Elsevier

CATENA

Volume 187, April 2020, 104351
CATENA

Flow resistance of overland flow on a smooth bed under simulated rainfall

https://doi.org/10.1016/j.catena.2019.104351Get rights and content

Highlights

  • The effect of rainfall on overland flow resistance is tested.

  • Theoretical flow resistance equation is calibrated and tested by available literature data.

  • Accurate estimate of the Darcy-Weisbach coefficient for an overland flow is obtained.

  • Darcy-Weisbach friction factor varies with rainfall intensity.

Abstract

In this paper a recently theoretically deduced flow resistance equation, based on a power-velocity profile, was tested using laboratory measurements by Yoon and Wenzel for an overland flow on a smooth bed under rainfall. These measurements of the Darcy-Weisbach friction factor, corresponding to a wide range of the flow Reynolds number (191–5700), were carried out for an overland flow under a simulated rainfall characterized by different intensity values ranging from 13 to 381 mm h−1.

At first, the available measured values of flow velocity, water depth, cross sectional area, wetted perimeter and bed slope were used to calibrate the relationship between the velocity profile parameter Γ, the slope steepness s and the flow Froude number F. Then the theoretical approach and the measurements in the investigated conditions allowed to state that the coefficients of the relationship (Eq. (8)) between Γ, s and F vary with rainfall intensity. Furthermore, the theoretical flow resistance equation (Eq. (9)) allowed an accurate estimate of the Darcy-Weisbach friction factor, which is characterized by errors less than or equal to ±10% for 91.3% of cases and less than or equal to ±5% for 84.8% of cases. Finally, the developed analysis stated that the effect of rainfall intensity on flow resistance is always negligible for turbulent conditions while for Re < 2000 the effect of rainfall impact becomes negligible for rainfall intensities greater than a critical value occurring in the range 31.75–95.25 mm h−1.

Introduction

Soil erosion due to rainfall and runoff is a process of detachment, transport and deposition of soil particles and it is one the main causes of landform modeling on earth’s surface.

Simple process-oriented models generally divide soil erosion processes in two components, one to represent interrill erosion and one to represent rill erosion (Toy et al., 2002). The major reason for applying the interrill-rill scheme is that interrill erosion behaves differently with respect to the variables controlling the physical process than does rill erosion. On interrill areas raindrop impact and interrill flow move sediment from these areas to rill channels. Most of the sediment delivered from interrill areas to rills are transported by the thin flow (overland flow) on the interrill area while soil detachment action is due to rainsplash (Mutchler and Young, 1975, Toy et al., 2002). In other words, according to this physical scheme (Water Erosion Prediction Project, WEPP), the raindrop impact is responsible for soil particle detachment while the overland flow, which is characterized by small values of both water depth and bottom shear stress, is able to transport the detached particles to rill areas.

In this context, the evaluation of overland flow velocity assumes a particular importance in estimating flow transport capacity and requires the knowledge of flow resistance under rainfall action. Flow resistance in overland flow is affected by soil roughness and a variable representative of raindrop impact as rainfall intensity.

Resistance of a free-surface flow over a smooth and sloping bed results from the internal fluid resistance and frictional resistance at the channel boundary (Katz et al., 1995). In a uniform laminar flow, for which the flow shear stress varies linearly with the bottom distance and the flow velocity is zero at the boundary, the local velocity increases parabolically to a maximum velocity at the free surface (Yalin, 1977). For laminar flow conditions the Darcy–Weisbach friction factor f assumes the following expression:f=K/Re

in which Re = V hk is the flow Reynolds number, νk is the water kinematic viscosity and K is a constant equal to 24 for uniform flow on a smooth bed. Emmett (1970) defined the overland flow under rainfall as a ≪disturbed laminar flow≫. The impact of raindrops on a flow causes both an increase of flow turbulence and a retarding effect on flow velocity.

For Re greater than 2000 the rainfall intensity effects on flow resistance can be considered negligible (Yoon and Wenzel, 1971), while for Re < 2000 flow resistance always increases with rainfall intensity. The rainfall produces a retarding effect on local flow velocity and this effect is more appreciable near the free surface. The analysis of flow velocity profiles measured for different rainfall intensities (Yoon and Wenzel, 1971) allowed to state that the surface velocity is less than the maximum value (dip effect) which is located below the free surface (y = 0.8–0.9 h, in which y is the distance from the channel bottom and h is the water depth) (Ferro and Giordano, 1993, Ferro and Baiamonte, 1994). In other words, the rainfall retarding effect is limited to a near surface region of the velocity profile, having a small thickness of 0.1–0.2 h, in which the power distribution should not be strictly applicable.

