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Analyzing Melts and Fluids from Ab Initio Molecular Dynamics Simulations with the UMD Package

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Lagrangian averaging with geodesic mean.

Marcel Oliver¹

¹School of Engineering and Science, Jacobs University, 28759 Bremen, Germany.

Proceedings. Mathematical, Physical, and Engineering Sciences

|December 12, 2017

Summary

This summary is machine-generated.

This study derives the Lagrangian averaged Euler (LAE) equations using a novel definition of the averaged flow map. This approach clarifies the mathematical underpinnings of fluid dynamics modeling.

Keywords:

Euler equations Lagrangian averaging Taylor hypothesis generalized Lagrangian mean

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Area of Science:

Fluid Dynamics
Mathematical Physics
Turbulence Modeling

Background:

The Euler-α equations are a class of turbulence models.
Previous derivations of these equations have varying assumptions.
A clear, intrinsic derivation is needed for theoretical advancement.

Purpose of the Study:

To revisit and clarify the derivation of the Lagrangian averaged Euler (LAE) equations.
To introduce an intrinsic definition of the averaged flow map.
To establish a rigorous foundation for Euler-α models.

Main Methods:

Utilizing an intrinsic definition of the averaged flow map as a geodesic mean on the volume-preserving diffeomorphism group.
Assuming statistical isotropy of first-order fluctuations.
Applying averaging to the kinetic energy Lagrangian of an ideal fluid within a Euclidean domain.

Main Results:

The derivation yields the LAE Lagrangian under stated assumptions.
The method provides a new perspective on the structure of averaged fluid equations.
The derivation is valid for unbounded Euclidean domains.

Conclusions:

The intrinsic definition of the averaged flow map offers a more fundamental approach to deriving LAE equations.
This work provides a clearer theoretical basis for Euler-α turbulence models.
Future work could explore extensions to bounded domains or different fluid regimes.