Estimating flow resistance in steep slope rills

Recent research recognized that the slope of 18% can be used to distinguish between the ‘gentle slope’ case and that of ‘steep slope’ for the detected differences in hydraulic variables (flow depth, velocity, Reynolds number, Froude number) and those representatives of sediment transport (flow transport capacity, actual sediment load). In this paper, using previous measurements carried out in mobile bed rills and flume experiments characterized by steep slopes (i.e., slope greater than or equal to 18%), a theoretical rill flow resistance equation to estimate the Darcy‐Weisbach friction factor is tested. The main aim is to deduce a relationship between the velocity profile parameter Γ, the channel slope, the Reynolds number, the Froude number and the textural classes using a data base characterized by a wide range of hydraulic conditions, plot or flume slope (18%–84%) and textural classes (clay ranging from 3% to 71%). The obtained relationship is also tested using 47 experimental runs carried out in the present investigation with mobile bed rills incised in a 18%—sloping plot with a clay loam soil and literature data. The analysis demonstrated that: (1) the soil texture affects the estimate of the Γ parameter and the theoretical flow resistance law (Equation 25), (2) the proposed Equation (25) fits well the independent measurements of the testing data base, (3) the estimate of the Darcy‐Weisbach friction factor is affected by the soil particle detachability and transportability and (4) the Darcy‐Weisbach friction factor is linearly related to the rill slope.

both detached by rill flow and delivered from the interrill areas to the rill channels (Bagarello et al., 2015;Bagarello & Ferro, 2004Bruno et al., 2008;Di Stefano et al., 2013;Di Stefano et al., 2015;Govers et al., 2007;Peng et al., 2015). Modelling of rill flow is necessary for an adequate simulation of rill erosion  and accurate rill measurements at plot scale allow both to understand the physical process and model rill erosion correctly (Wirtz et al., 2010(Wirtz et al., , 2012. Rill erosion strictly depends on hydraulic characteristics of the flow moving within the rill (Foster et al., 1984) and for this reason to study and model rill erosion processes, in addition to flow discharge, other hydraulic variables as surface width, water depth, mean flow velocity and roughness coefficient must be defined (Gilley et al., 1990).
A rill is the print resulting from changes in width, depth and bed roughness due to the interaction between the erodible wetted perimeter of the channel and the flow. The rill morphology and the development of channelized erosion processes are affected by detached bed material and actual sediment transport occurring within a rill (Wang et al., 2015).
Rills are small, steep sloping (Nearing et al., 1997;Peng et al., 2015) and ephemeral channels in which shallow flows move.
Therefore, rill hydraulics should be governed by physical laws quite different from those of open channel flows (Foster et al., 1984) which are characterized by a water depth much higher than the roughness representative size and a slower morphological evolution than rills. Govers et al. (2007) highlighted that equations obtained for alluvial rivers continue to be applied, while specific relationships for rill hydraulics are available Di Stefano, Ferro, Palmeri, & Pampalone, 2017c;Foster et al., 1984;Gilley et al., 1990;Govers, 1992a;Hessel et al., 2003;Line & Meyer, 1988;Takken et al., 1998). Neglecting the actual difference between rills and rivers, classical hydraulic equations developed for fixed bed channels, such as Manning's and Chezy's equations, have been often used in physically based soil erosion models (Ferro, 1999;Govers et al., 2007;Nouwakpo et al., 2016;Powell, 2014;Strohmeier et al., 2014). In principle, the interaction among rill flow velocity, erosion of the rill wetted perimeter and sediment transport within the rill could hinder the applicability of the uniform open channel flow equations (Di Stefano et al., 2018a;Nearing et al., 1997).
Many authors (Peng et al., 2015;Stroosnijder, 2005;Wirtz et al., 2012) recognized that rill flow experiments can be useful both to overcome the scientific gap in rill hydraulics and to test the applicability of concepts and equations which are currently used in soil erosion modelling (Govers et al., 2007;Wirtz et al., 2010Wirtz et al., , 2012Wirtz et al., , 2013. When a theoretical distribution can be applied to the velocity profiles measured in different verticals of the cross section (Ferro & Baiamonte, 1994), and the relationships that establish how the coefficients of the theoretical velocity profile distribution (Baiamonte et al., 1995;Ferro, 2003) vary from one vertical to another can be established, the flow resistance law can be theoretically obtained by an integration procedure.
