Determination of Skin Friction Factor in Gravel Bed Rivers: Considering the Effect of Large-Scale Topographic Forms in Non-Uniform Flows

Abstract

Determination of skin friction factor has been a controversial topic, particularly in gravel-bed rivers where total flow resistance is influenced by the existence of small-scale skin roughness and large-scale topographic forms. The accuracy of existing models predicting skin friction factors in conditions where small-scale skin roughness and large-scale topographic forms exist is very low. The objective of this study is to develop a modified model that improves the accuracy of the determination of skin friction factors in gravel-bed rivers. To this end, 100 velocity profile data obtained from eight gravel-bed rivers were utilized to develop an analytical method that considers the momentum thickness of the boundary layer and its deviation in large-scale topographic bedforms in a 1D force-balance model. The results show that the accuracy of the skin friction factors is enhanced when (1) the model is in the form of an exponential function of energy slope, and (2) the deviation of momentum thickness is considered in the model. The proposed model results in high accuracy of the predicted skin friction factors for energy slopes between 0.001 and 0.1, which exist in most gravel-bed rivers with different morphologies. Additionally, this study model was used to modify the classic Einstein–Strickler equation. The modified equation resulted in improved accuracy of the predicted skin friction factors in non-uniform flow conditions even when velocity profiles and energy slope were not available.

1. Introduction

Friction factor and its sources have always been a major research topic [1,2,3,4]. During the past decades, the decomposition of friction factor has been a prevalent approach in sedimentology. Usually, the total friction factor is assumed to be the sum of the skin friction factor and bedform friction factor [5,6]. Briefly, large-scale impediments, such as bedforms or submerged vegetation patches, are responsible for the formation of large-scale flow structures. These large-scale flow structures transfer large amounts of momentum in the upper and the middle region of flow [7,8]. On the other hand, skin friction is the key factor in the momentum transfer procedures in the near-bed region of rivers. To be brief, not only does the skin friction factor play a key role in the flow resistance, but also it is the main index of shear stress which governs sedimentation and erosion processes. While the determination of skin friction factor has been focused on in the field and experimental studies, the results can also be used in numerical modeling of sedimentation. Such models can be used to predict the changes in bed geometry under different flow scenarios [9].

In the main, velocity profile has been widely used to predict bed friction factor in natural waterways. However, in the presence of a highly rough bed, using of velocity profile may lead to some unexpected results. This condition arises from the multi-layer structure of the boundary layer above the coarse sediments. Indeed, the lower layer of flow (or roughness layer), which is in contact with the coarse sediments, may have different characteristics in closely similar bed materials of river beds with different geometries of bedforms [10]. For small values of h/D, where h is the water depth, and D is a geometry parameter representative of the bed roughness (generally a grain diameter), the streamwise velocity profile is not self-similar inside the roughness layer [11,12]. This characteristic can only be detected by highly sensitive and expensive velocimeter tools, such as acoustic Doppler velocity profiler (ADVP). On the other hand, in a wide range of engineering works, the accuracy and sensitivity of velocimetry are limited by common tools, such as rotary current meters with poor spatial resolution, particularly in the near-bed zone. Consequently, there is a fundamental need for the development of new approaches that be applicable in the absence of sensitive and expensive velocimeter tools.

2. Materials and Methods

Amongst the well-known friction and flow resistance indices, the Darcy–Weisbach friction factor is one of the most applied ones in hydraulic calculations. The total friction factor (f) can be written as the following [13,14]:

$$f=8{\left(\frac{u_{\ast }}{u_m}\right)}^2$$

(1)

where $u_m$ is the average streamwise velocity in a particular section of flow, and $u_{\ast }$ is the shear velocity which can be calculated via different methods. Application of the boundary-layer parameters based on the ASCE Task Force recommendation has led to a series of new approaches, such as the boundary-layer characteristics method (BLCM) [15]. Considering the BLCM, the shear velocity can be calculated as:

$$u_{\ast }=\frac{\left({\delta }^{\ast }-\theta \right)u_{max}}{c{\delta }^{\ast }}$$

(2)

where C is a constant with a value of 4.4 [16], ${\delta }^{\ast }$ is the boundary layer displacement thickness, $\theta$ is the momentum thickness of the boundary layer, and, u_max is the maximum velocity of a particular velocity profile.${\delta }^{\ast }$ and $\theta$ can be calculated as [16]:

$${\delta }^{\ast }={{\displaystyle \int }}_0^h\left(1-\frac{u}{u_{max}}\right)dy$$

