Lagrangian particle paths and ortho-normal quaternion frames

Abstract

Experimentalists now measure intense rotations of Lagrangian particles in turbulent flows by tracking their trajectories and Lagrangian-average velocity gradients at high Reynolds numbers. This paper formulates the dynamics of an orthonormal frame attached to each Lagrangian fluid particle undergoing three-axis rotations, by using quaternions in combination with Ertel's theorem for frozen-in vorticity. The method is applicable to a wide range of Lagrangian flows including the three-dimensional Euler equations and its variants such as ideal MHD. The applicability of the quaterionic frame description to Lagrangian averaged velocity gradient dynamics is also demonstrated.

9 pages, one figure, revised

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• Lagrangian analysis of alignment dynamics for isentropic compressible magnetohydrodynamics
• Lagrangian dynamics and regularity of the spin Euler equation
• The gradient of potential vorticity, quaternions and an orthonormal frame for fluid particles