an alternative approach for large-scale roughness

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Resistance coefficients for artificial and natural coarse-bed channels – an

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alternative approach for large-scale roughness

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Nian-Sheng Cheng

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School of Civil and Environmental Engineering, Nanyang Technological University, Nanyang

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Avenue, Singapore 639798. Email: [email protected]

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Abstract Traditional Manning-Strickler and Keulegan formulas underestimate open channel

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flow resistance in the presence of large scale roughness. How to theoretically evaluate

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resistance coefficients for large scale roughness remain challenging in spite of significant

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efforts made in recent decades. The present study provides an alternative understanding of

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energy losses associated with large scale roughness. This yields a new resistance formula, of

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which two empirical coefficients were calibrated with laboratory and field data available in

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the literature. The results show that the new formula applies for both shallow and deep

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flows and also agrees well with the best of previously-proposed formulas.

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Keywords: open channel; large scale roughness; resistance; friction factor; shallow flow

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Introduction

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Classical resistance coefficients for open channel flows include Chezy C, Manning’s n and

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Darcy-Weisbach friction factor f. They are related to each other as follows: C kuh1/ 6 8   f g n g

(1)

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where h is the flow depth, ku = units factor (= 1 for SI and 1.486 for US Customary), and g is

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the gravitational acceleration. Different from f that is dimensionless, C has a dimension of

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L1/2T-1 and n has a dimension of L-1/3T-1. Manning’s n can be normalised as St 

n g kuk1/ 6

(2)

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where k is the representative roughness length and St is referred to as Strickler number

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(Garcia 2008, page 28). Generally St varies, e.g. from 0.08 to 0.15 as summarised by Yen

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(1993), but can be approximated to be 0.12 for very wide channels (Garcia 2008). Resistance

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coefficients can be evaluated by rewriting the Manning-Strickler formula as 1/ 6

8 1 h    f St  k  34

(3)

or Colebrook-White type formula for h/k > 0.1 (Keulegan 1938),

8  12h   2.5ln   f  k 

(4)

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Eqs. (3) and (4) have been successfully applied for deep flow conditions, i.e. when the flow

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depth is at least one order of magnitude greater than the roughness length scale. By plotting

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(8/f)1/2 against h/k with Eqs. (3) and (4), it can be found that Eq. (3) (with St = 0.12) is

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comparable to Eq (4) for h/k = 6 – 300, the difference varying within ±5%. However, under

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shallow flow conditions or in the presence of large-scale roughness, the flow resistance 2

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predicted using Eq. (3) or Eq. (4) could deviate significantly from measurements (Bathurst

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1978; Ferguson 2007; Froehlich 2012; Katul et al. 2002; Lawrence 2000; Smart et al. 2002).

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Bed roughness elements are considered large if their length scale k is comparable to

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the flow depth h. Different definitions of large-scale roughness are available in the literature,

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but generally suggesting that the flow depth is less than about four to ten times the

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representative roughness length, i.e. h/k < 4-10 (e.g. Katul et al. 2002; Recking et al. 2008b;

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Rickenmann and Recking 2011). Some definitions appear more restrictive. For example,

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Bathurst et al. (1981) considered the bed roughness to be large when h/D84 < 1.2, where D84

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is the grain diameter for which 84% of the sediment is finer.

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How to extend Manning and Keulegan formulas to shallow flows have been studied

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for decades. Excellent contributions are due to Hey (1979), Bathurst et al. (1981), Katul et al.

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(2002), Ferguson (2007), Rickenmann and Recking (2011), Papanicolaou et al. (2011),

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Papanicolaou et al. (2012), Hajimirzaie et al. (2014), among others. From these studies and

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other classical literature (e.g. Chow 1959), it follows that (1) traditional boundary layer

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theories are inapplicable for open channel flows subject to large-scale roughness, (2) large

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roughness elements generate turbulent eddies and enhance mass and momentum transfer,

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and (3) in comparison with skin friction, energy dissipation is dominated largely by form drag

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(for isolated elements), wake interferences (for closely placed elements) and wave drag (for

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emergent elements).

