Guillaume Le Bihan

Olivier Payrastre

Eric Gaume

David Moncoulon

Frederic Pons

Regional hydrological models for distributed flash-floods nowcasting: towards an estimation of potential impacts and damages

Flash floos monitoring systems developed up to now generally enable a real-time assessment of the potential flash-floods magnitudes based on highly distributed hydrological models and weather radar records. The approach presented here aims to go one step ahead by offering a direct assessment of the potential impacts of flash floods on inhabited areas. This approach is based on an a priori analysis of the considered area in order (1) to evaluate based on a semi-automatic hydraulic approach (Cartino method, Pons et al. 2014) the potentially flooded areas for different discharge levels, and (2) to identify the associated buildings and/or population at risk based on geographic databases. This preliminary analysis enables to build a simplified impact model (discharge-impact curve) for each river reach, which can be used to directly estimate the importance of potentially affected assets based on the outputs of a distributed rainfall-runoff model. This article presents a first case study conducted in the Gard region (south eastern France). The first validation results are presented in terms of (1) accuracy of the delineation of the flooded areas estimated based on the Cartino method and using a high resolution DTM, and (2) relevance and usefulness of the impact model obtained. The impacts estimated at the event scale will now be evaluated in a near future based on insurance claims data provided by CCR (Caisse Centrale de Réassurrance).

Introduction

Hydrometeorological forecasting is an essential component of real-time flood management. The information it provides is crucial for crisis managers to anticipate and quantify the floods which will hit the territories at risk. In the particular case of flash floods which may affect very small watersheds spread over the territory, suitable forecasting systems are still currently under development. {examples et bibliographie à developper: US, UK.} In France, some approaches devoted to flash-flood forecasting have also been developed in a close from operational framework: AIGA system \citep{Lavabre2005, Javelle2014}, PreDiFlood project \citep{Naulin2013}. These developments often rely on highly distributed hydrological models and on radar based QPEs or rainfall nowcasts as input information. They provide indications on possible flood magnitudes, but are still rarely designed to directly account for the possible associated impacts on society. The translation of flood magnitude in impacts requires, indeed, a detailed knowledge of the vulnerability of the considered areas, which is generally well developed and shared at a very local scale (local authorities, inhabitants of flood areas,..), but much more difficult to assess and incorporate at the much larger scale at which hydrometeorological forecasting systems are generally implemented. Considering that these systems are designed to monitor a very large number of small rivers spread over large territories, a large number of simultaneous alarms may be generated in case of a significant rainfall event. Thus, providing directly an information on local vulnerabilities and associated possible impacts would probably be helpful for coordination managers to have a faster evaluation of the situation and to focus their actions on the most problematic situations.

The approach developed in this paper aims to enable a direct evaluation of flash flood impacts on inhabited areas based on an hydrometeorological simulation chain. The question of impact prediction has already been addressed in previous works conducted at IFSTTAR \citep{Versini2010, Naulin2013}, and were up to now mainly focused on road inundation risks. The aim of the new approach presented herein is now to account for risks of flooding in settlements. The methodology developped is based on an comprehensive analysis of the study area in order to build an impacts model on each river stream incorporated in the hydrometeorological simulation chain. This analysis is based on simplified 1-D hydraulic simulations to evaluate the extent of the flooded areas for different discharge levels. The impacts models are then used to directly translate the discharges simulated by the rainfall-runoff model into possible associated impacts.

Even if the approach developed may appear relatively straightforward, its application on a very detailed stream network including small watersheds prone to flash floods, may be too much time consuming to enable an application at a large scale (more than 100000 km²). Therefore, the challenge has been here to define a very simplified and automatic procedure for the building of the impacts model, with the objective to limit its implementation time to a reasonable level. This objective of simplification has been considered as a priority, placed before the accuracy of flooded areas and associated impacts evaluated. In other words, a decrease of the quality of the impacts model has been accepted to guaranty its applicability at a large scale including a very detailed stream network. {Bibliographie à développer sur l'hydraulique appliquée à grande echelle}

The content of the article presents the application of the method on a preliminary case study, and a first evaluation of the quality of the impacts model obtained, provided the relative large level of simplification of the estimation procedure. The article is organised as follows: the first section presents the application case study and the datasets involved in the study, the next section presents the methodology developped for the fast computation of flood areas and the associated building of impacts model. This section also describes the procedure used for the evaluation the results. The results are presented in section 4 and discussed in section 5. Lastly, section 6 presents conclusions of this work.

Presentation of the case study

The region of Alès in the Cévennes area, south eastern France

The study has been focused here on a territory of 1990 km² including the town of Alès in south eastern France. This area is located in the core of the Cévennes region, well known to be prone to intense flash floods. Moreover, this area has been identified during the implementation of the EU flood directive as one of the areas having the most significant vulnerability to floods over the whole French territory. For this reason, it was included in the list of Areas with Potential Significant Flood Risk (APSFR) selected for the application of the flood directive. This vulnerability is mainly related to the presence of the town of Alès, but also to other highly vulnerable small towns such as Anduze (see figure1). This region is represented on figure 1, showing the exact limits of the APSFR of Alès. This territory is part of two main watersheds: the Gardon d'Alès and the Cèze rivers. These two main rivers have their upstream course in the Cévennes mountains, and reach in their downstream part a plateau zone with limited slopes. The APSFR of Alès is located just in the transition zone between the mountainous and plateau areas. Therefore, this case study includes a large variety of river beds configurations including very steep and narrow valleys, up to relatively flat and wide plains. Some statistics about the river beds characteristics are provided in table 1. One part of the river network in this territory is already covered by the French national flood forecasting system. However, significant damages associated with flash floods also frequently occur on the secondary river network which is currently not monitored. A detailed analysis of the features the river network, based on a DTM treatment, shows that it includes 400 km of river streams having an up to 5 km² upstream catchment surface, divided in 192 river reaches (a river reach being defined as the portion of river located between two confluences), among which only 70 reaches (132 km) are covered by the current flood forecasting system (see main network figure 2. This illustrates both the relevance and the difficulty associated with the implementation of complementary flash flood forecasting systems on the very dense stream networks prone to flash floods. Even if the extent of the case study considered here is not very large, the quantity of rivers courses to be integrated in the It is noteworthy that a elementary catchment surface of 5 km² is represented here, and that decreasing this surface would lead to highly increase the quantity of river courses to be considered. This finally illustrates the importance to define very simple and efficient approaches for the implementation of flash flood forecasting, as far as the objective is to apply such approaches in areas of relatively large extent (largely exceeding the extent of the case study considered herein). This point is particularly important herein since the main objective is the definition of a flash flood impact model based on hydraulic simulations, which may represented a much time consuming task.

Available data

Digital Terrain Model

The implementation of the impacts model has been based on a 20m resolution DTM produced in 2007, which altimetric accuracy is less than 20 cm in clear areas, and less than 1 m in forest areas. This DTM was interpolated on a 5 m resolution grid for the purpose of the study. Therefore, the quality of the terrain information used herein remains relatively limited: the use of a real 5 m resolution DTM obtained fro Lidar measurements would probably lead to results of better accuracy ({references a inclure sur les effets de la resolution des MNT}).

Rainfall and discharges

The region is equipped with relatively dense streamgauge and raingauge networks, complemented with three hydrological radars. The positions of stream gauges are presented on figure 2. Thanks the OHMCV observatory, this data is carefully checked and can be considered as exceeding conventional quality standards. However, given the limited possibilities to make direct flow measurements during intense flash floods, the rating curves are often extrapolated, with consequently a large decrease of the quality of estimated discharges for high water levels.

