Testing a theoretical resistance law for overland flow on a stony hillslope

Overland flow, sediments, and nutrients transported in runoff are important processes involved in soil erosion and water pollution. Modelling transport of sediments and chemicals requires accurate estimates of hydraulic resistance, which is one of the key variables characterizing runoff water depth and velocity. In this paper, a new theoretical power–velocity profile, originally deduced neglecting the impact effect of rainfall, was initially modified for taking into account the effect of rainfall intensity. Then a theoretical flow resistance law was obtained by integration of the new flow velocity distribution. This flow resistance law was tested using field measurements by Nearing for the condition of overland flow under simulated rainfall. Measurements of the Darcy–Weisbach friction factor, corresponding to flow Reynolds number ranging from 48 to 194, were obtained for simulated rainfall with two different rainfall intensity values (59 and 178 mm hr⁻¹). The database, including measurements of flow velocity, water depth, cross-sectional area, wetted perimeter, and bed slope, allowed for calibration of the relationship between the velocity profile parameter Γ, the slope steepness s, and the flow Froude number F, taking also into account the influence of rainfall intensity i. Results yielded the following conclusions: (a) The proposed theoretical flow resistance equation accurately estimated the Darcy–Weisbach friction factor for overland flow under simulated rainfall, (b) the flow resistance increased with rainfall intensity for laminar overland flow, and (c) the mean flow velocity was quasi-independent of the slope gradient.     

Flow resistance equation for rills

In this paper, a new flow resistance equation for rill flow was deduced applying dimensional analysis and self‐similarity theory. At first, the incomplete self‐similarity hypothesis was used for establishing the flow velocity distribution whose integration gives the theoretical expression of the Darcy–Weisbach friction factor. Then the deduced theoretical resistance equation was tested by some measurements of flow velocity, water depth, cross section area, wetted perimeter, and bed slope carried out in 106 reaches of some rills shaped on an experimental plot. A relationship between the velocity profile, the channel slope, and the flow Froude number was also established. The analysis showed that the Darcy–Weisbach friction factor can be accurately estimated by the proposed theoretical approach based on a power–velocity profile.     

Testing slope effect on flow resistance equation for mobile bed rills

In this paper, a recently theoretically deduced rill flow resistance equation, based on a power‐velocity profile, is tested experimentally on plots of varying slopes in which mobile bed rills are incised. Initially, measurements of flow velocity, water depth, cross‐sectional area, wetted perimeter and bed slope conducted in 106 reaches of rills incised on an experimental plot having a slope of 14% were used to calibrate the flow resistance equation. Then, the relationship between the velocity profile parameter Γ, the channel slope, and the flow Froude number, which was calibrated using the 106 rill reach data, was tested using measurements carried out in plots having slopes of 22% and 9%. The measurements carried out in the latter slope conditions confirmed that (a) the Darcy–Weisbach friction factor can be accurately estimated using the proposed theoretical approach, and (b) the data were supportive of the slope independence hypothesis of rill velocity stated by Govers.     

Overland runoff erosion dynamics on steep slopes with forages under field simulated rainfall and inflow

Although numerous studies have acknowledged that vegetation can reduce erosion, few process-based studies have examined how vegetation cover affect runoff hydraulics and erosion processes. We present field observations of overland flow hydraulics using rainfall simulations in a typical semiarid area in China. Field plots (5 × 2 m²) were constructed on a loess hillslope (25°), including bare soil plot as control and three plots with planted forage species as treatments—Astragalus adsurgens, Medicago sativa and Cosmos bipinnatus. Both simulated rainfall and simulated rainfall + inflow were applied. Forages reduced soil loss by 55–85% and decreased overland flow rate by 12–37%. Forages significantly increased flow hydraulic resistance expressed by Darcy–Weisbach friction factor by 188–202% and expressed by Manning's friction factor by 66–75%; and decreased overland flow velocity by 28–30%. The upslope inflow significantly increased overland flow velocity by 67% and stream power by 449%, resulting in increased sediment yield rate by 108%. Erosion rate exhibited a significant linear relationship with stream power. M. sativa exhibited the best in reducing soil loss which probably resulted from its role in reducing stream power. Forages on the downslope performed better at reducing sediment yield than upslope due to decreased rill formation and stream power. The findings contribute to an improved understanding of using vegetation to control water and soil loss and land degradation in semiarid environments.     

