Two new methods for vertically averaged velocity computation are presented,
validated and compared with other available formulas. The first method
derives from the well-known Huthoff algorithm, which is first shown to be
dependent on the way the river cross section is discretized into several
subsections. The second method assumes the vertically averaged longitudinal
velocity to be a function only of the friction factor and of the so-called
“local hydraulic radius”, computed as the ratio between the integral of the
elementary areas around a given vertical and the integral of the elementary
solid boundaries around the same vertical. Both integrals are weighted with a
linear shape function equal to zero at a distance from the integration
variable which is proportional to the water depth according to an empirical
coefficient

Computation of vertically averaged velocities is the first step of two major calculations in 1-D shallow water modelling: (1) estimation of the discharge given the energy slope and the water stage and (2) estimation of the bottom shear stress for computing the bedload in a given river section.

Many popular software tools, like MIKE11 (MIKE11, 2009), compute the
discharge

The uniform flow formula almost universally applied in each subsection is still the Chezy equation (Herschel, 1897). The advantage of using the Chezy equation is that the associated Manning coefficient has been calibrated worldwide for several types of bed surface and a single value can be used for each application. However, it is well known that the Chezy equation was derived from laboratory measurements taken in channels with a regular, convex cross-sectional shape. When the section results from the union of different parts, each with a strongly different average water depth, one of two options is usually selected. The first option, called single channel method (SCM) is simply to ignore the problem. This leads to strong underestimation of the discharge, because the Chezy formula assumes a homogeneous vertically averaged velocity and this homogeneous value provides strong energy dissipation in the parts of the section with lower water depths. The second option, called divided channel method (DCM) is to compute the total discharge as the sum of the discharges flowing in each convex part of the section (called subsection), assuming a single water level for all parts (Chow, 1959; Shiono et al., 1999; Myers and Brennan, 1990). In this approach, the wet perimeter of each subsection is restricted to the component of the original one pertaining to the subsection, but the new components shared by each couple of subsections are neglected. This is equivalent to neglecting the shear stresses coming from the vortices with vertical axes (if subsections are divided by vertical lines) and considering additional resistance for higher velocities, which results in overestimation of discharge capacity (Lyness et al., 2001).

Knight and Hamed (1984) compared the accuracy of several subdivision methods for compound straight channels by including or excluding the vertical division line in the computation of the wetted perimeters of the main channel and the floodplains. However, their results show that conventional calculation methods result in larger errors. Wormleaton et al. (1982) and Wormleaton and Hadjipanos (1985) also discussed, in the case of compound sections, the horizontal division through the junction point between the main channel and the floodplains. Their studies show that these subdivision methods cannot assess well the discharge in compound channels.

The interaction phenomenon in compound channels has also been extensively studied by many other researchers (e.g. Sellin, 1964; Knight and Demetriou, 1983; Stephenson and Kolovopoulos, 1990; Rhodes and Knight, 1994; Bousmar and Zech, 1999; van Prooijen et al., 2005; Moreta and Martin-Vide, 2010). Their studies demonstrate that there is a large velocity difference between the main channel and the floodplain, especially at low relative depth, leading to a significant lateral momentum transfer. The studies by Knight and Hamed (1984) and Wormleaton et al. (1982) indicate that the vertical transfer of momentum between the upper and the lower main channels exists, causing significant horizontal shear able to dissipate a large part of the flow energy.

Furthermore, many authors have tried to quantify flow interaction among the subsections, at least in the case of compound but regular channels. To this end, turbulent stress was modelled through the Reynolds equations and coupled with the continuity equation (Shiono and Knight, 1991). This coupling leads to equations that can be analytically solved only under the assumption of negligible secondary flows. Approximated solutions can also be obtained, although they are based on some empirical parameters. Shiono and Knight developed the Shiono–Knight method (SKM) for prediction of lateral distribution of depth-averaged velocities and boundary shear stress in prismatic compound channels (Shiono and Knight, 1991; Knight and Shiono, 1996). The method can deal with all channel shapes that can be discretized into linear elements (Knight and Abril, 1996; Abril and Knight, 2004).

Other studies based on the Shiono and Knight method can be found in Liao and Knight (2007), Rameshwaran and Shiono (2007), Tang and Knight (2008) and Omran and Knight (2010). Apart from SKM, some other methods for analysing the conveyance capacity of compound channels have been proposed. For example, Ackers (1993) formulated the so-called empirical coherence method. Lambert and Sellin (1996) suggested a mixing length approach at the interface whereas, more recently, Cao et al. (2006) reformulated flow resistance through lateral integration using a simple and rational function of depth-averaged velocity. Bousmar and Zech (1999) considered the main channel/floodplain momentum transfer proportional to the product of the velocity gradient at the interface times the mass discharge exchanged through this interface due to turbulence. This method, called EDM (exchange divided method), also requires a geometrical exchange correction factor and turbulent exchange model coefficient for evaluating discharge.

A simplified version of the EDM, called interactive divided channel method
(IDCM), was proposed by Huthoff et al. (2008). In IDCM, lateral momentum is
considered negligible and turbulent stress at the interface is assumed to be
proportional to the spanwise kinetic energy gradient through a dimensionless
empirical parameter

An alternative approach could be to simulate the flow structure in its
complexity by using a 3-D code for computational fluid dynamics (CFD). In
these codes flow is represented both in terms of transport motion (mean flow)
and turbulence by solving the Reynolds-averaged Navier Stokes (RANS)
equations (Wilcox, 2006) coupled with turbulence models. These
models allow for closure of the mathematical problem by adding a certain number
of additional partial differential transport equations equal to the order of
the model. In the field of the simulation of industrial and environmental
laws, second-order models (e.g.