Flood quantiles

The regional SHYREG database ({reference a inclure}), providing flood quantiles of different return periods (from 2 to 1000 years) on all the river network, has been used herein for the implementation of the impact model. This database relies on the combined regionalization of parameters of a rainfall stochastic generator and of a distributed hydrological model. It has to be noticed here that both the work of regionalization and some specificities of watersheds such as kartsn urban areas, snowmelt, or hydraulic structures (dams) may reduce the reliability of SHYREG data.

However, the accuracy of information on flow frequency is not crucial herein for the implementation of the impact model (only the discharge values are used and not the information on frequency): it just enables to derive flood maps for discharges values of relatively homogeneous magnitude in all the considered area.

Reference flood maps

Thanks to the recent application of the EU flood directive in the considered case study, a great effort was put on flood cartography, enabling to produce detailed maps of probable flood extent for three reference events: a common event (30 year return period), a medium event (300 year return period), and a large event (exceeding a 1000 year return period). Therefore, 3 reference flood maps are available, which have been carefully validated using all the available information (including the extent of the exceptional september 2002 flood). These maps will be used for the validation of the flood areas evaluated for the purpose of the flash flood impacts model implementation. Unfortunately these maps, presented on figure 2, were produced on one part only of the river network considered herein which limits the validation possibilities: the river network covered represents 192 km (among 400 km included in the case study) and includes 101 river reaches (among 192).

Methodology

As indicated in introduction, the approach developed herein is based on a pre-evaluation of the extent of flooded areas for different discharge levels. This evaluation should be obtained in an as simple and automatic as possible way in order to enable its implementation on a large number of river reaches. Therefore, it has been based on a 1D hydraulic model run in permanent mode, and implemented automatically based on a high resolution DTM. This work has been based on the Cartino software (developed by Cerema), enabling to build the structure and run the model in a iterative way until satisfactory results are obtained. The Cartino parameters have been adapted to the objectives purchased herein, and some additional post treatments have been added to remove the most identifiable inconsistencies in the outputs.

The whole procedure will be presented hereafter in the following way: first, the overall prinicples of the Cartino software will be presented; then the implementation details (parameters, etc) will be provided; the additionnal post treatments introduced will be presented; and lastly, a short discussion about the main remaining sources of errors will be proposed.

The Cartino software

The Cartino software enables to build and run a 1-D hydraulic model on a river network based on a high resolution DTM, to finally obtained a simplified evaluation of potential flooded areas for a considered discharge value. The computation is achieved in 3 main steps presented in figure 3: 1 -a pretreatment module for the definition of the position of cross sections, 2 -a module running the 1-D hydraulic simulation for the computation of water elevations, and 3 -a post treatment for the estimation of the extent of flooded areas and associated water levels.

The first two modules are run in an iterative way to optimize commonly the couple width -distances between cross-sections, with the objective to finally ensure that all cross sections are wide enough for the considered discharge and that no overlapping is present between two successive cross sections. This procedure is initiated based on the position of the river reaches and a first possible extent of the flooded area (provided as input) which is used to estimate a default width of each cross section. The associated distances between profile are then defined as a proportion of each profile widths (to be defined as input parameter). After each run, the width of each profile is checked according to the computed water elevation, and it is enlarged if necessary in a proportion defined as input parameter. Distances between profiles are adapted in consequence. The roughness coefficient is also defined as an input parameter, and the same value is used for all computations.

The choice of the input information (default extent of flooded area, parameters values) have a great impact on the results and on the computation times. For instance, a too large default flood area or a too fast increase of profile widths may lead to incorporate in the model some depressions located nearby the river bed, but not connected to it for the considered discharge (see figure 4): this leads both to simulate flooded areas of too large extent at the considered profiles, and to decrease the altitude of the simulated water level, with a repercussion on the simulations results at the upstream profiles (possible under estimation of water level). On the other hand, a too narrow initial flood area and/or too slow incerase of profile width will highly increase the computation times.

To cope with these difficulties, the software offers the opportunity to change the position and/or modify the widths of any profile after a first iterative run.

Specificity of application of Cartino for the purposes of the study

For the purpose of building a comprehensive impact model on detailed river network considered in the case study, the objective is here to obtain a catalogue of flooded areas corresponding to different discharge quantiles. A wide range of quantiles has been selected to represent the entire panel of possible flood magnitudes: based on the SHYREG database, discharges corresponding to return periods of 2, 5, 10, 20, 50, 100, 200, 500, 1000 years have been selected for each river reach. For a given return period, the flooded areas has been computed for all river reaches in the region.

Depending on the flood quantile, the simulated flooded areas may be limited to the river bed or extend up to the entire floodplain. One crucial aspect is to determine as accurately as possible the discharge level at which the inundation of the floodplain begins. For this purpose, the Cartino software was applied in the following way:

-application first for the smallest discharge values (first run for T=2 years), using a narrow initial river profile widths and progressive increase in profiles widths to be able to estimate as accurately as possible the limits of the flooded river bed.

-application to the next discharge quantile based on the results obtained for the immediately lower discharge as initial river bed width. -A careful check of the results after each computation also enables to eliminate some errors related to wrong estimations of the shape of some cross sections, mainly associated with the limits of DTM information used as input: bridges still appearing in the DTM, remaining noise due to dense vegetation. This verification was based on the comparison of the wetted area of the successive cross sections, with an automaticremoval of cross sections appearing as inconsistent with the immediate downstream and upstream cross-sections..

Additional post-treatments

Despite the aforementioned precautions, some evident inconsistencies still remain locally in the computed flooded areas: flooded areas disconnected from the river bed, some areas included in the estimated flood extent for a given discharge and not for the larger one. Thus, a simple post-treatment was applied to the catalogue of flood maps to ensure a better overall consistency of the results. This enabled to:

-systematically remove disconnected flooded areas (see figure 4).

-systematically include each flooded area in the ones computed for larger discharges.

Remaining sources of uncertainty

Although some precautions were taken to enhance the quality of the results, important limits and sources of errors remain, affecting the quality of the results. These remaining errors are mainly related to the limits of the procedure applied herein, which was to guarantee a fast application and to accept the associated unavoidable errors. These errors are mainly due to:

-wrong representation of topology and characteristics of river bed due to the automatic definition of position of profiles and also to the limits of topographic information , -absence of representation of friction losses due to bridges and other singularities -choice of a fixed Manning roughness coefficient, equal to 0.05 -absence of representation of dynamic effects due to the systematic computation in steady state For these reasons, it is important to keep in mind that the information produced here cannot correspond exactly to the real extent of flooded areas. It represents a simplified estimation giving on order of magnitude of the level of flooding and enabling some comparisons at a regional scale.

Evaluation of the results

Comparison with reference flooded areas

The remaining error levels on flooded areas can be evaluated thanks to the availability of maps produced for the application of the EU flood directive, which were used as reference information.

