Turbulence Dynamics based on Lagrange Mechanics and Geometrical Field Theory of Deformation

TL;DR

This paper introduces a geometrical field theory of fluid deformation based on Lagrange mechanics, providing a new framework to describe turbulence as micro deformation on a curvature space, leading to novel turbulence wave equations.

Contribution

It develops a geometrical field theory of turbulence using Lagrange mechanics and deformation tensors, offering a new perspective beyond traditional Navier-Stokes models.

Findings

Turbulence is modeled as micro deformation on a curvature space.

Derived new turbulence wave equations with inward-traveling waves.

Compared results with Navier-Stokes and Reynolds stress models.

Abstract

The turbulence field is stacked on the laminar flow. In this research, the laminar flow is described as a macro deformation which forms an instant curvature space. On such a curvature space, the turbulence is viewed as a micro deformation. So, the fluid flow is described by the geometrical field theory of finite deformation. Based on the Lagrange mechanics and the deformation energy concept, using the Least Action Principle, the Euler-Lagrange motion equations are obtained. According to A E Green formulation, the stress concept is introduced by deformation tensor. The fluid motion is described by the multiplication of a macro deformation tensor and a micro deformation tensor. By this way, the geometrical field of fluid motion is well constructed. Then, the spatial derivative of deformation energy is expressed by the gradient of deformation tensors. By this way, the deformation energy related items in the Euler-Lagrange motion equations are expressed by the stress tensor and deformation tensor. The obtained Euler-Lagrange motion equations, then, are decomposed into average deformation equations and turbulence equations. For several special cases, the new results are compared with the conventional Navier-Stokes equation with Reynolds stress modification.The comparisons also show that the Bernoulli Equation is a natural precondition for the conventional Navier-Stokes equation.Generally, the turbulence wave is an inward-traveling wave. Unlike the normal outward-traveling wave related with the average deformation, the inward-traveling wave is the intrinsic feature of turbulence. So, the turbulence is well defined by the equations obtained in this research. For several typical cases, the simplified turbulence wave equations are given out with simple discussion.