Liutex-based Modified Navier-Stokes Equation

The Navier-Stokes (NS) partial differential equations, as the governing equation of fluid dynamics, are based on particles with zero volume. In such a condition, conservation law of moment of momentum is automatically satisfied, and thus NS equations only contain conservation of mass, momentum and energy. Because of the difficulty to get an analytical solution of NS equations, scientists develop finite element method (FEM) and finite volume method (FVM) to obtain approximate solutions. However, these methods are dependent on the size of finite volumes, which is not zero. Considering the size of the finite volume, conservation law of moment of momentum is no longer automatically satisfied and it is necessary to develop a new control equation to take effect of momentum into account. In this paper, new relations of reciprocal shear stresses are derived from conservation law of moment of momentum, and so are new constitutive relations and modified NS equations. The Liutex based NS equation may be the universal governing equations for both laminar and turbulent flow which is dominated by vortices and conservation of moment of momentum must be satisfied.