2.1 Field site

The Paglia River, in the central part of Italy (Fig. 1a), is a tributary of Tiber River, subject to severe flooding and high sediment transport. The reach of interest, near the town of Orvieto, is across the Adunata bridge (Fig. 1b) along the Paglia River (basin area of about 1200 km², average discharge of 10 m³ s⁻¹, flood discharge up to 2500 m³ s⁻¹). The Adunata bridge connects the settlements of Orvieto Scalo and Ciconia, as part of the Italian State Road no. 71 (Fig. 1c). It is a masonry multi-arch bridge, with five arches ending at four piers on the river bed. On the right-hand side, an abutment sustains the bridge and separates it from the floodplain; on the left-hand side, the bridge deck is supported by the main levee. Close to the bottom, the piers have a roughly elliptical shape with the major axis, aligned with the flow, 15 m long, and the minor axis, orthogonal to the flow, 5.7 m wide. At the bottom, each pier is sustained by an elliptical plinth, whose profile is 2.0 m larger than the pier. The center distance between the piers is 23.2 m. The pier width increases approaching the deck because of the arches; the deck width is approximatively 10 m.

Figure 1(a) Location of the field site; (b) downstream view of the Adunata bridge on the Paglia River during normal flow condition (11 November 2021); (c) digital terrain model (DTM) nearby the Adunata bridge (dotted line), with the domain of the 3D CFD model (black line); and (d) location of the level gauge and of the radar sensor with the field of view (FOV) in an aerial image (© Google Earth, 2023).

The main thread of the flow is on the right-hand side of the river, and a large depositional area forms on the left-hand side just downstream of the bridge (Fig. 1b). The main channel axis is characterized by a significant curvature, bending to the left at the bridge section (Fig. 1c).

2.2 Available data

At the downstream side of the Adunata bridge, a water level gauge and a radar sensor for measuring the water surface velocity are located at the center of the first and second arch, respectively (Fig. 1d). The time resolution of both the sensors is 10 min. In addition, a number of flow rate measures and cross-sectional velocity distributions were provided by the Umbria Region Hydrological Service. The flow rate data were collected using a current meter by wading a few tens of meters downstream of the Adunata bridge in the period 2009–2011 (flow rate ranging between 3.3 and 14.3 m³ s⁻¹), and from the bridge in the period 1995–2010 (flow rate ranging between 16.8 and 147 m³ s⁻¹); additional flow rate data were collected using an acoustic Doppler current profiler (ADCP) some hundreds of meters upstream of the bridge in the period 2014–2019 (flow rate ranging between 0.37 and 45 m³ s⁻¹). The official rating curve for the Adunata bridge, provided by ARPA Lazio, is based on these measurements.

As detailed in the following sections, the rating curve derived from current meter and ADCP data, the water levels, and the free-surface velocity data collected by the sensors mounted on the Adunata bridge were used to validate the hydrodynamic numerical models (Sect. 2.3 and Appendix A). The cross-sectional velocity distributions measured with the current meter just downstream of the bridge were used to further assess the spatial variability of the entropy-based velocity distributions, as detailed in Sect. 3.1.

2.3 Numerical model

The commercial CFD software STAR-CCM+ (Siemens) was used for the numerical simulations. It implements the finite-volume method to compute the flow field on unstructured, Cartesian computational grids. The software has been used and validated in several applications, including complex flows over deformed bathymetry, in the presence of obstacles (Chang et al., 2013; Kirkil et al., 2009; Lazzarin et al., 2023c, 2024b) and channel bends (Constantinescu et al., 2011, 2013; Koken et al., 2013). In the present application, the two-phase volume-of-fluid (VoF; Hirt and Nichols, 1981) method was used to track the water–air interface within the computational domain (Horna-Munoz and Constantinescu, 2018; Lazzarin et al., 2023b; Li and Zhang, 2022; Luo et al., 2018; Yoshimura and Fujita, 2020). STAR-CCM+ was used to solve the Reynolds-averaged Navier–Stokes (RANS) equations, in which the stress tensor in the momentum equations is related to the mean flow quantities by adopting the Boussinesq approximation. The eddy viscosity, μ_T, was determined by solving transport equations for the turbulent kinetic energy, k, and dissipation rate, ε, according to the realizable k–ε turbulence model (Shih et al., 1995), suitable for large-scale complex flows in natural rivers (e.g., Horna-Munoz and Constantinescu, 2018).

