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Research ArticleInvestigation of Velocity Distribution in Open Channel FlowsBased on Conditional Average of Turbulent Structures

Yu Han1 Shu-Qing Yang2 Muttucumaru Sivakumar2 and Liu-Chao Qiu1

1College of Water Resources amp Civil Engineering China Agricultural University Beijing 100083 China2School of Civil Mining amp Environmental Engineering University of Wollongong NSW 2522 Australia

Correspondence should be addressed to Yu Han yh916uowmaileduau and Liu-Chao Qiu qiuliuchaocaueducn

Received 4 January 2017 Accepted 24 April 2017 Published 29 May 2017

Academic Editor Jian G Zhou

Copyright copy 2017 Yu Han et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We report the development of a new analytical model similar to the Reynolds-averaged Navier-Stokes equations to determinethe distribution of streamwise velocity by considering the bursting phenomenon It is found that in two-dimensional (2D) flowsthe underlying mechanism of the wake law in 2D uniform flow is actually a result of updown events A special experiment wasconducted to examine the newly derived analytical model and good agreement is achieved between the experimental data in theinner region and the modelrsquos prediction The obtained experimental data were also used to examine the DML-Law (dip-modified-log-law) MLW-Law (modified-log-wake law) and CML-Law (Colersquos wake law) and the agreement is not very satisfactory in theouter region

1 Introduction

Velocity distribution in open channel flows offers a widerange of applications in the fields of hydrometry sedimenttransport river restoration power plant design and so forth[1ndash3] Accurate investigation of the velocity distribution inopen channel flows has been conducted over the past centuryThe law of the wall developed for pipe flows has often beenapplied to uniform flows in open channel flows [4] thisldquouniversal wall functionrdquo or log-law of the wall is expressed as

119906119906lowast =

1120581 ln

1199101199100 (1)

and herein 120581 is the von Karman constant 119906lowast is the meanfriction velocity and 119910 is the vertical direction In (1) theboundary condition is 119906 = 0 at 119910 = 1199100 [5] and the parameter1199100 is the constant of integration that must be determinedusing the boundary condition [6] It is expected that theparameter 1199100 in (1) should be a variable as well [7] for asmooth boundary 1199100 should be related to the thickness ofthe local viscous sublayer that is

1199100 = ]119888119906lowasts (2)

where 119888 is the coefficient to be determined experimentally ]is the kinematic viscosity and 119906lowast119904 denotes the local frictionvelocity which may be different from 119906lowast

The velocity vertical profile is well described by theclassical log-law in the inner region 119910ℎ lt 02 but the log-law normally deviates from the experimental data in the outerration 119910ℎ gt 02 [8] Since this method is unable to representthe outer region of a velocity profile Coles [9] suggested animprovement called the wake law Many researchers havediscussed the validity of such a law [10 11] and have improvedboth the log and wake laws for smooth or rough flows [12 13]

Coles [9] extended the log-law by introducing a purelyempirical correction function He developed an additionalterm (ie the wake term) as a supplement to the log-law toaccount for the velocity deviation caused bywakeThismodelis known as the wake law

119906119906lowast =

1120581 ln

1199101199100 +

2Π120581 sin2 (1205871199102ℎ ) (3)

where Π denotes the ldquoprofilerdquo parameter Coles [9] gave Π= 055 For open channel flows Coleman [14] obtained anaverage Π value of 019 Nezu and Rodi [15] yielded a Πvalue of 0 to 020 while Kirkgoz [16] reported a value of

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 1458591 9 pageshttpsdoiorg10115520171458591

2 Mathematical Problems in Engineering

01 Cardoso et al [10] obtained a Π value of minus0077 over asmooth bed Kironoto and Graf [11] found Π value in therange of minus008 to 015 for water flows over a gravel bedHowever the values of Π are established by experimenta-tion with large variations reported in the literatures Whilesignificant advances have been made by using Colersquos wakelaw the mechanism of Colersquos wake law (CWL-Law) and theassociated wake strength parameter are not fully understoodThese investigations often ascribe the large variations toeither smooth or rough boundaries and few researchers havetaken into account the velocity-dip phenomenon caused bysecondary currents [17] (Guo 2015)

Different from the positive deviation ofmeasured velocityfrom the log-lawrsquos prediction or the CWL-Law Yang et al [18]discussed the negative deviation of measured velocity fromthe log-lawrsquos prediction or the dip phenomenon in whichthe maximum longitudinal velocity occurs below the watersurface They suggested that Colersquos wake law is not able todescribe the entire velocity profile when the dip phenomenonexists Yang et al [18] proposed a dip-modified-log-law(DML-Law) based on the analysis of the Reynolds-averagedNavier-Stokes (RANS) equations This law involving twologarithmic distances one from the bed (ie the log-law)and the other from the free surface has the advantage that itcontains only one parameter 120572 for dip-correctionThe DML-Law reverts to the classical log-law for 120572 = 0 Yang et al[18] modified the log-law by adding a term to express the dipphenomenon instead of Colersquos wake law based on Reynoldsequations

119906119906lowast =

1120581 ln

1199101199100 +

120572120581 ln(1 minus 119910

ℎ) (4)

120572 = 13 exp(minus 1198872ℎ) (5)

Guo and Julien [12] proposed a modified-log-wake law(MLW-Law) which fits velocity profiles with a dip phe-nomenon However this law cannot be used for predictiveapplications since it requires fitting the near-free-surfacevelocities to the parabolic law to obtain dip position andmaximum velocity As Guo and Julien indicate the MLW-Law can be used only in flow measurements since it requiresmeasured velocities that is

119906119906lowast =

1120581 ln

1199101199100 +

2Π120581 sin21205871205892 minus 1205893

3120581 (6)

where 120589 = 119910120575 and 120575 is the distance from the bed to the pointwhere the velocity is maximum

Nezu and Nakagawa [19] found that in the outer regionthe mean velocity data deviate systematically from the log-law distribution at sufficiently high Reynolds numbers thisdeviation cannot be neglected in the free-surface region thatis 119910ℎ gt 06 It is thus worth investigating the underlyingmechanism of velocity deviation from the log-law Bonakdariet al [20] also proposed a new formulation of the verticalvelocity profile in open channel flows based on an analysisof the Navier-Stoke equations On the basis of this law and ofthe suggestions of Absi [21] a new analytical procedure waspresented by Pu et al [22]

