INCM derives from the IDCM idea of evaluating the turbulent stresses as proportional to the gradient of the squared averaged velocities, leading to Eqs. (7) and (11). Observe that the dimensionless coefficient α, in the stress computation given by Eq. (7), can be written as the ratio between αH and the distance between verticals i and i+1. For this reason, coefficient αH can be thought of as a sort of mixing length, related to the scale of the vortices with horizontal axes. INCM assumes the optimal αH to be proportional to the local water depth, because water depth is at least an upper limit for this scale, and the following relationship is applied: αH=ξH, where ξ is an empirical coefficient to be further estimated.

LHRM derives from the observation that, in the Manning equation, the average velocity is set equal to V=R2/3nS0 and has a one-to-one relationship with the hydraulic radius. In this context the hydraulic radius has the meaning of a global parameter, measuring the interactions of the particles along all the section as the ratio between an area and a length. The inconvenience is that, according to Eq. (14), the vertically averaged velocities in points very far from each other remain linked anyway, because the infinitesimal area and the infinitesimal length around two verticals are summed to the numerator and to the denominator of the hydraulic radius independently from the distance between the two verticals. To avoid this, LHRM computes the discharge as an integral of the vertically averaged velocities in the following form: q=∫0LhyUydy, where U is set equal to U=ℜ12/3nS0, and ℜ1 is defined as local hydraulic radius, computed as ℜ1y=∫abhsNy,sds∫abNy,sds2+dz2,a=max⁡0,y-βh,b=min⁡L,y+βh, where z is the topographic elevation (function of s), β is an empirical coefficient and L is the section's top width. Moreover N(y,s) is a shape function where Ny,s=-y-βh(y)-sβh(y)ifa<s<y,y-βh(y)-sβh(y)ifb>s>y,0otherwise. Equation (18) shows how the influence of the section geometry, far from the abscissa y, continuously decreases up to a maximum distance, which is proportional to the water depth according to an empirical positive coefficient β. After numerical discretization, Eqs. (15)–(17) can be solved to get the unknown q, as well as the vertically averaged velocities in each subsection. If β is close to zero and the size of each subsection is common for both formulas, LHRM is equivalent to DCM; if β is very large, LHRM is equivalent to the traditional Manning formula. In the following, β is calibrated using experimental data available in the literature. A sensitivity analysis is also carried out to show that the estimated discharge is only weakly dependent on the choice of the β coefficient, far from its possible extreme values.

INCM and LHRM parameters were calibrated by using data selected from six series of experiments run at the large-scale Flood Channel Facility (FCF) of HR Wallingford (UK) (Knight and Sellin, 1987; Shiono and Knight, 1991; Ackers, 1993), as well as from four series of experiments run in the small-scale experimental apparatus of the Civil Engineering Department at the University of Birmingham (Knight and Demetriou, 1983). The FCF series were named F1, F2, F3, F6, F8 and F10; the Knight and Demetriou series were named K1, K2, K3 and K4. Series F1, F2, and F3 covered different floodplain widths, while series F2, F8, and F10 kept the floodplain widths constant but covered different main channel side slopes. Series F2 and F6 provided a comparison between the symmetric case of two floodplains and the asymmetric case of a single floodplain. All the experiments of Knight and Demetriou (1983) were run with a vertical main channel wall but with different B/b ratios. The series K1 has B/b=1 and its section is simply rectangular. The B/b ratio, for Knight's experimental apparatus, was varied by adding an adjustable side wall to each of the floodplains either in pairs or singly to obtain a symmetrical or asymmetrical cross section. The geometric and hydraulic parameters are shown in Table 1; all notations of the parameters can be found in Fig. 1 and S0 is the bed slope. The subscripts mc and fp of the side slope refer to the main channel and floodplain, respectively. Perspex was used for both main flume and floodplains in all tests. The related Manning roughness is 0.01 m-1/3s.

The experiments were run with several channel configurations, differing mainly for floodplain geometry (widths and side slopes) and main channel side slopes (see Table 1). The K series were characterized by vertical main channel walls. More information concerning the experimental setup can be found in Table 1 (Knight and Demetriou, 1983; Knight and Sellin, 1987; Shiono and Knight, 1991).

Four series, named F1, F2, F3 and F6, were selected for calibration of the β coefficient using the Nash–Sutcliffe (NS) index of the measured and the computed flow rates as a measure of the model's performance (Nash and Sutcliffe, 1970).

