[Paper Review] The anisotropic averaged Euler equations

Jerrold E. Marsden, Steve Shkoller|ArXiv.org|May 3, 2000

Computational Fluid Dynamics and Aerodynamics15 references17 citations

TL;DR

This paper introduces the anisotropic averaged Euler equations, a new model for turbulent fluid flow that captures small-scale fluctuations via an advected symmetric tensor, enabling well-posedness and a corrector for the macroscopic velocity. The method uses variational principles and semidirect product geometry to derive equations with smooth solutions and convergence to the inviscid limit, even with boundaries.

ABSTRACT

The purpose of this paper is to derive the anisotropic averaged Euler equations and to study their geometric and analytic properties. These new equations involve the evolution of a mean velocity field and an advected symmetric tensor that captures the fluctuation effects. Besides the derivation of these equations, the new results in the paper are smoothness properties of the equations in material representation, which gives well-posedness of the equations, and the derivation of a corrector to the macroscopic velocity field. The numerical implementation and physical implications of this set of equations will be explored in other publications.

Motivation & Objective

• To derive a new set of equations that model large-scale fluid flow while incorporating small-scale fluctuation effects without requiring a separate closure model.
• To establish the geometric and analytic foundations of the anisotropic averaged Euler equations using variational principles and semidirect product structures.
• To prove the well-posedness of classical solutions in material (Lagrangian) representation for the anisotropic averaged Euler equations.
• To derive a corrector term that improves the macroscopic velocity field by accounting for unresolved fluctuations.
• To demonstrate convergence of viscous solutions to the inviscid limit, even in the presence of boundaries, ensuring robustness of the model.

Proposed method

• Uses ensemble averaging over small spatial scales to represent material particle sampling, combined with asymptotic expansions in the variational principle.
• Applies Lagrangian semidirect product theory to derive the anisotropic averaged Euler equations from a variational formulation.
• Introduces a symmetric tensor field $ F $ as a fluctuation variable that is advected by the mean velocity, capturing anisotropic small-scale effects.
• Derives the equations in material representation, ensuring smoothness and well-posedness of classical solutions for initial data in Sobolev spaces $ H^s $ with $ s > n/2 + 2 $.
• Implements a viscous regularization via $ u \Delta u^\nu $ to ensure existence and regularity of solutions, with convergence to the inviscid case as $ \nu \to 0 $.
• Establishes the connection to the Navier-Stokes equations through a viscous version of the anisotropic averaged equations, preserving the geometric structure.

Experimental results

Research questions

• RQ1 How can small-scale fluctuations in turbulent fluid flow be systematically incorporated into a macroscopic model without requiring additional closure assumptions?
• RQ2 What is the geometric and analytic structure of the anisotropic averaged Euler equations derived from a variational principle?
• RQ3 Can classical solutions to the anisotropic averaged Euler equations be shown to exist and depend smoothly on initial data in a Lagrangian framework?
• RQ4 What is the role of the fluctuation tensor $ F $ in correcting the macroscopic velocity field, and how is this corrector derived?
• RQ5 Does the viscous regularization of the anisotropic averaged equations converge to the inviscid solution as $ \nu \to 0 $, even with boundary conditions?

Key findings

• The anisotropic averaged Euler equations are derived via a variational principle using semidirect product structures, ensuring a geometric formulation analogous to Arnold’s geodesic flow on diffeomorphism groups.
• Classical solutions exist in material representation for initial data $ (u_0, F_0) \in H^s_\mu \times C^\infty(T^{2,0}) $ with $ s > n/2 + 2 $, and depend smoothly on initial conditions.
• A corrector term is derived that improves the macroscopic velocity field by accounting for unresolved fluctuations, providing a systematic correction beyond the mean flow.
• Solutions to the viscous anisotropic averaged Navier-Stokes equations converge regularly to the inviscid solutions of the anisotropic averaged Euler equations as $ \nu \to 0 $, with the time interval of existence independent of $ \nu $.
• The viscous Lagrangian flow $ \eta^\nu $ converges in the $ H^s $ topology to the inviscid Lagrangian flow $ \eta^0 $, confirming the robustness of the inviscid limit even with boundaries.
• The model avoids the need for additional closure models, unlike Reynolds averaging or Large Eddy Simulation, and naturally incorporates fluctuation effects through the tensor $ F $.