The computational domain reproduced a ∼1100 m long reach of the Paglia River (Fig. 1c), centered at the Adunata bridge. The average size of the grid elements was 1.0 m. Starting 100 m upstream of the bridge and up to 300 m downstream of the bridge, the grid was refined using elements with average length of 0.5 m. To capture the near-wall boundary layer well, a prism layer refinement with three layers was used to reduce the wall-normal thickness of the grid cells close to solid boundaries (i.e., the riverbed and bridge structure). The final computational grid was made of ∼4 million elements. A rough-wall, no-slip condition was imposed at the solid boundaries by means of a wall function (roughness height of 0.1 m at the bottom, and of 0.01 m at the bridge surfaces). The upper boundary of the computational grid was treated as a symmetry plane (i.e., slip condition) for the airflow. The water elevation at the outlet (i.e., downstream section) was kept fixed in time by imposing a suitable hydrostatic-pressure distribution. The value of the downstream level, for each of the simulated scenarios, was derived from an auxiliary two-dimensional (2D), depth-averaged hydrodynamic model calibrated on available data; the 2DEF model has been used for this purpose (see Appendix A for details on the model and its calibration/verification). A constant-in-time, logarithmic velocity distribution was imposed as the upstream boundary condition for the water fraction. For the air fraction (upper part of the numerical domain), zero velocity and zero pressure were imposed at the inlet and at the outlet, respectively. The simulations were advanced in time with an implicit, first-order discretization, until steady-state conditions were reached.

The 3D-CFD model was validated by comparing the surface velocity computed by the model with that measured by the radar sensor located downstream of the bridge (see the yellow bullets in Fig. A2c and d).

2.4 Flood events considered in the study

Three different steady flow conditions have been simulated with the 3D-CFD model STAR-CCM+, which correspond to the peak flow conditions of flood events that occurred in 2012, 2019, and 2022, as provided by the rating curve for the Adunata bridge (Table 1). In all three of the flow conditions, water flowed in the main channel and over the sediment bars that are dry in the low-flow condition of Fig. 1b and d. During the most severe flood of 2012, water also flowed on the floodplains adjacent to the main river and caused the incipient pressurization of flow below the bridge arches. The preliminary simulation carried out with the 2DEF depth-averaged model showed that, at the peak of the 2012 flood event, 700 m³ s⁻¹ flowed through the floodplain, overflowing the bridge access roads, and 1800 m³ s⁻¹ flowed within the main channel; this last value was used in the 3D-CFD simulation, which considered only the main channel of the river. The flood events of 2019 and 2022, although being quite ordinary, were the largest floods that occurred after the installation of the radar sensor for the surface velocity (thus, surface velocity data were not available for the 2012 flood).

Table 1Simulations performed in the present work. The value in brackets indicates the total discharge with consideration of the flow over floodplains, which is not considered in the 3D simulations.

Download Print Version | Download XLSX

2.5 Entropy theory

Entropy theory deals with physical systems that may have a large number of states from a probabilistic point of view. The concept of entropy is used for statistical inference, to determine a probability distribution function when the available information is limited to some average quantities, defined as constraints such as mean and variance. For the application of entropy to streamflow measurements, the pioneer was Chiu (1987), who developed a probabilistic formulation of the cross-sectional velocity distribution in open channels, in which the expected value of the point velocity is determined by applying the maximum entropy principle (Chiu, 1987, 1988, 1989). Using this probabilistic formulation, the velocity distribution is given analytically as a function of the cross-sectional geometry; of the dimensionless entropy parameter, M; and possibly of the depth at which the maximum velocity occurs (the so-called dip, h). There is a one-to-one correspondence between M and the ratio of mean to maximum velocity in the cross-section, which is defined as the entropic function, ϕ(M) (Chiu, 1991). In general, for a given river site, the magnitude of M and, in turn, of ϕ(M), mainly depends on hydraulic parameters such as roughness and hydraulic radius, whereas they are poorly affected by the flow discharge (Chiu and Murray, 1992; Moramarco and Singh, 2010). Moreover, ϕ(M) is consistently found to be nearly constant at different cross-sections through gauged river sites for different flow conditions (Moramarco and Singh, 2010; Ammari et al., 2022). This is because the value of ϕ(M) is associated with geometric and hydraulic characteristics that tend to vary smoothly within a river system (Ammari et al., 2022).

