Reach-averaged flow resistance in gravel-bed streams

Previous studies about flow resistance in gravel-bed streams mostly use the log-law form and establish the relationship between the friction factor and the relative flow depth based on field data. However, most established relations do not perform very well when applied to shallow water zones with relatively large roughness. In order to clarify the hydraulic variables defined in the single crosssection, and find the reasons that reflect the instability of flow and uneven boundaries of the river, the concepts of hydraulic variables, such as hydraulic radius, are re-defined in the river reach in the paper. The form drag in the river reach is solved based on a reach-averaged flow resistance model which is developed by force balance analyzing of the water body in the given river reach. The reachaveraged form drag relation is then formulated by incorporating the Einstein flow parameter and a newly derived roughness parameter defined in the river reach. A large number of field data (12 datasets, 780 field measurements) is applied to calibrate and validate the form drag relation. The relation is found to give better agreement with the field data in predicting flow velocity in comparison with existing flow resistance equations. A unique feature of the reach-averaged resistance relation is that it can apply to both deep and shallow water zones, which can be treated as a bridge to link the flow hydraulics in plain rivers and mountain streams.


INTRODUCTION
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 ). 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 ; Nikora et al. , ).
Early studies focused on deep water zones with relatively small roughnesses, such as plain rivers. The Manning formula (Manning ) is an empirical relation that accounts for the flow velocity, hydraulic radius, and channel slope in plain rivers. Keulegan ()  He then established the relation between the form drag coefficient f″ and a dimensionless Einstein flow parameter d/R 0 . 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 ; Schmeeckle & Nelson ) 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 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 crosssectional 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 the volumetric hydraulic radius to calculate bed load transport. Yang () 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 reachaveraged 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 esti-

DATA AND THEORY Dataset
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 (). 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 where U is the average flow velocity, U * is the bed shear velocity, R is the hydraulic radius, and d 84 is the grain diameter for which 84% of the sediment sample is finer.

Ferguson () developed a variable equation for gravel
and boulder bed streams by merging the Manning-Strickler formula for deep water zones and roughness-layer theory for shallow water zones together based on field data at relative submergences from 0.1 to more than 30: where a 1 ¼ 6.5 and a 2 ¼ 2.5.

Rickenmann & Recking () also proposed a variable
flow resistance equation with a large field dataset using a dimensionless variables analysis approach: where h is the water depth.
In addition, Einstein & Barbarossa () put forward a dimensionless Einstein flow parameter ψ and proposed a form drag relation: where ρ s and ρ f are the density of sediment particles and water, d is the sediment grain diameter and taken as d 90 here as large particles affecting the formation of bed forms or structures and the energy dissipation more, R 0 is the hydraulic radius related to skin friction, S is the river slope.
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.

Definition of reach-averaged hydraulic variables
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.
For a uniform flow, the hydraulic radius R equals the ratio of cross-sectional area A cs to the wetted perimeter of one cross-section χ cs . For a non-uniform flow, we obtain a volumetric hydraulic radius in the following form by extending the concept of cross-sectional hydraulic radius:  Further R can be written in integral and discrete form as follows: where A csi is the area of the ith cross-section, χ csi is the wetted perimeter of the ith cross-section, l i is the length of the ith micro-reach, L is the length of the reach, A cs and χ cs are the mean values of the cross-sectional area and wetted perimeter along a given river reach, respectively. 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 () 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.
The extension of the relative roughness to the river reach scale can be expressed as follows: 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.

Quantifying of reach-averaged form drag
For a given river reach with bed forms/structures, the force balance and resistance partitioning between two cross-sections separated by a distance L can be expressed as follows: where ρ is the water density, g is the gravitational acceleration, S is the river slope, τ 0 is the skin friction, F D is the form drag resistance, and τ 0 is the averaged skin friction in the river reach.
Then, one can obtain According to the Einstein-Barbarossa hydraulic radius partitioning approach (Einstein & Barbarossa ), the averaged skin friction can be expressed as follows: where R 0 is the hydraulic radius of the river reach which is related to the reach-averaged skin friction.

