Analysis of the local truncation error in the pressure-free projection method for incompressible flows: a new accurate expression of the intermediate boundary conditions

摘要

The numerical integration of the Navier-Stokes equations for incompressible flows demands efficient and accurate solution algorithms for pressure-velocity splitting. Such decoupling was traditionally performed by adopting the Fractional Time-Step Method that is based on a formal separation between convective-diffusive momentum terms from the pressure gradient term. This idea is strictly related to the fundamental theorem on the Helmholtz-Hodge orthogonal decomposition of a vector field in a finite domain, from which the name projection methods originates. The aim of this paper is to provide an original evaluation of the local truncation error (LTE) for analysing the actual accuracy achieved by solving the de-coupled system. The LTE sources are formally subdivided in two categories: errors intrinsically due to the splitting of the original system and errors due to the assignment of the boundary conditions. The main goal of the present paper consists in both providing the LTE analysis and proposing a remedy for the inaccuracy of some types of intermediate boundary conditions associated with the prediction equation. Such evaluations will be directly performed in the physical space for both the time continuous formulation and the finite volume discretization along with the discrete Adams-Bashforth/Crank-Nicolson time integration. A new proposal for a boundary condition expression, congruent with the discrete prediction equation is herein derived, fulfilling the goal of accomplishing the closure of the problem with fully second order accuracy. In our knowledge, this procedure is new in the literature and can be easily implemented for confined flows. The LTE is clearly highlighted and many computations demonstrate that our proposal is efficient and accurate and the goal of adopting the pressure-free method in a finite domain with fully second order accuracy is reached.