Flood risk in urbanized areas raises increasing concerns
as a result of demographic and climate changes. Hydraulic modelling is a key
component of urban flood risk analysis; yet, detailed validation data are
still lacking for comprehensively validating hydraulic modelling of
inundation flow in urbanized floodplains. In this study, we present an
experimental model of inundation flow in a typical European urban district
and we compare the experimental observations with predictions by a 2-D
shallow-water numerical model. The experimental set-up is 5 m

Floods are the most common natural hazard. In Europe, they caused around EUR 100 billion of damage between 1986 and 2006 (de Moel et al., 2009). Many cities were built along rivers, deltas or in coastal areas and are therefore partly located in the floodplains. Urban flood risk management is of particularly high relevance as urbanization is growing at an unprecedented pace and hydrological extremes tend to increase in magnitude and frequency (Domeneghetti et al., 2015; Merz et al., 2012; Vorogushyn and Merz, 2013). Reliable predictions of flood hazard are a prerequisite to support contemporary flood risk management policies. This includes the accurate estimation of inundation extents, water depths, discharge partition and flow velocity in urbanized floodplains, since these parameters are critical inputs for flood impact modelling (Brazdova and Riha, 2014; Kellermann et al., 2015; Kreibich et al., 2014).

Models based on the shallow-water equations are the most common approach for detailed inundation modelling (Costabile and Macchione, 2015). They are considered as state-of-the-art for large-scale real-world applications and have been the focus of much research over the last 2 decades. They also benefit from widely available detailed datasets obtained from remote sensing techniques, such as laser altimetry (Dottori et al., 2013). Difficulties remain nonetheless for estimating the roughness parameters which may vary significantly in space, particularly in floodplains.

Different state-of-the-art hydraulic models exist to compute the inundation characteristics. However, their validation for urban flood configurations remains incomplete as reference data from the field are relatively scarce and difficult to obtain (Dottori et al., 2013; El Kadi Abderrezzak et al., 2009; Neal et al., 2009). Water marks and aerial imagery provide some relevant information but they remain affected by high uncertainties. They are also far from sufficient to reflect the whole complexity of inundation flows in densely urbanized floodplains, the proper description of which requires information on the velocity fields and discharge partitions.

To address this lack of validation data, scale model studies are particularly valuable, since they deliver accurate measurements of flow characteristics under controlled hydraulic conditions (e.g. known distribution of inflow discharge and downstream boundary conditions). Several experimental studies of relevance have been conducted, mainly during the last decade. They have involved different degrees of complexity and realism in the considered flow configurations, ranging from isolated street intersections up to complete urban districts with surface and underground flow.

The flow behaviour in intersections is an essential component of urban flooding since an urban district may be seen as a network of streets and intersections. A limited number of experimental studies investigated specifically the flows at street intersections.

Weber et al. (2001) presented detailed 3-D flow and turbulence measurements at
a 90

Mignot et al. (2008) studied a junction of four branches, with supercritical inflow from two branches. They identified three different flow regimes, depending on the location of hydraulic jumps (either in the upstream channels or within the junction).

Based on 220 experiments, Rivière et al. (2011) studied subcritical flow in a four-branch open-channel intersection. They derived an empirical correlation for the flow partition as a function of the inflow discharges and the height of downstream weirs.

El Kadi Abderrezzak et al. (2011) and Rivière et al. (2014) studied transcritical flow at three- and four-branch intersections. Unlike previously assumed, the existence of a critical section in the lateral channel was shown not to ascertain that flow in the intersection is decoupled from downstream flow conditions. They correlated the partition of outflow discharge with the Froude number and the upstream critical depth. All these studies were conducted for a single intersection only and considering right angles as well as channels of equal width.

Mignot et al. (2013) studied the flow in a three-branch bifurcation with one or several obstacles representing, for instance, pieces of urban furniture. Nine obstacle configurations were considered, which lead to very different influences on the flow. Bazin (2013) confirmed the importance of including such small-scale obstacles within urban flood models explicitly.

