Advances in Geosciences, 2, 279–284, 2005 SRef-ID: 1680-7359/adgeo/2005-2-279 European Geosciences Union © 2005 Author(s). This work is licensed under a Creative Commons License.
Advances in Geosciences
New urban area flood model: a comparison with MIKE11-quasi2d A. Sole1 and G. Zuccaro2 1 Environmental 2 Hydrodata
Engineering and Physics Department, Univ. of Basilicata, via dell’Ateneo Lucano 10, 85100 Potenza, Italy S.p.a., via Pomba 23, 10123 Torino, Italy
Received: 11 November 2004 – Revised: 19 June 2005 – Accepted: 27 June 2005 – Published: 9 August 2005
Abstract. Recent hydrogeological events have increased both public interest and that of the Scientific Community in a more accurate study of flooding in urban areas. The present project proposes a new model which offers an optimal integration of two models, one for flood wave propagation in riverbeds and the other for flooding in urban areas. We consider it necessary to not only treat the modelling of the outflow in riverbeds and outside riverbeds.together but to integrate them thoroughly. We simulate the propagation in riverbed of the flood event with a model solving the equations of De Saint Venant with the explicit scheme at the finite differences by McCormack. The propagation outside the riverbed is simulated using an algorithm proposed by Braschi et al. (1990). This algorithm is based on a local discretization of the urban territory, divided in a series of “tanks” and “channels”. Each tank is associated with an area of an extension related to the position of the other tanks and the quantity of buildings, modelled as insurmountable obstacles. The model facilitates the simultaneous performance of the two simulations: at each instant, the quantitiy of water overflow, depending on the piezometric level in every section, is calculated as a function of the dimensions of the weirs (the banks), assuming it passes through the critical state. Then, it is transferred to the tanks placed in the surroundings of the overflow points. Those points are the starting nodes for the propagation of the flood because they are connected to the network of tanks in which the surrounding land has been schematised. In this paper, we present a comparison of one of the most powerful models of inundation simulation in urban and nourban areas. The field area is the city of Albenga (SV, Italy) and the simulated event is the inundation of the 1994 (return period of about 25 years).
Correspondence to: A. Sole (
[email protected])
1
Introduction
Recent climatatic events, which have caused crises in rivercrossed areas, have renewed interest in hydrogeological risks and methods of flood event prevention. The complex morphology of stressed urban areas makes it difficult to apprehend the real degree of vulnerability. Hence, it is necessary to create a model in which the in-bed propagation and the out-bed inundation are interrelated in a GIS environment. In the present paper, we propose a model that integrates the phenomenon of the in-bed propagation and the dynamic of the out-bed inundation and takes into account the diminishing of the discharge downward from an esundation point and the eventual re-entries at the end of the flood. Thus, it explains the mutual interaction between the water-course and the surrounding urban territory and it simulates the inundation event in a reasonable time. 2
Model description
The proposed model (Sole and Zuccaro, 2001 , 2002, 2003) is composed of several modules. They interact through an intense data exchange, until the graphical representation of the progressive advance of the flood either into the water-bed or outside is completed. These modules are: 1. the GIS module (input/output); 2. the in-bed flow module; 3. the out-bed inundation module. The first one acquires all the information about the urban territory (such as DTM, position of squares and streets, etc.. . . ) from files obtained in a GIS environment and it is able to represent the results. The second one simulates the flood-flow into the river-bed using a numerical finite difference scheme plus a TVD scheme for the attenuation of the numerical fluctuations. The last module simulates the propagation of the flood volumes using the network schematisation.
To simulate the flood propagation inside the potentially affected area, we set a series of representative points of the storing capacity of the open spaces. These A. Sole and G. Zuccaro: A comparison with MIKE11-quasi2d points, as shown in figure 2, are connected both to each other and to the main river by several rectangular channels describing the road network.
