Elementary course in Lagrange's equations and their by N. W Akímoff

By N. W Akímoff

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1800 ω 2 structures and 1920 s2 structures were extracted from the initial isotropic state and ordered by intensity. The structures contribute 39% and 28%, respectively, to the global variance of vorticity and strain rate while only occupying 13% and 10% of 26 global volume, reflecting the concentrated nature of the fluctuations within. In the sheared state 800 ω 2 and 810 s2 structures were extracted contributing 50% and 38%, respectively to global variance, and 16% and 13% to the volume. The variance-to-volume-ratio is 3 to 1 in both isotropic and sheared states, the average volume of ω 2 and s2 structures increases by 300%, indicating major amalgamation and growth of structures by shear.

Figure 6. 4. The extracted subvolume at each time is shown by the box. 6, then to AR ≈ 1 at St = 1, shortly after which the tube begins a transition to a hairpin vortex. As shown in Figs. 6(c)-(f), the process involves the elongating and flattening of the vortex tube and the bending over and formation of a rather flattened head that is distinct Dynamics of small-scale vorticity and strain-rate structures 29 from a single tube-like leg. As the vortex elongates and bends, the leg of the hairpin remains tube-like with little overlap with the more concentrated strainrate fluctuations.

Shear causes amalgamation of structures and suppresses strain-rate structures between enstrophy structures. Shear enhances “passive” strain-rate fluctuations—strain rate kinematically induced by local vorticity concentrations with negligible enstrophy production—relative to “active,” or vorticity-generating, strain-rate fluctuations. Enstrophy structures separate into “active” and “passive” based on the second eigenvalue of fluctuating strain rate. The time evolution of a shearinduced hairpin enstrophy structure was analyzed.

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