Ondulations secondaires en front d'intumescences et ondes solitaires

Abstract

The results presented in this paper follow the excellent work by Lemoine and Serre and more recent research by Sandover and Taylor, an account of which latier has already been published in La Houille Blanche. A distinction is made between two principal transient canal flow research theories, depending on whether the motion considered occurs en bloc or otherwise. The first of these theories, established by Boussinesq and then developed by Serre and Sandover, assumes nun-uniform distribution of the velocities-which no longer run parallel-and therefore allows for stream line curvature. Like Sand over, the present authors consider the flow as having relative motion with respect to a reference system accompanying the wave front, the entrainment velocity being the absolute celerity of the first wave peak. The apparent rate of flow in the relative motion is constant. Momentum and specific energy in a cross-section of a rectangular canal are given by equations (1) and (2). Allowing for the continuity equation and the Eulerian differential equations (4), equations (5) and (6) are obtained by integration of relationships (1) and (2). If frictional effects and canal slope are assumed to cancel out, momentum and specific energy conservation considerations lead to two second-order differential equations (7) and (11), which when integrated, yield the same equation (10) for the wave profile. The latter equation is also the one for a solitary wave, and enables some of the basic characteristics of this type of wave to be determined. The maximum rise above the initial water level is given by expression (15). The total wave volume is given by expression (16). The apparent rashness of some of the assumptions made for the determination of the secondary undulation characteristics is shown to be partly contradicted by experimental data. The secondary undulations are assumed to form from a train of solitary waves following each other a finite distance apart. Knowing the local volume of the solitary wave and a celerity relationship, the wave length can be determined, but the authors have preferred relying on a semi-empirical method for this purpose. Inspection of experimental results put forward by several authors shows that, for well-developed undulations, the dimensionless parameter te resulting from expression (14) has a roughly constant value, viz. te = 0.956. The wave length is thus given by expression (17), based on the above assumption as regards, the value of te. The mean wave height -or weighted height- is then equivalent to half the maximum rise in the water level. The authors have also considered an approximate approach to the effects of head lasses in the light of these assumptions (loss of head varying stepwise), the results of which can, however, only give a qualitative idea of the phenomenon. Extreme depth variations with Fraud number for relative motion and head loss parameter are shown on Plate 4. A systematic experimental study has enabled the validity of the assumptions made to be assessed. Comparison between theoretical and experimental data shows satisfactory agreement between both sets for the highest water levels (Fig. 10) and average wave heights (Fig. 12). The dimension less wave height (Fig. 11) is always between 5 and 10, usually about 8. Figures 7, 8 and 9 show experimental and theoretical wave profile variations for a large initial depth of water and a Froude number φ0 for absolute motion. A similar investigation has also been carried out for trapezoidal flumes, the theoretical results of which show less pronounced rises in water level than for the rectangular case.