There are a plethora of flow resistance formulae available for sand-bed river channels, but river scientists and engineers have been facing the very difficult problem of selecting a convincing one in practical problem solving (Zhang et al., 2020). Although the following Manning resistance formula (Manning, 1890; Herschel, 1897) was developed a long time ago to calculate the resistance to flow in fixed-bed open channels, it still is widely applied to determine the flow resistance of sand-bed river channels: V = 1 n R 2 / 3 J 1 / 2 (1) where V is the average velocity of channel flow, R is the hydraulic radius of the channel, J is the energy slope of flow, and n is the Manning roughness coefficient.

Because the roughness of a riverbed can be reflected by the size of the sediments composing the bed, the Manning roughness coefficient n in Eq. 1 has frequently been determined using the following relationship: n = D 1 / 6 / A (2) where D and A are a representative bed sediment size and a roughness parameter, respectively.

The sediments composing a riverbed, however, are hardly uniform, and there has been no consensus of opinions on what size of the non-uniform sediments can represent the role of D in Eq. 2. While the study by Chang (1939) granted support to the use of D ₅₀, nevertheless, Meyer-Peter and Muller (1948) argued that it is more appropriate to use D ₉₀ to represent D and consequently found A = 26. However, Maynord (1991) used a large number of flume data collected from various sources and identified that the use of particle size D ₉₀ gave slightly better results than the use of D ₅₀, with A taking 21.6 and 22.8 respectively for D ₅₀ and D ₉₀. In the reformulation of the Meyer-Peter and Muller (1948) equation, Huang (2010) also discovered that utilizing D ₅₀ or D ₉₀ had no effect on the accuracy of flow resistance calculations.

The roughness of a channel can be divided into grain roughness and form roughness for a riverbed with bedforms (Einstein, 1952; Engelund, 1961). While it has been assumed that a representative size of riverbed sediments, such as D ₉₀ or D ₆₅, may accurately reflect the grain roughness of a flat riverbed (Engelund and Hansen, 1967; van Rijn, 1982; Kamphuis, 1974; Zhang et al.,2012a; Zhang et al., 2012b; Niazkar et al., 2019), form roughness is closely related to the strength of flow acting on a channel bed, usually reflected with the Shields parameter (Yalin, 1963; van Rijn, 1984). Using data from the Compendium of Solids Transport Data compiled by Brownlie (1981), nevertheless, Peterson and Peterson (1988) made a comparison of the bed roughness values between observational data and the results calculated from different flow resistance formulae and commented that the commonly applied formulae with the Shields parameter as the sole factor were insufficient. Importantly, they demonstrated that the roughness of a mobile bed is also a function of the Froude number (Fr) and a dimensionless settling velocity (Zhang et al., 1998). On the basis of both physical reasoning and dimensional analysis, Wu and Wang (1999) proposed a relationship to determine the roughness of a mobile bed by relating the roughness parameter A in Eq. 2 to a non-dimensional shear stress parameter and Fr. Importantly, they tested the relationship with a large number of data observed in experimental flumes and fields collected from different sources and demonstrated that the values of n computed using their relationship yielded results consistent with the measured data at a much higher level than when using the other methods. Table 1 presents the selection of a representative sediment size D and the corresponding value of roughness parameter A by different investigators.

In contrast to the approach of selecting a representative sediment size D in Eq. 2, many studies determined the value of the Manning roughness coefficient n in Eq. 1 directly from a statistical analysis of field observations. Using field observations from the lower Yellow River, Chien and Wan (1983) analyzed the variation of n at various flow and sediment conditions and consequently recognized that n is closely related to both of the relative strength of tractive force and sediment size D ₆₅. Zhao and Zhang (1997) presented a detailed physical analysis of the effects of the rough thickness of channel bed on flow velocity, and a relationship expressing n as a complex function of flow depth, sediment size D ₅₀, and Fr received a high level of validation with a larger number of field observations obtained from the lower reach of the Yellow River. By reanalyzing the flume experimental data provided by Guy et al. (1966) and field observations at stations located in the middle and lower reaches of the Yellow River, however, Deng et al. (2007) identified that n had a poor relationship with flow discharge and depth, but increased initially and then decreased with an increase in sediment concentration. Importantly, they found that n had a good relationship with Fr and that a power function was shown to be fitting the collected data well.

