On a new method for numerical solution of the Navier-Stokes equations

Abstract

The method presented in this paper refers to the known system of partial differential equations of Fluid Mechanics, consisting of the continuity equation as well as of the Navier-Stokes equations, and describing unsteady flow of viscous incompressible fluid. In the plane case this system contains three unknown functions, i.e. two velocity components and pressure. At every instant all these functions have to be univalent with respect to space variables. Advantage is taken of this property in order to eliminate pressure from the Navier-Stokes equations. Because integration is applied for this purpose instead of the customary differentiation, the order of the resulting system of equations is not higher in comparison with the original one. Consequently, no need arises for additional boundary conditions. By means of the method of finite differences as well as the method of splines applied to space variables in the system of so obtained equations, which already do not contain pressure, a set of ordinary differential equations is derived-with time as the only independent variable. Two velocity components at every node of the computational mesh represent a set of unknown functions, their initial values being naught. The initial problem so determined has been solved by means of the Runge-Kutta method for the test case represented by the driven cavity problem. Workability of the method has been demonstrated.