YANG Sheng-fa, HUJiang

Department of River and Ocean, Chongqing Jiaotong University, Chongqing 400074, China,

E-mail: ysf777@163.com

LI Dan-xun, WANG Xing-kui3

State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing 100084, China GUO Yakun

School of Engineering, University of Aberdeen, Aberdeen, AB24 3UE, UK

SOME NEW DATA AND FORMULAS FOR RESISTANCE FLOW IN FLUVIAL OPEN CHANNELS*

YANG Sheng-fa, HU5Jiang

Department of River and Ocean, Chongqing Jiaotong University, Chongqing 400074, China,

E-mail: ysf777@163.com

LI Dan-xun, WANG Xing-kui3

State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing 100084, China GUO Yakun

School of Engineering, University of Aberdeen, Aberdeen, AB24 3UE, UK

Flow resistance in fluvial open channels, especially in steep gravel-bed channels, still presents challenges to researchers and engineers. This article presents some new data from both the flume experiments and field measurements. Data analysis using the divided hydraulic radius approach shows that the relative roughness plays a significant role in the bed form resistance. A new set of formulas that incorporate the relative roughness are proposed. As compared with several existing formulas, the proposed formulas can be used to better estimate the bed form resistance.

open channel flow, flow resistance, relative roughness, steep gravel-bed, flume experiments, field measurements

Introduction

An accurate prediction of the water stage is of great importance for hydraulic design and flood forecast. In natural rivers, the water stage at a station is influenced jointly by the flow discharge, the energy slope, the geometry of the channel, and the configuration of the river bed. At stable river reaches, the flow discharge and the energy slope are closely correlated following a relatively fixed pattern, as can be fairly understood. Therefore, the accurate prediction of the water stage for any incoming flow discharge largely depends on the estimation of the flow resistance of the channel.

The research on flow resistance dates back a long time. The classical experiments of Nikuradse reveal a relationship between the resistance coefficient and the Reynolds number[1]. In fluvial open channel flows, thebed resistance is complicated due to the presence of various bed forms, such as vegetation, ripples, dunes, and anti-dunes. Recently, many studies were carried out to understand the flow resistance caused by unsubmerged and submerged vegetation[2-7]. The flow resistance changes with the development of bed forms[1,8]. The overall bed resistance consists of two components: the resistance due to the skin friction and the resistance due to the form drag. Thus, one possible approach to better understand the overall bed resistance is to separately investigate these two components. Einstein and Barbarossa[9], among others, pioneered the separate study of the two resistances by using the divided hydraulic radius approach. Following this approach, for example, Shen proposed an equation with the parameter ωD/ν[10].

Similar to the divided hydraulic radius approach, the divided energy slope approach assumes that the total energy slope can be separated into two components related to the grain friction and the bed form resistance, respectively. Along this line, Yang et al.[11]presented a set of empirical formulas for calculating flow resistance. In their analysis, the equivalent roughness size was taken as 2-D without full justifica-tions.

Fig.1 Some steep gravel-bed rivers in the west of China

Though many studies were carried out to understand the movable bed resistance of open channel flows, their results involve great discrepancies. The resistance in open channel flows remains as an issue[12]. In particular, the studies related to the movable bed resistance in steep gravel rivers are few and far between. With the rapid economic development in China, the energy is in an ever greater demand. The exploitation of hydro energy, particularly, in the west of China, is one option. In the west of China, most rivers are steep with gravel beds. The main features of such rivers include[13,14]: (1) wide and shallow with relatively high gradient, rough grain size of bed sediment and comparatively thick cover layer of cobblegravel (see Fig.1), (2) relatively small unit-width discharge and water depth in flood seasons, but very high flow velocity, (3) extreme destruction to river structures (e.g., bridges) and high risk of severe disasters.

This study provides some new laboratory and field data to complement the existing data, and proposes a new formula for better estimating the bed form resistance. The analysis in this study follows the divided hydraulic radius approach, and the major factors that influence the bed form resistance are examined. The performance of the proposed formula is compared with those of the existing formulas.