The retarding effect of flow velocity, which is more evident at low Reynolds numbers where the flow would be laminar without rainfall, can be explained taking into account the concept of momentum transfer. If raindrops fall down approximately normal to the water surface, the component of momentum carried by raindrops in the mean flow direction is negligible. Therefore, a part of mean flow momentum has to be transferred to accelerate the raindrop mass. The transfer of mean flow momentum is greatest near the surface and consequently the velocity retardation diminishes from the surface downward. Since the mean flow momentum increases with Re and the rate of raindrop impact is constant for a given intensity and drop size, the quote of momentum transfer decreases with increasing Re.

At this time, limited information is available on the effect of rainfall on the steady flow resistance on a smooth bed (Ferro and Nicosia, 2019) even if the retarding effect of rainfall impact and the increase of flow turbulence are recognized effects.

After the pioneering paper by Yoon and Wenzel (1971), very few studies were carried out on flow resistance of an overland flow on a smooth bed subjected to rainfall impact. Further experiments were carried out by Shen and Li (1973), in a 0.6 m wide and 18.3 m long sloping flume with plexiglass walls and a stainless steel bottom and a rainfall simulator with 15 modules above the flume, using rainfall intensities ranging from 190 mm h−1 to 444 mm h−1. For Re < 900, Shen and Li (1973) established that the friction factor depends on both the flow Reynolds number and the rainfall intensity while for Re values higher than 2000 the measurements confirmed that flow resistance is independent of the rainfall intensity.

Other experiments were carried out for testing the effect of various components of roughness (e.g., stone cover, vegetation elements, rainfall) (Gilley et al., 1992, Rauws, 1988) and the overall Darcy-Weisbach friction factor for a rough surface condition was obtained combining linearly the sub-factors corresponding to different friction components.

Katz et al. (1995) carried out some laboratory investigations in a flume, 4.87 m long and 1 m wide, characterized by three slope values (4, 6 and 8%) and having a rough bed obtained by glued sand grains. Rainfalls, having an intensity ranging from 40.9 to 115.1 mm h−1, were applied by an oscillating simulator. The measurements demonstrated that the coefficient K of Eq. (1) is affected by the channel bed friction and the rainfall-induced roughness. Katz et al. (1995) determined values of the K constant higher than the value (K = 24) corresponding to a smooth, uniform laminar flow and increasing with the roughness height and the rainfall intensity.

For characterizing the hydraulic roughness, Kaiser et al. (2015) carried out field experiments, in some sites in Germany and Brazil, simulating rill and interrill conditions by a 1 × 1 m2 plot. Physical roughness of the plot surface was determined applying Structure from Motion algorithms (Frankl et al., 2015, Javernick et al., 2014, Seiz et al., 2006, Di Stefano et al., 2017b) which are able to produce high-resolution topography model.

Some experiments were carried out to highlight the influence of the water application method during the experiments for measuring hydraulic resistance (Govers et al., 2000, Mügler et al., 2011, Dunkerley, 2018). Trickle flow experiments are adequate for simulating flow resistance on a surface receiving overland flow from upland areas, while a rainfall simulation is required when the upslope contribution of overland flow is negligible (Govers et al., 2000). Parsons et al. (1994) compared these two different experimental setups and concluded that friction factors measured using runoff due to artificial rainfall may be an order of magnitude higher than those obtained by trickle flow-induced runoff experiments.

Even if field studies commonly use Darcy-Weisbach friction factor to model resistance (Smith et al., 2007), most of hydrological and soil erosion models apply Manning’s coefficient to model overland flows (Hessel et al., 2003). These models present an extreme simplistic vision of overland flow resistance (Smith et al., 2007) and many researchers have questioned the application of a simple friction coefficient. The challenge is to develop a representation of roughness, more sophisticated and yet simple, that can be incorporated into a large scale erosion model (Smith et al., 2007).