The estimation of cross-sectional velocity distribution, U(x,y), developed by Chiu (1989), was later simplified by Moramarco et al. (2004). Using this approach, one can divide the wet cross-sectional area into N_v verticals and determine the entropy-based velocity profile along each vertical as

where U is the time-averaged velocity, U_max(x_i) is the maximum value of U along the ith vertical, x_i is the distance of the ith sampled vertical from the left bank, h(x_i) is the dip (i.e., the depth of U_max(x_i) below the water surface), D(x_i) the flow depth, and y is the vertical distance from the bed. The relationship between the entropic parameter, M, and the entropic function, ϕ(M), is (Chiu, 1989)

in which U_m and U_MAX are the average and maximum flow velocities within the entire cross-section. It is worth mentioning that U_m represents the expected value of velocity that can be different from the observed mean velocity (Marini and Fontana, 2020). These two values are quite similar in the case of wide rivers (aspect ratio larger than 5). In the present research, considering the large aspect ratio for all cross-sections (Table 2), this hypothesis is valid.

Table 2Flow data for the cross-sections of Fig. 2 and the three considered flood events of Table 1. The values of the entropic function, ϕ(M), and parameter, M, are obtained from the 3D-CFD velocity distributions and estimated according to Eq. (4).

Download Print Version | Download XLSX

Introducing the variable $\mathit{\delta }\left(x_i\right)=Dt\left(x_i\right)/\left[D\right(x_i)-h(x_i\left)\right]$, the velocity dip, h(x_i), is estimated according to Yang et al. (2004) from the spanwise distribution of δ(x_i), which is given as

in which x_min is the spanwise distance of the x_i vertical from the nearest bank. Note that h(x_i)=0 and δ(x_i)=1 when the maximum velocity occurs at the free surface.

In the case of gauged cross-sections, ϕ(M) can be inferred from measured mean and maximum flow velocities (e.g., with ADCP). For ungauged sites, ϕ(M) can be estimated as (Moramarco and Singh, 2010)

where y₀ is the vertical coordinate, taken from the bottom, where the velocity is zero; k is the von Karman constant; R is the hydraulic radius; n is the Manning roughness; D is the maximum water depth; and h is computed with Eq. (3) at the thalweg, i.e., where the water depth is maximum. According to van Rijn (1982), y₀=0.065ξd₉₀, where d₉₀ is the 90th percentile for grain size and ξ a parameter ranging from 1 to 10 (Ferro, 2003; Moramarco and Singh, 2010).

When only the surface velocities, U_surf(x_i), are available at a river site, then U_max(x_i) can be estimated as (Fulton and Ostrowski, 2008)

For the current research, the methodological steps to estimate the cross-sectional velocity distribution (and hence the flow discharge) using entropy theory are as follows. The input data are the river-wide velocity distribution at the free-surface, U_surf, provided by the 3D-CFD model. When only the maximum value of U_surf is used as input, corresponding to the hypothetical case in which only point-sensor data are available, the spanwise distribution of U_surf is obtained by applying either a parabolic or an elliptical spanwise distribution (Bahmanpouri et al., 2022a). The velocity dip is computed using Eq. (3). The cross-sectional velocity distribution is then obtained using an iterative procedure, in which p denotes the iteration. At the first iteration, the entropic function, ϕ(M)_p=1, is computed with Eq. (4), and M_p=1 is computed with Eq. (2). After computing the maximum velocity for each vertical, U_max(x_i)_p=1, with Eq. (5), Eq. (1) allows the entropic velocity distribution in the whole cross-section, $U(x_i,y{)}_{p=\mathrm{1}}$, to be estimated. The following iteration starts by computing the average and the maximum flow velocities, U_m and U_MAX, from the velocity distribution obtained in the previous iteration and then $\mathit{\varphi }(M{)}_p=U_m/U_{\mathrm{MAX}}$, M_p using Eq. (2), U_max(x_i)_p with Eq. (5), and the velocity distribution U(x_i,y)_p with Eq. (1). The iterative procedure continues until the difference ϕ(M)_p-ϕ(M)_p−1 becomes lower than 0.01. For more details, the reader is referred to Moramarco et al. (2017).

