Research Article Extreme Velocity Fluctuations below Free Hydraulic …downloads.hindawi.com/journals/je/2013/678064.pdf · nition sketch of free hydraulic jump. uctuations on a ow

Hindawi Publishing CorporationJournal of EngineeringVolume 2013 Article ID 678064 5 pageshttpdxdoiorg1011552013678064

Research ArticleExtreme Velocity Fluctuations below Free Hydraulic Jumps

Rauacutel Antonio Lopardo

Instituto Nacional del Agua AU Ezeiza-Canuelas Km 165 1804 Ezeiza Argentina

Correspondence should be addressed to Raul Antonio Lopardo rlopardoinagobar

Received 5 December 2012 Accepted 20 May 2013

Academic Editor Michael Fairweather

Copyright copy 2013 Raul Antonio Lopardo This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The internal flow of hydraulic jump is essentially an unsteady flow subjected to macroturbulent random fluctuations and it wasnot known enough Then for the fluctuating motion interpretation the experimental research on the associated turbulence mustbe necessary The author developed in the past extensive laboratory research for the instantaneous pressure field determination bymeans of pressure transducers and new introductory experiments on velocity fluctuations by means of the ADV technique Theexperimental study of the instantaneous pressure field was based on the knowledge of several statistical parameters of amplitudesand frequencies as functions of the Froude number but for this paper the maximum instantaneous negative of pressure amplitudeson the floor is considered in order to estimate the extreme maximum positive velocities near the bottom A useful relationshipbetween turbulence intensity and the pressure fluctuation coefficient 1198621015840

119901was proposed from ADV velocity fluctuation for low

incident Froude numbers By means of this relationship the value 119906101584001 (instantaneous positive semiamplitude with 01 of

probability to be surpassed) can be considered for the determination of the turbulent extreme velocity near the bottom undera free hydraulic jump stilling basin with incident Froude number 3 lt 119865

1lt 6

1 Introduction

Many researchers have studied hydraulic jump but most ofthe works have focused on an integral analysis and little isknown about internal flow and turbulence characteristics Forphysical knowledge of the hydraulic jump on a horizontalstilling basin both the mean flow and the fluctuating motionare needed The mean flow analysis allows the determinationof the general pattern of the flow such as water levels thejump length the macroscopic quantification of energy lossand the mean pressure field

The internal flow of hydraulic jump is essentially anunsteady flow subjected to macroturbulent random fluctua-tions and it was not totally known enoughThe author devel-oped an extensive laboratory research for the instantaneouspressure field determination by means of pressure transduc-ers and introductory experiments on velocity fluctuations bymeans of the ADV technique

Early research on the turbulence characteristics of thehydraulic jump was carried out by Rouse et al [1] usinghot-wires techniques in an air model Acoustic instrumentsmethods were used in hydraulic jumps of low Froude number

by Liu et al [2] They used a microacoustic Doppler velocitymeter (ADV) and presented data ofmean velocity turbulenceintensities and Reynolds shear stresses along with energydissipation rate and Kolmogorov length scale calculations

Previous studies showed that ADVs are capable of report-ing accurate mean values of water velocity in three directions[3] even in low flow velocities In addition ADVs haveproved to yield a good description of turbulencewhen certainconditions are satisfied [4] These restrictions are related tothe instrument configuration (sampling frequency and noiseenergy level) and flow conditions (convective velocity andturbulence scales in the flow)

On the other hand former experimental studies on pres-sure fluctuations induced by hydraulic jump were publishedin English fifty years ago [5] and later by other authors [6 7]

Experimental research about pressure fluctuations belowhydraulic jumps is not concluded but the field of instan-taneous pressure can be measured without technologicalproblems For the measurement of instantaneous velocitiesthe excess of turbulence in the hydraulic jump and the airbubbles incorporation induced certain questions about theaccuracy of the instantaneous data obtained with ADV For

2 Journal of Engineering

this reason it was demonstrated [8] the possibility to usepressure fluctuations values to estimate turbulence intensitiesin the zone near the bed of a stilling basin

Even if there is available bibliography about the turbu-lence decay along the hydraulic jump length the experi-mental parameters were obtained by means of the standarddeviation of the velocity near the bottom and certainly theturbulence intensity is well represented but it is not possibleto know the real extreme instantaneous maximum velocitynear the bottom