The remaining three series, named F2, F8 and F10, plus four series from Knight and Demetriou (1983), named K1, K2, K3 and K4, were used for validation (no.) 1, as reported in the next section. NS is given by NS=1-∑j=1,2∑i=1,NJ∑K=1,MNJqi,j,kobs-qi,j,ksim2∑j=1,2∑i=1,NJ∑K=1,MNJqi,j,kobs-qi,j,kobs‾2, where Nj is the number of series, MNj is the number of tests for each series, qi,j,ksim and qi,j,kobs are, respectively, the computed and the observed discharge (j=1 for the FCF series and j=2 for the Knight series; i is the series index; and K is the water depth index). qi,j,kobs‾ is the average value of the measured discharges.

Both ξ and β parameters were calibrated by maximizing the NS index, computed using all the data of the four series used for calibration. See the NS versus ξ and β curves in Fig. 2a and b.

Calibration provides optimal ξ and β coefficients, respectively, equal to 0.08 and 9. The authors will show in the next sensitivity analysis that even a one-digit approximation of the ξ and β coefficients provides a stable discharge estimation.

We carried out a discharge sensitivity analysis of both new methods using the computed ξ=0.08 and β=9 optimal values and the data of the F2 and K4 series. Sensitivities were normalized in the following form: Is=1qINCMΔqΔξ,Ls=1qLHRMΔqΔβ, where Δq is the difference between the discharges computed using two different β and ξ values. The assumed perturbations “Δβ” and “Δξ” are, respectively, Δβ=0.001β and Δξ=0.001ξ.

The results of this analysis are shown in Table 2 for the F2 series, where H is the water depth and Qmeas the corresponding measured discharge.

They show very low sensitivity of both the INCM and LHRM results, such that a one-digit approximation of both model parameters (ξ and β) should guarantee a computed discharge variability of less than 2 %.

The results of the sensitivity analysis, carried out for series K4 and shown in Table 2, are similar to the previous ones computed for F2 series.

A first validation of the two methods was carried out by using the calibrated parameter values, the same Nash–Sutcliffe performance measure and all the available experimental series. The results were also compared with results of DCM and IDCM methods, the latter applied using the suggested α=0.02 value and five subsections, each one corresponding to a different bottom slope in the lateral y direction. The NS index for all data series is shown in Table 3.

The DCM results are always worse and are particularly bad for all the K series. The results of both the IDCM and INCM methods are very good for the two F series not used for calibration but are both poor for the K series. The LHRM method was always the best and also performed very well in the K series. The reason is probably that the K series tests have very low discharges and the constant α=0.02, the coefficient adopted in the IDCM method, does not fit the size of the subsections, and Eq. (13) is not a good approximation of the mixing length αH in Eq. (12) for low values of the water depth. In Fig. 3a and b the NS curves obtained by using DCM, IDCM, INCM and LHRM, for series F2 and K4, are shown.

Although rating curves are available in different river sites around the world, field validation of the uniform flow formulas is not easy for at least two reasons.

The average friction factor f and the related Manning coefficient are not known as in the lab case and the results of all the formulas need to be scaled according to the Manning coefficient to be compared with the actually measured discharges.

It is well known in the parameter estimation theory (Aster et al., 2012) that the uncertainty of the estimated parameters (in our case the roughness coefficient) grows quickly with the number of parameters, even if the matching between the measured and the estimated model variables (in our case the water stages in the downstream section) improves. The use of only one single parameter over all the computational domain is motivated by the need of getting a robust estimation of the Manning coefficient and of the corresponding discharge hydrograph.

Although the accuracy of the results is restricted by several modelling assumptions, a positive indication about the robustness of the simulation model (and the embedded relationship between the water depth and the uniform flow discharge) is given by (1) the match between the computed and the measured discharges in the upstream section, and (2) the compatibility of the estimated average Manning coefficient with the site environment.

The area of interest is located in the Alzette River basin (Grand Duchy of Luxembourg) between the gauged sections of Pfaffenthal and Lintgen (Fig. 4). The river reach length is about 19 km, with a mean channel width of ∼ 30 m and an average depth of ∼ 4 m. The river meanders in a relatively large and flat plain about 300 m, with a mean slope of ∼ 0.08 %.

