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.     

Comment on “Overland runoff erosion dynamics on steep slopes with forages under field simulated rainfall and inflow by C. Li and C. Pan”

Managing sloping landscapes to control soil erosion processes due to rainfall and runoff is a relevant problem, especially when the vegetation is absent or sparse. The aim of this paper was to investigate the applicability of a theoretically resistance law for overland flow under simulated rainfall, based on a power-velocity profile, using field measurements carried out by Li and Pan for three plots with planted forage species (Astragalus adsurgens, Medicago sativa and Cosmos bipinnatus).The relationship between the velocity profile parameter Γ, the flow Froude number and the rain Reynolds number was calibrated using the data by Li and Pan. The obtained overland flow resistance law was also verified by independent field measurements carried out by Li and Pan in the same plots with the same forage species subjected to three different treatments (intact grass control IG, no litter or leaves NLL (only the grass stems and roots were reserved) and only roots remaining OR). The theoretical approach and the measurements carried out in the investigated conditions allowed to state that a) the Darcy-Weisbach friction factor can be accurately estimated using the proposed theoretical approach, b) the Darcy-Weisbach friction factor varies with rainfall intensity and c) for the investigated forage species, the vegetation treatment (IG, NLL, OR) does not significantly affect the flow resistance in laminar regime.     

ADV measurements of velocity distributions in a gravel‐bed flume

Velocity measurements carried out by an acoustic doppler velocimeter (ADV) in a rectangular laboratory ?ume having a gravel bed are presented. The velocity pro?les are measured in six verticals of the channel cross‐section having an increasing distance (from 4 to 38·5 cm) from the ?ume wall. The experimental runs are carried out for ?ve different bed arrangements, characterized by different concentrations of coarser elements, and for the two conditions of small‐ and large‐scale roughness. For both hydraulic conditions, the velocity measurements are ?rst used to test the applicability of the Dean pro?le and of the logarithmic pro?le corrected by a divergence function proposed in this paper. Then, for each value of the depth sediment ratio h/d₈₄, the non‐dimensional friction factor parameter is calculated by integration of the measured velocity distributions in the different verticals of the cross‐section. Finally a semi‐logarithmic ?ow resistance equation is empirically deduced. Copyright © 2003 John Wiley & Sons, Ltd.     

Evaluating overland flow sediment transport capacity

An equation for evaluating the sediment transport capacity of overland flow is a necessary part of a physically based soil erosion model describing sediment detachment and transport as distributed processes. At first, for the hydraulic conditions of small-scale and large-scale roughness, the sediment transport capacity relationship used in the WEPP model is calibrated by Yalin and Govers' equation. The analysis shows that the transport coefficient K_t depends on the Shields parameter, Y, according to a semi-logarithmic (Yalin) or a linear (Govers) equation. The reliability of the semi-logarithmic equation is verified by Smart's, and Aziz and Scott's experimental data. Then the Low's formula, whose applicability is also proved by Smart's, and Aziz and Scott's data, is transformed as a stream power equation in which a stream power coefficient, K_SP, depending on Shields parameter, slope, sediment and water-specific weight, appears. A relationship between transport capacity and effective stream power is also proposed. Finally, the influence of rainfall on sediment transport capacity and the prediction of critical shear stress corresponding to overland flow are examined. © 1998 John Wiley & Sons, Ltd.     

A full-scale study of Darcy-Weisbach friction factor for channels vegetated by riparian species

In this article, an open channel flow resistance equation, deduced applying dimensional analysis and incomplete self-similarity condition for the flow velocity distribution, was tested using measurements carried out in a full-scale channel equipped with three types of riparian plants (Salix alba L., Salix caprea L. and Alnus glutinosa L.). In the experimental channel, having banks lined with boulders, the vegetation branches were anchored in a concrete bottom. For each species, the measurements were carried out with plants having different amounts of leaves, different plant density and plant area index. The relationship between the scale factor Γ of the velocity profile and the Froude number was separately calibrated by measurements carried out without and with vegetation. The component of Darcy-Weisbach friction factor corresponding to the riparian vegetation f_v was calculated as the difference between the measured friction factor value (channel grain roughness + vegetation) and that calculated for the channel without vegetation in the same hydraulic conditions. Using these f_v values, the relationship between the scale factor Γ and the Froude number was calibrated. In this last relationship, a scaling coefficient a varying with the investigated vegetation type was introduced. This coefficient, as expected, gives the highest friction factor values for vegetation having branches with leaves. The theoretical flow resistance law, coupled with the relationship for estimating the Γ function having a scaling coefficient different for each investigated vegetation type, allowed an accurate estimate of the Darcy-Weisbach friction factor (errors less than or equal to 20% for 82.6% of the investigated cases). Finally, for the investigated vegetation species that are characterized by a condition with few leaves or leafless, the scaling coefficient a resulted strongly related to the bending stiffness. This analysis demonstrated that the highest Darcy-Weisbach friction factors correspond to vegetation species characterized by the highest values of bending stiffness. The friction factor values calculated for this last condition are characterized by errors that were less than or equal to ±20% for 90.6% of cases.     

