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 magneto-hydrodynamics. The applicability of the quaterionic frame description to Lagrangian averaged velocity gradient dynamics is also demonstrated.

The three-dimensional Euler equations: Where do we stand?

“…(26) has its origins in the divergence-free condition Tr M = 0 and is an economical way of writing ∆ p = −u i, j u j,i . Several attempts have been made to model the Lagrangian-averaged pressure Hessian by introducing a constitutive closure -see [78] for a summary. The idea goes back to Léorat [79], Vieillefosse [80], Novikov [81] and Cantwell [82].…”

“…Other models of interest include the tetrad model of Chertkov, Pumir and Shraiman [88] which has recently been developed by Chevillard and Meneveau [89]. More ideas regarding the modelling of the pressure Hessian through a transformation from Eulerian to Lagrangian coordinates using a Lagrangian flow map have recently been discussed in [78].…”

“…Fluid particles not only take complicated trajectories but they also rotate in motion. Recent work has shown that Hamilton's quaternions are applicable to this type of problem [78,[100][101][102][103]. In his lifetime Hamilton's ideas did not meet with the approval of his contemporaries [104][105][106] but in the context of modern-day problems the crucial property that quaternions possess -that they represent a composition of rotations -has made them the technical foundation of modern inertial guidance systems in the aerospace industry where tracking the paths and the orientation of satellites and aircraft is critical [107].…”

“…Another example that could be cast into this format are the equations of ideal MHD in Elsasser form (see [78,100,101] although the existence of two material derivatives requires some generalization. In Section 5.3 it will be shown how the quartet in (35), based upon the pair of Lagrangian evolution equations (33) and (34), can determine the evolution of an ortho-normal frame for a fluid particle in a trajectory.…”

A geometric interpretation of coherent structures in Navier–Stokes flows

“…These results were based on earlier work by McIntyre and Roulstone, and were reviewed by them in McIntyre & Roulstone (2002). It has also been shown that the three-dimensional Euler equations has a quaternionic structure in the dependent variables (Gibbon 2002) and that this idea can be used to discuss the evolution of orthonormal frames on particle trajectories: see Gibbon et al (2006), Gibbon (2007Gibbon ( , 2008a and Gibbon & Holm (2007). The use of different sets of dependent and independent variables in geophysical models of cyclones and fronts has facilitated some remarkable simplifications of otherwise hopelessly difficult nonlinear problems: see Hoskins & Bretherton (1972).…”