The methodology was applied to a river reach 13 km long, between two instrumented sections, Pfaffenthal (upstream section) and Hunsdorf (downstream section), in order to have no significant lateral inflow between the two sections.

Events of January 2003, January 2007 and January 2011 were analysed. For these events, stage records and reliable rating curves are available at the two gauging stations of Pfaffenthal and Hunsdorf. The main hydraulic characteristics of these events, namely duration (Δt), peak water depth (Hpeak) and peak discharge (qpeak), are shown in Table 4.

In this area a topographical survey of 125 river cross sections was available. The hydrometric data were recorded every 15 min. The performances of the discharge estimation procedures were compared by means of the Nash–Sutcliffe criterion.

The results of the INCM and LHRM methods were also compared with those of the DCM and IDCM methods, the latter applied by using α=0.02 and an average subsection width equal to 7 m. The computed average Manning coefficients nopt, reported in Table 5, are all consistent with the site environment, although they attain very large values, according to DCM an IDCM, in the 2011 event.

The estimated and observed dimensionless water stages in the Hunsdorf gauged site for the 2003, 2007 and 2011 events are shown in Figs. 5–7.

Only the steepest part of the rising limb, located inside the coloured window of each figure, was used for calibration. The falling limb is not included, since it has a lower slope and is less sensitive to the Manning coefficient value.

A good match between recorded and simulated discharge hydrographs can be observed (Figs. 8–10) in the upstream gauged site for each event.

For all investigated events the Nash–Sutcliffe efficiency NSq is greater than 0.90, as shown in Table 6.

The error obtained between measured and computed discharges, with all methods, is of the same order of magnitude as the discharge measurement error. Moreover, this measurement error is well known to be much larger around the peak flow, where the estimation error has a larger impact on the NS coefficient. The NS coefficients computed with the LHRM and INCM methods are anyway a little better than the other two.

The vertically averaged velocities computed using DCM, IDCM, INCM and LHRM were compared with the results of the well-known ANSYS 3-D code, named CFX, which solves the RANS equations, applied to a prismatic reach with the irregular cross section measured at the Hunsdorf gauged section of the Alzette River. The length of the reach is about 4 times the top width of the section.

In the homogeneous multiphase model adopted by CFX, water and air are assumed to share the same dynamic fields of pressure, velocity and turbulence and water is assumed to be incompressible. CFX solves the conservation of mass and momentum equations, coupled with the air pressure–density relationship and the global continuity equation in each node. We denote α1, ρ1, μ1 and U1, respectively, as the volume fraction, the density, the viscosity and the time-averaged value of the velocity vector for phase l (l= w (water), a (air)), i.e. ρ=∑l=w,aα1ρ1,μ=∑l=w,aα1μ1, where ρ and μ are the density and the viscosity of the “averaged” phase. The air density is assumed to be a function of the pressure p, according to the state equation: ρa=ρa,0eγp-p0, where the subindex 0 marks the reference state values and γ is the air compressibility coefficient.

The governing equations are the following: (1) the mass conservation equation, (2) the Reynolds-averaged continuity equation of each phase and (3) the Reynolds-averaged momentum equations. Mass conservation implies ∑l = w, aα1=1. The Reynolds-averaged continuity equation of each phase l can be written as ∂ρ1∂t+∇⋅ρ1U=S1, where S1 is an external source term. The momentum equation instead refers to the “averaged” phase and is written as ∂ρU∂t+∇⋅ρU⊗U-∇⋅μeff∇U+∇UT+∇p′=SM, where ⊗ is the dyadic symbol, SM is the momentum of the external source term S, and μeff is the effective viscosity accounting for turbulence and defined as μeff=μ+μt, where μt is the turbulence viscosity and p′ is the modified pressure, equal to p′=p+23ρk+23μeff∇⋅U, where k is the turbulence kinetic energy, defined as the variance of the velocity fluctuations and p is the pressure. Both phases share the same pressure p and the same velocity U.

To close the set of six scalar equations (Eqs. 23–25), we finally apply the k–ε turbulence model implemented in the CFX solver. The implemented turbulence model is a two equation model, including two extra transport equations to represent the turbulent properties of the flow.

Two-equation models account for history effects like convection and diffusion of turbulent energy. The first transported variable is turbulent kinetic energy, k; the second transported variable is the turbulent dissipation, ε. The “k–ε” model has been shown (Jones and Launder, 1972; Launder and Sharma, 1974) to be useful for free-shear layer flows with relatively small pressure gradients. Similarly, for wall-bounded and internal flows, the model gives good results but only in cases where the mean pressure gradients are small.