This paper deals with experimental data of positive peakamplitudes of the longitudinal velocity component in macro-turbulent flow induced by free hydraulic jumps downstreamsluice gates Amplitudes with 01 to occurrence probabilityof be surpassed with more positive amplitudes were used as amanner of giving a statistic value to those peaks

2 Experimental Setup andExperimental Method

Free hydraulic jumps were generated downstream a sluicegate in a rectangular horizontal flume with both acrylicbottom and sidewalls 065m wide 100m deep and 1200mlong The tailwater depth was controlled by a vertical tailgatelocated at the end of the flume Two experiments wereperformed and Figure 1 includes the main details of theseexperiments All experiments were conducted with a gateopening equal to 007m A usual gage was used to measurethe subcritical water depth 119911

2located 6m downstream from

the sluice gateThemean velocity119880

1at location 119911

1was varied in order to

obtain two different incident Froude numbers 1198651= 303 and

1198651= 488 The incident Reynolds number Re

1= 1198801sdot 1199111]

(where ] is the cinematic viscosity) was in the order of Re1=

105The random pressure components were measured with

bidirectional pressure transducers with fully active strain-gage bridge Data were processed and analyzed in terms ofdiscrete sample valuesThe sampling interval in time adoptedwas 001 seconds according to the sampling theorem forrandom recordsThe number of samples taken at every pointwas 16384 (32 blocks of 512 values each one) The experi-mental methodology for measurement and data analysis ofpressure fluctuations below free hydraulic jumps was basedon a previous research [9]

3 Pressure Fluctuations andTurbulence Intensity

The intensity of isotropic turbulence in open channel flows isusually expressed by the relationship between the root meansquare of the velocity fluctuation and themean velocity in thepoint considered

119868 =

radic11990610158402

119880

(1)

On the other hand for structural purposes the turbulenceintensity can also be defined as a function of pressure

Q

z1

z2

U1

U2

s

z

xo = 0xo

Figure 1 Definition sketch of free hydraulic jump

fluctuations on a flow boundary usually [7] by means of thenondimensional parameter 1198621015840

119901

1198621015840

119901=

radic11990110158402

120588 (1198802

12) (2)

By means of the dimensional analysis for a free steadyhydraulic jump it is possible to obtain the following param-eters for turbulence intensity at the level 119911 = 1 cm (due toADV accuracy restriction) and pressure fluctuations on thebed (119911 = 0) respectively

radic11990610158402

1198801

= Φ1[119909

(1199112minus 1199111) 1198651] (3)

1198621015840

119901= Φ2[119909

(1199112minus 1199111) 1198651] (4)

The pressure fluctuation amplitude in a turbulent flow canto be obtained from the instantaneous velocity field by theseparation of themean and fluctuating variables in the knownPoisson equation [10] If the turbulence can be considered ashomogeneous and isotropic it can be expressed

radic11990110158402 = 12057212058811990610158402 (5)

where 120588 is the fluid density and 120572 is a nondimensionalcoefficient [11]

Taking into account that the mean velocity 119880 in eachpoint of measurement is connected with the mean velocitiesin the jump incident section 119880

1and in the downstream

section 1198802 it is possible to propose the following functions

for calculations

11990610158402

1198801

= 120582radic1198621015840119901 (6)

11990610158402

1198802

= (120582

2)(radic1 + 8119865

2

1minus 1)radic1198621015840

119901 (7)

In order to have a good agreement between turbulence in-tensity direct measurements with calculated values frompressure fluctuations determinations for two different inci-dent Froude numbers (119865

1= 3 y119865

1= 5) the nondimensional

coefficient proposed is 120582 = 06 [8]

Journal of Engineering 3

0002004006008

01012014016018

0 2 4 6 8 10 12 14 16 18 20

F1 = 3

F1 = 4

F1 = 5

F1 = 6

F1 = 3

radic u9984002

U1

x(z2 minus z1)