In this study, two new methods aimed at representing subsection interactions in
a compound channel are presented. The first method, named “integrated
channel method” (INCM), derives from the Huthoff formula, which is
shown to give results depending on the way the river cross section is
discretized in subsections. The same dynamic balance adopted by Huthoff is
written in differential form, but its diffusive term is weighted according
to a

The second one, named “local hydraulic radius method” (LHRM), derives from the
observation that, in the Manning formula, the mean velocity per unit energy
gradient is proportional to a power of the hydraulic radius. It should then
be possible to get the vertically averaged velocity along each vertical by
using the same Manning formula, where the original hydraulic radius is
changed with a “local” one. This “local” hydraulic radius should take into
account the effect of the surrounding section geometry, up to a maximum
distance which is likely to be proportional to the local water depth,
according to an empirical

The present paper is organized as follows: two of the most popular
approaches adopted for computation of the vertically averaged velocities are
explained in details along with the proposed INCM and LHRM methods. The

In the DCM method the river section is divided into subsections with uniform
velocities and roughness (Chow, 1959). Division is made by vertical lines and
no interaction between adjacent subsections is considered. Discharge is
obtained by summing the contributions of each subsection, obtained by
applying the Manning formula:

In order to model the interaction between adjacent subsections of a compound
section, the Reynolds and the continuity equations can be coupled (Shiono
and Knight, 1991) to get

In order to reduce to one the number of empirical parameters (in addition to

Integration of Eq. (3) over each

Turbulent stresses are modelled quite simply as

Following a wall-resistance approach (Chow, 1959), the friction factor

Equations (6) forms a system with an order equal to the number

IDCM has the main advantage of using only two parameters, the

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

LHRM derives from the observation that, in the Manning equation, the average
velocity is set equal to

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

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).

Geometric parameters of a compound channel.

Geometric and hydraulic laboratory parameters of the experiment series.

Four series, named F1, F2, F3 and F6, were selected for calibration of the

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

Both

Calibration provides optimal

NS versus

We carried out a discharge sensitivity analysis of both new methods using the
computed

The results of this analysis are shown in Table 2 for the F2 series, where

They show very low sensitivity of both the INCM and LHRM results, such that a
one-digit approximation of both model parameters (

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.

Sensitivities

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

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

Nash–Sutcliffe efficiency for all (calibration and validation) experimental series.

Estimated discharge values against HR Wallingford FCF measures for
F2

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

River bed roughness does change, along with the Manning coefficient from one water stage to another (it usually increases along with the water level).

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

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 (

In this area a topographical survey of 125 river cross sections was
available. The hydrometric data were recorded every 15

The results of the INCM and LHRM methods were also compared with those of the
DCM and IDCM methods, the latter applied by using

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.

The Alzette study area.

Main characteristics of the flood events at the Pfaffenthal and Hunsdorf gauged sites.

Optimum roughness coefficient,

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 NS

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.

Observed and simulated stage hydrographs at the Hunsdorf gauged site in the event of January 2003.

Observed and simulated stage hydrographs at the Hunsdorf gauged site in the event of January 2007.

Observed and simulated stage hydrographs at the Hunsdorf gauged site in the event of January 2011.

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.

Nash–Sutcliffe efficiency of estimated discharge hydrographs for the analysed flood events.

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

Observed and simulated discharge hydrographs at the Pfaffenthal gauged site in the event of January 2003.

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

To close the set of six scalar equations (Eqs. 23–25), we finally apply the

Observed and simulated discharge hydrographs at the Pfaffenthal gauged site in the event of January 2007.

Observed and simulated discharge hydrographs at the Pfaffenthal gauged site in the event of January 2011.

Two-equation models account for history effects like convection and diffusion
of turbulent energy. The first transported variable is turbulent kinetic
energy,

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,

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).

Boundary conditions assigned in the CFX simulation.

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.

Computational domain of the reach of the Alzette River.

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

A mesh section along the inlet surface.

Hunsdorf river cross section: subsections used to compute the vertically averaged velocities.

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

Simulated mean velocities in each segment section using 1-D hydraulic models with DCM, IDCM, INCM, LHRM and CFX, and the corresponding differences.

Streamwise vertical profile along the longitudinal axis of the mean channel.

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
(

Two new methods computing the vertically averaged velocities
along irregular sections have been presented. The first method, named INCM,
develops from the original IDCM method and it is shown to perform better than
the previous one, with the exception of lab tests with very small discharge
values. The second one, named LHRM, has empirical bases and gives up the
ambition of estimating turbulent stresses but has the following important
advantages.

It relies on the use of only two parameters: the friction factor

The

The sensitivity of the results with respect to the model

Comparison between the results of the CFX 3-D turbulence model and the LHRM model shows a very good match between the two computed total discharges, although the vertically averaged velocities computed by the two models are quite different near to the banks of the river.

Moreover, the estimation of the velocity profiles in each of the considered
subsections could be used in order to evaluate the vertical average
velocity and thus the shear stresses at the boundary of the whole cross
section. In fact, it is well known that bedload transport is directly
related to the bed shear stress and that this is proportional in each point
of the section to the second power of the vertically averaged velocity,
according to Darcy and Weisbach (Ferguson, 2007):

The authors wish to express their gratitude to the Administration de la gestion de l'eau of the Grand-Duché de Luxembourg and the Centre de Recherche Public Gabriel Lippmann for providing hydrometric and topographical data of the Alzette River. Edited by: R. Moussa