Roughness effect on the correction factor of surface velocity for rill flows

Flow velocity is one of the most important hydrodynamic variables for both channelized (rill and gullies) and interrill erosive phenomena. The dye tracer technique to measure surface flow velocity Vs is based on the measurement of the travel time of a tracer needed to cover a known distance. The measured Vs must be corrected to obtain the mean flow velocity V using a factor αv = V/Vs which is generally empirically deduced. The Vs measurement can be influenced by the method applied to time the travel of the dye‐tracer and αv can vary in different flow conditions. Experiments were performed by a fixed bed small flume simulating a rill channel for two roughness conditions (sieved soil, gravel). The comparison between a chronometer‐based (CB) and video‐based (VB) technique to measure Vs was carried out. For each slope‐discharge combination, 20 measurements of Vs, characterized by a sample mean Vm, were carried out. For both techniques, the frequency distributions of Vs/Vm resulted independent of slope and discharge. For a given technique, all measurements resulted normally distributed, with a mean equal to one, and featured by a low variability. Therefore, Vm was considered representative of surface flow velocity. Regardless of roughness, the Vm values obtained by the two techniques were very close and characterized by a good measurement precision. The developed analysis on αv highlighted that it is not correlated with Reynolds number for turbulent flow regime. Moreover, αv is correlated neither with the Froude number nor with channel slope. However, the analysis of the empirical frequency distributions of the correction factor demonstrated a slope effect. For each technique (CB, VB)‐roughness (soil, gravel) combination, a constant correction factor was statistically representative even if resulted in less accurate V estimations compared to those yielded by the slope‐specific correction factor.


| INTRODUCTION
Flow velocity is one of the most important hydrodynamic variables controlling channelized (rill and gully) and interrill erosion processes and process-based soil erosion models can be developed and tested by its knowledge and measurement (Takken et al., 1998).
This technique is based on the measurement of the travel time of a tracer (water marker, salt, magnetic material, water isotope) (Berman et al., 2009;Dunkerley, 2003;Olivier et al., 2005;Ventura Jr. et al., 2001) needed to cover the known distance from the injection point (Chen et al., 2017) to a given section. This measured surface flow velocity V s must be corrected with a correction factor α v to obtain the mean flow velocity V (Zhang et al., 2010): Indeed, for open channel flows local velocity varies along the vertical, is equal to zero at the bed and reach the maximum value at or below the water surface, depending on whether the effect of channel walls is negligible or not, respectively (Ferro & Baiamonte, 1994). The correction factor depends upon the form of the vertical velocity profile.
For a laminar flow on a smooth surface the parabolical velocity profile can be determined theoretically (Powell, 2014). The presence of roughness elements can modify the shape of the vertical velocity profile (Ferro, 2003;Powell, 2014). Accordingly, a good accuracy of the mean flow velocity measurement should be achieved by setting an appropriate correction factor α v for different hydraulic conditions.
Many investigations have been carried out exploring different conditions and determining different correction factor values. Horton et al. (1934) suggested α v = 0.67 for an infinitely wide laminar flow on a smooth and rigid bed. For transitional flows, Emmett (1970), carrying out flume experiments, found that α v increases with flow Reynolds number Re = Vh/ν k , in which h is the water depth and ν k is the kinematic viscosity, and α v is equal to 0.8 for turbulent flow. Luk and Merz (1992) determined experimentally an α v value equal to 0.75 for transitional and turbulent flow. Li et al. (1996), carrying out flume experiments with transitional and turbulent flows on a mobile sand bed having a slope s ranging from 4.7% to 17.7%, empirically determined an inverse relationship of α v with s (Li & Abrahams, 1997) and a direct relationship of α v with Re.
For sediment-free overland flows, Li and Abrahams (1997) found that for laminar flows α v decreases with the roughness height, increases rapidly with Re for transitional regime and more slowly with Re for turbulent flows, and is not affected by slope. For a sedimentfree flow, Zhang et al. (2010)  For sediment-laden flows, an inverse relationship between the correction factor and the sediment load was obtained by Li and Abrahams (1997) and Zhang et al. (2010). Ali et al. (2012), carrying out experiments in a mobile bed flume, established that the correction factor α v increases as the size of the transported particles increases. Di Stefano et al. (2018b), using measurements of surface flow velocity (Di Stefano et al., 2017b, 2018a, 2018b in rills incised on plots having a mean slope equal to 9, 14 and 22%, found that, to estimate the Darcy-Weisbach friction factor, α v can be indifferently assumed equal to 0.665 or 0.80. Di Stefano et al. (2020) proposed a theoretical relationship, based on the power velocity distribution, to estimate α v which was tested with flume measurements for sediment-free flow on a rough bed (Ferro & Baiamonte, 1994) and sediment-laden flow on a smooth bed (Coleman, 1986). The authors stated that the correction factor increases with the roughness height for the sediment-free flow while an inverse relationship between α v and the sediment load can be established for the sediment-laden flow. Chen et al. (2017) Polyakov et al. (2021), carrying out several experiments on overland flows in semiarid rangelands, suggested that the velocity correction factor is a dynamic, site specific property. These authors proposed a linear model to estimate the correction factor based on predictor variables as travel distance, unit discharge, and surface velocity.
The available literature findings corroborate the idea that establishing an appropriate α v value for correcting the surface velocity is a significant achievement to study rill flow hydraulics.
Since most of the available investigations regarding the correction factor were carried out for flume and overland flow, there is a scientific need to widen the existing knowledge (Chen et al., 2017;Yang et al., 2020) for rill flow scale. Di