Figure 2 Turbulence intensity along free hydraulic jump

4 Experimental Results

The turbulence intensity determination as defined in (6) (bydirect velocity measurements or through indirect measure-ments by pressure fluctuations) represents the ratio betweenmean square values of velocity fluctuation and the incidentmean velocity of the hydraulic jump Figure 2 where the ver-

tical axis isradic119906101584021198801and the abscissa is 119909(119911

2minus1199111) presented

in a previous publication [8] allows appreciation some resultsreferring to (3) for free hydraulic jumps The experimentaldatawas exposed as a function of the incident Froude number1198651 The curve was plotted for the lower incident Froude

number tested 1198651= 3 The length usually used to define the

length of stilling basin protection horizontal is119909(1199112minus1199111) = 6

but it is observed that in this point the rms value of velocityfluctuation reaches about 10 of the mean incident velocity1198801As noted in the case of pressure fluctuations below

hydraulic jump stilling basins the maximum positive pres-sure fluctuating amplitude on the floor of a hydraulic jump ismuch greater than the mean square value which is usuallyconsidered for conventional analysis of turbulence decayFor practically all authors it is a suitable parameter for thispurpose

However when it is desired to know the maximumvalue of instantaneous velocities on the bed for example todesign rock protections downstream stilling basins it is morerelevant to take into consideration the maximum positiveamplitude of velocity fluctuation as the value with 01of probability to be overtaken by another registered value(1199061015840max) assuming that this is the order of magnitude for themaximum detectable instantaneous velocity on experimentalmeasurements

Figures 3 and 4 show the variation of the parameter1199061015840

max1198801 as a function of 119909(1199112minus 1199111) for the four incident

Froude numbers considered It can be seen that for anyincident Froude number extreme amplitudes 1199061015840max1198801 havea peak around the abscissa 119909(119911

2minus 1199111) = 3 and the values

0

005

01

015

02

025

03

035

0 2 4 6 8 10 12 14

u998400 m

axU

1

F1 = 6

F1 = 4

F1 = 5

x(z2 minus z1)

Figure 3Maximum velocities below hydraulic jumps near the floorfor 1198651ge 4

0

005

01

015

02

025

03

035

0 2 4 6 8 10 12 14

u998400 m

axU

1

F1 = 4

F1 = 3

x(z2 minus z1)

Figure 4Maximum velocities below hydraulic jumps near the floorfor 1198651le 4

growwith the incident Froude numberThe above conclusionis clearly stated in Figure 5 which also shows that the peaksoccur in the vicinity of 119909(119911

2minus 1199111) = 3 For 119909(119911

2minus 1199111) le

4 the maximum amplitudes occur for 1198651asymp 4 while for

119909(1199112minus1199111) gt 4 amplitudes decrease with the value of 119865

1 This

assumption is clearly stated in Figure 4 which shows that theconclusion is only true for 119865

1= 3 in the upstream region

because from 119909(1199112minus 1199111) asymp 5 the maximum amplitudes data

exceed the values for 1198651= 4 This abscissa where a change

of curvature of the lines plotted can be seen is closed withthe value 119909(119911

2minus 1199111) asymp 497 which was demonstrated as the

point for the boundary layer separation on the floor below ahydraulic jump [12]

Briefly it is interesting that the maximum positive veloc-ity amplitudes measured for the range 3 lt 119865

1lt 6 are in the

order of 25 of the incident velocityTaking into account that always the maximum velocities

needed to scour process calculations are those near thebottom they are considered in the present analysis located


0

005

01

015

02

025

03

2 25 3 35 4 45 5 55 6 65F1

u998400 m

axU

1

x(z2 minus z1) = 1

x(z2 minus z1) = 2

x(z2 minus z1) = 3

x(z2 minus z1) = 4

x(z2 minus z1) = 5

x(z2 minus z1) = 6

x(z2 minus z1) = 7

x(z2 minus z1) = 8

Figure 5Maximum velocities below hydraulic jumps near the flooras a function of 119865

1

002040608

112141618

2

0 2 4 6 8 10 12 14

u998400U

2

F1 = 4

Lipay and Pustovoit [13]rms

x(z2 minus z1)

Figure 6 Extreme velocity near the bottom in free hydraulic jumpfor 1198651= 4

in the downstream zone of the hydraulic jump where erosionproblems would be present Then from the first historicbibliographic reference detected [13] the extreme velocityfluctuation near the bottom was related to the downstreamjump velocity 119880

