Flow resistance in smooth rectangular open-channels with low aspect ratios

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Flow resistance in smooth rectangular open-channels with low aspect ratios

Nguyen Ba Tuyen1 and Nian-Sheng Cheng2

1 Research Student, School of Civil and Environmental Engineering, Nanyang Technological

University, Nanyang Avenue, Singapore 639798. Email: [email protected]

2 Associate Professor, School of Civil and Environmental Engineering, Nanyang Technological

University, Nanyang Avenue, Singapore 639798. Email: [email protected]

Abstract

This note shows an approach that is useful for evaluating the resistance to flows in narrow open

channels. A series of experiments were first carried out in a smooth rectangular open channel of

which the aspect ratio varied from 0.12 to 1.53. Experimental data collected were then used to

validate a resistance formula that was proposed previously by modifying Prandtl’s friction law.

Introduction

For many practical applications, open channel friction could be estimated by treating the

channel as an equivalent pipe with a diameter four times the mean hydraulic radius. However,

open channel flows are generally considered more complicated than circular pipe flows, due to

the existence of secondary currents, a free surface, and the non-uniform distribution of

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boundary shear stress. Therefore, the treatment of open channel resistance simply by analogy

with pipe friction is often found inappropriate (e.g. Montes 1998, Cheng et al. 2011).

Recently, by modifying Prandtl’s friction law, Cheng et al. (2011) developed an approach

for evaluating the friction of smooth rectangular channels. They expressed the open channel

friction factor, fch, as a function of the Reynolds number and the aspect ratio as follows:

−−⎡ ⎤⎛ ⎞+

= ⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

21210.33 log 0.11Re w

ch chwf e

w

(1)

where ( )ν=Re 4ch hR V is the channel Reynolds number, V is the average velocity, Rh is the

hydraulic radius, ν is the kinematic viscosity of the fluid, α= / 2w for α ≥ 2 and α2 / for α < 2,

α (= B/h) is the aspect ratio, B is the channel width, and h is the flow depth. Cheng et al. (2011)

showed that Eq. (1) performs much better than several other formulas available in the literature,

which usually were derived by reconciling the resistance difference between circular pipes and

rectangular channels. They verified Eq. (1) with their own experimental data and also those

presented by Tracy and Lester (1961) and Knight et al. (1984), of which α varies from 1.07 to 15

and Rech from 7.7×103 to 5.5×105. It should be mentioned that the data employed for the

verification only included a few measurements for low aspect ratios, say, α ≤ 2. In addition, it

also is noted that the information for experiments conducted for low aspect ratios is also very

limited in the literature. Therefore, to further verify the applicability of Eq. (1) for low aspect

ratios, additional data were collected from narrow open channel experiments that were

purposely designed in this study. The study results may also be useful for engineering

applications such as two-dimensional approximation studies of open channel flows and

validation of numerical models.

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Experiments

Experiments were conducted in a tilting narrow glass channel, 1 cm in width, 20 cm in height

and 200 cm in length, as illustrated in Fig. 1. Both the channel wall and bed were smooth. The

channel slope could be adjusted using a screw-jack, and a digital protractor was used to

measure the bed slope to a resolution of 0.05° (or 0.087%). When used in conjunction with the

screw-jack, the precision of slope measurement was increased by two times, i.e. 0.025°. Water

was circulated by a centrifugal pump and controlled by a mechanical valve, providing a steady

discharge in the range of 0-0.6 l/s. The flow rates, measured with a mechanical flow meter, were

then calibrated by volumetric measurement using a collecting device and a stop watch.

Meanwhile water temperature was measured using a mercury thermometer with an accuracy of

0.1oC. The following empirical formula was employed to estimate the kinematic viscosity of

water (Cheng et al. 2011),

ν −⎛ ⎞= ⎜ ⎟+⎝ ⎠

1.45660

1040T

(2)

where T is the temperature in degrees Celsius and ν is in m2/s. Eq. (2) applies for T = 0-100oC.

All the measurements were made under steady and uniform flow conditions. Water

entered the channel smoothly after passing through a filter. The surface elevation of the flow

then dropped quickly, following with a gradual decrease that approached a uniform flow profile.