1. Laboratory experiments and field measurements

1.1Laboratory experiments

Laboratory experiments were conducted in a hydraulic flume of 24 m long and 7 m wide. From the upstream to the downstream, the flume channel consists of four consecutive sections: (1) a 2 m long section at the entrance for sediment feeding, (2) an 8 m long section for the flow development, (3) an 8m long measurement/observation section, and (4) a 6 m long section leading the flow to the flume exit (see Fig.2). The channel bed is fixed and smooth in Section (1), it is erodible sand-bed in other three sections. The bed slope is adjustable. The flow is supplied through a water tower with a maximum capacity of 1.0 m3/s. The flow rate, monitored by an electromagnetic flow meter, is set by a computer-controlled motor-operated valve.

The experiments were carried out under constant flow discharge and sediment transport. The channel bed was initially covered with sediment. The flow rate was gradually increased to the prescribed value and then was kept constant. When the flow intensity exceeded a limit, the sediment on the bed started to move, and the external sediment was successively fed to the flow from Section (1) of the flume. The sediment added to the flow was the same as that used to pave the channel bed. The sand-feeding rate was determined such that no apparent deposition or scour occurred on the channel bed. The equilibrium condition of flow and sediment transport at a prescribed discharge was maintained during each run of experiment until all measurements were made.

Water levels were monitored and measured at four cross-sections spaced 2 m apart in the observation section of the flume (Section 3, see Fig.2). The surface slope was calculated based on the measured water level data. The water depth and the topography of the channel bed in the observation section were measured with a specially designed device. Six equally spaced point measurements were made along each cross-section, i.e., the neighboring verticals were spaced 1 m apart. In total, 24 measurements were madein the observation section and the average was computed accordingly to represent the value at the section.

Fig.2 Sketch of the experimental setup

Table 1 Characteristics of the sediment used in the experiments

Fig.3 Size distributions of the two types of sediment

Two types of natural sediments were used in the experiments (see Table 1). Sediment A is fine gravel whereas Sediment B is coarse sand. The specific weight is 2.65 for both sediments. Their size distributions were determined through sieving (see Fig.3). A total of 52 experimental runs were performed with the first 30 runs using Sediment A and the last 22 runs using Sediment B. The flow rate per unit width ranges from 0.006 m3/s·m to 0.025 m3/s·m. The bed slope takes the value of 0.5%, 1.0% and 1.5% (see Table 2). The values of other variables, such asΨ′,RbandRb′are also included in Table 2 (their calculations are discussed in Section: “Data Analysis and Results”). Figure 4 shows typical ripples and dunes observed for experiment 16 listed in Table 2.

1.2Field data from the Hutubi River

Field data related to water level, flow discharge and sediment transport were obtained from measurements on the Hutubi River, a typical gravel-bed river located at the northern part of Xingjiang Uigur Autonomous Region in north-west of China. The bed slope of the river is 0.9%-1.4% and the width of the river at the measurement reach is about 240 m. The bed materials are composed mainly of gravels. The measurements were conducted mainly in August 2002. Water depths were measured with a typical wading rod. The velocity was measured with a propeller-type current meter. As the water in the river was relatively shallow (0.30 m-0.80 m), the 0.6-depth method was used to measure the vertical mean velocity. The average velocity and water depth of the cross-section were calculated based on the measurements spacing 2 m apart. The bed materials were sampled and analyzed, and the characteristics are shown to bed35=20.0mm ,d50=33.2mm ,d65=44.9mm , andγs=2.68. A total of 26 runs of measurements under various flow conditions were made (see Table 3).

1.3Data from other investigators

Data from published literature were also collected for use in this study. They include flume data[9,15-17]and field data[8,18]. In all, 319 data points are used in this study, covering a broad range of sediment size and flow parameters, as summarized in Table 4.

2. Data analysis and results

The collected laboratory and field data are analyzed based on the divided hydraulic radius approach.

2.1Hydraulic radius approach

RadiusRbis linearly divided intoRb′ andRb′′, related to the grain friction and the bed form resistance, respectively, i.e.,Rb=Rb′+Rb′′. One way for estimating the hydraulic radius is to use the crosssection average velocity. Einstein and Barbarossa[9]recommended the following equation forRb′

Table 2 Summary of the experiments

Fig.4 Typical bed forms observed in the experiments

whereχis a correction factor related toKs/δ,δ=11.6ν/u?, andνis the kinematic viscosity of water. WithKs=D65,D65being the sediment size for which 65% of sediment by weight is finer, the value ofRb′can be obtained using Eq.(1). They introduced a special parameter,Ψ′, to describe the bed form resistance

Table 3 Field data measured in the Hutubi River

Table 4 Summary of the collected data

wheresγandγare the specific weights of sediment and water, respectively,D35is sediment size for which 35% of sediment by weight is finer. Using field data from 10 sand-bed US rivers, they presented a form-resistance plot, which can be approximated by

whereωis the fall velocity of sediment in still water, andDis the particle diameter.