The computational domain was divided using both tetrahedral and prismatic elements (Fig. 11). The prismatic elements were used to discretize the computational domain in the near-wall region over the river bottom and the boundary surfaces, where a boundary layer is present, while the tetrahedral elements were used to discretize the remaining domain. The number of elements and nodes in the mesh used for the specific case are of the order of, respectively, 4×106 and 20×106.

A section of the mesh is shown in Fig. 12. The quality of the mesh was verified by using a pre-processing procedure by ANSYS^® ICEM CFD^™ (Ansys Inc., 2006).

The six unknowns in each node are the pressure, the velocity components, and the volume fractions of the two phases. At each boundary node, three of the first four unknowns have to be specified. In the inlet section a constant velocity, normal to the section, is applied, and the pressure is left unknown. In the outlet section the hydrostatic distribution is given, the velocity is assumed to be still normal to the section and its norm is left unknown. All boundary conditions are reported in Table 7.

The opening condition means that that velocity direction is set normal to the surface, but its norm is left unknown and a negative (entering) flux of both air and water is allowed. Along open boundaries the water volume fraction is set equal to zero. The solution of the problem converges towards two extremes: nodes with zero water fraction, above the water level, and nodes with zero air fraction, below the water level.

On the bottom boundary, between the nodes with zero velocity and the turbulent flow, a boundary layer exists that would require the modelling of microscale irregularities. CFX allows using, inside the boundary layer, a velocity logarithmic law, according to an equivalent granular size. The relationship between the granular size and Manning's coefficient, according to Yen (1992), is given by d50=n0.04746, where d50 is the average granular size to be given as the input in the CFX code.

Observe that the assumption of known and constant velocity directions in the inlet and outlet sections is a simplification of reality. A more appropriate boundary condition at the outlet section, not available in the CFX code, would have been given by zero velocity and turbulence gradients (Rameshwaran et al., 2013). For this reason, a better reconstruction of the velocity field can be found in an intermediate section, where secondary currents with velocity components normal to the mean flow direction can be easily detected (Peters and Goldberg, 1989; Richardson and Thorne, 1998). See in Fig. 13 how the intermediate section was divided to compute the vertically averaged velocities in each segment section. These 3-D numerical simulations confirm that the momentum Γ, proportional to the derivative of the average tangent velocities and equivalent to the left-hand side of Eq. (2), cannot be set equal to zero if a rigorous reconstruction of the velocity field is sought after.

To compute the uniform flow discharge, for a given outlet section, the CFX code is run iteratively, each time with a different average longitudinal velocity in the inlet section, until the same water depth as in the outlet section is attained in the inlet section for steady-state conditions. Using the velocity distribution computed in the middle section along the steady-state computation as upstream boundary condition, transient analysis is carried on until pressure and velocity oscillations become periodic.

In order to test the achievement of the fully developed state within the first half of the modelled length, the authors plotted the vertical profiles of the streamwise velocity components for 10 verticals equally spaced along the longitudinal axis of the main channel. See in Fig. 14 the plot of four of them and their locations. The streamwise velocity evolves longitudinally and becomes almost completely self-similar starting from the vertical line in the middle section.

The stability of the results was finally checked against the variation of the length of the simulated channel. The dimensionless sensitivity of the discharge with respect to the channel length is equal to 0.2 %.

See in Table 8 the comparison between the vertically averaged state velocities, computed through the DCM, IDCM, INCM and LHRM formulas (uDCM, uIDCM, uINCM, uLHRM) and through the CFX code (uCFX). Table 8 also shows the relative difference, Δu, evaluated as Δu=u-uCFXuCFX⋅100. As shown in Table 8, both INCM and LHRM perform very well in this validation test instead of DCM, which clearly overestimates averaged velocities. In the central area of the section, the averaged velocities calculated by the INCM, LHRM and CFX code are quite close with a maximum difference of ∼ 7 %. By contrast, larger differences are evident close to the river bank, in segments 1 and 9, where INCM and LHRM underestimate the CFX values. These larger differences show the limit of using a 1-D code. Close to the bank the wall resistance is stronger and the velocity field is more sensitive to the turbulent exchange of energy with the central area of the section, where higher kinetic energy occurs.