Flow Measurement and Instrumentation 32 (2013) 107–110

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Flow Measurement and Instrumentation

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journal homepage: www.elsevier.com/locate/flowmeasinst

Similarity of the velocity profile in geometrically distorted flow model

Gensheng Zhao a,b,(n), Paul J. Visser b, Jinyou Lu c, Johannes K. Vrijling b

a College of Water Conservancy and Hydropower Engineering, Hohai University, Chinab Department of Hydraulic Engineering, Faculty of Civil Engineering and Geosciences, Delft University of Technology, the Netherlandsc Changjiang River Scientific Research Institute, China

a r t i c l e i n f o

Article history:Received 6 July 2012Received in revised form21 March 2013Accepted 21 April 2013Available online 3 May 2013

Keywords:SimilarityProfile of vertical distributionGeometrically distorted physical models1-Point method

86/$ - see front matter & 2013 Elsevier Ltd. Ax.doi.org/10.1016/j.flowmeasinst.2013.04.005

responding author at: Department of Hydrgineering and Geosciences, Delft Universityrlands. Tel.: +31 152783426.ail address: g.zhao@tudelft.nl (G. Zhao).

a b s t r a c t

A method is presented to measure the depth-averaged horizontal velocity in distorted physical models,i.e., the depth-averaged horizontal velocity can be equivalent to a point velocity measured atapproximately 0.6h below the water surface (h is the water depth). Further a scale for the Chézycoefficient in distorted models is deduced, and the velocity distributions in the vertical in prototype anddistorted models are analyzed. By comparing data of the vertical variation of the horizontal velocitymeasured in a physical distorted model (with distortion ratio of 4.0) with prototype data, inaccuracies viathe one-point method, the three-point method and the five-point method are analyzed. It is concludedthat it is more accurate to measure the depth-averaged velocity via the three-point method than via theone-point method in the distorted model. What's more, the point velocity at 0.6h below the watersurface approximates the depth-averaged velocity in the distorted physical model and the one-pointmethod is proposed to be used in practice.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

In distorted physical models, there are deviations not only inthe geometric similitude but also in the kinematic and thedynamic similitude [1]. Usually physical models of rivers are setup with the Froude similitude. According to the similarity princi-ple, the dissimilarities of the vertical geometrics in distortedmodels affect the similarities of the hydraulic factors in the verticaldirection.

The Chézy coefficient is a very important factor in hydraulicsand river mechanics [2,3], so also in physical models. According tothe principle of similitude for physical models, the scale of theChézy coefficient in undistorted models must be equal to unity.However, that is not the case in distorted models. There the scaleof the Chézy coefficient depends on the distortion ratio, the waterdepth and the bed roughness.

The depth-averaged horizontal velocity is measured in physicalmodels and in prototype via 3-point and 5-point methods, i.e. thepoint velocities at the relative depths of 0, 0.2, 0.6, 0.8 and 1.0 or(0.2, 0.6, 0.8) are used to obtain the depth-averaged velocities[4,5]. Sometimes the 1-point method is used to measure thedepth-averaged velocity as a matter of convenience. This paperfocuses on the issue whether the 1-point method can be used tomeasure the depth-averaged velocity in distorted models.

ll rights reserved.

aulic Engineering, Faculty ofof Technology, 2628 CN, the

2. Depth-averaged velocity in a distorted model and inprototype

A flow in a river is almost always in the rough-turbulentregime [3], so the flow in physical models must be in the sameregime. Sometimes the models are geometrically distorted in order tomeet the condition of rough-turbulent regime.

According to Prandtl [3], the horizontal velocity in water-courses is logarithmically distributed in the vertical

uuðnÞ

¼ 1κln

yks

� �þ Bs ð1Þ

where u is the point velocity, κ is the Von Karman constant, uðnÞ isthe bed shear stress velocity, y is the distance from the river-bed,ks is the height of the bed roughness and Bs is a function of theReynolds number ReðnÞ, Bs is equal to 8.5 when the flow is in therough-turbulent regime. Then Eq. (1) can be written as

uuðnÞ

¼ 1κln As

yks

� �ð2Þ

where As[3] is equal to eκBs .In practice the depth-averaged velocity is the main hydraulic

parameter in the flow. And the depth-averaged velocity can becalculated by integration and expressed as follows:

v¼ 1h

Z h

0udy¼ 1

h

Z h

0

1κuðnÞln As

yks

� �¼ 1

κuðnÞ ln As

hks

� �−1

� �ð3Þ

where h is the water depth, and v is the depth-averaged velocity,others have been stated above.

