Computation of rapidly varied unsteady flows in open channels and comparison with physical model and field experiment

 

¹) Wei Zhang, ²) Wolfgang Summer

 

²) Civil Engineering Bureau Summer Vienna, Austria

¹) College of Harbor, Waterway and Coastal Engineering, Hohai University

1 Xikang Road, 210024 Nanjing, China

Tel.: +86-25-3713777-50611, Fax: +86-25-3735375, E-mail: zhangw@hhu.edu.cn

 

 

Abstract

Computer programs for the simulation of unsteady open channel flows are becoming increasingly available and practical. One of them is FLORIS (Flood-Routing River Systems), developed by the Federal Technical University of Zurich. It is based on the one-dimensional Saint Venant equations, and numerically solved by a box implicit difference method (IDM) and an implicit method of characteristics (IMOC). The most attracting feature of the program FLORIS is that subcritical and supercritical mixed flows can be implicitly computed by the IMOC built in the program. In this paper, FLORIS was applied to compute a surge wave and a dam-break wave, and compared with field experiment and physical model. The comparison between numerical solutions and experiment data is satisfactory, and this also verifies the applicability of FLORIS to simulate rapidly varied unsteady flows in irregular and nonprismatic channels.

 

Keywords: Surge waves, dam-break waves and numerical simulation

 

INTRODUCTION

Mathematical models to simulate unsteady flows in open channels have been common and practical tools for civil engineers, and computer programs related to these aspects are becoming increasingly available. In such a situation, more attentions should be paid now to study advantages/disadvantages as well as applicable limitations of available computer programs.

 

Rapidly varied unsteady flows in open channels include surge waves, stationary or movable hydraulic jumps and dam-break waves. They are characterized by flows with steep waver fronts and/or mixed-flow regimes. Most commercial hydrodynamic software cannot handle the circumstance with rapidly varied unsteady flows, especially mixed-flow situations. The reason may be lack of practically applicable methods. Almost all explicit methods are inappropriate for a commercial program due to the requirement of the numerical stability expressed by the Courant condition. Many implicit algorithms, such as the widely used Preissmann scheme, are usually not applicable for a change from subcritical to supercritical flow or conversely (Cunge et al. 1980; Jin and Fread, 1997).

 

FLORIS is a one-dimensional hydrodynamic model developed by the Federal Technical University of Zurich. The program has many technique merits, among which, the most attracting feature is one that unsteady open channel flows can be implicitly simulated by the implicit method of characteristics (IMOC), even though a mixed-flow regime occurs. Since numerically solving the Saint Venant equations by an implicit method has been a concern of many software developers and researchers, an investigation or a verification of the IMOC built in the program FLORIS seems of interest.

 

The program FLORIS had been analyzed by three samples of rapidly varied open channel flows, for which analytical solutions are available (Zhang and Summer, 1994). The situations used in that analysis were simplified and idealized, such as channels were assumed to be regular, prismatic and/or frictionless. In this paper, FLORIS will be applied to simulate practical problems and numerical solutions will be compared to field experiment and physical model. The emphasis will be focused on the IMOC built in the FLORIS.

 

BASIC EQUATIONS

The program FLORIS uses the one-dimensional Saint Venant equations to describe unsteady flows in open channels. Under some assumptions, such as hydrostatic pressure distribution and small channel bottom slope, the Saint Venant equations can be derived from the conservation of mass and momentum, and may be written as follows (Kuehne, 1992):

 

(1)

 

(2)

 

where x expresses the distance measured positive in the downstream direction, t the time, A the wetted cross-section area, Q the flow discharge, q the lateral inflow or out flow per unit length, b the coefficient of non-uniform velocity distribution, g the gravity acceleration, z the water surface level, J_f the energy slope and v_x the velocity component of the lateral inflow in the flow direction.

 

The Saint Venant equations are a set of quasi-linear, hyperbolic partial differential equations, for which analytical solutions are normally unavailable. Therefore, they have to be solved numerically.

