Development of finite analytic method for unsteady three-dimensional Navier-Stokes equations

Unsteady 1D, 2D and 3D incompressible Navier-Stokes equations are numerically analyzed by a numerical scheme called the finite analytic method. The basic idea of the finite analytic method is the incorporation of a local analytic solution in the numerical solution of linear and nonlinear partial differential equations. The local analytic solutions for unsteady 1D, 2D and 3D convective transport equations are obtained from locally linearized governing equations by specifying suitable initial and boundary conditions for each local element. When the local analytic solution is evaluated at a given nodal point, it gives an analytic algebraic relationship between a nodal value in a local element to its neighboring nodal points. The solution of the problem is then achieved by solving the system of algebraic equations. Depending on the boundary and initial functions chosen to represent the boundary and initial conditions for each local element, a number of local analytic solutions are derived. The results show that the boundary approximation based on the combination of exponential and linear function is the best one since the boundary function thus constructed is the natural solution of the governing equation. The finite analytic coefficients thus obtained are shown to be relatively simple and do give the correct asymptotic behavior for both diffusion and convection dominated cases.