Estimation of flow resistance in gravel-bedded rivers: A physical explanation of the multiplier of roughness length

The need to estimate velocity and discharge indirectly in gravel-bedded rivers is a commonly-encountered problem. Semilogarithmic friction equations are used to estimate mean velocity using a friction factor obtained from depth and grain size information. Although such equations have a semi-theoretical basis, in natural gravel-bed channels, an empirical constant (6.8 or 3.5) has to be introduced to scale-up the characteristic grain size (D₅₀ or D₈₄) to represent the effective roughness length. In this paper, two contrasting approaches are used to suggest that the multiplier of characteristic grain size is attributable to the effect of small-scale form resistance, reflecting the occurrence of microtopographic bedforms in gravel-bedded environments. First, spatial elevation dependence in short, detailed bed profiles from a single gravel-bedded river is investigated using semivariogram and zero-crossing analyses. This leads to objective identification of two discrete scales of bed roughness, associated with grain and microtopographic roughness elements. Second, the autocorrelation structure of the three-dimensional near-bed velocity field is examined to identify regularities associated with eddy shedding and energy losses from larger grains and microtopographic bedforms. Apart from improving the capacity to determine friction factors for velocity and discharge estimation, the findings have implications in general for the initial motion of gravelly bed material.     

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.     

Non-linear channel–shoal dynamics in long tidal embayments

The dynamics of finite-amplitude bed forms in a tidal channel is studied with the use of an idealized morphodynamic model. The latter is based on depth-averaged equations for the tidal flow over a sandy bottom. The model considers phenomena on spatial scales of the order of the tidal excursion length. Transport of sediment mainly takes place as suspended load. The reference state of this model is characterized by a spatially uniform M₂ tidal current over a fixed horizontal bed. The temporal evolution of deviations from this reference state is governed by amplitude equations: these are a set of non-linear equations that describe the temporal evolution of bed forms. These equations are used to obtain new morphodynamic equilibria which may be either static or time-periodic. Several of these bottom profiles show strong similarity with the tidal bars that are observed in natural estuaries. The dependence of the equilibrium solutions on the value of bottom friction and channel width is investigated systematically. For narrow channels (width small compared to the tidal excursion length) stable static equilibria exist if bottom friction is slightly larger than r_cr. For channel widths more comparable to the tidal excursion length, multiple stable steady states may exist for bottom friction parameter values below r_cr. Regardless of channel width, stable time-periodic equilibria seem to emerge as the bottom friction is increased.Responsible Editor: Jens Kappenberg     

Flow resistance and hydraulic geometry in bedrock rivers with multiple roughness length scales

Many models of incision by bedrock rivers predict water depth and shear stress from discharge; conversely, palaeoflood discharge is sometimes reconstructed from flow depth markers in rock gorges. In both cases, assumptions are made about flow resistance. The depth–discharge relation in a bedrock river must depend on at least two roughness length scales (exposed rock and sediment cover) and possibly a third (sidewalls). A conceptually attractive way to model the depth–discharge relation in such situations is to partition the total shear stress and friction factor, but it is not obvious how to quantify the friction factor for rough walls in a way that can be used in incision process models. We show that a single flow resistance calculation using a spatially averaged roughness length scale closely approximates the partitioning of stress between sediment and rock, and between bed and walls, in idealized scenarios. Both approaches give closer fits to the measured depth–discharge relations in two small bedrock reaches than can be achieved using a fixed value of Manning's n or the Chézy friction factor. Sidewalls that are substantially rougher or smoother than the bed have a significant effect on the partitioning of shear stress between bed and sidewalls. More research is needed on how best to estimate roughness length scales from observable or measurable channel characteristics. © 2019 John Wiley & Sons, Ltd.     