3.1 Variability of the entropy function

Some relevant parameters that characterize the flow field (e.g., aspect ratio, average and maximum velocity) at the selected cross-sections are presented in Table 2 for the peak flow condition of the three flood events. The values of the entropic function, ϕ(M)_CFD, were first computed as the ratio of average to maximum velocity within the cross-section provided by the 3D-CFD model. Then, assuming the site as being ungauged, ϕ(M)_Eq. (4) was estimated using Eq. (4) with d₉₀=0.01 m (Pilbala et al., 2024) and a Manning parameter, n, equal to 0.035 m${}^{-\mathrm{1}/\mathrm{3}}$s at the upstream (−50 m) and far downstream sections (+100 and +200 m) and equal to 0.055 m${}^{-\mathrm{1}/\mathrm{3}}$s just downstream of the bridge (+50 m cross-section), where larger energy losses are expected because of the wakes generated by the bridge piers. The values of ϕ(M)_Eq. (4), reported in Table 2 and corresponding with the points marked with dashed lines in Fig. 3b, were obtained using ξ=5 to compute y₀ (Sect. 2.5 just after Eq. 4), and the gray band was obtained by varying ξ in the range [1, 10]. Finally, the values of the entropic parameter associated with the different values of ϕ(M) are computed using Eq. (2).

Figure 3Entropic function ϕ(M), for the different simulated scenarios, as a function of the distance from the bridge (positive downstream), (a) computed from the 3D-CFD flow fields and (b) estimated with Eq. (4), where the lines refer to the average values, and the gray band is obtained by varying the reference height y₀ in Eq. (4) within the expected range. Green circles refer to data derived from velocity distributions measured with the current meter just downstream of the bridge.

Download

Since the entropic function is typically assumed to be constant for all flow conditions at a given cross-section, it is of interest to analyze its actual variation by exploiting the flow fields provided by the 3D-CFD model and to see the effectiveness of their first-guess estimates obtained using Eq. (4). The values of ϕ(M) reported in Table 2 are plotted in Fig. 3 as a function of the downstream distance from the bridge. At the first cross-section downstream of the bridge (i.e., cross-section 2), although referring to different flow conditions, the values of the entropic function computed with the 3D-CFD and the current meter velocity distributions show the same magnitude, further confirming the reliability of the 3D-CFD model. The first-guess estimates of ϕ(M) in Fig. 3b, although they have a marginal role on the entropy-based computations, show a similar trend to the 3D-CFD estimates (Fig. 3b), provided that the increased Manning parameter is used at the section just downstream of the bridge. The need to calibrate such an increased Manning parameter complicates efforts in the case of disturbed flows.

Figure 4Flood event of 2019, cross-section 2 (50 m downstream of the bridge). Velocity distributions provided by (a) the 3D-CFD model and (b) the entropy model forced with the river-wide distribution of the free-surface velocity. Comparison of vertical distributions of velocity at 0.2 B (c), 0.5 B (d), and 0.8 B (e), where B is the width of the cross-section.

Download

For each flood event, at cross-sections 1 and 4, i.e., where the flow field is not characterized by the wakes generated by the bridge piers, the entropic function assumes similar values, which can be identified as “undisturbed” values. The variability of such undisturbed values of ϕ(M) with the flow rate is relatively small, as all the values fall in the range $\mathrm{0.65}<\mathit{\varphi }\left(M\right)<\mathrm{0.75}$, in agreement with the range found by Bahmanpouri et al. (2022b) for similar European rivers. By contrast, at cross-sections 2 and 3, just downstream of the bridge, the values of ϕ(M) are consistently reduced due to the effect of the bridge. In the largest flood event of 2012, which produced near-pressure flow conditions at the bridge with marked localized increasing of the flow velocity, ϕ(M)_CFD equals 0.415 at cross-section 2, corresponding to $M_{\mathrm{CFD}}=-\mathrm{1.03}$. The low value of ϕ(M) and the negative value of M attest the markedly non-uniform distribution of the velocity (i.e., the maximum velocity in this cross-section is much higher than the average velocity). A sensible reduction is still present 100 m downstream of the bridge (cross-section 3). For the moderate peak flows of 2019 and 2022 events, the entropic function recovers undisturbed values already at cross-section 3, i.e., 100 m downstream of the bridge.