Using 64 sets of data observed in experimental flumes and more than 1,000 sets of data measured during 1958–1959 at six hydrological stations located in the lower reach of the Yellow River, Ma et al. (2017) evaluated the performance of three roughness formulae proposed respectively by van Rijn (1984), Zhao and Zhang (1997), and Qin et al. (1995). They identified that the formula of Zhao and Zhang fitted best the observations from natural rivers, while the van Rijn formula performed well in laboratory flumes. In addition, Ma et al. (2017) reanalyzed the collected data, and their statistical regression results demonstrated that n showed a very close relationship with Fr. Recently, Zhang et al. (2020) presented a systematic review of the progress on the determination of the roughness of a moving bed channel and highlighted the need to treat Fr as a primary parameter in the establishment of a practically useful flow resistance formula. Consequently, they proposed two formulae to determine n. The first formula is in a relatively simple form considering the effect of Fr only, while the second formula is in a comprehensive form taking into account the effects of Fr, sediment size D ₅₀ , roughness parameter A, and the von Kármán constant that reflects the effect of sediment concentration on flow energy consumption. While complex in form, the second formula had a much higher level of accuracy. The validation of the two roughness formulae clearly showed that the results generated by both were in a good agreement with a large number of data measured from natural rivers, although deviating slightly from observations in experimental flumes.

It is apparent that most flow resistance formulae were derived with reference to the data observed from experimental flumes and/or from natural rivers collected or conducted by original author(s), supplemented with some data obtained by one or several other researchers. Furthermore, the performance of a formula has often been assessed tentatively based on its agreement with a limited amount of data collected. In addition, there is considerable overlap of the data employed in the development of many formulae, with data incorporated in the subsequent evaluations of their performance. All of these contributed to the proliferation rather than the consolidation of flow resistance formulae for mobile-bed channels (Mark Powell, 2014; Ferro, 2018a; Ferro, 2018b; Di Stefano et al., 2020; Carollo and Ferro, 2021; Nicosia et al., 2021; Yadav et al., 2022). So far, none of the previously developed formulae has been either universally accepted or recognized as being especially appropriate for practical application. Although the performance of many formulae had been evaluated under a relatively wide range of hydraulic conditions, either from laboratory and/or from the field (e.g., Huang et al., 2004; Zhang, 2012a; Zhang, 2012b; Zhang et al., 2020), a collection of more observational datasets has become possible recently, and it is necessary to evaluate which flow resistance formula can give better predictions. For this purpose, this study selected flow resistance formulae for mobile-bed channels that have been most quoted in the literature and collected observational data wide in varying ranges and large in number from various sources. A set of statistical indicators was then used to evaluate the performance of the selected formulae against a large observational dataset collected in this study.

The selection of formulae for evaluation in this study was mainly in terms of popularity and the number and range of data used for developing them. From a detailed literature review of the progress on the development of the formulae, those developed by Wu and Wang (1999), Deng et al. (2007), Ma et al. (2017), and Zhang et al. (2020) satisfied these criteria and so were selected.

The flow resistance formula developed by Wu and Wang (1999) takes a form of: l g A g 1 / 2 F 1 / 3 = 0.911 − 0.273 l g T − 0.051 ( l g T ) 2 + 0.135 ( l g T ) 3 (3) where g is the acceleration due to gravity, Fr is the Froude number ( = V / g h , where h is the flow depth), and   T is the non-dimensional shear stress of flow acting on the riverbed sediments.

To obtain the value of T in Eq. 3, the following relationships need to be applied:       T = τ ′ / τ c 50 ;         τ ′ = ( n ′ n ) 3 / 2 τ ;         n ′ = D 50 1 / 6 20 ;       τ   =   g R J (4) where   τ ′ , τ c 50 , n ' , and τ are respectively the shear stress of flow acting on the riverbed sediments, the critical shear stress with D ₅₀ representing the size of bed sediments, the grain roughness coefficient, and the shear stress of flow acting on the channel boundary, in which the critical shear stress τ c 50 was determined from the following relationships given by Chien and Wan (1983) after their modification of the Shields curve:   τ c 50 ( γ s − γ ) D 50 = 0.126 D ∗ − 0.44 ;         D ∗ = D 50 ( γ s − γ γ g ν 2 ) 1 / 3 (5) where ν is the kinematic viscosity coefficient in square centimeters per second cm²/s.

