United friction resistance in open channel flows*

LI Shuo-lin （李碩林）, SHI Hao-ran （史浩然）, XUE Wan-yun （薛萬云）, HUAI Wen-xin （槐文信）

State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China, E-mail: Kesslerlee@whu.edu.cn

United friction resistance in open channel flows*

LI Shuo-lin （李碩林）, SHI Hao-ran （史浩然）, XUE Wan-yun （薛萬云）, HUAI Wen-xin （槐文信）

State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China, E-mail: Kesslerlee@whu.edu.cn

(Received April 16, 2015, Revised May 25, 2015)

It is now over half a century since Keulegan conducted his open channel flow experiments. Over the past decades, many empirical formulae were proposed based on his results, however, there is still not a combined expression to describe the effects of friction over all hydraulic regions in open channel flows. Therefore, in this letter, based on the analysis of the implicit model and the logarithmic matching method, the results of Keulegan (for authentic experiment data are no longer available, here we adopt the analytical solutions given by Dou) are rescaled into one monotone curve by combining the Reynolds number and the relative roughness of the river bed. A united expression that could cover the entire turbulence regions and be validated with Dou’s analytical solutions is suggested to estimate the friction factor throughout the turbulent region in open channel flows, with higher accuracy than that of the previous formulas.

open channel flows, frication factor, united formula, implicit model, logarithmic matching method

The friction factorfis a very important parameter in the river engineering and the environmental hydraulic engineering. It is a critical parameter, among other things, in the calculations of the flow resistance and the average velocity. Generally speaking, there are two distinct types of flow regimes-the pipe flow and the open channel flow-which are controlled by different physical mechanisms. In the case of the pipe flow, the flow is mainly driven by the water pressure. The open channel flow, on the other hand, is usually driven mainly by gravity. Yet in spite of their differences, the flow resistance in both flow regimes can be described by a friction factor. For example, the head loss in both types of flows can be determined bythe Darcy–Weisbach formula h=f( L/4R)( U2/f 2g)[1], whereU is the mean velocity of the flow,g is the gravitational acceleration,hfis the friction head loss,Lis the length of the flow path, andRis the hydraulic radius.

Nikuradse determined the friction factor in pipes with artificial (but measurable) roughness and presented a graph to describe the way in which this roughness influences the pipe flows[2]. While in open channel flows (without vegetation, as in vegetated flows, the flow structure is usually affected by vegetation[3]), the irregularities or the roughness of the riverbed bring about resistance. In order to disclose the congruence between the pipe flows and the open channel flows, Keulegan conducted simulation experiments in artificial channels[4]. As a result, a graph (Keulegan graph) similar to that for the pipe flows was produced. To better explain the inner relationship, Dou gave a theoretical description of Keulegan’s curves, which fitted well with the experimental data shown in Fig.1[5].

For open uniform laminar channel flows,f Re= 24 can be obtained from basic physical equations[2]. For the turbulent flows near a solid channel wall，the flow field can usually be divided into three regions:the hydraulically smooth region, the transition region and the hydraulically rough region[2]. In the hydraulically smooth region, Blasius suggested a formula for the pipe flows[6]

Equation (1) is also applicable for the open channel flows as we could see the line with a gradient of 1/4 in Fig.1. For the hydraulically rough region, Strickler proposed another formula for the pipe flows in the form of a power law and dependent on the relative roughness height r/ R , wherer is the equivalent height of irregularities on the bed surface[7]and it is expressed as follows

Previously justified by Gioia and Bombardelli[8], Eq.(2) is equally valid for the turbulence flows in the hydraulically rough region of open channel flows.

Given f Re as a constant in the laminar regime, integration is made as a whole in both the laminar and turbulence regimes. In the low Reynolds number region (i.e., with Reynolds number between 200 and 160 000) where the friction factor has no relationship with the relative roughness, analysis shows that we may assume that r/ R→0. Therefore we could obtain f Re=F( Re3/4)according to Eq.(1), where F( x)is an implicit function of x.

In the high Reynolds number region, it is appropriate to apply the formula proposed by Goldenfeld[9], so we havebased on Eq.(2). According to Tao[10], naturally, we could use Eq.(3) combined with the two boundary conditions to cover both laminar and turbulence regimes

whereαand CSare two constants to be determined in order to ensure that all six curves will merge to form one single curve.

Therefore, by processing the experimental data directly extracted from Fig.1 (since the original data of Keulegan’s experiments are no longer available),

Fig.1 Dou’s analytical solutions for Keulegan graph in open channel flows[5]

we obtain α=4and C=2× 10-9. Thus we obtain

S the explicit form of Eq.(4) as

In Fig.2, where Eq.(4) is used, all experimental data from Fig.1 fall into one monotone curve. A further analysis shows that, if we only consider the part of the turbulence flow, the related part in Fig.2 will possess two asymptotic solutions with slopes of 1 and 1/4 (in a log-log plot), which can be written specifically as

where K2=1/4,C2=1.324, and x=Re3/4+ CSRe4(r/ R)4/3.

Following Guo[11], adopting the method of logarithmic matching, we can merge the above two equations into a single expression to match the curve in Fig.2

where x0is derived from the equation lgx0=(C1-C2)/(K2-K1)，and for x<>x0,lg[1+(x/ x0)β]approaches to βlgx-βlgx0.

And the value of the transitional shape parameter βcould be found by a collocation method suggested by Griffiths and Smith[12]. That is to say, when x=x0, the value of lg(f Re)obtained from Eq.(7) is equal to that from experiments.

Then we obtain the accurate form of Eq.(7) as

from which, we have,

For the previous research findings of hydraulic resistance in open channels, for Re>25000, the Colebrook-White type formula is often used[13]

where m1,m2and m3are experiential coefficients.

Note that the fraction factor fappears on both sides of Eq.(10), which must be solved by iterations. Yen suggested a corresponding but more explicit formula for 2-D wide open channels whenRe>30 000 and r/ R<0.05[13]

In order to see the accuracy of Eq.(9), a compari-son between Eq.(9) and Eq.(11) is made, and the result is shown in Fig.3.

Figure 3 further shows the accuracy of covering the relationships among f,Reand r/ Rwith the formula suggested as Eq.(9) and a comparison between Eq.(9) and Eq.(11). It is indicated that in the scope of applications of both formulae, Eq.(9) is more accurate. At the same time, it has a more extensive application range in the whole turbulence region which starts fromx=2.41, while the application range of Eq.(11) starts fromx=5.81.

For both the open channel flows or the pipe flows, the friction factorf is an important physical quantity that describes the resistance to the flows in terms of the channel parameters. Both the Nikuradse curve and the Keulegan curve use the Reynolds number(Re)as a parameter and the relative roughness r/ Ras a variable parameter to describe the variation off. Through analysis using physical and mathematical models, this letter proposes an empirical formula that combines various influential factors pertaining to the friction factorf into a monotone curve and proposes an empirical formula to estimate the friction factor throughout the entire turbulence region. The use of a general relationship that may be applied to the whole flow profile might be convenient. Because the formula presented in this letter incorporates the Reynolds numbers and the shape factors (here we only consider one length scale,r/ R) for all hydraulic regions, it reflects the variation of the friction factor across the entire vertical flow profile for open channels.

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* Project supported by the National Natural Science Foundation of China (Grant Nos. 51479007, 11172218 and 11372232), the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20130141110016).

Biography: LI Shuo-lin (1994-), Male, Ph. D. Candidate

HUAI Wen-xin,

E-mail: wxhuai@whu.edu.cn