COMPARISON BETWEEN EXPERIMENTAL AND NUMERICAL RESULTS OF 2D FLOWS DUE TO LEVEE

Abstract: Results of a laboratory investigation about the 2D flow field induced by a levee-break are compared with numerical results. A lateral breach is instantaneously opened in a tilting flume 10 m long subject to a constant discharge. This causes the inundation of an initially planar 2D dry surface.

Measurements of water levels are made at various places around the breach, whereas the 2D velocity field is detected by means of ten acoustic transducers. Total discharge outflowing from the breach is also measured under stationary conditions.

The same test conditions were numerically simulated using a mathematical model based on the 2D shallow water equations and solved by means of the well known McCormack finite difference scheme. Artificial dissipation terms have been introduced in order to avoid non physical shocks and oscillations around discontinuities, whilst retaining the second order accuracy of the original scheme.

Comparison with experimental data confirms that the model can be reliably used to describe 2D flows that could occur in nature due to dam or levee-breaking.

Keywords: levee-break, two dimensional flow, experimental data, mathematical modelling, McCormack scheme

1 INTRODUCTION

Two dimensional flows consequent to the breaking of a dam or a levee often show a very complex behaviour due to discontinuous initial conditions at the breach site, to the propagation on a dry bed and to transitions between subcritical and supercritical flow, sometimes with the formation of shocks. In situations where a 2D schematisation is acceptable, mathematical modelling of these phenomena can be accomplished by means of 2D shallow water equations written in conservation form (Molinaro and Natale 1994). Validation of the capabilities of numerical schemes is often performed comparing numerical results only with 1D analytical solutions (Ritter 1892; Stoker 1948). However, for 2D cases analytical solutions are not available even for a simple geometry and in the absence of the source terms of the equations. Numerical solutions have therefore to be validated by comparison with experimental results (Bechteler et al. 1992; Bellos et al. 1992; Fraccarollo and Toro 1995; Soares Frazão and Zech 1999).

The paper compares the results of a laboratory investigation about the 2D flow field induced by a levee-break with numerical results. A lateral breach is instantaneously opened in a flume subject to a constant discharge. This causes the inundation of an initially planar 2D dry surface. The same test conditions were numerically simulated using a mathematical model based on the 2D shallow water equations and solved by means of the well known McCormack finite difference scheme.

2 EXPERIMENTAL SETUP

The experimental facility is composed of a tilting laboratory flume (10 m long and 0.30 m wide) in which one of the side walls was replaced by two separate plates in order to leave a lateral opening of a fixed width, where a manual opening gate was located. A lateral plane (Fig.1), on which the 2D flood caused by the sudden opening of a lateral breach develops, was added at the flume. Two different breach widths (b = 0.28 and b = 0.10 m) were considered. Inflow discharge and slope were maintained equal to Q_s= 0.035 m³s^-1 and I = 0.1% for the two situations considered. The Manning roughness coefficient for the bottom and walls of the laboratory flume was evaluated by means of several tests in steady state conditions as n=0.009. Water depths and velocity profiles inside the flume just upstream of the breach section were measured under steady state. In this condition it was also possible to measure the total outflow discharge from the breach by means of a sharp crested weir placed at the end of the gutter that surrounds the plate.

Water depths were collected by means of a point gauge, displaced through an apposite instrument carrier along the instrument rails of the flume. Velocity profiles have been collected by means of an acoustic Doppler velocimeter (Signal Processing DOP 1000) using ten 1 MHz ultrasonic transducers. In order to obtain value and direction of velocity vectors in the chosen grid points the transducers were positioned in two separate carriers; the first was placed outside the flume opposite to the breach section (y components), the second was positioned inside the flume (x components) downstream of the breach, in a region of supercritical flow in order to avoid any disturbance to the current. At steady state 1000 velocity profiles for each transducer and for both the experimental configurations were recorded. After the acquisition the profiles were time averaged, in order to remove fluctuations caused by turbulence or by an inflow discharge from the pump which was not perfectly constant.

3 NUMERICAL MODEL

Under the usual shallow water hypotheses, the governing two dimensional equations for open channel flow may be written in conservation form by

where U represents the vector of conserved variables (h is the water depth, uh and vh are unit discharges in the x and y directions), F and G are the fluxes of the conserved variables and S is the source term; also, t indicates time and g is the gravitational acceleration while S₀_x, S₀_y, S_fx, and S_f_yare, respectively, the bottom slopes and frictional slopes along the x and y axes.

