Canonical Fractional-Step Methods and Consistent Boundary Conditions for the Incompressible Navier–Stokes Equations

An account of second-order fractional-step methods and boundary conditions for the incompressible Navier–Stokes equations is presented. The goals of the work were (i) identification and analysis of all possible splitting methods of second-order splitting accuracy, and (ii) determination of consistent boundary conditions that yield second-order-accurate solutions. Exact and approximate block-factorization techniques were used to construct second-order splitting methods. It has been found that only three canonical types (D, P, and M) of splitting methods are nondegenerate, and all other second-order splitting schemes are either degenerate or equivalent to them. Investigation of the properties of the canonical methods indicates that a method of type D is recommended for computations in which zero divergence is preferred, while a method of type P is better suited to cases where highly accurate pressure is more desirable. The consistent boundary conditions on the tentative velocity and pressure have been determined by a procedure that consists of approximation of the split equations and the boundary limit of the result. It has been found that the pressure boundary condition is independent of the type of fractional-step methods. The consistent boundary conditions on the tentative velocity were determined in terms of the natural boundary condition and derivatives of quantities available at the current time step (to be evaluated by extrapolation). Second-order fractional-step methods that admit the zero-pressure-gradient boundary condition have been derived by using a transformation that involves the “delta form” pressure. The boundary condition on the new tentative velocity becomes greatly simplified due to improved accuracy built into the transformation.