General Synthesis of the Numerical Results

The resume of the benchmark cases, in the rigid-rigid (R-R) and rigid-free (R-F) cases is done by focusing on the main requested quantities: U*, V*,Ψ* and their locations. We recall that U*=maximum of | u(y) | at x=0.5; V*=maximum of | v(x) | at y=1 in R-R case and of | v(y) | at x=1 in R-F case; Ψ*=maximum of | ψ |. The first goal of this resume is to extract the most relevant computed solutions from the synthesis of the four different families of methods, carried out respectively by Behnia, Henkes & Segal, Sani and Deville, for F.D.M., F.V.M., F.E.M. and S.M.. Obviously, the computations needed to be performed with a high accuracy which is ideally obtained in such a convenient (simple) geometry through spectral approximations. But even these methods needed a series expansion with quite a large number of polynomials. (See Le Quere.) The other methods were also able to produce accurate results with an appropriate resolution (very high near the boundaries). Examples of such accurate results obtained by Behnia & de Vahl Davis with a finite difference method using a regular grid of 81x321 points, are given for all mandatory cases (Pr=0) in Figs. 1–4 and Figs.6–8. The iso-ψ patterns given in these figures contain 12 lines corresponding to ψi = i Δψ, for i = 0,1,.., 11; the Δψ value being mentioned in each figure. Consequently the value of ψmax is equal to 12 Δψ.