Using the same experimental facility as Mignot et al. (2013), Bazin (2013) also considered the influence of sidewalks in a three-branch bifurcation with subcritical flow. Comparing the experimental observations with results of 2D modelling, he points out that, on the scale of a single crossroad with subcritical flow, a bias in the average ground elevation leads to errors in the estimation of discharge partition in the streets. Based on a physical model of an urban drainage system, Bazin (2013) and Bazin et al. (2014) also validated an analytical model predicting the discharge exchange between surface flow and underground pipe flow.

Instead of focusing on a single street intersection, other experimental studies analysed the flow field in a whole urban district. They are, however, very rare and we could only identify five significant contributions.

Zech and Soares-Frazão (2007) investigated
transient flow in an idealized urban district located on a scale model of
Toce Valley in Italy. Two different layouts of 20 building blocks were
considered (aligned vs. staggered). Water depths were measured by electrical
conductivity gauges at some locations around and within the urban district.
However, they do not provide a truly distributed view of the flow pattern
over the whole urban district. Considering a simplified geometric setting
with a flat topography, similar tests were conducted by
Soares-Frazão and Zech (2008) under transient
flow conditions in an experimental flume. The idealized urban district was
made of 5

Ishigaki et al. (2003) used a scale model of a 1 km

A scale model of 17th Street of New Orleans was used by Sattar et al. (2008) to study the urban flooding induced by a dike breach during hurricane Katrina in 2005. The model was designed to evaluate the viability of using sandbags in a multibarrier configuration to effectively close the breach. The flow in the urbanized floodplain was represented to properly reflect its influence on the tailwater at the breach location. Water depths were measured at about 800 points using mechanical gauges, and flow velocity was measured by a micro acoustic Doppler velocimeter, at a limited number of grid points, mainly outside the urbanized area. The outflow discharge was measured in each street.

Paquier et al. (2009) set up two laboratory models (scales 1 : 100 and 1 : 24) to investigate the effects of non-structural measures such as urban development enabling temporary flood storage or car park regulations, which contribute to the preservation of the optimal flow conveyance in the streets during flood events.

Lipeme Kouyi et al. (2010) describe experiments in an idealized urban district. The district consists of seven streets aligned from north to south, crossing seven other streets aligned from west to east. Most streets are not straight; therefore complex geometric configurations arise at the intersections. The scale factor is 100 horizontally and 25 vertically. Only steady flows were considered and the set-up did not enable direct control of the partition of the inflow discharge between the different streets. Walls were positioned west and east of the district, so that only a single main flow direction could be analysed.

In the present study, we consider the Icube Laboratory model of a typical European district, similar to the configuration considered by Lipeme Kouyi et al. (2010). However, here, the boundary conditions allow for the flow distribution to be varied continuously between west–east and north–south directions. The present set-up enables the inflow discharge to be controlled independently in each street. We focus on steady flow conditions, which constitute a realistic approximation for long-duration floods.

The study provides new insights into the distribution of water depths and the discharge partition in a more general urban setting than considered in previous research. The experimental dataset is available for testing numerical models and we provide here a comparison with the results of one specific model, namely WOLF 2-D, which has been widely used for inundation mapping (Erpicum et al., 2010a) and in flood risk analysis research (Beckers et al., 2013; Bruwier et al., 2015; Detrembleur et al., 2015; Ernst et al., 2010).

The experimental and numerical models are presented in Sect. 2. The influence of varying the total inflow discharge is analysed in Sect. 3, together with a sensitivity analysis of the computed results considering different roughness parameters, grid spacing and turbulence models. Section 4 details how the partition of the inflow discharge influences the results. In Sect. 5, the upscaling of the experimental findings to real-world applications is discussed, as well as the enhancements brought by an improved modelling of the streets' geometry and the possible extension of the results to unsteady flow conditions. Conclusions and perspectives are given in Sect. 6.

