By E. John Finnemore, Joseph B. Franzini

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**Example text**

30d) appears the following main dimensionless parameters: which are well known and are, according to the order of the writing: Reynolds, Mach, Strouhal, and Prandtl numbers. 49) are indexed by stands “c” which holds for “characteristic value” of the indexed quantity. If the body force f (per unit mass) is assimiled with the gravity force (as in meteorological problems) then we have also the following parameter (a so-called ‘Boussinesq number’, according to Zeytounian (1990)): where g, is the magnitude of the acceleration due to gravity.

We do, however, feel that when such a procedure is feasible it should be undertaken. As a matter of fact, the application of this approach implies that the approximate, asymptotic-limit, model is associated with an asymptotic expansion procedure which, in principle, makes it possible to improve the approximation obtained with the model used by progressing through the hierarchy of approximations - going to higher-order terms in the asymptotic expansion. This is the rational and consistent basis of asymptotic modelling.

But the singular nature of the Mach number expansion which requires two matched asymptotic expansions was recognized, first, sixteen years later by Lauvstad, and more thoroughly discussed by Crow, Viviand and Obermeier; matching with the Navier incompressible and viscous limit model is a necessary step for the obtaining a ‘three-region’ significant asymptotic model. Further comments on the far flow may be found in Zeytounian and Guiraud (1984) and also in the recent thesis by Sery-Baye. The consistency of the MMAE (in relation to the far field) was extensively investigated by Leppington and Levine and by Tracey, and pushed up to in the inner region, checking consistency with the outer one, ,in the thesis by Sery-Baye.