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The previous studies have resulted in several useful formulas for the prediction of

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resistance induced by large scale roughness. For example, Hey (1979) proposed a Colebrook-

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White type equation for gravel beds for h/D84 > 0.3,

 3.36h  8  2.5ln   f  D84 

(5)

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By applying a mixing layer analogy for the inflectional velocity profile in the roughness layer,

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Katul et al. (2002) derived the following resistance equation for h/D84 = 0.2-7,   cosh 1  h / D84    8 1  4.5 1  ln     f cosh 1    h / D84 

(6)

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Ferguson (2007) expressed the deep-flow Manning-Strickler relation as (8/f)1/2 = a1(R/D84)1/6

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and the shallow-flow asymptote as (8/f)1/2 = a2(R/D84), where a1 and a2 are empirical

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coefficients. It is noted that a1 = 1/St by comparing the deep-flow relation with Eq. (3). By

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assuming that the two extreme f-relations are additive for a general coarse-bed stream,

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Ferguson (2007) obtained

a1a2r / D84 8  5/3 f a12  a22 r / D84 

(7)

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where r is the hydraulic radius, a1 = 6.5 and a2 = 2.5. Eq. (7) was fitted to 376 sets of field

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data with r/D84 from 0.1 to 26 (with one value of 87). Using an approach similar to that

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developed by Ferguson (2007), Rickenmann and Recking (2011) also conducted a

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dimensional analysis with a much larger field database (2890 sets in total), which yielded

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1.618     (8) h 1       1.283D84   In particular, Rickenmann and Recking (2011) reported that Eq. (7) performs the best in the

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prediction of flow resistance, in comparison with other formulas including Eq. (8). Fig. 1

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shows a comparison of Eqs. (3) to (8) by replacing r in Eq. (7) with h and plotting Eq. (4) with

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St = 0.12. It can be seen that for 0.5 < h/D84 < 7, Eqs. (5) to (8) are close to each other, but

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they differ from the two traditional formulas [Eqs. (3) and (4)].

1.904

 h  8  4.416   f  D84 

1.083

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The present study aims to provide an alternative physical reasoning about flow

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resistance induced by large-scale roughness. A resistance equation is thus developed, with

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two coefficients to be calibrated with laboratory and field data available in the literature. 4

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Theoretical consideration

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The analysis presented in this section is based on the following assumptions: (1) The channel

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is wide and the channel bed is planar and fully rough; (2) The flow is steady, turbulent, fully

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developed and uniform; (3) The variation in the bed elevation is in the order of sediment

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grain diameter D; and (4) The flow depth h is not large in comparison with D (e.g. 0 < h/D
.

1/ 2

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To extend the above analysis to a mixed-size sediment bed, it is necessary to know

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what sediment diameter should be selected to be a representative sediment size. Such a

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selection is made generally by arguing that the representative size is greater than the

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median diameter (Leopold et al. 1964). This is because grains of larger diameters protrude

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above the average bed level and expose greater volume, and thus exert most of the

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resistance to the flow (Whiting and Dietrich 1990). Perhaps for statistical reasons, the

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representative diameter is often taken as D84 (for which 84% of sediment grains are finer), as 9

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seen in various field data analyses (e.g. Bathurst 1985; Ferguson 2007; Hey 1979; Limerinos

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1970; Rickenmann and Recking 2011; Whiting and Dietrich 1990). Therefore, in the following

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analysis, D in Eq. (18) will be replaced with D84. In addition, h will be changed to the

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hydraulic radius, r. In particular, by noting that laboratory experiments may be affected

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significantly by sidewalls, a sidewall correction procedure will be applied to laboratory data,

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which yields a corrected hydraulic radius, rb.

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Calibrations

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The two constants ( and ) in Eq. (18) can be evaluated using laboratory and field data. In

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comparison with field studies, laboratory experiments are usually preformed under well-

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controlled flow and bed conditions. First, Eq. (18) is fitted with laboratory data that meet the

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requirements given in the foregoing derivation. As summarised in Table 1, in total 416 sets of

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laboratory data were compiled from six sources, i.e. Paintal (1971), Bathurst et al. (1981), Ho

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(1984), Cao (1985), Recking et al. (2008a) and Jordanova (2008). These data cover a range of

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D84 (= 2.2-58 mm) and h/D84 (= 0.2-39.9). Bathurst et al. (1981) obtained their experimental

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data for five different fixed roughness beds, each bed being one element thick. Paintal

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(1971), Ho (1984), Cao (1985) and Recking et al. (2008a) have measured flow resistance with

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and without bedload transport. The data of Paintal (1971), Cao (1985) and Recking et al.