With the advent of nonintrusive instrumentation visu-alisation techniques reveal that turbulence is dominated byforceful structures that are well organised and ordered eddieswith a certain lifespan now termed as coherent structuresCoherent structures can be placed into two categories burst-ing phenomena that occur in the wall region and large-scalevertical motion in the outer region For coherent structuralanalysis the method of quadrants has been developed basedon measured velocity fluctuations that is quadrant I 1199061015840 gt 0and V1015840 gt 0 quadrant II 1199061015840 lt 0 and V1015840 gt 0 quadrant III 1199061015840 lt 0and V1015840 lt 0 and quadrant IV 1199061015840 gt 0 and V1015840 lt 0 [23] Cellinoand Lemmin [24] pointed out that it would be meaningful toconcentrate on V1015840 gt 0 and V1015840 lt 0 in quadrants because themost significant feature of turbulence is fluctuations in thewall-normal direction They also developed an approach tocalculate conditionally averaged upwarddownward velocitybased on velocity fluctuations However the linkage of quad-rant analysis and upwarddownward events with velocitydeviation from the log-law has not been established

The objective of this research is to present a new analyticalapproach to determine the streamwise velocity profile andestablish a relationship between the log wake function andupdown events This leads to the present research aim todevelop a universal model to express the velocity profilein uniform flows Such a novel analytical approach will beused to explain that streamwise velocity profiles in the outerregion depart from the universal logarithmic law of thewall The proposed analytical approach and those previouslyproposed mathematical methods will be examined by usingexperimental data

2 Theoretical Consideration

The RANS momentum and continuity equations can bewritten for steady state flow as

119906(120597119906120597119909) + V(120597119906120597119910) + 119908(120597119906120597119911)

= 119892119878 + 120597120597119909 (minus11990610158401199061015840) + 120597

120597119910 (minus1199061015840V1015840) + 120597120597119911 (minus11990610158401199081015840)

+ ](12059721199061205971199092 +12059721199061205971199102 +

12059721199061205971199112)

(7)

120597119906120597119909 + 120597V

120597119910 + 120597119908120597119911 = 0 (8)

where 119906 V and119908 denote the mean velocity in the streamwise(119909) lateral (119910) and vertical (119911) directions respectively 119892 isthe gravitational acceleration 119878 is the energy slope andminus11990610158401199081015840are theReynolds stress tensor components For a uniform andfully developed flow (7) becomes

V(120597119906120597119910) + 119908(120597119906120597119911) = 119892119878 + 120597120597119910 (minus1199061015840V1015840) + 120597

120597119911 (minus11990610158401199081015840)

+ ](12059721199061205971199102 +12059721199061205971199112)

(9)

Mathematical Problems in Engineering 3

Alternatively (9) becomes

120597 (119906V minus 120591119909119910120588)120597119910 + 120597 (119906V minus 120591119909119911120588)120597119911 = 119892119878 (10)

120591119909119910 = 120583120597119906120597119910 minus 1205881199061015840V1015840 (11)

120591119909119911 = 120583120597119906120597119911 minus 12058811990610158401199081015840 (12)

where 120588 is the density of the fluid 120583 is the dynamic viscosityof the fluid and 1199061015840 V1015840 and 1199081015840 are the turbulent velocityfluctuation in the 119909 119910 and 119911 directions For 2D flow (10) canbe simplified as

120597 (119906V + 1199061015840V1015840 minus ] 119889119906119889119910)120597119910 = 119892119878 (13)

Integration of (13) yields

]119889119906119889119910 minus 119906V minus 1199061015840V1015840 = minus119892119878119910 + 119862 (14)

where 119862 is the integration constant Using the boundarycondition 119910 = 0 119906 = V = 0 1199061015840V1015840 = 0 and ]119889119906119889119910 = 1199062lowast andthus 119862 = 1199062lowast For water surface 119910 = ℎ V = 0 and 1199061015840V1015840 = 0and then the shear velocity can be rewritten as

1199062lowast = 119892119878ℎ (15)

Therefore the momentum equation has the following formby dividing (15)

119889119906+119889119910+ minus 119906+V+ minus

1199061015840V10158401199062lowast = 1 minus 119910

ℎ (16)

where 119906+ = 119906119906lowast and V+ = V119906lowastCellino and Lemmin [24] realised that it would be mean-

ingful to develop an approach to calculate the distributionof streamwise velocity by considering the bursting phe-nomenon However the above time averaged method cannotbe used to express the bursting phenomenon Yang [25]proposed an approach to calculate the conditionally averagedupwarddownward velocity based on velocity fluctuationsHe first defines the occurrence probabilities 119903 based on thedirection of vertical velocity For upflow (subscript 119906) anddownflow (subscript 119889) 119903 is expressed as

119903119906 = sum1198791199060 Δ119905119906119879 (17)

119903119889 = sum1198791198890 Δ119905119889119879 (18)

where Δ119905 is the time interval between two consecutivevelocity acquisitions in an experiment119879 is the total observedtime period that issumΔ119905119906 +sumΔ119905119889 = 119879119906 +119879119889 = 119879 The upflow

and downflow velocity definitions are given in the followingway

119906119906 = 1119879119906 int119879119906

0119906119889119905 (19)

119906119889 = 1119879119889 int119879119889

0119889119889119905 (20)

V119906 = 1119879119906 int119879119906

0V119906119889119905 (21)

V119889 = 1119879119889 int119879119889

0V119889119889119905 (22)

where 119906119906 and 119906119889 denote upflow and downflow eventsrespectively and the symbol ldquosimrdquo represents instantaneousvelocities Yang [25] obtained the Reynolds shear stress forupflow and downflow as follows and both deviate from thestandard linear distribution that is

minus1199061015840119906V10158401199061199062lowast119906 = (1 minus 119910ℎ) + 119906+119906V+119906

minus1199061015840119889V10158401198891199062lowast119889

= (1 minus 119910ℎ) + 119906+119889V+119889

(23)

where 1199061015840119906 = minus 119906119906 1199061015840119889 = minus 119906119889 119906+119906 = 119906119906119906lowast119906 and 119906+119889 = 119906119889119906lowast119889 ℎ is water depth and 119906lowast119906 and 119906lowast119889 are friction velocitiesfor upflow (subscript 119906) and downflow (subscript 119889) respec-tively