Plane view of the idealized urban district considered in the experiments. Adapted from Araud (2012).

The experiments considered here were conducted by Araud (2012) at the laboratory ICube in Strasbourg (France). Compared to previous studies, the experimental set-up achieves a relatively high degree of realism by involving streets of various widths and intersections of different types (both normal branches and branches of different inclinations). The study focused on long-duration and extreme events; i.e. a steady state was considered and the flow through the underground networks was assumed negligible and was not reproduced in the model. Only an overview of the experimental set-up and procedure is given here, while all details were described by Araud (2012).

The experimental set-up represents an idealized urban district. It extends
over 5 m

The street inlets are located along the north and west faces of the model, while the outlets are on the south and east faces. Fourteen pumps were used to control the inflow discharge into each street individually. The discharge was distributed between the streets of each face proportionally to their widths. The model was fed with water, assuming no sediment transport and no debris in the flow. The outlets enable free-flow conditions thanks to chutes.

The outflow discharge in each street was determined from the rating curve of calibrated weirs located downstream of the street outlets. An ultrasound sensor was used for measuring the water level upstream of each weir. With a measurement window of minimum 40 s, the uncertainty of the outflow discharge was shown to remain below 3.5 % (Araud, 2012). Given the calibration procedure of the regulation system of the pumps, the uncertainty of the inflow discharge is the same as for the outflow discharge (3.5 %).

The water depths along the centreline of the streets were measured by an
optical gauge fixed on an automatic traverse system. The gauge detects the
phase (air vs. water) in which it is located. Due to the relatively slow
operation of this system, water depths could only be measured along the
centreline of the streets, at about 600 different locations in total. The
measurement uncertainty results mainly from the accuracy of the motor, which
was estimated at

Two main series of experiments were considered here. In the first series,
the total inflow discharge was varied from 10 up to 100 m

The outflow discharge was measured downstream of each street for all the
tests. In addition, for the tests of the first series of experiments (except
for 10 m

The reproducibility of the experiments was tested by comparing water depth and discharge measurements on experiments repeated at several day intervals. The upstream and downstream boundary conditions were identical for all the repetitions. For instance, for test Q020-050, a difference of less than 1 mm was found on 90 % of replicate measurements of water depth. This difference corresponds to the resolution of the measuring device. 96 % of the differences were lower than 2 mm, which corresponds roughly to the fluctuations of the free surface for this test. The outflow discharges in all the streets were measured 13 times. The observed differences were also of the order of the measurement uncertainties (Araud, 2012).

Test series dedicated to testing the influence of the total inflow of the inflow partition between the west and north faces.

In the laboratory experiments, the Reynolds numbers

The ability of a standard shallow-water model to predict the observed
discharge partitions and water depths was tested using the numerical model
WOLF 2-D. This model has been developed by the research group HECE of the
University of Liege (Belgium). It solves the fully dynamic shallow-water
equations on multiblock Cartesian grids and a two-length scale

In 2003, the model WOLF 2D was selected by the regional authorities in Belgium to perform all detailed 2-D flow simulations to support official inundation mapping, including in the framework of the European Floods Directive. Since then, it has been routinely applied for inundation modelling.

In the applications considered here, the bottom shear stress was estimated
using the Darcy–Weisbach formulation and the friction coefficient was evaluated
by the Colebrook formula as a function of a roughness height

The geometry of the scale model was implemented in the numerical model by
using the building hole method (Schubert and Sanders,
2012). The cell size is uniform and was taken equal to 1

The inflow discharge was prescribed as a boundary condition upstream of each street, while a free flow was considered at the downstream boundaries. The boundary conditions for the turbulence model were set according to Camnasio et al. (2014) and Choi and Garcia (2002).