(3)

$$\theta ={{\displaystyle \int }}_0^h\frac{u}{u_{max}}\left(1-\frac{u}{u_{max}}\right)dy$$

(4)

Gravel bed rivers have recently been investigated in a distinct topic in which the friction factor can be divided into two parts [13,17,18,19,20]:

$$f=f^{\prime }+f^{″}$$

(5)

where $f^{\prime }$ is the skin friction factor (emerged from grains) and $f^{″}$ is the form friction factor (emerged from large-scale topographic forms of the river bed). For plane beds, bed shear stress is the representative stress of the skin friction factor. Figure 1 shows a typical velocity distribution in which bed shear stress can be defined via momentum thickness of the boundary layer:

$${\tau }_{bed}=\frac{1}{b}\frac{dD}{dx}=\frac{d}{dx}\left(\rho U^2\theta \right)$$

(6)

where D is the drag force emerged by the bed surface and b is the width of the channel. On the other hand, for a wide channel, the shear stress can be calculated based on the Darcy–Weisbach equation:

$${\tau }_{bed }=\gamma h{s}_{f }=\gamma h\left(\frac{1}{8}{f}^{\prime } \frac{{\overline{U}}^2}{hg}\right)=\gamma f^{\prime } \frac{{\overline{U}}^2}{8g}$$

(7)

where $\gamma$ is the specific gravity of water, h is the flow depth, g is the gravity acceleration, and $\overline{u}$ is the weighted average velocity. However, in the above-mentioned equations, the roughness of the bed surface is not considered directly. For a relatively plane bed and uniform flow, Einstein suggested a modified form of Stickler’s Equation to relate the bed roughness (n_b) to the median grain size (d₅₀) [21].

$$n_b=\frac{d_{50}{}^{1/6}}{24}$$

(8)

By considering Einstein’s Equation, the Darcy–Weisbach friction factor can be calculated as:

$$f^{\prime }=\frac{g}{72}{\left(\frac{d_{50}}{d}\right)}^{1/3}$$

(9)

where d is the flow depth in a wide channel. In the same approach to relative roughness, Keulegan (1938) [22] developed a relatively simple but practical equation:

$$f^{\prime }={\left[2.03log\left(\frac{12.2h}{k_s}\right)\right]}^{-2}$$

(10)

In this equation, h is the depth of flow and k_s is Nikuradse equivalent roughness size. Determination of k_s is a controversial issue and there is a wide range of estimations from 1.23d₃₅ to 3d₉₀ or 6.6d₅₀ [23,24,25]. In another approach, the Shields parameter can be applied in the Keulegan Equation via a series of calculations. Critical shields parameter (τ*_cr) can be defined as:

$${\tau }_{cr}^{\ast }=\frac{\tau }{\gamma \left(SG-1)\right)d_{50}}=\frac{h{S}_f}{\left(SG-1\right)d_{50}}$$

(11)

where ϒ is the specific gravity of water, and SG is the ratio of sediments’ specific gravity to water specific gravity. This equation is valid for both uniform and non-uniform flow if $S_f$ presents the true gradient of the energy loss. On the other hand, τ*_cr can be written in association with $S_f$ as the following [14,26]:

$${\tau }_{cr}^{\ast }=0.15S_f^{0.25}$$

(12)

Assuming k_s = d₅₀ in Equation (5), and by comparing Equations (6) and (7),$f^{\prime }$ can be calculated via logarithm rules. The final equation can be written as [14]:

$$f^{\prime }={\left[0.9742-1.5225log\left(S_f\right)\right]}^{-2}$$

(13)

While Einstein’s Equation (Equations (12) and (13)) is based on field and empirical studies on uniform flow and relatively plane bed, derivation of $f^{\prime }$ in Equation (13) is not restricted to those conditions. This fundamental difference will lead to considerably different results when the two approaches are applied to a unique case.

3. Motivation and Objective

Generally, skin friction factors calculated via classic approaches (e.g., [21]) are developed based on uniform flow conditions where plane bed morphology is assumed and where no emergent or submerged vegetation and obstacles exist. Accordingly, existing models for predicting the skin friction factors for riverbeds where large-scale topographic forms and vegetation exist yield inaccurate results. Determination of friction factor in non-uniform flow is a relatively complicated task that is very far from the classic approaches and needs advanced algorithms and methodologies [4,15,19].

The objective of this study is to develop a modified model that improves the accuracy of the determination of skin friction factors in gravel-bed rivers. Accordingly, prevalent equations have been modified to increase their accuracy in the presence of large bedforms and non-uniform flow.