280
Figure 1. Urban territory scheme Figure 2. Tanks and channels Fig. 1. Urban territory scheme. anks represent the storage capacity of a portion of urban territory. The We compute the dimension of the relevant area, for each node, with the method Fig. 2. Tanks and channels. of Thiessen (par. 2), of thereafter we subtract from it the block of the buildings to ant area of each tank is automatically calculated thanks to the method obtain the clean area that represents the store capacity (fig. 3). sen polygons. 2.1 Functioning model scheme
where “msd” is the longitudinal variation of the static momentum, “i” is the bed slope and “j ” is the friction slope. n-bed flowThe module proposed model has a graphical interface able to receive The model computes, for each time step, the values for the input data necessary for the hydrodynamic simulation, in the discharge and the wetted area for each cross section, as n-bed deflow module solves the equations of territory De Saint Venant with the particular: the water-bed geometry, the urban geomfunctions of the values assumed in the previous time step in etry, the hydrogram, factors (both in-bed and outcit finite difference schemetheoffriction McCormack plus a TVD scheme the [6]. sameThe section and in the two adjacent sections. When bed) and the and time integration ions are expressed inspatial the following vectorialinterval. form: These data adopting the method of McCormack (Garc`ıa-Navarro et al., are stored in a database and can be modified and refreshed ∂F by an external program. Using the GIS module, the urban 1992a), in order to calculate the new vector “U ”, it is neces+ = G (1) sary to go through the “predictor” and “corrector” steps. ∂x territory has been discretized in a series of tanks (generally The user, thanks to a button on the graphical interface, can the centre of the squares) linked by channels (Fig. 1). add a TVD scheme to the scheme of McCormack (Garc`ıae “U” is thein vector of the unknown variables: The tanks represent the storage capacity of a portion of uret al., 1992a; Garc`ıa-Navarro et al., 1992b; Hirsch, Ω ban territory. The relevant area of each tank is automatically Navarro 1990; Macchione and Morelli, 1996), to reduce the numeri (2) calculated thanks to the method of Thiessen polygons. cal oscillations. Q The discharge length that outflows Figurefor 3. unit Evaluation of clean form the river 2.2 wetted In-bedarea flow module e “Ω” is the and “Q” is the discharge; “F” is the vector of the is obtained from Eq. (5). areas This quantity has been computed ”: whenever the piezometric level is higher than the two bank The in-bed deflow module solves the equations of De The Sainthydrogram (fig. 4), rebounded by CIMA, corresponds to a flood event with levels:of about 25 years, a duration of 20 hours and a peak discharge of a return period QVenant with the explicit finite difference scheme of McCor q q 2 (3)a TVD scheme (Garc`ıa-Navarro et al., 1992a). 1600 m³/s. In the fluvial sections, also supplied by the CIMA, there is a bridge mack plus 3 q = akqnarrowing · 2 · g · (h −the zlxsection · 2there · g · (h − zrx (5)of hydraulic ) + kq but )3 which causes of is no evidence Q + gThe equations are expressed in the following vectorial form: ⋅ ms work. where “kq ” is the discharge coefficient (equal to 0.385), “h” Ω ∂F ∂U is the “G” piezometric level and “zlx ” and “zrx ” are the bank lev+ = G of gravity and “ms” is the static momentum; (1) e “g” is the∂tacceleration is ∂x els of the left and right respectively. source” vector: The first upstream boundary condition is due to the hydrowhere “U ” is the vector of the unknown variables: 0 gram and the other is calculated in one of the sequent meth (4) ods: U= (2) g ⋅ msd + ΩQ⋅ i − j – in case of local subcritical flow, using the method of e “msd” iswhere the longitudinal variation theisstatic momentum, bed “” is the wetted area andof“Q” the discharge; “F ” “i” is the characteristics and, in particular, the “negative” characand “j” is isthe friction slope. the vector of the “flux”: teristic; model computes, for each !time step, the values for the discharge and the Q