R. Della Morte, V. Iavarone and D. Pianese
Hydraulic and Environmental Department ²G. Ippolito” Via Claudio n. 21, Naples, Italy.
Tel. +39+81+7683429; fax +39+81+5938936; E-mail: dellamor@unina.it
Abstract:
In the present paper, a simplified numerical model LPM is used to estimate the
peak discharges and maximum flow depths in natural or man-made drainage networks
with a variational criteria. This provide a useful tool using the rainfall data
normally available.
Keywords: Variational approach, numerical model, dreinage networks, unsteady flow, free surface flow
The available rainfall data for the estimation of the peak discharges and maximum flow depths that might be present in a drainage network- natural or man-made- consist just of relative rainfall data, regarding to the maximal values of the total precipitation observed in given periods of duration d. Alternatively to the approaches which use the concept of a design hydrograph, such as the Chicago Storm (Keifer & Chu, 1957), the estimation of the maximal peak discharge and maximum flow depths in natural and man-made drainage networks are, normally, determined with the help of a variational approach. The method consists of i) the research of the maximal time-averaged peak discharges that could flow in each reach of the network in an interval of time D and ii) in the successive estimation, making vary the interval D and the corresponding time-averaged peak discharge, by mathematical simulation of wave propagation in the network of the maximum instantaneous discharge. The application of the variational approach has been verified just for particular conditions, in which the dynamic component of the propagation could be neglected, like in the case of detention storage (Penta, 1983; Pianese e Rossi, 1986) and by using an unsteady flow model in the case of drainage networks characterized by mild slopes. The aim of the present paper is to verify the effectiveness of the applicability of the variational approach with the help of a simplified unsteady flow model in order to determine the maximum flow depths and the instantaneous maximum discharges in the branches of a drainage network.
The analysis of the unsteady flow phenomena that could be present in open flow channel network is usually accomplished by the de Saint-Venant equations written in full or simplified form and numerically solved using one of the available algorithms. The mathematical models proposed for the simulation of wave propagation in networks of great extension present two different problems: the arise of numerical instability and the usage of a lot of time for the calculations. That’s why it’s important to apply methods that could avoid the latter problem, and that at the same time could point out physical point of view of the unsteady flow phenomena.
The LPM (Linearized Parabolic Model) is a numerical scheme, that compared with others, has the advantage of being easy to comprehend and to apply. Furthermore it is very fast and gives results that are very close to the results obtained with the Full Dynamic Model (FDM) (Todini & Bossi, 1986; Franchini & Todini, 1989 ).
The fundamental hypothesis of this approach is based on the statements that the inertial terms present in the momentum equation are of the same order of magnitude but with opposite sign and consequently could be neglected.
Since the integration in the time is carried out in an analytical way, the obtained numerical scheme is always stable. However the Courant’s stability condition allows to maintain the precision of the results. Under these circumstances, if slight inaccuracies and diffussive phenomena are considered, then it’s possible to accomplish the numerical integration of the equations using large times steps .
A parabolic model, not linearized, was already used in the past by Yen (1981) to simulate the propagation wave phenomena in drainage networks. However, because it was a full model although slightly simplified, that model is not able to simulate the waves propagation in very complex networks.
In the present paper the first step will be the description of the mathematical model proposed and used for the unsteady flow analysis. Then once the examined study cases will be shown, a comparison between the results obtained by using the FDM (Della Morte et.al. 2000) and the LPM coupled to the variational approach. Following this kind of approach, different rectangular hydrographs, respecting the time-averaged discharge vs. duration curves that correspond to each input hydrograph are used.
The de Saint-Venant equations, applied to an elementary reach of lenght dx, could be written as:
where:x: abscissa; t: time; Q: flow discharge; V: mean flow velocity; h: flow depth; Zf: bottom elevation; B: surface width; H: specific energy; J: specific energy loss per unity of length; q : lateral flow per unity of length (positive if q is inflowing).
When the longitudinal bed slope is small, the inertial terms of the momentum equation could be neglected, basically, because they are smaller than the slope of the channel. Then, we can write the equation (2) as:
(3)
in which So represents the longitudinal slope of the reach.