Comparing flow resistance law for fixed and mobile bed rills

Rills caused by run‐off concentration on erodible hillslopes have very irregular profiles and cross‐section shapes. Rill erosion directly depends on the hydraulics of flow in the rills, which may differ greatly from hydraulics of flow in larger and regular channels. In this paper, a recently theoretically deduced rill flow resistance equation, based on a power–velocity profile, was tested experimentally on plots of varying slopes (ranging from 9% to 26%) in which mobile and fixed bed rills were incised. Initially, measurements of flow velocity, water depth, cross‐section area, wetted perimeter, and bed slope, carried out in 320 reaches of mobile bed rills and in 165 reaches of fixed rills, were used for calibrating the theoretical flow resistance equation. Then the relationship between the velocity profile parameter Γ, the channel slope, and the flow Froude number was separately calibrated for the mobile bed rills and for the fixed ones. The measurements carried out in both conditions (fixed and mobile bed rills) confirmed that the Darcy–Weisbach friction factor can be accurately estimated using the proposed theoretical approach. For mobile bed rills, the data were supportive of the slope independence hypothesis of velocity, due to the feedback mechanism, stated by Govers. The feedback mechanism was able to produce quasicritical flow conditions. For fixed bed rills, obtained by fixing the rill channel, by a glue, at the end of the experimental run with a mobile bed rill, the slope independence of the flow velocity measurements was also detected. Therefore, an experimental run carried out by a rill bed fixed after modelling flow action is useful to detect the feedback mechanism. Finally, the analysis showed that, for the investigated conditions, the effect of sediment transport on the flow resistance law can be considered negligible respect to the grain roughness effect.     

Rill flow resistance law under equilibrium bed‐load transport conditions

In this paper, a recently deduced flow resistance equation for open channel flow was tested under equilibrium bed‐load transport conditions in a rill. First, the flow resistance equation was deduced applying dimensional analysis and the incomplete self‐similarity condition for the flow velocity distribution. Then, the following steps were carried out for developing the analysis: (a) a relationship (Equation  13 ) between the Γ function of the velocity profile, the rill slope, and the Froude number was calibrated by the available measurements by Jiang et al.; (b) a relationship (Equation  17 ) between the Γ function, the rill slope, the Shields number, and the Froude number was calibrated by the same measurements; and (c) the Darcy–Weisbach friction factor values measured by Jiang et al. were compared with those calculated by the rill flow resistance equation with Γ estimated by Equations  13 and 17 . This last comparison demonstrated that the rill flow resistance equation, in which slope and Shields number, representative of sediment transport effects, are introduced, is characterized by the lowest values of the estimate errors.     

Effects of sediment load on hydraulics of overland flow on steep slopes

Eroded sediment may have significant effects on the hydraulics of overland flow, but few studies have been performed to quantify these effects on steep slopes. This study investigated the potential effects of sediment load on Reynolds number, Froude number, flow depth, mean velocity, Darcy–Weisbach friction coefficient, shear stress, stream power, and unit stream power of overland flow in a sand‐glued hydraulic flume under a wide range of hydraulic conditions and sediment loads. Slope gradients were varied from 8·7 to 34·2%, unit flow rates from 0·66 to 5·26×10^?3 m² s^?1, and sediment loads from 0 to 6·95 kg m^?1 s^?1. Both Reynolds number (Re) and Froude number (Fr) decreased as sediment load increased, implying a decrease in flow turbulence. This inverse relationship should be considered in modeling soil erosion processes. Flow depth increased as sediment load increased with a mean value of 1·227 mm, caused by an increase in volume of sediment‐laden flow (contribution 62·4%) and a decrease in mean flow velocity (contribution 37·6%). The mean flow velocity decreased by up to 0·071 m s^?1 as sediment load increased. The Darcy–Weisbach friction coefficient (f) increased with sediment load, showing that the total energy consumption increased with sediment load. The effects of sediment load on f depended on flow discharge: as flow discharge increased, the influence of sediment load on f decreased due to increased flow depth and reduced relative roughness. Flow shear stress and stream power increased with sediment load, on average, by 80·5% and 60·2%, respectively; however, unit stream power decreased by an average of 11·1% as sediment load increased. Further studies are needed to extend and apply the insights obtained under these controlled conditions to real‐world overland flow conditions. Copyright © 2010 John Wiley & Sons, Ltd.     