For this purpose, intermediate inundation maps were added to the catalogue using the same flood discharge values as for the EU directive cartography, corresponding to return periods of respectively 30 years and 300 years. They were then compared with the reference flooded areas on each river reach based on the following procedure: the common surface (Sc) between estimated surface (ES) and reference surface (RS) was first considered (see figure 5), enabling to define the excess (Se) and default (Sd) surfaces as follows:

ES = Sc+Se

(1)

RS = Sc+Sd

(2)

A synthetic incoherent surfaces proportion (ISP) was then computed, representing the relative proportion of Se and Sd if compared to reference surface:

ISP = (Se+Sd)/ (Sc+Sd) (3)
It has here to be noted that the portion of the river bed being permanently wet (represented on figure 5) may increase the value of Sc when this area is not crucial for flood risks evaluation (absence of vulnerability). For this reason, this area was systematically removed from estimated (ES) and reference (RS) surfaces before the computation of Sc, Se, Sd, ISPs, and therefore the evaluation proposed here is strictly limited to nonpermanently inundated areas.

Comparison with rating curves established at stream gauges locations

The presence of stream gauge stations with calibrated rating curves offers the opportunity to evaluate the accuracy of stage-discharge relation estimated by the model in the associated cross sections. This comparison was achieved after a careful selection of cross sections which better correspond to the location of the stream gauges. However, considering that the distances between profiles are variable and may reach up to 100 m, a perfect correspondence could not systematically be achieved {alternative: an additionnal profile was added if necessary at the exact location of the gauging station}.

Comparison of impacts curves obtained

The last evaluation step was based on the computation and comparison of impact curves based on the inundation map catalogue obtained for each river reach. The impacts curves were computed herein in a simple way by counting the number of buildings present in the estimated flooded areas according to the IGN BDCarto database. These impact curves cannot be directly validated, but they were nevertheless compared beteen river reaches in order to verify whether the preliminary objective of the methodology developed herein -i.e. providing a hierarchy of possible impacts on a relatively large area -was achieved by obtaining significantly different impact curves depending on the locations of the considered river reaches.

Results

Extent of flooded areas

Figure 6 presents the distributions of ISPs computed for the 101 rivers reaches on which the reference inundation maps are available. This figure illustrates first the limited ISPs values obtained in a large majority of cases: ISPs rarely exceed 30 %. If we notice here that the ISP cumulates both default and excess areas, the real differences of surfaces of the estimated flood areas will be much more limited than the ISPs values: this suggests that the errors in the estimation of impacts will also be limited. These figures also illustrate significant differences depending on the magnitude of the simulated floods: the results obtained for the 300 year flood appear much more accurate, with ISPs rarely exceeding 50 %. This can be mainly attributed to the fact that the flood plains are almost entirely flooded for this range of discharges, leading to limit the errors associated with the description of local flow conditions. On the other hand, the 30 year return period flood extent appears much more difficult to appraise, with a significant part of very large errors: ISPs exceeding 100 % in almost 10 % of the considered reaches. These large errors are mainly related to the crucial importance of the description of the characteristics of the river bed and of the flow conditions to correctly simulate the beginning of inundation: errors in crosssection widths or position, on roughness values, absence of description of head losses, will all result in large errors in this range of discharges. Therefore it appears completely logical that the simplified procedure used herein leads to results of significantly lower quality for moderate return period floods.

The analysis of repartition of ISPs depending on the surface of the reference flooded area (RS), provided on figure 6b, confirms the same tendency: ISPs are much more important for low RS values, illustrating the difficulty to appraise correctly the beginning of inundation. This tendency is also present for the 300 year flood but in a much limited extent: in this case, a limited value of the reference surface RS often implies a very deep and narrow river valley, highly limiting the risks of errors.

Water levels at gauged river sections

The results of comparison of stage discharge relationships are presented on figure 7. This figure illustrates contrasted situations which are detailed hereafter.

The case of the Mialet station appears as an ideal situation, in which the shape of the cross-section is very well represented in Cartino, and the water levels computed don't seem to be significantly affected by other sources of errors (choice of roughness coefficient, width of considered profiles, etc ..). This may be explained by the simple configuration of the valley (deep and relatively narrow), the limited presence of vegetation in the river bed, and the significant slope. A sensitivity test to the value of roughness coefficient can show in this case that the associated errors are of secondary order, provided that the cross section is well described. The case of the Banne station is much more problematic since the topography of the river cross section is absolutely not well represented in the Cartino model. This may be attributed to local errors in the DTM used, maybe associated with the presence of dense vegetation. However, even if a significant error is made in the estimated elevation of the water level, this error will not necessarily be very problematic for the computation of inundated areas, since the same DTM including the same elevation bias will be used for this purpose.

The case of the Alès station is a more surprising one. Again in this case, the topography of the river bed appears well represented in Cartino. But the shape of the stage discharge relation appears very chaotic if compared to the rating curve of the station. A sudden decrease of water levels is simulated for discharge levels of 20 to 30 years return periods. The configuration of the river bed is different here from the previous cases. The section is located on the main Gardon d'Alès river, with a wide permanent river bed (about 100 m), and a very flat floodplain including some large depressions. This represent a typical case in which the width of the profiles included in Cartino will have a great impact on the results. Figure 8 illustrates in detail what happens in this case: for limited discharge levels the profiles defined by Cartino ends at the riverbanks, leading to very consistent results; but for the T=20 years discharge, a wide profile is incorporated in the hydraulic model, located just downstream the position of the station (see figure 8a). As a consequence, a wide flooded area (disconnected from the riverbed, see figure 4) is considered by the 1-D model, leading to a significant decrease of simulated water levels in the considered section and also the upstream sections. In such a situation, the use of a 2-D hydraulic model could help in improving the delineation of the flooded areas but at the price of a dramatic increase of computing times.

Impacts curves

The characteristics of the impacts curves obtained are summarized on figures 9 and 10. Figure 9 represents the geographical repartition of the maximum impact estimated at each river reach (number of buildings located in the 1000 year estimated flooded area). This figure confirms that the main urban areas, representing the main potential impacts are correctly identified by the model: a limited number of river reaches with very large estimated impacts. In addition, the impacts model also provides an interesting hierarchy of the possible impacts in other locations: figure 10.a shows that the model identifies a large majority of river reaches for which the potential impacts are very limited, and therefore enables to discriminate the more limited number of river reaches for which significant impacts may be observed, even if remaining much more limited than in urban areas. It is likely that this kind of information will be of great help for the interpretation of forecasts provided in real time by distributed hydro-meteorological models.

Moreover, figure 10 also shows that the impact model identifies an interesting hierarchy in the discharge return periods for which a significant part of the exposed assets begin to be affected. The model seem to be able to distinguish river reaches for which the inundation begins for relatively low discharges, or on the other hand for large discharges, providing interesting information about the flood levels (in terms of frequency) leading to the first potential damages. Of course, considering the relatively high uncertainty of estimated flooded areas in the beginning phase of the inundation, this information will probably be less accurate than the previous one (hierarchy of impacts for large discharges), and will probably include significant errors. But it is nevertheless interesting information for the management of floods of intermediate magnitude, for which the location of damages do not necessarily follow the repartition of damages presented on figure 9.

Discussion and further validations based on the CCR database

The first validation results presented herein are focused on the accuracy of estimated flooded areas. These results show that if very large errors may be observed in a limited number of cases (typically less than 10% of the river reaches) for which the results should still be improved, in a large majority of cases the errors associated with the important simplifications introduced in the computation procedure remain much more limited (ISPs not exceeding 20 to 30 %). Therefore the procedure applied herein combines the advantage of its computational efficiency, enabling to generate estimated flooded areas on a very detailed river network and for a wide range of discharge levels, and of acceptable error levels considering the objectivesof this study: giving a first estimation of the magnitude and hierarchy of impacts at a large regional scale.

