Reach-averaged flow resistance in gravel-bed streams Open Access

Flow resistance is important for predicting floods and sediment transport in plain and mountain streams. It also plays an increasingly significant role in bank stability, protective engineering design, and aquatic ecosystems (Ferguson 2007). In essence, flow resistance consists of skin friction, which is caused by the fluid viscosity and is proportional to the contact area of the interface between the flow and the boundaries, and form drag, which is mechanically the pressure differences due to flow separation around bed forms or bed structures and is related to the scales of the obstacles (Giménez-Curto & Lera 1996; Nikora et al. 2001, 2004).

Early studies focused on deep water zones with relatively small roughnesses, such as plain rivers. The Manning formula (Manning 1891) is an empirical relation that accounts for the flow velocity, hydraulic radius, and channel slope in plain rivers. Keulegan (1938) derived the famous log-type resistance formulas for open channel flow based on the theoretical investigations of Prandtl and von Karman and the experimental data of Nikuradse. The resistance relation can be explicitly related to the characteristic roughness height in the log-type formula for fully developed turbulent boundary layers under the condition of high submergence (Katul et al. 2002). Subsequent studies mostly used the log-law formula structure. Considering plain rivers with large bed forms, van Rijn (1984) divided the bed roughness into the grain roughness and the bed form roughness. He related the bed form roughness to the scales of the bed forms (e.g., wavelengths and heights of dunes) and developed resistance relations for this case. However, theoretical solutions of bed forms are still limited due to the complex characteristics of the bed surface and the difficulty of making precise measurements. Einstein & Barbarossa (1952) addressed the idea that bed forms are a function of sediment transport along the river bed. Sediment transport is itself a function of the skin friction τ′ on the sediment grains and of the representative grain diameter d. He then established the relation between the form drag coefficient f″ and a dimensionless Einstein flow parameter d/R′. Although Einstein's form drag relation was initially developed for plain rivers, their thinking on form drag production is also applicable to the mountain streams.

Quantifying flow resistance becomes more difficult under the conditions of shallow water zones with relatively large roughnesses, such as gravel-bedded streams, which feature low submergence, steep slopes, wide grain size distributions, different kinds of bed structures (e.g., step-pool system, cascades) and sharp variations in the streamwise direction. Under these conditions, the flow is extremely unsteady and is affected by wakes caused by sediment particles (Nowell & Church 1979; Schmeeckle & Nelson 2003) within the boundary layer, which is underdeveloped. To our knowledge, no unified theory for the velocity distribution in the roughness layer has been developed (Smart et al. 2002; Lamb et al. 2008). Several studies (Byrd & Furbish 2000; Wohl & Thompson 2000; Mclean & Nikora 2006; Ferguson 2007) have demonstrated that the velocity distribution deviates from the logarithmic form in the gravel-bed streams. Katul et al. (2002) developed a flow resistance model for shallow water using a mixing layer analogy for the inflectional velocity profile within and just above the roughness layer. The mixing layer theory can apply to the prediction of the roughness of gravel-bed streams. However, the hyperbolic tangent velocity profile is not appropriate for steep slopes. Many researchers (Hey 1979; Smart 1984; Bathurst 1985; Maxwell & Papanicolaou 2001) have used the logarithmic form of the Keulegan formula (1938) to derive semi-empirical resistance relations based on field data. Ferguson (2007) proposed a variable power resistance equation that is asymptotic to the Manning–Strickler formula for large relative flow depth and roughness-layer formulations for small relative flow depth. This equation can be applied to both deep and shallow water zones.

Previous studies have mainly focused on empirically quantifying the form drag in the shallow gravel-bed streams as the skin friction is nearly negligible, and the form drag makes up almost all of the total flow resistance (MacFarlane & Wohl 2003; Rickenmann & Recking 2011). Several researchers (Raju et al. 1983; Shields & Gippel 1995; Green 2006) have applied the ratio of the frontal area of an object to the cross-sectional flow area as the blockage ratio to measure form drag. Aberle et al. (1999) used the standard deviation of the bed surface elevation to predict the friction factor. Pagliara & Chiavaccini (2006) incorporated the effect of the arrangement, shape, and distribution density of boulders into a parameter to measure form drag. Qin & Ng (2012) used a semivariogram approach to derive a steepness parameter to express form drag. Other researchers (Wohl & Thompson 2000; Maxwell & Papanicolaou 2001; Curran & Wohl 2003; Church & Zimmermann 2007) have focused on step-pool channels and established resistance relations based on the step geometry. In addition, Wilcox et al. (2006) established a resistance relation for large woody debris using the cylinder drag method.