Volumetric displacement of flow depth by obstacles,and the determination of friction factors in shallow overland flows

Large roughness elements such as stones or plant stems (obstacles) influence the depth of overland flows in two ways. The first effect is a dynamic one, involving frictional retardation of the flow and associated reduction in flow speeds. The second influence is static, and arises from the upward volumetric displacement of flow depth because of the submerged volume of the obstacles. Depending upon the distribution of submerged obstacle volume with height above the soil surface, the proportion of the flow volume occupied (and so, the perturbation of flow depth arising from volumetric displacement) can vary irregularly or systematically with flow stage. Furthermore, the amount of volumetric displacement of flow depth would vary among surfaces carrying different cover fractions of identical obstacles. Consequently, estimates of the change in friction factors arising from the drag on flow traversing varying obstacle cover fractions are confounded with the parallel shift volumetric displacement. To understand the true frictional drag arising from obstacles, a correction must be made for the volumetric displacement. A method for making this correction is outlined. New laboratory experiments provide precise observations of depths and friction coefficients in laminar flows passing fields of regular obstacles. After making the proposed correction for volumetric displacement, increases of 40 to 75 per cent in the derived value of the Darcy–Weisbach friction factor, f, are found for an obstacle cover of 20 per cent. Many published studies of friction coefficients in shallow overland flows, such as those on stone‐covered dryland soils, involve larger obstacle cover fractions, and evidently involve the significant confounding effect of volumetric displacement. Copyright © 2002 John Wiley & Sons, Ltd.     

The impact of flow discharge on the hydraulic characteristics of headcut erosion processes in the gully region of the Loess Plateau

Headcut erosion is associated with major hydraulic changes induced by the gully head of concentrated flow. However, the variation in the hydraulic characteristics of the headcut erosion process is still not clear in the gully region of the Loess Plateau. A series of rainfall combined scouring experiments (flow discharges ranging from 3.6 to 7.2 m³ hr⁻¹, with 0.8 mm min⁻¹ rainfall intensity) were conducted on experimental plots to clarify the variation in the hydraulic parameters induced by gully head and erosion processes under different flow discharges. The results showed that concentrated flows in the catchment area and gully bed were turbulent (Reynolds number ranging from 1,876 to 6,693) and transformed between supercritical and subcritical (Froude number ranging from 0.96 to 3.73). The hydraulic parameters, such as the flow velocity, Reynolds number, shear stress, stream power, Darcy–Weisbach friction factor, and unit stream power in the catchment area were 0.45–0.59 m s⁻¹, 2086–6693, 1.96–5.33 Pa, 0.89–2.86 W m⁻², 0.08–0.16, and 0.023–0.031 m s⁻¹, respectively. When the concentrated flows dropped from the gully head, the hydraulic parameters in the gully bed decreased by 3.39–26.07%, 1.49–29.99%, 65.19–67.14%, 67.25–74.96%, 28.53–61.31%, and 67.82–77.14%, respectively, which contributed to the flow energy consumption at the gully head. As flow discharge increased, Reynolds number, shear stress, and stream power increased, while flow velocity, Froude number, unit stream power, and Darcy–Weisbach friction factor did not. The flow energy consumption at the gully head was 9.66–10.13, 13.25–13.74, 15.68–16.41, and 19.28–20.25 J s⁻¹, respectively, under different flow discharges and accounted for 60.58–68.50% of the flow energy consumption of the experimental plots. Generally, the sediment discharges increased rapidly at the initial stage, then increased slowly, and finally reached a steady state condition, which showed a significant declining logarithmic trend with experimental duration (P<.01) and increased with increasing flow discharge. Accordingly, the flow energy consumption was significantly correlated with the sediment yield. These findings could improve our understanding of the hydraulic properties and flow energy characteristics of headcut erosion.     

