A simple and versatile iterative implicit solver for incompressible flows with particular emphasis on fixed and free curved boundaries.

Abstract

An existing iterative implicit flow solver based on direct divergence-linked pressure correction is adapted for finite difference discretization with a modified form of Crank-Nicolson time integration and a co-located mesh system. The latter is judged to be more suitable for curved and free boundary applications. The convergence of such a flow solver is investigated with a novel combined experimental and theoretical analysis based on linear stability theory. The analyses showed that the critical relaxation factors depend on local flow and mesh parameters. The modified Crank-Nicolson time interation makes use of a backward-forward coefficient <italic>w</italic>. A scheme for choosing a variable <italic>w</italic> is proposed which, while still maintains second-order accuracy, was shown to be well behaved in test computations with suddenly-started initial conditions. As part of this investigation, the method of high-order Lagrange extrapolation for the pressure on the boundary was found to be both accurate and easy to implement. For the treatments of curved boundaries, our computational methods explicitly define curved boundaries and implement pressure boundaries and Neumann condition on curved boundaries with second order accuracy. The basic concept is to define the boundary by a set of boundary nodes on the chosen mesh and evaluate the flow variables on the boundary through the boundary conditions of the basic differential equations. The proposed method is validated by test computations for the 2-D spin-up problem and the flow past a circular cylinder. For free surface flow problems, we developed a front tracking procedure tracking the interfacial markers explicitly and maintaining the sharp property discontinuities and jump conditions on the free interface. The residuals of normal interfacial momentum equations are employed to correct guessed interface positions until all governing equations and boundary conditions are satisfied. This front tracking procedure is validated by 2-D droplet cylinder oscillation and homogeneous bubble growth problems.