By reanalyzing the flume experimental data provided by Guy et al. (1966) and field observations at stations located in the middle and lower reaches of the Yellow River, Deng et al. (2007) obtained the following power–function relationship between n and Fr by using a statistical regression method: n = 0.005 F r − 0.7336 (6) Using 64 sets of data observed in experimental flumes and more than 1,000 sets of data measured during 1958–1959 at six hydrological stations located in the lower reach of the Yellow River, based on data from both experimental flumes and field measurements in the Lower Yellow River, Ma et al. (2017) developed the following regression formula: n = − 0.0124 l n F r + 0.0009 (7)

Zhang et al. (2020) proposed two flow resistance relationships: the simple one, referred to as formula 1 of Zhang et al. in this study, takes the form n = 0.01 0.1 + 1.85 F r (8)

When the influences of bed materials and sediment concentration on flow energy expenditure were taken into account, Zhang et al. (2020) presented a more comprehensive formula, referred to as formula 2 of Zhang et al. in this study, taking the form   n = 0.9 D 50 1 / 6 ( κ / κ 0 ) 1 / 5 A 50 ( 0.1 + 1.85 F r ) (9) where κ is the Karman coefficient of flow, κ 0 = 0.4 . κ has a relationship with sediment concentration S V in the form κ = κ 0 ( 1 − 4.2 S V ( 0.365 − S V ) ) (10) The sources and other details pertaining to the development of the five formulae selected above are summarized in Table 2.

To evaluate the performance of the five formulae against the large datasets collected in this study, several statistical parameters were adopted in this study. Table 4 shows the expressions of these parameters. RMSE is the root mean square error between the observed data and computed results, CD is the square of Pearson’s correlation coefficient that reflects the proportion of the total variance in the observed data, and NA is the Nash coefficient that evaluates the agreement between the computed and observed values, with NA = 1 indicating perfect agreement between the computed and observed values. Furthermore, ARE (absolute relative error), computed as the ratio of the absolute error between the prediction and observation results to the observed value, was used to depict whether the predicted value was overestimated or underestimated. RE (relative error) was also used, and an MSE = 0 means that the computed value agreed perfectly with the observed value on the whole, while RE > 0 or RE < 0 indicate respectively that an overestimation or an underestimation occurred. Moreover, the percentages of data with relative errors of ≤50% and 25% (P ₅₀ and P ₂₅, respectively) and the percentages of the data with an overestimated error (POE) were also deployed (Nash and Sutcliffe, 1970; Bjerklie et al., 2005; López et al., 2007).

The AREs of n (computed n − Observed n)/Observed n) by all five selected formulae were computed for the three datasets, i.e., the laboratory data and field data compiled by Brownlie (1981) and the data from the drainage basins of the Yellow and Yangtze Rivers collected in this study. The distribution of AREs in the six groups of 0–0.05, 0.05–1, 0.1–0.2, 0.2–0.5, 0.5–1, and >1 is presented in Table 9 and shown in Figure 5. In the laboratory data of Brownlie, the AREs in the group 0.2–0.5 by all five selected formulae occupied around 60%, and there was a very small difference among the percentages in the group for the five formulae. The AREs in the group 0.5–1 occupied around 20%, while the AREs of each of the other groups occupied a very small percentage (Figure 5A).

In the field data of Brownlie, however, the AREs in the groups of 0.2–0.5 and 0.5–1 by all five formulae occupied large and nearly equal percentages of around 35%, while those in each of the other groups occupied only a very small percentage (Figure 5B). In the data from the Yellow and Yangtze Rivers, the AREs in the two groups of 0.1–0.2 and 0.2–0.5 by all five formulae occupied large and nearly equal percentages of slightly larger than 30%, while the AREs in the two groups of 0.05–0.1 and <0.05 occupied nearly equal percentages of about 18% (Figure 5C). Overall, it was obvious that the relative errors in the range of 0.2–0.5 accounted for the biggest percentage, with each of the nearby groups of 0.5–1 and 0.1–0.2 accounting for a sizable portion as well. This clearly manifests that the AREs by all five flow resistance formulae occupied a very large percentage almost in the same range of 0.1–1; hence, the five formulae needed to improve their predicting ability. Importantly, it is noticeable from Table 9 and Figure 5 that the AREs by each of the five selected flow resistance formulae were considerably large among the three datasets used.