The features of the impact model show that it contains useful information about the location of the vulnerable settlements, and could probably usefully complement the outputs of a rainfall runoff model to support crisis management and rescue operations. However, the combination of this impacts model with a rainfall-runoff model will introduce additional sources of errors such as: errors in discharge estimations, interpolation between two estimated flooded areas for the estimation of the associated impacts. The final accuracy and usefulness of information on possible impacts will highly depend on the combination of these sources of errors. Finally, it clearly appears that this kind of approach still needs further validation to estimate its real efficiency in a close to real world situation. For this purpose, one objective is now to valuate insurance claims data collated by the CCR in the XXX database. This information, indeed, may provide an interesting synthesis of the location of the main damages to settlements at the event scale (Moncoulon et al. 2014), to be compared with the impacts estimated by our model. This kind of validation at the event scale is in progress, but we present here some preliminary results showing the overall consistency between the nature and representativeness of information included in the CCR database and our impact model. Figure 11 presents for instance the correspondence between number of buildings identified in the impacts model and the number of insurance policies of individual houses included in the CCR database in 2014 (limited here to the well geocoded policies). This graph shows a good overall consistency of both information sources witha proportion policies/buildings of about xxx%.

Figure 12 shows the proportion of insurance claims/policies computed on the whole observation period covered by the CCR database (1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006)(2007)(2008)(2009)(2010)(2011)(2012)(2013)(2014), within the flooded areas used for the definition of the impacts model. This figure shows again a good consistency of information, with a decreasing proportion of claims/policies ratios as the return period of the considered flooded area increases. It can also be noted that the ratios are significantly higher within the identified floodplains than outside.

Figure 1 .

 1 Figure 1. Location of the APSFR of Alès territory in the Gardon and Cèze watersheds.

Figure 2 .

 2 Figure 2. River network considered in the APSFR of Alès (5 km² upstream catchment surface), coverage of reference flood maps and position of available streamgauges.

Figure 3 .

 3 Figure 3. Overall principle of the computation of flood maps based on CartinoPC software: a) input information (position of river streams and approximated extent of flood area), b) position of profiles integrated in the 1-D model, c) computation of water levels (1-D hydraulic model), c) map of water heights obtained after post-treatment.

Figure 4 .

 4 Figure 4. Illustration of additional post treatments applied to the Cartino results: a) initial Cartino results, b) after removal of disconnected areas.

Figure 5 .

 5 Figure 5. Illustration of comparison between estimated and reference inundated areas: definition of common surface (Sc), excess surface (Se), and default surface (Sd).

Figure 6 .

 6 Figure 6. Synthesis TSIs scores computed for the 75 river reaches on which reference inundation areas are available: a) overall distributions, b) repartition in function of reference inundation surfaces.

Figure 7 .

 7 Figure 7. Examples of comparison of cross sections and water levels at three stream gauges: a) Mialet station, b) Banne station, c) Alès station.

Figure 8 .

 8 Figure 8. Illustration of the impact of profile widths on the simulated water levels nearby the Alès station: a) position of cross sections for different discharge levels, b) longitudinal profiles (B) and computed water levels (H).

Figure 9 .Figure 10 .

 910 Figure 9. Map of the maximum impacts estimated for each river reach (number of buildings in the 1000-year flood area). a)

Table 1 .

 1 Characteristics of the river network considerered in the case study (extracted from SYRAH database).

		Bed slope (%)	River bed width (m)	Floodplain width (m)
	Location
		average	min -max	average	min -max	average	min -max
	Main network	0.48	0.22 -1	37	16 -84	470	120 -1670
	Secondary network	3.41	0.17 -20	7	2 -34	430	60 -3130

Based on these first results, the challenge is now to use the claim data for the validation at the event scale of the entire simulation chain including both hydrological rainfall runoff model and impacts model.

Conclusions

The case study presented herein aimed to illustrate the potential benefits of a simplified impact model to complement the real time information provided by hydrological models for the monitoring and forecasting of flash floods. The impacts model proposed is based on 1-D hydraulic computations achieved during a preanalysis phase : important assumptions are made, enabling to derive in a limited computation time a catalog of inundation maps for different discharge quantiles.

The evaluation of this catalogue versus reference flood maps shows that the level of errors, even if significant, remains acceptable for the monitoring and forecasting objectives . Moreover, this catalogue leads to a clear hierarchy of the exposure of the considered territory, including interesting information about the discharge levels above which the assets begin to be significantly affected.

Based on these promising results, the challenge is now to combine this impact model with a hydrological model to assess if the information on impacts obtained at the event scale can be valuable for crisis management operations. The validation of these results should be possible based on the insurance claims included in the CCR database.

Here are some examples: Prévision distribuée 1. Versini P-A., Berrenguer M., Corral D. and Sempere-Torres D. ( 2014). An operational flood warning system for poorly gauged basins : demonstration in the Guadalhorce basin (Spain


Giacomo Rossitto
email: giacomo.rossitto@ensma.fr

Christophe Sicot

Vale ´rie Ferrand

Jacques Bore

Fabien Harambat

Aerodynamic performances of rounded fastback vehicle

Keywords: Rounded edges, drag, vorticity, wake analysis, crosswind, spoilers, underbody diffuser

Experimental and numerical analyzes were performed to investigate the aerodynamic performances of a realistic vehicle with a different afterbody rounding. This afterbody rounding resulted in a reduction to drag and lift at a yaw angle of zero, while the crosswind performances were degraded. Rounding the side pillars generated moderate changes to the drag and also caused important lift reductions. A minor effect on the drag force was found to result from the opposite drag effects on the slanted and vertical surfaces. The vorticity distribution in the near wake was also analyzed to understand the flow field modifications due to the afterbody rounding. Crosswind sensitivity was investigated to complete the analysis of the aerodynamic performances of the rounded edges models. Additional tests were conducted with geometry modifications as spoilers and underbody diffusers.

Introduction

For the last decade car manufacturers have been facing the challenging task of reducing fuel consumption and CO 2 emissions. In response to this, optimization algorithms have been applied to generate new vehicle shapes to minimize the aerodynamic drag. Since the obtained optimum shapes have no brand differentiating details, nowadays stylists are trying to give back a brand signature by proposing ''non-conventional shapes. In that framework, the important rounding of the rear pillars becomes a differentiation strategy. The current study suggests to quantify the influence of such afterbody rounding on the flow field and on the drag development over a fastback vehicle.

Very few papers have addressed the question of the curvature of the rear edges in aerodynamic vehicle performances. One of the first works was presented by Gilhaus et al. [START_REF] Gilhaus | Drag and driving-stabilityrelated aerodynamic forces and their interdependenceresults of measurements on 3/8-scale basic car shapes[END_REF] and Howell. [START_REF] Howell | Shape features which influence crosswind sensitivity, vehicle ride and handling conference[END_REF] Thanks to balance measurements, it was found that rounded pillars reduced both drag and rear lift but yawing moment had a pronounced increase. Only recently, were advanced techniques used to achieve better understanding of the afterbody rounding. Thacker et al. [START_REF] Thacker | Effects of suppressing the 3D separation on the rear slant on the flow structures around an Ahmed body[END_REF] showed that rounding the edge between the roof and the rear slant of the Ahmed body results in a 10% drag reduction. Authors attributed this reduction to the fully attached flow over the backlight and the downstream shift of the rotating structures developing in the near wake. Fuller et al. [START_REF] Fuller | The importance of rear pillar geometry on fastback wake structures[END_REF] analyzed the benefits of rounding the rear side pillars on the Davis model. They observed that rounded edges generate a different wake structure dominated by the interaction between the longitudinal vortices and the separated region. The rounded edges model resulted in a drag and lift reduction of 11% and 25% respectively. The impact of afterbody rounding was mentioned also during the development of the Tesla Model S by Palin et al. [START_REF] Palin | The aerodynamic development of the Tesla model S-Part 1: Overview[END_REF] Important curved side pillars were avoided to reduce the highly dynamic wake, which caused large variation in the base pressure.