2 theoretically calculated by the application

of the Belanger equationDue to the previous considerations Figure 6 shows the

experimental results of 1199061015840max1198802 as a function of 119909(1199112minus 1199111)

for the incident Froude number condition 1198651= 4 The figure

allows to observe that themaximum instantaneous amplitudefor the ldquoclassical lengthrdquo of the hydraulic jump 119909(119911

2minus1199111) = 6

is 12 times themean downstream velocity1198802 which explains

the important local scour always detected downstream thestilling basins concrete protection

Historical values obtained by Lipay and Pustovoit areclearly below those proposed in the present research andreasonably near to the authorrsquos values calculated from the rootmean square amplitudes perhaps because these are the valuesusually measured from instantaneous velocity measurementsbelowhydraulic jumps Recently other authors [14] publishedADV observations on the turbulence in the transition regionfrom the end of jump to open channel flow for 119865

1= 4 and

1198651= 7 showing Reynolds normal stresses and shear stress

also useful in assessing the bed erodibility downstream thehydraulic jump stilling basins

5 Conclusion

Turbulence intensity is defined as the relationship betweenthe rms velocity fluctuation 11990610158402 and the mean velocity ofthe considered point 119880 For design purposes as the meanvalue is not known at any point of the hydraulic jump theuse of either the upstream velocity 119880

1or the downstream

velocity 1198802is needed On the other hand the maximum

positive amplitude of velocity fluctuation can be definedas the positive amplitude with 01 of probability to beovertaken by another registered value (1199061015840max) This value canbe obtained just over the floor from pressure fluctuationexperimental tests by means of a useful relationship betweenturbulence intensity and the pressure fluctuation coefficient1198621015840

119901Following previous experimental conclusions all the tests

for velocities and pressure were accomplished for incidentReynolds number119877

1up to 100000 and sluice gate openings

larger than 3 cm and for incident Froude numbers between1198651= 3 and 119865

1= 6 The magnitude of instantaneous actions is

analysed in order for example to estimate the representativedimensions of rip-rap for fluvial beds protection downstreamthe hydraulic jump stilling basins

References

[1] H Rouse T T Siao and S Nagaratnam ldquoTurbulence char-acteristics of the hydraulic jumprdquo Journal of TransportationEngineering vol 124 pp 926ndash950 1959

[2] M Liu N Rajaratnam and D Z Zhu ldquoTurbulence structure ofhydraulic jumps of low Froude numbersrdquo Journal of HydraulicEngineering vol 130 no 6 pp 511ndash520 2004

[3] N C Kraus A Lohrmann and R Cabrera ldquoNew acousticmeter for measuring 3D laboratory flowsrdquo Journal of HydraulicEngineering vol 120 no 3 pp 406ndash412 1994

[4] C M Garcıa M Cantero Y Nino and M H Garcıa ldquoTur-bulence measurements with acoustic doppler velocimetersrdquoJournal of Hydraulic Engineering vol 131 no 12 pp 1062ndash10732005

[5] N A Preobrazhenski ldquoLaboratory and field investigation offlow pressure pulsation and vibration of large damsrdquo in Proceed-ings of the 7th CongresoMundial de Grandes Presas (ICOLD rsquo58)1958

[6] V O Vasiliev and B V Bukreyev ldquoStatistical characteristics ofhydraulic jumprdquo in Proceedings of the 12th IAHR Congress vol2 pp 1ndash8 Fort Collins Colo USA 1967


[7] R A Lopardo J C de Lio and G F Vernet ldquoPhysical modelingon cavitation tendency formacroturbulence of hydraulic jumprdquoin Proceedings of the International Conference on the HydraulicModeling of Civil Engineering Structures pp 109ndash121 BHRACoventry UK 1982

[8] R A Lopardo and M Romagnoli ldquoPressure and velocity fluc-tuations in stilling basinsrdquo in Advances in Water Resources ampHydraulic Engineering C Zhang and H Tang Eds vol 6 pp978ndash973 Springer 2009

[9] R A Lopardo and R E Henning ldquoExperimental advances onpressure fluctuations beneath hydraulic jumpsrdquo in Proceedingsof the 21st IAHR Congress vol 3 pp 633ndash638 MelbourneAustralia 1985

[10] A Favre L S G Kovasnay R Dumas J Gaviglio and MCoantic La Turbulence en Mecanique des Fluides Gauthiers-Villars Paris France 1976

