stateof knowledge. Based on Eqs. �7� and �8�, different values of nresult from different values of � /ks and � �Fig. 4�b��. The results
Fig. 4. �a� Comparison of velocity profiles from ���. The law of thewall �Eq. �1� neglecting the law of the wake�, ��� Eq. �1� with �=0.2 and �—� Eq. �4�, for � /ks=50 and 300; �b� Power-law velocityparameters n and � as function of � /ks.
are useful from a practical point of view, because the power law is
automatically adjusted in shape �given by its parameters�, depend-ing on the boundary layer thickness and the wake strength, alonga chute.
Further questions may be raised up in connection of using ��
to define the equivalent power-law profile, whereas it might bepossible to use the momentum or the energy thickness. The con-sistency of using �� may be explained in connection to chute flowcomputations, where the free surface profile h=h�x� is obtainedfrom the energy equation as �Montes 1998; Chanson 2004�
H = z + h +q2
2g�h − ���2 �10�
where H=total energy head and z=bottom elevation. Thus, whatis needed is an equivalent power-law velocity profile that yieldsthe same �� than the log-wake velocity profile.
A comparison of Eq. �1� with Eq. �4� for the novel power-lawapproximation is shown in Fig. 4�a�, for various values of � /ks
using �=0.41, B=8.5 and �=0.2, resulting in a good agreementof the power-law velocity profile with the log-wake law, regard-less of � /ks. The velocity at the outer edge of the boundary layeris the same for both profiles, as they produce the same flow re-sistance. The power-law profile underestimates the flow velocity,roughly for y /��0.5, whereas it is overestimated in the lowerhalf. This slight asymmetry of the power-law produces the samedisplacement thickness as the complete flow profile. The differ-ences, however, are extremely small �Fig. 4�a��. Eq. �1� neglect-ing the law of the wake is also plotted therein, but it poorly fitsthe complete turbulent velocity profile for y /��0.2. Thus, thenovel approach for the power-law is a close approximation of thecomplete turbulent velocity profile within a developing turbulentshear layer, with better results than the logarithmic law of thewall. Fig. 4�b� shows the power-law parameters n and � versus� /ks, using Eqs. �7�–�9�. Neither n and � remain constant alongthe shear layer, but their values increase with � /ks.
Eq. �4�, with the power-law coefficients of Eqs. �7� and �8�, isvalidated in Fig. 5 against the data of Bormann �1968�. The agree-ment of the power-law profiles with the test data are reasonable.Another comparison was made using Bauer’s test data, with �=0.4, as previously deduced in Fig. 3 from the velocity defectprofiles. An important error was found in Bauer’s roughnessvalue, as also noted by Campbell et al. �1965�. Bauer proposedks=2.74 mm, based on uniform flow tests. Eqs. �1� and �4� werescaled with this roughness value to plot the velocity profiles u /u*
�y /h� in Fig. 6�a� together with Bauer �1951� experimental data.Fig. 6�a� shows that Bauer data significantly disagrees with bothEqs. �1� and �4�, thereby implying that the assumed roughnessvalue ks=2.74 mm is erroneous. An accurate roughness heightdefinition requires not only the roughness itself, but also the vir-tual origin of the law of the wall, from which the y-coordinate ismeasured. Bauer �1951� reported the y-coordinate from the me-dium plane of the screen, which is rather an arbitrary choice. Forsimplicity, to compare the accuracy of Eq. �4� rather than to de-scribe the real test conditions of Bauer �1951�, the reference planeis maintained at the screen center. The value of ks is then deter-mined by matching the velocity profiles measured by Bauer�1951�, keeping in mind that this roughness value includes addi-tional factors. An appropriate value was fitted to ks=1.22 mm�Fig. 6�b��. Using this value as the effective roughness height, alltest data of Bauer �1951� are compared in Fig. 7 with the power-law approximation for chute flow, resulting in a good agreement.Eqs. �7� and �8� for the power-law parameters may then be used
with the energy equation to predict the free surface profile of the
developing chute flow zone. This information is required forchute design to determine the cavitation inception index, the firstaerator position, and the point of air inception �Hager and Blaser1998�.
