By Radyadour Kh. Zeytounian

For the fluctuations round the capacity yet relatively fluctuations, and showing within the following incompressible approach of equations: on any wall; at preliminary time, and are assumed recognized. This contribution arose from dialogue with J. P. Guiraud on makes an attempt to push ahead our final co-signed paper (1986) and the most suggestion is to place a stochastic constitution on fluctuations and to spot the massive eddies with part of the likelihood area. The Reynolds stresses are derived from one of those Monte-Carlo technique on equations for fluctuations. these are themselves modelled opposed to a method, utilizing the Guiraud and Zeytounian (1986). The scheme is composed in a suite of like equations, regarded as random, simply because they mimic the massive eddy fluctuations. The Reynolds stresses are bought from stochastic averaging over a relatives in their suggestions. Asymptotics underlies the scheme, yet in a slightly free hidden manner. We clarify this in relation with homogenizati- localization approaches (described in the §3. four ofChapter 3). Ofcourse the mathematical good posedness of the scheme isn't recognized and the numerics will be bold! even if this try will motivate researchers within the box of hugely complicated turbulent flows isn't really foreseeable and we have now wish that the belief will end up precious.

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**Asymptotic Modelling of Fluid Flow Phenomena (Fluid Mechanics and Its Applications)**

For the fluctuations round the ability yet fairly fluctuations, and showing within the following incompressible method of equations: on any wall; at preliminary time, and are assumed recognized. This contribution arose from dialogue with J. P. Guiraud on makes an attempt to push ahead our final co-signed paper (1986) and the most suggestion is to place a stochastic constitution on fluctuations and to spot the big eddies with part of the likelihood house.

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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.