Nomenclature

u point velocityκ Von Karman constantuðnÞ bed shear stress velocityy distance from the river-bedks height of the bed roughnessReðnÞ Reynolds numberBs function of ReðnÞe natural logarithm numberAs eκBs

h water depthv depth averaged velocityy′=h¼ ðh−yÞ=h relative water depthλv velocity scaleupð0:6hÞ point velocity at the relative water depth y′=h¼ 0:6 in

the prototypeumð0:6hÞ point velocity at the relative water depth y′=h¼ 0:6 in

the model

umax velocity at the free surface of flowξ¼ v=uðnÞ dimensionless friction factorλξ scale of dimensionless friction coefficient ξC Chézy coefficientCm Chézy coefficient in the flow modelCp Chézy coefficient in the prototypehm water depth in the flow modelhp water depth in the prototypeksm height of bed roughness in the flow modelksp height of bed roughness in the prototypeBsm function of Reynolds number in the flow modelBsp function of Reynolds number in the prototype.λh water depth scaleλks scale of the height of bed roughnessg gravitational accelerationλC Chézy coefficient scaleλx horizontal scaleλy vertical scaleη distortion ratio, equal toλy=λx

G. Zhao et al. / Flow Measurement and Instrumentation 32 (2013) 107–110108

If the depth-averaged velocity is set to equal some pointvelocity at the same vertical line, then

1κuðnÞ ln As

hks

� �−1

� �¼ 1

κuðnÞln As

yks

� �ð4Þ

From Eq. (4), y¼ h=e is obtained, where e is the natural logarithmnumber. Hence, the velocity at y¼ h=e is equal to the depth-averaged velocity. In practice, the relative water depthy′=h¼ ðh−yÞ=h is often used. When the relative water depth isy′=h¼ ð1−1=eÞ≈0:6 , the velocity equals to the depth-averagedvelocity.

If the flow is in the rough-turbulent regime, the velocity aty′=h¼ 0:6 can represent the depth-averaged velocity, both inphysical models (distorted/normal) and in prototype. So thedepth-averaged velocity in prototype can be expressed by themodel velocity as

vp ¼ upð0:6hÞ ¼1λv

vm ¼ 1λv

umð0:6hÞ ð5Þ

where λv is the velocity scale, upð0:6hÞ is the point velocity at therelative water depth y′=h¼ 0:6 in the prototype, umð0:6hÞ is thepoint velocity at the relative water depth y′=h¼ 0:6 in the model.

So the depth-averaged velocity in the prototype can beexpressed with the velocity umð0:6hÞ and the scale of the depth-averaged velocity, regardless the model is undistorted or distorted.

3. Similitude of Chézy coefficient

The depth-averaged velocity v is related to the maximumvelocity umax at the free surface of the two-dimensional flow by [1]

vuðnÞ

¼ umax

uðnÞ−2:5 ð6Þ

where umax is the velocity at the free surface.

umax

uðnÞ¼ 1

κln

hks

þ Bs ð7Þ

Substitution of (7) into (6) gives

vuðnÞ

¼ 1κln

hks

þ Bs−2:5 ð8Þ

or when ξ¼ v=uðnÞ is the dimensionless friction factor [1]

ξ¼ 1κln

hks

þ Bs−2:5 ð9Þ

Then the scale λξ of the dimensionless friction coefficient ξ can beexpressed by

λξ ¼Cm

Cp¼

1κ ln

hmksm

þ Bsm−2:51κ ln

hpksp

þ Bsp−2:5¼

1κ ln

hpksp

λhλks

þ Bsm−2:5

1κ ln

hpksp

þ Bsp−2:5ð10Þ

where Cm is the Chézy coefficient in the flow model, Cp is theChézy coefficient in the prototype, hm is the water depth in theflow model, hp is the water depth in the prototype, ksm is theheight of bed roughness in the flow model, ksp is the height of bedroughness in the prototype, Bsm is the function of Reynolds numberin the flow model, and Bsp is the function of Reynolds number inthe prototype.