Lots of researchers (Bray ; Parker & Peterson ;
MacFarlane & Wohl ) argued that the Manning-Strickler formula can be used to estimate grain-induced resistance in gravel-bed streams. Thus, we use the Manning-Strickler formula in the following form to calculate the hydraulic radius R 0 that is related to skin friction where K s 0 is the equivalent roughness height that is related to skin friction, and U * 0 is the shear velocity that is related to skin friction. Here, d 50 was used to represent the grain roughness K s 0 as was suggested by Keulegan () and MacFarlane & Wohl ().
Then, the relation of R 0 is obtained as follows: We referenced the formula of the Darcy-Weisbach friction factor to define the form drag coefficient f″ by the ratio of the form drag to the hydraulic drag force in a river reach as follows: According to Equations (8)- (10) and (13) it can be written as follows: 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.

Impact factor of reach-averaged form drag
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 0 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 () 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.
The parameter r y should be modeled further for application. According to Yang (), the volume of the separation water zone can be expressed as follows: where k 0 and k 1 are coefficients related to the shapes of irregularities, and A p is the projected area of the irregularities perpendicular to the flow direction.
For the irregularities, the form drag can be expressed as follows: where C d is the drag coefficient, and can always be taken as 0.4 for the immobile particles.
By substituting (15a), (15b), (16), and (9) into (7), one can obtain In a word, (R À R 0 )/R is a measurement of the separation water zone and can depict the characteristics of different bed forms/structures. Therefore, d 90 /R 0 and (R À R 0 )/R are the two controlling factors of form drag production. The paper held the view that the two effects function together and proposed the new form drag relation by merging the two parameters together as follows: where α and β are the coefficients that need to be solved.
Combining Equations (14) and (18), the new-derived reach-averaged form drag relation can be expressed as follows: One can obtain the new form drag relation by calibrating and validating the coefficients in Equation (19) by applying the reach-averaged flow data.

Calibration of reach-averaged form drag relation
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  where Z ¼ (d 90 =R 0 ) 0:24 Á [(R À R 0 )=R] 0:70 , as was described above, and the hydraulic radius that is related to the skin friction R 0 can be calculated by Equation (12), which is written as follows: We In Figure 7, the relative flow depth R/d 90 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/d 90 is roughly negatively related to the parameter Z. From the correlation of R/d 90 against Z, one can know that when Z is smaller than 0.35, R/d 90 is larger than 9; when Z is larger than 3.5, R/d 90 is smaller than 0.8.
In other words, Equation (20) applies to the deep water zone, that is, the relative flow depth R/d 90 is larger than 9. Equation (21) applies to the shallow water zone, that is, the relative flow depth R/d 90 is smaller than 0.8.

Validation of reach-averaged form drag relation
In order to test the applicability of the form drag  (14) and (12) using data Part B. Then, the relationship of (8/f″) 0.5 against the parameter Z is plotted in Figure 8.    The averaged skin friction in the river reach The hydraulic radius of the river reach which is related to the reach-averaged skin friction The equivalent roughness height that is related to skin friction The shear velocity that is related to skin friction  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.  were computed and listed in Table 3.
The discrepancy ratio R d is equal to the ratio of the predicted flow velocities U p to the measured ones U m . The mean normalized error MNE is defined as follows: The average geometric deviation AGD is defined as follows:

Discussion of the computational accuracy
As is illustrated above, the resistance relation for all datasets is clearly divided into three segments, which are applicable to the deep water zone, the transition zone, and the shallow water zone, respectively. To be specific, when Z < 0.35, that is R/d 90 > 9, the form drag relation Equation (20b) can be rewritten into the following form according to Equation (14): The two-segmented flow resistance relations that were derived in this study for deep and shallow water zones approximate to those in Ferguson (). This is also the reason that the results of the form drag relation of this study and the resistance equation of Ferguson () are close. Figure 13

CONCLUSIONS
(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 (). The paper analyzed the relation between the form drag production and river (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.

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