What is it about?
Beltrami flow is a flow with vorticity, i.e., the curl of velocity, proportional to velocity at any local point in space and time. It has been known that the Beltrami flows with constant proportionality coefficient exist. In this paper, it is shown that three-dimensional Beltrami vortex is formed when some conditions are fulfilled. One of the sufficient conditions is the presence of the external force that fulfills some other condition.
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Why is it important?
Studying Beltrami vortex is particularly important because (1) owing to a spontaneous breaking of the scaling invariance, its structure will be different from the ones known as the exact solutions to the Navier-Stokes equation, and (2) the above Beltramian property is observed to emerge in numerical simulations. Beltrami flows with constant proportionality coefficient are theoretically allowed to exist. However, it is difficult to imagine a situation that generates such a perfect uniformity in the ratio of vorticity to velocity in the whole space and time. Rather, isn't it natural that the ratio varies from place to place? This work shows that such velocity configurations are really possible.
Perspectives
External force will generally be required for generating Beltrami vortex, albeit it is not known whether the external force is a prerequisite for the process. Since the sufficient conditions for Beltrami vortices have been identified, we may be able to track theoretically how a vortex turns to a Beltrami vortex after one of the conditions is switched on or how a Beltrami vortex alters its nature after some of the conditions are switched off. Comparing the results with, e.g., meteorological observations will be intriguing.
Dr. Koichi Takahashi
Reasearch Institute for Data Science, Tohoku Gakuin University
Read the Original
This page is a summary of: Three-Dimensional Unsteady Axisymmetric Viscous Beltrami Vortex Solutions to the Navier–Stokes Equations, J — Multidisciplinary Scientific Journal, August 2023, MDPI AG,
DOI: 10.3390/j6030030.
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