The recurring unavailability of measured velocity distribution in different verticals of a cross-section and the complexity of the integration procedure justify the use of the following empirical flow resistance equations (Ferro, 1999;Powell, 2014): in which V (m s À1 ) is the cross-section average velocity, C (m 1/2 s À1 ) is the Chezy coefficient, n (m À1/3 s) is the Manning coefficient, f is the Darcy -Weisbach friction factor, s is the channel slope, R (m) is the hydraulic radius (m) and g (m s À2 ) is the acceleration due to gravity.
Most field and laboratory studies on overland and rill flow use the Darcy-Weisbach friction factor, whilst the use of Manning's n is widespread in open channel flows (Hessel et al., 2003). However, this differentiation is not well-defined because "Manning's n is likely to behave in the same way as f" (Hessel et al., 2003;Takken & Govers, 2000). The Darcy-Weisbach friction factor is, as an example, currently used in WEPP (Water Erosion Prediction Project) (Foster et al., 1995;Govers et al., 2007;Nicosia et al., 2019).
Slope affects considerably the flow transport capacity T c (Ali et al., 2011;Ferro, 1998), which in turn influences the actual sediment transport. According to Jiang et al. (2018), the hydraulic mechanisms of soil erosion for steep slopes are different from those for gentle slopes. Recent experimental research on transport capacity for slopes steeper than 17%-18% (Ali et al., 2013;Wu et al., 2016;Zhang et al., 2009) established that T c relationships developed for gentle slopes (<18%) are unsuitable to be applied for steep slopes (17%-47%). Also, Peng et al. (2015) noticed that "there has been little research concerning rill flow on steep slopes (e.g., slope gradients higher than 10 )". For slope values greater than or equal to 18%, the measurements by Peng et al. (2015) showed that the flow is supercritical (1.09 ≤ F ≤ 2.33, where F = V/( gh) 0.5 is the Froude number of the flow and h (m) is the water depth) with a flow regime varying from transitional to turbulent (1140 ≤ Re ≤ 7629, where Re = Vh/ν k is the Reynolds number and ν k (m 2 s À1 ) is the water kinematic viscosity). On slopes less than 18% Peng et al. (2015) established that the friction factor decreases gradually as Re increases, while on steeper slopes (21%-27%) f increases gradually with the flow Reynolds number.
According to these authors, these different f-Re trends indicate that slope gradient plays a fundamental role in determining roughness in rills. Peng et al. (2015) also suggested that, when Re increases, on gentle slopes the effect of roughness reduction due to the flow depth increase is greater than that of roughness increase caused by sediment load and rill morphology. On the contrary, on steep slopes, this last effect, due to the intensified channelized erosion, prevails over the relative roughness reduction.
The slope of 18% could be used to distinguish between the "gentle slope' case from that of 'steep slope' for the possible difference in hydraulic (flow depth, velocity, Reynolds number, Froude number) and sediment transport variables (flow transport capacity, actual sediment load).
How the simultaneous increase of roughness and slope gradient affects flow resistance in a mobile bed rill continues to be a scientific debated topic (Di Stefano, Ferro, Palmeri, & Pampalone, 2017c;Foster et al., 1984;Giménez & Govers, 2001;Govers, 1992a;Nearing et al., 1997;Torri et al., 2012). Govers (1992a) proposed a 'feedback mechanism' in which the expected increase of flow velocity with slope gradient is counterbalanced by a simultaneous increase of bed roughness, due to the increase of erosion rate with slope (Torri et al., 2012;Xinlan et al., 2015). The result of this 'feedback mechanism' is that, in a mobile bed rill, the flow velocity tends to be independent of slope (Nearing et al., 1997(Nearing et al., , 1999Takken et al., 1998).