The boundaries are always non-uniform in plain and mountain streams. The hydraulic variables defined in the cross-sections can not reflect the non-uniformity of the boundaries on the flow. Moreover, in shallow water zones with relatively large roughnesses, the flow is unsteady, and the boundary varies dramatically. The concept of cross-sectional resistance is actually unreal. In addition, the pressure differences in the streamwise direction are uneven. Therefore, it is needed to define the hydraulic radius, friction factor, and roughness parameter at the river reach scale to quantify the reach-averaged flow resistance.

Water flow in gravel-bed streams is extremely non-uniform due to various river bed forms/structures, such as step-pools, riffle-pools, cascades, and isolated large particles. It is critical to explore the effects of the spatial heterogeneity and boundary non-uniformity of a given river reach on the fluid hydraulics. The extension of the hydraulic variables from cross-sectional scale to reach scale can reflect the flow unsteadiness and boundary non-uniformity. Mclean & Nikora (2006) accounted for the spatial heterogeneity in the near-bed region of sand- or gravel-bed rivers by double-averaging (in space and in time) the continuity and momentum equations, but there are still several difficulties when the double-averaged hydrodynamic equations are applied in the field. Smart et al. (2002) defined the ratio of the overlying water volume to the plan area of the bed as the volumetric hydraulic radius within a reach. Yager et al. (2007) used the volumetric hydraulic radius to calculate bed load transport. Yang (2013) defined the hydraulic radius by the ratio of the water volume to the wetted area within a reach to describe the three-dimensional hydraulics. He argued that the form drag is related to the separation zone or dead zone after the bed forms or structures and proposed the ratio of the volume of the separation zone to the water volume as the new form drag roughness parameter in three-dimensional hydraulics.

In the paper, we develop an alternative resistance relation to predict the mean flow velocity for both deep and shallow water zones in gravel-bed streams. The river reach is taken as the research object and the concepts of hydraulic elements and flow resistance which were defined in cross-sections are extended to river reach. A reach-averaged resistance model is established in consideration of boundary non-uniformity and flow heterogeneity. The cross-sectional hydraulic elements are re-defined in river reach, such as hydraulic radius, friction factor, and roughness parameter. Flow resistance is partitioned into skin friction and form drag according to their different production mechanism by force balance analysis of the water body in the river reach. The skin friction in the reach is estimated by the Manning–Strickler formula and the form drag is modeled by a modified Einstein form drag parameter. 780 field data measurements are applied for calibration and validation of the new proposed resistance relation. The multiple regression analysis method is used to derive the coefficients of the reach-averaged resistance relation. Validation results show the reach-averaged resistance relation fit the data well and behave best among the existing widely used flow resistance relations.

The datasets used in this study, which are collected from 12 different sources, are summarized in Table 1. Parts of these data have been used by Rickenmann & Recking (2011).

Details about data sources are described below. Data of Church & Rood (1983) were from essentially sinuous and meandering channels. Colosimo et al. (1988) measured the data in 43 reaches with quasi-uniform flow conditions. Bathurst (1978) obtained the data from three straight and uniform boulder reaches. Bathurst (1985) measured the data in straight riffles-pool reaches or over plane beds without pools. Thorne & Zevenbergen (1985) monitored the data in two uniform boulder streams. Jarrett (1984) got the data from straight and uniform channel reaches with minimal vegetation. Orlandini et al. (2006) obtained the data in step-pool streams. Griffiths (1981) measured the data in 72 straight wide trapezoidal reaches with little vegetation in New Zealand rivers. Hey & Thorne (1986) obtained the data in the self-formed channel in erodible material with riffle-pools and bank vegetation cover. Wohl & Wilcox (2005) obtained the data in step-pool streams. Zhang et al. (2013) monitored the hydraulic elements in Longxi River, Sichuan Province, China. Zhang (2012) measured the data in Yunnan Province, China.

Inevitably, some data may not be correct due to either measurement error or somewhat human factors. It is a heavy workload to review the details of all the data. Rickenmann & Recking (2011) gave out the criteria for a rough selection of the data, which were that friction values 30% higher than values predicted by Keulegan law or 30% lower than values predicted by the resistance law of Recking et al. (2008) would be removed. Then, the data which did not satisfy the continuity equation were also excluded. In the paper, we will use the retained data to do our further analysis.