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(2008a) were tabulated in Recking (2006). The present analysis is limited to the datasets

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with zero or negligible transport rate. Jordanova (2008) used hemispheres to simulate

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roughness elements and measured flow resistance with different roughness densities for

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submerged and emergent roughness conditions. Employed in this study is only part of 10

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Jordanova’s data, which were measured under the condition of submerged, closely packed

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hemispheres. In addition, D84 is taken as the diameter of hemispheres.

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Of the 416 sets of data, 11% were collected under the narrow channel condition of

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B/h < 5, where B is the channel width. To prepare the data for the calibration, the sidewall

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correction was applied to all the laboratory data, which yields a corrected hydraulic radius rb

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that is bed-related. The sidewall correction procedure employed here is the same as that

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developed by Vanoni and Brooks (1957). With the cross-sectional average flow velocity V

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and energy slope S, the friction factor f (= 8grS/V2) and Reynolds number R (= 4rV/) are first

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calculated, where r [=Bh/(2h+B)] is the hydraulic radius and  is the kinematic viscosity of

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fluid. Then the sidewall friction factor is evaluated using the relation, fw  31 ln1.3R / f 

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(Cheng 2011b). Next, the bed friction factor is calculated with fb  2h  B  f  2hfw  / B .

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Finally, the bed hydraulic radius is obtained as rb  rfb / f .

2.7

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Plotted in Fig. 4 is the comparison of Eq. (18) with the data summarised in Table 1.

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Three sets of - and -values are used for plotting Eq. (18), which describes the upper

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bound of the data with  = 0 and  = 0.1, the lower bound with  = 0.1 and  = 0.3 and the

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trend line with  = 0,  = 0.2. The trend line was obtained by comparing with the

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experimental data of (8/f)1/2 with the results predicted using Eq. (18) for a series of  and 

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combinations. The comparison shows that the average of the absolute error minimizes when

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taking  = 0 and  = 0.2. Also shown in Fig. 4 are Manning-Strickler formula [Eq. (3) with St =

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0.12], Keleugan formula [Eq. (4)] and Ferguson’s formula [Eq. (7)], the latter representing

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field measurements. Fig. 4 shows that Manning-Strickler formula overestimates the value of

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(8/f)1/2 for low submergence, the upper bound of the laboratory measurements of (8/f)1/2 is

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close to Keulegan formula, and the lower bound is close to Ferguson’s formula. 11

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Next, Eq. (18) is compared with field data (376 sets in total), which were compiled by

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Ferguson (2007) from nine different sources. The same data was used by Ferguson (2007) for

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fixing the two empirical coefficients in Eq. (7). Fig. 5 shows that the field data are generally

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confined between the upper bound [Eq. (18) with  = 0 and  = 0.1] and the lower bound

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[Eq. (18) with  = 0.4 and  = 0.6], and the trend line of the data can be described well by

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Eq. (18) with  = 0.1 and  = 0.25. It can be also observed that the trend line agrees well

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with Eq. (7), which is expected because of the same database used for calibration. It should

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be mentioned that due to different choices of St, there is an offset between Ferguson’s

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formula and Eq. (18) at the deep-flow side of the plot.

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In addition, by comparing Figs. 4 and 5, it is noted that the trend line derived from

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the laboratory data is slightly different from that from the field data. For a given r/D84, the

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value of (8/f)1/2 observed in field is generally lower (and thus the resistance is larger) than

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the laboratory measurement. In other words, for a given flow depth, the same friction factor

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can be observed for both laboratory and field conditions provided that a greater roughness

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element is employed in a laboratory experiment than in field. This is understandable by

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noting bed surface irregularities that are usually higher in field than in a laboratory setup

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even for a planar channel bed. With the two trend lines shown in Figs. 4 and 5, it can be

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estimated that for a given flow depth, about 50% increase in D84 is needed in a laboratory

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experiment to achieve the same friction factor as in field for r/D84 = 0.6-10.