According to Equation (1) the feedback mechanism can occur only if the Darcy-Weisbach friction factor f increases linearly with slope gradient. The feedback mechanism does not occur in fixed bed rills, for which experiments by Foster et al. (1984) demonstrated that flow velocity increases with slope as roughness is slope-invariant.
Soil surface roughness is the key variable synthesizing the effects of soil surface irregularities due to soil particle size, rock fragments, vegetation cover and land management (Thomsen et al., 2015;Zhang et al., 2016). Grain resistance acting along the rill wetted perimeter is In previous papers (Carollo et al., 2021;Di Stefano et al., 2018a;Di Stefano, Ferro, Palmeri, & Pampalone, 2017c;Palmeri et al., 2018), the Π-Theorem of the dimensional analysis and the self-similarity theory (Barenblatt, 1979(Barenblatt, , 1987 allowed to deduce a theoretical rill flow resistance equation based on the integration of a power velocity distribution. The applicability of this theoretical flow resistance equation was tested by measurements carried out in rills (Di Stefano et al., 2018a;Di Stefano, Ferro, Palmeri, & Pampalone, 2017c;Palmeri et al., 2018), shaped on experimental plots having different slope values s p (9%, 14%, 22%, 24% and 26%) and soil textures, joined with literature data (Jiang et al., 2018;Peng et al., 2015;Strohmeier et al., 2014).
Previous experiments (Jiang et al., 2018;Peng et al., 2015) induce to consider that additional work must be done on hydraulics of rills on steep slopes (i.e., slope gradient greater than or equal to 18%). In this paper, using experimental measurements carried out in recent studies (Di Stefano et al., 2018a;Huang et al., 2020;Jiang et al., 2018;Palmeri et al., 2018;Yang et al., 2020) the applicability of the theoretical rill flow resistance equation is further tested for the steep slope condition. In particular, the data base constituted by rill (Di Stefano et al., 2018a;Palmeri et al., 2018) and flume measurements (Huang et al., 2020;Jiang et al., 2018) carried out for steep slopes (Table 1)

| DEDUCING VELOCITY PROFILE AND RILL FLOW RESISTANCE LAW
For an open channel flow the local flow velocity distribution v(y) along a given vertical is expressed by the following functional relationship (Barenblatt, 1987(Barenblatt, , 1993Di Stefano, Ferro, Palmeri, & Pampalone, 2017c;Ferro, 1997): in which ϕ is a functional symbol, y is the distance from the bottom, is the shear velocity, ρ is the water density and μ is the dynamic water viscosity.
Using as dimensional independent variables y, u * and μ, and applying the Π-Theorem of the dimensional analysis the following dimensionless groups are obtained: Note: s p , plot or flume slope; clay, clay fraction of the investigated soil; silt, silt fraction of the investigated soil; sand, sand fraction of the investigated soil.
Barenblatt (1987) also underlined that "In some cases, it turns out to be convenient to choose new similarity parametersproducts of powers of the similarity parameters obtained in the previous step". In other words, Barenblatt (1987) suggested to combine the original dimensionless groups to obtain new similarity parameters Π i .
From Equations (4) and (5) it follows: Coupling Equations (7), (5) and (9) the following equation is obtained: while from Equations (8) and (4) it follows The functional relationship (2) can be rewritten in the following form: where ϕ 1 is a functional symbol.
Introducing into Equation (12) the expression of each dimensionless group, the functional relationship can be rewritten as follows: where ϕ 2 is a functional symbol.
Rearranging Equation (14) results in where ϕ 4 is a functional symbol.
