Immersed boundary method with non-uniform distribution of Lagrangian markers for a non-uniform Eulerian mesh

This study presents a technique to incorporate spheres in a channel flow that uses a non-uniform Eulerian grid using immersed boundary methods with direct forcing. An efficient algorithm is presented which distributes the Lagrangian markers non-uniformly to match the fluid grid and keep the number of markers optimized. Also a novel method to calculate the area weights of the Lagrangian markers is given. It is observed that even the best available algorithms for uniform distribution of markers on a sphere result in a finite error. Using vector spherical harmonics, this error is quantified and reduced to machine precision. A series of simulations of a stationary and moving sphere in a periodic channel at Reynolds number range of 1–100 are presented. Results for a sphere in an ambient shear flow in close proximity of a wall are also shown, where the present non-uniform distribution offers an order of magnitude reduction over uniform distribution of Lagrangian markers. Simulations of a random cluster of 640 monodisperse spherical particles show a 77% reduction in Lagrangian markers with an error of 0.135% in computing the total drag.