2.2Determination ofRb′′

For the determination ofRb′′(=Rb?Rb′), it is necessary to calculate the values ofRb′ andRb. To facilitate analysis, a fitted equation for calculating the correction factor is used in this study. Einstein and Barbarossa[9]used the log-law to describe the velocity distribution, i.e.,

whereUyis the flow velocity at a distance ofyabove the channel bed,Bis constant,κis Karman constant. From Eq.(5) one obtains

IfBis known, the value ofχcan be easily calculated. This article proposes the following equation forB

The value ofRbis obtained by using the method proposed by Einstein[1]with the Manning coefficientnbeing divided into two parts,nbandnw, related to the channel bed and the bank, respectively:

whereP(wettedperimenter)=Pw(bank)+Pb(bed). The value ofnis calculated from the velocity formulas for uniform flows. Assumenw=0.010,Pw= 2handPbis channel width, andRbcan be obtained by first solving Eq.(6) fornband then substitutingnbinto the following equation

Fig.5 The relationship betweenU/u?′′ andΨ′

Relevant results are listed in Tables 2 and 3. It is worth noting that the Einstein method has been found to give relatively lower values of bed shear stresses than other formulas.

2.3New formulas fortheu?′′-Ψ′relationship

For comparison, the results from Eq.(4) are also plotted in Fig.6, which generally fall below the measured data of this study, whereas Eq.(6) gives results far above the data of Einstein and Barbarossa (1952). The results of Eq.(11) lie in between those of Eqs.(4) and (6), showing some improvement in representing the data, but it is still insufficient to represent such scattered data.

Fig.6 The relationship betweenU/u?′′ and /50Hd

The analysis of data shows that the value ofU/u?′′ continuously increases withH/d50for constantΨ′, indicating that the relative roughness size,H/d50, has a significant effect on the bed form resistance (see Fig.6). Therefore, to better represent the data, a new parameter,ζ, containingH/d50, is introduced, and the regressed best-fit curve takes the following form

2.4Performance and comparison of various formulas

The performances of equations proposed by Einstein (1952), Shen (1962), and this article are evaluated against the collected data, as summarized in Table 5.

Table 5 Prediction accuracy of the four equations

Note that only a part of the data (273 out of 319) are used because the variableU/u?′′ can not be properly defined whenRb′′=0 as is the case for some data with no sediment transport. For an appropriate evaluation, the accuracy of the various equations in predicting the flow depth, H, is compared. For Eqs.(3), (4), and (12),H=Rb′+Rb′′. The percentages of calculated and measured flow depths,HcandHm, falling into certain intervals, are given in Table 5, where it is shown that the predicting performances of Eq.(3) and Eq.(4) are not satisfactory, with only 49.5% and 78.0% ofhc/hmfalling into the interval of [0.5, 2], respectively. In comparison with Eqs.(3), (4), the predicting accuracy of Eq.(12) is much better. The comparisons between the calculated and the measured results are plotted in Fig.7. It is seen that Eq.(12) gives good predictions.

Fig.7 Comparison of Eq.(3) and Eq.(12)

3. Conclusion

The flow resistance in fluvial open channels is ofgreat importance for hydraulic design and flood forecast. This article presents some new data from both flume experiments and field measurements. Other data from previous researches are also used. The data cover a broad range of bed slope (J=0.015%-1.5%) and relative roughness (H/d50=5-15000). Based on the divided hydraulic radius approach pioneered by Einstein and Barbarossa (1952), the data are analyzed to show that the relative roughness has a significant influence on the bed form resistance. A new set of formulas incorporating the relative roughness are proposed, which, as compared with several existing formulas, perform better in estimating the bed form resistance in fluvial open channel flows.

Acknowledgement

This work was supported by the CSTC 2011 is also gratefully acknowledged.

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July 16, 2009, Revised June 15, 2011)

* Project supported by the National Natural Science Foundation of China (Grant No. 50779082), the National Basic Research Program of China (973 Program, Grant No. 2007CB407202).

Biography: YANG Sheng-fa (1970-), Male, Ph. D., Professor

2011,23(4):527-534

10.1016/S1001-6058(10)60146-1