As can be seen, a total of 780 field measurements are compiled in Table 1. These data cover wide ranges of slopes (0.004–28.7%), discharges (0.04–16,696 m³/s), d₉₀ (0.6–1,687.5 mm), and the relative flow depth R/d₉₀ (0.16–17,935.92). The data represent field flow data measurements of gravel-bed streams with different bed morphologies.

Among these data, flow velocity was measured directly by salt dilution methods or a current meter, and could also be concluded from the continuity equation. The sediment grain size distribution was obtained by the Wolman method or sieving of several representative samples. Actually, the existing general approaches for observing and processing field data (Bathurst 1985; MacFarlane & Wohl 2003; Comiti et al. 2007; Nitsche et al. 2012; Zhang et al. 2013) are to use the average values of the measured hydraulic elements of several cross-sections that are preselected in the reach to represent the reach-averaged values. In other words, most of the collected flow data are reach-scaled by averaging several cross-sectional values.

The paper divides the data into two parts, including 512 data measurements for Part A, which cover slopes of 0.004–28.7% and R/d₉₀ values of 0.17–17,935.92, and 268 for Part B, which cover slopes of 0.009–28.7% and R/d₉₀ values of 0.16–74.94. Part A is used for calibration of the new-derived form drag relation in the following paragraph, and Part B is used for validation of the new-derived form drag relation in the following paragraph.

Great efforts have been made to extend the Manning and Keulegan formula to shallow water zones, including representative studies from Bathurst (1985), Ferguson (2007), and Rickenmann & Recking (2011).

Details about the studies of the representative flow resistance formulae are as follows:

To evaluate the performance of the representative equations listed above and give a visualized display, comparison of these equations is plotted against the dataset in Part B.

Figure 1 shows that these equations fit the data well in the deep water zone but badly in the shallow water zone. The theoretical foundations and structures of these equations are basically based on the logarithmic law of the Keulegan formula (1938) for open channel rough flows. Hence, there is no doubt that these classical equations approach each other in the range of large relative depth R/d. But with R/d smaller than 10, they diverge as the hydraulic mechanics change in mountain streams and the boundaries are more complex.

The datasets in Figure 2(a) are scattered and it illustrates that the Einstein form drag relation cannot apply in gravel-bed streams very well. Considering slope S may have a significant correlationship with d, the paper did the step-wise regression analysis on d/R′ and S by the SPSS Statistics software. The result shows that the slope S can be excluded and the relative roughness height d/R′ is reserved. Figure 2(b) shows that the performance of the Einstein form drag relation improves especially in the range of d/R′ larger than 1, which is nearly the shallow water zone. While in the range of d/R′ smaller than 1, which is nearly the deep water zone, the data points are still scattered. This implies that the form drag relation proposed by Einstein & Barbarossa (1952) is more suitable for the mountain streams than the plain rivers. This may be ascribed to the fact that the boundary conditions of plain and mountain streams are different. In mountain rivers with small relative flow depth, the sizes of the large particles are similar to the dimensions of the bed structures (Chin 1989, 1999; Yager et al. 2007). Large particles are the dominant nodes in the channel morphology process and have significant impacts on the development of bed structures, such as step-pool systems, cascades, and riffle-pools (Church & Zimmermann 2007). These bed structures are the primary sources of flow energy dissipation. However, in plain rivers with relatively high submergences, the sizes of individual particles are much smaller than the dimensions of the bed forms, which are responsible for a large proportion of the energy dissipation. Consequently, the sediment grains cannot dominate the dimensions and development of bed forms and the energy dissipation in plain rivers. Therefore, Einstein's form drag relation behaves better in mountain streams than in plain rivers.

On account of this, this paper attempts to propose an alternate resistance relation covering from the lower values to the higher values of R/d in gravel-bed streams.

Figure 3 illustrates the procedure of the methodology of deriving the form drag relation in gravel-bed streams. The paper defines the hydraulic variables in the river reach scale and solves the reach-averaged form drag. Considering the mechanism of the formation of bed forms or structures, the Einstein flow parameter and a relative roughness parameter by Yang (2013) are taken as the main impact factor of the reach-averaged form drag. Then, a reach-averaged form drag relation is obtained by applying the flow data.

In order to describe the effect of boundary non-uniformity on flow resistance, the paper defines the reach-averaged hydraulic variables in a given river reach.