Deriving the (1) respect
to x, the (3) respect to t
and taking from (1) the expression
, dividing by B could be obtained:
(4)
Deriving J
respect to t and substituting the expr ession
taken from (1), we can obtain the
expression that corresponds to the complete parabolic model. That is:
(5)
with
diffusivity
and
celerity
The (5) is a differencial equation of the second order whose result could be obtained by the “linearization”, considering the coefficients C and D as constants in an interval Dt, and updating the value at each next time variation.
If the equation is a linear one, it’s possible to apply the ²superposition principle” of the effects. It is possible to figure out the relative solution of an impulse and to express any other by means of a convolution integral. The solution of the (5) is (Dooge, 1973; Natale & Todini, 1975):
(6)
under the following conditions: t≥0; C≥0; D≥0.
For each stretch Dx, uDx (t) gives, at time t, the outflow discharge that corresponds to an “impulsive” inflow discharge originated upstream the stretch. Hence, the outflow from the stretch could be expressed at time t, in terms of the input hydrograph Q (x,t), as:
(7)
This integral is normally discretized supposing that at each interval Dt, the functions that expressed the input and the output assume constant values. If we define:
(8)
the expression of the Instantaneous Unit Hydrograph is:
(9)
Substituting the expression (6) in the (8), and solving the integral in the case that the inflow increment is impulsive, it could be obtained (Todini & Bossi, 1986; Franchini & Todini, 1989):
(10)
in which
represents the Normal Standard
Distribution Function.The expression that corresponds to the
discretized convolution integral could be written as follows:
(11)
in which k and i are related to the time t and to Dt by the relationships:
If the space distribution of the discharges is known at any new interval t, it is possible to reproduce the water surface profile under steady flow conditions by integrating respect to the space, the Bernoulli equation or the momentum equation, considering the instantaneous steady flow hypothesis.
In the past, the linear parabolic model has been applied with success only in the case of a single channel (Todini & Bossi, 1986; Franchini & Todini, 1987).The reliability of the results obtained by the application of LPM on man-made drainage networks case characterized by small bottom slopes has been tested by the authors (Della Morte & al., 2000).
The verification of a successful use of the LPM
coupled with a variational criteria, in order to determinate the maximum flow
depths that might occur in a drainage network, was accomplished on an artificial
network, consisting of an “Y”
configuration, with two channels (n. 3 and n.2) in which converges another with
a 30° angle (n.1). Although
the model is able to consider any type of cross section, also if it’s not a
prismatic one, and any type of flow, supercritical or subcritical, in this paper
were considered prismatic sewers with rectangular
cross section and small slopes with the following widths: B1=B3=0.5 m; B2=1.0 m.
The cases of study are indicated on Table 1. On the accomplished tests were varied (i) the shape of the input hydrographs by means of linear functions (triangle) and gamma functions; (ii) the parameters, i.e., the peak discharge Qc, the peak time tc, the parameter a (corresponding to gamma hydrographs) and the overall flood duration td (corresponding to the linear hydrographs).
To evaluate the effect of the network’s geometry, were varied both, the length of the different reaches of the network and their bottom slopes. The objective was to verify how effective was the model and, because of that, were accomplished some tests in which were considered input hydrographs that were very rare in nature (cases 24, 26, 27, 28, 29, 30). Indeed for these cases were expected larger errors.
The procedure used for the verification has been developed as follows:
Phase 1: Evaluation of the maximum flow depths and maximum instantaneous discharges by means of the FDM model (Della Morte & al., 2000) applied on the described hydrographs in which was considered a section 50 m downstream of the confluence;
Phase 2: Evaluation, by integrating the most upstream inflow discharges over the durations d, of the maximum time-averaged discharges in given duration;
Phase 3: Estimation of the maximum flow depths and of the maximum instantaneous discharges using the LPM model and applying as inputs rectangular hydrographs in which duration and discharge are consistent with values obtained in phase II. For each of the input hydrographs inflowing in the two upstream reaches this phase was developed as follows:
(1) It was first elaborated “curve of reduction of the flood peak with the duration D”:
in which w is the instant where it starts period of duration D, and ti and td are, respectively, the instant when initiates the flood and when it finishes;
(2) For each value of D, were considered as inputs the rectangular hydrographs that correspond to a duration D in the specific ²curve of reduction of the flood peak with the duration D”.
