Computation of two-dimensional unsteady supercritical flows in open channel contraction of spillway chutes
29 April 2009
Supercritical flow through open channel contractions often generates disturbances and instabilities at the surface, which appear as transverse waves. Contraction of channel, often called transitions, can be found in several hydraulic structures but mainly in free surface spillway chutes. Saint Venant’s equations governing two-dimensional unsteady free surface flows can be successfully applied to this kind of problem by making some simplified assumptions, which in turn lead to a non-linear hyperbolic system of equations for which an analytical solution is not yet available. Thereafter, the equations of motion obtained are generalized with cases of unspecified bottom slopes to take into account the effect of a variable bottom slope.
In this paper, a second order two step MacCormack explicit finite differences scheme is applied, in conjunction with a geometric transformation of the irregular physical domain, to a simpler computational one of rectangular shape. The integration time steps are adjusted at each incrementation time according to Courant-Friedrichs-Lewy’s stability condition. An existing example of horizontal channel contraction is reproduced herein, and the obtained results of flow pattern and water surface profiles are compared to previous experimental and numerical studies. A rectangular channel with steep slopes will be studied as a second application in order to see if the elaborate model can simulate the flow in a channel with a high bottom slope. The numerical results obtained will be compared with experimental measurements. After this a third application, in which a supercritical flow in a recti-linear contraction of channel with variable bottom slopes, is presented. The results obtained here will be compared with the results corresponding to a horizontal contraction.
