By Radyadour Kh. Zeytounian

for the fluctuations round the capacity yet quite 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 thought 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 a type of Monte-Carlo approach 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 acquired 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 methods (described in the §3. four ofChapter 3). Ofcourse the mathematical good posedness of the scheme isn't recognized and the numerics will be bold! no matter if this test 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 useful.

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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. Indeed, we have the following relation 36 where CHAPTER 2 is the Froude number based on the vertical length and in this case is the ‘long wave or hydrostatic’ parameter. In the case of atmospheric motions it is necessary to take into account also the so-called Rossby number, which characterizes the effect of the earth’s rotation on the atmospheric motions: with the constant Coriolis parameter, where is the magnitude of the vector of the earth’s rotation and a constant value of the algebraic latitude.

1) is written) and are the components of second - order tensors, while are the components of a third - order tensor. The precise statement of the conservation law (balance equations) is: if n (with the Cartesian components is the exterior normal unit vector and D/Dt the material derivative operator. We assume, now, that our fixed domain is, in fact, the sum of two subdomains, and adjacent to a common boundary G, located inside having a continuously varying tangent plane at each of its point P. Call W(t, x) the velocity field of the points of G, N the normal unit vector at P, to the surface G, pointing inside and U(t, x), with the Cartesian components the velocity of the particle located at P at time t.

In wing theory, the flow in the vicinity of either the leading, the trailing, the side or the round-edge of the planform may be considered as understandable only on the grounds of asymptotic modelling. Laminar separation, a challenge to fluid mechanicists for seven decades, has been understood only recently as an application of the triple deck (with Sychev’s proposal) asymptotic model. Interfaces endowed with material properties or simply too thick to be considered as pure discontinuities are best understood when considered as thin layers embedded within a large-scale flow.