An immersed boundary method for nonuniform grids

Abstract

We propose a new direct forcing immersed boundary method for simulating the flow around an arbitrarily shaped body in nonuniform grids. A new formulation of the weight function for the distribution of the immersed boundary forcing with an efficient strategy that distributes the Lagrangian points on the surface of the immersed body is developed. We validate our method for various flows induced by a translational or angular motion of a sphere by measuring the lift, drag, and moment coefficients. For an arbitrary shape particle, we test our method for flow around an ellipsoid in the uniform flow. All tests show good agreement with available references. Finally, we investigate the interaction between a tilted ellipsoid and the wall in the uniform flow and the Poiseuille flow.

Introduction

The immersed boundary method (IBM) based on the direct forcing scheme proposed by Uhlmann [1] is one of the popular numerical methods used for the simulation of a moving body in Cartesian grids. This kind of IBM that utilizes the discrete delta function [2] requires some modification for the force and moment conservation property between the Lagrangian point and Eulerian grids [3] when applied to nonuniform grids in the near-wall region. Furthermore, the uniform distribution of the Lagrangian points on the surface of the body typically adopted in uniform Eulerian grids can cause problems. A distribution of the Lagrangian points relative to the nearby Eulerian grids that is too sparse results in inaccuracy, while a distribution that is too dense wastes computing time.

For the application of IBM to nonuniform grids, Pinelli et al. [3] adopted reproducing kernel particle method (RKPM) and Toja-Silva et al. [4] used the radial basis function. Recently, Akiki and Balachandar [5] proposed an IBM for computing the flow around a sphere in near-wall nonuniform grids. They used the method of Pinelli et al. [3] for interpolation/distribution functions and modified the method of Saff and Kuijlaars [6] to distribute the Lagrangian points on the immersed surface of a sphere in nonuniform grids. In particular, they used the vector spherical harmonics to calculate the surface area for each Lagrangian point with very high accuracy. However, this distribution method [6] and the vector spherical harmonics are restricted only to a sphere, and a nonuniform distribution of the Lagrangian points based on the spiral distribution does not guarantee the optimum number of the Lagrangian points per Eulerian grid cell in nonuniform grids.

In this study, we propose a simple strategy to distribute the Lagrangian points over an arbitrarily shaped body. We also propose an adjustment of the weight functions on nonuniform grids based on the reproducing polynomial particle method (RPPM) [7], which is simple to use and requires no arbitrary parameter to avoid singularity in the moment matrix on weight points needed in RKPM [3]. The proposed method performs better than the method of Akiki and Balachandar [5] with a lower number of Lagrangian points when applied to the flow around a sphere in the near-wall shear flow.

This paper is organized as follows. In Section 2, the formulation of our IBM including the weight function calculation and a new strategy to distribute the Lagrangian points over the immersed surface with validation through the total surface area, drag and torque calculation is provided. In Section 3, our method is validated by various examples such as a settling sphere, a freely rotating sphere in the shear flow, a moving sphere in the near-wall region and a tilted ellipsoid in the uniform flow. Then, the interaction between a tilted ellipsoid and the wall is investigated. Section 4 concludes our study.