The expression of eddy viscosity ]119905 is]119905119906lowastℎ = 120581120585 (1 minus 120585) (24)

where 120585 = 119910ℎ and (24) can be expressed as

120581120585 (1 minus 120585) 119889119906+119906119889120585 = (1 minus 120585) + 119906+119906V+119906 (25)

The velocity gradient in upflow and downflow can then beexpressed by

119889119906+119906119889120585 = 1120581120585 +

119906+119906V+119906120581120585 (1 minus 120585) 119889119906+119889119889120585 = 1

120581120585 +119906+119889V+119889120581120585 (1 minus 120585)

(26)

Similarly the gradient of mean velocity can be expressed by

119889119906+119889120585 = 119903119906 119906lowast119906119906lowast

119889119906+119906119889120585 + 119903119889 119906lowast119889119906lowast119889119906+119889119889120585 (27)

Inserting (26) into (27) yields

119889119906+119889120585 = 1

120581120585 (119903119906119906lowast119906119906lowast + 119903119889 119906lowast119889119906lowast )

+ 119903119906 V+119906120581120585 (1 minus 120585)119906lowast119906119906lowast (1 minus

119906+119889119906+119906 )119906+119906 (28)


Table 1 Summary of the main hydraulic parameters in rectangular channel

Slope Water depth ℎ (m) Width 119887 (m) 119877 (m) Aspect ratio 119887ℎ 119876 (m3s) 119906lowast = radic119892119877119878 (ms) 119906lowast = radic119892ℎ119878 (ms)Run 1 0001 0065 0300 0045 4615 0008 0021 0025Run 2 0001 0085 0300 0054 3529 0012 0023 0029Run 3 0001 0100 0300 0060 3000 0015 0024 0031Run 4 0001 0110 0300 0063 2727 0017 0025 0033Run 5 0001 0150 0300 0075 2000 0025 0027 0038Run 6 0001 0160 0300 0077 1875 0028 0028 0040

where 119903119906(119906lowast119906119906lowast) + 119903119889(119906lowast119889119906lowast) = 1 and then

119889119906+119889120585 = 1

120581120585 + 119903119906V+119906120581120585 (1 minus 120585)

119906lowast119906119906lowast (1 minus119906+119889119906+119906 )119906+119906 (29)

Here we denote

119860 = 119903119906V+119906 119906lowast119906119906lowast (119906+119906 minus 119906+119889) (30)

Then

119889119906+ = 1120581120585119889120585 +

119860120581120585 (1 minus 120585)119889120585 (31)

Integration of (31) with respect to 120585 yields119906+ = 1

120581 ln 120585 + 119860120581 ln 120585 minus 119860

120581 ln (1 minus 120585) + 1198621 (32)

In nonslip boundary conditions 119906 = 0 and then 120585 = 1205850 = 1199100ℎand thus 1198621 = minus(1120581)ln(1199100ℎ)

Finally we obtained the corrected log equation as

119906+ = 1120581 ln

1199101199100 +

119860120581 ln( 119910ℎ

1 minus 119910ℎ) (33)

The value of 119860 used in (30) corrects the deviation of velocityfrom the log-law by considering upward and downwardmotion adequately Thus we can find the existence of thewake function from the newly derived log-law Upward anddownward motion can decompose turbulent flow into twodistinct events and it also indicates that if 119906+119906 = 119906+119889 then theclassic log-law can be obtained Many researchers determinethe different 120581 and119861parameters in119906+ = 1120581 ln 119906lowast119910]+119861 fromtheir experiments Equation (33) implies that themain reasonfor such different results is not from different experimentalconditions but it is rather from upflow and downflow effectsIt is necessary to verify (33) using experimental data

3 Experimental Set-Up

The experiments were undertaken in a large-scale flow loopat the University of Wollongong Australia in which waterwas supplied from a head tank The main components ofthis flume were the head tanks tail tank glass water channelrecirculation pipe system and two pumps as shown inFigure 1

The head tank and the tail tank were constructed usingstainless steel The head tank was aligned centrally to the

Pump

LDA

Tank Tank Tank

Y

X

X

Z

Flume length

Figure 1 General view of the flume

flume and symmetrical about the centre line of the channelso that the flow at the entrance of the channel was asuniform as possible A honeycomb section was also locatedat the entrance to the flume Through the addition of thishoneycomb the velocity distribution within the channelcross-section became increasingly uniform The water depththrough the flume was controlled by an adjustable tailgatein the downstream tail tank The relative height of thistailgate was adjusted by rotating the attached handle Thiswas installed at the entrance of the tail tank From the datarecorded at these locations it was possible to visually analysethe development of uniform flow in the channel prior toexperiments

The experimental flume was 105m long 03m wide and045m high with a slope set to a certain value for differentexperimental conditions Experimental conditions are givenin Table 1 A series of velocity profiles were measured incentral line with different 119887ℎ values The flow depth ℎ waschanged in order to examine the effect of aspect ratio in thisstudy Hence a comprehensive set of high quality data suchas velocities turbulence intensities Reynolds stresses andboundary shear stresses of the 3D turbulence structure werealso obtained

Throughout the experimental process in this studythe point velocities were measured using a Dantec two-component LDA system It is a two-component configurationwith a 60mm optical fibre probe and a front lens witha 400mm focal length The system consists of a 300mWcontinuous wave Argon-Ion laser and transmitting opticsincluding a beam splitter Bragg-cell and signal processorsGreen (5145 nm) and blue (488 nm) components were usedto measure the horizontal and vertical components of the


velocity respectively The optics for 2D LDA was arranged inoff-axis backscatter modeThe backscatter mode of receivinga signal from a particle refers to the reflected light beingcaptured by the receiving optics on the same side as theincident laser beams with reflected light going back intothe prove head through the front lens The dimension ofthe measurement volume was approximately 0189 times 0189 times397mm for both colours when measured in air

Hence no calibration for LDA was necessary prior tothe experiments It is important to note that during themeasurement of velocity 119906 and V by LDA data acquisitionof the components needed to be coincidental in time for allthe data pairs 119906(119905) and V(119905) The signal processor is a DantecBurst SpectrumAnalyser (BSA) connected to an oscilloscopeand a PC The BSA converts the electrical signals whichare processed by the oscilloscope into velocity data whichagain are monitored online by a PC Raw data exportedfrom BSA can be processed into statistical values like meanvelocities root mean squared values of velocity fluctuationsand Reynolds stresses