This transition measured about 0.5-1.25 m in length, depending on flow discharge and channel

bed slope. After the transition, the water depth remained almost constant until about 25 cm

upstream of the channel outlet. The measurement area was located from 140 to 160 cm

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measured from the beginning of the channel. For supercritical flow conditions (Fr > 1), this

observation area was far away from the outlet to ensure a constant water level in this section.

On the other hand, for subcritical flows (Fr < 1), a tailgate, equipped at the end of the flume, was

used to establish the uniform flow profile.

Due to the small scale of the experimental set-up, special attention was paid to the

accuracy of the experimental data. To investigate the longitudinal water surface profile, the flow

depths were measured first using semi-transparent rulers attached to the side wall of the glass

channel, and then checked with a point gage and/or images captured by a CCD camera

positioned perpendicular to the flow. The rulers were fixed to the side wall at a 10-cm interval for

the measurement area and a 25-cm interval for the rest of the flume (see Fig. 1). In addition, an

auxiliary ruler was also used to ensure the eye level (point of observation) was at the same level

as the water surface elevation before a reading was taken. This auxiliary ruler was mounted on

a rigid movable frame such that it was parallel to the ruler fixed on the glass wall. By matching

the two rulers, the water surface was measured exactly in the direction perpendicular to the side

wall, and, thus the flow, as illustrated in Fig. 2.

Given the narrow channel, water surface tension yielded a concave surface profile in the

lateral direction [see Fig. 3(a)]. From the observation point (side view), the water surface

appeared to be a triple-line, i.e., two bright lines [lines 1 and 3, see Fig. 3(b)] and a dark line in

the middle (line 2). Line No. 1 was induced by the light scattered by fluctuating water on the side

walls (due to water surface tension). Line 3 was caused by the light penetrating into the flow

through the fluctuating water surface, which visualized the lower part of the concave surface

profile. Line 2 represents the image of the channel bottom (which was made from black Perspex)

being reflected by the curved side of the water surface profile (acting like a convex mirror). With

those observations, the actual water level was taken as the elevation at the lower edge of line 2.

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Such measurements were further verified using point gage readings taken at accessible points

across the channel. For most of the cases, the sharp edges of line 2 existed and, thus, the

water level reading could be made with sub-millimeter accuracy. For the case of large Froude

numbers, the water surface was subject to shock-like disturbances, and flow depth was taken

as an average over a duration of about 30 seconds. This value was also checked by the image

captured with a CCD camera that was positioned at the reading elevation.

In total, 200 tests were completed for a wide range of experimental conditions, where the

flow depth h varies from 0.006 to 0.076 m, the aspect ratio α varies from 0.12 to 1.53, the bed

slope S varies from 0.017 to 0.176, the Reynolds number varies from 5×103 to 3×104, and the

Froude number varies from 0.48 to 4.28. The flow characteristics are summarized in Table 1

and the data are listed in Error! Reference source not found..

Comparisons

A comparison was first made between the friction factor f estimated from flow measurements, fe,

and the prediction made using Eq. (1), denoted the computed friction factor, fc. fe was calculated

from the flow shear velocity * hu gR S= and the cross-sectional average velocity V = Q/(B⋅h), as

( )2 2*8 / 8 /e hf u V gR S V= = (3)

where g is the gravitational acceleration, Rh = B⋅h/(B+2h) is the hydraulic radius, and S is the

energy slope that is the bed slope for steady, uniform flow. The goodness of agreement was

assessed using the relative prediction error,

100(%)c e

e

f fprediction error

f−

= × (4)

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As shown in Fig. 4, the computed friction factors generally agree with the measurements,

in spite of some slight underestimates. The average prediction error is 1.32%. This level of

deviation is deemed acceptable, since the difference was within the measurement errors.

Second, fc was also used to compute the average flow velocity = 8c h cV gR S f . This

predicted velocity is then compared with the cross-sectional average velocity Vm calculated as

Q/(B⋅h). The results are plotted in Fig. 5 and show good agreement.

