(1)
Sopta L.
Professor D. Sc., Faculty of Engineering, University of Rijeka
Vukovic S.
Assistant Professor D. Sc., Faculty of Engineering, University of Rijeka
Kranjcevic L.
Faculty of Engineering, University of Rijeka
Petras J.
Professor D. Sc., Faculty of Civil Engineering, University of Zagreb
Holjevic D.
Croatian Waters, Rijeka
Plisic, I., M. Sc.
Civil Engineering Institute of Rijeka
Address for correspondence: Faculty of Engineering, University of Rijeka, Vukovarska 58, 51000 Rijeka, Croatia, tel.: +385-51-651494, fax: +385-51-651490, senka.vukovic@ri.tel.hr, hrvatska-vodoprivreda@ri.tel.hr, jpetras@master.grad.hr
Abstract: This paper presents a CFD (computational fluid dynamics) approach applied in flood forecasting and prevention planning in the case of the lower part of the river Rjecina (Croatia, Europe). Rjecina is a torrential river, which in its lower part flows through the city of Rijeka and into the Adriatic Sea. Key flood risk factors for the urban area surrounding the watercourse are possible sediment settlement, bridges that narrow the primary riverbed profiles, high tides etc. In order to design flood prevention measures acceptable in the urban environment a computer model was developed. This model is based on the 2D shallow water (i.e. St. Venant) equations. Since these equations are hyperbolic conservation laws with a significant source term, Q-schemes developed by Bermudez, Dervieux, et al. were applied in the numerical computation. In this way the resulting computer application can simulate nonstationary river flow and flooding, dam breaks, supercritical flow, hydraulic jumps etc. First step in the application of this approach on the case of the lower part of the river Rjecina was digitalization of geodetic data and development of the numerical mesh. The model was then verified by comparison with measurements. In this paper some of the numerous simulation results are presented: concerning the possible sediment accumulation and concerning influence of the bridges. The main advantage of the developed computational model are verified computational accuracy and the simplicity with which hypothetical changes in the riverbed and surrounding urban area can be introduced. Obviously, the computer application can and will be applied to other flood prediction case studies.
Keywords: mathematical model, Q-schemes, computer simulations, river estuary, flood forecasting and prevention
1 INTRODUCTION
The river Rjecina is 18.3km long, with an orographic catchment area of the size of 218km2 and it is located in the western part of Croatia (Europe). Its flow is torrential with oscillations in the discharge from 0m3/s up to 400m3/s during the year. In its lower part it flows through the city of Rijeka and into the Adriatic Sea. The downtown and the most economically significant industries are situated next to the watercourse of river Rjecina and its estuary. In the history this river endangered the city many times by its floods. The greatest flood was recorded in the autumn of 1898, when the entire downtown was flooded and the most recent large flood was recorded in the autumn of 1952.
The upper part of the river Rjecina is characterized by major longitudinal slope, canyon parts and unstable riverbank zones with an occurrence of significant torrential sediment quantities, which accumulate in the lower riverbed section. In addition, increased flood risk factors are bridges because they narrow the primary riverbed profiles. Tidal wave propagation conditions in this area furthermore complicate flooding problems. Traditional principles of riverbed regulation and protective flood-dam construction are only partially acceptable as flood prevention measures because of the urbanized environment. For the purpose of solving these problems, Company “Croatian waters”, which is the state enterprise in charge of water management and implementation of flood protection and control measures, directed its activities toward flood prediction and key factor analysis of flood occurrence and intensity. To that purpose, a CFD approach was applied and a mathematical model of the river flow and flooding was created. In the following paragraphs this model and its application are presented.
2 THE CFD APPROACH
2.1 2d shallow water equations
The mathematical model for the flooding simulations are, as generally accepted, two-dimensional shallow water equations, i.e. St. Venant equations ([2]):
(1)
,
(2)
,
,
(3)
Here
represents time,
space coordinates,
computational domain,
water depth,
,
discharge,
velocity,
gravity acceleration,
bottom elevation and
Manning’s roughness coefficient.
Together with initial conditions
,
, and boundary conditions this system forms an initial-boundary value problem.
2.2 Q-schemes
The numerical schemes applied to solve the initial-boundary value problem were Q-schemes ([2]). These schemes were chosen because system (1) is a system of hyperbolic conservation laws with an additional difficulty in numerical modeling – the source term ([1]-[7]). So a scheme that equally well models numerical flux and source term was needed.
