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The turbulence field is stacked on the laminar flow. In this research, thelaminar flow is described as a macro deformation which forms an instantcurvature space. On such a curvature space, the turbulence is viewed as a microdeformation. So, the fluid flow is described by the geometrical field theory offinite deformation. Based on the Lagrange mechanics and the deformation energyconcept, using the Least Action Principle, the Euler-Lagrange motion equationsare obtained. According to A E Green formulation, the stress concept isintroduced by deformation tensor. The fluid motion is described by themultiplication of a macro deformation tensor and a micro deformation tensor. Bythis way, the geometrical field of fluid motion is well constructed. Then, thespatial derivative of deformation energy is expressed by the gradient ofdeformation tensors. By this way, the deformation energy related items in theEuler-Lagrange motion equations are expressed by the stress tensor anddeformation tensor. The obtained Euler-Lagrange motion equations, then, aredecomposed into average deformation equations and turbulence equations. Forseveral special cases, the new results are compared with the conventionalNavier-Stokes equation with Reynolds stress modification.The comparisons alsoshow that the Bernoulli Equation is a natural precondition for the conventionalNavier-Stokes equation.Generally, the turbulence wave is an inward-travelingwave. Unlike the normal outward-traveling wave related with the averagedeformation, 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 givenout with simple discussion.

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