Quantifying flow resistance in mountain streams using computational fluid dynamics modeling over structure-from-motion photogrammetry-derived microtopography

Flow resistance in mountain streams is important for assessing flooding hazard and quantifying sediment transport and bedrock incision in upland landscapes. In such settings, flow resistance is sensitive to grain-scale roughness, which has traditionally been characterized by particle size distributions derived from laborious point counts of streambed sediment. Developing a general framework for rapid quantification of resistance in mountain streams is still a challenge. Here we present a semi-automated workflow that combines millimeter- to centimeter-scale structure-from-motion (SfM) photogrammetry surveys of bed topography and computational fluid dynamics (CFD) simulations to better evaluate surface roughness and rapidly quantify flow resistance in mountain streams. The workflow was applied to three field sites of gravel, cobble, and boulder-bedded channels with a wide range of grain size, sorting, and shape. Large-eddy simulations with body-fitted meshes generated from SfM photogrammetry-derived surfaces were performed to quantify flow resistance. The analysis of bed microtopography using a second-order structure function identified three scaling regimes that corresponded to important roughness length scales and surface complexity contributing to flow resistance. The standard deviation σ_z of detrended streambed elevation normalized by water depth, as a proxy for the vertical roughness length scale, emerges as the primary control on flow resistance and is furthermore tied to the characteristic length scale of rough surface-generated vortices. Horizontal length scales and surface complexity are secondary controls on flow resistance. A new resistance predictor linking water depth and vertical roughness scale, i.e.  H/σ_z, is proposed based on the comparison between σ_z and the characteristic length scale of vortex shedding. In addition, representing streambeds using digital elevation models (DEM) is appropriate for well-sorted streambeds, but not for poorly sorted ones under shallow and medium flow depth conditions due to the missing local overhanging features captured by fully 3D meshes which modulate local pressure gradient and thus bulk flow separation and pressure distribution. An appraisal of the mesh resolution effect on flow resistance shows that the SfM photogrammetry data resolution and the optimal CFD mesh size should be about 1/7 to 1/14 of the standard deviation of bed elevation. © 2019 John Wiley & Sons, Ltd.     

The vertical structure of tidal currents

Abstract

The accuracy to which the vertical structure of tidal currents can be predicted is examined. Theoretical models for current structure are developed employing (a) a constant eddy viscosity E = ε and (b) an eddy viscosity varying linearly with height above the sea bed z; E(z)=βz. By requiring these models to satisfy the commonly accepted quadratic friction law, the condition ε>½k is deduced where k is the bed friction coefficient, W a representative velocity and D the depth.

The sense of rotation of a current ellipse is shown to be related to the configuration of co-tidal charts. The vertical structure of the current ellipse is illustrated from the theoretical models and the sensitivity of this structure is examined for the following variables: (a) eddy viscosity ε or βz, (b) the bed friction parameter kW, (c) rotation of the prescribed pressure gradients and (d) tidal period. While reasonable agreement between observed and calculated current profiles may often be reported, precise agreement is shown to depend upon accurate specification of both eddy viscosity and the bed stress condition.     

The effect of bed mobility on resistance to overland flow

When sediment grains are transported as bed load in overland flow, there is a net transfer of momentum from the flow to the grains. When these grains collide with other grains, whether on the bed or in the flow, streamwise flow velocity decreases and resistance to flow increases. Resistance to flow generated in this manner is termed bed‐load transport resistance. Resistance to flow f over a plane bed may be partitioned into grain resistance f_g and bed‐load transport resistance f_bt. We use the symbols f_btf and f_btm to denote f_bt for flows over fixed beds and over mobile beds, respectively, and we compute the effect of bed mobility on flow resistance f_mob by subtracting f_btf from f_btm. The data for this study come from 54 flume experiments with fixed beds and 38 with mobile beds. On average f_mob is approximately equal to half of f_btm, which is about one‐quarter of f. Hence, f_mob is about one‐tenth of f. Predictive equations are developed for f_btf, f_btm and f_mob using dimensional analysis to identify the relevant independent variables and regression analysis to evaluate the coefficients associated with these variables. Values of f_mob are always positive which implies that mobile beds offer greater resistance to flow than do fixed beds. Evidently bed‐load grains colliding with mobile beds lose more momentum to the bed than do grains colliding with fixed beds. In other words, grain collisions with mobile beds are less elastic than those with fixed beds. Copyright © 2005 John Wiley & Sons, Ltd.     