Figure 4 shows water flow in a non-uniform river reach with bed forms or structures. Figure 5 shows water separation after bed forms or structures. U is the average flow velocity in the river reach, V_w is the volume of water within the reach, and A_w is the wetted area of the interface between water and the boundary. V_dz is the maximum volume of the separation zone after the irregularities, and L_dz is the length of the separation zone from the irregularities to the reattachment point.

Some emerging field measurement techniques, such as LIDAR (Feurer et al. 2008), stereo imaging systems (Asada et al. 1999), and underwater vehicles (Galceran et al. 2015), can be applied to obtain three-dimensional boundary information on a given region or the river-immersed topography to obtain the volumetric hydraulic radius R. For the reach-averaged flow velocity U, salt dilution methods (Lee & Ferguson 2002) are always applied to measure it. If traditional cross-sectional flow hydraulic variables are measured, their averaged values can be taken as reasonable approximations of reach-defined hydraulic variables as long as the measured cross-sections in a river reach are sufficient.

In quasi-uniform flow, the relative flow depth R/d is always taken as a measurement of bed roughness. Whereas for the non-uniform flow, the sediment grain size can not reflect the non-uniformity of the river bed surface. The paper addressed the idea of Yang (2013) and defined a relative roughness parameter in the river reach which is related to the volume of the separation water zone after the bed forms/structures.

Obviously, the relative roughness parameter r_y means the energy dissipation caused by the irregularities and is an indication of form drag. The modeling of r_y can be seen in the ‘Impact factor’ section.

Then, we can quantify the reach-averaged form drag coefficient f″ using Equations (14) and (12) by applying the reach-averaged flow velocity, hydraulic radius, water surface gradient, and sediment grain size.

As we know, form drag is related to the bed forms or structures in the river reach. The paper held the view that form drag is a function of the formation and scales of bed forms or structures. Two parameters are applied to describe the two impact factors. The Einstein flow parameter d/R′ means skin friction acting on the sediment grains and is an indication of bed forms or structures formation. The relative roughness parameter in the river reach r_y proposed by Yang (2013) means energy dissipation caused by bed forms or structures and is a function of the shape, size, and distribution of the irregularities on the river bed surface.

In a word, (R − R′)/R is a measurement of the separation water zone and can depict the characteristics of different bed forms/structures.

One can obtain the new form drag relation by calibrating and validating the coefficients in Equation (19) by applying the reach-averaged flow data.

According to the ‘Dataset’ Part, the whole data are divided into Part A and Part B (Table 1). Part A is used to calibrate the new form drag relation Equation (19) and Part B is for the validation.

In order to solve the coefficients α and β in Equation (19), the paper firstly established the optimum correlation between and ⁠. We did logarithm operation to and and then use the multiple linear regression analysis in the SPSS Statistics software by applying the dataset of Part A.

It can be obtained that α = − 0.24, β = − 0.70, and the correlation coefficient R² is 0.96. The values of α and β are both negative. This accords with the law of flow resistance varying with hydraulic variables.

We set and then plot versus Z in Figure 6.

It is observed from Figure 6 that the new form drag relation is consistent with the data. The entire relation can be approximately decomposed into three zones. By applying the slope clustering method to analyze the relation, we determine that the two turning points of the three zones are at approximately Z = 0.35 and 1.5.

We then apply the regression analysis method without considering the few scattered points to obtain the two log-linear relations as follows:

Analysis of the exponential coefficients of the hydraulic variables on both sides of the two rewritten relations (Equations (20b) and (21b)) demonstrates that the form drag relations are not auto-correlated.

We now obtain the generalized relation of the form drag which can be applicable to different regions.

In open channel flow, three basic hydraulic parameters dominate the flow mechanics: the flow velocity U, hydraulic radius R, and river slope S, under the condition that the characteristic sediment grain size distributions or other topographic information are additionally known. Then, the full flow information can be solved based on two of the three basic parameters by applying the generalized relation (Equation (22b)). Therefore, Equation (22b) is actually a type of hydraulic geometric relation of gravel-bed streams.

The parameter is not a simple and direct indicator. In order to ensure the application ranges of the two log-linear resistance relations, by considering the formulation structure of the parameter Z, we attempt to search for the relation between the parameter Z and the relative flow depth R/d₉₀.

In Figure 7, the relative flow depth R/d₉₀ versus the parameter Z is plotted by applying the datasets of Part A.

As can be seen in Figure 7, the relative flow depth R/d₉₀ is roughly negatively related to the parameter Z. From the correlation of R/d₉₀ against Z, one can know that when Z is smaller than 0.35, R/d₉₀ is larger than 9; when Z is larger than 3.5, R/d₉₀ is smaller than 0.8.