The well known McCormack explicit shock capturing scheme (McCormack 1969) was used to solve (1). The finite difference algorithm consists of a predictor-corrector sequence and is of the second order of accuracy both in time and space. The shock capturing capability of the numerical scheme allows a simple description of the discontinuities that could occur in the water surface without any special treatment. The discontinuities are also captured in a few spatial intervals. The scheme has the form (Chaudhry 1993):

The x and y directions are designated by the subscripts i and j respectively, while the time level is indicated by the superscript n; the notations Ñ_x and Ñ_y, D_x and D_y have been used to indicate the backward and forward space differences in the x and y directions while D_x and D_y represent the Dt/Dx and Dt/Dy ratios.

In (3) D_xUⁿ e D_yUⁿ represent the components along the principal axes of a dissipative operator D having the role of an artificial viscosity. This term is necessary to smooth the numerical oscillations around shocks and for Fr »1, typical of second order finite difference schemes and to avoid unphysical shocks that otherwise could occur in the numerical solution. The implementation of the term D is based on the procedure suggested by Jameson (1982):

where e is a parameter defined from a normalized form of the gradients of the quantity h=z+h as

where z is the bottom elevation of the points involved in the calculations (measured above z_i,j). The introduction of h instead of h ensures that if water is initially at rest inside a closed area it remains undisturbed in the absence of external forces, regardless of any irregularities in the bottom surface. The parameter a, used to regulate the amount of dissipation, is set to 0.2.

4 COMPARISON BETWEEN NUMERICAL AND EXPERIMENTAL RESULTS

The experimental tests were simulated by means of the a.m. numerical model. Reflective boundary conditions (free-slip) have been incorporated at the side walls, while along inflow boundaries (subcritical flow) the values of unit discharges uh and vh have been assigned. Along outflow boundaries (at the downstream end of the flume and along lateral plane contours) zero gradients on each of the dependent variables were introduced. Since this kind of boundary condition is properly used only in the case of supercritical flow, it was applied at the end of short high slope slides introduced in the numerical description of the domain geometry. This allows to represent a free water outflow, obtaining supercritical flow conditions. Measured and computed total discharges flowing from the breach at steady state are summarised in Table 1.

Breach width	Discharge Q (10^-3 m³/s)
b (m)	exp.	num.
0.10	2.80	3.00
0.28	6.73	7.10

The numerical results slightly overestimate the measured discharge outflowing from the breach; these differences could be due to the strong water surface curvatures in the breach region which induce a non-hydrostatic pressure distribution. However the differences are small (from 5 to 7 %) and comparable with the measure approximations.

Since the main characteristics of the phenomenon are similar for the two considered configurations, only the results referring to the case with b=0.28 m are shown here. Details of the case with b=0.10 m are given by Aureli (1999). After a few seconds from the lateral opening, the outflow from the breach settles around 7.1×10^-³ m³/s, while the discharge travelling downstream is about 30.0×10^-³ m³/s; it is obvious that the amount of the two discharges leaving the flume is greater than the total inflow (35.0×10^-³ m³/s).

This can be explained by considering the Figure 2 that shows the time evolution of the computed water surface on the lateral plane and the portion of the flume in front of it. Immediately after the breach a bore origins just at the end of the lateral opening and starts to travel downstream very slowly, leaving the flume after about 50 seconds. Once steady state conditions are reached everywhere, the flow downstream of the breach is supercritical throughout and the water surface is characterised by a stationary cross-wave train originated at the downstream end of the opening. Figure 2 also shows that the flood does not cover the entire plane, leaving a dry region in front of the upstream wall of the flume. Video recordings taken during the experiences clearly show the same behaviour.

Figure 3 shows the comparison between numerical and experimental velocity profiles along the x and y directions in a region inside the flume in front of the opening. Note that the measured velocity profiles were acquired placing the transducers 0.02 m far from the bottom, while the numerical results refer of course to depth-averaged values.