If Reynolds number ReðnÞ > 70, Bsm ¼ Bsp ¼ 8:5 [1], κ¼ const≈0:4then

λξ ¼ð1=0:4Þlnðhp=ksp Þðλh=λks Þ þ 6

ð1=0:4Þlnðhp=ksp Þ þ 6¼ lnðhp=ksp Þðλh=λks Þ þ 2:4

lnðhp=ksp Þ þ 2:4¼ 1

þ lnðλh=λks Þlnð0:875ðhp=ksp ÞÞ

ð11Þ

where λh is the water depth scale, λks is the scale of the height ofbed roughness, others have been stated above.

The relationship between the Chézy coefficient and the dimen-sionless coefficient [1] is

C ¼ ξffiffiffig

p ð12Þ

where C is the Chézy coefficient and g the gravitationalacceleration.

So the Chézy coefficient scale λC can be also expressed by

λC ¼ 1þ lnðλh=λks Þlnð0:875ðhp=ksp ÞÞ

ð13Þ

In the flow model, Chézy coefficient scale λC is related with waterdepth scale, height of bed roughness scale, water depth in theprototype as well as the height of bed roughness in the prototype.

G. Zhao et al. / Flow Measurement and Instrumentation 32 (2013) 107–110 109

In the distorted model, as λh ¼ λy; λks ¼ λ4y=λ3x then,

λC ¼ 1þ lnð1=ηÞ3lnð0:875ðhp=ksp ÞÞ

ð14Þ

where λx is horizontal scale, λy is vertical scale, η is the distortionratio that is equal to λy=λx.

In the distorted model, the Chézy coefficient scale λC is notequal to unity but expressed by Eq. (14). So λC is related to thedistortion ratio η, the water depth in the prototype hp and theroughness height ks. When η is equal to 1, i.e. the model isundistorted, λC is 1: the Chézy coefficient is the same in the modeland in the prototype. However, when the distortion ratio is notequal to 1, the Chézy coefficient is different in the model and in theprototype. What's more, λC varies with hp and ks.

0

0.2

0.4

0.6

0.8

11 1.5 2 2.5 3

(h-y

)/h

Jing 78# in PrototypeJing 78# in ModelJing 66# in PrototypeJing 66# in ModelJing 53# in PrototypeJing 53# in Model

4. Case study of the vertical distribution of horizontal velocity

In order to reveal the differences in vertical distribution of thehorizontal velocity in prototype and in the distorted model, threerepresentative cross-sections in the Yangtze River [4,6,7] arechosen, i.e. Jing53, Jing66 and Jing78. The model is geometricallydistorted with a distortion ratio of 4. The discharges in these cross-sections are 39,000 m3/s and 19,600 m3/s, respectively. All thevelocities were measured with 5-point method both in the modeland prototype.

The vertical distributions of the horizontal velocities in thedistorted model are different from the ones in prototype, as can beseen from Figs. 1 and 2. The point velocities near the river-bed areless than the ones in the prototype; however, the point velocitiesnear the free surface in the distorted model are larger than thosein prototype with an exception that the velocities in the prototypeand model are more or less the same. The deviations between thepoint velocities in the model and those in prototype decrease fromthe free surface to approximately the relative depth of 0.6 andincrease all the way to the bottom. At a relative depth ofapproximately 0.6, the point velocities in prototype are equal tothe point velocities in the distorted model, taking into account thevelocity scale of the depth-averaged velocity.

The data analysis is consistent with Eq. (5). The depth-averagedvelocity can be obtained by measuring the point velocity at therelative water depth y′=h¼ ð1−1=eÞ≈0:6. So the depth-averagedvelocity in prototype can be obtained by measuring the pointvelocity at the relative water depth y′=h¼ ð1−1=eÞ≈0:6 in thedistorted model, taking into account the model scale of thedepth-averaged velocity.

