Dam-Break flows in presence of abrupt bottom variations

 

F. AURELI^*, P. MIGNOSA^** and M. TOMIROTTI^***

 

(*) Ph.D. Student, Dept. of Civil Engineering, Parma University, Italy.

(**) Associate Professor, Dept. of Civil Engineering, Parma University, Italy.

(***) Post-Doctoral Fellow, D.I.I.A.R., Milan Polytechnic, Italy.

 

Prof. P. Mignosa, Dept. of Civil Eng., Parma Univ., Viale delle Scienze 1,

43100 Parma, Italy.

Tel: int+0521+905925; Fax: int+0521+905924;

EMail: mignosa@parma1.eng.unipr.it

 

 

Abstract

In the present work numerical results of a 1D mathematical model for Dam-Break flows, based on MacCormack shock-capturing scheme, are compared with experimental tests realised in presence of abrupt bottom variations. The laboratory investigation has been carried out in a tilting flume in which bumps of relevant height, if compared to water depth at the dam section, were placed; this causes shock formation, strong surface curvatures and reverse flow.

Numerical results show quite a satisfactory agreement with experimental data, even if the considered configurations violate, almost locally, some St. Venant hypotheses. This confirms the applicability of the numerical model even in limit conditions that could occur in nature.

 

Keywords: Dam-Break, St. Venant equations, MacCormack scheme, shock waves, experimental data

 

Introduction

Floods resulting from the sudden collapse of a dam (Dam-Break) are frequently accompanied by shock waves due to valley contractions, irregular bed slopes and non-zero tailwater depth. It is commonly accepted that, except for extremely high slopes and curvatures, mathematical description of these phenomena can be accomplished by means of 1D St. Venant equations written in conservation form:

 

, (1)

 

in which Q=discharge, A=wetted area, x=distance along the channel (positive downstream), t=time, g=gravitational constant, =bottom slope, =friction slope calculated according to Manning equation. The terms I₁ and I₂ are:

 

(2)

 

with h= water depth and =width of the cross-section at height above the bottom.

Numerical schemes for the integration of (1) are often verified comparing computed results with analytical solutions (Ritter, Stoker) which consider simple situations where Sº0 (Molinaro & Natale, 1994). If the source term S is not negligible, as in natural valleys with highly irregular topography, the validation of the model can only be achieved by comparison with laboratory tests.

Both experimental results available in literature (Chervet & Dalléves, 1970; Bellos et al., 1992) and new data obtained by the Authors themselves have yet been considered in some previous works (Aureli et al., 1998a, 1998b, 1998c). The aim was to validate a numerical model based on the well known MacCormack shock-capturing scheme in complex situations in which shock formation and propagation, reverse flow and wetting and drying conditions occurred in the flow field.

In the present work the experimental verification is extended to situations in which some of the St. Venant hypotheses are - almost locally - violated. Then a laboratory investigation has been carried out in a tilting flume placing bumps of relevant height at the dam section and downstream. The examined configurations could be considered representative of a partial breach along the vertical axis and of a steep rising in the bathymetry. The considerable bump height was able to induce strong curvatures, shock formation and reverse flow along the channel.

 

 

Figure 1. Channel geometry and main symbols.

 

Table I. Test conditions.

 

Experimental setup and test conditions

Tests were carried out in a tilting laboratory flume at Department of Civil Engineering of Parma University. The flume was rectangular in section, 1.0 m wide, 0.5 m high and 7.0 m long. The instantaneous dam failure was simulated by means of the sudden (less than 0.05 s) removal of a gate controlled by an oil pressurised circuit. To minimise disturbances in the flow field, the gate is mounted on a frame completely separated from the rest of the flume, without any guide rails on bottom and sides of the dam section.

Measurements of water depth versus time were made at four sections along the flume, including the dam site, based on video recordings of the flow. Velocity measurements were accomplished with an Acoustic Doppler Velocity meter (ADV Nortek).

The examined test cases are summarised in Table I and Figure 1. In all the test cases the dam was placed at x=2.25 m from the head of the reservoir (x=0). x_A and x_B are the abscissas of the upstream ends of the bumps A and B respectively, while i refers to the channel bed slope. Only smooth conditions were considered with a Manning's roughness coefficient n=0.01.

 

Numerical simulation

Numerical solution of eqns. (1) was based on the well known MacCormack (1969)

predictor-corrector finite difference scheme:

 

(3)

 

in which i and n are the spatial and temporal grid levels, and the superscripts 'p' and 'c' denote the variables at predictor and corrector steps, respectively. q=0 and q=1 allow to interchange, also cyclically, the order of backward and forward differentiation in the scheme.

