By Victor N. Nikolaevskiy
Turbulence concept is without doubt one of the such a lot fascinating elements of fluid mechanics and plenty of impressive scientists have attempted to use their wisdom to the advance of the idea and to provide priceless techniques for answer of a few sensible difficulties. during this monograph the writer makes an attempt to combine many particular ways into the unified thought. the elemental premise is the straightforward concept that a small eddy, that's a component of turbulent meso-structure, possesses its personal dynamics as an item rotating with its personal spin pace and obeying the Newton dynamics of a finite physique. a few such eddies fills a coordinate cellphone, and the angular momentum stability needs to be formulated for this spatial cellphone. If the mobilephone coincides with a finite distinction aspect at a numerical calculation and if the exterior size scale is huge, this basic quantity could be regarded as a differential one and a continuum parameterization needs to be used. Nontrivial angular stability is a end result of the asymmetrical Reynolds tension motion on the orientated facets of an simple quantity. before everything look, the averaged dyad of pace parts is symmetrical, == even though, if averaging is played over the airplane with general nj, the main of commutation is misplaced. hence, the tension tensor asymmetry j depends upon different components that perform the angular momentum stability. this is often the one danger to figure out a rigidity in engineering.
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Additional info for Angular Momentum in Geophysical Turbulence: Continuum Spatial Averaging Method
Averaging over the cross-section of a flow or of a body is also the conventional procedure that yields simple and efficient engineering calculations, for example, in hydraulics . In the theory of turbulence itself the proper two procedures of introduction of mean flow parameters (for some volume interval and/or for some time interval) were shown even in the Reynolds initial work . The turbulent structure created in a fluid flow by interaction with other objects at high Reynolds numbers can be accounted for on the basis of continuum principles.
21) 0ipO jl we finally obtain the following equation  for an incompressible fluid: E.. aUk di = _[CP. aU i IJk dt J ax. J ax. 22) J This means that the first right-hand side term of the Friedman vortex equation for a perfect fluid: aUk + aUp+~E. 23) is responsible for the inertia moment change, as it was mentioned also by G. Batchelor . 11) take the form of the angular momentum balance (in the Euler and Lagrange representation respectively) for differential volume. In the Cartesian coordinates system the Euler inertia moment of incompressible fluid contained in a cubical cell V=l 3 has the form independent of both time and coordinate: p Iij = P [2 J ij /12 In the Lagrange case, the inertia moment iij is associated with fluid contained in Vat the instant t.
17) = < (}'.. 2). 2. 29) for the elementary macrovolume Ltv. 1) SPATIAL AVERAGING AND MACROEQUATIONS aat JP ~v Ui dV + iat JP (E + Ui Ui ~v = J (Tij Ui JP Ui Uj 2 dS - IlS J dS j = IlS (Tij )dV + JP (E + Ui Ui) AS Jq J dS j + ~v IlS j dS j + IlS 2 . AV. Due to continuity of all variable fields in a viscous fluid, we can apply the Ostrogradskiy-Gauss theorem. 4) a < 0'.. >. 7) The elementary volume LiV itself, as well as the area of these limiting crosssections, are assumed to be big enough to include the ensemble of all random realizations of turbulent mesovortexes.