According to Ferro (2018), the flow Froude number takes also into account the ratio h/d and integrating Equation (15) the following power velocity distribution is obtained: in which C i is integration constant. According to experimental results (Barenblatt & Prostokishin, 1993;Butera et al., 1993;Ferro & Pecoraro, 2000) the integration constant C i can be assumed equal to zero and Equation (16) can be rewritten as follows: in which Γ(s, Re, F) is a function of channel slope, Reynolds number and flow Froude number to be defined by velocity measurements and the exponent δ can be calculated by the following theoretical equation (Barenblatt, 1991;Castaing et al., 1990) Integrating the velocity distribution (Equation 17), the following expression of the Darcy-Weisbach friction factor f is deduced (Barenblatt, 1993;Ferro, 2017;Ferro & Porto, 2018): From Equation (17) the following estimate Γ v of Γ function (Ferro, 2017;Ferro & Porto, 2018) can be obtained by setting y = α h, being α h the distance from the bottom at which the local velocity is equal to the cross-section average velocity V: The coefficient α is less than 1 and takes into account that: (a) the average velocity V is located below the water surface and, (b) the mean velocity profile in the cross-section is considered (i.e., the velocity profile is obtained by averaging for each y the v values measured in different verticals, and its integration gives the cross-section average velocity). The coefficient α has to be calculated by the following theoretical relationship deduced by Ferro (2017): Considering that, according to Equation (17), Γ theoretically depends only on channel slope, Reynolds number and flow Froude number (Ferro, 2018), Γ v can be estimated by s, Re and F using the following power equation: where a, which summarizes the effect of the soil characteristics on the flow resistance law, b, c and e are coefficients to be determined from experimental measurements.
For considering the effect of soil texture (Carollo et al., 2021), the following equation to estimate the scale factor Γ v of the velocity profile can be applied: in which a o , m and p are coefficients to be determined by measurements carried out in rills incised in soils having different texture and CLAY and SILT are the percentage of clay and silt of the investigated soil, respectively. In Equation (23)  were used to carry out these experiments. Each plot was 2 m wide and 7 m long. Table 1 reports the textural fractions, and the plot slope values s p of this data base, which is characterized by clay fractions ranging from 42% to 71%, silt of 19.9%-26.4%, sand in the range 9.1% -34.5% and slope values s p varying from 18% to 26%. The experiments were carried out using a constant inflow discharge ranging from 0.11 to 1 L s À1 . Each rill was manually incised along the plot maximum slope direction and then further shaped using a clear flow discharge of 0.1 L s À1 which was applied for 3 min. Each rill was divided into 9 longitudinal segments defined by two cross-sections having a distance of 0.624 m. A rill reach is defined as the channel portion between a given measurement cross-section and the rill end.
Measurements of bed slope, discharge, water depth, cross section area, wetted perimeter and flow velocity were carried out in 216 rill reaches.
A set of 70 photographs of the plot area and a 3D-photo reconstruction technique were used to obtain the 3D-DTM, applying the image-processing software Agisoft Photoscan Professional (version 1.1.6, Agisoft, Russia) (Frankl et al., 2015;Javernick et al., 2014;Seiz et al., 2006). The rill thalweg, extracted by 3D-DTM, was used to calculate the slope of each rill segment.
The water depth in a cross-section was measured by a microhydrometer located in the rill thalweg, constructed by a small aluminium rod (Figure 1), having a measurement accuracy of ±1 mm. The measured water depth and the geometric cross-section, obtained by 3D-DTM, allowed to calculate the cross-section area and wetted perimeter.