0

0.2

0.4

Velocity (m/s)

0.6

0.5 1 1.5

0.8

(h-y

)/h

Jing 78# in Prototype

Jing 78# in Model

Jing 66# in Prototype

Jing 66# in Model

Jing 53# in Prototype

Jing 53# in Model

Fig. 1. Comparison of vertical variation of velocity in prototype and model ofYangtze River with a discharge of 39,000 m3/s.

5. Comparison of the methods for measuring the depth-averaged velocity in a distorted model

Generally speaking, the approach of measuring the depth-averaged velocity with the 5-point method can be considered togive a good approximation of the depth-averaged velocity [8], bothin the model and in prototype. However, usually the 5-pointmeasurement method is not used, whereas the 3-point methodand 1-point method are quite often applied.

In the representative cross-sections in the distorted model withdistortion ratio of 4, the depth-averaged velocities are measuredvia 5-point method, 3-point method and 1-point method, asshown in Table 1 [6]. It is assumed that the depth-averagedvelocity measured by 5-point method will give the actual depth-averaged velocity in the distorted model. Then the maximumrelative deviation of the depth-averaged velocities measured by 3-point method is 3.50, whereas the maximum relative deviation ofthe depth-averaged velocities measured by 1-point methodis 5.66%.

Therefore, the 1-point method could replace both the 5-pointmethod and 3-point method for the measurements in the dis-torted model. Furthermore, the velocity measured by the 1-pointmethod in the distorted model can be used to approximate thedepth-averaged velocity in prototype, that is to say, to the extentpermitted by the experimental accuracy.

Velocity (m/s)

Fig. 2. Comparison of vertical variation of velocity in prototype and model ofYangtze River with a discharge of 19,600 m3/s.

Table 1Relative errors of average vertical velocity in distorted model (%).

Discharge(m3/s)

Cross-section

Average vertical velocity (m/s) Relative error (%)

5-Pointmethod

3-Pointmethod

1-Pointmethod

3-Pointmethod

1-Pointmethod

39,000 Jing53 1.98 2.03 2.02 2.59 1.92Jing66 2.55 2.63 2.69 3.17 5.66Jing78 1.90 1.93 1.89 1.54 −0.74

19,600 Jing53 1.18 1.21 1.21 2.72 2.72Jing66 1.18 1.22 1.19 3.50 0.68Jing78 1.39 1.40 1.40 1.25 1.01

G. Zhao et al. / Flow Measurement and Instrumentation 32 (2013) 107–110110

6. Conclusions

Due to the distortion in the physical model, there are deviationsnot only in the geometric similitude but also in the kinematic anddynamic similitude. The differences in vertical distributions of thehorizontal velocity in the distorted model and in the prototypehave been analyzed and the primary conclusions are as follows:

(I)

The depth-averaged velocity equals the point velocity at therelative water depth y′=h¼ ð1−1=eÞ≈0:6 both in prototype andundistorted model and distorted model. The point velocitymeasured in the distorted model at this relative water depthcan be used to determine the depth-averaged velocity inprototype.

(II)

The Chézy coefficient scale λC is not equal to the unit in thedistorted model. λC is related to the distortion ratio η, thewater depth in the prototype hp and the roughness height ksand can be expressed as

λC ¼ 1þ lnð1=ηÞ3lnð0:875ðhp=ksp ÞÞ

ð15Þ

The Chézy coefficient is different in the distorted model andin prototype. When the model is undistorted, the Chézycoefficient scale λC is equal to 1 and the Chézy coefficient isthe same in the undistorted model and in prototype.

(III)

In the measurements in the distorted model, 1-point methodcan replace both the 3-point method and the 5-point methodto obtain the depth-averaged velocity in prototype, that is to

say, to the extent permitted by the accuracy of themeasurement.

(IV)

It is recommended to verify both 1- point method and 3-point method with data from distorted models with differentdistortion ratios.

Acknowledgments

Many thanks to support from CRSRI Open Research Program(CKWV2012301/KY) and the National Natural Science Foundationof China (50879006).

References

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