In other words, Equation (20) applies to the deep water zone, that is, the relative flow depth R/d₉₀ is larger than 9. Equation (21) applies to the shallow water zone, that is, the relative flow depth R/d₉₀ is smaller than 0.8.

In order to test the applicability of the form drag relation Equation (22b), the datasets of Part B are used. The parameter Z can be calculated by applying Equation (12) using datasets of Part B. The form drag coefficient f″ can be calculated by applying Equations (14) and (12) using data Part B. Then, the relationship of (8/f″)^0.5 against the parameter Z is plotted in Figure 8.

Figure 8 shows that the form drag relation Equation (22b) derived from the datasets of Part A fits the datasets of Part B very well.

Now, Table 2 lists the important variables of the equations in the paper.

In order to evaluate the performance of the present resistance relation and existing representative resistance formulae, we examined the applicability of predicting flow velocity by Bathurst (1985), Ferguson (2007), Rickenmann & Recking (2011), and the new-derived form drag relation by applying the datasets of Part B.

Then, the datasets of Part B were used to verify the form drag relation Equation (22b) by comparing the predicted flow velocities with the measured ones. Considering the fact that flow velocities cannot be calculated from Equation (22b) explicitly, we programmed an iteration procedure to solve flow velocities. The iteration code is provided in the paper. Figure 9 shows that the predicted flow velocities U_p fit with the measured flow velocities U_m well, and the correlation coefficient R² = 0.87. Most data points are distributed in the range of R_d = 0.5–2. The examination indicates that the generalized form drag relation derived in this study for gravel-bed streams is convincing for predicting the flow velocity.

The other three flow resistance relations are also tested by applying the datasets of Part B as a comparison.

Figure 10 plots the predicted flow velocities calculated by the flow resistance relation of Bathurst (1985) versus the measured ones. The validation results indicate that the measured flow velocities U_m are consistent with the calculated ones U_p. The correlation coefficient R² = 0.76.

Ferguson (2007) developed a resistance relation for both deep and shallow water zones, and Rickenmann & Recking (2011) concluded that it has the highest accuracy in predicting the average flow velocity among the many resistance equations that they reviewed. Figure 11 shows the validation results of the Ferguson (2007) resistance relation with the datasets of Part B. Graphically, the performance of Ferguson (2007) is close to that of the present form drag relation Equation (22b). The correlation coefficient R² = 0.84.

Figure 12 demonstrates the predicted flow velocities by flow resistance relation of Rickenmann & Recking (2011) versus the measured ones. The verification indicates that the measured flow velocities U_m are in accordance with the predicted ones U_p. The correlation coefficient R² = 0.79.

It can be seen from the figures above that the validation results of the existing flow resistance formulae are similar to those of the resistance relation proposed in this study. To quantitatively compare the performance of these resistance relations, their discrepancy ratio (R_d), mean normalized error (MNE), and average geometric deviation (AGD) were computed and listed in Table 3.

Generally, the formula performs better if the discrepancy ration R_d in each range is bigger. Conversely, the formula performs better if the parameters MNE and AGD are smaller.

One may notice that the present form drag relation performs best in every index except the discrepancy ratio R_d in the range of 0.8–1.25. Such is the case with the Ferguson (2007) resistance relation: it performs best among the other three existing resistance formulae, except for the indexes of R_d in ranges of 0.8–1.25 and 0.67–1.5. Generally speaking, the resistance relation of Rickenmann & Recking (2011) behaves a little better than that of Bathurst (1985).

It can be seen from the figures that Bathurst (1985) behaves best enough with the datasets of Griffiths (1981), Hey & Thorne (1986), and Bathurst (1985), while worse with the datasets of Zhang et al. (2013) and Wohl & Wilcox (2005). The R_d in ranges of 0.8–1.25 and 0.67–1.5 of Bathurst (1985) are also sufficiently bigger and this means the data gather tightly nearby R_d = 1.0.

One may figure out the reason from the formula structure of the Bathurst (1985) resistance relation and the relative flow depth R/d₈₄ ranges easily. The Bathurst (1985) resistance relation is basically based on the logarithmic law of the Keulegan formula (1938) for rough open channel flows and therefore a better fit for the larger relative flow depth.