Flow velocity was measured by the dye tracing technique Di Stefano et al., 2018b;Di Stefano et al., 2020a;Govers, 1992a;Line & Meyer, 1988) using a Methylene F I G U R E 1 View of an experimental plot and micro-hydrometer used in this investigation blue solution. A correction factor equal to 0.8 was used in the present experiments to convert the surface velocity V s to the mean flow velocity V (Di Stefano et al., 2018bStefano et al., , 2020aLi & Abrahamas, 1997;Luk & Merz, 1992;Zhang et al., 2010). However, few studies have been carried out to estimate mean flow velocity in rills (e.g., Abrahams et al., 1996;Di Stefano, Ferro, Palmeri, & Pampalone, 2017c;Rodrigo-Comino et al., 2017). To the best of our knowledge, the available studies on rill flow resistance (i.e., Govers, 1992a;Li et al., 1996)  The experiments were carried out by a colluvial soil, using seven slope gradients (Table 1) and five discharges in the range 0.0672-0.528 L s À1 . The investigated hydraulic conditions were characterized by transitional and turbulent flows (639 ≤ Re ≤ 5529) and supercritical flows (1.57 ≤ F ≤ 8.14) (Figure 2). Further details on the experimental apparatus and the measurement techniques are reported in the original paper. Huang et al. (2020) The experiments by Huang et al. (2020)  were performed by a silt loam soil (23.8% sand, 64.6% silt and 11.6% clay), using three slope gradients (Table 1)

| Experiments by Yang et al. (2020)
In this investigation the rill flume had a rectangular cross-section, 0.1 m wide and 0.4 m high, with an erodible rough bed and fixed smooth sides. The experimental runs were performed by a soil containing 4.35% sand, 69.78% silt and 25.87% clay particles and using four slope gradients (Table 1)

| Experiments carried out in this investigation
In this investigation the measurement of discharge, water depth, 3.6 | Characteristics of the data bases used for calibrating and testing the flow resistance law
Coupling Equations (24) and (9) the following flow resistance law was obtained: Equation (25)  The good agreement between the measured Darcy-Weisbach friction factor values of the calibrating data base and those calculated by the theoretical flow resistance equation (Equation 25) is shown in Figure 6.
The friction factor values calculated by Equation (25) are characterized by errors E = 100 (f À f m )/f m that are less than or equal to ±20% for 98.1% of cases and less than or equal to ±10% for 78.1% of cases.   (20) and (21), and those calculated applying Equation (24) for the independent testing data set (i.e., data by Yang et al. (2020) and this investigation, Table 2). ) for the testing dataset is shown in Figure 9.
The latter are characterized by errors E less than or equal to ±20% for 87.7% of cases and less than or equal to ±10% for 32.1% of cases.
Into Equation (25) (20) and (21) and those calculated by Equation (24) (Gao & Abrahams, 2004). Furthermore, the impact of the overall (i.e., bed load and suspended) sediment transport on the rill flow resistance should be significantly lower than that of the grain roughness (Di Stefano, Nicosia, Palmeri, et al., 2019).
The actual sediment transport is the result of the interaction between the flow characteristics, which are represented by the sediment transport capacity T c (Ferro, 1998), and soil particle properties.
These are described by the attitude of soil particles to be detached from the rill wetted perimeter (detachability) and transported (transportability). The properties of the eroded particles including grain size, particle shape, specific weight, organic matter content, mineralogy and aggregate stability influence the sediment delivery processes. Soil particles detached by water erosion processes result in a mixture of primary particles and aggregates and are transported as compound particles (Liu et al., 2019). Results presented by Meyer, Foster, and Nikolov (1975);Meyer, Foster, and Römkens (1975) indicated that about 15% of the particles transported in rill flow from a tilled soil were larger than 1 mm and almost 3% of the sediment eroded was larger than 5 mm, indicating that rill flow should be able to transport large particles (Di Stefano & Ferro, 2002).
Several research efforts have been made to detect the influence of different hydraulic parameters, such as unit discharge, mean flow velocity and slope on T c (Abrahams et al., 2001;Ali et al., 2011Ali et al., , 2012Beasley & Huggins, 1982;Everaert, 1991;Ferro, 1998;Finkner et al., 1989;Govers, 1990Govers, , 1992bGovers & Rauws, 1986;Guy et al., 1990;Julien & Simons, 1985;Prosser & Rustomji, 2000;Wu et al., 2016;Xiao et al., 2017;Zhang et al., 2009). The results obtained by Ali et al. (2011Ali et al. ( , 2012 for erodible beds clearly showed that slope gradient has a strong impact on sediment transport capacity and this circumstance was attributed to the increase of the tangential component of the gravity force with slope gradient. Many studies demonstrated that sediment transport capacity increases as a power function of slope gradient (Xiao et al., 2017) and its exponent varied within a range of 0.9-1.8 (Prosser & Rustomji, 2000;Zhang et al., 2011). Previous research demonstrated that the relationship between transport capacity and unit discharge is always dependent on slope (Ali et al., 2011;Prosser & Rustomji, 2000;Zhang et al., 2009) and that the effect of unit discharge and slope on T c is affected by erodible to non-erodible bed conditions (Zhang et al., 2009). In particular, for given hydraulic and sediment conditions, the roughness of nonerodible beds is always less than that of erodible beds and for this last condition the effect of slope on transport capacity is higher than the effect of unit discharge (Everaert, 1991;Govers, 1990).