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Discussions

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In the model derivation, the variation in the channel bed elevation is considered only with a 12

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streamwise-vertical slice, as shown in Fig. 3. It should be noted that similar variations exist

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also in the direction across the channel. For a shallow flow over such a ‘corrugated’ channel

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bed, the water surface would not be planar, and the flow acceleration/deceleration would

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be somewhat less than what has been calculated in the derivation. For example, if the bed

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consists of an array of hemispherical stones, some lateral deflection of flow would occur

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through the gaps between obstacles, which may result in less acceleration/deceleration over

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their tops. Therefore, the head loss involved in the model derivation should be generally

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replaced with a width-averaged head loss. This is worthy of further efforts in the future.

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Eq. (18) fits well to both laboratory and field data. This can be attributed, in part, to

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the use of the adjustable coefficients,  and . However, on the other hand, from the

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derivation, it follows that both  and  are physically linked to geometrical properties of the

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formed channel bed. For the regular bed surface, as sketched in Fig. 3, D measures the

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average distance from the tops of obstacles to the mean bed level, while D quantifies the

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average of the largest variations in the bed surface elevation. Therefore, for some two-

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dimensional but regular bed configurations, it is possible to theoretically fix the two

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coefficients based on the bed geometry. However, in the presence of irregular bed surfaces

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as in real river beds, it may be necessary to apply a statistic or probabilistic approach to the

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evaluation of the two coefficients. Recently, Coleman et al. (2011) reported that for water-

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worked gravel beds, the crest-to-trough roughness height, hct, scales with the standard

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deviation of bed elevations, b. They assumed that hct is equivalent to the median sediment

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diameter and then obtained that hct = (1.8-2.4)b with b = 0.22D84. If taking hct = βD84, β can

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be estimated to be 0.40-0.53. This range of β appears greater than β (= 0.2) used for the

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laboratory data trend in Fig. 4 and β (= 0.25) for the field data trend in Fig. 5.

13

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In addition,  and/or  may vary with the sediment size distribution. In the forgoing

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calibration, D84 is used to be the representative sediment diameter. If D84 is replaced by a

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diameter with another percentile, both  and  may have different values in the fitting of

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Eq. (18) to the data. This implies that one or both of the two coefficients may also depend

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on the sediment size distribution.

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Finally, it should be mentioned that the proposed approach to the evaluation of

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coarse-bed resistance applies only for the condition of planar rough beds. However, there

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still exist other factors that may affect flow resistance significantly, particularly in natural

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channels. For example, cross sections of natural stream channels can vary significantly in size

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and shape, even within a short reach, which increases resistance to flows (Wohl 2010).

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Further increases in flow resistance in natural coarse-bed streams occur in the presence of

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pools and riffles (Maxwell and Papanicolaou 2001), boulders (Papanicolaou et al. 2012),

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vegetation (Cheng 2011a; Darby and Thorne 1996), and large woody debris (Manga and

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Kirchner 2000; Montgomery and Buffington 1997). Therefore, when fitting the coefficients α

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and β to flow resistance measurements, the fitted values account for the additional effects

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only in an average sense. This also explains why the large spreading of the data exists about

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the “trend line”, particularly for values of r/D84 < 1. For example, Fig. 5 shows that the

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measurement of 8 f at r/D84 = 0.3 varies from 0.03 to 2, a wide range in comparison to

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0.75 calculated according to the proposed trend line.

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Given the uncertainty in the prediction of resistance for large-scale roughness in

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natural channels, the present approach may be further improved by taking into account the

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other factors as mentioned above. For example, for a gravel bed subject to boulders, the bed

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resistance may be partitioned into gravel-bed and boulder-affected components so that they

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can be evaluated individually. Such an improvement could be made based on characteristics 14

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of open channel flows over boulder-affected gravel beds, such as those presented by Thanos

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Papanicolaou’s group in their recent studies (e.g. Hajimirzaie et al. 2014; Papanicolaou et al.