Flow resistance in steep mountain streams

Resistance to flow at low to moderate stream discharge was examined in five small (12–77 km² drainage area) tributaries of Chilliwack River, British Columbia, more than half of which exhibit planar bed morphology. The resulting data set is composed of eight to 12 individual estimates of the total resistance to flow at 61 cross sections located in 13 separate reaches of five tributaries to the main river. This new data set includes 625 individual estimates of resistance to flow at low to moderate river stage. Resistance to flow in these conditions is high, highly variable and strongly dependent on stage. The Darcy–Weisbach resistance factor (ff) varies over six orders of magnitude (0·29–12 700) and Manning's n varies over three orders of magnitude (0·047–7·95). Despite this extreme range, both power equations at the individual cross sections and Keulegan equations for reach‐averaged values describe the hydraulic relations well. Roughness is divided into grain and form (considered as all non‐grain sources) components. Form roughness is the dominant component, accounting for about 90% of the total roughness of the system (i.e., form roughness is on average 8.6 times as great as grain roughness). Of the various quantitative and qualitative form‐roughness indicators observed, only the sorting coefficient (σ = D₈₄/D₅₀) correlates well with form roughness. Copyright © 2008 John Wiley & Sons, Ltd.     

Testing a new rill flow resistance approach using the Water Erosion Prediction Project experimental database

In this paper, a recently theoretically deduced rill flow resistance equation, based on a power‐velocity profile, was tested using the Water Erosion Prediction Project database. This database includes measurements of flow velocity, water depth, cross section area, wetted perimeter, and bed slope that were made in rills shaped on experimental sites distributed across the continental United States. In particular, three different experimental conditions (only rainfall, only flow, and rain with flow) were examined, and for each condition, the theoretically based relationship for estimating the Γ function of the power velocity profile was calibrated. The results established that (a) the Darcy‐Weisbach friction factor can be accurately estimated using the proposed theoretical approach, and (b) the flow resistance increases with the effect of rainfall impact.     

Effect of saltating sediment on flow resistance and bed roughness in overland flow

The acceleration of saltating grains by overland flow causes momentum to be transferred from the flow to the grains, thereby increasing flow resistance and bed roughness. To assess the impact of saltating sediment on overland flow hydraulics, velocity profiles in transitional and turbulent flows on a fixed sand-covered bed were measured using hot-film anemometry. Five discharges were studied. At each discharge, three flows were measured: one free of sediment, one with a relatively low sediment load, and one with a relatively high sediment load. In these flows from 83 to 90 per cent of the sediment was travelling by saltation. As a result, in the sediment-laden flows the near-bed velocities were smaller and the velocity profiles steeper than those in the equivalent sediment-free flows. Sediment loads ranged up to 87·0 per cent of transport capacity and accounted for as much as 20·8 per cent of flow resistance (measured by the friction factor) and 89·7 per cent of bed roughness (measured by the ratio of the roughness length to median grain diameter). It is concluded that saltating sediment has a considerable impact on overland flow hydraulics, at least on fixed granular beds. Saltation is likely to have a relatively smaller effect on overland flow on natural hillslopes and agricultural fields where form and wave resistance dominate. Still, saltation is generally of greater significance in overland flow than in river flow, and for this reason its effect on overland flow hydraulics is deserving of further study. © 1998 John Wiley & Sons, Ltd.     

Terrestrial laser scanning soil surfaces: a field methodology to examine soil surface roughness and overland flow hydraulics

Conventional roughness–resistance relationships developed for pipe and open‐channel flows cannot accurately describe shallow overland flows over natural rough surfaces. This paper develops a new field methodology combining terrestrial laser scanning (TLS) and overland flow simulation to provide a high‐resolution dataset of surface roughness and overland flow hydraulics as simulated on natural bare soil surfaces. This method permits a close examination of the factors controlling flow velocity and a re‐evaluation of the relationship between surface roughness and flow resistance. The aggregate effect of flow dynamics, infiltration and depression storage on retarding the passage of water over a surface is important where runoff‐generating areas are distant from well‐defined channels. Experiments to separate these effects show that this ‘effective resistance’ is dominated by surface roughness. Eight measurements of surface roughness are found to be related to flow resistance: standard deviation of elevations, inundation ratio, pit density (measured both perpendicular and parallel to the flow direction), slope, median depth, skewness of the depth distribution and frontal area. Hillslope position is found to affect the significant roughness measures. In contrast, infiltration rate has little effect on the velocity of water fronts advancing over the soil surfaces examined here and the effect of depression storage is limited. Overland flow resistance is depth dependent where complex microtopographic structures are progressively inundated. Copyright © 2010 John Wiley & Sons, Ltd.     