This first analysis suggests that assuming constant values of ϕ(M) can be reasonable in undisturbed river reaches; however, in the case of irregular flow fields induced by the interaction with in-stream structures, the entropic function ϕ(M) can vary with respect to undisturbed values, and, in addition, it can show significant variations with the flow rate.

Table 3Comparison between 3D-CFD outputs and entropy-based estimations forced with the river-wide distribution of the free-surface velocity.

Download Print Version | Download XLSX

Figure 5Flood event of 2019, cross-section 3 (100 m downstream of the bridge). Velocity distributions provided by (a) the 3D-CFD model and (b) the entropy model forced with the river-wide distribution of the free-surface velocity. Comparison of vertical distributions of velocity at 0.2 B (c), 0.5 B (d), and 0.8 B (e), where B is the width of the cross-section.

Download

Figure 6Flood event of 2019, cross-section 4 (200 m downstream of the bridge). Velocity distributions provided by (a) the 3D-CFD model and (b) the entropy model forced with the river-wide distribution of the free-surface velocity. Comparison of vertical distributions of velocity at 0.2 B (c), 0.5 B (d), and 0.8 B (e), where B is the width of the cross-section.

Download

3.2 Entropy model forced with the river-wide profile of free-surface velocity

The efficacy of the entropy model is tested here for the case in which the surface velocity is known for all the width of the cross-section. This could be the case in which the river-wide surface velocity is estimated from imaging techniques (e.g., Eltner et al., 2020; Schweitzer and Cowen, 2021). The results, in terms of cross-sectional velocity distributions, are presented for brevity only for the intermediate peak flow of the 2019 flood event and for the most challenging cross-sections just downstream of the bridge, where the flow field is disturbed by the pier wakes. The same results, for the peak flows of 2012 and 2022 events, are provided in the Supplement.

Figure 7Flood event of 2012, cross-section 1 (50 m upstream of the bridge). Cross-sectional velocity distribution computed with the 3D-CFD model (a). Entropy theory with parabolic spanwise velocity distribution (b) and entropy theory with elliptic spanwise velocity distribution (c).

Download

Figure 4 presents the cross-sectional velocity distribution 50 m downstream of the bridge (cross-section 2). As shown by the 3D-CFD flow field (Fig. 4a) and reflected in the low value of ϕ(M) for this cross-section (Table 2 and Fig. 3), the effect of the piers is very strong, such that there is a clearly uneven distribution of the cross-sectional velocity because of the wakes developing downstream of the piers. Just downstream of the bridge, due to the presence of the bridge arches, the flow field provided by the 3D-CFD model is configured as a sort of partial orifice flow that increases the vertical uniformity of the velocity distribution compared to a uniform shear flow. Of course, the entropy model cannot capture such localized flow features, which entails some difference in the patchiness of the physics-based and the entropy velocity distributions (Fig. 4a–e). Despite that, using the river-wide distribution of the surface velocity provided by the CFD simulation as input, the entropy model can reliably capture the salient features of the cross-sectional velocity distribution. Figure 4c–e highlight the comparison of 3D-CFD and entropy flow velocities along three verticals located at 0.2, 0.5, and 0.8 B (where B is the channel width). Compared to the results of the 3D-CFD model, the entropy approach underestimates the velocity close to the bed. Since the velocities and the volumetric fluxes are still relatively small near the bed, these discrepancies marginally affect the estimation of the section-averaged velocity and, consequently, of the total discharge (Table 3). The percentage error is larger (7.6 %) for the very high-flow condition of the 2012 event (see Supplement), due to the accentuation of orifice-flow conditions associated with the higher water levels.

Figure 8Flood event of 2012, cross-section 1 (50 m upstream of the bridge). Spanwise distribution of the surface velocity (a) and comparison of vertical distributions of velocity at 0.2 B (b), 0.5 B (c), and 0.8 B (d).