In eqns. (3) d is an artificial dissipation term that must be added to the original form of the MacCormack scheme in order to avoid spurious oscillations and non-physical discontinuities. Two versions of the model have been implemented: in the first d was calculated according to the classical theory (Jameson, 1982), whereas in the second d is obtained directly from the discretisation of the equations following the TVD approach and adopting the Van Leer limiter function (Harten, 1983; Hirsh, 1990, Garcia Navarro & Alcrudo, 1992).

No boundary conditions are necessary when h vanishes at both ends of the computational domain. When the wetting front reaches the downstream end, the channel was extended further assuming a steep slope (10%); this avoids the introduction of an unrealistic boundary condition and prevents from downstream influences. When the water touches the vertical wall at the upstream end of the channel a fictitious node inside the solid wall is introduced; the channel geometry is extended symmetrically imposing a reflection boundary condition.

A pointwise approach has been adopted to treat source terms S of the eqns. (3) (Aureli et al., 1998a). Care was taken in order to avoid the introduction of artificial source terms in the momentum equation if a water body initially at rest inside some closed area is considered (Nujic, 1995).

As h(r)0 friction becomes very large, causing numerical instability in the solution. To avoid this non-physical effect grid points are included in the computational domain only if calculated depths are greater than a threshold value (Bellos & Sakkas, 1987, Bechteler et al., 1992).

 

Results

For lack of space only test cases 2,3 and 4 are presented in the following. The results of test case 1 are similar to case 2 until the shock wave originated by the bump at the end of the flume reaches the considered section.

Figure 2 shows the comparison between experimental data and numerical results for test 2. Measured and computed hydrographs at x=1.40 m, x= 2.25 m and x= 3.40 m, together with velocity along the x axis at x=1.40 m, are plotted.

 

 

Figure 2. Computed and measured stage hydrographs for test case N.2.

 

Computed water levels are in a good agreement with experimental data, with slight differences between the Jameson and the TVD version of the numerical scheme.

The computed velocity hydrograph at x=1.40 m is not correctly time located and does not match very well experimental data. However it must be noticed that numerical velocities are cross-sectional averages while measured data are local velocities along the symmetry axis at small distances from the bottom (this is due to the necessity of maintaining the instrument transmitter submerged).

Figure 3 shows stage hydrographs for test 3, in which only the bump B at the end of the channel is present. The results of the two versions of the numerical model are almost identical except around discontinuities that are more sharply captured by the TVD scheme (see detail). Computed and measured water levels are in good agreement as a whole and also the shock heights are well predicted; only a moderate shift in time location of the discontinuities is still present.

Figure 4 refers to test 4, which differs from case 3 for a non-zero channel slope. In this configuration the most upstream sections dry out and then the shock originated by the bump at the end of the channel vanishes travelling upstream. Still the differences between the computed results of the two versions of the numerical model are negligible and the agreement with measured data is good, except for the beginning of the hydrograph at dam location which is slightly overestimated.

 

 

CONCLUDING REMARKS

Even if the experimental configurations examined violate, almost locally, some of the St. Venant hypotheses (small bottom slopes and curvatures, hydrostatic pressure and uniform velocity distribution in the cross section) the numerical results agree satisfactorily with experimental data. Shocks are well predicted even in complex geometrical situations that cause reflections and partial transmissions as in test 2. The only significant disagreements between numerical and experimental results originate in those situations in which strong water surface curvatures occur at the dam section, as in tests 1 and 2. The differences between the results obtained by two versions of the numerical model are almost negligible everywhere, except at dam location in test 1 and 2 and around the discontinuities which are more sharply captured by the TVD version.

The computed results essentially confirm the applicability of the numerical model even in limit conditions that could occur in nature.

 

REFERENCES

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Aureli, F., Belicchi, M., Maione, U., Mignosa, P. & Tomirotti, M. (1998b), Unsteady Flows due to Dam-Breaking. Part II: Experimental Investigation and Numerical Modelling in Presence of Shock Waves (in Italian), L'Acqua, 5, 27-36.

Aureli, F., Mignosa, P. & Tomirotti, M. (1998c), Numerical simulation and experimental verification of Dam-Break flows with shocks, XII Int. Conf. on Computational Methods in Water Resources, June 15-19, Crete, 387-394.

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