From the literature review, it appears that rounding afterbodies considerably affects the aerodynamic loads and flow development, but a systematic investigation into the effects of changing the radii of the side backlight edges was not reported. To complement the recent study proposed by the same authors [START_REF] Rossitto | Influence of afterbody rounding on the pressure distribution over a fastback vehicle[END_REF] over a simplified car model (Ahmed body), the present study addresses this question on a realistic car model equipped with various rear pillars curvatures. Special care is taken to understand how the modified flow on the backlight interacts with the near wake and how this promotes drag and lift changes.

Four rear ends were analyzed by combining PIV data to balance surface pressure measurements to fully characterize the flow structures and the associated aerodynamic forces. The zero yaw case was first explored before focusing on the crosswind effects. Complementary numerical simulations were systematically applied to complete the experimental data and to help the physical analysis. Geometry modifications, by means of spoilers and underbody diffusers, were investigated to understand their sensitivity to the side edges rounding.

Experimental and numerical set-up

The model and its relevant dimensions are reported in Figure 1. Four rear ends are tested; they differ by their side pillars curvature. The curvature radius is given as a percentage of the model span, i.e. 300 mm. The model equipped with sharp pillars having 0% radius is referred to as S 0 and it is considered as the reference case. The others models are S 8 , S 20 , and S 40 . All of the rear ends have the same curvature at the end of the roof to avoid flow separation. The corresponding radius is chosen to maximize the room for the rear passengers for a fixed backlight angle of 23 degrees. The model features a realistic non symmetric underbody with an exhaust line. It does not have open front air intake. The horizontal projection of the slanted surface, j = 440 mm, will be used as a reference length. Starred spatial coordinates are normalized by the reference length.

The experimental results reported in this work were obtained from tests conducted in the PSA Groupe wind tunnel of La Ferte´Vidame. The Eiffel wind tunnel is 52 meters long and has a test section which is 2 meters high, 5.2 meters wide and 6 meters long. It has a maximum free stream velocity of 53 m/s. The wind tunnel blockage ratio was 1.4%. The wind tunnel experiments were conducted with a fixed ground and the model is placed over a false floor. All of the data were obtained at 40 m/s which gives a Reynolds number based on the length of the model of 2.6e6. Starred velocities are normalized by the free stream velocity. A six components balance was used to measure the aerodynamic forces acting on the model. The drag and lift coefficients were calculated as follows

C d, l = F d, l 1
2 rU 2 0 S
where F is the force measured by the balance, r the density of the air, and S the frontal surface area of the model. The precision of the balance was 0.001 for the drag coefficient Cd and 0.002 for the lift coefficient Cl.

Static surface pressure was recorded by 40 pressure probes over the vertical surface of the rear end. Thirtyfive probes were located on the driver side and the rest were located on the passenger side to check the symmetry of the static surface pressure over the vertical base. Even thought the non-symmetric underbody should have induced asymmetry in the flow the comparison between the pressure probes, not reported here for brevity, resulted in negligible differences. Those probes were connected to a SCANdaq 8000 acquisition system. The acquisition rate was 40 Hz for 3000 samples giving 75 s of time recording. The static surface pressure coefficient at one point i was computed from the expression

C P (i) = P(i) À P 0 1 2 rU 2 0
where P(i) is the static pressure of point (i), P 0 the static pressure measured upstream of the model. The static accuracy of the system is 613 Pa, i.e. 0.015 Cp. Furthermore, PIV measurements were performed in the wake of the model. The laser sheet was set by a 2*120 mJ Nd:Yag Quantum Big Sky Laser. The Dantec Flowsense 4M mkII camera (2024 pixels by 2024 pixels) was equipped with 105 mm lenses which generated a 462 mm by 462 mm field of view. 2D PIV was performed on the symmetry (x,z) plane Y*=0. For all the PIV measurements, post-processing was performed with a final interrogation window of 16*16 pixels, after an initial window of 32*32 pixels, with an overlap of 50% in horizontal and vertical directions. The grid spacing was 1.82 mm. An SNR filter was applied to remove vectors with SNR \ 1:2. 900 images were recorded with a 7 Hz trigger rate. With this setting, the 95% confidence limit represents approximately 63% of the mean velocity. [START_REF] Benedict | Towards better uncertainty estimates for turbulence statistics[END_REF] The numerical simulations presented in this work were computed with the commercial code PowerFLOW Ó (version 5.0c) based on the lattice Boltzmann model. Applications of the lattice Boltzmann method to vehicles aerodynamics can be found in the literature. [START_REF] Fares | Unsteady flow simulation of the Ahmed reference body using a lattice Boltzmann approach[END_REF][START_REF] Duncan | Numerical simulation and spectral analysis of pressure fluctuations in vehicle aerodynamic noise generation[END_REF] The simulation setting is reported in Table 1. The model size for the numerical simulations corresponds to a real vehicle scale. At Reynolds 1.04e7, 324,000 time-steps were computed (3 seconds of real time). Local variable refinement regions VRs were defined to locally allow the coarsening of the grid by a factor of 2 starting from the Minimum Level of VR. [START_REF] Fares | Unsteady flow simulation of the Ahmed reference body using a lattice Boltzmann approach[END_REF] The minimum value of VR (Table 1) corresponds approximately to 300 wall units. The computational grid consisted of 1.2 million volume elements and 0.6 million surface elements. The computational domain was chosen according to PSA methodology. The model is positioned 17.5L from the domain inlet and the total length of the computational domain is 35L (L is the model length). In the following, temporally averaged velocity and pressure fields will be shown. These averages are computed over approximately 25 convective time scales L=U 0 where U 0 is the external velocity.

This setting allowed for agreements between the numerical and experimental results, as it is possible to examine in Figure 3 which reports a comparison of mean streamwise velocity profiles.

Base geometry flow analysis

This section analyzes the mean properties of the flow field modifications due to afterbody rounding. First, the zero yaw flow field and forces are analyzed. At the second stage, the crosswind performances are investigated.

Table 2 reports the variation of aerodynamic coefficients and the averaged base pressure with respect to the reference case S 0 . Increasing the side radius generates a small drag reduction and a significant lift reduction. For the S 0 configuration, most of the rear lift is generated over the rear end by the low value of the pressure over the curved roof edge and from the low pressure foot-print of the longitudinal vortices, the so-called C-pillar vortices. These vortices are generated by the interaction between the flow coming from the side of the model and the flow over the rear window. Since the roof radius is constant among the four models, lift reductions are caused by the C-pillar vortices intensity modifications due to side rounding.