 

NUMRICAL METHODS

The program FLORIS solves the Saint Venant equations by two numerical methods: a four-point or box implicit difference method (IDM) and an implicit method of characteristics (IMOC). The IDM can be regarded as a version of the Preissmann scheme, which can give satisfactory results for many practical applications, provided that a subcritical and supercritical mixed flow occurs. To handle such circumstance, the IMOC was added to FLORIS.

 

For commercial hydrodynamic models the box implicit difference method is most widely applied scheme and becoming the standard approach. Generally, it uses a double-sweep algorithm to solve simultaneous equations. During sweep procedure two boundary conditions will be separately picked up at the upstream and downstream end of a channel, i. e., one boundary condition at the upstream end in the foreword sweep and another one at the downstream end in the return sweep. The application of the double-sweep method requires that the flow in the whole computed channel must be subcritical. If a mixed-flow regime occurs, the method doesn't work any more and the program falls down.

 

The implicit method of characteristics is one of the specified time interval scheme on a fixed-grid network. In a traditional scheme, such as the Hartree scheme, the time step cannot be selected freely, it is subject to the Courant constraint. To free the flow computation from this limitation, Schmitz (1980) developed the IMOC. Based on this concept, Lai (1988) suggested a comprehensive method of characteristics, which combines both implicit and explicit method into one. The model FLORIS uses the comprehensive method described as Lai, even though we still call it IMOC. The IMOC can easily distinguish a subcritical and supercritical flow according to the directions of characteristic curves, and will pick up appropriate conditions to make the flow computation foreword. Naturally, it can be used to handle circumstances with a mixed- flow regime.

 

Both the box implicit difference method and the implicit method of characteristics are available in the program FLORIS. The selection of the numerical method in a modeling procedure depends on the flow situation. The IDM is appropriate for subcritical flow. The IMOC can be, by way of contrast, applied to both subcritical flow and supercritical flow. Due to enormous executive time, it is recommended by the program developer that the IMOC should be used only to simulate a subcritical and supercritical mixed flow.

 

Characteristic methods belong to the first effort to numerically solve the Saint Venant equations. However, they are seldom adopted in commercial hydrodynamic models due to their complexities or due to that their numerical solutions may violate mass conservation even under simple boundary conditions, such as under constant and equal inflow and outflow conditions (Stelkoff and Falvey, 1993). Therefore, the IMOC built in the program FLORIS should be carefully tested before practical applications. Many approaches can be applied to verify computational models. These include as follows (Wang, 1992): (1) analytical solutions, (2) physical model data, (3) field data, and (4) some others. As indicated below, we will apply both field data and physical model data to verify the IMOC. Besides, we will compare computational results from the IMOC to those from the four-point implicit difference method.

 

NUMERICAL SIMULATIONS

Two numerical examples were selected and simulated by the IMOC. They can be briefly described as: (1) field experiment on surge wave propagation following a sudden closing of turbines in a hydropower plant, and (2) physical model experiment on flood wave propagation induced by partial dam breach. In the first case, the flow is subcritical, for which the IMOC is not recommended by the program developer. This example aimed at investigating the numerical behaviors of the IMOC, and comparing the results by both numerical methods built in the program FLORIS.

 

Experiences have showed that although the time step for an implicit method is no more restricted by the Courant condition, but it will still have effect on numerical accuracy. Since the Courant number is usually used as reference for the selection of a time step, we define the Courant number C_r as:

 

(3)

 

where Dt expresses the time step, u the flow velocity, g the gravity acceleration, A the wetted cross-section area, B_s the water surface width, and Dx the space step.