4. Methodology and Technical Approach

Since many classic equations of roughness and friction factor were developed based on the uniform flow condition, considering the fundamental assumptions in the application of the uniform flow is essential. According to (1/7)th power velocity profile law of Prandtl, the velocity distribution of a uniform flow can be described as [27,28]:

$$\frac{u}{U}={\left(\frac{y}{\delta }\right)}^{1/7}$$

(14)

where δ is the boundary layer which is defined as the region adjacent to a surface over which the velocity changes from zero to the free-stream velocity (0.99 U) [29]. The momentum thickness is also calculable as:

$$\theta =\frac{7}{72}\delta =\frac{7}{72}y{\left(\frac{U}{u}\right)}^7$$

(15)

By considering the definition of $\delta$, the momentum thickness can be defined as a linear function of flow depth:

$$\theta =\frac{7}{72}y{\left(\frac{U}{0.99U}\right)}^7\cong 0.1043y$$

(16)

where $y$ is the vertical elevation in which u = 0.99 U and is typically equal to the total depth of flow. Inserting Equation (16) into Equation (6) results in a differential form of bed shear stress for uniform flow:

$${\tau }_{Uniform}=\frac{d}{dx}\left(0.1043\rho U^{2}y\right)$$

(17)

Assuming a condition in which skin friction factor is calculated via a classic approach (such as Einstein–Strickler, which relies on uniform flow) and also via a boundary layer characteristic approach such as in Equation (13), the ratio of skin friction factors can be written as:

$$\frac{f_{bed}^{\prime }}{f_{Uniform}^{\prime }}=\frac{{\tau }_{bed}}{{\tau }_{Uniform}}$$

(18)

where $f_{bed}^{\prime }$ is the calculated value of skin friction factor via momentum thickness and $f_{Uniform}^{\prime }$ is the skin friction factor calculated via Einstein’s Equation. By inserting Equations (6) and (17) into Equation (18), the ratio of skin friction factors can be obtained as:

$$\frac{f_{bed}^{\prime }}{f_{Uniform}^{\prime }}=\frac{\frac{d}{dx}\left(\rho U^2\theta \right)}{\frac{d}{dx}\left(0.1043\rho U^{2}y\right)}\cong \frac{9.587}{y_{0.99}}\theta$$

(19)

Subsequently, the skin friction in a non-uniform flow can be estimated as:

$$f^{\prime }=\frac{9.587}{y}\theta f_{Uniform}^{\prime }$$

(20)

Nonetheless, there is a fundamental assumption in this equation: all the momentum thickness of the boundary layer is produced by skin friction. However, despite uniform and even quasi-uniform flow, it is a controversial assumption for non-uniform flow. Figure 2 shows a conceptual scheme of momentum thickness above a large-scale topographic form. Owing to Equation (5), the momentum transition is affected by skin friction and form friction in a complex process. Consequently, a portion of the momentum thickness, which is not restricted to one as the upper limit, must be considered in Equation (20). This phenomenon can be included in Equation (20) in the form of:

$$f^{\prime }=\frac{9.587}{y}(\phi \theta )f_{Uniform}^{\prime }$$

(21)

and

$$\frac{\phi \theta }{d}=0.1043 \frac{f^{\prime }}{f_{Uniform}^{\prime } }$$

(22)

where $\phi$ is an indicator representing the portion of momentum thickness which is shaped by the skin friction and d is the flow depth in relatively shallow rivers. Equation (22) and the value of $\frac{\phi \theta }{d}$ are investigated via data gathered in 100 velocimetry stations of eight gravel-bed rivers, including four reaches in Iran and four in Italy (

Table 1. The rivers and profiles used to derive Equation (24).

River	Location	No. of Profiles	Avg. Depth (cm)	Avg. Umean (cm/s)	Avg. d50 (mm)	Avg. Fr	Avg. f	Average f’ (Keulegan)	Avg. Sf
Melodari	Italy	14	13.6	64.9	20	0.554	1.204	0.101	0.038
Cerasia	Italy	15	19.5	56	47	0.416	0.78	0.072	0.016
Valanidi	Italy	8	15.8	61.4	35	0.502	1.089	0.086	0.026
Gallico	Italy	13	24.5	72.2	52	0.470	1.182	0.088	0.028
Zayanderud	Iran	5	73	77	10	0.291	0.092	0.044	0.001
Kaj	Iran	8	27	63.9	10	0.396	0.153	0.042	0.004
Gamasyab	Iran	24	30	84.2	19	0.287	0.107	0.043	0.003
Marbor	Iran	13	22	92.6	17	0.433	0.116	0.050	0.006

). Total and skin friction factors were respectively calculated via BLCM (Equations (1)–(4)) and the Keulegan approach (With a wide range of K_s values). The equivalent skin friction of uniform flow was also calculated via the Einstein–Strickler Equation (Equations (8) and (9)) by applying the value of d₅₀.