For a moderate value of C-pillars rounding, the origin of the longitudinal vortices is displaced towards the base of the body along the C-pillars. [START_REF] Rossitto | Influence of afterbody rounding on the pressure distribution over a fastback vehicle[END_REF] For larger rounding, three-dimensional flow separation does not occur anymore on the side of the model and thus the side vortices disappear. [START_REF] Rossitto | Wake structure and drag of vehicles with rounded rear edges[END_REF] The weakening of the C-pillar vortices allows for a pressure recovery over the rear window reducing its contribution to drag and lift. However, the variation of the average base pressure over the vertical surface, reported in Figure 2 and in Table 2, gives evidence that the base pressure is reduced when the curvature side radius is increased. For S 40 , the base pressure is 17% lower than for S 0 , resulting in an important local drag increase. This effect counter-balances the local drag reduction of the rear window and can explain the weak effect of the pillars rounding on the global drag (Table 2).

The mean velocity fields in the symmetry plane are reported in Figure 4. It was chosen not to report the intermediate configurations, i.e. S 8 and S 20 , to focus on the main differences between the most rounded rear pillar model S 40 and the sharpest one S 0 . The symmetry plane is far from sufficient to describe the entire topology of the tridimensional near wake, nevertheless it is very useful to get a first hint of the changes made by side rounding. Over the rear window, no major differences are noticed between the two models. The flow is attached to the slanted surface due to the important roof radius for both models. For S 0 , the average separated zone in the close wake is characterized by two counter-rotating structures. The mean recirculating length, identified by the saddle point of the 2D mean streamlines, is at X*= 0.33 and X*= 0.23 respectively for S 0 and S 40 . Adding pillar curvature reduces the length of the near wake and modifies the topology of the enclosed structures. For S 40 , the mean curvature of the near wake is increased due to the negative normal velocity. Regarding the flow organization inside the near wake, the upper structure is not evidenced anymore and the lower structure seems to be reduced. Such a strong modification of the near wake properties in the symmetry plane, may result from a 3D modification of the flow field due to side rounding.

To understand this complex mechanism, cross-flow planes are displayed in Figure 5. The data are issued from the numerical simulations. Each picture is divided in two: the left side represents S 0 and the right side S 40 . At X*= 0.02, right after the rear slant, the longitudinal vorticity (O x à = O x Ãj U 0 ) is concentrated in the C-pillar vortex for S 0 (left part of top of Figure 5). At the same X plane, a diffuse horizontal small area of longitudinal vorticity is present for S 40 located in the shear layer (right part of top of Figure 5). Two-dimensional lines tangent each point to the vector (V,W) are also drawn in Figure 5 to underline the different organization of flow fields due to the afterbody rounding. At X*= 0.23, i.e. the S 40 saddle point location, the overall longitudinal rotation of the S 40 near wake is showed in the bottom of Figure 5. The high spanwise and vertical velocity associated to the transversal flow in S 40 , therefore, results in a global longitudinal rotation of the flow in the near wake. To quantify the rotation of the flow, the longitudinal evolution of the mean circulation Gà = Ð Sà O à x dS was analyzed as in the work of Rossitto et al. [START_REF] Rossitto | Influence of afterbody rounding on the pressure distribution over a fastback vehicle[END_REF] The domain of the calculation included all of the numerical domain presented in Figure 5. Despite the high levels of vorticity of the C-pillar vortices for S 0 , the overall circulation is higher for S 40 . At X*= 0.23, Gà since S 40 is 30% higher than S 0 . The flow overcoming the rounded edge results in a notable 3D organization characterized by a large scale rotation around the longitudinal axis. The longitudinal rotation, as reported in Figure 6, generates a lower pressure for S 40 with respect to S 0 , which is consistent with the surface pressure measurements analyzed previously in Figure 2 relating to the local base drag increase.

To help to visualize the tridimensional flow modifications due to afterbody rounding, streamlines from numerical simulations are reported in Figure 7. Consistently with the cross-flow visualizations, it can be noticed the presence of the C-pillar vortices for the S 0 thanks to the low pressure foot-print of the vortices. The sharp side pillars force the lateral flow to separate, generating longitudinal vortices. As analyzed by the same authors, [START_REF] Rossitto | Influence of afterbody rounding on the pressure distribution over a fastback vehicle[END_REF] it is suggested that for S 0 , the sharp lateral edges ''shield the wake separation region at the base and enable a ''two-dimensional separation, ''2D meaning that the mean streamlines at separation are more parallel to the orientation of the flow over the window, while longitudinal vorticity is concentrated in the C-pillar vortices. On the contrary, for rounded edges the flow is drawn in from the sides toward the center over the slanted surface. The separation at the base is then notably 3D for S 40 , associated with high spanwise and vertical velocity components.

Crosswind

To gain further insight on the influence of afterbody rounding on aerodynamic performances, crosswind tests were performed. A sketch of the crosswind aerodynamic characteristic is drawn in Figure 8. In the present experiments, the yaw angle b varied from À308 to 308 and the wind tunnel speed was 40 m/s. All of the aerodynamic coefficients are scaled by the same frontal surface area as is usually done in the study of crosswind characteristic. [START_REF] Hucho | Aerodynamics of road vehicles[END_REF] Since the model is symmetric, only the results in [À308; 08] are reported. The choice of such large range of angles is motivated in what follows. Due to meteorological phenomena, small angles of crosswind have a non negligible probability of exceedance. As reported by Palin et al., [START_REF] D'hooge | Alternative simulation methods for assessing aerodynamic drag in realistic crosswind[END_REF] the majority of possible yaw angles are within the band of zero to 10 degrees, which covers more than 99% of the probability distribution. The study of this interval permits then to complete the aerodynamic performance analysis of the model. Angles in [À308; À208] are tested mostly for safety reasons. Even though the natural probability of exceedance of such angles is extremely rare, they can occur due to driving conditions such as overtaking [START_REF] Noger | Investigation of the transient aerodynamic phenomena associated with passing maneuvers[END_REF] or extreme weather conditions. [START_REF] Baker | Wind-induced accidents of road vehicles[END_REF] The yawing moment is generated by the front and rear contribution of the side forces. It is computed by using the middle of the wheelbase and the truck as a reference. Since the modification of the side pillars radii is in the model rear end the analysis will be focused on the generation of the rear forces, i.e. Cy rear and Cl rear . The overall drag will be discussed as well. Figure 9 reports experimental aerodynamic coefficients in [À30 8; 08] yaw angles. The starred coefficients are normalized by the value of S 0 at À308. As a general trend, drag experiences almost a 30% increase in [À158; 08]. At À5 8 and À108 yaw angle S 40 drag is respectively 3.5% and 4.5% lower compared to S 0 . The drag increase due to yaw angle increasing is more important for the sharp side pillar. Moreover while drag is constant in [À308; À208] for the rounded models, it continues to increase for S 0 .

Since yaw angles do not have the same probability of occurrence, many authors have performed their performance analysis in terms of wind average drag coefficient as Buckley et al. [START_REF] Buckley | Comparisons of effectiveness of commercially available devices for the reduction of aerodynamic drag on tractor-trailers[END_REF] and Cooper [START_REF] Cooper | Truck aerodynamics reborn: Lessons from the past[END_REF] . Taking into account the yaw angles probability distribution proposed, [START_REF] D'hooge | Alternative simulation methods for assessing aerodynamic drag in realistic crosswind[END_REF] , S 40 averaged drag in [À10 8; 08] which was 3% lower than S 0 .