Without consideration of the data of Bathurst (1985), the relative flow depth R/d₈₄ of the datasets of Griffiths (1981) and Hey & Thorne (1986) reaches nearly two orders of magnitude. This indicates that these data are from the deep water zone and the log-law could apply too. However, the relative flow depths R/d₈₄ of the datasets of Zhang et al. (2013) and Wohl & Wilcox (2005) are smaller than 10 and too shallow to apply the log-law.

For the resistance relation of Ferguson (2007), Rickenmann & Recking (2011), and the present study, they all try to cover the ranges of deep and shallow water zones. As a result, they may behave similarly for data with different ranges of relative flow depth.

Considering the relatively complicated form of the new form drag relation Equations (20b) and (21b), we attempt to simplify Equations (20b) and (21b) to explore the physical essence behind the new form drag relation.

Because Z > 1.5 and R/d₉₀ < 0.8, the form drag relation is suitable for zones of large roughness and shallow water, as in the mountainous region. Skin friction is always considered to be negligible in mountain streams; thus, the hydraulic radius that is related to the skin friction R′ is also much smaller than the hydraulic radius R.

Now we derive two simple form resistance relations based on the relative flow depth (R/d₉₀) parameter for deep and shallow water zones, respectively.

It is known that the resistance equation of Ferguson (2007) is obtained by adding the resistance relations of the deep and shallow water zones together. The resistance relations for the deep and shallow water zones are as follows:

The two-segmented flow resistance relations that were derived in this study for deep and shallow water zones approximate to those in Ferguson (2007). This is also the reason that the results of the form drag relation of this study and the resistance equation of Ferguson (2007) are close. Figure 13 exhibits the comparison results of the segmented resistance relations for deep and shallow water zones of the present study and Ferguson (2007). It can be concluded that in the deep water zone, the flow resistance calculated by the form drag relation of this study is a little bigger than that calculated by Ferguson (2007). In the shallow water zone the opposite is the case.

This paper uses the idea of form drag production that was proposed by Einstein & Barbarossa (1952) and then develops a new form drag relation by incorporating the Einstein flow parameter and the relative roughness parameter in a river reach by Yang (2013). The validation demonstrates that the new relation fits the field data well. In contrast to previous resistance equations, this study mainly contributes an alternate resistance equation that makes improvements in predicting the flow velocity. The idea of modeling form drag that is used in this paper is significant. The paper shows a new river reach-scaled perspective to investigate the flow resistance within a reach and propose integral and discrete forms of the hydraulic variables in a river reach.

In addition, this paper extends the understanding of the production of form drag by Einstein & Barbarossa (1952) to both plain and mountain rivers. The scale relationship between the sediment grain sizes and the bed forms or bed structures in plain rivers is different from that in mountain rivers. This difference makes the form drag impact factors of plain and mountain rivers a little different. In essence, the difference in the relative flow depth R/d is the underlying reason for the different flow regimes in plain and mountain rivers. We developed a unified form drag relation that can be applied to both plain and mountain rivers and the new form drag relation can be considered an attempt to link the flow hydraulics in mountain streams with those in plain rivers.

• (1)

  This paper uses large sets of field data with slopes ranging from 0.004 to 28.7% to develop a new alternate form drag relation that is appropriate for both plain and mountain rivers. The concepts of traditional hydraulic elements are extended from cross-sectional to river reach scale. The hydraulic radius, relative roughness height, flow resistance, and form drag resistance coefficient are defined in a given river reach to consider the non-uniformity of the boundary. The reach-averaged flow resistance model is founded by force balance analysis of the water body in the river reach.

• (2)

  Skin friction and form drag were solved separately based on their different production mechanisms. Reach-averaged form drag is formulated by applying the Einstein flow parameter and the relative roughness parameter in a river reach by Yang (2013). The paper analyzed the relation between the form drag production and river bed topography in plain rivers and mountain streams based on the idea of Einstein & Barbarossa (1952). The characteristics of the flow and river bed surface differ remarkably in plain rivers and mountain streams. This distinguishes the impact factors of flow resistance in plain rivers from those in mountain streams. It is concluded that the shapes, sizes, and distributions of the bed forms/structures have significant effects on fluid hydraulics and should be considered.

• (3)

  A new form drag relation is derived from a multivariable regression analysis by applying a large set of field data. The validation results show that the new obtained form drag relation is consistent with the field data and has good agreement in both deep and shallow water zones, which demonstrates the inherent uniformity in the resistance mechanism of plain and mountain rivers.

All relevant data are available from an online repository or repositories (https://github.com/luowengg2020/Reach-averaged-flow-resistance-in-gravel-bed-streams-data).