(3) with the help of the LPM it was accomplished a simulation that represented the unsteady flow phenomena derived from these inputs and was determined, for the cross section, hD, defined as follows:
being hD(t) the function that represented the trend of the flow depths in time corresponding to the duration D and discharge QD;
(4) then it was varied the duration D, repeating the procedure on points (2) and (3) and, hence, constructing the curve:
hD = h(QD)=h[QD(D)]
(5) it was estimated the value of the maximum flow depth, for the assigned cross
(6) section, considering the maximum value of the height hD for each duration D:
The results obtained are showed in Table 2.
Table 1 Examined study-cases
Table 2 Calculated values
From the Table 2 it was concluded that:
l It is possible to apply the variational approach using LPM as a convenient wave propagation model, because the errors that arose when applying the model to determine the maximum flow depths were around the 8.4%, acceptable from a technical point of view.
l The cases 24, 26, 27, 28, 29 e 30, characterized by sharp decrease of the hydrograph generated results with errors (on the maximum flow depths) that were between 6.6% and 12.7%.
l If we exclude those cases, then the mean error obtained for the maximum flow depths corresponded to 1.4%, while the one that is associated with the maximum discharges was 3.36%.
l From the analysis of the times that corresponded to the maximum flow depths, and if we exclude the mentioned cases, it might be observed a tendence of the variational approach when it’s coupled with the LPM, to underestimated that means that if it is used together with a forecast system, might generate unreliable results.
The estimation of the maximum flow depths and of the maximum instantaneous discharges that could be observed on natural or man-made drainage network establishes an important element for both, the design and regulation phases.
In the present paper, it was proved that the estimation of the mentioned unknowns, could be accomplished by means of a variational criteria coupled with the LPM model.
The results obtained suggest that in the future the research will be addressed on the transformation of the rainfall into the inflow volume that should be the input of the network.
References
Akan, O. A.,
Yen, B.: Diffusion-wave flood routine in channel networks. Journal of Hydraulic
Division, ASCE, Vol. 107, pp. 719-732, June, 1981.
Della Morte, R., Matarazzi, C., Pianese, D.: Sulla possibilità di utilizzazione dell’approccio variazionale per l’individuazione delle massime portate defluenti nelle reti di drenaggio.Atti del XXVII Convegno di Idraulica e Costruzioni Idrauliche, Genova, settembre 2000.
Della Morte, R., Iavarone, V:, Brangi, S.,
Pianese, D.: Applicazione di un modello idraulico semplificato per l’analisi
dei fenomeni di moto vario nelle reti di drenaggio. Submitted to “L’Acqua”,
dicembre 2000.
Dooge, J.C.I.:
Linear Theory of Hydrologic System, Technical Bulletin N.1468, U.S. Dept.
of Agriculture, Washington, 1973.
Franchini, M., Todini, E.: Pabl : A Parabolic and Backwater scheme with lateral inflow and outflow. Università di Bologna, Facoltà di Ingegneria – Istituto di Costruzioni Idrauliche, pubblicazione n° 10, Maggio, 1989.
Franchini, M.,
O’ Connell, P. E.: An analysis of dynamic component of the geomorphologic
instantaneous unit hydrograph. Journal of Hydrology, Vol. 175, pp. 407-428,
1996.
Keifer, C. J.,
Chu, H. H.: Synthetic storm pattern for drainage design. Journal of Hydraulic
Division, ASCE, Vol. 83, pp. 1332-1352, August, 1957.
Natale, L.,
Todini, E.: A Constrained Parameter Estimation Techniques for Linear Model in
Hydrology. Mathematical Models for Surface Water Hydrology, Wiley & Sons,
1975.
Todini, E.,
Bossi, A.: Pab (Parabolic and Backwater) an unconditionally stable flood routine
scheme particularly suited for real time forecasting and control. Journal of
Hydraulic Research, Vol. 24, 1986, No. 5.