4 Results and Discussion

The velocity profiles for a smooth rectangular channel areplotted in Figures 2ndash7 and the velocity profiles predicted byDML-Law MLW-Law and CML-Law are used for compari-son to experimental dataThese figures show that the log-lawis able to predict all experimental data in the inner regionIt can be clearly found that the wake function indeed existsin uniform flow However DML-Law MLW-Law and CML-Law are not able to agree with the experimental data properlyin the outer region that is 119910ℎ gt 06 It is observed that themeasured values for all runs are relatively higher than thecalculated results from DML-Law in the outer region It isimpossible to improve the velocity profiles predicted by theDML-Law by adjusting parameter 120572 so further improvementof (5) is recommended The DML-Law seems to be able topredict the velocity-dip phenomenon but with inaccuratemaximum velocities Accurate predictions of the velocity-dipphenomenon therefore require larger values of 120572

It can be seen from Figures 2ndash4 that the MLW-Lawand CML-Law significantly overestimate the velocity profilein the outer region Both of these two methods seem notto be a suitable predictor for channels with 119887ℎ gt 3 inthe outer region In particular the CML-Law is importantfor an inadequate prediction of the dip phenomenon with119887ℎ gt 3 and Π = 055 seems to be too small to improvepredictions Figures 5ndash7 show that the MLW-Law and CML-Law are more accurate in predicting the velocity profiles for119887ℎ lt 3 The effect of each term in the MLW-Law and CML-Law is not significant In general the average differencesbetween the MLW-Law CML-Law and DML-Law theoriesand the current measurements give relatively large averaged119864 values of 71 59 and 57 respectively The averagedifferences between measurements and the current approachgive a relatively small average 119864 value of 10

Figures 2ndash7 present comparisons of predicted velocityprofiles obtained by (33) with experimental data and wenote that the parameter 119860 in (33) is an important value

68

10121416182022242628303234

MeasurementsTheoretical equationDML-law

MLW-lawCML-law

001 002 003 004 005 006 007

u+

y

Figure 2 Comparison of velocity profiles obtained from theoreticalequation (see (33)) DML MLW and CML with experimental data(Run 1)

00112

14

16

18

20

22

24

26

28

30

32

34


MLW-lawCML-law

u+

y


in improving the predicted velocity profiles Thus to plotthe velocity distribution in the entire flow region using theproposed theoretical equation it was necessary to determinethe value of 119860 in (30) accurately The values of 119903119906 and V+119906 in(30) can be easily calculated using (17) and (21) respectivelyfrom a given data set of upflow velocities We also note that119906lowast in (30) is the mean shear stress that is 119906lowast = radic119892119877119878Furthermore in a plot ofmeasured upflowdownflowvelocityagainst log119910 for a profile along the central line the unknowns119906lowast119906 and 119906lowast119889 can be evaluated from its slope respectively



MLW-lawCML-law

12

14

16

18

20

22

24

26

28

30

32

34

001 01

u+

y

Figure 4 Comparison of velocity profiles obtained from theoreticalequation (Eq (33)) DML MLW and CML with experimental data(Run 3)

5

10

15

20

25

30

35

40

u+

001 01y


MLW-lawCML-law


The values of 119903119906 V+119906 119906lowast 119906lowast119906 and 119906lowast119889 used in (30) are thencalculated and included in Tables 2 and 3

The experimental data of 119906119906 and 119906119889 scale with 119906lowast119906 and119906lowast119889 and then 119906+119906 and 119906+119889 can be calculated respectively In(30) the ratio of 119906+119889 and 119906+119906 is larger than 1 in the outer regionas shown in Figure 8 thus the values of 119860 are all negative inthe outer region Hence the wake function (119860120581) ln(119910ℎ(1minus119910ℎ)) in (33) would also be a negative value in the outerregion implying that the velocity from (33) must be lowerthan that predicted by the log-law or there is a velocity

68

10121416182022242628303234363840

Measurements

u+

Theoretical equationDML-law

MLW-lawCML-law

001 01y


10

15

20

25

30

35

Measurements

u+

001 01y


MLW-lawCML-law


dip near the water surface Then it is clearly indicated thatdownward motions will mainly cause the wake strength Inother words in 2D flows there indeed exists a wake functionwhich is caused by variations in the wall-normal velocityThese different 119860 values have been applied in the proposedtheoretical equation and have amounted to approximately15 errors from the velocity encountered in the experimentsThey are much smaller than the corresponding errors forother comparisons It will be seen that the value of 119860 usedin (30) adequately accounts for the errors in comparison


Table 2 Summary of the main parameters in (30) (Run 1 Run 3 and Run 4)

ℎ = 0065m ℎ = 0100m ℎ = 0110m119906lowast119906 119906lowast119889 119906lowast 119906lowast119906 119906lowast119889 119906lowast 119906lowast119906 119906lowast119889 119906lowast0021 0020 0022 0027 0026 0026 0029 0029 0025119910ℎ 119903119906 V+119906 119910ℎ 119903119906 V+119906 119910ℎ 119903119906 V+1199060131 0460 1182 0085 0504 1375 0077 0496 12740154 0477 0911 0100 0503 0790 0091 0477 08630185 0492 0935 0150 0482 0768 0136 0507 09090231 0520 0855 0200 0505 0824 0182 0505 08980308 0458 1089 0250 0470 0750 0227 0469 08700385 0457 0617 0300 0488 0680 0273 0494 07960462 0496 0621 0400 0478 0423 0364 0458 07610538 0446 0636 0500 0513 0559 0455 0458 06950615 0443 0547 0600 0489 0382 0545 0493 05580692 0489 0523 0700 0487 0423 0636 0476 05330769 0514 0696 0800 0489 0393 0727 0474 03680846 0471 0832 0900 0504 0364 0818 0512 03120877 0471 0752 0864 0473 0302