Consequently, considering $\frac{\phi \theta }{d}$ as a nonlinear multivariable function of a series of non-dimensional parameters, this parameter can be defined as:

$$\frac{\phi \theta }{d}=m\ast F{r}^{c1}\ast {\left(\frac{d_{50}}{d}\right)}^{c2}\ast S_f^{c3}\ast R{e}_{grain}^{c4}$$

(23)

Subsequently, the modified momentum thickness (${\theta }_{skin}$) which is formed by the skin friction would be:

$${\theta }_{skin}=\phi \theta =\left(m\ast F{r}^{c1}\ast {\left(\frac{d_{50}}{d}\right)}^{c2}\ast S_f^{c3}\ast R{e}_{grain}^{c4}\right)\ast d$$

(24)

Using the conjugate gradient approach to minimize the root-mean-square deviation (RMSD) for the 100 velocity profile, “m”, “c1”, “c2”, “c3”, and “c4” were respectively calculated for different K_s values. The results are presented in

Table 2. Calculated parameters to determine modified momentum thickness emerged by skin friction for different equivalent roughness in the Keulegan Equation.

${\theta }_{skin}=\phi \theta =\left(m*F{r}^{c1}*{\left(\frac{d_{50}}{d}\right)}^{c2}*S_f^{c3}*R{e}_{grain}^{c4}\right)*d$
ks	m	c1	c2	c3	c4	Pearson Correlation Coefficient	Equivalent Equation for Skin Friction Factor
D50	0.013	0	−0.8	0	0	0.89	$0.017\ast {\left(\frac{d_{50}}{d}\right)}^{-0.47}$
1.5 D50	0.014	0	−0.85	0	0	0.93	$0.018\ast {\left(\frac{d_{50}}{d}\right)}^{-0.52}$
2 D50	0.015	0	−0.9	0	0	0.94	$0.019\ast {\left(\frac{d_{50}}{d}\right)}^{-0.57}$
Lamb–Shields Method (Equation (13))	0.225	0	−0.33	0.33	0	0.80	$0.3\ast S_f^{0.33}$

. RMSD should always be a non-negative value including 0, which represents the perfect fit, and generally, the lower value of RMSD is considered the better result.

While none of the classic variants of the Keulegan Equation contain flow parameters, the Lamb–Shields-based form of the Keulegan Equation (Equation (13)) was considered as the basis for comparison (Figure 3). Normalized root-mean-square deviation (NRMSD) was calculated for this comparison as the following:

$$NRMSD=\frac{1}{\overline{{\theta }_{skin}} \left(Keulegan-Shields\right)}\sqrt{\frac{{{\displaystyle \sum }}_1^N{\left({\theta }_{skin} \left(Keulegan-Shields\right)-{\theta }_{skin}\right)}^2}{N}}$$

(25)

In order to describe the application of the developed equation in a practical situation, a field case study was established to evaluate the skin friction factor where bed topography was captured with high resolution.

5. Field Study and Data Collection

The Marbor River is located in the central Zagros Mountains in the Dena region of the Isfahan province in Iran. A relatively straight reach of a local branch was selected for this study. The bed materials consist of gravel size grains with sparse round cobbles and negligible sand-size grains in the banks. The reach width varies from 3 to 4.5 m and the length of the study area was 15.2 m. Grain size distribution and the flow velocity were captured in five sections through 13 stations. The grain size distribution was calculated by using the Wolman method in all stations [30]. Figure 4 shows the general grain size distribution of the reach. Water surface elevation and topographic map of the bed were produced via accurate surveying. The topography of the bed, river plan, and the location of the sections are shown in Figure 5. The topography and velocimetry results represent a completely non-uniform flow condition in the selected reach. Figure 6 shows the profiles of the riverbed and water surface and also velocity distributions in the centroid of the reach. Bed slope has been calculated via accurate surveying and the 2D gravity projections on global coordinates have been included in the water fluxes [31,32]. All in all, data from 13 stations were used for the evaluation of skin friction factor by different methods. The displacement and momentum thicknesses of the boundary layer were calculated for each station through the 2D velocity profile (

Table 3. Measured and calculated flow parameters in the Marbor River.