The calculated values agree satisfactorily with the measured data as a whole; the comparisons at y=0.24 m and x=1.14 m are noteworthy. At x=1.22 m and x=1.30 m the computed velocity component v_y slightly overestimates the measured velocity; at y=0.28 m, close to the breach section, the computed profile shows quite a different behaviour from that measured near the downstream end of the opening. Here the presence of the solid wall normal to the opening induces strong curvatures in the water surface with a vertical pressure distribution far from the hydrostatic assumption. Moreover, because of the imposed free-slip boundary condition, the influence of the wall cannot be taken properly into account (Molls et al. 1998).

Figure 4 shows the computed and measured water surface and velocity field, for a region near the breach, under steady state conditions.

Fig. 4 Water depths and velocity vector maps at t = 29.5s; (a) computed, (b) measured

Inside the flume and downstream of the opening section, water surface is characterised by a train of stationary cross-waves which is adequately represented by the numerical model; also the agreement between measured and computed water depths is good with only a slight underestimation at the downstream wall normal to the opening.

5 CONCLUSIONS

The agreement between computed results and experimental data for the 2D flow field induced by a levee-break demonstrates that the proposed model is capable to reproduce the main characteristics of the phenomenon studied.

The numerical model can then be reliably used to describe 2D rapidly varying flows that could occur in nature (dam break flows, levee-breaks) and even in complex situations in which the underlying assumptions at the basis of the mathematical description are not completely satisfied.

Alcrudo, F. and Garcia-Navarro, P., 1993: A high resolution Godunov-type scheme in finite volumes for the 2D shallow water equations. Int. J. Numer. Meth. Fluids, 16, P. 489-505.

Anastasiou, K. and Chan, C.T., 1997: Solution of the 2D shallow water equations using the finite volume method on unstructured triangular mesh. Int. J. Numer. Meth. Fluids, 24, P.1225-1245.

Aureli, F. 1999: Modellazione numerica e verifica sperimentale di fenomeni di moto rapidamente vario mono e bidimensionali. Ph. D Thesis, Milan Polytechnic, (in Italian).

Bechteler, W., Kulisch, H. and Nujic M., 1992: 2-D Dam-Break Flooding Waves: Comparison between Experimental and Calculated Results. In Floods and Flood Management, ed. Saul, A.J., Kluwer Academic Publishers, Dodrecht, P. 247-260.

Bellos, V., Soulis, J.-V. and Sakkas J.-G., 1992: Experimental investigations of two dimensional dam-break-induced flows. Journ. of Hydr. Res., IAHR, 30(1), P. 47-63.

Chaudhry, M.-H. 1993: Open-Channel Flow. Prentice Hall, Englewood Cliffs, New Jersey.

Fennema, R.-J. and Chaudhry, M.-H.,1990: Explicit Methods for 2D Transient Free-Surface Flows. Journal of the Hydraulics Division, ASCE, 116(8), P. 1013-1034.

Fraccarollo, L. and Toro, E.-F., 1995: Experimental and numerical assessment of the shallow water model for two-dimensional dam-break type problems. Journal of Hydraulic Research, IAHR, 33(6), P. 843-864.

Hirsch, C. 1990: Numerical Computation of Internal and External Flows, vv.1-2, Wiley, Chichester.

Jameson, A., 1982: Transonic Airfoil calculations using the Euler equations. In Numerical Methods in Aeronautical Fluid Dynamics, ed. P.-L. Roe, Academic Press, New York.

McCormack, R.-W. 1969: The effect of viscosity in hypervelocity impact cratering, AIAA Paper 69-354.

Molinaro, P. and Natale, L. 1994: Modelling of Flood Propagation over Initially Dry Areas. Proc. of the Specialty Conf. held in Milan, 29 June-1 July 1994, ASCE, New York.

Molls, T., Zhao, G. and Molls, F., 1998: Friction slope in depth-averaged flow, Journal of the Hydraulics Division, ASCE, 124(1), P. 81-85.

Ritter, A. 1892: Die Fortplanzung der Wasserwellen. Zeitschrift des Vereines Deutscher Ingenieure, 36(3), P. 947-954.

Soares Frazão, S. and Zech, Y., 1999: Effects of a sharp bend on Dam-Break flows. Proceedings of XXVIII IAHR Congress, 22-27 August 1999, Graz, Austria.

Stoker, J.-J., 1948: The formation of breakers and bores. Communications on Pure and Applied Mathematics, 1, P. 1-87.