1. Introduction
Estimating bed roughness with adequate precision is essential in understanding the dynamic and morphological processes (e.g., [
1,
2]) of rivers and in guiding channel engineering (e.g., [
3,
4]). In particular, Chang [
5] indicated that the parameters of channel morphology, such as the stable channel width, can be obtained through a combination of the minimum stream power concept, the water continuity equation, the sediment transport equation and a proper flow resistance equation (expression of bed roughness). The determination of the stable channel width is crucial in river training, especially for the determination of a river training strategy for the Lower Yellow River (LYR), which is well known for its high sediment concentration [
6]. Currently, the wide-river training strategy (leaving adequate space between the main channel and the Grand Levees (the outer boundary of the LYR) for flood detention and sediment deposition) [
7] is adopted for the LYR, but the debate regarding whether to keep the wide-river training strategy or change to the narrow-river training strategy (constructing various river training structures in the vicinity of the outer boundary of the main channel to maintain a narrow river channel for flood conveyance and sediment transportation) [
7] continues (e.g., narrow-river training strategy: Zhang et al. [
8], and Hu [
9]; wide-river training strategy: Wang and Liu [
10]). The existing method of Chang [
5] applies to channels with low flow discharge and sediment discharge (sediment concentration) only, and it fails in the calculation of the stable channel width for large sand-bed rivers such as the LYR; in the case of the LYR, this failure is due to the inapplicability of the flow resistance equations and sediment discharge equations recommended by Chang [
5], as noted by Ma [
11]. Therefore, finding an analytically accurate flow resistance equation will contribute greatly to the determination of the stable channel width of the LYR and thus, may provide theoretical support for the training of the LYR.
Numerous flow resistance equations have been developed based on various theoretical/empirical schemes and flume/field data (for a detailed review, see Niazkar et al. [
4]). According to the selection of representative variables for flow resistance and solution methods, these equations can be classified into three types: (1) equations based on the calculation of the Manning’s roughness coefficient corresponding to the grain roughness; (2) equations based on the integral solutions of velocity distribution equations; and (3) equations based on a regression analysis of hydraulic and sediment factors. The SI units are used in this paper.
For the first type of flow resistance equation, Strickler [
12] derived an equation used to calculate Manning’s roughness coefficient corresponding to the grain roughness by combining
D (the size of the bed-material load) and an empirical roughness parameter
A (
A is set to 21.1) when grain roughness plays a leading role in the total bed roughness. This equation is suitable for a flat bed with large particle sizes, and the calculated value by this equation is smaller than the actual value for small particle sizes [
13]. Therefore, Chang [
14] adjusted
A from 21.1 to 19. For the stationary flat bed,
A is related to the gradation, shape and distribution patterns of bed materials, and efforts have been made to adjust the
A and representative
D values according to different riverbed conditions (e.g., [
15,
16]). Generally,
A is calculated by Equation (5) hereinafter when
D is set to the median size of bed-material load
D50 [
17]. Notably, Wu and Wang [
18] asserted that the Froude number
F, grain bed shear stress
and critical bed shear stress
should be considered in the expression of
A. They developed an empirical flow resistance equation suitable for the flow regime from stationary flat beds to moving plane beds by a third-order polynomial curve fitting based on flume and field data.
The second type of equation is established based on the integral solutions of velocity distribution equations. For example, Keulegan [
19] obtained the expressions of the ratio of the mean velocity
V to the friction velocity
for smooth and rough boundaries, respectively, by integrating the logarithmic velocity distribution equation. Einstein [
20] unified the above two expressions by introducing the corrective parameter
, a function of the ratio of Nikuradse roughness height
ks to viscous sublayer thickness
. Accordingly,
V/
is related to the hydraulic radius
R and the Nikuradse roughness height
ks. Then, Chezy’s coefficient
C can be represented by a function of
R,
, and
ks. Consequently, the bed roughness can be obtained by solving
ks. Li and Liu [
21] expressed
ks in the logarithmic velocity distribution equation as the product of
D50 and a proportionality coefficient
, and derived the graphical relation between
and
(
Vc is the incipient velocity of the bed-material load, an indicator of bed material movement) based on field data including the Yellow River, the Yangtze River and the Ganjiang River. They used the relation between
and the relative velocity
V/
Vc to reflect the influence of the flow intensity and movement of the bed material on flow resistance. van Rijn [
22] investigated bed-form height and length based on flume and field data and developed an equation for
by adding grain roughness height
and form roughness height
.