0909 0481 02950927 0496 0253


ℎ = 0085m ℎ = 015m ℎ = 016m119906lowast119906 119906lowast119889 119906lowast 119906lowast119906 119906lowast119889 119906lowast 119906lowast119906 119906lowast119889 119906lowast0026 0024 0024 0026 0025 0025 0028 0028 0027119910ℎ 119903119906 V+119906 119910ℎ 119903119906 V+119906 119910ℎ 119903119906 V+1199060100 0491 0915 0057 0508 1015 0055 0485 09890118 0475 0877 0067 0500 1214 0065 0482 10180141 0470 0900 0100 0468 1248 0097 0522 11790176 0483 0792 0133 0497 1271 0129 0451 12680212 0496 0862 0167 0514 1248 0161 0545 12290235 0478 0862 0200 0535 1286 0194 0503 12070294 0500 0750 0267 0478 1347 0258 0482 12180353 0467 0685 0333 0513 1267 0323 0502 12180376 0478 0735 0400 0521 1401 0387 0485 11860412 0479 0712 0467 0513 1450 0452 0484 11710471 0473 0538 0533 0476 1355 0516 0473 10460529 0510 0427 0600 0449 1294 0581 0443 09040588 0483 0527 0667 0456 1172 0645 0481 08320647 0463 0488 0733 0496 1137 0710 0489 08390706 0468 0415 0800 0449 1218 0774 0501 07210765 0450 0454 0867 0519 1065 0839 0514 04460824 0478 0288 0947 0609 1477 0929 0500 09570847 0496 0250

Similarly 120573 = (119906+119906 minus 119906+119889) can be determined fromexperimental data The above analysis applies to (30) inwhich all parameters are found and thus the relationshipbetween 119860 and 120573 can then be plotted in Figure 9 Thedimensionless profiles of 120573 collapse reasonably well betweenthe experiments indicating that the relationship between 119860and 120573 for different water depths are similar Therefore an

empirical equation can be fitted using data119860 and 120573 as shownin Figure 9

The parameter 119860 in (33) is an important value forimproving the predicted velocity profiles To simplify thecalculation of 119860 in practice Figure 8 depicts the fitting of aregression equation For a particular channel (constant aspectratio) it is easier to check the value of 119906+119889119906+119906 from Figure 8 to


09

10

11

Run 1Run 2Run 3Fitting equation

00 02 04 06 08 10

u+ du

+ u

yℎ

u+du+u = 103(yℎ)005

R2 = 09

Figure 8 Variation of 119906+119889119906+119906 with 119910ℎ

00

05

10

15

20

25

30

Run 1Run 2Run 3Run 4

Run 5Run 6Fitting equation

0 1 2 3 4minus1

A

120573

R2 = 09543

y = 00574x2 + 04315x

Figure 9 Variation of 120573 with parameter 119860

obtain the value of 120573 thus facilitating obtaining the value of119860 in Figure 9

5 Conclusions

In this study we theoretically investigated conditionallyaveraged turbulent structures for upward and downwardmotions in a 2D uniform flowThe connection between wakelaw and updown events was also explored The followingconclusions can be drawn from the foregoing analysis

(1) In 2D flows the wake function is caused by theadditional momentum flux By considering updown

events it explores the underlying mechanism ofColesrsquo wake law and establishes a relationshipbetween the wake function and updown events

(2) Applying the proposed theoretical equation to exper-imental measurements good agreement is achievedbetween the theoretical equation and the measure-ments The theoretical results show that the momen-tum flux caused by the conditional updown eventsleads to deviation of velocity from the log-law

(3) The DML-Law MLW-Law and CML-Law modelsrecorded in the literature were examined in this studyFigures 2ndash7 show that all of these models are ableto predict all experimental data in the inner regionHowever DML-Law MLW-Law and CML-Law arenot able to agree with the experimental data properlyin the outer region It is clearly found that the wakefunction indeed exists in uniform flow

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors express gratitude for the financial support fromthe National Key Research and Development Plan (Grant no2016YFC0400207) the National Natural Science Foundationof China (Grant no 51509248) the Chinese UniversitiesScientific Fund (Grant nos 2015QC025 2017SL001) andthe Scientific Research and Experiment of Regulation Engi-neering for the Songhua River Mainstream in HeilongjiangProvince P R of China (Grant no SGZLKY-12)

References

[1] H Afzalimehr and F Anctil ldquoVelocity distribution and shearvelocity behaviour of decelerating flows over a gravel bedrdquoCanadian Journal of Civil Engineering vol 26 no 4 pp 468ndash475 1999

[2] Y Han S-Q Yang N Dharmasiri and M Sivakumar ldquoEffectsof sample size and concentration of seeding in LDA mea-surements on the velocity bias in open channel flowrdquo FlowMeasurement and Instrumentation vol 38 pp 92ndash97 2014

[3] T Stoesser R McSherry and B Fraga ldquoSecondary currentsand turbulence over a non-uniformly roughened open-channelbedrdquoWater (Switzerland) vol 7 no 9 pp 4896ndash4913 2015

[4] Y Han S-Q Yang N Dharmasiri and M Sivakumar ldquoExperi-mental study of smooth channel flow division based on velocitydistributionrdquo Journal of Hydraulic Engineering vol 141 no 4Article ID 06014025 2015

[5] S-Q Yang S-K Tan S-Y Lim and S-F Zhang ldquoVelocitydistribution in combined wave-current flowsrdquo Advances inWater Resources vol 29 no 8 pp 1196ndash1208 2006

[6] R Absi ldquoAn ordinary differential equation for velocity distri-bution and dip-phenomenon in open channel flowsrdquo Journal ofHydraulic Research vol 49 no 1 pp 82ndash89 2011

[7] A J Smits B J McKeon and I Marusic ldquoHigh-Reynoldsnumber wall turbulencerdquo Annual Review of Fluid Mechanicsvol 43 pp 353ndash375 2011


[8] I Nezu ldquoOpen-channel flow turbulence and its researchprospect in the 21st centuryrdquo Journal of Hydraulic Engineeringvol 131 no 4 pp 229ndash246 2005

[9] D Coles ldquoThe law of the wake in the turbulent boundary layerrdquoJournal of Fluid Mechanics vol 1 pp 191ndash226 1956

[10] A H Cardoso W H Graf and G Gust ldquoUniform flow in asmooth open channelrdquo Journal of Hydraulic Research vol 27no 5 pp 603ndash616 1989

[11] B A Kironoto W H Graf and Reynolds ldquoTurbulence charac-teristics in rough non-uniform open-channel flowrdquo Proceedingsof the Institution of Civil Engineers - Water and MaritimeEngineering vol 112 no 4 pp 336ndash348 1995

[12] J Guo and P Y Julien ldquoModified log-wake law for turbulentflow in smooth pipesrdquo Journal of Hydraulic Research vol 41 no5 pp 493ndash501 2003