For an open channel flow, Ferro, 2017, Ferro, 2018 stated that dimensional analysis and self-similarity (Barenblatt, 1979, Barenblatt, 1987) can be usefully employed to theoretically deduce a power flow velocity profile. This distribution can be integrated for obtaining an expression of the Darcy-Weisbach friction factor f. The applicability of this theoretical flow resistance equation was already tested for flows moving in mobile bed rills shaped by a clear flow discharge (Di Stefano et al., 2017c, Di Stefano et al., 2018, Palmeri et al., 2018). Ferro and Nicosia (2019), using the results of a laboratory investigation by Jiang et al. (2018) focusing on rill erosion, positively tested the applicability of this theoretical flow resistance law for rill flows hit by raindrops. In particular, the theoretical relationship (Ferro, 2018) between the velocity profile parameter Γ and flow Froude number F and slope s (see Eq. (8)) was calibrated by measurements carried out for three different rainfall intensities. Notwithstanding the power velocity profile should not be applicable to a flow under rainfall action, the Darcy-Weisbach friction factor resulted accurately estimated by the theoretical approach of Ferro, 2017, Ferro, 2018 assuming a parameter Γ varying with rainfall intensity.

Recently, Nicosia et al. (2019) tested the theoretical approach of Ferro, 2017, Ferro, 2018 using the Water Erosion Prediction Project data-base for the condition of a rill flow subjected to rainfall. The analysis demonstrated the applicability of the theoretical approach, the effect of rainfall on Γ parameter and the increase of flow resistance with the effect of rainfall impact. Therefore, loosening the hypothesis on the velocity distribution (i.e. a power velocity profile can be applied for a flow under rainfall action) an accurate estimate of the Darcy-Weisbach friction factor can be obtained by the theoretical approach proposed by Ferro, 2017, Ferro, 2018.

The main aim of this paper is testing the applicability of the theoretically deduced flow resistance equation (Di Stefano et al., 2017a, Di Stefano et al., 2017c, Ferro, 2017, Ferro, 2018, Ferro and Porto, 2018, Palmeri et al., 2018) based on a power-velocity profile, using laboratory measurements by Yoon and Wenzel (1971) for an overland flow on a smooth bed under rainfall. In particular, these measurements of the Darcy-Weisbach friction factor, corresponding to a wide range of the flow Reynolds number (191–5700), are used i) to assess the relationship between the velocity profile parameter Γ, the channel slope, the flow Froude number and rainfall intensity, ii) to assess the applicability of the theoretical flow resistance equation for an overland flow on a smooth bed disturbed by rainfall impact and iii) to test only the influence of the rainfall intensity on the overland flow resistance.

Section snippets

The applied theoretical flow resistance equation

According to Di Stefano et al., 2017c, Di Stefano et al., 2018 the Π-Theorem of the dimensional analysis and self-similarity theory (Barenblatt, 1979, Barenblatt, 1987, Barenblatt, 1993, Ferro, 1997, Ferro, 2017, Ferro, 2018, Yalin, 1977) can be usefully applied to deduce the power velocity profile yielding to flow resistance equation for a uniform open channel flow.

The velocity distribution along a given vertical can be expressed by the following functional relationship (Barenblatt, 1987,

Measurements of overland flow resistance under rainfall used in this investigation

Yoon and Wenzel (1971) investigated the resistance effect of rainfall in a steady sheet flow on a smooth surface, carrying out experiments in a 0.91 m wide and 7.31 m long flume, with a plate glass bottom and plywood walls coated with epoxy paint. One pump supplied water from a reservoir to the rainfall modules and another one from the reservoir to an head box for base flow. Four different intensities (R = 12.7, 31.75, 95.25 and 381 mm h−1) were simulated by a framework with 12 modules

Results and discussion

For testing the effect of rainfall on flow resistance law, the theoretical relationship (Eq. (8)) between the measured velocity profile parameter Γ (obtained by Eqs. (6), (7) and the measurements of V, h, R, s), the slope steepness, and the flow Froude number was calibrated for each rainfall intensity. The values of the estimated coefficients a, b and c of Eq. (8) are listed in Table 2.

The comparison between the measured Γv values and those calculated by Eq. (8) with the coefficients a, b and c

Conclusions

A theoretical flow resistance law, deduced by the Π-Theorem of dimensional analysis and self-similarity theory, was applied to the overland flow data by Yoon and Wenzel (1971) obtained by flume measurements. Using the measurements carried out with four different rainfall intensities and two different values of channel slope, the theoretical relationship between the velocity profile parameter Γ, the slope, and the flow Froude number was calibrated and tested by independent data. The three

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

All authors set up the research, analyzed and interpreted the results and contributed to write the paper. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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