[11] G K Batchelor ldquoPressure fluctuations in isotropic turbulencerdquoMathematical Proceedings of the Cambridge Philosophical Soci-ety vol 47 no 2 pp 359ndash374 1951

[12] R A Lopardo and J M Casado ldquoBoundary layer separationbeneath submerged jump flowsrdquo in Proceedings of the 32ndIAHR Congress vol 2 p 584 Venecia Italy 2007

[13] I E Lipay and V G Pustovoit ldquoOn the vanishing of intensivmacroturbulence in open channel below hydraulic structurerdquoin Proceedings of the 12th IAHR Congress vol 2 pp 362ndash369Fort Collins Colo USA 1967

[14] A T M H Zobeyer N Jahan Z Islam G Singh and NRajaratnam ldquoTurbulence characteristics of the transition regionfromhydraulic jump to open channel flowrdquo Journal of HydraulicResearch vol 48 no 3 pp 395ndash399 2010

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Navigation and Observation



DistributedSensor Networks



this reason it was demonstrated [8] the possibility to usepressure fluctuations values to estimate turbulence intensitiesin the zone near the bed of a stilling basin

Even if there is available bibliography about the turbu-lence decay along the hydraulic jump length the experi-mental parameters were obtained by means of the standarddeviation of the velocity near the bottom and certainly theturbulence intensity is well represented but it is not possibleto know the real extreme instantaneous maximum velocitynear the bottom

This paper deals with experimental data of positive peakamplitudes of the longitudinal velocity component in macro-turbulent flow induced by free hydraulic jumps downstreamsluice gates Amplitudes with 01 to occurrence probabilityof be surpassed with more positive amplitudes were used as amanner of giving a statistic value to those peaks

2 Experimental Setup andExperimental Method

Free hydraulic jumps were generated downstream a sluicegate in a rectangular horizontal flume with both acrylicbottom and sidewalls 065m wide 100m deep and 1200mlong The tailwater depth was controlled by a vertical tailgatelocated at the end of the flume Two experiments wereperformed and Figure 1 includes the main details of theseexperiments All experiments were conducted with a gateopening equal to 007m A usual gage was used to measurethe subcritical water depth 119911

2located 6m downstream from

the sluice gateThemean velocity119880

1at location 119911

1was varied in order to

obtain two different incident Froude numbers 1198651= 303 and

1198651= 488 The incident Reynolds number Re

1= 1198801sdot 1199111]

(where ] is the cinematic viscosity) was in the order of Re1=

105The random pressure components were measured with

bidirectional pressure transducers with fully active strain-gage bridge Data were processed and analyzed in terms ofdiscrete sample valuesThe sampling interval in time adoptedwas 001 seconds according to the sampling theorem forrandom recordsThe number of samples taken at every pointwas 16384 (32 blocks of 512 values each one) The experi-mental methodology for measurement and data analysis ofpressure fluctuations below free hydraulic jumps was basedon a previous research [9]

3 Pressure Fluctuations andTurbulence Intensity

The intensity of isotropic turbulence in open channel flows isusually expressed by the relationship between the root meansquare of the velocity fluctuation and themean velocity in thepoint considered

119868 =

radic11990610158402

119880

(1)

On the other hand for structural purposes the turbulenceintensity can also be defined as a function of pressure

Q

z1

z2

U1

U2

s

z

xo = 0xo

Figure 1 Definition sketch of free hydraulic jump

fluctuations on a flow boundary usually [7] by means of thenondimensional parameter 1198621015840

119901

1198621015840

119901=

radic11990110158402

120588 (1198802

12) (2)

By means of the dimensional analysis for a free steadyhydraulic jump it is possible to obtain the following param-eters for turbulence intensity at the level 119911 = 1 cm (due toADV accuracy restriction) and pressure fluctuations on thebed (119911 = 0) respectively

radic11990610158402

1198801

= Φ1[119909

(1199112minus 1199111) 1198651] (3)

1198621015840

119901= Φ2[119909

(1199112minus 1199111) 1198651] (4)

The pressure fluctuation amplitude in a turbulent flow canto be obtained from the instantaneous velocity field by theseparation of themean and fluctuating variables in the knownPoisson equation [10] If the turbulence can be considered ashomogeneous and isotropic it can be expressed

radic11990110158402 = 12057212058811990610158402 (5)

where 120588 is the fluid density and 120572 is a nondimensionalcoefficient [11]