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2011; Papanicolaou et al. 2012). Papanicolaou et al. (2012) investigated effects of a fully

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submerged, wall-mounted boulder placed on a flat rough bed on mean and turbulent flow

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fields. Their results show that the form roughness could be up to two times larger than the

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skin roughness in the near-wake region of the boulder. Papanicolaou et al. (2012) also

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reported that in comparison to a flat rough bed, the location of the maximum turbulence

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intensity shifted away from the bed due to the vortices generated by the boulder. As a

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result, the presence of boulder would enhance the vertical mixing near the bed and also

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cause significant energy dissipation. For example, the average reach-averaged bed shear

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stress increased by 30% when 40 isolated boulders were placed over a gravel bed, covering

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only 2% of the bed area (Papanicolaou et al. 2011).

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Summary

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In this study, the rough bed surface is simplified as a periodic boundary to quantify energy

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losses induced by the large scale roughness in open channel flows. The obtained analytical

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resistance formula [Eq. (18)] is applicable for both deep and shallow flow conditions. It

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reduces to the traditional Manning-Strickler formula when the flow depth is much greater

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than the sediment size. The new formula was calibrated separately using laboratory and field

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data. It is consistent with the best of previously-developed formulas in the prediction of flow

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resistance for r/D84 = 0.1-10.

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Acknowledgements

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The author is grateful to the comments and additional references, provided by the reviewers

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and editors.

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Notation

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The following symbols are used in this paper:

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B

= channel width;

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C

= Chezy coefficient;

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D

= sediment grain diameter;

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D84

= grain diameter of which 84% of the sediment is finer;

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f

= Darcy-Weisbach friction factor;

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fb

= bed friction factor;

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fw

= sidewall friction factor;

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g

= gravitational acceleration;

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H

= nominal flow depth;

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HC

= head loss due to contraction;

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HE

= head loss due to expansion;

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Hf

= head loss due to friction;

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HL

= total head loss;

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h

= average flow depth;

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h1,h2,h3 = local flow depth;

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hct

= crest-to-trough roughness height of sediment beds; 16

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k

= representative roughness length;

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L

= wave length;

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n

= Manning coefficient;

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q

= flow rate per unit width;

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r

= hydraulic radius;

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R

= 4rV/ = Reynolds number;

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S

= energy slope;

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St

= Strickler number or normalised Manning’s n;

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U1,U2 = local depth-averaged velocity;

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V

= cross-sectional average flow velocity;

360



= coefficient;

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

= coefficient;

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

= kinematic viscosity of fluid;

363



= h/D = relative flow depth; and

364

b

= standard deviation of the bed elevation.

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References

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510

20

Table Click here to download Table: Table 1.docx

Table 1. Summary of laboratory data used for calibration Investigator

Slope

Froude

Reynolds

D84

number

number

(mm)

1/2

Paintal (1970)

0.00117-

U/(gh)

4Uh/

0.43-0.98

7.410 -

0.02-0.08

dataset

2.9-24

4.0-39.9

34

11.5-58

0.3-6.1

88

2.2-15

1.1-35.9

168

14.3-54

0.8-10.1

50

2.4-9.6

1.2-34.7

64

47

0.2-2.6

12

9.910 0.19-1.93

3

4.410 5

(1981) Ho (1984)

Number of

5

0.0103 Bathurst et al.

4

h/D84

2.410 0.002-0.1

0.55-1.72

3

2.710 5

2.110 Cao (1985)

0.005-0.09

0.44-1.51

4

4.710 6

1.310 Recking (2008)

0.01-0.05

0.49-1.25

4

1.210 5

3.210 Jordanova

0.0011-

(2008)

0.0021

0.07-0.28

3

1.010 5

1.310

Figure1 Click here to download Figure: fig 1.pdf

10

8   f

1

Manning‐Strickler, Eq. (3) Keulegan, Eq. (4) Hey, Eq. (5) Katul et al., Eq. (6) Ferguson, Eq. (7) Rickenmann & Recking, Eq. (8)

0.1 0.1

1

      Fig. 1    Comparison of previous formulas   

 

 

10

h/D84 

100

 

Figure2 Click here to download Figure: fig 2.pdf

                     

h 

    Fig. 2    Open channel flow subject to large‐scale roughness (0