A study on preconditioning iterative methods and relativity of pressure in the numerical calculation of fluid flow

Abstract

This article presents the effect of preconditioning iterative methods on boundary conditions of the pressure-correction in the numerical computation of fluid flow with known velocity components on all boundaries using the SIMPLE algorithm. In such computation, a set of solutions of the pressure-correction is indefinite, because only the Neumann condition is imposed on all boundaries. However, solutions become unique if the value of pressure-correction is fixed at least on one boundary point, and the Dirichlet condition is additionally imposed. Though both conditions must give exactly the same velocity and temperature fields, this problem arises from the relativity of the pressure. The mathematical illustration for this problem is provided using the numerical computation of the natural convection in an enclosure. It is concluded that the preconditioner adopted and the condition that only the Neumann condition on all boundaries is given are effective to reduce the number of iterations in solving the linear system of equations of the pressure-correction at the computation of the natural convection.