Interpolation between Darcy–Weisbach and Darcy for laminar and turbulent flows

An equation describing flow in an open channel with obstacles is derived, following the conservation of momentum approach used by Bélanger and St. Venant. When the obstacles are all submerged the result yields the Darcy–Weisbach equation for turbulent flow in pipes and open channels. When the obstacles are only partially submerged the result leads to the governing equation in a porous medium. If the flow is turbulent the square of the velocity is proportional to the hydraulic gradient and if the flow is laminar, which is the usual case, the velocity is proportional to the hydraulic gradient. This last result is in agreement with Darcy’s law in porous media. Thus our equation interpolates between and reduces to, the two fundamental results of Darcy. In general our equation should prove useful in practice for open flow in a channel with both submerged and emerging obstacles.     

Hydraulics of overland flow influenced by litter incorporation under extreme rainfall

Water erosion on hillslopes is a worldwide environmental problem, which is a rainfall‐induced process, especially extreme rainfall. The great intensity of extreme rainfall strongly enhances the power of overland flow to detach soil and transport sediment. Plant litter is one of the most important constituents of ecosystems that often covers the soil surface and can be incorporated into topsoil. However, little attention has been paid to its effect on flow hydraulics owing to the veiled nature. This study aimed to examine the effects of incorporated litter on the hydraulic properties under extreme rainfall condition. To reach this goal, six litter rates of 0, 0.05, 0.10, 0.20, 0.35, and 0.50 kg m^?2 and four litter types collected from deciduous trees, coniferous trees, shrubs, and herbs were incorporated into topsoil. Then, simulated rainfall experiments were performed on five slope gradients (5°, 10°, 15°, 20°, and 25°) with an extreme rainfall intensity of 80 mm h^?1. The results showed that Froude number and flow velocity of the overland flow decreased, whereas flow resistance increased exponentially with litter incorporation rate. Litter type had an influence on flow hydraulics, which can mainly be attributed to the variations in surface coverage of the exposed litter and the litter morphology. Flow velocity and Darcy–Weisbach coefficient increased markedly with slope gradient. However, the variation of slope gradient did not modify the relationships between flow hydraulics and incorporated litter rate. The random roughness, resulting from heterogeneous erosion due to the uneven protection of surface exposed litter, increased linearly with litter incorporated rate. As rainfall proceeded, flow hydraulics varied with incorporated litter rate and slope gradient complicatedly due to the increases in flow rate and coverage of the exposed litter and the modification of soil surface roughness.     

Generalized conceptual modeling of dimensionless overland flow hydrographs

The rising and recession limbs of conceptual dimensionless overland flow hydrographs are calculated for specific values of the rating exponent in the range 1 ≤ m ≤ 3, including a linear reservoir (m = 1); 100% turbulent Chezy friction (m = 3/2); 100% turbulent Manning friction (m = 5/3); 67% turbulent Chezy (or 75% turbulent Manning) (m = 2); and 100% laminar flow (m = 3). These conceptual overland flow hydrographs show finite amounts of diffusion, increasing with decreasing rating exponent, unlike the kinematic wave hydrograph, which is nondiffusive.     

A unifying framework for watershed thermodynamics: constitutive relationships

The balance equations for mass and momentum, averaged over the scale of a watershed entity, need to be supplemented with constitutive equations relating flow velocities, pressure potential differences, as well as mass and force exchanges within and across the boundaries of a watershed. In this paper, the procedure for the derivation of such constitutive relationships is described in detail. This procedure is based on the method pioneered by Coleman and Noll through exploitation of the second law of thermodynamics acting as a constraint-type relationship. The method is illustrated by its application to some common situations occurring in real world watersheds. Thermodynamically admissible and physically consistent constitutive relationships for mass exchange terms among the subregions constituting the watershed (subsurface zones, overland flow regions, channel) are proposed. These constitutive equations are subsequently combined with equations of mass balance for the subregions. In addition, constitutive relationships for forces exchanged amongst the subregions are also derived within the same thermodynamic framework. It is shown that, after linearisation of the latter constitutive relations in terms of the velocity, a watershed-scale Darcy's law governing flow in the unsaturated and saturated zones can be obtained. For the overland flow, a second order constitutive relationship with respect to velocity is proposed for the momentum exchange terms, leading to a watershed-scale Chezy formula. For the channel network REW-scale Saint–Venant equations are derived. Thus, within the framework of this approach new relationships governing exchange terms for mass and momentum are obtained and, moreover, some well-known experimental results are derived in a rigorous manner.     