CY front , not shown here, does not have important differences among the four rear ends. On the contrary, CY rear experiences significant changes. S 40 side rear force is 20% and 30% lower than S 0 respectively at À108 and À308. It is important to notice that CY rear acts as a counter-rotating force to the yawing moment (Figure 8). This means that, reducing the side rear force, induces the increase of the yawing moment. The value of the yawing moment CM Z , grows with an increasing C-pillar radius. At À308, S 40 yawing moment is 21% higher than S 0 . It is interesting to notice that S 8 , featuring a very small side radius, resulted in a 12% CM Z increase when compared to S 0 . It results then that the yawing moment has a pronounced increase as the side edges feature a small radius. [START_REF] Howell | Shape features which influence crosswind sensitivity, vehicle ride and handling conference[END_REF] Howell et al. [START_REF] Howell | A relationship between lift and lateral aerodynamic characteristics for passenger cars[END_REF] analyzed the relationship between lift and lateral aerodynamics. According to them, when the rear lift is reduced, there is often a reduction in yawing moment. However, the use of rounded edges does not follow this pattern since the model with the lowest lift at zero yaw angle had the highest yawing moment. The rear lift in [À108; 08] is similar to a 08 yaw angle: the S 40 model has lower rear lift compared to S 0 due to the C-pillar vortices and their induced flow elimination over the rear window (-27% at 0 8 yaw angle). Interestingly, an inversion of the behavior appears for yaw angles bigger than À158. The rounded models experience a sudden growth of rear lift ( + 47% at À308 yaw angle) due to the flow acceleration around the rounded rear pillar, creating a strong suction on the windward side of the model. [START_REF] Cairns | Lateral aerodynamic characteristics of motor vehicles in transient crosswinds[END_REF] On the contrary, the windward side sharp edge of S 0 , promotes flow separation over the rear window, resulting in moderate growths of rear lift when compared to the rounded model. The differences in the yawing moment can be explained by the different pressure distributions over the model, reported in Figure 10.

The top view of the numerical pressure distributions at 0 yaw angle are reported to show the rear pressure characteristics. The C-pillar vortices pressure foot-prints is clearly remarked over the side sharp pillar for S 0 (Figure 10(a)). At b = À 308, C-pillar vortices disappear due to the flow separation imposed by the windward sharp edge. Strong acceleration over the windward rounded edge is visible for S 40 thanks to the important pressure decrease over the edge (Figure 10(d)).

Side views are reported in Figures 10(e) to 10(h) to complete the pressure distribution analysis. In fact, not only the windward side, i.e. the suction over rounded Cpillar, contributes to the overall yawing moment. It is interesting to notice that, on the leeward side, the reduced pressure foot-print of the vortex generated at the forebody and the pressure recovery over the rounded C-pillar contribute to the yawing moment increase.

Figure 11 reports a horizontal pressure profile over the vehicle as in the work of Howell. [START_REF] Howell | The side load distribution on a Rover 800 saloon car under crosswind conditions[END_REF] Those plots quantify the effects of the rear slant pressure to the yawing moment. Values outside the solid black line are negative. For S 0 , the contribution to the yawing moment derives mainly from the leeward front part. Moreover, this contribution is constant for both models. For S 40 , the windward rear side low pressure is visible due to the rear peak outside the solid black line. The differences between the two models plots underline the origin of the increased yawing moment for S 40 .

Geometry modifications

Spoilers

Rear spoilers are undoubtedly efficient devices used to increase vehicles downforce in racecars. [START_REF] Katz | Aerodynamics of race cars[END_REF] Nowadays, small spoilers are used in commercial vehicles to modify In this test, three different spoilers at the rear slant end were used. The size of the spoilers, their placement, and their shape are chosen by the PSA style department. In fact, small devices that extend the rear slant line are an interesting solution for both style and aerodynamic performances. The position of the spoilers and their numbering is reported in Figure 12.

Regarding the flow modifications generated by the use of such devices a pressure recovery is expected upstream of the spoiler. It results in a rear slant drag reduction. This effect, coupled with the high pressure over the spoiler itself, contributes to the overall drag and lift reductions. The spoiler also allows for base drag reduction due to modification of the near wake close to the vehicle base. If the mechanisms associated with rear spoilers are well known and understood, [START_REF] Fukuda | Improvement of vehicle aerodynamics by wake control[END_REF] the interactions with the flow modifications generated by rounded side pillars have not been investigated yet.

The normalized aerodynamic coefficients relative to the different rear spoiler configurations are reported in Figure 13. They are normalized by the values of the model without a spoiler, referred to as NoSpoiler. Spoiler1 generates the same drag reduction for both S 0 and S 40 , i.e. 3.5% with respect to NoSpoiler. Increasing the size of the spoiler does not further reduce drag for S 0 . On the contrary, S 40 with spoiler3, experiences a drag reduction of up to 4.5% when compared to S 40 without a spoiler. Regarding the relative comparison between the two models with spoiler3, drag is 3% lower for S 40 when compared to S 0 . As previously explained, rounding the edges resulted in a local increase of base drag. Furthermore, in the 2D analysis of the flow field, Figure 4, the shortening of the near wake, imposed by the side flow downwash, was related to the base drag increase. In terms of drag, the use of spoilers placed at the rear slant end, is then more effective when the near wake length is not optimal as for S 40 .

Regarding the lift, spoiler1 strongly impacts S 0 more so than S 40 . In fact, spoiler1 generates a 30% lift reduction for S 0 and only a 20% reduction for S 40 . The S 0 model with spoiler3, experiences a lift reduction of up to 40% when compared to the same model without a spoiler. Despite this important reduction, the overall lift coefficient for S 40 is still 15% lower than that of S 0 . As a matter of fact, the lift without a spoiler for S 40 is 27% lower than S 0 as reported in Table 2. The higher lift reduction for S 0 can be related to the presence of the C-pillar vortices due to the sharp side edges. The spoilers, promoting a pressure recovery over the rear window, directly impacts the strength of those vortices, resulting in an important lift reduction.

As a general trend, spoilers are able to reduce the negative local effects generated by the side pillar edges. They are then more efficient in terms of drag reduction for models with rounded rear edges. Regarding the lift, spoilers are suitable passive devices to improve lift reduction for sharp rear edge models.

Diffuser

An interbody diffuser is commonly used in racing cars to develop large downforce. Recently, diffusers have become a good strategy for drag and lift reduction even for passenger cars. [START_REF] Marklund | Performance of an automotive under-body diffuser applied to a sedan and a wagon vehicle[END_REF] Three different mechanisms contribute to produce high levels of downforce and drag reduction in a passenger car. The first mechanism is due to the ground interaction. By placing the body in ground proximity, the acceleration of the flow underneath the body will reduce the underbody pressure resulting in a downforce (i.e. negative lift). [START_REF] Barlow | Experimental parametric study of rectangular bodies with radiused edges in ground effect[END_REF] The second mechanism generating downforce is related to the upsweep of the rear underbody. The angle of the diffuser will camber the flow resulting in a local pressure reduction around the diffuser edge. The last principle, is commonly referred to as ''diffuser pumping''. As the ratio of the inlet to outlet area increases, pressure recovery is performed in the diffuser area. Given that the near wake pressure remains constant, the pressure upstream to the diffuser decreases and the underbody flow rate increases. These effects result in a further decrease in underbody pressure, which produces the ''pumping down''. [START_REF] Cooper | The aerodynamic performance of automotive underbody diffusers[END_REF] The diffuser used in these experimental tests was produced by means of 3D printer. A flat underbody was used here instead of the realistic underbody used for the previous tests. The underbody featured already a diffuser angle of Y=15 degrees. The schematics of the underbody diffuser are drawn in Figure 14. The underbody used was made from a thin cambered plate reproducing the original diffuser shape. Its angle variation had several limitations. Since the original diffuser was not flat but featured a convex shape, reducing the angle Y increased the frontal surface of the model. For that reason, the minimum angle tested was 7 degrees which results in a frontal area variation of approximately 2%. All of the aerodynamic coefficients are rescaled to take into account the area variation. Moreover the maximum angle was limited to 12 degrees due to the hinge limitation of the diffuser. The diffuser was fitted with side plates as in the works of Cooper et al. [START_REF] Cooper | Selecting automotive diffusers to maximise underbody downforce[END_REF] to prevent the flow from overcoming the side of the diffuser and to provide fixation for the diffuser sides.