Station	IA	IC	IIA	IIB	IIC	IIIA	IIIB	IIIC	IVA	IVB	IVC	VA	VB
d50 (mm)	20	22	21	17	23	17	13	16	20	16	19	19	16
d (cm)	21	23	21	21	17	23	21	17	29	23	11	33	25
Umean (m/s)	1.26	1.22	0.94	1.11	0.58	1.16	1.08	0.94	0.94	0.76	0.46	0.88	0.71
u* (m/s)	0.16	0.17	0.14	0.162	0.092	0.098	0.086	0.074	0.095	0.094	0.036	0.14	0.075
Fr	0.88	0.81	0.65	0.77	0.45	0.77	0.75	0.73	0.56	0.51	0.44	0.49	0.45
${\mathit{\delta }}^{\ast }$ (m)	0.034	0.043	0.049	0.048	0.038	0.024	0.024	0.014	0.049	0.033	0.008	0.082	0.043
$\mathit{\theta }$ (m)	0.019	0.023	0.025	0.027	0.018	0.016	0.017	0.01	0.028	0.022	0.006	0.043	0.028
$\mathit{f}$ (-)	0.123	0.153	0.175	0.168	0.199	0.056	0.050	0.049	0.082	0.123	0.048	0.195	0.090
${\mathit{f}}^{\prime }$ (-)	0.066	0.067	0.061	0.067	0.050	0.047	0.045	0.044	0.040	0.040	0.034	0.052	0.040
${\mathit{f}}^{″}$ (-)	0.057	0.086	0.114	0.101	0.148	0.009	0.005	0.005	0.042	0.083	0.014	0.143	0.049
${\mathit{f}}^{\prime }$Uniform (-)	0.062	0.062	0.063	0.059	0.070	0.057	0.054	0.062	0.056	0.056	0.076	0.053	0.055

).

Subsequently, the skin friction factor values were evaluated using the following four methods:

a: Applying the friction factor of the equivalent uniform flow and the measured momentum thickness (Equation (20)): In this approach, the momentum thickness must be calculated via velocity profile. The results are comparable to the exact values of the flow characteristics method with equivalent Lamb–Shields parameter (Figure 7a). While there is a general correlation between these values, the deviation is also clear because of the $\phi$ parameter.

b: Applying the friction factor of the equivalent uniform flow and the modified momentum thickness (Equation (21)): Using the $\phi$ parameter in this approach leads to an independent equation that only relies on the energy slope, and the results are very close to the exact values. This approach is independent of velocity profile which makes it a suitable method in engineering works (Figure 7b). The applied equation is shown in Table 2 ($f^{\prime }=0.3\ast S_f^{0.33}$). The calculated values of $\phi$ (using Equation (24)) and the measured values of this parameter are also shown in

Table 4. Comparison between measured and calculated φ.

Section	I		II			III			IV			V
Station	Left	Right	Left	Cent.	Right	Left	Cent.	Right	Left	Cent.	Right	Left	Cent.
Calculated $\phi$	1.21	1.14	0.85	0.95	0.72	1.23	1.07	1.30	0.85	0.90	0.81	0.82	0.68
Measured $\phi$	1.18	1.11	0.83	0.93	0.71	1.22	1.07	1.30	0.78	0.77	0.87	0.81	0.69

. Investigation of the values of $\phi$ at the 37 stations of the two gravel-bed rivers (Marbor and Gamasyab) shows that for the majority of the velocimetry stations, $\phi <1$. For the centerline of the explored reach of the Marbor River, higher values of $\phi$ ($\phi \ge 1)$ are observed in the local contractions (Sec. I and Sec III), while the bed profile seems to have a much lower impact on the values of $\phi$ (Figure 8). The higher values of $\phi$ in the contractions might be related to the morphological drag of walls. Essentially, wall drag is the main source of scattering of results, particularly in natural channels [33]. However, Lamb et al. (2008) [26] mentioned that the effect of wall drag increases in the high bed slops (S > 0.02). In this study, this phenomenon was observed for two sections, although more data sets are required to evaluate the value of $\phi$ for various ranges of contractions. Figure 9 shows the surface waves near the wall of a contraction zone, which clearly shows the development of a horizontal boundary layer.

c: Applying modified friction factor of the equivalent uniform flow: Modified Einstein—Strickler Equation was used (by considering K_s = D₅₀. See Table 2). This method is also independent of the velocity profile. The results contain a higher level of errors in comparison with the two previous methods, although it is less scattered in comparison with the classic Einstein–Strickler Equation (Figure 7c).

d: Applying friction factor of the equivalent uniform flow: The classic Einstein–Strickler Equation according to Equation (9) (Figure 7d).