The time step used in the computations is optimized based on the
Courant–Friedrichs–Lewy (CFL) stability condition (e.g. Bates et al.,
2010). It takes values of the order of 10

In a first series of tests, the total inflow discharge was varied from 10 to
100 m

We present hereafter the experimental and numerical results in terms of
discharge partition between the south and east faces, as well as at the
street level. We also describe the observed and computed water depths, both
at the district and at the street levels. In terms of numerical results,
this section only discusses the results obtained by using a cell size of
1 cm, the

Distribution of outflows between the downstream faces in the observations and for different model characteristics.

The circular markers in Fig. 2 represent the observed partition of the outflow discharge between the east and south faces of the urban district for five different total inflow discharges. Overall, about 60 % of the outflow discharge crosses the south face and 40 % the east face. This first distinctive result of the experiments is partly explained by the total flow width available in the north–south direction compared to the west–east direction. Indeed, only one “wide” street (street 4) is aligned in the west–east direction, whereas two of them (streets C and F) convey the flow in the north–south direction. As a result, the total flow width in the west–east direction is 42.5 cm, which is lower than the total flow width in the north–south direction (50 cm). Consequently, the available flow width along the east face is 46 % of the total outflow width, whereas it is 54 % for the south face. This difference goes in the same direction as the difference in outflows (40 % vs. 60 %).

Besides, the observed outflow partition between the south and east faces
(60 % vs. 40 %) remains virtually independent of the total inflow
discharge. Indeed, for a total inflow discharge varying by 1 order of
magnitude, from 10 up to 100 m

The shallow-water model succeeds in predicting the overall discharge partition between the south and east faces (Fig. 2). The difference between the observed and computed values remains in the range 0.02–2.16 %. So, the numerical model also reproduces the quasi-independence of the outflow discharge partition with respect to the total inflow.

Observed and computed contributions of each street to the total outflow discharge for five different inflow discharges.

Figure 3 details the partition, street by street, of the outflow discharge for five different inflow discharges. The highest outflow discharges correspond to the widest streets, particularly those which are straight and start more upstream in the model i.e. closer to the north-west corner (streets C and 4).

At the outlet of street C, which is 2.5 times wider than streets A and B, the observed outflow discharge is about 2.3 to 2.5 times higher than the corresponding discharges in streets A and B. This leads to similar unit discharges in the different streets and may results from the similar configurations of streets A, B and C in terms of shape (all three are straight) and encountered types of intersections. In contrast, street 4 collects between 2 and 3 times more discharge than streets 1, 2 and 3, while the ratio of the street widths is also 2.5. These larger deviations may result from the different configurations of streets 1, 2 and 3 compared to street 4, since the latter is straight while the former are curved, leading to different types of intersections.

Another example of influence of the shape of the streets and intersections may be noticed by comparing streets 1 and A. Their inflow discharges are the same; but street A has an observed outflow discharge between 60 and 70 % higher than the observed outflow in street 1. The number of intersections is the same for both streets. However, street A has mostly right-angle intersections, while all intersections in street 1 have different angles, which seems to promote more flow to be diverted towards the lateral streets. The difference in the outflow discharges results most likely from this difference in the shapes of the streets. Similarly, street F, which is as wide as street C, discharges at the outflow only about 55–58 % of the discharge from street C, as street F is curved and located further from the “upstream” corner (north-west).

Similarly as for the discharge partition between the faces, the observed
portion of outflow discharge in each street remains essentially independent
of the total inflow (Fig. 3). For a total inflow
varying by 1 order of magnitude (from 10 to 100 m

In the computed results, the outflows from the streets with the highest discharges (4 and C) are overestimated by 10–30 % compared to the experimental results. The opposite is observed for some of the streets with the lowest discharges (1–3, D–F), while the outflows from streets 5–7 and G are fairly well represented by the numerical model. The outflow discharges from the streets with intermediate discharges (A, B and F) are also generally well predicted by the model.

