A High Order Numerical Method Of Incompressible Navier-stokes Equation

A high-order numerical method of Incompressible Navier-Stokes equation is presented in this paper. Not only in Cartesian coordinate system but also in curvilinear coordinate system, Incompressible Navier-Stokes equation is calculated by the high-order numerical method. In order to validate the high-order numerical method, Two-Dimension and Three-Dimension examples have been calculated numerically and its numerical results are compared with the analytic solutions or with others’numerical results. As a result, the numerical method has been confirmed as high accuracy and high resolution.High-order compact schemes and upwind compact schemes are used to calculate space derivative; And multi-stage Runge-Kutta scheme is used as time schemes. To solve the pressure Poisson equation, a high-order compact iterative scheme is adopted.Poiseuille flow in Two-Dimension and Three-Dimension is simulated by Incompressible Navier-Stokes equation in Cartesian coordinate system. The result has compared with the analytic solution. And unsteady cavity drive flow is also simulated with Reynolds number from100to10000. The result has compared with others’numerical results. All of the above, it is confirmed that the numerical method of Incompressible Navier-Stokes equation is presented in this paper has the high accuracy and high resolution and the ability to simulate the high Reynolds number turbulence flow.To extend our numerical method to simulate in complex geometry mesh, Incompressible Navier-Stokes equation n curvilinear coordinate system is adopted as the governing equation. Three examples with analytic solutions have been calculated already. They are Poiseuille flow on non-uniform mesh, Circular motion between coaxial cylinders on a polar mesh and Hagen-Poiseuille flow on a cylindrical mesh. All of them have been passed the high accuracy test.By calculating on different kinds mesh, the numerical method of Incompressible Navier-Stokes equation presented in this paper is shown as a high-accuracy and high-resolution method which can be widely used in Cartesian coordinate system and curvilinear coordinate system.