is calculated by representative diameters and some alternative equations were also developed (e.g., [
23,
24]).
is calculated by geometric parameters of the bed-form, but they are hard to observe in natural rivers. Niazkar et al. [
4] considered that
is closely related to the settling velocity,
F, Shields parameter
and particle size, and they obtained the expression of
based on a large amount of flume and field data.
Manual adjustments are often required when studying flow resistance based on the logarithmic velocity distribution equation due to invalidation of the logarithmic velocity distribution equation near the boundary regions caused by the neglection of viscous shear impact. To address this issue, Zhao and Zhang [
25] derived a flow resistance equation widely applicable in the Yellow River by vertically integrating the velocity distribution equation proposed by Zhang et al. [
26], which is valid near the boundary regions and also has the ability to reflect the variation of velocity distribution caused by sediment concentration, and introducing a hydraulic frictional depth
. The equation proposed by Zhao and Zhang [
25] reflects the effects of variation of hydraulic and sediment factors and additional roughness in natural channels since the equation builds connections with the Froude number, particle size and sediment concentration.
The third type of equation is established based on a regression analysis of hydraulic and sediment factors. For example, Ma and Huang [
27] proposed three types of empirical flow resistance models based on data from gauge stations in the LYR. The first kind of model considers the water depth
H and slope
J, the second kind of model also considers the channel width, and the third kind of model also considers the channel width and sediment concentration
S. Ma et al. [
28] established a linear relationship between Manning’s roughness coefficient
n and ln
F suitable for the LYR. Notably, by introducing
(
g is the gravitational acceleration) in Manning’s roughness equation,
n is expressed in a form that has a positive correlation with
R1/6J1/2 and a negative correlation with
F. Actually, the data of
n used to fit the equation proposed by Ma et al. [
28] are calculated by Manning’s roughness equation. Under the condition where
R1/6J1/2 generally have a little change in sand-bed rivers, the equation proposed by Ma et al. [
28] naturally has a good performance in the LYR, and limited information about flow resistance thus can be determined from the equation.
Many valuable advances in river dynamics research and river management applications have been made based on these types of flow resistance equations, but some issues remain to be addressed. The first issue is that these equations, with the exception of the work of Zhao and Zhang [
25] and the third kind of model proposed by Ma and Huang [
27], do not reflect the influence of
S on bed roughness. Moreover, the validation results of the third model proposed by Ma and Huang [
27] (the one involving
S) show that the model produces large uncertainty [
27]. The second issue lies in the applicability of the existing equations. In our validation process of the various flow resistance equations based on data samples from different rivers and canals, we found that the equations proposed by Wu and Wang [
18] and Niazkar et al. [
4] achieve good performance in predicting
n for sediment-laden flows under only low-concentration scenarios and that the equations proposed by Zhao and Zhang [
25] and Ma et al. [
28] achieve good performance in predicting
n for sediment-laden flows under only high-concentration scenarios. Therefore, this study aims to develop a proper flow resistance equation suitable for different sediment concentration scenarios in sand-bed rivers. In addition, this study quantifies the stable channel width corresponding to the minimum
n in the typical wandering reach of the LYR under the current channel-forming discharge through a combination of the proposed flow resistance equation and the suspended load carrying capacity equation. Details of the research data used in this paper and the establishment of the proposed flow resistance equation are introduced in
Section 2. The validation of the equation and the quantification of the stable channel width in the LYR are delineated in
Section 3. Conclusions are presented in
Section 4.