As the obtained discrepancies are maximum in curved streets (1, 2 and 3), it is likely that they partly result from the Cartesian grid used, which relies on a “staircase” approximation of the obstacles not aligned with the grid. A Cartesian grid remains, however, of high relevance in practice (Kim et al., 2014), as it makes it generally straightforward to handle contemporary gridded data obtained from remote sensing technologies (e.g. Light Detection And Ranging, lidar).

Another possible explanation for the discrepancies stems from the complexity of the actual flow fields at the intersections, involving different flow regimes, hydraulic jumps and waves as described in the literature cited in the Introduction section. Here, it is, however, difficult to identify which intersection is responsible for the main discrepancies as they all interact with each other and the experimental flow partition between the streets is only available at the downstream end of each street and not between all the intersections.

A closer look at the computed results reveals that the higher the total
discharge, the higher the outflow from the widest straight streets 4 and C
(

Observed water depths in streets C and 4 for inflow discharges
varying between 20 and 100 m

Observed and computed water depths

Observed and computed water depths

Bias and root mean square error (RMSE) on the computed water depths in streets C and 4.

For four different inflow discharges, maps of the observed and computed
water depth distributions over the whole district are provided as
Supplement 2, both in absolute values and scaled by the district-averaged
water depth (Figs. S1 and S2). The observed district-averaged water depth is
shown to increase from 3.4 up to 9.0 cm when the inflow discharge is
varied from 20 to 100 m

The computed values are in excellent agreement with the observations for the
lowest inflow discharge; but they deviate by about 11 % for the highest
inflow discharge (100 m

District-averaged water depths for inflow discharges between 20 and 100 m

The profiles of observed water depths along the centreline of the streets are displayed in Fig. 4 for the two widest streets (4 and C). The most significant streamwise variations in the water depths take place locally, in the vicinity and immediately downstream of the intersections, particularly close to the intersections of two wide streets such as streets C and 4, or streets 4 and F. In contrast, between the intersections, the water depths remain fairly constant. Therefore, friction is expected to play a minor part, as discussed in Sect. 3.2.

As shown in Figs. 5 and 6, the computed results reproduce the main features of the water depth profiles
qualitatively, which are characterized by sudden drops near the intersections and remain almost
constant between the intersections. From a quantitative perspective, the
computed results are relatively accurate for the lowest discharge
(20 m

Although experimental measurements of water depths could only be performed
along the streets' centreline, we used the numerical simulations to
appreciate to which extent the flow depth varies within a cross section. As
shown by the shaded area (

Root mean square error of the outflow discharges in each street (expressed in percentage of the total inflow), as a function of the total inflow and the modelling characteristics.

Computed vs. measured outflow discharges in each street (in
percent of the total inflow) considering different cell
sizes

To appreciate the sensitivity of the results to different modelling choices, a sensitivity analysis was performed to investigate the influence of roughness, mesh refinement and the turbulence model.

The reference simulations were carried out with a roughness height equal to
zero (

The distribution of the outflow discharges between the different streets
remains virtually unchanged when the roughness height is varied between 0
and 1 mm (Fig. S3). For

The influence of the mesh refinement on the results was tested by using cell
sizes of 1 cm and 5 and 2.5 mm. For total inflows of 10, 20, 60, 80 and
100 m

Froude numbers

For a total inflow of 20 m

The sensitivity of the computed water depths with respect to the grid resolution was also assessed. As shown in Fig. 7, refining the cell size to 5 mm instead of 1 cm reduces the RMSE and the bias by about 20 % in the case of the highest inflow discharges, for which these errors are maximum. This reduction is in agreement with a decreased influence of the staircase approximation of the obstacles' geometry when a finer grid resolution is used.