Download

Figure 5 depicts the cross-sectional velocity distributions at a larger distance from the bridge, i.e., at cross-section 3, placed 100 m downstream of the bridge. The visual comparison with Fig. 4 suggests that the effects of the piers on the flow field are reduced because of the increased distance, and the cross-sectional distribution provided by the 3D-CFD model (Fig. 5a) appears to be more regular. The statistical analysis confirms that in this case the entropy model (Fig. 5b) is able to simulate the velocity profiles with a higher accuracy.

Figure 6 shows the cross-sectional velocity distributions of 3D-CFD and entropy models for cross-section 4, located 200 m downstream of the bridge. Compared to cross-section 3, the effect of the bridge piers is further reduced because of both the distance and the more compact shape of the cross-section. Since the effect of the bridge piers is minimum, the statistical analysis shows a better agreement of the entropy model results with the CFD-based data. Though areas with relatively high velocities are still visible in simulations with higher values of the discharge (i.e., events of 2012 and 2019), for the high-flow conditions of 2022, the effect of the bridge pier has completely vanished. Therefore, the lower the flow discharge, the lower the distance from the bridge to recover undisturbed flow conditions.

The results presented here show that, when the river-wide distribution of the free-surface velocity is provided, the entropy method provides good estimations of the cross-sectional velocity distribution even when the influence of bridge piers, and thus the unevenness of the flow field, is relevant. The main discrepancies are observed in low-velocity regions, which slightly affect the estimation of the flow discharge. Table 3 lists some statistics and error percentages for the depth-averaged velocity and discharge estimations for all cross-sections and the three events considered. The estimations provided by the entropy method are in good agreement with results of CFD model, both upstream and downstream of the Adunata bridge. Though the accuracy is slightly reduced downstream of the bridge, the results are also reliable in the vicinity of the structure (i.e., at cross-section 2), suggesting the applicability of the entropy model to estimate the flow discharges, even in the case of irregular distributions of the cross-sectional velocity, provided that the river-wide distribution of the surface velocity is used as input data.

Figure 9Flood event of 2022, cross-section 4 (200 m downstream of the bridge). Cross-sectional velocity distribution computed with the 3D-CFD model (a). Entropy theory with parabolic spanwise velocity distribution and (b) entropy theory with elliptic spanwise velocity distribution (c).

Download

Figure 10Flood event of 2022, cross-section 4 (200 m downstream of the bridge). Spanwise distribution of the surface velocity (a) and comparison of vertical distributions of velocity at 0.2 B (b), 0.5 B (c), and 0.8 B (d).

Download

3.3 Entropy model forced with a single value of free-surface velocity

In this section, the results are presented considering only a single value of the surface velocity as input for the entropy model, which corresponds to the maximum surface velocity predicted by the 3D-CFD model. Two different spanwise velocity distributions are enforced in the entropic model, namely a parabolic spanwise distribution (PSD) and an elliptic spanwise distribution (ESD). Of course, applying the entropy model using a unique value of the velocity is particularly sensitive to this value and supposes a unimodal velocity distribution in the spanwise direction. For this reason, this kind of approach cannot be used in the cross-sections immediately downstream of the bridge, where the spanwise velocity distribution is markedly irregular (see e.g., Fig. 4). Herein, the results are presented for cross-section 1, located 50 m upstream of the bridge for the high-flow condition of the 2012 event, and for cross-section 4, located 200 m downstream of the bridge, for the modest peak flow condition of the 2022 event, where the effect of bridge piers on the velocity distribution wears off in a shorter distance.

Figure 7 shows the distribution of the surface velocity based on the 3D-CFD outputs and both the PSD and ESD entropy models. The agreement of both the PSD and the ESD is generally good in the central and the right parts of the channel and less good in the left part of the channel. Here, due to the irregular bathymetry (i.e., gravel deposit), the 3D-CFD model predicts localized stagnation zones that cannot be captured by the entropy model based on a single value of the surface velocity. This is confirmed by Fig. 8, which shows the cross-sectional distribution of the surface velocity and three vertical profiles. In the perspective of estimating the flow discharge, the lateral discrepancies represent a minor limit, as the central part of the cross-sections conveys the largest part of the total discharge.