[13] S-Q Yang S-Y Lim and J A McCorquodale ldquoInvestigationof near wall velocity in 3-D smooth channel flowsrdquo Journal ofHydraulic Research vol 43 no 2 pp 149ndash157 2005

[14] N L Coleman ldquoEffects of Suspended Sediment on theOpen-Channel Velocity Distributionrdquo Water ResourcesResearch vol 22 no 10 pp 1377ndash1384 1986

[15] I Nezu and W Rodi ldquoOpen-channel flow measurements witha laser doppler anemometerrdquo Journal of Hydraulic Engineeringvol 112 no 5 pp 335ndash355 1986

[16] M S Kirkgoz ldquoTurbulent velocity profiles for smooth andrough open channel flowrdquo Journal of Hydraulic Engineering vol115 no 11 pp 1543ndash1561 1989

[17] S-Q Yang and J A McCorquodale ldquoDetermination of bound-ary shear stress andReynolds shear stress in smooth rectangularchannel flowsrdquo Journal of Hydraulic Engineering vol 130 no 5pp 458ndash462 2004

[18] S-Q Yang S-K Tan and S-Y Lim ldquoVelocity distributionand dip-phenomenon in smooth uniform open channel flowsrdquoJournal of Hydraulic Engineering vol 130 no 12 pp 1179ndash11862004

[19] I Nezu and H E D Nakagawa Turbulence in Open-ChannelFlows Balkema Rotterdam The Netherlands 1993

[20] H Bonakdari F Larrarte L Lassabatere and C JoannisldquoTurbulent velocity profile in fully-developed open channelflowsrdquo Environmental Fluid Mechanics vol 8 no 1 pp 1ndash172008

[21] R Absi ldquoComments on ldquoturbulent velocity profile in fully-developed open channel flowsrdquordquo Environmental Fluid Mechan-ics vol 8 no 4 pp 389ndash394 2008

[22] J H Pu H Bonakdari L Lassabatere C Joannis and FLarrarte ldquoProfil de vitesses turbulent une nouvelle loi pour lescanaux etroitsrdquo La Houille Blanche no 3 pp 65ndash70 2010

[23] H Afzalimehr R Moghbel J Gallichand and J Sui ldquoInves-tigation of turbulence characteristics in channel with densevegetationrdquo International Journal of Sediment Research vol 26no 3 pp 269ndash282 2011

[24] M Cellino and U Lemmin ldquoInfluence of coherent flow struc-tures on the dynamics of suspended sediment transport inopen-channel flowrdquo Journal of Hydraulic Engineering vol 130no 11 pp 1077ndash1088 2004

[25] S-Q Yang ldquoConditionally averaged turbulent structures in 2Dchannel flowrdquo Proceedings of the Institution of Civil EngineersWater Management vol 163 no 2 pp 79ndash88 2010

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Journal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


01 Cardoso et al [10] obtained a Π value of minus0077 over asmooth bed Kironoto and Graf [11] found Π value in therange of minus008 to 015 for water flows over a gravel bedHowever the values of Π are established by experimenta-tion with large variations reported in the literatures Whilesignificant advances have been made by using Colersquos wakelaw the mechanism of Colersquos wake law (CWL-Law) and theassociated wake strength parameter are not fully understoodThese investigations often ascribe the large variations toeither smooth or rough boundaries and few researchers havetaken into account the velocity-dip phenomenon caused bysecondary currents [17] (Guo 2015)

Different from the positive deviation ofmeasured velocityfrom the log-lawrsquos prediction or the CWL-Law Yang et al [18]discussed the negative deviation of measured velocity fromthe log-lawrsquos prediction or the dip phenomenon in whichthe maximum longitudinal velocity occurs below the watersurface They suggested that Colersquos wake law is not able todescribe the entire velocity profile when the dip phenomenonexists Yang et al [18] proposed a dip-modified-log-law(DML-Law) based on the analysis of the Reynolds-averagedNavier-Stokes (RANS) equations This law involving twologarithmic distances one from the bed (ie the log-law)and the other from the free surface has the advantage that itcontains only one parameter 120572 for dip-correctionThe DML-Law reverts to the classical log-law for 120572 = 0 Yang et al[18] modified the log-law by adding a term to express the dipphenomenon instead of Colersquos wake law based on Reynoldsequations

119906119906lowast =

1120581 ln

1199101199100 +

120572120581 ln(1 minus 119910

ℎ) (4)

120572 = 13 exp(minus 1198872ℎ) (5)

Guo and Julien [12] proposed a modified-log-wake law(MLW-Law) which fits velocity profiles with a dip phe-nomenon However this law cannot be used for predictiveapplications since it requires fitting the near-free-surfacevelocities to the parabolic law to obtain dip position andmaximum velocity As Guo and Julien indicate the MLW-Law can be used only in flow measurements since it requiresmeasured velocities that is

119906119906lowast =

1120581 ln

1199101199100 +

2Π120581 sin21205871205892 minus 1205893

3120581 (6)

where 120589 = 119910120575 and 120575 is the distance from the bed to the pointwhere the velocity is maximum

Nezu and Nakagawa [19] found that in the outer regionthe mean velocity data deviate systematically from the log-law distribution at sufficiently high Reynolds numbers thisdeviation cannot be neglected in the free-surface region thatis 119910ℎ gt 06 It is thus worth investigating the underlyingmechanism of velocity deviation from the log-law Bonakdariet al [20] also proposed a new formulation of the verticalvelocity profile in open channel flows based on an analysisof the Navier-Stoke equations On the basis of this law and ofthe suggestions of Absi [21] a new analytical procedure waspresented by Pu et al [22]

With the advent of nonintrusive instrumentation visu-alisation techniques reveal that turbulence is dominated byforceful structures that are well organised and ordered eddieswith a certain lifespan now termed as coherent structuresCoherent structures can be placed into two categories burst-ing phenomena that occur in the wall region and large-scalevertical motion in the outer region For coherent structuralanalysis the method of quadrants has been developed basedon measured velocity fluctuations that is quadrant I 1199061015840 gt 0and V1015840 gt 0 quadrant II 1199061015840 lt 0 and V1015840 gt 0 quadrant III 1199061015840 lt 0and V1015840 lt 0 and quadrant IV 1199061015840 gt 0 and V1015840 lt 0 [23] Cellinoand Lemmin [24] pointed out that it would be meaningful toconcentrate on V1015840 gt 0 and V1015840 lt 0 in quadrants because themost significant feature of turbulence is fluctuations in thewall-normal direction They also developed an approach tocalculate conditionally averaged upwarddownward velocitybased on velocity fluctuations However the linkage of quad-rant analysis and upwarddownward events with velocitydeviation from the log-law has not been established