Changing the grid size also leads to local changes in the flow pattern. As an example, Fig. 9 shows the details of the flow field near the downstream end of street 4 in the case of test Q100-W050 computed with cell sizes of 1 cm and 5 mm. The discretization of the model geometry on a Cartesian grid induces local discontinuities in the street widths as long as these streets are not perfectly aligned with the grid, as is the case here particularly because the “as-built” coordinates of the obstacles were used. These sudden changes in the street width lead in some cases to the development of flow structures (such as cross waves), which as a matter of fact are mesh-dependent (Fig. 9a and b). Their impact remains, however, very limited further upstream in the domain, where the flow patterns are extremely similar for the two grid sizes (Fig. 9).

Considered grid sizes and corresponding resolution obtained on the field scale.

Despite the better results obtained with the cell size of 5 mm, in most of
the simulations performed hereafter, the 1 cm grid was kept nonetheless
because in our opinion this choice is the most consistent with grid
refinements reasonably accessible for inundation mapping in practice. It
corresponds indeed to five cells over the width of the narrow streets and
about 12 cells over the width of the wide streets (4, C and F, as
detailed in Table 4). At the field scale, it leads
to a grid spacing of 2 m (Sect. 5.2). Using finer
cells (5 or 2.5 mm) would not be realistic compared to typical grid
refinements used for real-world flood hazard mapping (e.g. Bazin, 2013).
The 1 cm grid may also be considered as a reasonable trade-off between
accuracy and computational burden since opting for the 5 mm grid would also
decrease the time step by a factor of 2 due to the Courant–Friedrichs–Lewy (CFL)
stability condition (e.g. Bates et al., 2010) and, therefore, it
would lead to an increase in the computational cost by almost 1 order of
magnitude (

Most flood hazard mapping in practice is conducted with depth-averaged
models which do not incorporate a proper turbulence model, apart from a
friction term which lumps all dissipative effects. Here, we specifically
tested the influence of activating the

Velocity fields computed with

The computations reveal that the turbulence model hardly influences the
outflow discharges. This result applies when the outflows are examined by
face (Fig. 2) and also when they are disaggregated at the street level (Fig. 8). At
the face level, switching the turbulence model on or off induces variations
in the outflow discharge not exceeding 1.5 % (Fig. 2). At the street level, the relative
differences may reach 2 %; but these differences are much lower than
those observed when changing the grid size from 1 cm to 5 mm. For the whole
range of considered total inflows, the influence of the turbulence model on
the root mean square error of the outflow partition between the streets is
of the order of 0.1 % of the total inflow, both for the 1 cm grid and for
the 5 mm grid (Tables 3 and S2). This remains about 5 times lower than the influence of the
grid size, but similar to the influence of the roughness parameter

As regards the influence of the turbulence model on the computed water depths, the simulations without the turbulence model perform slightly worse than when the turbulence model is used (Figs. 5 and 6). This is also confirmed by an increase in the RMSE and in the bias for most considered discharges (Fig. 7).

Like the cell size, the turbulence model also has a substantial influence on
some local features of the flow field. For tests Q020-W050 and Q100-W050,
Figure 10 shows the computed velocity fields. As
also noticed in the experiments, flow recirculations and vena contracta are observed
downstream of the intersections. This is consistent with flow descriptions
available in literature (e.g. Weber et al., 2001; Neary et al., 1999) and
with the significant crosswise variations in the water depths shown in
Figs. 5 and 6. Figure 10 also reveals that the
recirculation lengths in the streets differ significantly between the
results computed with and without the turbulence model. In the former case,
the recirculation length

Froude numbers

Such changes in the flow structure between simulations with and without the turbulence model may be related to local variations in the discharge partition. For test Q100-W050, Fig. 11 details the flow field at the intersection between streets 4 and F. Besides variations in the recirculation lengths as already highlighted in Fig. 10, Fig. 11 also reveals a change in the recirculation width in the southern branch. While the width reaches 6–7 cm when the turbulence model is used (Fig. 11a and c), it is restricted to 3–4 cm without the turbulence model. As a result, the share of discharge maintained in street 4 in the simulation with the turbulence model is about 9 % higher than in the simulation without the turbulence model. This figure was obtained with a grid size of 5 mm, while the change is estimated at 6 % based on the 1 cm grid size. In turn, this change in the flow partition alters the shape of the control sections in the downstream part of street 4 (Fig. 11a and b), and also the overall distribution of water depths. Figure 11a and b also highlight the presence of control sections in the vena contracta, which is consistent with e.g. Fig. 5 in Rivière et al. (2014).