The error bars presented in Figs. 4 and 5 were estimated as follows. First, the errors (δ)

of the component parameters that were directly measured (i.e. the precisions of these

measurements) were evaluated. They are δQ = 0.001 l/s; δB = 0.1 mm; δh = 0.2 mm; δS =

0.025°; and δT = 0.1°C. For composite quantities, f and V, that depended on the component

parameters and were indirectly measured, their errors were estimated with the errors of all

component parameters. Since f = 8gRhS/V2, where Rh=Bh/(B+2h) and V = Q/(Bh), applying the

theory of errors (Herschy 2009), it can be shown that

1/22 2 2( ) ( ) ( ) ( )u V u Q u B u h⎡ ⎤= + +⎣ ⎦ (4)

1/22 2 2 2( ) 10 ( ) 10 ( ) 4 ( ) ( )u f u B u h u Q u S⎡ ⎤= + + +⎣ ⎦

(5)

where u(x)= 100·δx/x stands for the percentage error of quantity x. From the measurements, the

percentage errors of Q, B, h, and S were calculated for each test. With those values, the

percentage errors of V and f were then evaluated using Eqs. (4) and (5), respectively. The

average percentage error of f is 6.09% and that of V is 2.05%.

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Conclusions

In this study, a series of experiments were conducted for steady uniform open-channel flows to

validate the formula previously proposed by Cheng et al. (2011) for evaluating smooth channel

resistance for the case of low aspect ratios. Comparisons show that the predicted friction factor

and mean flow velocity agree well with the values derived from the measurements. Together

with the results in Cheng et al. (2011), the results of both studies have verified that the proposed

formula is applicable for flows with the aspect ratio ranging from 0.12 to 15, Reynolds number

ranging from 5×103 to 5.4×105, and Froude number ranging from 0.24 to 4.28. This suggests

that the proposed formula is applicable for both wide and narrow open-channels.

References

Cheng, N. S., Nguyen, H. T., Zhao, K., and Tang, X. (2011). “Evaluation of flow resistance in

smooth rectangular open-channels with Modified Prandtl Friction Law.” Journal of Hydraulic

Engineering, ASCE, 137(4), 441-450.

Herschy, R. W. (2009). Streamflow Measurement, Routledge, London ; New York.

Knight, D. W., Demetriou, J. D., and Hamed, M. E. (1984). "Boundary shear in smooth

rectangular channels." Journal of Hydraulic Engineering, ASCE, 110(4), 405-422.

Montes, S. (1998). Hydraulics of open channel flow, ASCE press, Reston, VA.

Tracy, H. J., and Lester, C. M. (1961). "Resistance coefficients and velocity distribution, smooth

rectangular channel." U.S. Geological Survey Water-Supply Paper No. 1592-A.

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Notations

The following symbols are used in this paper:

B channel width

f friction factor

Fr Froude number

g gravitational acceleration

h flow depth

Rech Channel Reynolds number (=4RhV/ν)

Rh hydraulic radius

S energy or bed slope

T temperature, in degree Celsius

V cross-sectional average velocity

α aspect ratio (= B/h)

ν kinematic viscosity

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Table 1. Experimental conditions and flow characteristics

Parameter Dimension Magnitude

Water flow rate (Q) m3·s-1 0.03×10-3 - 0.57×10-3

Flow depth (h) m 0.006 - 0.076

Channel width (B) m 0.01

Aspect ratio (α) - 0.12 - 1.53

Bed slope (S) - 0.017 - 0.176

Reynolds number (Rech) - 5×103 - 30×103

Froude number (Fr) - 0.48 - 4.28

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Table 2. Experimental data

Test No.