 

SURGE WAVE

The program FLORIS was used to simulate the propagation of a surge wave in the power canal of the Rosegg/St. Jakob hydropower plant, located in the middle reach of the river Drau in Austria. The canal is some 3.5 km long, connecting the weir St. Martin and the power station Rosegg (see Fig. 1). A field experiment of the surge wave was conducted by the Oesterreichische Draukraftwerke A. G. (Baumhackl, 1991). In the experiment, a rapid shut-down of the turbine gate at the downstream end was carried out, which decreased the discharge from 425 m³/s to zero within 6 seconds. Consequently, a major surge wave developed and propagated in the power canal against the flow direction. Having reached the upper end of the canal, the wave was partially reflected and a negative surge traveled in the opposite direction downstream. The surge wave was monitored at several sites, such as at the site M₄ and M₆, which are located at about 2075 m and 940 m from the inflow structure, respectively.

 

 

The power canal has nonprismatic cross-sections. Thirteen profiles of the canal were surveyed. The distances between the measured cross-sections vary from 15 m to 910 m. Additional cross-sections were interpolated due to the large difference of the distances between the profiles actually surveyed. The profile distances after the interpolation vary from 15 m to 21 m. The Manning friction coefficient of the canal is set to be 0.015, which reflects a very smooth surface, mainly consisting of asphalt cover.

 

To test the mass-conservation capacity of the IMOC, a constant water stage and a constant outflow were imposed at the upstream end and downstream end of the power canal, respectively. The program computed the flow with a roughly estimated initial condition. After about 35 hours, a steady flow was established. The discharges at every cross-section in the canal were equal to the outflow at the downstream end. This may preliminarily prove that numerical solutions from the IMOC can maintain the mass conservation for a nonprismatic channel.

 

Fig. 2 and Fig. 3 show water stage hydrographs computed by both the IDM and the IMOC at the site M₄ and M₆ along with the data from the field experiment. The time step in the computation was selected equal to Dt = 0.001 h, for which the Courant number near the surge wave front approximated to 2.0. The computation results from both numerical methods are satisfactory, they agree well with the experiment data.

 

 

 

Expectedly, the program FLORIS cannot reproduce the observed disintegration of the wave into several wavelets and the wave run-up on an inclined embankment, since it is based on the assumption of one-dimensional flow. Regarding the problem of the wave run-up, it may be indicated that the modeling results can be used to obtain order-of-magnitude estimates (Zhang et. al., 1993) according to the guidelines introduced by Benet and Cunge (1971).

 

DAM-BREAK WAVE

In 1995, Hohai University carried out a physical model to predict dam-break flood waves. The model dealt with a 21.1 km long reservoir and a 10.6 km long river reach downstream of the dam. The reservoir is long and narrow, more like a normal river. The downstream river widens relatively rapidly, it varies from about 400 m wide at the dam site to 2000 m at the lower end (See Fig. 4).

 

 

The dam is composed of earth-rock. It is 400 m long, 30 m high. The reservoir has capacities of 136,000,000 m³ and 194,000,000 m³ corresponding to a normal water stage elevation of 4315 m and to a dam top elevation of 4318 m, respectively. Several dam-break types were simulated, one of which will be in the following introduced. The dam was assumed to collapse partially and rapidly due to overflow. Within 1.5 hours the dam breach will develop to a trapezoid with a bottom width 110 m, a bottom elevation 4301.5 m and a side slope 0.7.

 

The mathematical model included the upstream reservoir and the downstream river reach. The dam breach was treated as an interior boundary condition, and simulated as a weir, the bottom of which can move down with the time. The Manning roughness coefficient of the reservoir and the river was set equal to 0.03, the same as that in the physical model.

 

The upstream reservoir and downstream river are irregular and nonprismatic. Altogether 31 cross-sections were surveyed, among them 1 was at the dam site, 21 in the upstream and 9 in the down stream. The average distance between two surveyed cross-sections was about 1.06 km. For the sake of numerical accuracy and stability, additional cross-sections were interpolated. The profile distances after the interpolation ranged between 80.0 m 100.0 m. In the simulation a time step was chosen to be Dt=0.01 h, for which the Courant number near the wave front approximated to 1.8.