section, as functions of the values assumed – inincase d area forFeach theof local supercritical flow, using a Q-h relation. = Qcross (3) 2 g · ms section and in the two adjacent sections. When ous time step in the+ same At the downstream end of the river, the first condition is deing the method of McCormack [6], in order to calculate the new vector termined using the method of the characteristics and, in parwhere “g” is the acceleration of gravity and “ms” is the static ticular, the “positive” curve that proceeds in the same direcit is necessary to go through “predictor” momentum; “G” is thethe “source” vector: and “corrector” steps. tion of the flow, but the second one is determined, like the first cross section upstream, in function of the local regime 0 G= (4) of the current. g · [msd + · (i − j )]
[
(
)]
Figure 2. Tanks and channels Figure 4. Hydrogram of the We compute the dimension of the relevant area, for each node, with the method studied event A. Sole and G. Zuccaro: A comparison with MIKE11-quasi2d 281 of Thiessen (par. 2), thereafter we subtract from it the block of the buildingsWe toreport the following results of the simulations: the extension of the inundated area (fig. 5a), the water depth for each time step and for each node and the obtain the clean area that represents the store capacity (fig. 3). maximum water depth for each node (fig. 5b).
(a)
(b) Inundated area extension and maximum water depths Fig. 5. Inundated area extension and maximum water depths. Figure 6 represents the inundation map, obtained by interpolation of the water depth in each node, and indicates the maximum values of the water depth during the whole simulation.
Figure 3.
Figure 5.
Evaluation of clean
Fig. 3. Evaluation of clean areas. areas
The hydrogram (fig. 4), rebounded by CIMA, corresponds to a flood event with a return period of about 25 years, a duration of 20 hours and a peak discharge of 1600 m³/s. In the fluvial sections, also supplied by the CIMA, there is a bridge which causes a narrowing of the section but there is no evidence of hydraulic work.
Figure 6.
Inundation map and maximum water depths Fig. 6. Inundation map and maximum water depths. The biggest inundations occur in the central section of the water course under examination , coinciding with the position of the bridge where the bed narrows and the banks are relatively lower. Other 3 inundations Model testtake place in the final part of the water course due to significantly reduced bank levels,particularly on the right side, in the mouth area the inundated .
Figure 4.
Hydrogram of the studied event e report the following results of the simulations: the extension of ea (fig. 5a), the water depth for each time step and for each nodeInand the the absence of observed data, the test phase was carried 2.3 Out-bed inundation module aximum water depth for each node (fig. 5b). out t by a comparison with a commercial model: the MIKE Fig. 4. Hydrogram of the studied event.
The out-bed inundation module simulates the flow of the flood volumes outflowing from the river-bed using an algorithm due to Braschi et al. (1990) and developed by the CIMA - Interuniversitary Environmental Monitoring Research Centre (Boni et al., 2002; Braschi et al., 1990; Crosta et al., 2001). This module verifies the existence for each node with a no-null water depth, of one or more connected nodes with a minor piezometric level. If such nodes exist, it calculates the exchange discharge with the following equation: s Q12 = χ · B ·
(z1 + y1 ) − (z2 + y2 ) · L12
y1 + y2 2
3 (6)
b) Figure 5. Inundated area “B” is the channel where “χ” is the Chezy friction factor, andwater maximum width and “y” and “z” extension represent the depth and the water depths height on the sea of the generic nodes 1 and 2. gure 6 represents theand inundation map, obtained by interpolation The outin-going volumes are functions of all the expth in eachchange node, and indicates the maximum values of the water discharges. Analogously, for all the time steps, the module is able to calculate the water depths for each node e whole simulation. and to delimit the inundated areas.