Channel width

B

Flow depth

h

Aspect ratio� α

Flow rate Q

Temperature

T

Bed slope

S

Friction factor

f

(cm) (cm) - (l/s) (0C) - - D1-1 0.96 7.63 0.13 0.302 28.25 0.017 0.036 D1-2 0.96 7.23 0.13 0.284 28.20 0.017 0.036 D1-3 0.95 6.38 0.15 0.249 28.20 0.017 0.036 D1-4 0.95 5.50 0.17 0.210 28.10 0.017 0.037 D1-5 0.95 5.00 0.19 0.189 28.05 0.017 0.037 D1-6 0.95 4.58 0.21 0.172 26.05 0.017 0.037 D1-7 0.95 4.28 0.22 0.160 28.00 0.017 0.037 D1-8 0.94 3.80 0.25 0.140 27.90 0.017 0.038 D1-9 0.94 3.39 0.28 0.123 27.80 0.017 0.038

D1-10 0.94 3.00 0.31 0.108 27.70 0.017 0.038 D1-11 0.94 2.79 0.34 0.099 28.30 0.017 0.038 D1-12 0.94 2.37 0.40 0.083 28.30 0.017 0.039 D1-13 0.94 2.12 0.44 0.072 26.20 0.017 0.040 D1-14 0.94 1.94 0.48 0.066 28.50 0.017 0.039 D1-15 0.94 1.93 0.49 0.065 26.25 0.017 0.040 D1-16 0.94 1.73 0.54 0.058 26.60 0.017 0.040 D1-17 0.94 1.67 0.56 0.055 28.50 0.017 0.040 D1-18 0.94 1.59 0.59 0.052 27.00 0.017 0.041 D1-19 0.93 1.13 0.83 0.034 27.40 0.017 0.043 D1-20 0.93 1.03 0.91 0.031 27.50 0.017 0.043 D2-1 0.94 2.85 0.33 0.151 25.90 0.035 0.035 D2-2 0.94 2.70 0.35 0.143 25.50 0.035 0.035 D2-3 0.94 2.51 0.37 0.132 26.00 0.035 0.035 D2-4 0.94 2.33 0.40 0.121 26.10 0.035 0.035 D2-5 0.94 2.07 0.45 0.105 26.25 0.035 0.035 D2-6 0.94 1.84 0.51 0.092 26.30 0.035 0.036 D2-7 0.94 1.74 0.54 0.086 26.40 0.035 0.036 D2-8 0.94 1.57 0.60 0.077 26.50 0.035 0.036 D2-9 0.94 1.50 0.62 0.073 26.60 0.035 0.036

D2-10 0.94 1.40 0.67 0.067 26.70 0.035 0.036 D2-11 0.93 1.37 0.68 0.065 26.75 0.035 0.037 D2-12 0.93 1.28 0.73 0.060 26.85 0.035 0.037 D2-13 0.93 1.13 0.83 0.052 26.90 0.035 0.038 D2-14 0.93 1.08 0.86 0.049 27.00 0.035 0.038 D2-15 0.93 1.05 0.89 0.047 24.75 0.035 0.038 D2-16 0.93 1.03 0.91 0.046 27.10 0.035 0.038 D2-17 0.93 0.93 1.00 0.040 24.95 0.035 0.040 D2-18 0.93 0.85 1.10 0.036 25.05 0.035 0.040 D2-19 0.93 0.79 1.18 0.033 25.15 0.035 0.041 D2-20 0.93 0.73 1.28 0.030 25.40 0.035 0.041 D3-1 0.95 4.23 0.22 0.299 27.65 0.052 0.031

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D3-2 0.94 3.90 0.24 0.274 27.65 0.052 0.031 D3-3 0.94 3.49 0.27 0.243 27.70 0.052 0.031 D3-4 0.94 3.10 0.30 0.213 27.75 0.052 0.032 D3-5 0.94 2.78 0.34 0.188 27.75 0.052 0.032 D3-6 0.94 2.49 0.38 0.166 27.80 0.052 0.032 D3-7 0.94 2.27 0.41 0.149 27.85 0.052 0.033 D3-8 0.94 1.99 0.47 0.128 27.85 0.052 0.033 D3-9 0.94 1.71 0.55 0.108 27.90 0.052 0.033