Flow Resistance
The analysis of velocity profiles resulted in �=0.41, B=8.5, and�=0.2. Eq. �1� then gives for the boundary shear coefficient
Cf−1/2 = 3.967 log� �
ks� + 6.7 �11�
Eq. �11� is compared in Fig. 8 with the test data of Bormann�1968� by integrating numerically the streamwise momentumequation using the measured velocity profiles. Eq. �11� is seen tobe a good approximation for Cf as compared to the data of Bor-mann �1968�. For �=0.4, the resistance law applied to the tests of
Fig. 5. Velocity profiles at successive chute stations from �—�power-law Eq. �4� and Bormann �1968� test data �� � � � � � ��; each symbol refers to a different chute station�: �a� ks=2 mm q=0.1606 m2 /s; �b� ks=2 mm q=0.246 m2 /s; �c� ks=0.33 mm q=0.1664 m2 /s; �d� ks=0.33 mm q=0.251 m2 /s; �e� ks=0.83 mmq=0.1665 m2 /s; �f� ks=0.83 mm q=0.2473 m2 /s; �g� ks
=2.35 mm q=0.1623 m2 /s; and �h� ks=2.35 mm q=0.2468 m2 /s
Bauer �1951� is finally
.136:447-452.
Dow
nloa
ded
from
asc
elib
rary
.org
by
V U
L P
ER
IOD
ICA
LS
on 0
5/01
/13.
Cop
yrig
ht A
SCE
. For
per
sona
l use
onl
y; a
ll ri
ghts
res
erve
d.
Cf−1/2 = 3.967 log� �
ks� + 7.39 �12�
which is compared in Fig. 8 with the test data of Bauer �1951�,indicating also agreement, but, as expected, with a slight discrep-ancy between Eqs. �11� and �12� due to the different � values foreach type of surface roughness. These results apply to the devel-oping flow zone of steep chutes, upstream of the air inceptionpoint. Accordingly, the effects of flow aeration are not accountedfor herein.
Conclusions
A detailed reanalysis of turbulent velocity profiles on steep chuteshas been made, resulting in mean values for the parameters of the
Fig. 6. Typical velocity profiles in Bauer �1951� experiments: �a�from �– – –� Eq. �1�, �—� Eq. �4� and Bauer �1951� test data ���,using ks=2.74 mm; �b� from �– – –� Eq. �1�, �—� Eq. �4� and Bauer�1951� test data ���, using ks=1.22 mm
Fig. 7. Velocity profiles in successive chute stations from �—�power-law Eq. �4� and Bauer �1951� test data �� �; a symbol changeis used to mark successive chute stations� for chute slope: �a� �=20°; �b� �=40°; and �c� �=60°
JO
J. Hydraul. Eng. 2010
complete turbulent velocity profile in the turbulent rough regime.The values obtained for the wake parameter � reproduce thevelocity defect profiles with a reasonable accuracy, with an im-perceptible effect of chute slope. However, the surface roughnesshas a slight effect on �. Further research is necessary here; forpractical purposes, a mean value of �=0.2 was found to be rea-sonable for concrete chutes.
From the complete mean turbulent velocity profile, accountingboth for the law of the wall and the law of the wake, a rationalapproach was proposed for a power-law velocity profile withinthe boundary layer. The power-law parameters were computed bydefining an equivalent shear layer producing identical flow resis-tance and mass deficit as the real shear layer. These power-lawvelocity profiles agree well with velocity profiles measured in asteep chute flow. The novel approach as defined in Eqs. �7� and�8� is free from any parameter adjustment. The results were usedto analytically determine the flow resistance on steep slopes,which also agrees well with test data.
Acknowledgments
The writer thanks Cornelia F. Mutel, IIHR Historian and Archi-vist, University of Iowa, for her helpful assistance in providing areprint of the original Ph.D. thesis of Dr. William John Bauer.
Notation
The following symbols are used in this technical note:B constant of the law of the wall �m/s�;
Cf skin friction coefficient;H total energy head �m�;h flow depth �m�;ks roughness height �m�;n exponent of power-law velocity profile;q unit discharge �m2 /s�;S shape factor of boundary layer;U free stream velocity �m/s�;u� shear velocity �m/s�;u time-averaged turbulent velocity parallel to
spillway bottom �m/s�;x streamwise distance along chute �m�;y distance normal to spillway face �m�;z bottom elevation �m�;
�� boundary layer displacement thickness �m�;
Fig. 8. Law of flow resistance on steep chutes from �—� Eq. �11�,�– – –� Eq. �12�, Bormann �1968� data for ��� ks=0.33 mm, ���ks=2 mm, ��� ks=0.83 mm, and ��� ks=2.35 mm, ��� Bauer�1951� data
� angle of channel bed with horizontal �deg�;�m boundary layer momentum thickness �m�;� von Karman constant;� coefficient of power-law velocity profile;� wake strength parameter;� water density �N /m3�; and
�o boundary shear stress �N /m2�.
