By Victor N. Nikolaevskiy
Turbulence idea is likely one of the such a lot exciting elements of fluid mechanics and lots of remarkable scientists have attempted to use their wisdom to the improvement of the idea and to supply invaluable ideas for resolution of a few sensible difficulties. during this monograph the writer makes an attempt to combine many particular methods into the unified concept. 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 speed and obeying the Newton dynamics of a finite physique. a couple of such eddies fills a coordinate mobilephone, and the angular momentum stability should be formulated for this spatial telephone. If the telephone coincides with a finite distinction point at a numerical calculation and if the exterior size scale is big, this ordinary 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 pressure motion on the orientated aspects of an straight forward quantity. at the start look, the averaged dyad of pace elements is symmetrical, == although, if averaging is played over the aircraft with general nj, the primary of commutation is misplaced. accordingly, the tension tensor asymmetry j is dependent upon different components that perform the angular momentum stability. this can be the one threat to figure out a tension in engineering.
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Additional info for Angular Momentum in Geophysical Turbulence: Continuum Spatial Averaging Method
3). 10 Ahmadi et al  mentioned that mean viscous stress tensor depended on the orientation of a cross-section. A. Zagustin  had noted the same for Reynolds stresses. 14). 17) j >. 19), the value of the total momentum flux Nij includes the stress tensors, averaged over the volume AV, but as a part of true macro-stress tensor. In accordance with the Cauchy concept (see [217, 287]), the latter corresponds exactly to a force, acting at the oriented area. Therefore, a volume-average stress tensor cannot be used as a macro-stress  instead of Nij in the impulse balance (compare with ).
AUk di jl + E.. i. 12) yields £ ijk . 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.
19) include the moments of velocity fields of higher order. They can be expressed in their turn by the balance equations for the moments of momentum of the next orders, etc. 2) by XkXm"', followed by the volume averaging. 21) They are averaged over all the oriented cross-sections: a a +.. ·-PUU;~k~ ... >]"• at
Angular Momentum in Geophysical Turbulence: Continuum Spatial Averaging Method by Victor N. Nikolaevskiy