Flow resistance in gravel‐bed channels with large‐scale roughness

A previously published mixing length (ML) model for evaluating the Darcy–Weisbach friction factor for a large‐scale roughness condition (depth to sediment height ratio ranging from 1 to 4) is brie?y reviewed and modi?ed (MML). Then the MML model and a modi?ed drag (MD) model are experimentally tested using laboratory measurements carried out for gravel‐bed channels and large‐scale roughness condition. This analysis showed that the MML gives accurate estimates of the Darcy–Weisbach coef?cient and for Froude number values greater than 0·5 the MML model coincides with the ML one. Testing of the MD model shows limited accuracy in estimating ?ow resistance. Finally, the MML and MD models are compared with the performance of a quasi‐theoretical (QT) model deduced applying the P‐theorem of the dimensional analysis and the incomplete self‐similarity condition for the depth/sediment ratio and the Froude number. Using the experimental gravel‐bed data to calibrate the QT model, a constant value of the exponent of the Froude number is determined while two relationships are proposed for estimating the scale factor and the exponent of the depth/sediment ratio. This indirect estimate procedure of the coef?cients (b₀, b₁ and b₂) of the QT model can produce a negligible overestimation or underestimation of the friction factor. Copyright © 2003 John Wiley & Sons, Ltd.     

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.     

Flow structure over square bars at intermediate submergence: Large Eddy Simulation study of bar spacing effect

Turbulent open-channel flow over 2D roughness elements is investigated numerically by Large Eddy Simulation (LES). The flow over square bars for two roughness regimes (k-type roughness and transitional roughness between d-type and k-type) at a relative submergence of H/k = 6.5 is considered, where H is the maximum water depth and k is the roughness height. The selected roughness configurations are based on laboratory experiments, which are used for validating numerical simulations. Results from the LES, in turn, complement the experiments in order to investigate the time-averaged flow properties at much higher spatial resolution. The concept of the double-averaging (DA) of the governing equations is utilized to quantify roughness effects at a range of flow properties. Double-averaged velocity profiles are analysed and the applicability of the logarithmic law for rough-wall flows of intermediate submergence is evaluated. Momentum flux components are quantified and roughness effect on their vertical distribution is assessed using an integral form of the DA-equations. The relative contributions of pressure drag and viscous friction to the overall bed shear stress are also reported.     

A rational sediment transport scaling relation based on dimensionless stream power

The concept of stream channel grade – according to which a stream channel reach will adjust its gradient, S, in order to transport the imposed sediment load having magnitude Q_b and characteristic grain size D_b, with the available discharge Q (Mackin, 1948 , Geological Society of America Bulletin 59 : 463–512; Lane, 1955 , American Society of Civil Engineers, Proceedings 81 : 1–17) is one of the most influential ideas in fluvial geomorphology. Herein, we derive a scaling relation that describes how externally imposed changes in either Q_b or Q can be accommodated by changes in the channel configuration, described by the energy gradient, mean flow depth, characteristic grain size and a parameter describing the effect of bed surface structures on grain entrainment. One version of this scaling relation is based on the dimensionless bed material transport parameter (W*) presented by Parker and Klingeman ( 1982 , Water Resources Research 18 : 1409–1423). An equivalent version is based on a new dimensionless transport parameter (E*) using dimensionless unit stream power. This version is nearly identical to the relation based on W*, except that it is independent of flow resistance. Both versions of the scaling relation are directly comparable to Lane's original relation. In order to generate this stream power‐based scaling relation, we derived an empirical transport function relation relating E* to dimensionless stream power using data from a wide range of stable, bed load‐dominated channels: the form of that transport function is based on the understanding that, while grain entrainment is related to the forces acting on the bed (described by dimensionless shear stress), sediment transport rate is related to the transfer of momentum from the fluid to the bed material (described by dimensionless stream power). Copyright © 2010 John Wiley & Sons, Ltd.     