The starred aerodynamic coefficients are normalized by the value at 7 degrees for each model. Drag evolution for the diffuser's angles variation is reported in the top of Figure 15. The diffuser's drag reduction is the same for both models.

Increasing the diffuser angle, from 7 to 12 degrees, results in drag reduction up to 10% for both models. At 12 degrees, S 40 had a drag 1.5% lower than S 0 . This is consistent with the base geometry trends reported in Table 2. The averaged base pressure, reported in the center of Figure 15, increases as well for larger diffuser angles, in accord with the drag reduction. It is quite striking to notice how the plots are ''parallel''. The gap between the lines seems to be imposed by the side radius and remain constant for all diffuser angles.

The bottom of Figure 15 reports the lift evolution. As expected by the use of the diffuser, increasing its angle results in downforce, i.e. lift reduction. The lift for both models is canceled out. The lift reduction is the same for both models from 7 to 9 degrees. For diffuser angles greater than 9 degrees the lift reduction is more effective for the rounded model. For S 40 , the upsweep of the diffuser should make the wake more symmetrical with respect to the flow structures organization reported in Figure 4. For S 0 , since the reference wake was already symmetric, it is suggested that a high diffuser angle tends to break the symmetry for S 0 resulting in a wake directed upwards. It could explain that high diffuser angles for S 0 has a moderate lift reduction with respect to S 40 . [START_REF] Marklund | Performance of an automotive under-body diffuser applied to a sedan and a wagon vehicle[END_REF]

Conclusions

Experimental and numerical analyzes were used to study the influence of afterbody rounding on the aerodynamic characteristics of a fastback vehicle model. At zero crosswind, rounded edges resulted having a minimal impact on drag due to opposite effects over the rear slant and the base. Important lift reductions were achieved thanks to the elimination of the C-pillar vortices. The impact of rounded edges on the near wake organization due to the different longitudinal vorticity distribution was highlighted. Unfortunately, rounded edges resulted in a detriment of the lateral stability, generating a severe increase of yawing moment and rear lift. Geometry modifications were also tested to study their efficiency variation due to side rounding. Spoilers resulted to recover the negative base drag effects of side rounding. The lift reduction was much more effective for a sharp rear edge. Regarding the use of diffuser, the overall drag modifications due to diffuser angles variations were not sensitive to afterbody rounding. Lift force is found to be more sensitive to high diffuser angles for the rounded edge model. This analysis helps for future vehicles efficient design giving the overall aerodynamic behavior of afterbody rounding. Innovative design should take into account the right balance between sharp and rounded edge since they have different performances with respect to yaw angles. Additional tests should be performed to study the effects of a non-constant side radius to find an optimal compromise in terms of drag, lift, and lateral stability.

Figure 1 .

 1 Figure 1. Fastback vehicle model and location of the rounded edges.

Figure 2 .

 2 Figure 2. Experimental surface pressure coefficient distribution over the vehicle base.

Figure 3 .

 3 Figure 3. Comparison between numerical and experimental results. Mean streamwise velocity profiles. 8 Experimental, -Numerical.

Figure 4 .

 4 Figure 4. Time averaged streamlines and mean streamwise velocity U* at Y*=0.

Figure 5 .

 5 Figure 5. Mean Streamwise Vorticity OÃ x . Top X*= 0.02, bottom X*= 0.23. Left S 0 , Right S 40 .

Figure 6 .

 6 Figure 6. Mean pressure coefficient Cp at X*= 0.23. Left S 0 , Right S 40 .

Figure 7 .

 7 Figure 7. Time averaged streamlines from numerical simulations. The streamlines and the surface of the body are colored by the pressure coefficient C p . (a) S 0 . (b) S 40 .

Figure 8 .

 8 Figure 8. Sketch of the crosswind aerodynamic characteristics.

Figure 9 .

 9 Figure 9. Normalized aerodynamic coefficients. From top to bottom: Cd à , CY à rear , CM à Z , Cl à rear .

Figure 10 .

 10 Figure 10. Surface pressure and streamlines visualizations from numerical simulations. (a) S 0 configuration at 0 8 yaw. (b) S 0 configuration at À30 8 yaw. (c) S 40 configuration at 0 8 yaw. (d) S 40 configuration at À30 8 yaw. (e) S 0 configuration at À30 8 yaw, windward side. (f) S 0 configuration at À30 8 yaw, leeward side. (g) S 40 configuration at À30 8 yaw, windward side. (h) S 40 configuration at À30 8 yaw, leeward side.

Figure 11 .

 11 Figure 11. Pressure distribution at Z* = 0.43: Top S 0 , Bottom S 40 . Values outside the solid black line correspond to negative pressure coefficient.

Figure 12 .

 12 Figure 12. Spoilers location over the model. Spoiler2 includes spoiler1 and spoiler3 includes spoiler2.

Figure 13 .

 13 Figure 13. Normalized aerodynamic coefficients for the different rear spoiler configurations. Top: drag coefficient, bottom: lift coefficient.

Figure 14 .

 14 Figure 14. Schematics of the underbody diffuser. The side view in the top shows the maximum and minimum diffuser angles tested.

Figure 15 .

 15 Figure 15. Normalized aerodynamic coefficients for underbody diffuser angles variation. Top: drag coefficient, center: averaged base pressure, bottom: lift

Table 1 .

 1 Numerical PowerFLOW Ó setting.

	Physical Time Scaling	1 timestep = 9:308310 À6 s
	Minimum Level of VR	VR 9 (2:50310 À3 m/voxel)
	Simulation time	324,000 timesteps (3 s)
	Reynolds	1:04402310 7

Table 2 .

 2 Variation of aerodynamic coefficients and averaged base pressure relative to S 0 .

		DCd	DCl	DCp base
	S 0	-	-	-
	S 8	0%	213%	212%
	S 20	21.5%	218%	215%
	S 40	22%	227%	217%

Acknowledgements

The authors are thankful to B. Tixier from the Exa company for performing the numerical simulations and to Y. Goraguer, the wind tunnel manager, for his time and efforts during the experimental tests. We acknowledge the PSA designer Yong Wook Sin for the fruitful collaboration. This work has been performed in the framework of the ''OpenLab Fluidics @Poitiers.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The present work results from a CIFRE agreement between the E ´cole Nationale Supe´rieure de Me´canique et d'Ae´rotechnique (ENSMA) and the research and innovation department of fluid mechanics of PSA Gropue.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.