La Houille Blanche
Number 3, Avril 1964
|Page(s)||389 - 395|
|Published online||24 March 2010|
Sur les propriétés mécaniques et thermiques des écoulements turbulents au voisinage des parois chauffées
Mechanical and thermal properties of turbulent flows near heated walls
Professeur à l'Ecole Supérieur d'Ingénieurs de Beyrouth (Liban), détaché du Centre de Recherche et d'Essais de Chatou.
Some interesting information can be obtained on mechanical and thermal flow properties by studying the structure of turbulent flows near walls with the aid of Mac Laurin expansions starting from a point at the wall. The validity of the Navier equations for the usual turbulent flow cases has been established beyond doubt. All the relationships at the wall show that there is a single average solution with which the special case of flow between two planes can be investigated. The dynamic solution for a fluid of constant physical properties is independent of the thermal solution. The latter can be broken down into elementary solutions and the dissipation and transfer terms thus separated. Turbulent flow between heated and cooled flat plates with opposite mean fluxes. The following assumptions were made for the calculations : validity of the Navier equations, negligible thermal effect of dissipation, mean flow independent of z, zero velocity at the wall. The indefinite equations of motion (system 1) and the boundary conditions (system I') are expressed non-dimensionally by relating velocity to u*, length to u/u*, time to u/u*2 and temperature to φ0φCpu*. The functions u+, v+, w+ and 0+ are expanded as a Mac Laurin series starting from y+ = 0. The instantaneous coefficients for the expansions (system II) are found by eliminating pressure and then identifying each term, whereupon the mean value of these coefficients call be found (system III). An additional assumption has to be made regarding the correlation between the velocity components u and w at two points in order to be able to proceed with the calculation. If this correlation is assumed to be so small as to be negligible, the mean coefficients assume a simplified form (system III'). A certain number of mean coefficients can then be calculated, as follow: a1 = 0 a2 = 1/R* a3 = 0 a4 ≈ 0 a5 ≠ 0 a6 = 0 c1 = 0 c2 = 0 c3 = 0 c4 = 0 c5 ≠ 0 c6 = 0 d1 = P d2 = 0 d3 = 0 d4 ≠ 0 d5 ≠ 0 .... Velocity and temperature profiles can then be defined in clear terms (system IV). The thermal and dynamic profiles dffer even at very high Reynolds numbers and a Prandtl number equal to 1. Diffusivity study. Expressions for the turbulence factors generally termed 'turbulent viscosity' (μt), 'turbulent conductivity' (λt) and 'turbulent Prandtl number' (μtCp/λt) can be obtained from the above coefficients. The equivalence of these factors near the walls shows that y + 4 increases for μt, y + 3 for λt and y + Pt (system V). An experimental verification can be attempted from the conventional velocity profile formulations and Sleicher and Corcaran's experimental data. Its results show fairly satisfactory agreement, but measurement accuracy is too poor to allow any reliable conclusions to be drawn. The suggested theoretical data could only be verified satisfactorily if direct measurement of the correlations near the walls were possible.
© Société Hydrotechnique de France, 1964
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