| MATERIALS AND METHODS
The experimental investigation was carried out using a sloping flume Experimental runs were carried out using two different roughness conditions. The first arrangement was obtained fixing, by a waterproof vinylic glue, a sieved soil ( Figure 1) to the flume bed and walls. Figure 2 shows the particle size distribution of the investigated soil.
The median diameter d 50 of the soil was equal to 0.014 mm.
The second arrangement was obtained fixing, by a waterproof The mean value, d m , of these three measurements was considered the representative diameter of the gravel element. Figure   A Methylene blue solution was used as a dye-tracer to measure the surface velocity V s (Figure 6a were also calculated. This last hydraulic variable was used to test the reliability of the relationship between α v (=V/V m ) and F s , which would allow to estimate the correction factor by the measured surface velocity.
For each slope s, the mean value of α v = V/V m , corresponding to different discharges, named α vm , was finally obtained.
For the soil and gravel arrangements and for each measurement technique, 480 and 600 measurements were carried out, respectively.    (Kirkman, 1996). This test, which was used to statistically compare each pair of distributions, considers the maximum vertical deviation between two cumulative empirical distributions and the null hypothesis of no differences between data sets is rejected if the calculated P value is small (Kirkman, 1996; P < 0.05 in this investigation). For each roughness condition, the V s /V m values were used to plot the empirical distribution for fixed slope (Figure 9). For both the investigated rough beds, the overlapping of the six empirical frequency distributions for both the applied techniques (VB and CB) and the results of the KS test suggested that V s /V m does not depend on slope. Since the V s /V m ratio was independent of slope and discharge, the sample corresponding to a fixed roughness belongs to a single population. For this reason, a single frequency distribution of the V s /V m ratio for each measurement technique was considered. Figure 10 shows, as an example for the VB technique and the soil arrangement   Table 3. Figure 15, which shows the s À α vm experimental pairs, highlights that α vm and s are not correlated for both the measurements techniques and investigated roughness conditions.

| Evaluating the correction factor for rill flows
Finally, in the light of absence of suitable α v predictors, the mean value of α v (=V/V s ) for each slope was calculated (   Table 4. Figure 16 shows, as an example for the CB technique, the frequency distributions of the relative errors. The latter, which are calculated as the difference between the forementioned mean values and the 480 (soil) or 600 (gravel) α v normalized using these measurements, are normally distributed with a mean almost equal to zero for both the examined techniques and roughness conditions. The same result was obtained using the slope-specific α v values.  Equations (3), (4) and (5) show that, for the three roughness conditions, the VB velocity measurement is on average greater than the CB one but of 3.3% at the most. Therefore, neglecting the differences among these equations, the following unique V mCB À V mVB relationship was calibrated ( Figure 11d):     (Dunkerley, 2001). In addition, the magnitude of this mixing effect does not seem to substantially affect the flow velocity profile, resulting in correction factors not significantly varying with Re. The same results were obtained for turbulent overland (Li & Abrahams, 1997) flows.
The analysis of the α v = V/V s values ( Figure 14) obtained with both techniques shows that the curves generally do not overlap, demonstrating the influence of the slope on the correction factor. Figure 14 and Table 3 show that, generally, both the α v range and its variability expressed in terms of CV depend on the slope. However, for both the measurements techniques and investigated roughness conditions, α vm and s are not correlated (Figure 15). Except for s = 8.7%, α vm values for the gravel arrangement are higher than those obtained for the soil arrangement. This result can be justified by the circumstance that the water depth measurement for the highest slope value (s = 8.7%) is more affected by water surface irregularities than the measurement performed on the lower slopes.
For fixed mean surface velocity V m and both the measurement techniques, the gravel arrangement is characterized by values of mean flow velocity V higher than those corresponding to the soil arrangement (Figure 17a,b). Figure 18, which shows the frequency distribution of the Strickler coefficient c (m 1/3 s À1 ) for both soil and gravel arrangement, highlights that the c median value (53 m 1/3 s À1 ) of the soil is lower than that (58 m 1/3 s À1 ) of the gravel. In other words, the gravel was hydraulically smoother than the soil. This last result justifies that the gravel is characterized by values of α v (Figure 15a,b) and V (Figure 17a,b) higher than those of the soil.
The circumstance that the gravel arrangement is smoother than the soil one can originate from the deployment of gravel particles, which are oriented with the smallest or intermediate dimension perpendicular to the flume bed and walls. Table 4 shows that using a constant value of α v gives a worse estimate of V compared to that yielded using the slope-specific α v value, especially for the case of flume covered with gravel. However, using a constant α v for each investigated roughness condition, the errors are randomly distributed around zero ( Figure 16) and it can be considered representative of the correction factor, accordingly.
Finally, being the mean estimate errors of mean flow velocity relatively low, dye-tracing can be considered a simple and reliable measurement method.

| CONCLUSIONS
The applicability of the dye-tracer technique needs an appropriate correction factor α v for different hydraulic and bed conditions to scale down the measured surface velocity V s . In this paper, experiments were performed to study the applicability of the dye-tracer technique in a small channel simulating a rill. Two techniques for measuring the travel time of the dye-tracer and two roughness conditions were used.
The analysis allowed to establish the following main results: ii. the mean value V m of the surface flow velocity, which is representative of the scale effect due to the discharge and slope, can be used to compare the two examined measurement techniques for 0 ≤ d 50 ≤ 4.7 mm; iii. for the examined rough flumes both the measurement techniques allow precise surface flow velocity measurements.
Furthermore, in accordance with previous results obtained with the same flume for the smooth bed condition, the developed analysis on the correction factor confirmed that α v is not correlated with Reynolds number for turbulent flow regime. The results also demonstrated that the correction factor is not correlated with the Froude number F s and the channel slope. However, the analysis of the empirical frequency distributions led to recognize a slope effect on the correction factor.

DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available on request from the corresponding author.