In the following, the turbulence model has been systematically used due to the higher realism that it provides for simulating the flow processes.

A distinctive feature of the present experimental set-up is that it enables
the partition of inflow to be varied continuously between the west and the
north faces. For a total inflow discharge of 60 m

Observed and computed portions of outflow discharge through the
east face (

As shown in Fig. 12, the observed outflow
discharge through the east face (noted

The computed outflow discharges through the east face slightly underestimate
the observed values (by 1.5 to 3.3 %) but they show a similar trend
as the observations:

The distribution of the outflow discharges at the street level is shown in
Fig. 13. Three groups of streets may be
distinguished.

In streets 1 to 4, which end up on the eastern face, the outflow discharge
declines steadily as

Contrarily, the outflow discharges rise in street C and to a lesser extent in streets A, B, D and F, which all end up on the southern face.

The other streets are less significantly influenced by the partition of outflow discharge.

Observed outflow discharges in each street for when the inflow
through the west face (

Computed vs. observed outflow discharges in each street for different partitions of the inflow discharge between the west and north faces.

As detailed in Sect. 3.1.1, the experimental
observations indicate that, for

Characteristic Reynolds number

To test this hypothesis of “attraction” effect of the wider streets
independently of the total inflow, we undertook additional simulations
corresponding to a single “equivalent” four-branch intersection, with the
north–south and west–east streets widths respectively equal to 0.5 and
0.425 m. These widths mimic the cumulative street widths in the
north–south and the west–east directions in the experimental model
(respectively equal to 0.5 and 0.425 m). We performed the simulations for
the two extreme discharges (20 and 100 m

Nonetheless, this geometric effect only partly explains the difference in the experimentally observed outflow discharges through the south and east faces (60 % vs. 40 %). We attribute the remaining difference to (i) the spatial distribution of the wider streets within the scale model, and (ii) the inclination of several streets. These two effects are not properly reflected in the single “equivalent” intersection considered here; but they are expected to further amplify the difference in the outflow discharges between the north–south and the west–east directions.

The simulations presented in Sect. 3.2 have
revealed that the values of the roughness parameter

First, all geometric characteristics of the laboratory model were magnified
by a factor

However, Prototype 1 is hardly realistic in terms of height to width ratio
in the streets. Most real-world urban floods are characterized by much
smaller water depths compared to the widths of the streets. Therefore,
upscaling of the laboratory model was also performed, assuming a distorted
model with a horizontal scale factor

In Sect. 3.1, the discrepancies between computed and observed results were, to a large extent, attributed to the intrinsic limitations of Cartesian grids to reproduce oblique boundaries. To investigate this effect further, we tested an extended shallow-water model involving anisotropic porosity parameters to improve the representation of complex boundaries in a Cartesian grid framework. This approach is similar to the “cut-cell” technique (An et al., 2015; Causon et al., 2000; Kim and Cho, 2011).

We used two types of porosity parameters (Fig. 15). First, a “storage porosity”

The partition of outflow discharges at the street level has been computed using the model with porosity. The results are displayed in Fig. 16, which should be compared to Fig. 3. While the outflows from streets 4 and C are overestimated by 10–30 % when the standard shallow-water model is used, this discrepancy is reduced here to around 10 %. Similarly, the outflow discharges through the narrower streets 1–3 and D–F were significantly underestimated by the standard model, whereas these outflows are now predicted with an error not exceeding 8 %. This leads to a root mean square error of the outflow discharge which is reduced from 19 to 6.6 % as a result of using the shallow-water equations with porosity.

