By Radyadour Kh. Zeytounian

For the fluctuations round the capability 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 notion is to place a stochastic constitution on fluctuations and to spot the massive eddies with part of the chance area. The Reynolds stresses are derived from one of those Monte-Carlo strategy on equations for fluctuations. these are themselves modelled opposed to a method, utilizing the Guiraud and Zeytounian (1986). The scheme is composed in a collection of like equations, regarded as random, simply because they mimic the massive eddy fluctuations. The Reynolds stresses are obtained from stochastic averaging over a kin in their suggestions. Asymptotics underlies the scheme, yet in a slightly unfastened hidden means. We clarify this in relation with homogenizati- localization techniques (described in the §3. four ofChapter 3). Ofcourse the mathematical good posedness of the scheme isn't recognized and the numerics will be ambitious! no matter if this try will motivate researchers within the box of hugely advanced turbulent flows isn't really foreseeable and we've got desire that the assumption will end up valuable

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2. Asymptotic modelling! The main goal of asymptotic modelling is to derive a simplified (approximate) set of equations, but also, initial and boundary conditions, which can be solved with less numerical effort (as a “model” problem) than the original full NS-F equations, with the corresponding initial and boundary conditions, related to a specific physical problem. In this way, my purpose, in the present book, is to initiate a process which does not seem to have sufficiently attracted the attention of scientists.

We do not, of course, assert that this is the only way, or even the most efficient one, for deriving such models. 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.

1) is related with the high Strouhal and Reynolds numbers when we consider low Mach number flow, such that: where is a similitude parameter. When tends to zero, then the dissipative coefficient (so-called the “Stokes number”) in the Burgers equation: 40 CHAPTER 2 tends also to zero. 1 in Chapter 10); and k° are the value of and k, at T = T°. In Zeytounian (1998) the reader can find a review of the problems encountered in the Bénard-Marangoni thermocapillaryinstability. 1. 1. Why asymptotic? Actually, the word “asymptotics” is often used in place of “asymptotic methods or analysis”.

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