A modified nodal scheme for the time-dependent, incompressible Navier–Stokes equations

Based on Poisson equation for pressure, a nodal numerical scheme is developed for the time-dependent, incompressible Navier–Stokes equations. Derivation is based on local transverse-integrations over finite size brick-like cells that transform each partial differential equation to a set of ordinary differential equations (ODEs). Solutions of these ODEs for the transverse-averaged dependent variables are then utilized to develop the difference scheme. The discrete variables are scalar velocities and pressure, averaged over the faces of brick-like cells in the (x,y,t) space. Cell-interior variation of transverse-averaged pressure in each spatial direction is quadratic. Cell-interior variation of transverse-averaged velocity in each spatial direction is a sum of a constant, a linear and an exponential term. Due to the introduction of delayed coefficients, the exponential functions are to be evaluated only once at each time step. The semi-implicit scheme has inherent upwinding. Results of applications to several test problems show that the scheme is very robust and leads to a second-order error. As expected in such coarse-mesh schemes, even relatively large size cells lead to small errors. Extension to three dimensions is straightforward.