La Houille Blanche
Number 1, Janvier 1964
|Page(s)||33 - 39|
|Published online||24 March 2010|
Intégration numérique des équations d'écoulement de barré de Saint-Venant par un schéma implicite de différences finies
Numerical integration of Barré de Saint-Venant's flow equations by means of an implicite scheme of finite differences. Applicants in the case of alternately free and pressurised flow in a tunnel
Société Grenobloise d'Etudes et d'Applications Hydrauliques (SO.GR.E.A.H.).
The study of free-surface flows involves the solution of hyperbolic partial differential equations. A numerical solution methodes described, in which the partial derivatives are replaced by finite differences so that, for a function f (x, t) : ∂f/∂x = 0 f (x + Δx,t + Δt) - f (x, t + Δt) / Δx + (1 - 0) f (x + Δx, t) - f (x, t) / Δx ∂f/∂x = f (x + Δx, t + Δt) + f (x, t + Δt) / 2 Δt - f (x + Δx, t) + f (x, t)/ 2 Δt where 0 is a weighting factor such that 1/2 < 0 ≤ 1. Furthemore, by assuming a linear relationship between level and velocity increases in the time increment and by using an algorism described by Richtmyer , the authors succeed in solving a system of concatenated linear equations clearly defined by the boundary conditions. The derivation and solution of the formulae are described for the first time in detail for the application of this implicit scheme of finite differences to hyperbolic equations. This method obviously requires electronic computing equipment for its application. In view of the completely satisfactory results obtained for river and canal wave problems, it was thought worth applying it to the problem of composite tunnel flow as well (i.e. varying between pressurised and free-surface flows), of which numerous cases occur in practice. For this, the authors use an artifice first thought of by Preissmann  to reduce pressurized flow to free-surface flow, in which a very narrow slot is provided in the upper part of the tunnel to act as a piezometer without affecting the tunnel cross-sectional area. This method is then used to calculate the rapid tail race tunnel and surge tank filling conditions produced when the turbines at Wettingen power station are opened up fully in five seconds. A comparison of the author's results with those of model tests at the Zürich Hydraulic Research Laboratory [ 8] and data calculated by Calame  gives the following figures for the two most important factors concerned : MODEL CALAME CALCULATION SOGREAH CALCULATION Time taken by the wave to arrive (seconds)... 30 28 32 Maximum water level in the surge tank (metres)... 4.05 3.93 3.96 The calculations are thus seen to be adequately representative of the phenomenon, certain differences being due to the fact that the boundary conditions chosen were not the same in all three cases, especially as regards the tunnel outlet level.
© Société Hydrotechnique de France, 1964