6. Discussion

The findings of this study indicate that the skin friction factor can be calculated via Equation (21) with high accuracy in comparison with other classic methods. Many studies have shown that the average velocity is not the only relevant velocity scale in determining bed friction, and the effect of flow characteristics needs to be considered [34,35,36,37,38]. Correspondingly, the presented approach does not rely on average velocity as a single variable. Actually, it contains the effects of momentum thickness and its deviations in topographic forms. This method calculates the skin friction factor as an exponential function of energy slope, which means the skin friction factor rises when the energy slope increases. Lamb et al. (2008) [26] explained this phenomenon as a result of backwater effects and an associated pressure differential, which increased the mobility of particles on steeper slopes. This method also eliminated the need for velocity profiles which play a significant role in many fluvial parameters. However, the concept of the $\phi$ parameter is not restricted to the measurements where energy slope is available. Applying this concept in the classic Einstein–Strickler method leads to a modified version of the method, evaluating skin friction factor without considering energy slope and velocity profiles.

In order to compare the accuracy of the methods which are developed based on the $\phi$ parameter, the relative error ((f’_BLCM − f’)/f’_BLCM) was calculated for 100 velocimetry points (Figure 10). This was done for a wide range of energy slopes (0.001–0.1). The results showed that Equation (21) contains less than 5% of error in comparison to the boundary layer flow characteristics method for energy slopes between 0.0014 and 0.056, considering the normal range of friction factor in gravel-bed rivers (Figure 10a). For the high energy slope (S_f > 0.06), the error increases linearly. The results also showed that for the conditions where the energy slope is not available, the modified Einstein’s Equation (based on the $\phi$ parameter’s concept) can reduce the error up to 50% on average (Figure 10b). In addition to the Marbor river, for other rivers with different energy slopes, the accuracy of results obtained from Equation (21) can be compared with the results obtained from the Einstein equation (Figure 11). According to the Figure, the Einstein equation is not appropriately accurate in low and high energy slopes. One reason can be attributed to the assumptions taken into account by Einstein. The normal or semi-normal flow regime is not prevalent in rivers with very steep energy slopes because of the tendency of flow to accelerate in such conditions. On the other hand, very low energy slopes are prevalent in the presence of flow blockage or decelerating flow. Despite the Einstein equation, Equation (21) takes energy slope into account. Consequently, the accuracy of the estimated skin friction factor increased on extreme slopes. However, it must be considered that the findings of this research are limited by the number of rivers where flow data are available. As presented in Figure 11a, the accuracy of results decreased sparsely in higher energy slopes. However, the accuracy of the results is still acceptable in comparison with classic methods. Despite the energy slope, relative errors do not comply with a predictable pattern for average velocity and relative roughness (Figure 12). The relative error remains below 6% in most cases. However, large relative errors have been observed in the presence of higher relative roughness. Consequently, it is recommended that future research studies focus on the accuracy of the proposed approach in rivers with higher relative roughness.

7. Conclusions

The skin drag in open-channel flow is determinable through the modified momentum thickness of the boundary layer. In order to derive a general equation, a 1D force-balance model was applied based on the momentum thickness of the boundary layer. The values of the initial model were compared to standard uniform flow models of skin friction. The deviation of the predicted values (known as $\phi$) was formulated through dimensionless parameters and an optimization procedure that used data sets of eight gravel-bed rivers with 100 velocimetry stations. Applying the $\phi$ parameter in the Einstein–Strickler Equation, the skin friction factor reveals an exponential function of the energy slope. The comparison with previous studies (e.g., [26]) shows a similar relation for the total friction factor and bed slope. This method was also considered for a field case study and the results showed a high correlation between the new method and the Keulegan method. The high correlation rate is also expectable for an energy slope range between 0.001 and 0.1, which involves the majority of gravel-bed rivers. The application of $\phi$ parameters can lead to engineer-friendly equations which are applicable in engineering works where velocity profile is not available. The results showed that for the conditions where the energy slope is not available, the modified Einstein’s Equation (based on the $\phi$ parameter’s concept) can reduce the error up to 50% on average.