11 model (Danish Hydraulic Institute Water & Environment) (DHI, year?). The considered event occurred, in 1994, in the city of Albenga (SV, Italy), on the Centa river, which has been known since the Roman age for its tendency to overflow. In this case we know the inundated area extension, but not the maximum water depth in some points of the urban territory. To simulate the flood propagation inside the potentially affected area, we set a series of representative points of the storing capacity of the open spaces. These points, as shown in Fig. 2, are connected each other Figure 7.both to Inundation mapand and to the main river by several rectangular channels describing the road network. maximum water depths Figure We 7 is compute a map of the historically subject to inundation.A theareas dimension of the relevant area, for comparison each with Fig. 5 and 6 reveals that the model has produced an overestimation of the node, with the method of Thiessen (par. 2), thereafter we extension of the flooded area which, however, refers to an event withsuba 20 year tract fromand it the of to thethebuildings to obtain the clean area return period, thusblock inferior event under consideration. In the absence of other data havecapacity utilized the MIKE that historical represents thewestore (Fig. 3). model 11 together with the The hydrogram (Fig. 4), rebounded by CIMA, corresponds to a flood event with a return period of about 25 years, of the water of 20 h and a peak discharge of 1600 m3 /s. In the a duration depth during fluvial sections, also supplied by the CIMA, there is a bridge which causes a narrowing of the section but there is no evidence of hydraulic work.
ination , coinciding with the position of the bridge where the bed narrows he banks are relatively lower. Figure 8. Inundation map and inundations take place in the final part of the water course due to water 282 A. Sole and G. Zuccaro: A maximum comparison withdepths MIKE11-quasi2d icantly MIKE reduced bank right of side, in the mouth area 11 GIS as alevels,particularly “model of reference” foron thethe validation the proposed according to MIKE 11 model..This model facilitates calculations of flood propagation in urban areas in quasi-bidimensional modality. The same idrogramma di ingresso was used for the test as that utilized in Fig 4. It represents a flood event with a 25 year return period. . The connections between the water-bed and the road network are modelled as “link channels”, more precisely as a series of short and wide-width channels with this Q-h relation:
MIKE 11 GIS model
Q = C q ⋅ Y ⋅ 2 ⋅ g ⋅ Y (8) L where “Q” is the discharge, “L” is the channel width, “Cq” is the discharge coefficient, “g” is the acceleration of gravity and “Y” is the water depth. Figure 8 shows the maximum water depths according to the MIKE 11 simulation and after the interpolation of the piezometric levels made by MIKE 11 GIS. Sono state osservate Differences in results were observed between the in-bed module of the proposed model and MIKE 11 as well as between the out -bed module and the coupled MIKE 11 – MIKE 11 GIS model. In the first case Figure 9. Absolute and percent maximum water depth differences of 8% were observed in the main inundation disagreement between area. In the second case, the differences between the two series of water depths,Fig. 9. Absolute and percent disagreement between the two models. the two models depicted in figure 9, are ordered in increasing size and are related to the Asgreater highlighted in the planimetric representation (fig. 10b), in right-hand areas corresponding percent differences. It can be noted that, near the the small. percent disagreements are about 15%, while at the west boundary of the disagreements (in absolute value), the percent differences are quite model has produced an overestimation of the extension of interested area the percent and absolute differences are quite relevant. The Figures 10a and 10b show this kind of disagreement in a planimetric manner. theis flooded which, however, refers to an event with a 20 water It is possible to observe an exceed variation on the left-side of the water-course difference due to aarea negative variation in the calculation of the maximum Figure 7. Inundation map and year return period, and thus inferior to the event under conand a negative variation on the right. The positive disagreement is certainly depth due into the river near the same area. By simply looking the left-side area, it is to the differences of the maximum piezometric level inside the water-sideration. In the the percent absencedisagreements of other historical data only we have Fig. 7. Urban(positive) areas historically subject to inundation. maximum water depths possible to observe that are relevant for a limited bed. The negative ones, instead, are negligible with respect to the waternumber depth inof utilized the MIKE model 11 together with the MIKE 11on GIS points beside the Centa river. The percent differences the whole aright map of the areas historically subject to inundation.A comparison areas: they are about 30 cm, but the water depth is about 2 m.