 

At the beginning of the dam breach, the flow in the whole computed reach was subcritical, and the implicit difference method was used. After 1.22 h a mixed subcritical and supercritical flow was detected in the downstream river reach. The model automatically selected the implicit method of characteristics to compute the flow in the downstream reach, and remained using the difference method in the reservoir due to its high efficiency

 

 

Fig. 5 shows both measured and computed discharge hydrographs at the dam site. The numerical result agrees well with the measurement. Fig. 6 to Fig. 8 illustrate stage hydrographs at profiles CS2, CS4 and CS8, which are located at 2.406 km, 4.501 km and 9.202 km downstream from the dam, respectively. It can be seen that the program FLORIS will give satisfactory results, if the flow presents more one-dimensional character (see Fig. 6 and Fig.7). For a two-dimensional flow, the FLORIS can only produce an approximate solution (see Fig. 8).

 

 

 

 

CONCLUSIONS

This investigation verifies the applicability of the implicit method of characteristics built in the program FLORIS to simulate rapidly varied unsteady flows in irregular and nonprismatic open channels. The IMOC was found to maintain the mass conservation and to provide adequate numerical solutions for both subcritical flows and mixed-flow regimes. For a one-dimensional flow numerical results from the FLORIS agreed well with physical model and field experiment. If a flow presents more two-dimensional characters and cannot be simplified to be one-dimensional, the FLORIS can only provide approximate numerical solutions.

 

REFERENCES

1.      Baumhackl, G. (1991). Schwallversuch am Oberwasserkanal des KW - Rosegg/St. Jakob. Report, Oesterreichische Draukraftwerke A. G., Klagenfurt, Austia.

2.      Benet, F. and J. A. Cunge (1971). Analyse d'expériences sur les ondulations secondaires dues aux intumescences dans les canaux trapézoidaux. Journal of Hydraulic Research, 9(1), 11-13.

3.      Cunge, J.A., F. M. Holly Jr. & A. Verwey (1980). Practical Aspects of Computational River Hydraulics. London, Pitman.

4.      Kuehne, A. (1992). Abflusssimulation mit einem hydrodynamisch-numerischen Modell. Wasser, Energie, Luft, 84(11/12), 340-342.

5.      Jin, M & D. L. Fread (1997). Dynamic flood routing with explicit and implicit numerical solution schemes. Journal of Hydraulic Engineering, ASCE, 123(3), 166-173.

6.      Lai, C.-T. (1988). Comprehensive method of characteristics models for flow simulation. Journal of Hydraulic Engineering, ASCE, 114(9), 1074-1097.

7.      Schmitz, G. (1980). Instationaere Eichung mathematischer Hochwasserablauf-Modelle auf der Grundlage eines neuen Loesungsprinzips fure hyperbolische Differentialglaichungs-Systeme. Bericht Nr. 46, Versuchsanstalt fuer Wasserbau der Technischen Universitaet Muenchen.

8.      Strelkoff, T. S, & H. T. Falvey (1993). Numerical method used to model unsteady canal flow. Journal of Hydraulic Engineering, ASCE, 119(4), 637-655.

9.      Wang, S.S.Y. (1992). On verifications of computational hydro-system models. HYDROCOMP '92, J. Gayer, Oe. Starosolszky & C. Maksimovic (eds), 163-169. Budapest, Vituki.

10. Zhang, W., B. H. Schmid & D. Gutknecht (1993). Surge waves in an Austrian power canal - Computation and field experiment. Proceedings of XXVth IAHR Congress, A-2-3, 57-64. Tokyo.

11. Zhang, W. & W. Summer (1994). The application of FLORIS to simulate surge waves, hydraulic jumps and dam-break flood waves - A program analysis. Hydroinformatics '94, A. Verwey, A. M. Minns, V. Babovic & C. Maksimovic (eds), 731-737. Balkema, Rotterdam.