References
Balachandar, R., Bakely, D., and Bugg, J. �2002�. “Friction velocity andpower law velocity profile in smooth and rough shallow open channelflows.” Can. J. Civ. Eng., 29, 256–266.
Bauer, W. J. �1951�. “The development of the turbulent boundary layer onsteep slopes.” Ph.D. thesis, Univ. of Iowa, Iowa.
Bauer, W. J. �1954�. “Turbulent boundary layer on steep slopes.” Trans.Am. Soc. Civ. Eng., 119, 1212–1233.
Bormann, K. �1968�. “Der abfluss in schussrinnen unter berücksichtigungder luftaufnahme.” Bericht 13, Versuchsanstalt für Wasserbau, Oskarvon Miller Institut, Technische Hochschule, München, Germany �inGerman�.
Cain, P., and Wood, I. R. �1981�. “Measurements of self-aerated flow ona spillway.” J. Hydr. Div., 107�11�, 1425–1444.
Campbell, F. B., Cox, R. G., and Boyd, M. B. �1965�. “Boundary layerdevelopment and spillway energy loss.” J. Hydr. Div., 91�3�, 149–163.
Chanson, H. �1997�. Air bubble entrainment in free-surface turbulentshear flows, Academic, London.
Chanson, H. �2004�. The hydraulics of open channel flows: An introduc-tion, Butterworth-Heinemann, Oxford, U.K.
Chen, C. L. �1991�. “Unified theory on power laws for resistance.” J.Hydraul. Eng., 117�3�, 371–389.
Dean, R. B. �1976�. “A single formula for the complete velocity profile ina turbulent boundary layer.” J. Fluids Eng., 98�12�, 723–726.
George, W. K. �2007�. “Is there a universal log-law for turbulent wall-bounded flows?” Phil. Trans. R. Soc. London, Ser. A, 365, 789–806.
Hager, W. H., and Blaser, F. �1998�. “Drawdown curve and incipientaeration for chute flow.” Can. J. Civ. Eng., 25�3�, 467–473.
Halbronn, G.�1952a�. “Étude de la mise en régime des écoulements surles ouvrages à forte pente: Application au problème de l’entrainementd’air.” Houille Blanche, 7�1�, 21–40 �in French�.
Halbronn, G.�1952b�. “Étude de la mise en régime des écoulements surles ouvrages à forte pente: Application au problème de l’entrainementd’air.” Houille Blanche, 7�3�, 347–371 �in French�.
Halbronn, G.�1952c�. “Étude de la mise en régime des écoulements surles ouvrages à forte pente: Application au problème de l’entrainementd’air.” Houille Blanche, 7�5�, 702–722 �in French�.
Halbronn, G. �1955�. “Discussion to turbulent boundary layer on steepslopes.” Trans. Am. Soc. Civ. Eng., 119, 1234–1240.
Kironoto, B. A., and Graf, W. H. �1994�. “Turbulence characteristics inrough uniform open channel flow.” Proc. Inst. Civ. Eng., Water Mari-time Energ., 106�4�, 333–344.
Krogstad, P. A., and Antonia, R. A. �1999�. “Surface roughness effects inturbulent boundary layers.” Exp. Fluids, 27, 450–460.
Krogstad, P. A., Antonia, R. A., and Browne, L. W. B. �1992�. “Compari-son between rough- and smooth-wall turbulent boundary layers.” J.Fluid Mech., 245, 599–617.
Montes, J. S. �1998�. Hydraulics of open channel flow, ASCE, Reston,Va.
Tachie, M. F., Bergstrom, D. J., and Balachandar, R. �2000�. “Rough wallturbulent boundary layers in shallow open channel flow.” J. FluidsEng., 122�3�, 533–541.
Tominaga, A., and Nezu, I. �1992�. “Velocity profiles in steep open-channel flows.” J. Hydraul. Eng., 118�1�, 73–90.
White, F. M. �1991�. Viscous fluid flow, McGraw-Hill, New York.
