| Unsteady 2D and 3D incompressible Navier-Stokes equations are numerically analyzed with a new numerical method called the "Explicit Finite Analytic Scheme". This new scheme is developed from the fact that the mathematical characteristic of unsteady incompressible Navier-Stokes equations begins to exhibit quasi-hyperbolic behavior rather than parabolic and elliptic behavior when the Reynolds number is large. The idea is that the convective transport equation is solved with a local analytic solution based on the characteristic method in space and time variables for the inviscid portion, such that the time dependent analytic solution is explicit in time, while the viscous diffusion and source term are approximated by standard finite differences. In this study a piecewise second-order polynomial is adopted to approximate the initial condition in each local element. Then this piecewise second-order polynomial is propagated along the characteristics according to the direction of velocity. When the local analytic solution is evaluated at a given nodal point, it gives an analytic algebraic relationship between the evaluated nodal value and the previous time step nodal points in a local element. The solution of the problem is then achieved by solving these explicit algebraic equations.; The 2D explicit finite analytic scheme is proven by the Fourier analysis to be stable when the CFL (Courant-Friedrichs-Lewy) condition is satisfied. Its truncation error is of first order accuracy in time and second order accuracy in space.; Explicit finite analytic scheme is then employed to solve several unsteady fluid flow problems. It is employed to solve unsteady 2D incompressible Navier-Stokes equations both in the vorticity in Vorticity-Stream Function Formulations and velocity components in Primitive Variable Formulations. Numerical solutions are obtained for starting cavity flows of Reynolds numbers of 100, 400, 1000, 5000, 10000. In the 3D case, it is employed to study a lid driven 3D cavity flow using Primitive Variable Formulations. Numerical results are obtained for starting cavity flows of Reynolds numbers of 400 and 1000. In all cases, numerical solutions are shown to converge rapidly, and to be stable and accurate. |
