On the application of the Helmholtz-Hodge decomposition in projection methods for incompressible flows with general boundary conditions

This paper is concerned with the analysis of the Helmholtz-Hodge decomposition theorem since it plays a fundamental role in the projection methods that are adopted in the numerical solution of the Navier-Stokes equations for incompressible flows. The paper highlights the role of the orthogonal decomposition of a vector field in a bounded domain when general boundary conditions are in effect. In fact, even if Fractional Time-Step Methods are standard procedures for de-coupling the pressure gradient and the velocity field, many problems are encountered in performing the decoupling with higher accuracy. Since the problem of determining a unique and orthogonal decomposition requires only one boundary condition to be well posed, thus either the normal or the tangential ones, result exactly imposed at the end of the projection. Numerical errors are introduced in terms of both the pressure and the velocity but the orthogonality of decomposition guarantees that the former does not contribute to affect the accuracy of the latter. Moreover, it is shown that depending on the meaning of the vector to be decomposed, i.e. acceleration or velocity, the true orthogonal projector can be defined only when suitable boundary conditions are verified. Conversely, it is shown that when the decomposition results non-orthogonal, the velocity accuracy suffers of other errors. The issue on the resulting accuracy order of the procedure is clearly addressed by means of several accuracy studies and a strategy for improving it is proposed. This paper follows and integrates the issues reported in Iannelli and Denaro (Int. J Numer. Meth. Fluids 2003; 42:399-437). Copyright (C) 2003 John Wiley Sons, Ltd.