According to these observations, for steep slopes (i.e., for slope gradients greater than or equal to 18%) the high sediment transport capacity is not a limiting factor for the actual sediment transport. Consequently, the actual sediment transport in rills on steep slopes can be limited only by the detachability of soil particles from the soil mass or the transportability of the eroded particles.
Equation (25) shows that the exponent of SILT fraction is almost equal to zero demonstrating that the effect of the soil particle detachability is negligible respect to that of the transportability.
Also, Equation (25) with STI values shown in Figure 10 allows to deduce that, for given slope, hydraulic conditions and sand fraction, f decreases as CLAY increases. Therefore, f is inversely related to transportability of soil particles in steep slopes. For the steep slope condition, in which T c is not a limiting factor of the actual sediment transport, the soil characteristics and in particular the soil transportability, exert the main influence on the energy dissipation due to sediment transport. When the clay fraction increases detached particles are easily transported, and this condition implies that a low rate of flow energy is used for sediment transport. Furthermore, clay particles are transported as suspended sediments and experimental results (Vanoni & Nomicos, 1959) indicate that the suspended load reduces the friction factor by damping flow turbulence. Vanoni and Nomicos (1959) suggested that when a sediment laden flow moves fast on a flat bed the friction factor is lower than that of a clear flow moving on a fixed bed of comparable roughness. Vanoni (1946) suggested that the Darcy -Weisbach friction factor decreases in presence of suspended sediment load and this hypothesis was recently confirmed by Di Stefano et al. (2020b) which tested the theoretical flow resistance Equation (19) using literature measurements carried out by flume investigations with suspended sediment-laden flows.
In agreement with previous studies (Di Stefano et al., 2018a;Di Stefano, Ferro, Palmeri, & Pampalone, 2017c;Palmeri et al., 2018), Figure 11 and Equation (27) demonstrate that the Darcy-Weisbach friction factor f increases with a power of slope gradient having an exponent close to 1. This result confirms, for the steep slope case, the hypothesis of Govers (1992a) (25) that flow velocity is quasi-independent of slope in mobile bed rills.
This result is physically explained for mobile-bed rills on steep slopes by the well-known 'feedback mechanism' which assures that when slope gradient increases the expected increase of flow velocity is counterbalanced by the increase of bed roughness due to a more active rill erosion process.
Notwithstanding the flow resistance Equation (25)  Previous data on mobile bed rill channels joined with measurements carried out in rill flumes, in which the sediment load of the flow was equal to its transport capacity, were used for testing the applicability of a theoretical rill flow resistance law. This database was constituted by measurements characterized by high slope values (s p ranging from 17.6% to 84%) and a wide range of textural fractions.
Considering that the actual sediment transport in rills on steep slopes is not limited by sediment transport capacity and can be limited only by the detachability of soil particles from the soil mass or the transportability of the eroded particles, the developed analysis established that Darcy-Weisbach friction factor is affected by silt and clay fraction of the investigated soil. The measurements confirmed that the Darcy-Weisbach friction factor can be accurately estimated by the proposed theoretical approach.
The developed analysis for the steep slope condition pointed out that the soil characteristics, and in particular the soil transportability represented by clay fraction, exert the main influence on the energy dissipation. Furthermore, considering that clay particles are transported as suspended sediments, the obtained results, in agreement with previous studies, indicated that the suspended load reduces the friction factor.
Finally, the analysis on the exponent of the slope gradient in the flow resistance equation confirmed the hypothesis of Govers (1992a) that flow velocity in mobile bed rills is quasi-independent of slope.