Taking into account that the mean velocity 119880 in eachpoint of measurement is connected with the mean velocitiesin the jump incident section 119880

1and in the downstream

section 1198802 it is possible to propose the following functions

for calculations

11990610158402

1198801

= 120582radic1198621015840119901 (6)

11990610158402

1198802

= (120582

2)(radic1 + 8119865

2

1minus 1)radic1198621015840

119901 (7)

In order to have a good agreement between turbulence in-tensity direct measurements with calculated values frompressure fluctuations determinations for two different inci-dent Froude numbers (119865

1= 3 y119865

1= 5) the nondimensional

coefficient proposed is 120582 = 06 [8]


0002004006008

01012014016018

0 2 4 6 8 10 12 14 16 18 20

F1 = 3

F1 = 4

F1 = 5

F1 = 6

F1 = 3

radic u9984002

U1

x(z2 minus z1)
















0

005

01

015

02

025

03

035

0 2 4 6 8 10 12 14

u998400 m

axU

1

F1 = 6

F1 = 4

F1 = 5

x(z2 minus z1)


0

005

01

015

02

025

03

035

0 2 4 6 8 10 12 14

u998400 m

axU

1

F1 = 4

F1 = 3

x(z2 minus z1)



2minus 1199111) = 3 For 119909(119911

2minus 1199111) le



1 This









1lt 6 are in the




0

005

01

015

02

025

03

2 25 3 35 4 45 5 55 6 65F1

u998400 m

axU

1

x(z2 minus z1) = 1

x(z2 minus z1) = 2

x(z2 minus z1) = 3

x(z2 minus z1) = 4

x(z2 minus z1) = 5

x(z2 minus z1) = 6

x(z2 minus z1) = 7

x(z2 minus z1) = 8


1

002040608

112141618

2

0 2 4 6 8 10 12 14

u998400U

2

F1 = 4


x(z2 minus z1)








2minus1199111) = 6




1= 4 and



5 Conclusion


1or the downstream









References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0002004006008

01012014016018

0 2 4 6 8 10 12 14 16 18 20

F1 = 3

F1 = 4

F1 = 5

F1 = 6

F1 = 3

radic u9984002

U1

x(z2 minus z1)
















0

005

01

015

02

025

03

035

0 2 4 6 8 10 12 14

u998400 m

axU

1

F1 = 6

F1 = 4

F1 = 5

x(z2 minus z1)


0

005

01

015

02

025

03

035

0 2 4 6 8 10 12 14

u998400 m

axU

1

F1 = 4

F1 = 3

x(z2 minus z1)



2minus 1199111) = 3 For 119909(119911

2minus 1199111) le



1 This









1lt 6 are in the




0

005

01

015

02

025

03

2 25 3 35 4 45 5 55 6 65F1

u998400 m

axU

1

x(z2 minus z1) = 1

x(z2 minus z1) = 2

x(z2 minus z1) = 3

x(z2 minus z1) = 4

x(z2 minus z1) = 5

x(z2 minus z1) = 6

x(z2 minus z1) = 7

x(z2 minus z1) = 8


1

002040608

112141618

2

0 2 4 6 8 10 12 14

u998400U

2

F1 = 4


x(z2 minus z1)








2minus1199111) = 6




1= 4 and



5 Conclusion


1or the downstream









References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

005

01

015

02

025

03

2 25 3 35 4 45 5 55 6 65F1

u998400 m

axU

1

x(z2 minus z1) = 1

x(z2 minus z1) = 2

x(z2 minus z1) = 3

x(z2 minus z1) = 4

x(z2 minus z1) = 5

x(z2 minus z1) = 6

x(z2 minus z1) = 7

x(z2 minus z1) = 8


1

002040608

112141618

2

0 2 4 6 8 10 12 14

u998400U

2

F1 = 4


x(z2 minus z1)








2minus1199111) = 6




1= 4 and



5 Conclusion


1or the downstream









References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article Extreme Velocity Fluctuations below Free Hydraulic …downloads.hindawi.com/journals/je/2013/678064.pdf · nition sketch of free hydraulic jump. uctuations on a ow