e 7 is as a “model of reference” the induces validation of thedifferences proposed into river, though quite small (less thanfor 8%), greater Fig. 5 and 6 reveals that the model has produced an overestimation of the urban areas. model. This model facilitates calculations of flood propagasion of the flooded area which, however, refers to an event with yearareas in quasi-bidimensional modality. tiona in20 urban period, and thus inferior to the event under consideration. In the absence The same input hydrograph was used for the test as that her historical data we have utilized the MIKE model 11 together with the 4. utilized in Fig.
the
The connections between the water-bed and the road network are modelled as “link channels”, more precisely as a series of short and wide-width channels with this Q-h relation: a)
p Q = Cq · Y · 2 · g · Y L
(7)
b) Figure 10. Absolute and percent where “Q” is the discharge, “L” is thebetween channel width, “Cq ” disagreement is the discharge coefficient, “g” is the acceleration of gravity the two models and “Y ” is the water depth. Figure 11 shows a part of the digital urban terrain model and the inundation Fig. 8. Inundation map and maximum water depths according to simulated by the two model (MIKE 11 in cyan, proposed model in green). We Figure 8 shows the maximum water depths according to MIKE 11 – MIKE 11 GIS model. register the the MIKE maximum discrepancy and in the areas. Theofheight 11 simulation afterinvolved the interpolation the of the
We report the following results of the simulations: the extension of the inundated area (Fig. 5a), the water depth for each time step and for each node and the maximum water depth for each node (Fig. 5b). Figure 6 represents the inundation map, obtained by interpolation of the water depth in each node, and indicates the maximum values of the water depth during the whole simulation. The biggest inundations occur in the central section of the water course under examination , coinciding with the position of the bridge where the bed narrows and the banks are relatively lower. Other inundations take place in the final part of the water course due to significantly reduced bank levels,particularly on the right side, in the mouth area . Figure 7 is a map of the areas historically subject to inundation.A comparison with Figs. 5 and 6 reveals that the
piezometric levels made by MIKE 11 GIS. Differences in results were observed between the in-bed module of the proposed model and MIKE 11 as well as between the out -bed module and the coupled MIKE 11 – MIKE 11 GIS model. In the first case maximum water depth differences of 8% were observed in the main inundation area. In the second case, the differences between the two series of water depths, depicted in Fig. 9, are ordered in increasing size and are related to the corresponding percent differences. It can be noted that, near the greater disagreements (in absolute value), the percent differences are quite small. Figures 10a and b show this kind of disagreement in a planimetric manner. It is possible to observe an exceed variation on the leftside of the water-course and a negative variation on the right. The positive disagreement is certainly due to the differences (positive) of the maximum piezometric level inside the waterbed. The negative ones, instead, are negligible with respect
depth into the river near the same area. By simply looking the left-side area, it is possible to observe that the percent disagreements are relevant only for a limited number of points beside the Centa river. The percent differences on the whole A. Sole and G. Zuccaro: MIKE11-quasi2d river, thoughA comparison quite smallwith (less than 8%), induces greater differences into the urban areas.
283
(a)
(b) Absolute and percent Fig. 10. Absolute and percent disagreement between the two models. disagreement between buildings is fixed the two modelsat 3 m to appreciate also the entity of these discrepancie relation to a base height. Figure 11right shows a they part are of about the digital urban terrain model and the inundation to the water depth in areas: 30 cm, but simulated the two model (MIKE 11 in cyan, proposed model in green). We the water depth is aboutby 2 m. register the maximum discrepancy in the involved areas. The height of the As highlighted in the planimetric representation Figure 10.
