Contribution à l'étude de la diffusion en canal pour un régime turbulent stationnaire

¹ Maître-assistant, Institut National des Sciences Appliquées, Villeurbanne.
² Chef de travaux, I.N.S.A., Villeurbanne.

Résumé

This article discusses a study of diffusion in turbulent stationary open-channel flow without temperature and density gradients. There are two possible theoretical approaches to the turbulent diffusion problem, i.e. : a) Starting out from an Eulerian description of the problem, a characteristic tensor Dij for the dispersive properties of the medium leads to the conventional diffusion equation : dC / dt= a /axi (Dij aC/axj) b) Starting out from a Lagrangian description and assuming a Gaussian displacement probability distribution for the diffusing particles, an expression is obtained directly for the concentration C. Both descriptions meet in the special case of homogeneous turbulence where, by identification, the Dij coefficients can be expressed, in terms of the Rij coefficients, the latter being characteristic coefficients for the relationships between the velocity components of a particle at two different times. In view of the difficulty of calculating the Rij coefficients, however, a more direct method of determining the diffusion tensor is sought ; by choosing a suitable axis system the tensor can be reduced to its diagonal form with only three main coefficients to be considered : Dx, Dy and Dz . The simplest assumption is to take constant diffusion coefficients, the values of which can be determined from the Reynolds analogy or from such semi-empirical relationships as Elder's for example. This gives an immediate solution for equation (1) for steady flow in a rectangular flume. A solution has also been obtained for variable diffusion coefficients based on the Reynolds analogy, the calculation of which was based on an assumed logarithmic velocity distribution in the flume. A computer programme has also been prepared for the solution of equation (1) for steady flow with any diffusion coefficients. The experimental part of the study was carried out in a 35 m x 0.50 m rectangular flume with a slope of 1/1000 and flows of up to 151/s. Once steady streaming flow was established, a rhodamine B solution was fed in at a given cross-section, usually at a single point, the dye concentration then being measured by spectro-photometry at arbitrary points downstream. Concentration profiles were then plotted across and along the flume for various dye input points and rates of flow and both smooth and rough flume sides (Strickler coefficients about 80 and 60 respectively). When the experimental and theoretical data were compared it was found that the simple constant diffusion coefficient assumption gave very close agreement where Dy had been calculated by an Elder-type relationship of the form Dy = KUo*h. where K is a coefficient depending on wall roughness and the flow Reynolds number and coefficients Dx and D2 are both equal to x/6 Uo*h. The digital computer calculation resulted in an improvement so small as to be negligible, in view of the accurate results obtained with the constant coefficient assumption. The presence of bends, a hydraulic jump or other special features was found to cause considerable additional spread, but the type of flow (i.e. shooting or streaming flow) did not seem to have any effect on diffusion.