La Houille Blanche
Number 7, Novembre 1967
|Page(s)||735 - 746|
|Published online||24 March 2010|
Quelques applications de modèles mathématiques à l'étude des écoulements non permanents dans un réseau ramifié de rivières ou de canaux
A few applications of mathematical models to the study of unsteady flows in a ramified river or canal network
Electricité de France, Direction des Etudes et Recherches : 6, quai Watier, 78 - Chatou, France.
Mathematical models are coming into increasing use as a means of investigating unsteady river and channel flow. Their potential scope is briefly described. The Saint-Venant equations for gradually varying unsteady flow constitue a hyperbolic quasi-linear system. They can be integrated by two basic numerical methods : explicit methods requiring a stability condition, and implicit methods which may be unconditionally stable. The choice of which method to use is very often given by the type of problem, for whereas the increment along the abscissa depends on river or channel bed geometry and water level definition accuracy, the time increment, on the other hand, must account for significant variations. In the case of a flood (a 'slow' phenomenon) the implicit method is the obvious one as it can cope with a comparatively large time increment. Very rapidly developing phenomena, on the other hand, require a short time increment, which satisfies the stability condition of explicit methods. Each problem is represented by a hydraulic scheme allowing for the various ramifications and featuring all the weirs, flow inputs, cross-sectional changes, diversions and other special points in the system. One or two conditions (hydrograph or level chart) will be given upstream from each branch, and a stage/discharge relationship or a level record downstream from the model, also an operating relationship for each singular point. The initial conditions also require defining. They can be calculated for steady flow, but if the phenomenon is already developing at the point at which the calculation starts, they have to be assumed ; they are usually of little importance, however, as their effects are no longer felt after a certain time. Reference is made to examples of a few applications to illustrate the potential usefulness of mathematical models in this field. Examples of flood-routing studies mentioned are a study of the Rhine at Karlsruhe and one of the Rhone above Lyons. The Rhine's hydraulic scheme features a number of hydro-power plants and that of the Rhone several weirs (Fig. 1 to 8). Under the heading of hydro-power development an example of a surge wave study arises in the problem of 'by the lockful' operation (Fig. 9 to 17). In this particular case, the implicit method could be used to reduce the computing time requirements, but it should be noted that the explicit one is preferable for cases involving more rapid operation. A scheme with power governed by water level is mentioned as a further example. It features a reservoir, a canal, a pressure tunnel with two surge tanks and a penstock. Turbine discharge is governed by reservoir level (Fig. 18.) Downstream and upstream boundary conditions remain permanently interdependent in this example. Mathematical models have also been used to study tidal propagation. Figures 19, 20 and 21 show the close agreement reached between observed and calculated data in a study of the Seine and the formation of a tidal bore which, if considered as a weak shock wave, can be dealt with by the explicit method. In a tidal propagation problem in a small river the downstream condition represents a floodgate system keeping the sea out at rising tide and opening at the ebb tide (Fig. 23). In a study of a cooling system for a thermal power station using water from the Loire and discharging it back to a shallow river arm the mathematical model features control sections at the ends of the river arm when the latter are high and dry at low water (Fig. 22). The various examples are merely mentioned as an illustration of the potential scope of mathematical models ; a brief discussion on numerical integration methods is given in an Appendix.
© Société Hydrotechnique de France, 1967