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D7-10 0.94 1.41 0.66 0.140 28.00 0.123 0.030 D7-11 0.93 1.29 0.72 0.127 28.05 0.123 0.030 D7-12 0.93 1.10 0.85 0.104 28.10 0.123 0.031 D7-13 0.93 1.20 0.78 0.117 28.20 0.123 0.030 D7-14 0.93 1.00 0.93 0.093 28.30 0.123 0.031 D7-15 0.94 2.97 0.32 0.331 27.40 0.123 0.028 D7-16 0.94 2.57 0.37 0.281 27.50 0.123 0.028 D7-17 0.94 2.16 0.43 0.229 27.55 0.123 0.029

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D7-18 0.94 1.69 0.55 0.174 27.65 0.123 0.029 D7-19 0.94 1.42 0.66 0.141 27.70 0.123 0.030 D7-20 0.93 1.15 0.81 0.110 27.75 0.123 0.031 D8-1 0.94 4.16 0.23 0.512 22.00 0.141 0.028 D8-2 0.94 3.84 0.25 0.466 22.25 0.141 0.028 D8-3 0.94 3.64 0.26 0.442 22.35 0.141 0.028 D8-4 0.94 3.27 0.29 0.396 25.20 0.141 0.027 D8-5 0.94 3.23 0.29 0.386 22.45 0.141 0.028 D8-6 0.94 2.94 0.32 0.348 25.50 0.141 0.028 D8-7 0.94 2.73 0.34 0.321 25.75 0.141 0.028 D8-8 0.94 2.44 0.38 0.284 25.80 0.141 0.028 D8-9 0.94 2.25 0.42 0.259 26.00 0.141 0.028


D9-10 0.93 1.27 0.74 0.142 25.10 0.158 0.030 D9-11 0.93 1.12 0.83 0.123 25.40 0.158 0.030 D9-12 0.93 1.04 0.90 0.112 25.50 0.158 0.030 D9-13 0.93 0.91 1.03 0.095 25.65 0.158 0.031 D9-14 0.93 0.78 1.20 0.077 25.80 0.158 0.032 D9-15 0.93 0.64 1.46 0.061 25.90 0.158 0.032 D9-16 0.94 1.41 0.66 0.161 26.00 0.158 0.029 D9-17 0.94 3.13 0.30 0.405 26.70 0.158 0.027 D9-18 0.94 4.07 0.23 0.541 26.50 0.158 0.027 D9-19 0.94 3.92 0.24 0.519 26.50 0.158 0.027 D9-20 0.94 3.64 0.26 0.478 26.50 0.158 0.027 D10-1 0.94 3.69 0.26 0.517 26.45 0.176 0.026 D10-2 0.94 3.41 0.28 0.471 26.45 0.176 0.027 D10-3 0.94 3.23 0.29 0.442 26.75 0.176 0.027 D10-4 0.94 3.10 0.30 0.423 26.45 0.176 0.027 D10-5 0.94 2.76 0.34 0.371 26.50 0.176 0.027

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D10-6 0.94 2.73 0.34 0.366 26.80 0.176 0.027 D10-7 0.94 2.51 0.37 0.333 26.55 0.176 0.027 D10-8 0.94 2.31 0.41 0.304 26.75 0.176 0.027 D10-9 0.94 2.18 0.43 0.284 27.45 0.176 0.028

D10-10 0.94 2.08 0.45 0.269 26.90 0.176 0.028 D10-11 0.94 2.06 0.46 0.266 26.85 0.176 0.028 D10-12 0.94 1.58 0.59 0.196 27.00 0.176 0.028 D10-13 0.93 1.32 0.71 0.159 27.00 0.176 0.029 D10-14 0.93 1.14 0.82 0.134 27.15 0.176 0.029 D10-15 0.93 1.06 0.88 0.122 27.00 0.176 0.030 D10-16 0.93 1.02 0.92 0.116 26.70 0.176 0.030 D10-17 0.93 1.00 0.93 0.113 27.05 0.176 0.030 D10-18 0.93 0.85 1.10 0.093 27.25 0.176 0.030 D10-19 0.93 0.73 1.28 0.076 27.20 0.176 0.031 D10-20 0.93 0.65 1.43 0.065 27.40 0.176 0.032

Flow resistance in smooth rectangular open-channels with low aspect ratios

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