The objective of this research is to present a new analyticalapproach to determine the streamwise velocity profile andestablish a relationship between the log wake function andupdown events This leads to the present research aim todevelop a universal model to express the velocity profilein uniform flows Such a novel analytical approach will beused to explain that streamwise velocity profiles in the outerregion depart from the universal logarithmic law of thewall The proposed analytical approach and those previouslyproposed mathematical methods will be examined by usingexperimental data

2 Theoretical Consideration

The RANS momentum and continuity equations can bewritten for steady state flow as

119906(120597119906120597119909) + V(120597119906120597119910) + 119908(120597119906120597119911)

= 119892119878 + 120597120597119909 (minus11990610158401199061015840) + 120597

120597119910 (minus1199061015840V1015840) + 120597120597119911 (minus11990610158401199081015840)

+ ](12059721199061205971199092 +12059721199061205971199102 +

12059721199061205971199112)

(7)

120597119906120597119909 + 120597V

120597119910 + 120597119908120597119911 = 0 (8)

where 119906 V and119908 denote the mean velocity in the streamwise(119909) lateral (119910) and vertical (119911) directions respectively 119892 isthe gravitational acceleration 119878 is the energy slope andminus11990610158401199081015840are theReynolds stress tensor components For a uniform andfully developed flow (7) becomes

V(120597119906120597119910) + 119908(120597119906120597119911) = 119892119878 + 120597120597119910 (minus1199061015840V1015840) + 120597

120597119911 (minus11990610158401199081015840)

+ ](12059721199061205971199102 +12059721199061205971199112)

(9)



120597 (119906V minus 120591119909119910120588)120597119910 + 120597 (119906V minus 120591119909119911120588)120597119911 = 119892119878 (10)

120591119909119910 = 120583120597119906120597119910 minus 1205881199061015840V1015840 (11)

120591119909119911 = 120583120597119906120597119911 minus 12058811990610158401199081015840 (12)


120597 (119906V + 1199061015840V1015840 minus ] 119889119906119889119910)120597119910 = 119892119878 (13)




1199062lowast = 119892119878ℎ (15)


119889119906+119889119910+ minus 119906+V+ minus

1199061015840V10158401199062lowast = 1 minus 119910

ℎ (16)



119903119906 = sum1198791199060 Δ119905119906119879 (17)

119903119889 = sum1198791198890 Δ119905119889119879 (18)



119906119906 = 1119879119906 int119879119906

0119906119889119905 (19)

119906119889 = 1119879119889 int119879119889

0119889119889119905 (20)

V119906 = 1119879119906 int119879119906

0V119906119889119905 (21)

V119889 = 1119879119889 int119879119889

0V119889119889119905 (22)



minus1199061015840119889V10158401198891199062lowast119889

= (1 minus 119910ℎ) + 119906+119889V+119889

(23)




120581120585 (1 minus 120585) 119889119906+119906119889120585 = (1 minus 120585) + 119906+119906V+119906 (25)


119889119906+119906119889120585 = 1120581120585 +

119906+119906V+119906120581120585 (1 minus 120585) 119889119906+119889119889120585 = 1

120581120585 +119906+119889V+119889120581120585 (1 minus 120585)

(26)


119889119906+119889120585 = 119903119906 119906lowast119906119906lowast

119889119906+119906119889120585 + 119903119889 119906lowast119889119906lowast119889119906+119889119889120585 (27)


119889119906+119889120585 = 1



119906+119889119906+119906 )119906+119906 (28)





119889119906+119889120585 = 1

120581120585 + 119903119906V+119906120581120585 (1 minus 120585)


Here we denote


Then

119889119906+ = 1120581120585119889120585 +

119860120581120585 (1 minus 120585)119889120585 (31)


120581 ln 120585 + 119860120581 ln 120585 minus 119860

120581 ln (1 minus 120585) + 1198621 (32)



119906+ = 1120581 ln

1199101199100 +

119860120581 ln( 119910ℎ

1 minus 119910ℎ) (33)





Pump

LDA

Tank Tank Tank

Y

X

X

Z

Flume length












68

10121416182022242628303234


MLW-lawCML-law

001 002 003 004 005 006 007

u+

y


00112

14

16

18

20

22

24

26

28

30

32

34


MLW-lawCML-law

u+

y





MLW-lawCML-law

12

14

16

18

20

22

24

26

28

30

32

34

001 01

u+

y


5

10

15

20

25

30

35

40

u+

001 01y


MLW-lawCML-law




68

10121416182022242628303234363840

Measurements

u+


MLW-lawCML-law

001 01y


10

15

20

25

30

35

Measurements

u+

001 01y


MLW-lawCML-law






0909 0481 02950927 0496 0253







09

10

11


00 02 04 06 08 10

u+ du

+ u

yℎ

u+du+u = 103(yℎ)005

R2 = 09


00

05

10

15

20

25

30



0 1 2 3 4minus1

A

120573

R2 = 09543

y = 00574x2 + 04315x



5 Conclusions








Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






120597 (119906V minus 120591119909119910120588)120597119910 + 120597 (119906V minus 120591119909119911120588)120597119911 = 119892119878 (10)

120591119909119910 = 120583120597119906120597119910 minus 1205881199061015840V1015840 (11)

120591119909119911 = 120583120597119906120597119911 minus 12058811990610158401199081015840 (12)


120597 (119906V + 1199061015840V1015840 minus ] 119889119906119889119910)120597119910 = 119892119878 (13)




1199062lowast = 119892119878ℎ (15)


119889119906+119889119910+ minus 119906+V+ minus

1199061015840V10158401199062lowast = 1 minus 119910

ℎ (16)



119903119906 = sum1198791199060 Δ119905119906119879 (17)

119903119889 = sum1198791198890 Δ119905119889119879 (18)



119906119906 = 1119879119906 int119879119906

0119906119889119905 (19)

119906119889 = 1119879119889 int119879119889

0119889119889119905 (20)

V119906 = 1119879119906 int119879119906

0V119906119889119905 (21)