(Fig. 10b), in right-hand areas the percent disagreements are about 15%, while at the west boundary of the interested area the percent and absolute differences are quite relevant. The difference is due to a negative variation in the calculation of the maximum water depth into the river near the same area. By simply looking the left-side area, it is possible to observe that the percent disagreements are relevant only for a limited number of points beside the Centa river. The percent differences on the whole river, though quite small (less than 8%), induces greater differences into the urban areas. Figure 11 shows a part of the digital urban terrain model and the inundation simulated by the two model (MIKE 11 Figure 11. Part of the inundation in cyan, proposed model in green). We register the maxiFig. 11. Part of the inundation maps. maps mum discrepancy in the involved areas. The height of the buildings is fixed at 3 m to appreciate also the entity of these 4 Conclusions cially when dealing with hydraulic discontinuities and abrupt discrepancies in relation to a base height. discharge or geometry variations. Therefore, the proposed model has to simulate the flood propagation eiThe proposed model hasgood the power advantage of using a quasi-bidimensional appr ther inor out-bed in the surrounding urban territory. 4 Conclusions for the simulation of flood propagation in urban areas. The synchr between the proposed model and the cou- the estima functioning of theThe in-differences and out-bed propagation modules facilitates The proposed model has the advantage of using a quasipledMIKE 11 - MIKE 11 GIS are probably due to the differof the inundation map in a more precise way by taking into account the re-en bidimensional approach for the simulation of flood propaences between the two in – bed hydrodynamic propogation inside of aspart of esundated volumes and reducing gation in urban areas. The synchronic functioning of the the in- water-bed models as well to a different flow modality in the channels computational time. form The use of thearea.The model first of Braschi [2] seems particu and out-bed propagation modules facilitates the estimation of which the urban differenceetisal. probably a suitable for theresult schematisation the of urban texture of and the inundation map in a more precise way by taking into acof the differentofforms interpolation theallows sectionsus to reac optimum between simplicity the 12 model and the leve count the re-entries inside the water-bed of part of esundatedcompromise using the two numericthe models, as shownof in Fig. (Sole and volumes and reducing the computational time. details. The use of Zuccaro, 2002, 2003). the model of Braschi et al. (1990) seems particularly In the propogation phase of the area the utilization Thesuitable out-bed inundation module is based onurban a steady state equation (thus, for the schematisation of the urban texture and subject allows us to to improvement), of the Eq. (5) inbut the proposed model is an obvious limit can correctly reproduce thewhich flood propaga reach an optimum compromise between the simplicity of the we are endeavouring to overcome. dynamic in a complex territory like a town. Moreover, the in-bed de However, we need to proceed cautiously and make further model and the level of details. propagation module exhibits comparisons of thegood resultsreliability of the modeland withrobustness, observed data.especially w The out-bed inundation module is based on a steady state dealing with hydraulic discontinuities and abrupt discharge or geom equation (thus, it is subject to improvement), but can corEdited by: L. Ferraris variations. rectly reproduce the flood propagation dynamic in a complex Therefore, the proposed model has good power to simulate the f Reviewed by: anonymous referees propagation territory like a town. Moreover, the in-bed deflow propa- either in- or out-bed in the surrounding urban territory. The differences between the proposed model and the coupledMIKE 11 - M gation module exhibits good reliability and robustness, espe-
11 GIS are probably due to the differences between the two in – hydrodynamic propogation models as well as to a different flow modality in channels which form the urban area.The first difference is probably a resu
284
A. Sole and G. Zuccaro: A comparison with MIKE11-quasi2d
Figure 12.
Comparison of different interpolation methods of sezioni trasversali
Fig. 12. Comparison of different interpolation methods of cross sections.