CORRECTION FACTORS IN THE DETERMINATION OF MEAN VELOCITY OF OVERLAND FLOW

The velocity of overland flow has been conventionally measured using tracers, but it is difficult to measure the mean flow velocity directly because the centroid of the tracer plume is not easily identified. Consequently, previous investigators have measured the velocity of the leading edge of the plume and multiplied it by a correction factor α to obtain an estimate of mean velocity. An alternative method is to measure the velocity of the peak concentration in the tracer plume and multiply this velocity by another correction factor β to estimate mean velocity. To investigate the controls of α and β and develop predictive models for these correction factors, 40 experiments were performed in a flume with a mobile sand bed. Multiple regression analyses reveal that both α and β vary inversely with slope and directly with Reynolds number. The derived regression equations may be used to calculate the mean velocity of other shallow overland flows, at least within the range of slope and Reynolds number for which the equations were developed. In the experiments, slope ranged from 2.7;° to 10° and Reynolds number from 1900 to 12 600.     

Hydrodynamic characteristics of rill flow on steep slopes

Rill erosion is a dominant sediment source on sloping lands. However, the amount of soil loss from rills on steep slopes is vastly more than that on gentle slopes because of differences in rill shape and hydraulic patterns. The aims of this paper are to determine the hydrodynamic characteristics of rills and the friction coefficients in steep slope conditions and to propose modifications of some hydraulic parameters used in soil loss prediction models. A series of inflow experiments was conducted on loess slopes. The results show that the geometric and hydraulic properties of rill on the steep loess slopes, which are characterized by the mean width of cross sections, mean velocity and mean depth of flow, are related to discharge and slope gradient in power functions. However, the related exponents to discharge are 0.26, 0.48 and 0.26, respectively, which are different from the exponents derived in previous studies, which were conducted on gentle slopes. The Manning roughness coefficient ranged from 0.035 to 0.071, with an average of 0.0536, and the Darcy–Weisbach friction coefficients varied from 0.4 to 1.9. The roughness coefficients are closely related to the Reynolds numbers and flow volumes; however, the correlations vary with slope gradient. The roughness coefficients are directly proportional to the Reynolds number and the flow volume on steep slopes, in contrast with the roughness coefficients found on gentle slopes, which decrease as the Reynolds number and flow volume increase. This difference is caused by the interactions among the hydraulics of the flow, the shape of the rills and the sediment concentrations on steep slopes. The results indicate that parameters used in models to predict rill erosion have to be modified according to slope gradient. Copyright © 2015 John Wiley & Sons, Ltd.     

An experimental study on the evolution law of roll wave parameters in overland flow

Roll waves commonly occur in overland flow and have an important influence on the progress of soil erosion on slopes. This study aimed to explore the evolution and mechanism of roll waves on steep slopes. The potential effects of flow rate, rainfall intensity and bed roughness on the laws controlling roll wave parameters were investigated. The flow rates, rainfall intensities and bed roughness varied from 5 to 30 L/min, 0 to 150 mm/h, and 0.061 to 1.700 mm, respectively. The results indicate that roll waves polymerize significantly along the propagation path, and bed roughness and rainfall affect the generation and evolution of roll waves. The wave velocity, length and height decreased with bed roughness, whereas the wave frequency increased with increasing bed roughness under fixed flow rate and rainfall intensity conditions. Rainfall increased the wave velocity and wavelength and decreased the wave frequency. The wave velocity, height and wavelength tended to increase with an increasing flow rate. Rainfall promoted the generation of roll waves, whereas bed roughness had the opposite effect. The generation of roll waves is closely related to the Froude number (Fr) and flow resistance. In this experiment, the range of the Reynolds number for the roll waves generated in the laminar region was 142–416, and the range of the flow resistance coefficient was 0.64–4.85. The critical value of the Fr for flow instability in the laminar region was approximately 0.57. Exploring the generation and evolution law of roll waves is necessary for understanding the processes and dynamic mechanisms of slope soil erosion.