New governing equations for fluid dynamics

Abstract

The difference in the governing equation between inviscid and viscous flows is the introduction of viscous terms. Traditional Navier–Stokes (NS) equations define stress based on Stokes’s assumptions. In NS equations, stress is supposedly proportional to strain, and both strain and stress tensors are symmetric. There are several questions with NS equations, which include the following: 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 is not clear, which is coordinate-dependent. 3. It is hard to measure strain and stress quantitatively, and viscosity is really measured by vorticity, not by symmetric strain. 4. There is no vorticity tensor in NS equations, which plays an important role in fluid flow, especially for turbulent flow. The newly proposed governing equations for fluid dynamics use the vorticity tensor only, which is anti-symmetric. The advantages include the following: 1. Both shear and stress are anti-symmetric, which are Galilean invariants 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 experiments, which use vorticity. 4. The vorticity term can be further decomposed into rigid rotation and anti-symmetric shear, which are important to turbulence research. 5. The computation cost for the viscous term is reduced to half as the diagonal terms are all zero and six elements are reduced to three. Several computational examples are tested, which clearly demonstrate both NS and new governing equations have exactly the same results. As shown below, the new governing equation is identical to NS equations in mathematics, but the new one has lower cost and the several advantages mentioned above, including the possibility to study turbulent flow better. It is recommended to use the new governing equation instead of NS equations. The unique definition and operation of vectors and tensors by matrix and matrix operation are also discussed in this paper.