Abstract
It is essential for a Navier-Stokes equations solver based on a projection method to be able to solve the resulting Poisson equation accurately and efficiently. In this paper, we present a new 2D Navier-Stokes equation solver based on a recently proposed fourth-order method, namely the generalized harmonic polynomial cell (GHPC) method, as the Poisson equation solver. The GHPC method is a generalization of the 2D HPC method originally developed for the Laplace equation. In the recent development of the HPC method, loss of accuracy on highly stretched or distorted grids has been reported when solving the Laplace equation, while the performance of the GHPC method on non-uniform grids is still not explored and discussed in the literature. Therefore, the local accuracy of the GHPC method is investigated in detail in the present study, which reveals that the GHPC method allows for the use of much larger grid aspect ratio than that for the original HPC method. Global accuracy of the GHPC method on stretched non-uniform girds is also thoroughly analyzed by considering cases with analytical solutions. Obvious advantages of using the GHPC method in terms of accuracy are demonstrated by comparing with a second-order central Finite Difference Method (FDM). The present Navier-Stokes equations solver uses second-order FDMs for the discretization of the diffusion and advection terms, which may be replaced by other higher-order schemes to further improve the accuracy. Meanwhile, an immersed boundary method [1] is used to study the fluid-structure-interaction problems. The Taylor-Green vortex and flow around a smooth circular cylinder are studied to confirm the accuracy and efficiency of the new 2D Navier-Stokes equation solver. The predictions show good agreements with the experimental and numerical results in the literature.