In the domain
with standard triangulation, for
each node
,
a dual cell
is considered. Let
be the approximation of the mean
value of the solution in the cell
at time
, where
is the time step of the numerical scheme. Shallow water equations are integrated
over each dual cell, and then following approximations are applied:
(4)
(5)
(6)
Here
represents dual cell area,
set of indexes of all the
neighboring nodes for the
node,
part of the edge of the dual cell
common with dual cell
,
vector normal to
with length equal to the length of
,
numerical flux,
subcell,
- subcell area and
numerical source term. Particularly in the case of Roe variation of the Q-scheme
numerical flux is calculated using expressions:
(7)
,
,
(8)
(9)
,
(10)
and numerical source term using expressions:
(11)
(12)
Here
represents distance between nodes
and
.
2.3 Computer model
On this basis an object oriented (C++ programming language) computer application for simulation of river flow and flooding was developed. The program was applied to the case of the lower part of the river Rjecina. First geodetic data for the riverbed and the surrounding terrain were digitized and numerical mesh of finite volumes (Fig.1) with the aid of ACAD and SMS softwares was developed. Thus the computer model of the lower part of river Rjecina and simulations of its flow and flooding was built. The model was successfully verified through comparison of simulation results vs. measurement data in riverbed.
3 SIMULATION RESULTS
The developed model for the flow and flooding of the lower part of river Rjecina was used to simulate various flooding scenarios and investigate influence of different factors like ebb and tide oscillations, sediment settlement in riverbed, bridges etc. For example, a hypothetical settlement of the sediment was introduced in the numerical mesh and simulations for various flow conditions with and without this hypothetical sediment were simulated. Results for the worst possible case - the thousand-year return period inflow discharge (Q = 466 m3/s) and highest tide (see level at 1,2 m) are presented in Fig.2 and Fig.3. All results show that riverbed sediment impact is significant on the downstream watercourse sections because it causes much earlier overflow occurrences. Another group of simulations was concerned with the influence of bridges. Again, in the numerical mesh all the 7 bridges over the lower part of river Rjecina were removed and various simulations with and without the bridges were computed. Results for the inflow discharge of the hundred-year return period (Q = 361 m3/s) and an average see level at 0.0 m are shown in Fig.4. This group of simulations demonstrates that bridges are a minor flood risk factor.
4 CONCLUSION
From the presented model and simulation results following conclusions can be drawn:
–– the CFD approach to flood problem in the lower part or river Rjecina resulted in an advanced computer tool for simulation of river flow and flooding,
–– the impact of various key factors (sediment settlement, ebb and tide, bridges, etc.) was easily and promptly investigated using this mathematical and computational model,
–– the results of simulations are and will be used in the flooding prevention for the urban areas surrounding the lower part of river Rjecina, where traditional structural flood prevention measures are not acceptable,
–– obviously, the developed computer application can and will be applied to other problems where simulation of river flow and flooding is needed.
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[2] Hubbard, M. E., Garcia-Navarro, P., Flux Difference Splitting and the Balancing of Source Terms and Flux Gradients, Numerical Analysis Report, The University of Reading, Department of Mathematics, 1999.
[3] Hudson, J., Numerical Techniques for the Shallow Water Equations, Numerical Analysis Report, The University of Reading, Department of Mathematics, 1999.
[4] LeVeque, R. J., Bale, D. S., Wave Propagation Methods for Conservation Laws with Source Terms, submitted to Proc. 7th International Conference on Hyperbolic Problems, Zurich, 1998.
[5] Paquier, A., 1-D and 2-D Models for Simulating Dam-Break Waves and Natural Floods, 1st CADAM Meeting, Walingford, UK, 1998.
[6] V«zquez-CendÙn, M. E., Improved Treatment of Source Terms in Upwind Schemes for the Shallow Water Equations in Channels with Irregular Geometry, Journal of Computational Physics 148, 497-526, 1999.
[7] Zhao, D. H., Shen, H. W., Tabios III, G. Q., Lai, J. S., Tan, W. Y., Finite-Volume Two-Dimensional Unsteady-Flow Model for River Basins, Journal of Hydraulic Engineering, Vol. 120, No. 7, 1994.
Fig.1 Numerical mesh segment for the urban area surrounding the lower part of the river Rjecina
Fig.2 Water levels without and with sediment, simulation results for Q=466 m3/s and sea level at 1.2 m
Fig.3 Water elevation (in colors as axplained in the legend) and velocity direction vectors in a section of a flooded urban area, simulation results for Q=466 m3/s and sea level at 1.2 m
Fig.4 Water levels without and with bridges, simulation results for Q=361 m3/s and sea level at 0.0 m