The model based on anisotropic porosity parameters described here is
certainly a viable approach for practical applications. For the considered
experiments, all porosity parameters were deduced directly from geometric data and
there was no calibration of these porosity parameters:

The present study relies on steady-state experiments and numerical simulations, while real-world floods are intrinsically transient. Whether a real flood wave can be approximated as a succession of steady states depends on the timescales of interest. Therefore, we undertook extra simulations in unsteady mode, with the purpose of identifying a characteristic timescale of the urban district considered here.

The initial condition corresponds to virtually no water in the model
(initial water depth

Storage porosity

Contributions of each street to the total outflow discharge for five different inflow discharges: observations vs. results computed with the shallow-water equations with porosity.

Computed water depths in the unsteady simulations, at the
intersections

Time series of computed water depths in the centre of the most upstream intersection (i.e. between streets 1 and A) and between the two wide streets 4 and C are displayed in Fig. 17. They reveal that the time necessary for reaching a steady state is of the order of 30–60 s on the scale of the laboratory model.

The scale factor for time is given by the ratio between the scale factor for
horizontal lengths (

For Prototype 1, this leads to a magnification factor
of

For the more realistic Prototypes 2 and 3, a magnification factor of

To investigate the flow characteristics in urbanized floodplains, here we
considered the 5 m

Two test series were considered. In the first one, 50 % of the inflow
discharge was injected through the west face of the model and 50 %
through the north face. The partition of the outflow discharge between the
downstream faces (east and south) did not change by more than 2 % as the
total inflow was varied by 1 order of magnitude (between
10 and 100 m

Drops in the free surface profiles were observed downstream of each intersection. Between the intersections, the water depths remain fairly constant. In general, the computed water depths overestimate the experimental observations, by 1 % (lowest total inflow) up to about 10 % on average (highest total inflow). This overestimation of the overall flow resistance is consistent with the staircase representation of complex geometries when using a Cartesian grid.

The roughness parameter has very little influence on the computed results.
Some features of the velocity field are predicted more realistically
when a depth-averaged

In a second series of tests, the partition of the inflow discharge between
the west and north faces was varied systematically by steps of 10 %
between

The upscaling of the present findings to real-world applications was discussed. In particular, the influence of the roughness parameter is expected to become significantly stronger on the prototype scale due to the combined effect of a lower height-to-width ratio and a relatively higher roughness height characterizing real urban settings.

To overcome the influence of the Cartesian grid on the flow computations, a subgrid model based on porosity parameters was implemented. It leads to substantially better predictions of the outflow discharges at the street level.

In the future, velocity measurements and more detailed water level measurements will be performed in the near-field of the street intersections. The present research will also be extended to investigate the hydrodynamic characteristics of unsteady inundation flow in urbanized floodplains, as well as the influence of the bottom slope. Several other processes have not yet been considered and should be addressed in subsequent studies, such as flow within the buildings, morphodynamic changes and the transport of debris by the inundation flow.

The Icube Laboratory experimental model was funded by the Alsacian network of laboratories in Environmental Engineering and Sciences and the Fluid Mechanics research team of the Icube laboratory. The authors are grateful to the Icube Laboratory technicians (Martin Fisher, Johary Rasamimanana and Abdel Azizi) and the PhD students (Hakim Ben-Slimane, Vivien Schmitt, Noëlle Duclos, Alain Petit-Jean and Sandra Isel) for their priceless contribution to the building of the experimental set-up. Special thanks are also expressed to Quentin Araud who conducted most of the experiments.

Part of this research was funded through the ARC grant for Concerted Research Actions, financed by the Wallonia-Brussels Federation. The authors gratefully acknowledge Mathieu Debaucheron who contributed to some of the numerical computations. Edited by: A. Günther Reviewed by: two anonymous referees