Aeolian sediment flux decay: Non-linear behaviour on developing deflation lag surfaces

Wind tunnel simulations of the effect of non-erodible roughness elements on sediment transport show that the flux ratio q/q_s, shear velocity U*, and roughness density λ are co-dependent variables. Initially, the sediment flux is enhanced by kinetic energy retention in relatively elastic collisions that occur at the roughness element surfaces, but at the same time, the rising surface coverage of the immobile elements reduces the probability of grain ejection. A zone of strong shearing stress develops within 0·03 to 0·04 m of the rough bed because of a relative straightening of velocity profiles which are normally convex with saltation drag. This positive influence on fluid entrainment is opposed by declining shear stress partitioned to the sand bed. Similarly, because the free stream velocity U_f is fixed while U_* increases, velocity at height z and particle momentum gain from the airstream decline, leading eventually to lower numbers of particles ejected on average at each impact. When the ratio of the element basal area to frontal area σ is approximately equal to 3·5, secondary flow effects appear to become significant, so that the dimensionless aerodynamic roughness parameter Z₀/h and shear stress on the exposed sand bed T_s decrease. It is at this point that grain supply to the airstream and saltation drag appear to be significantly reduced, thereby intensifying the reduction in U_*. The zone of strong fluid shear near the bed dissipates.     

Dimensionless critical shear stress evaluation from flume experiments using different gravel beds

Flume experiments were conducted using four different gravel beds (D₅₀ + 12–39 mm) and a range of marked particles (10–65 mm). The shear stresses were evaluated from friction velocities, when initial movement of marked particles occurred. Two kinds of equations were produced: first for the threshold of initial movement, and second for generalized movement. Equations of the type 0_c + a(D_i/D₅₀)^b, as proposed by Andrews (1983) are applicable even if the material is relatively well sorted. However, the values of a and b are lower (respectively 0·050 and -0·70) for initial movement. Generalized movement requires a higher shear stress (a + 0·068 and b + -0·80). D₉₀ of the bed material and y₀ (the bed roughness parameter) were also used as reference values in place of D₅₀. They produced lower values than in natural streams, mainly owing to the fact that the material used in the flume is better sorted: clusters are less well developed and the bed roughness is lower.     

Analysis of rill step–pool morphology and its comparison with stream case

In this paper, the morphology of step–pool features is analysed using rill measurements and literature data for streams. Close-range photogrammetry was used to carry out ground measurements on rills with step–pool units, shaped on a plot having slope equal to 14, 15, 22, 24 and 26%. Data were used to compare the relationships between H/L, in which H is the step height and L is the step length, and the mean gradient of the step–pool sequence, S_m, for streams or the slope of the step–pool unit, S, for rills. The relationship of H/L against S_m is widely used to test the occurrence of the maximum flow resistance condition in streams, which is associated with the range 1 ≤ (H/L)/S_m ≤ 2. Further analyses were carried out to compare both the formation process and the profile of the pool in rills with those related to streams. Moreover, for a single rill channel, an analysis of flow characteristics expressed in terms of Darcy–Weisbach friction factor and Froude number was developed. The results allowed us to state: (i) the relationships of H/L versus S_m and S are quite similar and the steepness ratio for streams, (H/L)/S_m, and for rills, (H/L)/S, generally ranges from 1 to 2; (ii) the formation process and the profile of the pool in rills are not consistent with those occurring in streams; (iii) in the rills, the longitudinal size of the pool is dominant with respect to the maximum scour depth; (iv) the presence of a sequence of step–pool units within a rill segment noticeably increases flow resistance compared to segments with a flat bed; (v) the Froude number of the flow over the sequence of step–pool units in rills is slightly below the range of 0.8–1 corresponding to the maximum flow resistance in step–pool units.     

Relationships between higher cumulants of channel response. I. Properties of the linear channel response

Abstract

The use of dimensionless cumulants to characterize the shape of the linear response of a semi-infinite prismatic channel is extended and simplified by means of the dimensionless coefficient (γ_R) linking the Rth order shape factor to the 3rd order shape factor raised to the power of (R—2). It is shown that, for the special structure of the cumulants of the linearized St. Venant equations, the dimensionless cumulant coefficient (γ_R) of any order R > 4 is a single-valued function of γ₄. It is also shown that the value of γ_R for any Froude number F₀ combined with any shape of section and any friction law (both characterized by the parameter m) is intermediate between the values for the two limiting cases of the diffusion analogy model (F₀ = 0) and the rapid flow model (F₀ = 1). The dimensionless cumulant coefficient γ_R is related to γ₄ through a parameter p(m, F₀) by a polynomial of order [R/2]—1.