The formula of Wu and Wang performed better in the laboratory and field data of Brownie, with ARE <30%. In the data from the Yellow and Yangtze Rivers, nevertheless, ARE >50% and the group of ARE >100% still accounted for 14.67% of the total. The formula of Deng et al. considerably yielded large values of 41.3% and 44.6% for ARE in the laboratory data and field data of Brownlie, respectively. In the data from the Yellow and Yangtze Rivers, nevertheless, the formula allowed ARE to take a relatively small value of 22.3%, and the group of ARE <0.5 occupied a very large percentage of 97.37% of the total. The formula of Ma et al. also yielded considerably large values of 44.37% and 33.1% for ARE in the laboratory data and field data of Brownlie, respectively. In the data from the Yellow and Yangtze Rivers, the formula allowed ARE to take a considerably small value of 18.78%, and the group of ARE <0.5 occupied 97.74% of the total. Hence, the formulae of Deng et al. and Ma et al. performed best in the data from the Yellow and Yangtze Rivers among the three datasets used. Formulae 1 and 2 of Zhang et al. yielded relatively small values of <40% for ARE in the laboratory data and field data of Brownlie, respectively. In the data from the Yellow and Yangtze Rivers, meanwhile, the formulae allowed ARE to take considerably small values of 19.32% and 17.96%, and the group of ARE <0.5 occupied 96.39% and 95.54% of the total. Hence, formulae 1 and 2 of Zhang et al. both performed well in the data from the Yellow and Yangtze Rivers among the three datasets used.

Because different formulae performed at considerably different levels in the different datasets used, ARE appears a suitable index to evaluate the performance of the five selected flow resistance formulae in the three datasets. Assuming that the performance of a formula can be ranked to the highest level “I” when 0 < ARE < 20%, to the relatively high level “II” when 20% < ARE < 30%, to the moderate level “III” when 30% < ARE < 40%, to the relatively low level “IV” when 40% < ARE < 50%, and to the lowest level “V” when ARE > 50%, the detailed ranks of the performance of the five selected flow resistance formulae in the three datasets are provided in Table 10.

It can be clearly seen from Table 10 that the formula of Ma et al. and formulae 1 and 2 of Zhang et al. gained the highest rank I in the data from the Yellow and Yangtze Rivers, while they gained the moderate rank III or the relatively low rank IV in the laboratory and field data of Brownlie. In contrast, the formula of Wu and Wang gained the relatively high rank II in the laboratory and field data of Brownlie, yet the lowest rank V in the data from the Yellow and Yangtze Rivers. Within the two relatively extremal cases, the formula of Deng et al. gained a relatively low rank IV in the laboratory and field data of Brownlie and also a relatively high rank II in the data from the Yellow and Yangtze Rivers.

Among the numerous flow resistance formulae for sand-bed channels, this study selected five for evaluation in terms of theoretical advances and the scope and quantity of the data used in the development, including those developed by Wu and Wang (1999), Deng et al. (2007), Ma et al. (2017), and Zhang et al. (2020). In order to cover flow conditions in sand-bed river channels as widely as possible, this study collected 1,636 sets of field measures from the hydrological stations in two large river systems of China (the Yellow and Yangtze Rivers), in addition to the data compiled by Brownlie (1981) that, in complete form, included 3,623 sets of laboratory data and 1,546 sets of field data from different nations and areas. The conclusion of this paper is generally used in alluvial rivers. A detailed statistical analysis of the performance of the five selected flow resistance formulae in yielding the values of the Manning resistance coefficient n using the very large number of datasets, 6,805 sets in total, led to the following important findings:

1) The formula of Wu and Wang yielded n values best fitting the laboratory and field data of Brownie, yet fitting the data from the Yellow and Yangtze Rivers at a relatively low degree. In contrast, the formula of Ma et al. and formulae 1 and 2 of Zhang et al. yielded n values fitting the data from the Yellow and Yangtze Rivers at a very high degree, yet fitting the laboratory and field data of Brownie at a moderate degree. The formula of Deng et al. yielded n values fitting the laboratory and field data of Brownlie and the data from the Yellow and Yangtze Rivers both at a relatively low degree.