New Governing Equations for Fluid Dynamics

TL;DR

This paper introduces new fluid dynamics equations based on vorticity tensor, offering advantages like Galilean invariance, clearer physical interpretation, reduced computational cost, and potential improvements in turbulence analysis.

Contribution

The paper proposes a new set of governing equations for fluid dynamics using vorticity tensor, addressing limitations of Navier-Stokes equations and enhancing turbulence research capabilities.

Findings

New equations are mathematically identical to Navier-Stokes.

Reduced computational cost by half due to tensor simplification.

Demonstrated equivalence of results between new and traditional equations.

Abstract

There are several questions with NS, which include: 1. Both symmetric shear terms and stretching terms in strain and stress are coordinate-dependent and thus not Galilean invariant; 2. The physical meaning of both diagonal and off-diagonal elements are not clear, which is coordinate-dependent; 3. It is hard to measure the strain and stress quantitatively, and viscosity is really measured by vorticity not by symmetric strain; 4. There is no vorticity tensor in NS, which plays important role in fluid flow especially for turbulent flow. The new proposed governing equations for fluid dynamics use vorticity tensor only, which is anti-symmetric. The advantages include: 1. Both shear and stress are anti-symmetric, which are Galilean invariant and independent of coordinate rotation; 2. The physical meaning of off diagonal elements is clear, which is anti-symmetric shear stress, 3. Viscosity coefficients are obtained by experiment which uses vorticity, 4. The vorticity term can be further decomposed to rigid rotation and anti-symmetric shear, which are important to turbulence research, 5. The computation cost for viscous term is reduced to half as the diagonal terms are all zero as six elements are reduced to three. Several computational results are made, which clearly demonstrate both NS and new governing equations have exactly same results. As shown below, the new governing equation is identical to NS in mathematics, but the former has lower cost and several advantages mentioned above including possibility to better study turbulent flow. There are reasons to use the new governing equations to replace Navier-Stokes equations. The unique definition and operation of vector and tensor by matrix and matrix operation are also discussed in this paper.