References
Hirsch, C.: Numerical computation of internal and external flows – volume 2: Computational methods for inviscid and viscous Boni, G.,In Crosta, Laverneda, S., Roth, G., Taramasso, C., and area flows, Wiley & Sons, theA.,propogation phase of theA.urban theJohn utilization of1990. the equation (5) in Versace, C.: U-flood (“urban flood”) modello di propagazione Macchione, F. and Morelli, M. A.: Propagazione delle piene conthe proposed model28◦isConvegno an obvious limit we are endeavouring to overcome. delle piene in ambiente urbano, di Idraulica e which seguenti alla rottura di sbarramenti murari, 18◦ Corso di aggiorCostruzioni idrauliche,we Potenza, 16–19 2002.cautiously and However, need toSettembre proceed make further of the namento in Tecniche per la comparisons difesa dall’ inquinamento, a cura di Braschi G., Gallati, M., and Natale, L.: La simulazione delle inonG. Frega, Editoriale Bios, Cosenza, 1996. results of the model with observed data. dazioni in ambiente urbano, CNR-GNDCI, 1990. Marchi, E. and Rubatta, A.: Meccanica dei fluidi, UTET, 1981. Crosta, A., Laverneda, S., Roth, G., Siccardi, F., and Versace, Sole, A. and Zuccaro, G.: Flooding modelling, 3rd EGS Plinius C.: Implementazione del modello di propagazione dell’ onda di Conference on Mediterranean Storms, 1–3 Ottobre 2001, Baja piena in ambiente urbano relativo al piano comunale da rischio Sardinia, Abstr. No. 27, 2001. idrogeologico del comune di Genova, CIMA, October 2001. Sole, A. and Zuccaro, G.: Urban areas flooding modelling, in: Risk DHI: Mike 11 version 3.11 – General Reference manual, Scientific Analysis III, edited by: Brebbia, C. A., WIT Press Southampton Documentation, Danish Hydraulic Institute, 2003. UK, 521–530, 15 pp., 2002. Garc`ıa-Navarro, P., Alcrudo, F., and Saviron, J. M.: 1-D open[1] Boni G., Crosta A., Laverneda S., Roth Taramasso A. C., Versace C.,channel U- flow Sole,G., A. and Zuccaro, G.: Comparison between open channel flow simulation using TVD-McCormack scheme, J. models in natural rivers, in: River Basin Management, edited by: flood (“urban flood”) modello di propagazione delle piene in ambiente Hydr. Eng., 118, 10, 1359–1372, October, 1992a. Brebbia, C. A., WIT Press Southampton UK, 66–768, 2003. 28°J.Convegno disIdraulica Garc`ıa-Navarro,urbano, P. and Saviron, M.: McCormack’ method for e Costruzioni idrauliche, Potenza, 16-19 the numericalsettembre simulation of 2002. one-dimensional discontinuous unsteady open channel flow, J. Hydr. Res., 30, 1, 95–105, 1992b.
References
[2] Braschi G., Gallati M., Natale L., La simulazione delle inondazioni in ambiente urbano, CNR-GNDCI, 1990. [3] Crosta A., Laverneda S., Roth G., Siccardi F., Versace C., Implementazione del modello di propagazione dell’ onda di piena in ambiente urbano relativo al piano comunale da rischio idrogeologico del comune di Genova, CIMA, october 2001. [4] DHI, Mike 11 version 3.11 - General Reference manual, Scientific Documentation, Danish Hydraulic Institute. [5] Garcìa-Navarro P., Alcrudo F., Saviron J. M., 1-D open-channel flow simulation using TVD-McCormack scheme, J. hydr. eng., vol. 118, n. 10, october 1992. [6] Garcìa-Navarro P., Saviron J. M., McCormack’ s method for the numerical simulation of one-dimensional discontinuous unsteady open channel flow, J. hydr. res., vol. 30, n. 1, 1992. [7] Hirsch C., Numerical computation of internal and external flows - volume 2: Computational methods for inviscid and viscous flows, John Wiley & sons, 1990. [8] Macchione F., Morelli M. A., Propagazione delle piene conseguenti alla rottura di sbarramenti murari, 18° Corso di aggiornamento in Tecniche per la difesa dall’ inquinamento, a cura di G. Frega, Editoriale Bios, Cosenza,
