Bounds for Euler from vorticity moments and line divergence

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Kerr, Robert M. (Robert McDougall) (2013)

The inviscid growth of a range of vorticity moments is compared using Euler\ud calculations of anti-parallel vortices with a new initial condition. The primary goal\ud is to understand the role of nonlinearity in the generation of a new hierarchy of\ud rescaled vorticity moments in Navier–Stokes calculations where the rescaled moments\ud obey Dm ≥ Dm+1, the reverse of the usual \ud Ωm+1 ≥ Ωm Hölder ordering of the original\ud moments. Two temporal phases have been identified for the Euler calculations. In the\ud first phase the 1 < m < ∞ vorticity moments are ordered in a manner consistent with\ud the new Navier–Stokes hierarchy and grow in a manner that skirts the lower edge of\ud possible singular growth with D2\ud m → � sup ӏωӏ ~ Am(Tc-t)-1 where the Am are nearly\ud independent of m. In the second phase, the new Dm ordering breaks down as the Ωm\ud \ud converge towards the same super-exponential growth for all m. The transition is\ud identified using new inequalities for the upper bounds for the -dD-2m/dt that are based\ud solely upon the ratios Dm+1/Dm, and the convergent super-exponential growth is shown\ud by plotting log(d log Ωm/dt). Three-dimensional graphics show significant divergence\ud of the vortex lines during the second phase, which could be what inhibits the initial\ud power-law growth.
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