Table 4Comparison between 3D-CFD and entropy-based outputs considering a single surface velocity.

Download Print Version | Download XLSX

Overall, the cross-sectional velocity distributions based on ESD seem more accurate than those based on the PSD: they provide similar results at the center of the channel, but the parabolic distribution generally underestimates the flow velocity close to the banks. Both cross-sectional and vertical distributions of the velocity profiles (Figs. 7a and 8c) highlight the existence of a velocity dip; i.e., the maximum velocity is below the water surface, particularly at the center of the channel. This is generally the consequence of secondary currents superposed on the main flow (Termini and Moramarco, 2020). Yang et al. (2004) and Moramarco et al. (2017) reported that for large aspect ratios of channel flow, $B/D$, the dip phenomenon appears primarily near the sidewall region, whereas for relatively low aspect ratios ($B/D=\mathrm{9.26}$ for cross-section 1) the velocity dip is generally located at the center of the channel (Bahmanpouri et al., 2022a, b; Kundu and Ghoshal, 2018; Moramarco et al., 2017; Termini and Moramarco, 2020). In this case, the 3D flow field from the CFD simulation shows that the dip depends on the counter-clockwise rotating secondary current generated by the upstream right-handed bend. Indeed, rotational inertia makes these curvature-induced helical flow structures propagate downstream for relatively long distances (Dominguez Ruben et al., 2021; Lazzarin and Viero, 2023; Thorne et al., 1985).

Figure 11Flood event of 2022. Color map of the instantaneous surface velocities computed with the 3D-CFD model for the Paglia River at the Adunata bridge (aerial image from © Google Earth, 2023).

The velocity distribution at cross-section 4 (200 m downstream of the bridge) is presented in Fig. 9 for the moderate peak flow condition of the 2022 event. For this cross-section, in the 3D-CFD results (Fig. 9a), the maximum surface velocity is located on the left side of the channel, rather than at its center (this aspect is discussed in the following). Forced with the maximum water surface velocity, the entropy model reproduces the velocity field in the central part of the riverbed well. Larger discrepancies are instead observed in the lateral part of the cross-section, with the elliptic spanwise distribution (ESD) that performs slightly better than the parabolic (PSD), particularly in the right side. Figure 10 shows the cross-sectional distribution of the surface velocity and the velocity distribution along three verticals. In terms of cross-sectional average velocity and flow discharge, both the PSD and ESD produce error that are lower than 10 % (Table 4), larger than those obtained using the river-wide surface velocity as input for the entropy model.

A last point worth discussing regards the unusual cross-sectional distribution of flow velocity in Sect. 4 (Fig. 9a). The reason that the 3D-CFD model locates the maximum velocity on the left of the thalweg is the alternate vortex shedding occurring downstream of the bridge piers, which propagates beyond the last considered cross-section. This is evident in the map of instantaneous surface velocity of Fig. 11. This particular occurrence poses interesting questions on the application of the entropy model to estimate the flow discharge downstream of in-stream structures. First, the spanwise location of the maximum surface velocity is subject to a periodical shift, which prevents its correct detection by means of a fixed sensor with a small-size field of view, like the one mounted on the Adunata bridge. Secondly, marked time-varying flow fields, which occasionally (or periodically) deviate from nearly uniform flow conditions, can hardly be captured by any preset velocity distribution. To alleviate the problem, the periodic signal of surface velocity can be filtered, which is equivalent to looking at time-averaged modeled flow fields; this requires knowing the frequency of vortex shedding.

The results shown in this section confirm the general accuracy of the entropy model in predicting the cross-sectional velocity distributions. As expected, when using a single value of velocity in place of the river-wide distribution of surface velocity, the accuracy of the method slightly decreases. Provided that using a single velocity is beyond the scope of the method when the spanwise velocity distribution is markedly irregular, the entropy approach can still be forced with a single surface velocity and produce accurate results, when there is no evidence of strong disturbances of the flow. Indeed, such an approach cannot capture marked unevenness in the flow field, as shown in the case of the lateral low-velocity regions at cross-section 1 for the 2012 event (Fig. 7) and in the time-varying flow field of cross-section 4 for the 2022 event (Fig. 9).