A second-order accurate immersed boundary method for fully resolved simulations of particle-laden flows

An immersed boundary method (IBM) with second-order spatial accuracy is presented for fully resolved simulations of incompressible viscous flows laden with rigid particles. The method is based on the computationally efficient direct-forcing method of Uhlmann [M. Uhlmann, An immersed boundary method with direct forcing for simulation of particulate flows, J. Comput. Phys. 209 (2005) 448–476] that is embedded in a finite-volume/pressure-correction method. The IBM consists of two grids: a fixed uniform Eulerian grid for the fluid phase and a uniform Lagrangian grid attached to and moving with the particles. A regularized delta function is used to communicate between the two grids and proved to be effective in suppressing grid locking. Without significant loss of efficiency, the original method is improved by: (1) a better approximation of the no-slip/no-penetration (ns/np) condition on the surface of the particles by a multidirect forcing scheme, (2) a correction for the excess in the effective particle diameter by a slight retraction of the Lagrangian grid from the surface towards the interior of the particles with a fraction of the Eulerian grid spacing, and (3) an enhancement of the numerical stability for particle–fluid mass density ratios near unity by a direct account of the inertia of the fluid contained within the particles. The new IBM contains two new parameters: the number of iterations Ns of the multidirect forcing scheme and the retraction distance rd. The effect of Ns and rd on the accuracy is investigated for five different flows. The results show that rd has a strong influence on the effective particle diameter and little influence on the error in the ns/np condition, while exactly the opposite holds for Ns. A novel finding of this study is the demonstration that rd has a strong influence on the order of grid convergence. It is found that for spheres the choice of rd = 0.3Δx yields second-order accuracy compared to first-order accuracy of the original method that corresponds to rd = 0. Finally, Ns = 2 appears optimal for reducing the error in the ns/np condition and maintaining the computational efficiency of the method.