V119889 = 1119879119889 int119879119889

0V119889119889119905 (22)



minus1199061015840119889V10158401198891199062lowast119889

= (1 minus 119910ℎ) + 119906+119889V+119889

(23)




120581120585 (1 minus 120585) 119889119906+119906119889120585 = (1 minus 120585) + 119906+119906V+119906 (25)


119889119906+119906119889120585 = 1120581120585 +

119906+119906V+119906120581120585 (1 minus 120585) 119889119906+119889119889120585 = 1

120581120585 +119906+119889V+119889120581120585 (1 minus 120585)

(26)


119889119906+119889120585 = 119903119906 119906lowast119906119906lowast

119889119906+119906119889120585 + 119903119889 119906lowast119889119906lowast119889119906+119889119889120585 (27)


119889119906+119889120585 = 1



119906+119889119906+119906 )119906+119906 (28)





119889119906+119889120585 = 1

120581120585 + 119903119906V+119906120581120585 (1 minus 120585)


Here we denote


Then

119889119906+ = 1120581120585119889120585 +

119860120581120585 (1 minus 120585)119889120585 (31)


120581 ln 120585 + 119860120581 ln 120585 minus 119860

120581 ln (1 minus 120585) + 1198621 (32)



119906+ = 1120581 ln

1199101199100 +

119860120581 ln( 119910ℎ

1 minus 119910ℎ) (33)





Pump

LDA

Tank Tank Tank

Y

X

X

Z

Flume length












68

10121416182022242628303234


MLW-lawCML-law

001 002 003 004 005 006 007

u+

y


00112

14

16

18

20

22

24

26

28

30

32

34


MLW-lawCML-law

u+

y





MLW-lawCML-law

12

14

16

18

20

22

24

26

28

30

32

34

001 01

u+

y


5

10

15

20

25

30

35

40

u+

001 01y


MLW-lawCML-law




68

10121416182022242628303234363840

Measurements

u+


MLW-lawCML-law

001 01y


10

15

20

25

30

35

Measurements

u+

001 01y


MLW-lawCML-law






0909 0481 02950927 0496 0253







09

10

11


00 02 04 06 08 10

u+ du

+ u

yℎ

u+du+u = 103(yℎ)005

R2 = 09


00

05

10

15

20

25

30



0 1 2 3 4minus1

A

120573

R2 = 09543

y = 00574x2 + 04315x



5 Conclusions








Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of








119889119906+119889120585 = 1

120581120585 + 119903119906V+119906120581120585 (1 minus 120585)


Here we denote


Then

119889119906+ = 1120581120585119889120585 +

119860120581120585 (1 minus 120585)119889120585 (31)


120581 ln 120585 + 119860120581 ln 120585 minus 119860

120581 ln (1 minus 120585) + 1198621 (32)



119906+ = 1120581 ln

1199101199100 +

119860120581 ln( 119910ℎ

1 minus 119910ℎ) (33)





Pump

LDA

Tank Tank Tank

Y

X

X

Z

Flume length












68

10121416182022242628303234


MLW-lawCML-law

001 002 003 004 005 006 007

u+

y


00112

14

16

18

20

22

24

26

28

30

32

34


MLW-lawCML-law

u+

y





MLW-lawCML-law

12

14

16

18

20

22

24

26

28

30

32

34

001 01

u+

y


5

10

15

20

25

30

35

40

u+

001 01y


MLW-lawCML-law




68

10121416182022242628303234363840

Measurements

u+


MLW-lawCML-law

001 01y


10

15

20

25

30

35

Measurements

u+

001 01y


MLW-lawCML-law






0909 0481 02950927 0496 0253







09

10

11


00 02 04 06 08 10

u+ du

+ u

yℎ

u+du+u = 103(yℎ)005

R2 = 09


00

05

10

15

20

25

30



0 1 2 3 4minus1

A

120573

R2 = 09543

y = 00574x2 + 04315x



5 Conclusions








Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of











68

10121416182022242628303234


MLW-lawCML-law

001 002 003 004 005 006 007

u+

y


00112

14

16

18

20

22

24

26

28

30

32

34


MLW-lawCML-law

u+

y





MLW-lawCML-law

12

14

16

18

20

22

24

26

28

30

32

34

001 01

u+

y


5

10

15

20

25

30

35

40

u+

001 01y


MLW-lawCML-law




68

10121416182022242628303234363840

Measurements

u+


MLW-lawCML-law

001 01y


10

15

20

25

30

35

Measurements

u+

001 01y


MLW-lawCML-law






0909 0481 02950927 0496 0253







09

10

11


00 02 04 06 08 10

u+ du

+ u

yℎ

u+du+u = 103(yℎ)005

R2 = 09


00

05

10

15

20

25

30



0 1 2 3 4minus1

A

120573

R2 = 09543

y = 00574x2 + 04315x



5 Conclusions








Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






MLW-lawCML-law

12

14

16

18

20

22

24

26

28

30

32

34

001 01

u+

y


5

10

15

20

25

30

35

40

u+

001 01y


MLW-lawCML-law




68

10121416182022242628303234363840

Measurements

u+


MLW-lawCML-law

001 01y


10

15

20

25

30

35

Measurements

u+

001 01y


MLW-lawCML-law






0909 0481 02950927 0496 0253







09

10

11


00 02 04 06 08 10

u+ du

+ u

yℎ

u+du+u = 103(yℎ)005

R2 = 09


00

05

10

15

20

25

30



0 1 2 3 4minus1

A

120573

R2 = 09543

y = 00574x2 + 04315x



5 Conclusions








Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







0909 0481 02950927 0496 0253







09

10

11


00 02 04 06 08 10

u+ du

+ u

yℎ

u+du+u = 103(yℎ)005

R2 = 09


00

05

10

15

20

25

30



0 1 2 3 4minus1

A

120573

R2 = 09543

y = 00574x2 + 04315x



5 Conclusions








Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





09

10

11


00 02 04 06 08 10

u+ du

+ u

yℎ

u+du+u = 103(yℎ)005

R2 = 09


00

05

10

15

20

25

30



0 1 2 3 4minus1

A

120573

R2 = 09543

y = 00574x2 + 04315x



5 Conclusions








Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of











Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Documents

Investigation of Velocity Distribution in Open Channel ...downloads.hindawi.com › journals › mpe › 2017 › 1458591.pdf · ResearchArticle Investigation of Velocity Distribution